This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

This paper provides a comprehensive derivation of the total energy equations for the atmospheric components of Earth System Models (ESMs). The assumptions and approximations made in this derivation are motivated and discussed. In particular, it is emphasized that closing the energy budget is conceptually challenging and hard to achieve in practice without resorting to ad hoc fixers. As a concrete example, the energy budget terms are diagnosed in a realistic climate simulation using a global atmosphere model. The largest total energy errors in this example are spurious dynamical core energy dissipation, thermodynamic inconsistencies (e.g., coupling parameterizations with the host model) and missing processes/terms associated with falling precipitation and evaporation (e.g., enthalpy flux between components). The latter two errors are not, in general, reduced by increasing horizontal resolution. They are due to incomplete thermodynamic and dynamic formulations. Future research directions are proposed to reconcile and improve thermodynamics formulations and conservation principles.

Earth System Models (ESMs) have numerous total energy budget errors. This article establishes the governing total energy equations for large‐scale ESMs and assesses the energy budget errors in real‐world simulations in a widely used climate model. To move towards a closed energy budget in ESMs, further research on total energy conserving discretizations (in the dynamical core), unified thermodynamics (through thermodynamic potentials/conserved variables) and missing processes is paramount. This research is especially important since some of the energy budget errors will not improve with higher spatial resolution and may even get worse.

Closing total energy budgets in Earth System Models without *ad hoc* fixers is a monumental task

Largest errors are from missing processes/terms, thermodynamic inconsistencies and dynamical core

Further research is needed on conservative discretizations, unified thermodynamics and missing processes

Earth System Models (ESMs) are extremely complex, and represent a wide range of physical processes in separate component models (atmosphere, ocean, land surface, and sea ice). In addition, each component has separate sub‐modules that compute the effect of unresolved physical processes, such as cloud microphysics and radiation in atmosphere models. The Navier–Stokes equations that serve as the foundation for the ESMs used for climate change projections satisfy conservation laws for mass, energy, and momentum. These conservation laws, combined with the second law of thermodynamics, impose key constraints on the climate system. It is believed that accurate satisfaction of these constraints is critical for the integrity of long‐term climate change simulations.

In practice, satisfying such constraints is complicated. Discretization schemes tend to introduce aliasing errors, phase errors, unphysical modes, and spurious extrema. These errors may introduce spurious sources and sinks of energy, erroneous spectral transfers between energy modes, make positive definite concentrations negative etc. Some numerical schemes and generalized vertical coordinates require interpolation, which may produce similar errors.

The use of certain physical approximations (such as the anelastic or Boussinesq approximations) alter the form of the conserved energy quantities and thermodynamic potentials in (sometimes) subtle ways that need to be properly understood to avoid the introduction of spurious sinks and sources. Understanding how to endow such approximations with consistent energetics and thermodynamics is relatively recent, but has been progressing rapidly over the past decade, for example, Ingersoll (2005), Pauluis (2008), Young (2010), Tailleux (2012), Brown et al. (2012), Vasil et al. (2013), and Dewar et al. (2016). Many of these advances, however, have yet to be widely adopted in ESMs. Additionally, the fact that physical parameterizations are developed independently may mean that different components of a model rely on inconsistent approximations for constants or conserved quantities. For example, until quite recently NCAR's Community Atmosphere Model (CAM, Neale et al., 2010) used a total energy formula that was inconsistent with the sigma‐pressure vertical coordinate (Williamson et al., 2015). Another example is when a physics subgrid scale model assumes that a certain quantity is conserved (e.g., heat based on liquid water potential temperature, *θ*_{l}) while the atmosphere model is expecting another quantity to be conserved (e.g., enthalpy). These sorts of inconsistencies can easily arise if the choices made by each sub‐module, or component as a whole, are not documented and coupled consistently.

Another source of energy errors is that different components of an ESM often operate on different temporal scales and are coupled asynchronously. This may produce spurious energy transfers. Also, the fact that models cannot resolve scales below the model grid scale means that assumptions have to be made about the energy transfers between the resolved and unresolved motions. The physical validity of these assumptions is hard to assess, although emerging literature is making significant strides towards a better and more physical understanding of grid and sub‐grid scale coupling (e.g., Gassmann & Herzog, 2015). The basic problem is that dissipative processes occur at the viscous scale which cannot be resolved with a finite grid size. Hence, energy propagates from resolved to unresolved scales (or vice versa) and is no longer visible to the resolved dynamics. It is therefore by no means straightforward to represent physically correct energy transfers. It is likely that designing prognostic sub‐grid models that are coupled consistently with the resolved‐scales is the only way to ensure physically realizable energy transfers for more complicated wave interactions.

Numerical dissipation, which is commonly used to stabilize fluid flow solvers, can produce physically unrealizable solutions. By physically unrealizable we mean that the second law of thermodynamics is not satisfied (i.e., there is “unphysical” up‐gradient mixing). If one argues that the second law should be satisfied for numerical dissipation operators, then they must be positive definite. This is, for example, not the case for standard hyperdiffusion operators, but is true for carefully implemented Smagorinsky type (second‐order) diffusion (Becker, 2001; Becker & Burkhardt, 2007; Schaefer‐Rolffs & Becker, 2018).

Even though they continue to neglect the complications of grid/sub‐grid scale coupling, research groups studying the Earth's energy imbalance using observations and re‐analysis products have started developing advanced energy equations that include, for example, enthalpy fluxes associated with precipitation (Kato et al., 2021; Mayer et al., 2017; Trenberth & Fasullo, 2018). The global modeling community has been lagging behind in terms of incorporating these processes in their modeling systems. Kato et al. (2021) noted

When models are constrained by ToA (Top of Atmosphere) radiation data products, models that conserve energy and water mass do not match an observed global mean precipitation rate (e.g. Held et al., 2019) because of, in part, the existence of a significant energy balance residual when satellite energy flux products are integrated.

Hence it is clear that even though the energy budget is closed, in part by using energy conserving formulations or the use of energy fixers where energy consistency is not possible, the energy balance still suffers from significant errors (Kato et al., 2011; L’Ecuyer et al., 2015). The causes of these errors are largely unknown, but are observed to be large over tropical oceans (Harrop et al., 2022; Kato et al., 2016; Loeb et al., 2014). Furthermore, Lucarini and Ragone (2011) analyzed pre‐industrial simulations from a wide range of IPCC climate models, and found that most climate models featured biases of the order of 1 W/m^{2} for the net global and the net atmospheric, oceanic, and land energy balances. They concluded that these imbalances are partly due to imperfect closure of the energy cycle in the fluid components.

As a result of these problems, demonstrating that ESMs are physically consistent is an enormously complicated problem that requires a holistic view and wide expertise that most individual scientists do not possess. In October 2019 we organized a week‐long workshop at the Banff International Research Station (BIRS) in order to address some of these big‐picture fundamental issues related to ESMs (BIRS, 2019). We brought together scientists involved in developing operational and research atmosphere models, physics packages, ocean models and couplers. Our meeting did not include representatives of the land surface and sea ice modeling communities, although they are clearly important. Participants included developers for the MOM6 (Adcroft et al., 2019), CROCO (Hilt et al., 2020) and NEMO (Madec et al., 2019) ocean models, the Community Atmosphere Model (CAM, Neale et al., 2010) which is part of the Community Earth System Model (CESM, Danabasoglu et al., 2020) from the National Center for Atmospheric Research (NCAR), the Energy Exascale Earth System Model (E3SM) (Golaz et al., 2019), the DYNAMICO (Dynamical Core on Icosahedral Grid, Dubos et al., 2015), the Unified Model (Walters et al., 2017), LFRic (S. Adams et al., 2019; Melvin et al., 2019) and ICON‐IAP (Icosahedral Nonhydrostatic model at the Institute for Atmospheric Physics, Gassmann, 2013) atmosphere models, and the CLUBB (Cloud Layers Unified By Binormals) parameterization of turbulence and clouds in the atmosphere (Larson, 2017). Participants also included developers of Adaptive Mesh Refinement (AMR) methods for geophysical flows via the CHOMBO library (M. Adams et al., 2019; Ferguson et al., 2016, 2019) as well as WAVETRISK (Dubos & Kevlahan, 2013; Kevlahan & Dubos, 2019). Since the group was deliberately heterogeneous, the workshop began with a series of overview talks on each component (atmosphere, ocean, physics, couplers). This ensured that all participants started from a common knowledge base for each component. In addition to reviewing the basic structure and assumptions of each component, overview talks considered specific topics such as total energy errors in dynamics, energy conversion and internal entropy production, energy conservation in cloud and turbulence parameterizations, and ocean‐atmosphere coupling.

The goals of the workshop were to identify the most important fundamental questions related to energy conservation in each component/sub‐module and in the coupled system, and to propose prioritized future directions of research. We also decided to make some specific recommendations for best practices which take into account all components of ESMs. These classifications and recommendations were developed through extensive focused discussions involving all workshop participants during and after the workshop. With such a broad interdisciplinary group, it was challenging to assemble a coherent manuscript, and many iterations were necessary to produce this version. At an early stage we decided to focus on the atmosphere component, leaving out very important discussions on other components in the Earth system. Attempting to write a paper encompassing all components and discussions during the workshop would require leaving out important details that are paramount for understanding the complexity of energy budgets in ESMs. Also, any attempt at a “unified discussion” of energetics of all components would lead to a paper written at an undesirably high level of mathematical abstraction.

A first step towards a more accurate and comprehensive closure of energy budgets in ESMs is to formulate comprehensive energy equations and diagnose the different terms involved in the energy budget. In the spirit of reaching the broader modeling community, we have chosen to start with well‐known large‐scale equation sets (and associated assumptions) and then add complexity incrementally. Please note that this paper focuses on documenting possible sources of energy non‐conservation, raising awareness about all the problems that may arise, as well as providing suggestions on how to address some of the issues identified. Hence, this paper is by no means an exhaustive analysis of the overall problem, but rather an attempt at assembling the building blocks of a future, more general, approach.

The paper is structured as follows:

**Theoretical energetics/budgets (Section** **2****)**

First, the total energy equations for traditional (large‐scale) climate models are derived. We start with the dry hydrostatic primitive equations (HPE) and gradually increase the thermodynamic complexity by first adding water vapor and then condensates to the HPE. Special attention is given to the derivation of enthalpy terms (and associated reference states), latent heat terms and surface flux terms. For these models, a detailed explanation of the approximations made in large‐scale models can be included rigorously. An in‐depth discussion is included of surface fluxes and the complications arising due to falling precipitation and/or water entering the atmosphere using a single‐component fluid approach.

**Energy budget errors of a climate model (Section** **3****)**

After deriving the theoretical energy budget, we use the NCAR CAM climate model to estimate the magnitude of various terms in the total energy equation. This example highlights which energy terms are important and illustrates why it is difficult to close energy budgets when using the assumptions that are standard for large scale models (e.g., that all components of air have the same temperature).

**Other energy budget errors (Section** **4****)**

In addition to the energy budget errors and imbalances discussed in Section 3, we discuss the following additional energy errors:

– Numerical truncation energy errors in dynamical cores (adiabatic).

– Physics–dynamics coupling errors due to spatial and temporal discretization errors. This excludes inconsistencies at the level of the continuous equations.

– Thermodynamic inconsistency energy errors:

⋆ As an illustration we discuss a specific example in some detail: coupling the CLUBB cloud parameterization package with the CAM climate model.

⋆ Mass “clipping” errors and energy.

⋆ Thermodynamic and vertical coordinate inconsistencies between dynamical core and parameterizations.

**Summary and future directions (Section** **5****)**

We end by summarizing the various energy budget imbalances discussed in this paper. This provides an opportunity to rank these errors in order of importance, and suggests where advances need to be made to reduce energy imbalances in a large‐scale (traditional) climate model setting.

Since the workshop participants also include experts in cutting edge mathematical areas that could lead to major advances in developing more energetically consistent models, we take the opportunity to provide accessible introductions to these new approaches. These include:

**Geometric mechanics formulations for Geophysical Fluid Dynamics (GFD)**

A framework for deriving energy conserving equation sets beyond the hydrostatic primitive equations is given in Section 5.2.1.

**Structure‐preserving discretizations**

Strategies for eliminating or reducing energy discretization errors in the fluid flow solver (i.e., the dynamical core) using mimetic discretizations.

**Thermodynamic Potentials**

Section 5.2.3 introduces the concept of a thermodynamic potential in some detail. While the use of thermodynamic potentials is standard in the ocean community, due to the more complex equation of state for seawater (TEOS‐10 standard;

Several discussions and details have been moved from the main text to Appendices, with the intention of a better flow of the discussions in the main text. These Appendices include comprehensive detail and we hope they will serve as a standard reference for expert readers. The Appendices are:

*A note on averaging: unresolved and unrepresented total energy.* This is a huge, complicated and immensely important subject. Although it is not tackled in this paper, we feel it is important to introduce the reader to this subject.

*Derivation of the single‐fluid energy equation and energy conservation equation in the primitive equation set with a generalized vertical coordinate in the presence of mass sources.* A detailed derivation of local energy equations for a single‐fluid is given. Among other things, this Appendix serves the purpose of discussing how the geopotential and internal energy terms are converted into surface geopotential and enthalpy terms. The two pairs can not be conflated. Their relationship to the surface fluxes is also presented.

*Enthalpy and energy formulas using different reference states.* The energy formulas can be derived using different reference states and different communities tend to use different reference states. For completeness, we include the energy formulas using all possible reference states. The Appendix is divided into several sub‐sections: (B1) Partial specific enthalpies for different reference states, (B2) Surface enthalpy flux, and (B3) Final energy equations with different reference states.

*Ocean enthalpy fluxes.* A brief discussion of the enthalpy boundary condition between atmosphere and ocean as well as their associated complications.

*Mechanical dissipation of kinetic energy and resultant heating.* The momentum exchange between the atmosphere and ocean is intriguing since it is not a material energy flux. The ocean and atmosphere exert mechanical work on each other through a stress tensor, but there is no transfer of kinetic energy via a flux. The energetics of that problem is discussed in some detail.

*Use of the barycentric velocity as a basis for consistently representing the frictional dissipation of rain in atmospheric models.* Large‐scale models have many missing processes associated with water entering and leaving the column. One missing process is the frictional heating of falling precipitation, which can be represented using the barycentric velocity framework. This is not explicitly discussed in the literature and so we add a discussion on this approach in this Appendix.

*CAM setup.* Details of the climate model setup we are using in this paper to assess the magnitude of different terms in the energy equation.

*Sponge‐layer diffusion and frictional heating.* This Appendix supports the discussion of sponge layers and energetics.

List of symbols appearing more than once in the main text to help the reader through the many equations.

From a modeling perspective an ESM is a collection of components (atmosphere, ocean, sea ice, land surface, …) that are coupled so that they exchange energy. The upper boundary of an ESM is the top of the atmosphere component and the lower boundary is either the sea floor or the lower boundary of the land component (see Figure 1). From first principles the Earth's total energy budget should be closed, meaning that the change in total energy of the Earth system, *ρ*^{(all)} is the total density (defined in detail later). Here, the direction of the fluxes is defined positive upwards. The subscript *esm* refers to ESM. The upper boundary flux of energy

The total energy of the ESM, *E*_{esm}, can be divided into atmosphere (“atm”), ocean (“ocn”), etc. components

The total energy conserved by the governing equations of motion and associated thermodynamics is referred to as the *fluid equations of motion energy*, *E*_{feom}. The fluid equations of motion and the thermodynamics are usually approximated. For example, the fluid equations of motion may make the hydrostatic assumption, which means that the total energy of the system *E*_{atm} includes energy associated with motions neglected in the fluid equations of motion (e.g., non‐hydrostatic motion, breaking gravity waves and turbulence). Other simplifications in the fluid equations of motion could include neglecting individual momentum equations for hydrometeors, and making the single temperature assumption, so that all components of moist air have the same temperature. Therefore, the total energy for the atmosphere may be divided into the energy associated with the fluid equations of motion and the energy associated with all motions and processes (such as radiation) not represented in the fluid equations of motion, *E*_{other}

Alternatively, one could describe *E*_{other} as *unrepresented* energy, as it is not represented in the fluid equations of motion and their associated thermodynamics. We obviously do not have an exact expression for *E*_{other}. However, some parameterizations include approximations for the energies associated with some of the processes that we know must be in the true unrepresented energy *E*_{other}. As models are run at higher and higher resolutions, and use less approximated equation sets and associated thermodynamics, more and more processes are being represented in the fluid equations of motion energy rather than in the “other” category.

In addition to this prior argument for the continuous equation of motion, there is an even more complex problem. It is not possible to run ESMs at the small scales necessary to resolve all fluid and physical processes. We must therefore homogenize (i.e., average) processes smaller than about 50–100 km in operational climate models, and roughly 0.5–3 km for cutting edge convection‐permitting global models. In particular, the smallest energetic turbulent scales of motion are *O*(10^{−3}) m, which means that the energy *E* will always have both a resolved and an unresolved component

Things now become complicated and less well understood. Since this topic, though immensely important, is not the main focus of this paper, a (brief) discussion of the treatment of resolved and unresolved scales is provided in Appendix A. Henceforth, we will focus mainly on the unaveraged energy equations and only, when necessary, distinguish between resolved and unresolved scales (e.g., when discussing frictional heating and sub‐grid‐scale energy reservoirs).

