This study examines the characteristics of tropical‐cyclone (T‐C) turbulence and its related predictability implications. Using the Fourier‐Bessel spectral decomposition for convection‐permitting simulations, it is shown that T‐C turbulence possesses different spectral properties in the azimuthal and radial directions, with a steeper power law in the radial‐wavenumber than that in the azimuthal‐wavenumber direction. This spectral difference between the azimuthal and radial directions prevents one from using a single wavenumber to interpret T‐C intensity predictability as for classical homogeneous isotropic turbulence. Analyses of spectral error growth for a high‐wavenumber perturbation further confirm that the spectral growth is more rapid for high azimuthal wavenumbers than for the radial wavenumbers, reaching saturation after ∼9 hr and ∼18 hr for the azimuthal and radial directions, respectively. This result highlights the key difficulty in quantifying T‐C intensity predictability based on spectral upscale error growth for future applications.

This study examines the characteristics of tropical cyclone (T‐C) turbulence and related predictability implications. We show that T‐C turbulence possesses different spectral properties in the azimuthal and radial directions. This spectral difference between the azimuthal and radial directions prevents one from using a single wavenumber to interpret T‐C intensity predictability as for classical homogeneous isotropic turbulence. Our analyses of spectral kinetic energy error growth at the quasi‐stationary stage reveal that the spectral growth is more rapid for azimuthal wavenumbers, reaching saturation after ∼9 hr as compared to ∼18 hr for the radial direction. This study highlights the difficulty in quantifying T‐C intensity predictability based on spectral upscale error growth for practical applications.

Tropical‐cyclone (T‐C) convective‐scale turbulence is not isotropic/homogeneous, thus possessing different predictability properties from classical turbulence

T‐C error‐energy spectra show distinct characteristics with a steeper power law in the radial direction than that in the azimuthal direction

T‐C error energy spectral growth is more rapid for the azimuthal wavenumbers than for radial wavenumbers, reaching saturation after ∼18 hr

The tropical cyclone (TC) is a multiscale nonlinear system in which any small‐scale perturbation could amplify and progressively influence larger scales, resulting in T‐C intensity variability. This type of upscale‐error growth, termed the “real butterfly effect” (Palmer et al., 2014), exists in various fluid‐flow systems and determines their intrinsic predictability. Unlike deterministic systems whose predictability is governed by Lyapunov exponents and attractor invariants (e.g., Alligood et al., 2000; Aurell et al., 1996; Boffetta et al., 1998; Goldhirsch et al., 1987; Lorenz, 1963; Palmer, 1993), quantifying predictability related to upscale growth in multi‐scale systems requires knowledge of the system's statistically stationary turbulence. As extensively examined in previous studies (e.g., Kraichnan, 1967; Leith & Kraichnan, 1972; Lorenz, 1969; Palmer et al., 2014; Rotunno & Snyder, 2007; Schneider & Griffies, 1999), such a stationary background contains key characteristics of turbulence flows and plays a fundamental role in atmospheric‐predictability research.

Given a background energy spectrum for a turbulent system, predictability is often defined as the time interval