The energy *E*_{feom} associated with the governing equations of motion is the energy of a fluid subject to gravity and rotation:

The various equation sets discussed below have different expressions for kinetic energy *K*, internal energy *U* and geopotential Φ, which depend on the particular assumptions made and the choice of thermodynamics. Commonly made assumptions for geophysical fluid models are both geometric (traditional shallow‐atmosphere, spherical‐geoid) and dynamical (quasi‐hydrostatic, Boussinesq). In what follows, our emphasis will be almost entirely on the form of internal energy *U* and the associated thermodynamics, since this is usually the place where significant approximations are made. It is also where inconsistency is easily introduced. An important assumption we will always make is that of *local thermodynamic equilibrium*, which ensures that mean fluid quantities such as temperature and pressure are well‐defined.

It is important to note here that in this section we discuss the total (conserved) energy for a given set of assumptions (e.g., hydrostatic, shallow atmosphere) for the fluid part of the atmosphere. If we consider only reversible dynamics, total energy in fact defines the associated processes that occur in the fluid (through the Lagrangian/Hamiltonian formulation). However, this same energy is also conserved if we introduce irreversible dynamics, which is a new set of processes, but with the same energy. For example, both the compressible Euler (inviscid) and compressible Navier–Stokes–Fourier equations (viscous) conserve the same total energy. However, the Navier–Stokes–Fourier equations have additional terms describing the irreversible (viscous) processes. This distinction is important since we are interested in physics‐dynamics coupling, which necessarily involves both reversible and irreversible dynamics.

A formal general derivation of the energy conservation equations for the primitive equations is given in Appendix B. Here we take a more constructive approach, for didactical purposes and for application to special cases.

We start by defining the energy for the atmosphere component *E*_{feom} for commonly used equation sets. The energy formulae are for the non‐averaged equations applying to all scales.

Many ESMs make use of the *primitive* equations on the sphere (essentially the Navier–Stokes equations in the hydrostatic and shallow atmosphere limits). We refer to this equation set as the hydrostatic primitive equations (HPE). The corresponding total energy is well‐established in text books and the literature*K* is *horizontal* specific kinetic energy (*U* is the specific internal energy and Φ is the geopotential. Specific total energy *K* + *U* + Φ differs from that of a non‐hydrostatic flow (obeying the standard Euler or Navier–Stokes equations) only in that vertical kinetic energy does not contribute. In the commonly used shallow atmosphere approximation Φ = *gz* where *g* is the (constant) gravitational acceleration. Note that there is an arbitrary choice of reference for the geopotential. The common reference for *z* is mean sea‐level so that *z* is elevation above sea level.

To begin, let us consider a single phase, single component fluid, that is, the dry atmosphere. Dry air is actually a mixture of many different gases, but it is assumed to have constant concentrations of each component, same temperature and velocity and thus is referred to as a single‐component fluid.

For an ideal perfect gas the internal energy (up to a constant reference discussed later) is given by

The energetics of the dry primitive equations should be unambiguous. However, we will highlight several sources of widespread ambiguities that arise even for these simple equations.

**Ambiguity 1**: It should be noted that specifying how *U* depends on *T* does not fully determine the thermodynamics used by the atmospheric model. In general, *U* is not a function of temperature alone. However, even when it is, the relationship *U* = *U*(*T*) must be complemented by extra information, such as the perfect gas law_{B} is the Boltzmann constant,

**Ambiguity 2**: Depending on the type of boundaries and the boundary conditions, total energy might not be conserved. Typically, in the atmosphere we assume a rigid lower boundary and either a rigid (constant‐height) or a pressure top upper boundary. In the case of a rigid upper boundary, total energy will be conserved up to boundary fluxes, with the following expression*dA* = *r*^{2} cos *φdλdφ* is an infinitesimal surface area element, *z*_{t} is the height at the model top and *ρ*^{(d)} is the mass of dry air per unit volume of air (i.e., density). However, in the case of a pressure top, total energy is not conserved, even in the case of no boundary fluxes. Instead, the fluid exchanges energy with the environment through the motion of the upper surface. There is a conserved quantity (up to boundary fluxes), sometimes referred to as the total enthalpy, given by_{t} is the geopotential height at the model top. The additional term accounts for the work done by the moving pressure top. Finally, Equation 12 can be integrated by parts to yield_{s} = *gz*_{s} for shallow atmosphere approximation, *z*_{s} is surface height, and

Apart from differences in the top boundary condition, a widespread ambiguity is that it is tempting to regard

The definition of energy per unit mass, or local energy, must include a term for internal energy, but specific enthalpy, which in the particular formulation above equals*α*^{(d)} is the specific volume of dry air.

The transformation that allows one to use *h*^{(d)} instead of *U*^{(d)} under the integral in Equation 13 is possible only for the HPE with specific boundary conditions. Modifying the boundary conditions or the specific assumptions underlying the HPE invalidates this transformation. For instance, this transformation is not valid for deep‐atmosphere equations, or shallow‐atmosphere based non‐traditional quasi‐hydrostatic equations. However, depending on the upper boundary condition used either Equation 11 or Equation 12 will hold.

We will largely ignore these complications, however, and use Equation 13 as our total energy expression in this section.

**Ambiguity 3:** The energy flux is NOT proportional to the energy density! For tracer transport, for example, a material quantity *ψ* is transported, and the flux of *ψ* is equal to *ψ* times mass flux so that the transport equation reads: *E*_{feom}. This is, however, incorrect. The local energy density

**Ambiguity 4:** When using a pressure‐based vertical coordinate system where we have just pointed out that the integrand in the energy Equation 13 is not local energy density, one may ask why most physics packages are formulated in terms of constant pressure and, if conserving energy, use (Equation 13)? The reason is that enthalpy is commonly used to redefine the first law of thermodynamics:*Q* is heating per unit mass. From Equation 15 and substituting Equation 18 it is observed that for enthalpy we have

Hence we see that when pressure is constant and there is no heating *dQ* = 0 then enthalpy is conserved (Vallis, 2006). This simplifies conservation of energy during phase transitions (once the definition of enthalpy is extended to include water). Since the total energy in the formulation (Equation 13) includes only boundary terms that do not change under the constant pressure assumption, using enthalpy as the conserved thermodynamic variable automatically conserves total energy of the model.

Aside: Using (Equation 18) the prognostic thermodynamic equation used in models can be written in an “internal energy form” (heating under constant volume)*w* is vertical velocity and *ω*^{(d)} ≡ *Dp*^{(d)}/*Dt*.

Henceforth, unless otherwise stated, we will assume a pressure‐based vertical coordinate, *η*, with the common assumption of constant pressure at the top boundary.

First, we remark that with a pressure‐based vertical coordinate, the energy equation can be written more generally in terms of specific enthalpy, *U*^{(d)} + *p*^{(d)}/*ρ*^{(d)}, which for an ideal perfect gas (up to a constant reference) is*ρ*^{(d)}*dz* despite using a pressure‐based vertical coordinate. The reason for using *ρ*^{(d)}*dz* is to avoid the ambiguity of using *p* as a mass integration variable. In addition to the differences associated with the hydrostatic and non‐hydrostatic approximations, *p* could either include or exclude all water, or all condensed water, or precipitating water etc. It seems better to avoid that ambiguity and extra complication by explicitly integrating over vertical length times all the relevant densities.

Large‐scale models will include, at least, the gas phase of water (i.e., water vapor) in its equations of motion and thermodynamics in addition to dry air. Many readers will likely view this as a solved or trivial problem. However, this seemingly modest increase in complexity leads to significant energy inconsistencies in many current large‐scale models (due to using the specific heat for dry air also for water vapor).

Most large‐scale models make the following assumptions:

**Temperature assumption**: Assumes that all constituents of the fluid (in this case, dry air and water vapor) have the same temperature.

**Momentum assumption**: Assume that dry air and water vapor move with the same horizontal velocity (at this point we are not considering phase changes, precipitation and evaporation, so we may call *gaseous* velocity vector).

With these assumptions the addition of water vapor does not modify the general expression *K* + *U* + Φ for total energy, but it does lead to two changes. One is the addition of water vapor mass to the mass of the atmosphere and the other is the thermodynamic effect. These will be discussed in the following two paragraphs.

**Mass effect**: Density now includes the effect of water vapor so that*m*^{(ℓ)} is the dry mixing ratio defined by*ρ*^{(ℓ)} is the mass of species *ℓ* per unit volume of moist air. Note that *m*^{(d)} ≡ 1. The mass of air now includes contributions from all species in moist air. If *M*^{(ℓ)}(*z*) is the mass of species *ℓ* in a column of air above height *z**z* is

Equations 24, 26, and 27 are completely general and not just applicable to moist air consisting of dry air and water vapor.

It is noted that one usually formulates the above equations using a moist basis in terms of specific contents

If we assume that dry air and water vapor move with the same velocity the advantage of using a dry basis instead of a moist basis is that the mixing ratios do not need to be re‐computed when, for example, water leaves the air parcel (precipitation). In this case *ρ*^{(all)} changes so *q*^{(ℓ)} must be recomputed. On the other hand, *ρ*^{(d)} remains constant so *m*^{(ℓ)} does not need to be re‐computed. This is discussed at length in Lauritzen et al. (2018). However, note that if using a barycentric velocity basis, the prognostic continuity equation for dry air *ρ*^{(d)} will have additional source/sink terms based on the diffusive precipitation and turbulent mixing fluxes (discussed in Appendix F2 as well as Wacker and Herbert [2003]). These terms are, however, small.

**Thermodynamic effect**: The second effect of water vapor is to modify the equation of state *p* = *p*(*T*, *ρ*^{(d)}, *ρ*^{(wv)}) and, more generally, all thermodynamic relationships. A simple way to take water vapor into account is to regard dry air and water vapor as ideal perfect gases, and moist air as an ideal mixture of them. From first principles:

The perfect gas law holds in the form*ℓ*, where *ℓ* = *d* (dry air) or *wv* (water vapor)). For non‐gasses (i.e., condensates) *R*^{(ℓ)} is zero.

The perfect gas law is often rewritten in the form*T*_{v} defined as

Specific heat at constant pressure:

Internal energy per unit mass

As a result, the primitive equations conserve the total energy (we reiterate that the integrand of Equation 35 is *not* total specific energy)

The above expression in Equation 35 decomposes the total energy into contributions from each component of moist air treated as an ideal mixture of perfect gases. For real moist air, for example, Hermann et al. (2009) and Feistel et al. (2010), the expressions

The effect of water vapor is not always included in the specific heat at constant pressure, that is, some models keep

While the above practice involves a slight loss of physical accuracy, it is energetically consistent since the equations conserve an energy. Note that we are considering a mixture of dry air and water vapor only, so there are no latent heat terms. When including condensates in moist air (and thereby phase transformations) the latent heat terms appear in the energy equation as discussed in detail in the next section. The energy inconsistency error occurring if dynamics uses (Equation 35) and physics (Equation 37) is discussed in Section 4.5.1.

Now let's add condensates to the mix! Condensates are not gases, so their inclusion in the thermodynamics is not straightforward. A series of assumptions are made in today's ESMs to keep the problem more tractable, rather than using a full multi‐component fluid formulation (Bannon, 2002).

We extend the assumptions from the previous section to air that contains condensates:

**Temperature assumption**: Assume that all constituents of the fluid have the same temperature.

**Zero volume assumption**: Assume the condensates do not take up any volume.

**Momentum assumption**: Assume that all components of moist air move with the same horizontal velocity

The hydrostatic mass effect of condensates is straightforward. As mentioned in Section 2.1.2, the equations for total density (Equation 24), species masses *ℓ* (Equation 26) and hydrostatic pressure (Equation 27) are the same when including condensates. Typical condensates included in ESMs are cloud liquid *cl*, cloud ice *ci*, rain *rn*, snow *sw* and graupel *gr*. In this case the components of moist air in equations for

Since

The specific heat of liquid forms of water are assumed the same

For notational convenience we refer to the density of liquid forms of water as *ρ*^{(liq)} = *ρ*^{(cl)} + *ρ*^{(rn)}, and similarly for frozen forms of water *ρ*^{(ice)} = *ρ*^{(ci)} + *ρ*^{(sn)} + *ρ*^{(gr)}. All mixing ratios represent mass of water species with respect to the mass of dry air. If the model's microphysics scheme requires more forms of water they can easily be added as long as they are characterized correctly in terms of liquid and ice. For notational purposes it is convenient to denote

Please note that even if we assume condensates to be incompressible, the partial pressures of dry air and water vapor are both affected by the weight (force exerted by the matter when it is in a gravitational field) of the condensates. We see this by integrating the hydrostatic balance equation*ρ*^{(all)} includes condensates) and using Dalton's law of partial pressures to observe that*M*^{(ℓ)} is mass per unit area for species *ℓ* defined in Equation 26. When there are no condensates present then

In the presence of condensates we can no longer pair terms on the left and right‐hand side of Equation 48 and separate the pressure into a dry and moist part (see, e.g., detailed discussion in Section 2.4.2 in Lauritzen et al. [2018]).

The perfect gas law and virtual temperature take the same form as presented in Equations 31 and 32, respectively, if one assumes that the physical volume occupied by condensates is negligible (for details on the validity of this approxiation see Equations 1.71–3 in Staniforth et al. [2006]). The virtual temperature is often approximated further

However, if this expression for *T*_{v} is used in the perfect gas law *p* = *ρ*^{(all)} *R*^{(d)} *T*_{v} then the resulting thermodynamics can be inconsistent, depending on how the equations are used. For example, diagnosing *p* from *p* = *ρ*^{(all)} *R*^{(d)} *T*_{v} using an approximated *T*_{v} is, in general, inconsistent. A general framework for making consistent approximations is described in Section 5.2.3.

Relaxing the momentum and temperature assumption (Section 2.1.2) leads to a significant increase in complexity. However, if we do make the momentum and single temperature assumptions, the total specific energy takes the same form, *K* + *U*^{(all)} + Φ, but now condensates contribute to *ρ*^{(all)}, the internal energy and enthalpy.

The specific enthalpy of an air constituent *ℓ* (we may call it a partial enthalpy) can be written in the form*ℓ* at *T*_{00} = 273.15 K = 0°C (it would be equally valid to use the triple point of water which is *T*_{00} = 273.16273 K). Note that for ocean models the complete expression for enthalpy is a nonlinear function of pressure, temperature and salinity rather than a simple linear function of temperature as used in atmosphere models, (Equation 52). The enthalpy of moist air can be written as

Expanding the sum on the right‐hand side of Equation 53 we get

Only enthalpy changes have physical meaning. Hence, we rewrite Equation 54 using Kirchhoff's equations for latent heats (note that latent heat release is when the heat is liberated to become sensible heat; here we are only talking about the physical measures and not yet the physical effect):

**Latent heat of vaporization (liquid** **→** **water vapor):**

**Latent heat of sublimation (solid** **→** **water vapor):**

**Latent heat of fusion (solid** **→** **liquid):***L*_{f}(*T*), may also be written in terms of latent heat of vaporization and sublimation

In Equation 54, there is some liberty in choosing the reference state in terms of reference enthalpy *T*_{00} (we highlight this dependency on the left‐hand side of the latent heat formulae in Equations 55–57). That said, the constants entering the definitions of the specific enthalpies of the different phases of water are not independent of each other but must be such that *h*^{(wv)}(*T*) − *h*^{(liq)}(*T*) = *L*_{v}(*T*) and *h*^{(liq)}(*T*) − *h*^{(ice)}(*T*) = *L*_{f}(*T*) as is shown next.

If we use the enthalpy of ice

The specific enthalpies for a liquid and water vapor reference states are given in Appendix C1. Substituting Equations 59–61 into the total enthalpy formula (Equation 54) and rearranging terms yields*E*_{feom} for other reference states in Appendix C1). We reiterate that *E*_{feom} is *not* total specific energy, that is, *E*_{feom} must be integrated over the entire domain to constitute a global energy equation.

The dry air constant term

We also note that if we expand the latent heat terms on the right‐hand side of Equation 63 using Equations 56 and 57 and re‐arrange terms, Equation 63 can be written as*L*(*T*).

Perhaps a natural choice of reference enthalpy is

We have, so far, not considered energy fluxes into and out of the atmosphere, in particular, at the surface. Given Equation 17 (cf., also Equations B18–B20), the total energy fluxes at the surface are divided into

Atmospheric models lose water vapor (dew, rime), liquid (rain) or ice water (snow, hail, …) to the land, land‐ice, sea‐ice and the ocean. Similarly, the atmosphere receives water vapor (through evaporation and sublimation), snow (through snow drift, except from the ocean) and other forms of water from the land, land‐ice, sea‐ice or ocean surface. Not all of these processes are included in ESMs (usually just evaporation from the surface and precipitation in either frozen or liquid form from the atmosphere).

When, for example, water evaporates from the ocean the atmosphere gains energy (and mass) which is compensated for by ocean cooling due to the latent heat flux. Hence, this process occurs without any net change in the total energy of the coupled system. In addition to the latent heat flux there is also an enthalpy flux associated with the mass transfer between components that depends on the temperature of water. For evaporation, this temperature can naturally be assumed to be that of the ocean or land surface. However, for falling water the temperature at landfall is harder to compute (since falling precipitation is not prognostic in our current framework). More on this later.

For a mathematical description of the surface enthalpy flux exchange, we distinguish between enthalpy transfer from the atmosphere to the surface, denoted with subscript *a* → *s*, and from surface to atmosphere, *s* → *a*. Similarly, let the temperature of the atmosphere at the surface be denoted *T*_{atm,s} and that of the surface itself be *T*_{surf,s}. Although there is a thin layer at the surface for which *T*_{surf,s} = *T*_{atm,s} we distinguish between the two, since models usually operate with layer mean temperatures, in which case, the surface layer temperatures will be different between components.

The enthalpy flux from atmosphere to surface depends on the form the water has before it hits the surface. This is also true for enthalpy fluxes from surface to atmosphere (although this is simpler, as discussed in the previous paragraph). Assuming local thermodynamic equilibrium, the flux from the atmosphere to the surface is the mass‐flux of each species *T*_{atm,s},

In reality each form of water has a different temperature in which case the terms in the sum on the right‐hand side of Equation 66 would be *ℓ*.

The superscript *h* refers to total enthalpy flux (*F* denotes flux). We are assuming that the dry air flux is zero,

Similarly, the flux from the surface to the atmosphere (exactly the same derivations as before but with *T* = *T*_{surf,s} and flux reversed; *s* → *a* instead of *a* → *s*)*T*_{surf,s} is the surface temperature in the surface. Water vapor can enter the atmosphere through evaporation and sublimation. The process with which liquid enters the atmosphere could be spray and ice can enter the atmosphere through snow drift.

The net flux from the surface to the atmosphere,

Note that the net fluxes of mass balance the mass change in the column*ℓ* due to falling precipitation or evaporation and not phase changes, *ℓ* in the column, and thus the terms cancel regardless of the choice of reference enthalpy *T*_{00}.

For simplicity we assume that the temperature at the interface is the same in both atmosphere and the surface,

The surface enthalpy flux formulas for other reference states can be found in Appendix C2. Note that assuming that all forms of water enter and leave the atmosphere at the same temperature is questionable (this assumption is discussed in more detail in Section 2.1.13). Also, this surface enthalpy interface condition does not consider a surface component using a different definition of enthalpy. An example from ocean‐atmosphere coupling is given in Appendix D. In this paper, we will not pursue this more complicated interface condition further, however, the reader should be aware of the simplifications made in this section.

The flux at the surface should specify the flux of kinetic, internal and potential energy. Thus far we have only discussed the enthalpy flux. The flux of horizontal kinetic energy due to falling precipitation and water entering the atmosphere can be written as*K*_{atm,s} is the specific kinetic energy at the surface from the atmosphere side (or surface layer as we will discuss in a moment) and *K*_{surf,s} is the horizontal specific kinetic energy of the water entering the atmosphere, respectively.

The complications associated with specifying the surface (horizontal) kinetic energy flux is discussed further in Section 2.1.13. For simplicity, let's assume that

Ideally the potential energy flux through the surface would require one to track the altitude at which each water molecule in the air evaporated from the surface, and subtract the geopotential when that water molecule left the atmosphere as precipitation. Thus precipitation takes potential energy out of the atmosphere *only* if it hits the ground at a different altitude than evaporation (analogous to hydropower in mountains). Lacking such information, our best guess is that the overwhelming majority of water has evaporated from the ocean and the flux can be approximated as

Another argument for Equation 74 is that most of the potential energy excess of precipitation with respect to Φ_{s} is dissipated aerodynamically when the drops/snowflakes fall through air (after conversion of that excess potential energy into kinetic energy of the droplets). Hence most of the Φ − Φ_{s} energy is in fact left in the air. A way to consistently include that process in models (that assume a single temperature) is given in Appendix F.

Note that we have implicitly chosen sea‐level as the reference height for computing potential energy. So, similarly to enthalpy, there is a reference state (height) for potential energy. No models, as far as the authors are aware, transfer potential energy between components.

In addition to the fluxes discussed so far, there is also a surface turbulent/sensible heat flux *F*^{(turb)} at the surface which come about by direct contact interaction with the surface. It is typically parameterized based on measurements of turbulent transport at a height, *z*, in the atmosphere:*St*_{H}, depending on underlying model, *z*, and the modeled states. Although these dependencies vary, flux differences are generally much less than the uncertainty of Equation 75, with the use of temperature difference

In addition to the flux terms and heating/cooling terms associated with phase transformations (that are already represented in the enthalpy terms), there is *bulk* heating and cooling within the atmosphere. In particular, heating/cooling that is associated with radiation. Others could be turbulent or wave dissipation incl., for example, ion friction in the mesosphere. The radiative heating/cooling is represented as the divergence of radiative fluxes and such a term should be included on the left‐hand side of the energy equation. Integration of this term over a column results in boundary terms (top of atmosphere, ToA, and surface terms). These terms can be written as*α*_{s} the surface albedo and

We re‐iterate that the radiative fluxes represent a bulk source/sink of atmospheric energy, not a surface (or ToA) boundary term; these result from integrating the divergence of the radiative flux over the column. For notational clarity (and since this term is not discussed further), the sensible/turbulent heat flux and the bulk heating/cooling terms are lumped into one term,

For notational brevity we assume that water entering/leaving the atmosphere have the same temperature, *T*_{atm,s} = *T*_{surf,s}, in the “final” energy equations (extending to the more general equation allowing for different temperatures for water entering or leaving the atmosphere is straight forward by using Equation 69 instead of Equation 71 for the net enthalpy flux). Adding all the terms described in the previous sections results in the following total energy equation

Final total energy equations using other reference states can be found in Appendix C3.

For the following discussions/observations the time‐change of water species *ℓ* is separated into local phase changes and changes associated with water entering or leaving the column

**Important observations to be made regarding the enthalpy reference state terms**: First of all, note that the constant enthalpy term

In Equation 80 we have decomposed *∂m*^{(ℓ)}/*∂t* into local phase changes and falling precipitation

Second, note that, by making the common simplification of assuming that all heat capacities are that of dry air, the reference temperature terms on the left‐ and right‐hand side of the energy Equation 78 cancel since

Third, if variable heat capacities are used then a similar equation only holds for falling precipitation and water entering the column*T*_{00} ≠ 0; it is in effect an enthalpy term asociated with the latent heats of water phases and implies no dependence on the chosen reference state (ice, liquid, vapor) or temperature *T*_{00}. For constant latent heats and phase‐independent water heat capacities (e.g., *T*_{00} so that effectively one would get the same “physical” result (i.e., the energy equation should not depend on reference state but the left‐hand side and right‐hand side separately do depend on reference state).

In the following we will discuss the total energy equations under various assumptions. The discussion has been split into several sections and will (in some sections) touch upon vast and complicated problems/subjects with extra detail diverted to Appendices. The sections are about energy conservation

(Section 2.1.10) under adiabatic conditions (dynamical cores),

(Section 2.1.11) under the presence of local phase changes due to (parameterized) microphysics (no falling precipitation, no re‐evaporation, no momentum changes due to other unresolved processes, etc.),

(Section 2.1.12) under the presence of momentum sources and sinks due to parameterized processes such as boundary layer turbulence, gravity waves, etc. and

(Section 2.1.13) for falling precipitation/evaporation and surface fluxes.

Consider an adiabatic dynamical core with inert water species that is, there are no phase transformations and/or fluxes so the global mass of each component of air is conserved. As a consequence, the latent heat terms and constant enthalpy terms integrate to zero in the total energy Equation 62

The total energy equation becomes

It is common practice in large scale models to assume that the latent heat terms are constant which, by examination of Equations 55–57, is equivalent to assuming that all components of moist air have the same specific heat,

Parameterizations are, in general, not formulated such as to satisfy the conservation law in Equation 62, but rather in terms of local thermodynamic variables that are conserved in the process under consideration (e.g., various forms of static energy; Emanuel, 1994). Nevertheless, parameterizations (e.g., micro physics) that rearrange water locally through phase transformations from one species to another, denoted

For notational clarity on the left‐hand side of Equation 92 we add the term _{s} terms do not appear in Equation 92. Also, the property that *∂p*(*t*, *η*)/*∂t* = 0, for example, the vertical coordinate is pressure‐based. Then Equation 92 is a special case of the first law of thermodynamics *dh*^{(all)} = *α*^{(gas)}*dp* = 0 when *dp* = 0.

Invoking the usual assumption/approximation that all forms of water have the same specific heat,

Any temperature change due to local phase changes (not falling precipitation or surface water sources) should satisfy this equation locally (in each grid point). A discussion of

Sources and sinks of momentum (e.g., gravity wave parameterization, boundary layer turbulence schemes or other drag parameterizations) affect kinetic energy, and enforcing total energy conservation in their presence is not straightforward due to its interaction with sub‐grid‐scales. In particular, we would like to point out that a “naive” closure of the energy budget by transferring kinetic energy change into heat is, in general, **not** physically correct*∂K*/*∂t* should in general be decomposed into a flux divergence and source, and *only* the latter is potentially convertible into heat. Let us explain in more detail.

Frictional dissipation of kinetic energy ultimately occurs at the molecular scale where it is turned into heating (a.k.a. frictional heating). However, at resolutions at which ESM's are run there are many processes resulting in a frictional force on momentum that are much more efficient than viscous effects for transferring momentum to/from sub‐grid scales (e.g., small‐scale turbulent eddies, “eddy friction” due to cumulus momentum mixing and breaking gravity waves). These processes are represented by a stress tensor in the equations of motion (e.g., Smagorinsky, 1963).

Since this discussion is focused on column physics, let's consider only frictional terms due to the vertical mixing of horizontal momentum (formally, the *z*‐component of the divergence of the stress tensor *τ*_{ij}, see e.g., p. 36 in Peixoto and Oort [1992]); that is, we are ignoring the cross terms in the stress tensor. The frictional force on momentum is the divergence of the stress tensor (in tensor notation

Equation 97 is the mean kinetic energy equation as it describes the evolution of resolved‐scale kinetic energy. The interaction between the sub‐grid scale kinetic energy (often referred to as **turbulent kinetic energy**, **TKE**, but could also be other un‐resolved wave interactions such as gravity waves) and resolved scales is represented by the stress tensor terms (right‐hand side of Equation 97).

Now lets integrate (Equation 97) over a layer. First it is noted that by repeated application of the chain rule for differentiation after expanding the material derivative and using the continuity equation for density, the left‐hand side of Equation 97 can be written as*D*/*Dt* = *∂*/*∂t* + *w∂*/*∂z* rather than

As a results the mean kinetic energy equation (Equation 97), integrated over a layer can be written as

The first term on the right‐hand side of Equation 99 is in flux‐form. Therefore, if we assume zero velocity or stress at the lower and upper boundaries (more on that in Section 2.1.5) then this term does not change kinetic energy in the column (only redistributes kinetic energy between layers). The second term represents the shear production of TKE, and is a term in the sub‐grid scale (turbulent) kinetic energy equations (Lumley & Panofsky, 1964).

The crux, in terms of energy conservation, is whether TKE produced by shear should be converted into a resolved‐scale temperature tendency or not. A more conceptual question is whether this conversion should be called viscous/frictional.

Regarding the first question, if subgrid energy reservoirs are neglected, total resolved energy should indeed be conserved, which can only be achieved by translating the shear production of TKE into an increase of temperature. Whether the underlying process is indeed viscous dissipation is more subtle. For example, by the second law of thermodynamics, conductive heat transfer in stably stratified turbulence increases entropy, and thus potential temperature, thereby increasing column‐integrated enthalpy at a rate linked to the parameterized turbulent heat flux. The energy needed for this increase comes from the TKE. Only the fraction of TKE not consumed by heat transfer is dissipated to heat by viscosity, and conversely the parameterized heat flux cannot exceed an upper bound controlled by the shear production of TKE (Akmaev, 2008).

Thus, although a loss of resolved kinetic energy must be compensated somehow by an increase of (potential) temperature, the underlying small‐scale processes are not purely viscous. An example of a rigorous closure in ocean models, in terms of energetics, is given in Eden (2016).

In the special case where the stress tensor represents an eddy momentum flux (down‐gradient diffusion typically found in turbulence and boundary layer schemes) and is parameterized as*ν*^{(all)} is positive (note also that eddy diffusion with *ν*^{(all)} > 0 fulfills the second law of thermodynamics [Schaefer‐Rolffs & Becker, 2018]) and can be included in the (resolved‐scale) thermodynamic equation for energy conservation (e.g., Bister & Emanuel, 1998):

Note that the local change in the winds *K* (first term on left‐had side of Equation 101) is missing.

In general, however, there is no guarantee that the last term on the right‐hand side of Equation 99 is negative; if positive (e.g., for sub‐grid‐scale gravity waves accelerating the flow) it would lead to “frictional cooling” if added as to the thermodynamic equation to close the energy budget, which makes no physical sense (Becker, 2001). Hence, in general, a sub‐grid model is needed to close the energy budget since artificially closing the energy budget at resolved‐scales can lead to unphysical cooling. This is related to the immensely complicated subject of the interaction between resolved and unresolved scales discussed briefly in Appendix A.

For a comprehensive discussion of entropy budgets in numerical models see Gassmann and Herzog (2015). The horizontal part of frictional heating is discussed in the context of dynamical cores in Section 4.1. Also, kinetic energy and the boundary condition (surface stress) is discussed in Section 2.1.5.

In Section 2.1.11 we discussed energy conservation for processes in the atmosphere not involving falling precipitation and evaporation. For water entering or leaving the vertical column the picture becomes much more complicated compared to, for example, having a closed energy budget for phase changes.

Making the assumption that falling precipitation (that does not re‐evaporate or undergo other microphysical interactions) instantly hits the surface after formation (i.e., that it does not interact with the atmosphere, so that it falls like an *insulated rain shaft*), the latent heat terms on the left and right‐hand side of Equation 77 should exactly match

This equation states that the temperature

For Equation 105 to be satisfied (i.e., closed energy budget) under the “insulated rain shaft” assumption then

While Equations 103–106 under the insulated rain shaft assumption are energetically consistent, important physical processes are omitted. For example, measurements and theory for the surface rainfall temperature show that it is approximately equal to the surface wet‐bulb temperature (Byers et al., 1949; Gosnell et al., 1995; Kinzer & Gunn, 1951), which, on average, will be warmer than the temperature at which condensation occurs. When the droplets hit the surface their temperature can be different than the surface ambient air temperature or temperature of the actual surface. For example, Anderson et al. (1998) observed that in the Indo‐Pacific warm pool precipitation is ∼5 K colder than the sea surface temperature (SST). This means that specifying

**Frictional heating of falling precipitation**: Let's first discuss the vertical component of kinetic energy of falling precipitation as this has received some attention in the literature without broad adoption in global models. The process is referred to as frictional heating of falling precipitation (Pauluis & Dias, 2012; Pauluis et al., 2000). As soon as a droplet starts to fall there is a loss of potential energy which is first converted into kinetic energy, which is afterward turned into internal energy by friction. Representation of this process is possible (it is actually automatic) using the barycentric velocity framework. This is described in detail in Appendix F. Like in the case discussed above where shear production is not the same as molecular dissipation, here the mixing dissipation represents the true Stokes dissipation (which is likewise invisible). In Gassmann and Herzog (2015) this reinterpretation for numerical models has been given by analyzing the local entropy production of this process. Using the barycentric framework this problem can, to some extent, be considered solved. That said, this formulation assumes additionally that all species have the same temperature. There is some concern in the community about that assumption. Since global models usually do make the constant temperature assumption, using the barycentric framework would be a first step towards adding missing processes associated with precipitation and evaporation.

The horizontal momentum of falling precipitation has not been addressed (to the authors knowledge) in the literature. The falling droplets will, in general, exert a drag in the horizontal (slowing down the horizontal winds) as they fall through the atmosphere. Also, when the droplets hit the surface they will have a horizontal momentum component (even though the wind may be zero in a very thin layer at the surface, the lower model level will not have zero winds). Possibly the common Lagrangian frame of reference as described in Appendix F could automatically treat this problem.

In conclusion, when making the insulated rain‐shaft assumption it is problematic to assign a temperature to the latent heat flux and enthalpy flux (right‐hand side of Equations 103 and 104, respectively). If one assumes that the heat capacities for all forms of water are the same,

Most large scale models do not rigorously account for the energy terms associated with falling precipitation and water entering the atmosphere, other than through (constant) latent heat terms. In terms of equations, this means that terms on the left‐hand side of the energy equation are not balanced by parameterizations in the atmosphere and right‐hand side flux terms:

The magnitude of these terms will be estimated in the next section.

While the previous section is quite general in its assumptions (e.g., single temperature and velocity), in this Section we discuss detailed issues that occur due to specific assumptions that are made in CAM in terms of energetics. Therefore details specific to CAM will be explained in some detail although they are not necessarily relevant to other modeling systems.

We present a concrete example of the energy budget, using simulation results from the CAM climate model. We focus on the energy budget in parameterization suites where the total energy budget should be closed in each column. In the dynamical core, where there is interaction between columns, the integrated energy equation only holds globally.

To keep the discussion tractable we assume a hydrostatic model with the total energy equation given by Equation 77 that uses an ice reference state. This energy equation holds for the continuous state variables

Before discussing the energy errors, it is insightful to first assess the magnitude of the energy budget terms in a real‐world climate simulation. Again, to keep the discussion tractable, we use the CAM model, and hence make assumptions specific to CAM. We will also diagnose energy budget changes associated with relaxing certain assumptions, for example, by introducing variable latent heats and fluxes associated with enthalpy, kinetic and potential energy.

Please note that in the following we focus on the vertically integrated energy budget in the CAM physics parameterizations. These represent all diabatic heat sources of the equations of motion. In a steady‐state, or long‐term time average, their vertical integral must match energy flowing in and out of the upper and lower boundaries in order for energy conservation to be satisfied. The horizontal transport we are going to neglect (and which is computed in the dynamical core of the model) in general gives by design a vanishing contribution to the global total energy budget, and its main role is simply to redistribute the heat of the diabatic sources, column by column, such as to yield a locally vanishing long‐term time average. Therefore, errors in the column energy budgets due to physics parametrizations directly impact the dynamics, which in response produces local errors equal in magnitude and opposite in sign. Note that the integrated energy equations derived in Section 2 can be applied to the partial (physics‐only) budgets in the column, since the tendencies associated with transport are neglected. Local energy conservation equations are presented in Appendix B2.

In terms of neglected terms and approximations to terms in the energy equation, CAM physics makes the following assumptions:

The latent heat terms are constant. This is equivalent to assuming that the specific heats of all species are the same (in the case of CAM *∀ℓ*).

The kinetic, enthalpy and surface‐geopotential surface flux terms are neglected so that only the sum

The ice phase represents zero reference enthalpy, *T*_{00} = 0°C (for reasons that will become clear that term is kept in the equations).

With these assumption the energy equation for each physics column can be written as

Only water vapor contributes to kinetic, potential and sensible heat: *K*, Φ_{s}, and *c*_{p}*T* terms multiplied by

In addition, the total water mass is assumed constant during physics updates. That means the water mass, *K*), potential (Φ_{s}), and non‐latent enthalpy (*c*_{p}*T*) is regarded as constant over the time‐step; however the water mass associated with latent heat is updated consistently with water mass conservation and phase changes. Keeping mass fixed is denoted with subscript *t* = *t*^{n}, that is, keeping the mass of water species *ℓ* constant at its initial value when initiating physics parameterization computations is denoted

These simplifications lead to the following energy equation that each CAM parameterization, in theory, should satisfy:

Since total water is kept fixed, the total water flux term

After the last parameterization the total water (in CAM's case

By subtracting Equation 111 from Equation 112 (and expanding the time‐derivative using the chain rule for differentiation), it can be shown that the imbalance of updating total water is

**Caution**: We remind the reader that the two last terms in the integrand on the right‐hand side of Equation 113 are NOT potential and internal energy (see ambiguity 3 in Section 2.1.1). These terms came about by integration by parts and application of a specific boundary condition used in most pressure‐based models. In this form the internal and geopotential energy terms are not separated. The two terms are enthalpy (which obviously has physical meaning) and (what we may refer to as) a surface geopotential term. If we consider falling precipitation; the change in geopotential energy (not surface geopotential term) is large (since Φ − Φ_{s} is large where falling precipitation forms) compared to the internal energy difference between where falling precipitation is formed and the ground value of the internal energy. But in Equation 113 the enthalpy term is the largest. Hence it is important not to conflate the internal/potential energy terms with enthalpy and surface geopotial energy terms. A complete mathematical description of this “caution” in the context of CAM is given in Appendix B1.

If we split *∂m*/*∂t* into phase changes/kinetic energy of condensates “staying in the atmosphere,”

The first term in the imbalance, Equation 115, would have been zero if all forms of water were included in

A global energy fixer computes the global integral of this imbalance

**An important observation**: Any total energy formula should be independent of reference state and, in particular, the reference temperature *T*_{00}. Since we are using constant latent heats, we see that the *T*_{00} terms on the left and right‐hand side of Equation 112 cancel regardless of the value of *T*_{00} (see derivations in Equation 84) if all forms of water are included in *E*_{feom}*F*^{(wv)}*L*_{s,00} + *F*^{(liq)}*L*_{f,00} and *F*^{(wv)}*L*_{v,00} − *F*^{(ice)}*L*_{f,00}, respectively, with associated reference temperatures. For variable latent heats it becomes much more complicated as discussed later.

A caveat for the above argument (particular to CAM) is that only water vapor is included in the *T*_{00} terms on the left and right‐hand side of Equation 112 so they do not exactly cancel since the left‐hand side term includes “spurious phase transformations” as described above. Hence there is a spurious dependence (although likely small) on the reference state in CAM due to only including water vapor in total water.

Next we would like to give the reader a sense of what the terms in the CAM total energy equation look like in real‐world climate simulations. Figure 2 shows spatial plots of 1‐year average values of terms in Equation 111. The terms are computed using inline diagnostics from an Atmospheric Model Intercomparison Project (AMIP)‐like simulation cycling over year 2000 SSTs (more details on the CAM setup is given in Appendix G). Row 1 shows 1‐year averages of the left‐ and right‐hand sides of Equation 111 vertically integrated in each column as well as the residual. That is, (a) is the average time‐tendency of total energy from all CAM parameterizations (assuming total water stays constant) and (b) shows the time‐averaged fluxes into each column. Since CAM physics has been constructed (in theory) so that each parameterization satisfies Equation 111, plot (a) and (b) should balance. This is shown in (c) to be true to within 10^{−6} W/m^{2}. The reader is reminded that we show terms in the total energy equation only for CAM physics processes, with vertically integrated energy tendencies. In steady‐state, or equivalently for long‐term time averages in climate equilibrium, the energy budget must be closed (Figure 2c), that is, the net energy flux into the column through its upper and lower boundaries (surface energy flux minus ToA energy flux) must be exactly balanced by the physics heat tendencies integrated over the column.

Figures 2d–2f show the break‐down of total energy flux terms. That is, latent heat fluxes (d,e) and turbulent/sensible and radiative flux (f), respectively. Note that individual latent heat flux terms depend on the reference state (compare the last two terms in Equations C5, C7, and C9), but the total latent heat balance (adding up left and right‐hand side latent heat terms) does not (physically, latent heats are measurable quantities and should therefore not depend on reference state).

While Equation 111 provides a closed energy budget, total water (which in the case of CAM is *K*, Φ_{s} and enthalpy ^{2} globally, and over ±30 W/m^{2} locally, as observed in previous atmosphere studies (Guo et al., 2019; Harrop et al., 2022; Lauritzen & Williamson, 2019) and forced ocean simulations (see Appendix A4 in Griffies et al. [2014]). While CAM does not have “explicit enthalpy fluxes” it is noted that they are effectively in the global energy fixer (but not necessarily with the correct sign since all the energy stays in the atmosphere and is not communicated/exchanged with the surface).

In the following we assess the energy tendencies associated with relaxing assumptions in the CAM total energy equation. A natural first step is to include all forms of water in the total energy Equation 111, so that there is no spurious phase change term. This leads to the following imbalance^{2} (see Figure 4a). Larger values are found in heavy precipitation zones. Note that the energy of this imbalance should stay in the atmosphere. Henceforth we will neglect

If we now compute the dry‐mass adjustment, including all water species in Equation 111, we get the following imbalance:

In the previous section estimates of the energy changes in the column that either should be compensated for by missing boundary fluxes or missing processes in the atmosphere have been given. First, to understand and analyze the energy fluxes at the surface, it is useful to plot the water fluxes at the surface (since they are paramount to the energy flux terms). Figure 5a shows the net water flux (with positive sign for water entering the atmosphere) and (b,c,d) show the breakdown of the flux into water vapor, liquid and ice fluxes, respectively. Not surprisingly, the liquid and water vapor fluxes dominate the net fluxes (on average).

In the CAM energy equation for each column the neglected flux terms for kinetic, geopotential and enthalpy, are

Note that in Section 2.1.6 the gepotential flux term was approximated by substituting Φ with Φ_{s}. Essentially that means that we assume that the “Φ − Φ_{s} energy” remains in the atmosphere. We argue that this assumption is physically defensible. For falling precipitation potential energy is converted to kinetic and then into heating as a result of microturbulence dissipation processes (frictional dissipation of falling precipitation; Appendix F).

As discussed in Section 2.1.13, it is problematic to specify the surface kinetic energy and temperature in Equation 120. For the sake of simplicity, considering that surface‐air kinetic energy is generally small, and that the surface air temperature often (not always!) is very close to the surface temperature, let's use the kinetic energy in the lowermost level in the atmosphere, and similarly for temperature. Note that splitting the net material (water) fluxes into downward and upward with different temperatures can easily be done—see Section 2.1.4 for the enthalpy flux.

Including these terms in the CAM energy equation, as well as using all forms of water in

It is important to note that these imbalance terms do not depend on reference states. Before discussing the physical meaning of the imbalance terms, we note that including the extra flux terms on the right‐hand side of Equation 121 drastically reduces the imbalance in areas where the imbalances without those terms were large (^{2} to ∼0.5 W/m^{2}.

Rows 1, 2 and 3 in Figure 7 shows the first and second term on the right‐hand side of Equation 122 (enthalpy), Equation 123 (kinetic energy) and Equation 124 (geopotential energy), respectively, as well as the difference (imbalance). Note that the separate terms depend on somewhat arbitrary reference states (so, on their own, they do not have physical meaning) whereas the residuals (column 3) are not (they have physical meaning).

Why are the residuals in Equations 122–124 not zero? A theoretical answer to this question was given in Section 2.1.13. We will repeat the argument here, but backed with values from our simulation. Let's start with the kinetic energy imbalance where, the “dry‐mass adjustment” (Figure 7d) is an order of magnitude larger (on average) than the surface kinetic energy term (Figure 7e). This is due to the falling precipitation being formed higher up in the atmosphere where the winds (i.e., kinetic energy) tend to be larger than the surface winds (that are used to compute the surface *K* flux in this experiment). In CAM it is assumed that precipitation instantly falls to the ground (as if it was falling through an insulated rain shaft). Assuming that the precipitation does indeed hit the surface with a horizontal velocity component equal to the ambient atmosphere (lower level winds) then

The interpretation is similar for **Caution** paragraph” under Equation 113, that this formulation approximates the geopotential flux in terms of Φ_{s} (see Section 2.1.6) and therefore assumes that the Φ − Φ_{s} energy of falling precipitation remains in the atmosphere (through conversion of potential energy to kinetic energy to frictional dissipation/heating).

We note that cleverly choosing a surface wind and temperature that exactly matches the “dry mass adjustment” for kinetic energy,

The same argument can be made for the reverse process (transfer of water mass from the surface to the atmosphere), for example, evaporation. At the air‐sea (or ‐lake, or ‐soil/‐canopy) momentum is exchanged in the mass transfer. In this case momentum transfer within the gas phase is extremely efficient and can be safely assumed to be complete.

The temperature of water vapor from the surface is likely close to the known surface temperature.

Now we move to the most complicated energy equation we will discuss here. Let's no longer assume constant latent heats. In this case the total energy equation becomes

First of all we note that Equations 129 and 130 are the same as

The difference between Equations 132 and 133 is the imbalance

The difference between these two equations is the enthalpy imbalance

The first integral on the right‐hand side of Equation 134 is the imbalance due to not using the “correct” heat capacities for all forms of water, combined with the imbalance of only using water vapor in total water. The second integral would have been zero if all water species had the same specific heat since total water is constant for phase transformations. Since a phase change with variable latent heats is “weighted” with different specific heats this extra term is non‐zero. Unfortunately, we cannot compute this term in our diagnostics since we are not able to separate phase‐changes from falling precipitation/evaporation. In general, however, the two terms are of opposite sign and of comparable magnitude. for example, for pure freezing of for example, supercooled cloud water, the first term is positive

The global average energy imbalance resulting from using variable latent heats when excluding falling precipitation/evaporation, ^{2} (see Figure 8a) which, as mentioned in the previous paragraph, is likely an overestimation. The imbalance is more than 1 W/m^{2} in heavy precipitation zones where there are large amounts of water vapor phase changing to cloud liquid. Since the specific heat of liquid water is about four times larger than the specific heat of dry air, it is not surprising that areas with large precipitation amounts have the largest signal. This imbalance should remain in the atmosphere and would be zero if CAM physics would have used variable latent heats.

Now let's consider the imbalance terms due to falling precipitation/evaporation. To compute the imbalance terms in Equations 129 and 131 we use that

Figure 8b shows the first term in the imbalance

Note that Figures 8b and 8c depend on the specific reference temperature used in CAM. Compared to CAM, the enthalpy terms on the left and right‐hand side of the energy equation are now weighted by their actual specific heats. Compared to ^{2} compared to 10–20 W/m^{2} in default CAM with constant (dry air) latent heats. For heavy precipitation zones the imbalance increases by roughly a factor of four, as expected. The mean is now −20 W/m^{2} as compared to 0.3 W/m^{2} in default CAM. This is again due to the relative weighting of the different phases. In Figure 5 one sees that evaporation and precipitation roughly balance in terms of water flux amount. However, precipitation is weighted twice as much as evaporation in the enthalpy term because the specific heat difference makes the imbalance negative.

Now to the imbalance *T* and *K*. For example, in the tropics the falling precipitation is on average colder than the

An example of a variable versus constant latent heat inconsistency in a coupled ESM is given in Section 4 of Golaz et al. (2019). The imbalance in this case was approximately 0.5 W/m^{2}. To avoid spurious energy imbalances in the coupled climate system, the local inconsistency was fixed globally by adding a correction term to the sensible heat flux (Appendix A in Golaz et al. [2019]).

The analysis above is based on annual average values and thereby omits temporal variations in the energy budget imbalances. Time dependence of energy imbalances will be discussed briefly in the following section in the context of the global energy fixer in CAM.

The global energy fixer in CAM restores total energy conservation by fixing energy imbalances introduced by the dry‐mass energy adjustment (^{2} and ∼1 W/m^{2}. The blue line on Figure 9b is the energy imbalance only for the dry‐mass adjustment (‐^{2} and −0.32 W/m^{2}, respectively (the latter is consistent with Figure 3a; note opposite sign since the Figure shows ‐^{2} and 0.44 W/m^{2}, respectively. In short, most of the imbalance is due to the dry‐mass adjustment approximation.

In the second run, we account for the enthalpy of precipitation and evaporation (the last term in Equation 120) in the total energy computation for the global fixer using temperature of the bottom atmospheric layer. Removing this term from the total energy of the atmosphere at the end of the physics step in CAM mimics the transfer of the enthalpy of precipitation from the atmosphere and transfer of the enthalpy of evaporation to the atmosphere. These transfers are currently missing from the model. In their absence, the global fixer recovers corresponding quantities within the atmosphere, with the net amount close to zero.

Although the global, averaged in time, amounts of enthalpies of precipitation and evaporation are close in magnitude, their variations in time are not small. These variations significantly affect the amount of energy distributed by the global fixer. As shown in Figure 9 (a) (red line representing the global energy fixer) and (b) (red line representing the dry‐mass adjustment), accounting for the transfer of the enthalpies significantly reduces the variability in the global fixer and dry‐mass adjustment, with standard deviations 0.13 and 0.10 W/m^{2}, respectively. The averages of these quantities, −0.35 and −0.25 W/m^{2}, respectively, are close to the default run, which confirms that the averaged effect of the transport of the enthalpies (or its absence) is small.

From Figures 9a and 9b we conclude that, on average, the amount of energy redistributed by the global fixer is close to the amount of the energy flux introduced by the dry‐mass adjustment. To confirm this, we perform a third run where we reset the global energy of the model to the energy after the dry‐mass adjustment. In other words, in this run the energy fixer corrects only energy errors or fluxes due to the discretizations in the dynamical core and forcings that are applied in the dynamical core (Lauritzen & Williamson, 2019), not due to the dry‐mass adjustment. In this case, the curve in Figure 9c of the global fixer has much smaller variability, with standard deviation of 0.07 W/m^{2} and average −0.10 W/m^{2}. Hence, the energy error associated with the dynamical core and physics–dynamics coupling combined is much smaller in magnitude and variability than the error associated with the dry‐mass adjustment.

This concludes the analysis of (CAM) physics energy terms and imbalances. We now consider other energy budget errors in ESMs.

In the absence of space‐time truncation errors, the dynamical core conserves the same energy that the continuous equations of motion conserve. In the case of the HPE system, the total energy equations were discussed in Section 2.1.10. However, no operational dynamical cores known to the authors conserve total energy, mostly due to explicit horizontal diffusion of the prognostic variables, filters and other sources of numerical energy error (e.g., Jablonowski & Williamson, 2011). At ∼1° resolution dynamical cores typically dissipate energy at a rate of 1 W/m^{2} (Lauritzen & Williamson, 2019). This energy dissipation can come from intrinsic dissipation (e.g., shape‐preserving finite‐volume operators) or explicit dissipation. The latter is discussed next.

Consider artificial Laplacian diffusion of momentum added to the momentum equations in a HPE model_{h} is applied along coordinate surfaces and *ν*_{2} is an empirical second‐order horizontal diffusion coefficient. In some models, this term is added to the equations of motion for numerical stability reasons to control spurious numerical noise at the grid scale. The diffusion term is formally similar to molecular viscosity (a physical process) although it is artificial. It differs from molecular viscosity in that it is not 3D and only applied along coordinate surfaces without a correction term to *z*‐surfaces.

The associated kinetic energy equation (similar to Equation 101) is

For higher‐order operators (e.g., ^{2} so it reduces the energy error by roughly ∼50%. Since it is not obvious how to separate its diffusive and dissipative parts, this “artificial frictional heating” is more like a local energy fixer than a correction derived from theory (i.e., deriving a symmetric stress tensor). Note that models that use implicit diffusion through limiters and filters in the numerical operators (e.g., Lin, 2004) do not have an explicit expression for diffusion/dissipation, and it is less obvious how to represent frictional heating due to momentum dissipation.

An aspect that needs attention in weather and climate models is the choice of all dissipation processes and, in particular, the treatment of the upper boundary in the atmospheric dynamical core. Most often, dynamical cores apply artificial sponge‐layer mechanisms near the model top to damp upwards traveling waves and reduce the wind speeds. These can grow substantially in the upper model domain which is, in particular, true for models with high model lids between 80 and 150 km. Examples of explicitly applied sponge‐layer dissipation mechanisms in the dynamical core portfolio of NCAR's CESM version 2.2 (CESM2.2) are Rayleigh friction, enhanced second‐order horizontal divergence damping, or an energy‐ and mass‐conserving 2‐Δ*z* filter in CAM's cubed‐sphere finite‐volume dynamical core FV3 (L. M. Harris et al., 2021). In addition, CAM's spectral transform Eulerian (EUL, Neale et al., 2010) and the Spectral Element (SE) dynamical cores (Lauritzen et al., 2018) apply enhanced second‐order horizontal diffusion near the model top. Some models like ICON, ICON‐IAP and MPAS use the gravity wave damping mechanism which is inherent in the vertical solver (Klemp et al., 2008).

Regardless of the chosen dissipation method, once it is applied to the velocity components or the related horizontal divergence and relative vorticity fields it reduces the kinetic energy of the flow. This triggers the question of whether lost kinetic energy should be converted to heat in an effort to conserve the total energy. And if yes, how? Various algorithms are possible for this energy conversion which can either target a local energy restoration or a global energy fix.

Here, we focus on the local energy restoration that is applied after an explicitly applied dissipation process. In general, the kinetic energy loss due to dissipation *δK* < 0 can be assessed via the approach*ρ*^{(all)} stands for the density and is treated as a constant. This is typically justified since the velocity dissipation mechanisms do not change the mass or pressure at a grid point. In addition, *δK* computation depends on the dissipation mechanism, time‐stepping approach and the dynamical core design, for example, whether a hydrostatic or nonhydrostatic design is chosen. The details are not important for the generic discussion here. However, a concrete example for Rayleigh friction is shown in Appendix H.

The kinetic energy change (−*δK* > 0) needs to get converted to heat which, as before, needs to take the design of the dynamical core into account. For example, when picking a hydrostatic HPE model with a pressure‐based vertical coordinate and the horizontal velocity vector *u* and *v* the conversion utilizes (assuming a pressure‐based vertical coordinate)

The resulting, slightly warmer temperature *T*_{conserve} then replaces the temperature *T* locally via*w*.

It is also possible to approach the problem differently and utilize the chain rule for differentiation to assess the kinetic energy change. This leads to

Equation 144 apparently differs from Equation 139. However, in models decisions need to be made about the time discretization, and

Whether frictional heating should be applied, especially at the local scale and not as a global total energy fixer, is debatable as it was outlined for the Laplacian diffusion operator above. When the dissipation is treated as a physical process one can find arguments for it. However, if the dissipation is simply used to prevent numerical artifacts, such as the reflection of waves at the model top, then frictional heating becomes questionable and might even lead to artificial temperature signals. This is demonstrated for a Rayleigh friction (RF) sponge layer in Figure 10 that shows the time‐mean and horizontal‐mean temperature of the FV3 dynamical core (L. M. Harris et al., 2021) in a dry Held–Suarez test case configuration without topography (Held & Suarez, 1994). In general, the Held–Suarez test replaces the physical parameterization package of an atmospheric model with a Newtonian temperature relaxation and Rayleigh friction at low‐lying levels below 700 hPa. This thereby mimics the effects of radiation and the planetary boundary layer mixing, and enables climate‐like model assessments of a dynamical core.

In the example below, FV3's resolution was set to C24 (400 km) with 64 vertical levels that reach up to about 100 km. The vertical grid placement utilized the level configuration of NCAR's Whole Atmosphere Model WACCM6 with 70 levels (Gettelman et al., 2019), but without reaching up to WACCM6's model top at about 140 km. Rayleigh friction can formally be represented by *t* is the length of the time step and *k* is the RF damping coefficient. In FV3 RF is activated above a user‐chosen cutoff pressure *p*_{c} and utilizes the user‐selected maximum RF timescale *τ*. We choose *p*_{c} = 10 Pa and the three *τ* timescales 1, 3 and 5 days for demonstration purposes. These settings activate the Rayleigh friction in the top 14 levels between 10^{−1} and 10^{−4} hPa with increasing strength. The vertical profiles of the damping strength *k* are shown in Figure 10b, and the underlying equation is documented in L. M. Harris et al. (2021) and Appendix H. Appendix H also highlights that a fully time‐implicit algorithm is used to compute the induced frictional heating in FV3.

In FV3, local frictional heating due to RF in the sponge layer is applied by default. Here, we turn the frictional heating either on or off to document its impact on the time‐mean horizontal‐mean temperature in the upper atmosphere. This is demonstrated in Figure 10a that shows the 6‐month temperature averages in the upper domain after an initial 12‐month spin‐up period. This spin‐up period is needed to reach an equilibrium with the Held–Suarez forcing when starting from an isothermal state at rest, as done here. Figure 10a documents the model response to three RF damping timescales with and without frictional heating. The figure shows that the damping timescale *τ* has a decisive impact on the mean temperature in the sponge layer above 0.1 hPa. In general, stronger RF damping with shorter *τ* timescales and thereby increased Rayleigh friction coefficients *k* (see Figure 10b) leads to enhanced frictional heating effects in FV3. This is reflected by the higher mean temperatures with increasing RF strength (dashed curves). This is expected behavior and not the focus of our discussion. Here we highlight the large temperature differences for an identical *τ* setting that are solely due to the application or non‐application of frictional heating. These are depicted by the dashed and solid curves with identical colors. Differences of up to 15–20 K are apparent near the model top. These temperature differences due to frictional heating become even more pronounced above 10^{−4} hPa. If, for example, WACCM6's full vertical domain with a model lid around 140 km is utilized temperature differences of over 100 K were found near the model top due to frictional heating (not shown). However, a high sensitivity to other FV3 sponge‐layer mechanisms like an enhanced second‐order divergence damping (used by default in FV3 in the two topmost layers) and an optional 2‐Δ*z* filter was also found. In the experiments here the 2‐Δ*z* filter is not activated and the enhanced second‐order divergence damping mostly affects the topmost kinks of the temperature profiles in Figure 10a. The general spread between the simulations with and without frictional heating remains unaffected by the top‐layer divergence damping.

None of the depicted simulations with and without frictional heating can be considered a reference solution for the Held‐Suarez test. The Held–Suarez test relaxes the temperature above 100 hPa to 200 K so that the closeness to the 200 K isotherm can be used to judge the temperature profiles. The FV3 simulations start deviating from this isotherm rather quickly once the Rayleigh friction is activated above 0.1 hPa. This illustrates the impact of frictional heating in FV3, raises awareness, and encourages continued discussions about energy conservation and the use of frictional heating processes in atmospheric models.

In conclusion, we would like to highlight that Rayleigh damping has no physical meaning since it is ad‐hoc in form and not justifiable on the basis of physical theory. Hence, we argue that there should be no physical heating/cooling term associated to it in the thermodynamic equation. Also, Laplacian damping introduced near the upper boundary to prevent reflection of waves is purely artificial (introduced due to an artificial upper boundary) and does not represent a physical dissipation process. Therefore, the momentum that is being dissipated should not be converted into a heating/cooling term in the thermodynamic equation.

In addition to artificial diffusion operators, the choice of discretization of the inviscid equations of motion often affect the energy budget. Nevertheless, energy conserving numerical methods, called mimetic discretizations, do exist (e.g., Dubos et al., 2015; Eldred et al., 2019; Gassmann, 2013; Taylor & Fournier, 2010; Taylor et al., 2020) (see Section 5.2.1). Briefly, one discretizes the Hamiltonian and the Poisson brackets themselves, rather than the associated discrete equations of motion. The discrete equations of motion are then derived from the discrete Hamiltonian and Poisson brackets. If the Poisson brackets are discretized in a way that preserves the key elements of their geometric structure (anti‐symmetry and some of the null space), then the resulting equations of motion will inherit the desired conservation properties. The resulting Poisson system of ODEs can then be integrated in time using an energy‐conserving time integrator, such as a discrete gradient method (Cohen & Hairer, 2011). This latter step is rarely taken (although see Eldred et al., 2019), since these integrators are in general fully implicit and therefore too expensive for practical atmospheric models. In addition, the energy time‐truncation errors for inviscid dynamics are negligible (∼−0.005 W/m^{2}), making it hard to justify the increased computation expense of energy‐conserving time‐stepping. In contrast, spatial mimetic discretizations are currently being used in some operational climate dynamical cores (e.g., Dubos et al., 2015; Gassmann, 2013; Melvin et al., 2019; Taylor, 2011; Taylor & Fournier, 2010; Taylor et al., 2020).

In the absence of time‐truncation errors, vertical remapping errors and explicit diffusion the spectral‐element model conserves total energy. It has been found necessary for practical applications to employ explicit diffusion operators that dissipate energy, because the nonlinear (inertial) energy cascade extends beyond the resolved scales. The advantage of mimetic methods is that they help distinguish between the reversible and irreversible parts of a model.

In contrast, energy‐conserving vertical remapping has been found to be problematic. Hence, at least for the immediate foreseeable future, dynamical cores (even those using mimetic discretizations) require a total energy fixer to close their energy budget. Nevertheless, mimetic discretizations can reduce the amount of spurious energy that must be handled in the fixer. Figure 9c shows the global energy fixer for the spectral‐element dynamical core (and physics dynamics coupling) as a function of time‐step. Compared to the dry‐mass adjustment, the spurious tendency from the dynamical core (and physics‐dynamics coupling) is much less (∼−0.1 W/m^{2}) in terms of standard deviation and mean.

Assuming that physics and dynamics are based on the same equations of motion and thermodynamics, how physics and dynamics are coupled has a significant impact on the energy and mass conservation properties. Adding physics tendencies throughout the dynamical core time‐integration can produce energy errors (discussed in detail in Lauritzen and Williamson [2019]). The applicability of parallel coupling of physics and dynamics is limited by mass and energy errors (Donahue & Caldwell, 2020). If physics tendencies are interpolated from a physics grid to a dynamical core grid (and the two grids differ), then the interpolation can lead to energy errors (Herrington et al., 2018). There can also be physics–dynamics coupling errors if the continuous equations of the dynamical core conserve a total energy that is different from the one conserved in the physics. This could be characterized as a consistency error. An example is given in Lauritzen and Williamson (2019). For an extensive discussion on physics–dynamics coupling please see Gross et al. (2018). An example of similar issues in water mass conservation can be found in Zhang et al. (2018). Since coupling errors have already been discussed in the literature, and the scope of this paper must be limited, these errors will not be discussed further.

Richard Feynmann remarked that (Feynman et al., 1989),

The subject of thermodynamics is complicated because there are so many different ways of describing the same thing …with respect to the internal energy *U*^{(all)}, we might say that it depends on the temperature and volume, if those are the variables we have chosen ‐ but we might also say that it depends on the temperature and pressure, or the pressure and volume, and so on.

Not surprisingly various different conserved variables have been used in moist thermodynamics (see, e.g., Emanuel, 1994), and parameterizations may adopt one rather than another based on what makes most sense for that particular parameterization. This may lead to inconsistencies with the total energy formula. These inconsistencies occur when a parameterization is based on a conserved variable which is different from that of the host model. From first principles both sets of conserved variables are correct, but they do not lead to the same energy conservation formula.

Confused? Below is an example from the CLUBB parameterization of turbulence and clouds (Larson, 2017), which has been implemented in CAM (Bogenschutz et al., 2013). This example illustrates the intricacies of this problem, which are likely to be present in some form or extent in many comprehensive parameterization suites (as far as the authors are aware). Please note that this is not a discretization error, but rather an error arising from the choice of conserved variables. Since this subject has received little attention in the literature, it will be discussed in some detail in the following section.

The CAM interface with parameterizations conserves Equation 111. That is, the total enthalpy plus kinetic energy in each column satisfies

(Williamson et al., 2015) where, for notational clarity, the right‐hand side terms in Equation 111 are denoted *θ*_{l} below, a water vapor reference state is assumed (despite CAM using an ice reference state). The thermodynamic potential (which in CAM is a conserved quantity) is used to update temperature as a result of heating in CAM parameterizations (assuming a water vapor reference state). This thermodynamic potential is

Also note that since CLUBB conserves water,

CLUBB is a model based on the anelastic approximation for the fluid equations (Larson, 2017). Rather than enthalpy, CLUBB's prognostic thermodynamic variable is liquid water potential energy, *θ*_{l}, which is conserved under adiabatic transformations in the relevant approximations. The use of a different thermodynamic variable in CLUBB (*θ*_{l}) than in CAM

First, the mapping between *θ*_{l} and *θ*_{l} violates conservation of *θ*_{l} and was only later ported to CAM. The same problem also occurs in the Energy Exascale ESM (E3SM, Golaz et al., 2019).

In detail, CLUBB transports an approximate form of the conserved moist potential temperature *θ*_{l} (see Cotton et al., 2011; Tripoli & Cotton, 1981), which is defined as*θ*_{l},*w* is vertical velocity.

Therefore, if CLUBB is advanced one time step, then the profiles of *θ*_{l} before and after the call to CLUBB are related by*θ*_{l} Equation 152 into Equation 155, and multiplying by

Now, consider the host model CAM. Its conserved prognostic thermodynamic variable is enthalpy, Equation 149, which should be updated according to Equation 148. Since CLUBB does not alter *m*^{(ice)} until CLUBB's diffusion acts on it in a different part of the code, we will omit the last term in Equation 148 and take

If CLUBB were to conserve *T* by using the definition of

What CLUBB does (see Equation 156) is not what is required for conservation of CAM's ^{−1}, which preferentially weights CLUBB's tendencies in the upper part of the atmosphere. The resulting energy imbalance in each column is shown in Figure 11c. The global average tendency is rather large ∼0.4 W/m^{2}, and similar in magnitude to other leading energy imbalances/errors in the system. Currently, this imbalance is restored in each column as a uniform heating increment (but only in the layers where CLUBB is active). To avoid this inconsistency in CAM, CLUBB could use *θ*_{l} as its thermodynamic potential.

Aside: Figure 11a shows the total imbalance and Figure 11b the kinetic energy imbalance in CAM‐CLUBB. The kinetic energy imbalance is non‐zero since CLUBB does not include frictional heating (also discussed in the context of dynamical cores). The kinetic energy budget would be closed if local heating were applied where momentum is dissipated (through mixing/turbulence). This missing heating is large (∼1.8 W/m^{2}).

Another thermodynamic inconsistency in CAM‐CLUBB is that the sensible heat flux, *F*^{(turb)}, provided by the host model, CAM, is passed incorrectly to CLUBB.

Neglect kinetic energy (i.e., assume for the moment that CLUBB does not alter winds), neglect radiation, and assume that there are no phase changes. Then CAM's energy equation reduces to:

In contrast, CLUBB conserves

That is, CLUBB conserves a potential temperature variable rather than temperature. In the absence of phase changes, Equation 161 becomes

In order for the sensible heat flux to result in the same temperature change in the column, then the CAM turbulent heat flux should be scaled by the surface Exner function, Π_{s}, so that^{2}], and so

In the current coupling of CLUBB with CAM, the following flux is erroneously passed to CLUBB_{s} is large. By using the more consistent formulation of the sensible heat flux (Equation 163), the imbalance over orography is significantly reduced. Since the spurious biases over high orography were negative the imbalance in terms of the global average is increased from ∼0.41 to ∼0.57 W/m^{2}. Discussing the impact of this change to the climate is beyond the scope of this paper.

Parameterizations (e.g., vertical diffusion) or the dynamical core may produce small negative mixing ratios of water vapor and/or other forms of water. Usually these negative value are clipped by simply setting the mixing ratios to zero. This is seen by the model as a spurious source of water and thereby impacts total energy. That said, these errors have been found to be very small, but will show up in an accurate budget analysis. The error appears as a bias (always positive), so it could accumulate over time in long runs. This topic is discussed further in Zhang et al. (2018).

The dynamical core may be based on equations of motion and thermodynamics different than those assumed in the parameterization suite.

NCAR's spectral‐element dynamical core uses the same HPE equations as CAM physics, except that it assumes variable latent heats. This means that the enthalpy terms, and therefore the total energy formula, differ as discussed in detail in Section 2.1.10. This leads to a ∼0.6 W/m^{2} imbalance. This inconsistency can be alleviated by scaling the temperature increments, Δ*T*, accordingly

Many non‐hydrostatic models are formulated using a height‐based vertical coordinate (e.g., Skamarock et al., 2012). Leaving aside the non‐hydrostatic effect, the total energy differs between a *z*‐based HPE system and pressure based HPE system due to the different boundary conditions (constant *z* and *p* at model top, respectively); see Equations 11 and 13. Heating increments, Δ*Q*_{k}, added under constant pressure and constant volume assumptions lead to different temperature increments (

A summary of total energy errors discussed in this paper is given below:

Dynamical core total energy errors (Sections 2.1.10 and 4.1) can be large (∼0.1–1 W/m^{2}) at large‐scale model resolutions (∼1°; Lauritzen & Williamson, 2019). Errors are expected to decrease as horizontal/vertical resolution increases. At least for lower resolution applications, further research in energy conserving numerical methods is highly recommended (Sections 5.2.1 and 5.2.2).

Temporal physics–dynamics coupling errors (where tendencies are added throughout the dynamical core time loop) can be large (∼0.4 W/m^{2}, Lauritzen & Williamson, 2019, see also Section 4.2), but can be reduced by clever physics–dynamics coupling methods (e.g., mass‐weighting of momentum and enthalpy tendencies; see Section 4.5). Errors, in general, are expected to decrease with shorter physics time‐steps and adequate weighting of the tendencies for energy conservation.

Spatial physics–dynamics coupling errors due to mapping tendencies between grids (if applicable) are small (∼−0.1 W/m^{2}, Herrington et al., 2018). This error will decrease with increased resolution (see Section 4.5).

Physics–dynamical errors due to the fact that the energy associated with the continuous equations of the dynamical core is different than the energy of the physics (e.g., due to different a‐principio approximations) can be large when the dynamical core uses variable latent heats and physics does not (∼0.5 W/m^{2}, Lauritzen & Williamson, 2019). These errors can be alleviated by clever scaling of tendencies (e.g., Equation 165; see Section 4.5). Other errors could be associated with constant volume dynamical core versus constant pressure physics (Section 4.5.2).

Enthalpy tendencies associated with falling precipitation and water entering the atmosphere are large (∼0.3 W/m^{2}) when using constant latent heats and even larger (∼1.1 W/m^{2}) when using variable latent heats. Locally, the errors can be orders of magnitude larger. This error, in general, is not expected to decrease with increased resolution. In fact, larger water contents at higher resolution may make the issue worse. This error is extensively discussed in Section 3.

Kinetic and potential energies associated with falling precipitation (and evaporation or other forms of water entering the atmosphere) are small (∼<0.02 W/m^{2}). This error is expected to increase with increased resolution for the same reasons as for enthalpy flux. This error is extensively discussed in Section 3.

Errors associated with not including all forms of water in pressure/mass are small (∼<0.01 W/m^{2}). Local errors could increase with increased resolution as the water content locally is larger at higher resolutions. This error is extensively discussed in Section 3.

Thermodynamic inconsistency errors in parameterizations: These imbalances are, of course, specific to the inconsistency in question. For example, we showed that the inconsistency between CLUBB using moist potential temperature *θ*_{l} as its conserved variable and CAM using enthalpy leads to ∼0.4 W/m^{2} imbalance. Similarly, inconsistencies in surface sensible heat flux between the host model and the parameterization (in this CLUBB) can lead to significant errors/imbalances in the total energy budget. This error is extensively discussed in Section 4.3.

For a closed energy budget that fully accounts for falling precipitation (and water entering the atmosphere) it is necessary to specify temperatures and kinetic energies of water entering/leaving the atmosphere (for the surface flux terms). This has also been noted in several other studies assimilating observations and diagnosing energy budgets (Kato et al., 2021; Trenberth & Fasullo, 2018). This is problematic (as discussed here in detail) since falling precipitation is formed away from surface and the temperature and winds can be quite different from those near the surface where, ultimately, falling precipitation leaves the atmosphere. Complicating the matter is that large‐scale models use a single temperature for all components of moist air. This is especially an issue for larger hydrometeors (e.g., hail) that tend to have appreciably different temperatures than the air around them and the supercooled droplets that they accrete. This is because riming releases latent heat that can't be diffused away quickly. As these large hydrometeors become included in advanced microphysics schemes, such as *PUMAS* (Gettelman & Morrison, 2015), the error of using the assumption of a single temperature may be worth considering. Also, climate models do not typically parameterize processes associated with falling precipitation (frictional heating of falling precipitation, drag exerted by falling hydrometeors, etc.).

The total energy budget errors associated with these processes can be large and we recommend that this issue be addressed as a high priority in climate and weather models. We have shown that including a surface enthalpy flux (though difficult to accurately compute) can reduce the variance of energy errors/imbalances significantly (see Figure 9) but, maybe, not entirely for the right reasons. A framework for representing frictional heating of falling precipitation can be achieved by using the barycentric velocity framework as outlined in detail in Appendix F. It still makes use of the single temperature assumption that most ESMs follow, so using the barycentric velocity framework could be a first “incremental” step towards a more realistic and better closure of the total energy budget.

Given these challenges in rigorously closing energy budgets, it is unlikely that the community will be able to entirely remove energy fixers in operational models anytime soon. We recommend that, if an energy budget cannot be closed locally, global energy fixers should be used.

Global fixers do not seem to affect the climate negatively, and they restore the global energy budget so that coupled climate simulations do not have spurious drifts. Trying to restore energy conservation with a local fixer can lead to incorrect solutions (e.g., Williamson et al., 2009). As a rule of thumb, a small global heating increment (i.e., a global energy fixer) seems to be less detrimental. That said, some local errors in current models (e.g., CAM) are entirely fixable. These include adding frictional heating to compensate for down‐gradient diffusion of kinetic energy. Note that this would not actually be a fixer, but would emulate a physical process.

In this section, we conclude the paper by suggesting possible future directions and associated major problems that have begun to receive some attention in the literature. This list is not exhaustive. It is based on countless discussion between the BIRS workshop participants that inherently could be biased due to the lack of participation from experts in all aspects of Earth System Modeling.

Global models are moving away from the HPE set to equations of motion that are less approximated, and even incorporate some irreversible processes into the dynamical core. As this transition is made, it becomes increasingly tedious and error‐prone to ensure that fundamental physical principles are satisfied. These physical principles include energy conservation and entropy conservation by reversible processes, and entropy generation by irreversible processes. Verifying these physical principles is especially challenging when approximations are made at the level of the equations of motion. In this section, we give a brief discussion of three related developments that aim to make this process simpler, and provide a unifying approach:

**Geometric mechanics formulations** (i.e., Lagrangian or Hamiltonian) to ensure physically consistent equations of motion.

**Structure‐preserving mimetic discretizations** based on these geometric structures. These discretizations are inherently energy‐conserving.

**Thermodynamic potentials** to ensure consistent thermodynamics.

This section also includes short notes on averaging operators and the available and non‐available components of energy.

Rather than working at the level of the equations of motion, a more powerful approach is to instead work with a geometric mechanics formulation, such as a Lagrangian (Holm et al., 1998, 2002) or Hamiltonian (Bannon, 2003; Dubos & Tort, 2014; Salmon, 1998) approach. This is commonly done in other areas of physics (it is at the heart of modern physics), and it can also be done for GFD.

Briefly, a variational approach to fluid dynamics states that the equations of motion arise from the minimization (through the calculus of variations) of the *action functional* *Lagrangian*.

By using the appropriate Lagrangian, one can recover all the consistent equation sets used in GFD for both single and multicomponent fluids. More specifically, it is possible to incorporate (Eldred & Gay‐Balmaz, 2021; Tort & Dubos, 2014):

*Geometric* (shallow/traditional atmosphere, spherical geoid, etc.) and *dynamical* (quasi‐hydrostatic, etc.) approximations.

Arbitrary *equations of state*, since the *thermodynamic potential* (see below) determines the internal energy *U*^{(all)} (in fact, internal energy is one of the possible choices for thermodynamic potential).

Sound‐proofing *constraints* that lead to *semi‐compressible* equations: anelastic, Boussinesq, pseudo‐incompressible, semi‐hydrostatic, etc.

*Free surface boundary surfaces*, such as a constant pressure upper boundary.

In all cases, by construction the resulting equations of motion conserve energy. The variational approach can also be used to derive the associated Hamiltonian formulation (Bannon, 2003; Dubos & Tort, 2014; Salmon, 1998), which is useful for (amongst other things) the development of numerical schemes that have no spurious energy conversions and therefore discretely conserve energy (see below).

In addition to providing a unified framework for reversible (entropy‐conserving) dynamics, geometric mechanics formulations (both variational (Gay‐Balmaz, 2019) and bracket‐based (Eldred & Gay‐Balmaz, 2020)) have recently been extended to incorporate irreversible (entropy‐generating) processes such as viscous dissipation. This includes pioneering work of Gassmann and Herzog (2015) and Gassmann (2018) on the treatment of physics parameterizations by analogy with molecular‐scale irreversible processes, so that they have the same functional form. This has provided, for the first time, a full model (dynamical core + physics) with a consistent treatment of the first and second laws of thermodynamics that does not require an energy fixer to correct mismatches between the dynamical core and the physics, and has a physically plausible treatment of entropy generation by physics parameterizations. Of course, treating subgrid processes by analogy with irreversible processes can have limitations and lead to an unphysical representation (despite being energetically and thermodynamically consistent), especially since many subgrid processes are actually unresolved reversible dynamics. Some key examples of this for the atmosphere are gravity wave drag and convection. Nevertheless, the work in Gassmann and Herzog (2015) and Gassmann (2018) represents an important first step towards energetically and thermodynamically consistent physics prameterizations.

Many of the important properties that the equations used in GFD possess, such as energy conservation, arise fundamentally from characteristics of the geometric structures that underlie them. For example, energy‐conservation can be associated with temporal symmetry of the Lagrangian in the variational formulation, or anti‐symmetry of the Poisson brackets in the Hamiltonian formulation. A structure‐preserving (or mimetic) discretization preserves these properties in the discrete sense, to produce numerical models that have the same properties as the continuous equations. This enables, for example, dynamical cores that inherently conserve energy for the correct physical reasons (e.g., reversible transformations between different energy reservoirs). Structure‐preserving discretizations have a long history in atmospheric dynamical cores, going back to the celebrated Arakawa and Lamb scheme (Arakawa & Lamb, 1981). Some recent work in this area for variational formulations is found in Gawlik and Gay‐Balmaz (2021), Bauer and Gay‐Balmaz (2019), Brecht et al. (2019) and for Hamiltonian formulations in Dubos et al. (2015), Gassmann (2013), Eldred and Randall (2017), Taylor et al. (2020), and Wimmer et al. (2020). It has even been extended to incorporate irreversible processes in Gassmann and Herzog (2015) and Gassmann (2018). Both research models such as ICON‐IAP and “production/operational” models such as CAM‐SE (used in this study), E3SM and DYNAMICO make use of this approach. In addition, both ICON‐IAP and the MPI‐M/DWD ICON model (Zängl et al., 2015) makes use of the barycentric velocity framework although a simplified lower boundary condition is used in ICON (*w* = 0; as a consequence ICON erroneously conserves the total mass rather than dry mass) and the diffusive fluxes of the non‐precipitating species are neglected.

Given the issues concerning thermodynamic consistency, we would like to point the reader to advances in this area as well. In the atmospheric dynamics literature it is common to work directly with equations of state like Equation 29 and potential temperature definitions. However, this approach becomes increasingly complicated and error‐prone as complexity increases, and it is easy to lose thermodynamic consistency if approximations are made to these expressions individually. This leads to violations of the first and/or second laws of thermodynamics, which is an undesirable scenario.

The correct approach is to start more fundamentally, with a *thermodynamic potential* (such as *internal energy*, *enthalpy* or *Gibbs function*) regarded as a function of *canonical state variables* (e.g., pressure, temperature and composition for the Gibbs function), and use this thermodynamic potential to derive the equations of state and other required thermodynamic equations or quantities (i.e., the thermodynamics of the model). Then, when making approximations, or introducing complications such as condensates, or using a temperature‐dependent

This thermodynamic potential approach is well‐known in the ocean modeling community, since realistic ocean models require an accurate approximation of the complex thermodynamics of seawater. This is achieved consistently through the definition of an approximate, but accurate, Gibbs potential (Feistel, 2008; Feistel et al., 2010; Hermann et al., 2009; Thuburn, 2017). This approach is less common in the atmospheric modeling community, but some recent work has investigated thermodynamic potentials for moist air. This includes an approach based on the Gibbs function for moist air with liquid water (Thuburn, 2017, 2018), a comprehensive study of multiple thermodynamic potentials for moist air with liquid and frozen water (including a consistent treatment of the commonly used constant *κ* approximation; Eldred et al., 2022), and an approach based on the internal energy for moist air with liquid and frozen water (that also includes non‐equilibrium thermodynamic processes; Bowen & Thuburn, 2022a; Bowen & Thuburn, 2022b). Bowen and Thuburn (2022a) found that using Gibbs function is viable for moist air containing vapor and liquid water but problematic to extend to include the ice phase (due to ambiguity at the triple‐point). They show that using internal energy potential resolves this issue.

Oceanographers have a new thermodynamic standard (TEOS‐10; *h*^{(W)}. Atmospheric modelers should be aware of how oceanographers have set the value of enthalpy for seawater, in order to define the value of the specific enthalpy for water vapor to ensure that *h*^{(wv)} − *h*^{(W)} is equal to an acceptable form of the latent heat of evaporation. How to achieve consistent definitions of liquid water and moist air enthalpies has been discussed in detail in Feistel et al. (2008).

An important issue highlighted in this paper is the need to understand the pathways to dissipation of mechanical energy. This is required to determine when the energy loss should contribute to the Joule heating associated with viscous dissipation in the temperature equation and when it should enhance the vertical turbulent transfer of heat and other tracers (resulting in change in the gravitational potential energy of the fluid, but with no contribution to viscous dissipation). In the oceanographic community, such an idea has motivated the development of “energetically consistent models,” for example, Eden (2016), whereby explicit descriptions of the sub‐grid‐scale energy transfers and of their interactions with the resolved energy reservoirs are sought. One possible way to achieve this is to develop separate evolution equations for the turbulent kinetic energy and the available potential energy. This approach is being explored in the atmospheric boundary layer community. Further research in this area is facilitated by the fact that a fully general local theory of available potential energy for multi‐component compressible stratified fluids is now available.

The energetics of a model depends a priori on the exact interpretation of the coarse‐grained prognostic variables. In particular, it depends on the averaging operator used to construct the coarse‐grained equations satisfied by the model. In practice, however, the precise form of this averaging operator is often unspecified. In oceanography, there is increased interest in coarse‐grained equations obtained from some form of Lagrangian averaging, such as the thickness‐weighed averaged equations, for example, Young (2012). In the atmosphere, however, Favre‐averaging may be a more natural way to account for the large variations in density. To a large extent, the present paper has approached the energetics of atmospheric models by treating the model variables in a similar way as their non‐averaged form. A more consistent and physically based approach would require to have a clear sense of the exact way in which the coarse‐graining has been performed.

For simplicity, the present paper considered only the total potential energy of the fluid. In particular, it did not attempt to develop separate budgets for its available and non‐available components as originally defined by Lorenz (1955). Physically, however, the available and non‐available forms of potential energy have very distinct signatures, so that it might be desirable in the future to extend the present analysis to the various forms of potential energy, by using the newly developed exact available potential energy frameworks of Novak and Tailleux (2018) and Tailleux (2018), although significant conceptual barriers remain to incorporating moisture in such frameworks, for example, B. L. Harris and Tailleux (2018).

A rigorous definition of the resolved and unresolved scales requires a complete equation set valid for all energetic scales of motion and processes present in the atmosphere. Unfortunately, we do not have such a set of equations, derived without approximation from first principles. Nevertheless, we can consider the most comprehensive system currently available. We then need to define an appropriate averaging operator to separate the resolved and unresolved scales. Unfortunately, there is not a unique choice for such an operator in general. The most developed example is large eddy simulation (LES), which assumes an averaging scale in the inertial range of fully developed turbulence (Lesieur & Metais, 1996). Reynolds‐averaged Navier—Stokes (RANS) models are less accurate, but are computationally cheap and easy to implement. They use Reynolds averaging to decompose a flow into its time‐averaged and fluctuating (turbulent) components and provide an approximate solution for the time‐averaged flow.

Fluid turbulence is a particularly straightforward case since we have a complete equation set (the Navier–Stokes equations) valid over all energetic scales of motion. We also have a numerical method, direct numerical simulation (DNS) to solve the unapproximated Navier–Stokes equations. DNS provides a benchmark for evaluating and validating various approximate models. In turbulence modeling an appropriate filter is used to partition the Navier–Stokes equations into resolved unresolved components. Because the Navier–Stokes equations are nonlinear, this filtering results in a residual Reynolds stress‐like term that represents the effect of the unresolved fluctuations on the resolved large scales. Modeling these Reynolds stresses is known as the “closure” problem, since an additional model is required to close the large scale equations. The best‐known closure model is Smagorinsky eddy viscosity, which was proposed in the 1960s for atmospheric flows (Smagorinsky, 1963). LES models are now well‐developed, and include a range of sophisticated filtering methods (e.g., dynamic LES, which uses the Germano identity (Germano, 1992) to measure energy flux across the cut‐off scale). However, LES requires that the cut‐off scale is in the scale‐free inertial range, and that the equations of motion are the same at all scales of motion. LES may not be directly applicable to atmospheric dynamics because of the interaction between fluid dynamics and thermodynamics.

We have a pretty good idea of how averaging and subgrid scale modeling (i.e., parameterization) works for fluid turbulence. However, when dealing with anything more complicated than the Navier–Stokes equations the correct approach is not obvious. Any attempt to describe the average state of the atmosphere from an approximate system of equations is bound to be problematic. The interaction between resolved and unresolved scales in multiphysics simulations is a vast, largely undeveloped challenge for ESMs.

In particular, developing appropriate averaging operators and subgrid scale models for thermodynamics is problematic. For simplicity, we assume that the average internal energy is the energy of the averaged variables substituted into the continuous formula

As grid resolution increases, we eventually cross a threshold where the basic nature of the filtering, and the resolved equations, changes. This happens because processes that had been completely unresolved become partially resolved. They are therefore associated with equations that must be solved, in addition to the unresolved scales that must still be parameterized.

One well‐known example of such a resolution threshold is the “gray zone” problem of deep convection problems, where convection that was completely unresolved (i.e., fully parameterized) at larger scales suddenly becomes partially resolved at scales of *O*(10) km. In addition, the statistical assumptions of some parameterizations are no longer satisfied. This means that in practice it is not possible to simply increase the resolution of ESMs as more powerful computers become available. A significant jump in resolution requires time‐consuming tuning of the parameterizations and, in some cases, adding new equations to the dynamical core. Had the parameterizations been formulated in terms of an averaging operator and a well‐defined closure then the transition would be seamless.

One way of dealing with changing resolutions is to use so‐called “scale aware” parameterizations (Arakawa & Wu, 2013; Arakawa et al., 2011; Freitas et al., 2018; Park, 2014). In this approach the parameterizations modify themselves based on information provided by the dynamical core about the current local scale.

Given the challenges of developing consistent averaged equations, henceforth we focus on the unaveraged total energy equations, with the exception of Section 2.1.12 on kinetic energy dissipation and heating which inevitably involves turbulent (sub‐grid scale) kinetic energy.

The fluid energy equation is obtained from the two separate equations for the enthalpy and the kinetic energy of the fluid:

Equation B1 expresses the first principle of thermodynamics, and Equation B2 is the scalar product of the Euler equations with the 3‐dimensional fluid velocity *α* =: 1/*ρ*, pressure *p*, and specific enthalpy *h*. Note that for a fluid composed of several, non interacting constituents, 1/*α* = *∑*_{ℓ}1/*α*^{(ℓ)}; likewise, *p* is the sum of the partial pressures. Expressions for *h* are given in Appendix C1. The r.h.s. term *Q* is the heat acquired by the fluid, per unit mass and per unit time, from external sources (e.g., radiation). In Equation B2, Φ is the geopotential (a combination of the Earth's gravitational potential and the centrifugal potential, calculated on geoid surfaces), and **F** represents other forces acting on the fluid (which might include viscous and other dissipative forces); in a multi‐constituent fluid, **F** = *∑*_{ℓ}**F**^{(ℓ)} where **F**^{(ℓ)} is the force acting on constituent *ℓ*. (Note that here we do not discuss cases when different constituents might have different three‐dimensional velocities, i.e., we adhere to a strict single‐fluid framework; this assumption is partially relaxed in Appendix F).

Summation of Equations B1 and B2 results in the evolution equation:

Note Equation B3 is not valid for every choice of coordinates: for example, the pressure force has to be a spatial gradient of the pressure. Hence Equation B3 is only valid for time‐fixed (time‐independent) coordinates for example, terrain‐following *z*‐based coordinates. Hence, the partial time derivatives in Equation B3 should be understood to be taken at fixed *z*. In a coordinate transformation where *z* is replaced by a generalized vertical coordinate (“GVC”),*η* is a *C*^{1} function of {*t*, *x*, *y*, *z*} strictly monotonic in *z*, these terms become (cf., Equation 3.6 in Kasahara [1974])

In this section, as in the rest of the paper, we make the assumption of hydrostatic equilibrium, that is, that the vertical (*z*−) acceleration of the fluid is always negligible in comparison to other inertial, pressure, viscous, turbulent or external forces. This assumption breaks down at the scales where buoyancy acceleration becomes important, for example, in topographic gravity waves and in local circulations driven by atmospheric convection. In global models, such motions are heavily parameterized and their associated energy is mostly unaccounted for (see also Appendix A and Appendix E). In a GVC system Equation B6, the hydrostatic condition reads (cf., Equation 3.20 of Kasahara [1974])_{η} is formed of partial derivatives taken at constant *η*, and *η*, related to the vertical velocity *w* as (cf., Equation 3.12 of Kasahara [1974]):

Frequently, hydrostatic pressure is used as a GVC, for example, in Equation 13. In such cases, Equation B8 is the *definition* of the pressure *coordinate*, which needs, in general, to be distinguished from the pressure *state variable*. Only if condition Equation B8 is *assumed* to be valid for the pressure variable, as we do here, may the two be confused and a single symbol, *p*, used for both.

Finally, we combine Equation B3 with the equation of mass conservation to obtain a conservative form of the energy equation. In local Cartesian coordinates, mass conservation is expressed as

In GVC, the divergence term on the r.h.s. of Equation B11 reads (cf., Equations 3.16 and 3.17 of Kasahara [1974])

Note that Equation B13 is valid (physically or mathematically) irrespective of the hydrostatic assumption. If we now make that assumption, then can multiply Equation B3 by *∂*_{η}*z α*^{−1} and sum it with Equation B13 multiplied by *ϵ* to obtain the total energy equation in formally conservative form:*η* from the partial time derivative operators. In the following, we will also drop them from the shorthands of three‐component vectors and operators. Equation B14 is formally valid for a general geopotential field Φ = Φ(*t*, *x*, *y*, *z*). However, consistency with the primitive equation set used here, which neglects spherical geometry terms in the momentum equations under the assumption of a thin‐shell atmosphere stratified approximately over equilibrium geoid surfaces, in fact requires the approximation that Φ be linearly proportional to *z*, Φ = *gz*. Together with the hydrostatic approximation, this implies*q* may be interpreted to be the specific humidity for either total water (in general) or for water vapor only (CAM assumptions) since in all cases dry air mass is conserved. Next, we explore various form of Equation B16 integrated over a global or limited horizontal domain and over the entire vertical domain.

We write out Equation B16 as*u*_{m} ≔ *ϵ* − *αp* is the total (kinetic + thermal + latent + potential) energy per unit mass of air.

Integration of Equation B17 between the top of the model domain, *η* = *η*_{t}, and the surface, *η* = *η*_{s}, and over a horizontal domain *dS* is the area element in *∂*_{t}Φ_{s} = 0, viz no flow out of the top of the vertical domain, and time‐invariant surface geopotential. In the case of a fixed lid, *∂*_{t}*p*_{t} = 0, Equation B18 can be also be written for the sum of kinetic energy and enthalpy *ϵ*_{m} ≔ *ϵ* − Φ, by using *∂*_{η}Φ; this yields:

Equation B19 is the form used in CAM to compute and check the atmospheric energy (rectius: enthalpy) budget. It is useful to gather the terms involving the geopotential Φ, by substituting

First, we note that, except for the term in Φ_{s}, the l.h.s. of all those equations express the l.h.s. of Equation B20 with different forms of the thermodynamic enthalpy *h* = *ϵ*_{m} − *K* of the fluid and with different decompositions of the mass term _{s} term is expressed as part of **F** ⋅**v** is not part of Equations 110, 121, and 125; it is discussed in Section 4.1 and in Appendix E. So we only have to discuss, on the r.h.s. of Equation B20, the surface integral of the sensible and latent heat fluxes *∂*_{η}*pdη* = *g*/*αdz*) of *Q* and of *Q*/*α* and *Q*/*α*, is associated with the convergence of radiation fluxes, −**∇** ⋅ **F**^{(rad)}, which in numerical models is approximated as *F*^{(turb)}(*z*_{s}), in the r.h.s. term

While *Q* might contain other diabatic heating terms, for example, mesospheric ion drag or the dissipation of vertically propagating gravity waves, it does not include latent heating terms, which are all accounted for in the expressions for the thermodynamic enthalpy of the fluid *h*. There are therefore no explicit terms associated with evaporation/sublimation or condensation/deposition processes in Equation B20. However, to understand, within the present formalism, the role of precipitation or other processes involving exchanges of material with the surface, which change the composition of the fluid (e.g., by only removing liquid water, but not dry air), we have to apply Equation B20 to individual fluid components under dynamic and thermodynamic equilibrium assumptions.

The resulting energy conservation equation is a sum of equations of the form Equation B20 with the pressure replaced by the partial pressure and the specific enthalpy replaced by the specific enthalpy of each substance, velocities and temperatures being shared by all components. The volume integral of *ℓ*, considered to be part of the fluid motion. This source term is associated with all the *local* enthalpy, kinetic energy and geopotential of the relevant fluid component as a fraction of its mass is considered to join or leave the atmospheric fluid.

The fluid equations thus only describe the process of “joining” or “leaving” the fluid, and the associated energy change. Additional processes that occur in the lead‐up, or as a consequence, of mass changes have to be described separately in an ad‐hoc fashion. Physically, the fractional mass source of each fluid component does not just appear or disappear. In the case of precipitation, it travels vertically and exits the atmospheric domain at its surface boundary, and similarly for water vapor that joins the atmosphere as a result of surface evaporation.

A zeroth‐order assumption is that there is no interaction whatsoever between the newly acquired, or newly lost, mass and that persisting in the atmospheric fluid. In that case the mass source term, *∂F*^{(ℓ)}*∂z*. If all the divergence were localized at one altitude *z*, the vertical integral*z*_{p} is where precipitation is formed. However, this would be an unrealistic assumption, because local thermodynamic equilibration of for example, rain drops with the surrounding air is generally faster than the drop crossing time as they fall to the ground. Similarly, aerodynamic friction slows the drops down to much less than the free‐fall speed (in vacuum).

Thus, a much more plausible physical approximation would allow rain drops to exchange with the atmosphere all of their energy that is in excess of the water enthalpy equilibrated at the temperature of the lowest atmospheric layer. Effectively, this represents an additional, ad‐hoc contribution to *Q*, say

Because now all precipitation is associated with the same, surface, specific enthalpy, its altitude of provenance no longer matters and Equation B22 holds without any assumption on the shape of

Summing Equation B22 for all atmospheric fluid species (*ℓ*) finally yields the r.h.s. of Equations 110, 121, and 125.

Note that assumptions like Equation B22 are not in any way necessary. If explicit expressions are used for

Here we give explicit expressions for the terms forming the enthalpy function of the air‐water fluid used in the thermodynamic Equation B1. In doing so, we adopt simplified expressions of *h* which are generally sufficient to account for the leading errors in energy conservation of numerical atmosphere models like CAM.

The most general formulation of the thermodynamic equations for a fluid of variable density, pressure, temperature and composition is by means of the Gibbs function, *G*, with total differential given by *dG* = −*sdT* + *αdp* + *μdr* where we indicate with *s* the fluid entropy and with *μ* and *r* the chemical potential and the mixing ratio of for example, water vapor (Bowen & Thuburn 2022a; Eldred et al., 2022; Feistel et al., 2010; Thuburn, 2017). Thermodynamic equilibrium states are characterized by a minimum of *G* on *T*, *p* isosurfaces (Adkins, 1968, Section 10.2). The differential of *G* shows conjugate variable pairs (*T*, − *s*), (*p*, *α*), and (*r*, *μ*), and *h* is formally obtained as the Legendre transform w.r.t. the first pair, *h* ≔ *g* + *Ts*. Hence, from general principles, *h* is a conserved quantity under isobaric, isothermal transformations.

We note that an advantage of using the form Equation B1 of the thermodynamic equation, compared to for example, an evolution equation for the entropy or the Gibbs function, is that there is no need to explicitly represent terms arising from changes in chemical potential associated with compositional changes.

This is of particular advantage when neglecting the effects of significant departures from thermodynamic equilibrium in for example, phase changes of water, a common situation in the atmosphere. Equation B1 is general and the source term *Q* only includes external heating sources such as radiation flux‐divergence or viscous (and, in the Reynolds‐average equation set, turbulent) dissipation. Therefore, the thermodynamic approximations made in this paper are contained within the expression used for *h*.

Following Thuburn (2017) and Bowen and Thuburn (2022a), we use a linearized form of the enthalpy function, based on the two assumptions:

specific heats are constants independent of temperature

the specific volume of condensates (liquid water and ice) is zero.

These approximations are generally very good in the range of temperatures and pressures typical of atmospheric conditions, and they allow us to avoid significant formal complications which could distract from the main scope of this paper. Please see Thuburn (2017), Bowen and Thuburn (2022a), and Bowen and Thuburn (2022b) for alternative, general formulations that incorporate non‐zero condensate volume but keep temperature‐independent specific heats. The resulting expressions follow below.

This Appendix section provides formulas for enthalpy and energy using different reference enthalpy states (see Section 2.1.3). Specific enthalpies are given by

Net enthalpy fluxes at the surface can be written as

If the interface temperature is the same in both atmosphere and the surface,

The total energy equation with different reference enthalpies are

For liquid and ice as a reference states the total energy equation takes the form

Let's briefly consider the interface condition for atmosphere‐ocean coupling:*h*^{(sw)} is the enthalpy of seawater and *h* is, in general, temperature dependent *h*^{(ℓ)} = *h*(*T*^{(ℓ)}), *ℓ* = “*wv*”, “*liq*”, “*ice*” and the temperatures of each species *T*^{(ℓ)} are, in general, different for different forms of water. The enthalpy of seawater, *h*^{(sw)}, is a much more complicated function involving pressure and a nonlinear dependence on temperature (TEOS‐10;

For reasons that will become clear, let's rewrite Equation D3 as

If we assume a single interface temperature and a liquid reference state, then*h*^{(liq)} = *h*^{(sw)}, then Equation D8 simplifies to

If one specifies sea surface water to have the temperature *T*^{(ocn)}, evaporated water vapor to have the same temperature as the sea surface *T*^{(ocn)} and all forms of precipitation to have the same temperature *T*^{(atm)}, then the more general formula (Equation D3) must be used

The winds in the atmosphere and the surface (e.g., currents in the ocean) exchange momentum through a surface stress

The total rate of change of kinetic energy in each atmospheric layer due to internal stresses was given in Equation 99. Integrating over the entire column we get

As discussed in Section 2.1.12, the last term represents dissipation of kinetic energy in the interior of the atmosphere due to Reynolds stresses (see e.g., Landau and Lifschitz [1968] Section 16, Peixoto and Oort [1992] Section 3.2.1, Duran et al. [2018]), but also includes unresolved turbulence or wave‐breaking processes, and possibly numerical diffusion terms (these various sources will determine the form of *τ*_{z} = 0 is applied at the top of the computational domain. That said, the lower the model top, the worse the approximation of vanishing stress at the model top becomes. For example, in the lower thermosphere local tendencies caused by gravity wave propagation are still important. However, even in this case the total work remains negligible in the global average. In numerical models the top layers usually are subject to increased artificial damping to avoid wave reflection from the model top. These artificial numerical stresses at the upper boundary may result in a significant non‐physical contribution to the column energy budget which is not balanced by physical terms (more on this in Section 4.1.2).

At the lower boundary of the atmosphere, in contrast, the corresponding term on the right‐hand side of Equation E1 vanishes only if the boundary condition *C*_{D} is a drag coefficient which depends on the sea‐state, near‐surface atmospheric quantities and *z* = *z*_{1} the thickness of the surface boundary layer. The thickness of the surface boundary layer is usually assumed to be the altitude of the lowest grid point of the atmospheric model (i.e.,

We also note that global atmosphere models normally do not include the interface layer between atmospheric domain and the surface explicitly in the computational domain. This has implications for the energy budget, because the lowest resolved air speed (whether in the dynamical core or in the sub‐grid parameterizations) is non‐zero even over solid ground — and in general it is different from the velocity of the surface (e.g., the ocean current). The surface stress therefore appears to perform work, which is lost from the global budget. This work is, in fact, entirely transformed into heat within the interface layer, which is defined as the thin boundary layer of air near the surface within which both heating and stress are zero and the corresponding vertical fluxes of heat and momentum constant. For momentum, the implication is that within this layer energy is dissipated according to the last term on the r.h.s. of Equation E1, which across the surface layer integrates to

This section makes use of vector notation using bold‐face fonts. For example, the barycentric velocity vector is denoted **v** and is a three‐dimensional velocity vector, whereas the main part of the paper is using

Pauluis et al. (2000) and Pauluis and Dias (2012) estimated the frictional dissipation due to Stokes friction between precipitation and gas to be globally comparable to the frictional dissipation due to the turbulent kinetic energy cascade. Locally in the tropics, the dissipation due to falling rain might dominate and could be as large as 10 W/m^{2}. This process must be represented in numerical models. It is argued here that this process is automatically included into the model if the barycentric (center of mass) velocity is used in the model equations.

When representing thermodynamic processes from the perspective of the resolved model equations, we have to take into account that normal and tangential stresses in the vicinity of falling precipitation are not resolved. Microphysics equations are valid for droplets of specific sizes for which the Stokes friction law can be accessed directly. Contemporary parameterization schemes for microphysics deliver an integrated precipitation flux **P** over all droplet sizes to the coarse grained model equations. The precipitation flux can be interpreted to be the difference between the bulk velocity of the precipitation, superscript *prec*, and the gaseous part of the air, superscript *gas*, hence **P** = *ρ*^{(prec)}(**v**^{(prec)} − **v**^{(gas)}). A bulk terminal velocity characterizing all droplets is then defined as − *V*^{T}**k** = **v**^{(prec)} − **v**^{(gas)}, where **k** is the vertical unit vector.

The terminal velocity of a droplet is obtained from the equilibrium of the Stokes friction force, the gravitational force and the buoyancy force. At microscales, when the shear zones of the droplets is resolved by a model, the friction force is larger than the buoyancy force. At the coarse grained macroscales, however, this Stokes friction force is not directly accessible. Therefore, the equilibrium is represented as a balance between the gravitational force and a buoyancy (pressure) force. This shift in interpretation correctly models the energy transfer processes. The lost potential energy of a droplet is first converted into kinetic energy and then converted into internal energy by friction. An alternative representation is that the conversion of kinetic energy into heat is also achieved by the pressure gradient force. In the following it will be shown that this process is indeed recovered when using the barycentric velocity of an air parcel and the associated kinetic energy.

The concept of the barycentric velocity is documented in de Groot and Mazur (1984), Zdunkowski and Bott (2004), Degrauwe et al. (2016), Catry et al. (2007), Gassmann and Herzog (2008), and Wacker et al. (2006). A macroscopic air parcel is considered to consist of diverse species *ℓ*, which may have different bulk velocities. The barycentric velocity is then defined to be*q*^{(ℓ)} = *ρ*^{(ℓ)}/*ρ*^{(all)} with **v** are usually assumed the same.

Diffusive fluxes are defined as the difference between the actual flux of a given species and the barycentric flux

Note that the diffusive flux in a barycentric velocity frame must not be confused with molecular diffusion fluxes, and also not with turbulent diffusion fluxes arising from Reynolds averaging/filtering. By definition, the sum of all diffusive fluxes is zero

It might seem that the sum of the diffusive fluxes should not vanish at the surface (Bott, 2008). If it rains, there, a vanishing flux sum would be associated with an artificial flux of dry air through the surface. In fact, the sum vanishes as well by definition at the surface. Rather the barycentric velocity is non‐zero there. When evaporation and precipitation take place at the same time, we can separate the velocity of water vapor into two distinct contributions, one which is equal to the dry air velocity, and a second part that differs from the dry air velocity and only appears if dry air and water vapor mix

Clearly, the surface dry air velocity vanishes, **v**^{(d)} = 0. The here given form is only chosen for the sake of making the definitions of **P** and **E** obvious, which are both always formally defined with the help of mass flux differences. In the above equation it is assumed that suspended condensates move with the dry air velocity and hence do not show up in the boundary condition. This boundary condition is applicable to all prognostic equations, especially also for the momentum equation, which means that momentum leaves the atmosphere if it rains.

The continuity equation for each specie can be written in two different flavors (note that these are three‐dimensional continuity equations. The gradient operator ∇ is three‐dimensional!)*S*^{(ℓ)} is the source/sink term. Note that total mass is conserved, so

We note that the constituent equation can also be written in Lagrangian form

Often, models do not use a prognostic form for the equations of falling precipitation. Rather, the precipitation is thought to fall down immediately. This is expressed as 0 = −∇ ⋅ **J**^{(ℓ)} + *S*^{(ℓ)}. Hence, the barycentric framework is easily applicable for this case, since the required rule

Summing the continuity equations for all components gives

As said above, at the surface, the barycentric velocity is the sum of the precipitation flux and the evaporation flux. Hence, the lower boundary conditions for the continuity equation are easily obtained when applying the barycentric framework. The general and intuitive form of the continuity Equation F7 allows to derive its primitive equation formulation as we are used to it. Specifically then, the total density is replaced with *ρ*^{(all)} = −*∂p*/*∂*Φ.

Specifically, we can also give the continuity equations for the species independent of the gas velocity, which is likewise the velocity of the other non‐precipitating species like cloud ice and cloud water. For comparison, we repeat here the two forms for the continuity equation of the non‐precipitating constituents, hence the sum of the gas and the non‐precipitating liquid and frozen parts**v**^{(gas)} to be known. The second equation assumes an available **v** and uses the diffusive flux of the non‐precipitating species as

Likewise, the possible prognostic equations for the precipitation are found to be

The second equation using the barycentric framework includes the diffusive precipitation flux as **J**^{(prec)} = *q*^{(non−prec)}**P**.

The different velocities are related according to

Therefore, modelers may choose either prognostic equation for the species in their numerical model. For instance, when modeling passive tracer transport, **v**^{(gas)} is a more natural choice for the advective velocity than performing a two‐step process consisting of an advection with the barycentric velocity and a correction with the diffusive velocity. A common counterargument against the use of barycentric velocities is that we cannot measure them directly. However, if the precipitation flux, the wind velocity—and hence the velocity of the non‐precipitating part—and the density are known, the recovery of the barycentric velocity is straightforward.

Let us write the resolved momentum equations for the gaseous part, the precipitating part and the cloud part independently**F** subsume all forces which are not tractable by the coarse‐grained model. The force **F**_{gp} is exerted with opposite signs to the moist air around the precipitation particles and the bulk precipitation velocity. The force **F**_{gc} accounts for a similar interplay between gas and cloud particles. Several possible reasons exist for such forces. The most obvious is the frictional force, but also, momentum might be exchanged because of condensation or evaporation. The forces **F** are not known, and they are not provided by any microphysics parameterization of the coarse‐grained model. At the microphysics level, the comparable Stokes friction depends on the radius of the droplets and the shear zones formed around these droplets. However, on the macroscopic level these radii are not known and the shear zones are not represented.

The assumption of a common—namely barycentric—advective velocity **v** is necessary here, because the forces ±**F** are relative forces between the gas and the droplets. Therefore, all species need a common Lagrangian frame of reference for their motion.

The forces **F** are not explicit in the barycentric momentum equation,**F** are not explicit means that using the barycentric velocity as the prognostic variable is an attractive choice. An apparent complication is that the barycentric velocity is needed for advection in this momentum equation.

Let us derive the kinetic energy change of an air parcel moving with the barycentric velocity. This is obtained by multiplying Equation F16 by the barycentric velocity

The first three terms on the right represent individual energy changes which are attributable to known physical processes. The first term is the pressure work term that converts kinetic energy into internal energy via the compression work −**v**^{(gas)} ⋅ ∇*p* = −∇ ⋅ (**v**^{(gas)}*p*) + *p*∇ ⋅ **v**^{(gas)}. The two next terms describe the exchange of kinetic energy with the potential energy, that is, the rising or falling air parcels. The fourth term represents the kinetic energy change due internal forces between the precipitation and the moist air. This term is identified as an effect of internal friction because the other physical processes are already accounted for. Forces between the cloud droplets and the gas do not result in a kinetic energy change, because the gas and the cloud droplets have the same velocity.

Let us now devote our attention to the last term in Equation F17. Assuming that the hydrostatic relation dominates,*V*^{T}**k** = **v**^{(prec)} − **v**^{(gas)}, the macroscopic effect of Stokes friction can be approximated as

It is then clear that this term represents dissipation of kinetic energy

Pauluis et al. (2000) denotes this as the frictional dissipation of rain. In conclusion, we note that this dissipation is only accessible if the barycentric velocity is chosen to be prognostic. This is because, otherwise, the particular energy budget Equation F17 would not have been obtained.

In a hydrostatic primitive equation modeling framework, the left hand side of Equation F17 would be set to zero. This means that kinetic energy of vertical motions is not stored in a kinetic energy reservoir, but pretends to be converted directly between potential energy and internal energy. Besides this, the physical processes are represented in the same manner in a primitive‐hydrostatic modeling environment as they are in a non‐hydrostatic modeling environment.

After having analyzed the loss of kinetic energy due to falling rain, we have to find/derive the terms where the respective gain of internal energy shows up as a positive temperature tendency. We will cover both the constant height formulation (internal energy) and the constant pressure formulation (enthalpy).

The internal energy Equation 20 using the barycentric velocity formulation can be written as*Γ* represents three terms not central for this discussion (divergences of radiation and sensible heat fluxes, turbulent shear production term) and *U*^{(all)} is the specific internal energy (see Equation 22)

The internal energy equation combines temperature and constituent changes and we are now after the temperature change. Before starting, let us replace the pressures in the above internal energy specification by their expression of the ideal gas law, hence *p*^{(d)}/*ρ*^{(all)} = *R*^{(d)}*q*^{(d)}*T* and *p*^{(wv)}/*ρ*^{(all)} = *R*^{(wv)}*q*^{(wv)}*T*. For gaseous components we have

Now, we use that to rewrite the left‐hand side of Equation F20

Now, it is clear that the last term in Equation F22 cancels with a part of Equation F21. Substituting Equation F21 on the left‐hand and the rewrite Equation F22 on the right hand side of the internal energy equation Equation F20 yields after rearranging, using again

The important terms are the compression term and the diffusive pressure flux term. For the latter we do not yet have an interpretation and, indeed, there is no stand‐alone interpretation as we shall see soon. We can only understand the frictional heating due to falling rain when we consider both terms together. To demonstrate this, recall the definitions of the previous sections: **P** = *ρ*^{(prec)}(**v**^{(prec)} − **v**^{(gas)}) = −*ρ*^{(prec)}*V*^{T}**k** and **J**^{(gas)} = −*q*^{(gas)}**P**. Using them to express the diffusive pressure flux yields

Now, use the definition of the barycentric velocity **v** = *q*^{(non−prec)}**v**^{(gas)} + *q*^{(prec)}**v**^{(prec)} in the compression term and differentiate the diffusive pressure flux term by parts. This gives after some cancellations and using *q*^{(non−prec)} + *q*^{(prec)} = 1

It is interesting to give an estimate of the relative orders of magnitude of the dissipative heating term due to precipitation and the last term in Equation F23, which accounts for the differential advection of the species besides the barycentric advection. When considering only rain fluxes below a cloud, it becomes

Assuming the worst case of a neutral stratification with a vertical temperature gradient of

What are the implications for a temperature equation in the height based coordinate system? Assume that the precipitation fluxes **P**^{(ℓ)} are the output of some parameterization scheme. Then, Equation F23 can be cast in the following form**J**^{(ℓ)} = −*q*^{(ℓ)}**P** for every non‐precipitating constituent and **J**^{(ℓ)} = *q*^{(non−prec)}**P** for a precipitating component. This must be done independently for different precipitation fluxes, say those of rain, snow, etc. There might also be fluxes of gaseous components due to turbulent mixing. These are represented in the second term under the divergence operator. Finally, the diffusive fluxes in the last term might be summed up from quite different contributions.

Writing the thermodynamic Equation 21 in enthalpy form—as it is convenient in the primitive equation's modeling framework—we get

It is remarkable that no additional term appears, compared with the existing formulation. This is because the barycentric vertical pressure velocity is not “just” the gas pressure vertical velocity when taking the barycentric framework into account

The last term is here the frictional dissipation term. In contrast to the internal energy form (Equation F25), the frictional heating due to falling precipitation is completely hidden in the pressure vertical velocity. This velocity becomes larger when precipitation is falling, indicating a stronger apparent subsidence.

The internal energy (Equation F25) and enthalpy (Equation F26) forms of the temperature equation assign the heating to all ingredients of the air parcel to the extent that they all have the same temperature, but different heat capacities and different mass concentrations. It lies at the very heart of our current understanding of non‐equilibrium thermodynamics that a single temperature is assigned to the air parcel consisting of the diverse species. This is an idealization which is heavily challenged, but an alternative would require multiple temperature equations for multiple species. Since we describe droplet size distributions with only bulk quantities and smaller droplets certainly have different temperatures than bigger droplets, an alternative approach quickly becomes confusing.

In Section 2.1.13 we speculated that the barycentric framework could possibly also handle horizontal velocity deviations/friction for falling precipitation. In that case, we would have to define a terminal velocity so that it not only has a vertical component, but also a horizontal component. The barycentric viewpoint already formally also accounts for the horizontal momentum of falling precipitation. The challenge is that in the microphysics parameterization scheme it would be unclear how to compute a terminal velocity in the horizontal direction as that is not governed by Stokes law and the empirical formula used there.

The simulations with CAM use the cam_energy_analysis branch of *cam*6_2_026 (tag in cam_developement branch of *my*_*scripts* directory.

Reproducibility information on the simulations for Figure 9 is stored in

The following constants in the energy equation are used in CAM:

This version of the spectral‐element dynamical core differs from the CESM2.2 release version (Lauritzen et al., 2018) in that an FV3 vertical remapping algorithm is used (which reduced vertical energy conservation due to higher‐order numerics) and reference states are subtracted from hyperviscosity operators to reduce spurious noise for flow over orography (Liu et al., 2022) and thereby also energy errors.

As discussed in Section 4, frictional heating is often implemented in the dynamical cores of ESMs after the application of second‐order or higher‐order horizontal diffusion processes and sponge‐layer dissipation mechanisms, such as Rayleigh friction, near the model top. These processes are represented on the right‐hand‐side of the momentum equation as also shown for the Laplacian diffusion in Equation 136. In addition, horizontal fourth‐order hyperdiffusion and Rayleigh friction are represented respectively by*ν*_{4} is a fourth‐order horizontal diffusion coefficient (see also the discussion in Jablonowski and Williamson [2011]) and *k* is the Rayleigh friction damping coefficient.

Section 4.1.2 highlighted a particular application of the sponge‐layer Rayleigh friction in the FV3 dynamical core (L. M. Harris et al., 2021). FV3 lies at the heart of the Unified Forecast System (UFS) developed by the National Oceanic and Atmospheric Administration (NOAA) and is an optional dynamical core in NCAR's CESM2.2 framework. We therefore provide additional insight into the implementation details to foster the reproducibility of the FV3 results. They were generated via CESM2.2's “Simpler Models” hierarchy which includes the dry Held‐Suarez test (Held & Suarez, 1994) as the “FHS94 compset” with analytic initial conditions. FV3's Rayleigh friction in the sponge layer is implemented in a time‐implicit way. When using the acceleration of the zonal wind component *∂u*/*∂t* = ⋯ −*ku* as an example this leads to*n* + 1 and *n* denote the updated (via the RF process) and current time levels, respectively. The time‐implicit update of the meridional wind component *v* follows the same principle and is given by

The loss of kinetic energy due to Rayleigh friction in the hydrostatic FV3 configuration with the horizontal velocity vector

This kinetic energy loss is applied as frictional heating (−*δK*) as shown in Equation 140. It leads to the expression*n* + 1. A dry specific heat is used since there is negligible moisture in the sponge layer domain. L. M. Harris et al. (2021) show the equivalent temperature update equation due to frictional heating (their Equation 8.17) for the nonhydrostatic FV3 configuration. It is*K*_{3D} is the kinetic energy that takes all three velocity components into account. This heat conversion utilizes the dry specific heat at constant volume

The choice of the Rayleigh friction damping coefficient in FV3 is*p* symbolizes the pressure at the chosen grid point. The symbol *p*_{c} stands for the user‐selected RF cutoff pressure, *p*_{t} is the pressure at the model top, and *τ* is the maximum RF timescale which needs to be expressed in units of seconds. For the FV3 C24L64 simulations in Section 4.1.2 the following settings were selected: *p*_{t} = 9.04 × 10^{−5} hPa, *p*_{c} = 0.1 hPa and *τ* = 1, 3, and 5 days.

Below is list of variables appearing more than once in the main part of this paper:*α*^{3}/kg*α*^{(d)}^{3}/kg*ℓ**ℓ**D*/*DT**E*_{x}*x* where *x* = “atm” atmosphere and *x* = “ESM” Earth System model*E*_{feom}*ℓ* into the atmosphere^{2}/s*ℓ* from atmosphere to surface^{2}/s*ℓ* from surface to atmosphere^{2}/s^{2}/s^{2}/s^{2}^{2}^{2}^{2}^{2}^{2}^{2}/s^{2}*g*^{3}/kg/s^{2}*h*^{(ℓ)}*ℓ**ℓ**k**K*^{2}/s^{2}*K*_{surf,s}^{2}/s^{2}*K*_{atm,s}^{2}/s^{2}^{2}/s^{2}*δK*^{2}/s^{2}*m*^{(ℓ)}*ℓ* (≡ *ρ*^{(ℓ)}/*ρ*^{(d)})*m*^{(all)}*M*^{(ℓ)}(*z*)*z* for component *ℓ*^{2}*L*_{f,00}*L*_{s,00}*L*_{v,00}^{2}/s^{2}_{s}^{2}/s^{2}*ρ*^{(ℓ)}*ℓ*^{3}*ρ*^{(all)}^{3}*p**p*_{t}*p*^{(d)}*p*^{(wv)}_{s}*q*^{(ℓ)}*ℓ**Q**R**R*^{(ℓ)}*ℓ* = ‘*d*′, ‘*wv*′*T**T**T*_{k}*k**T*_{00}*T*_{atm,s}*T*_{surf,s}*T*_{v}*δT**Θ**θ*_{l}*U*^{(all)}*U*^{(d)}*u*, *v*)*δu**δv***v***V*^{T}*ν*_{2}^{2}/s*w**ω**Dp*/*Dt*)*z**z*_{t}*η**I*^{2}

We are especially grateful that the Banff International Research Station (BIRS) approved our proposal for the week‐long workshop Physics‐Dynamics Coupling in ESMs (19w5153). Without this support, this paper would not exist. BIRS provided a beautiful location and ideal facilities for a true workshop that encouraged lively scientific discussion and debate on the fundamental challenges facing Earth Systems models. We thank the reviewers and editors for taking the time to review and handle this unusually long and complex paper. It has taken many iterations (and countless Zoom meetings during the pandemic) to cast our ideas/discussions/solutions into publishable form and we are extremely grateful that the reviewers and editors have appreciation for that. Initially we were also planning on covering ocean models which would have made the paper even longer and more complex! We thank BIRS participants who contributed to the many discussions at the BIRS meeting but ended up not contributing to this paper (A.J. Adcroft, H. Johansen, D.A. Randall and R. Kumar). We thank Richard Neale (NCAR) for suggesting the *insulated rain‐shaft* terminology. P.H. Lauritzen acknowledges funding support by NSF under Cooperative Agreement No. 1852977. Computing resources (

The data for the energy analysis is available here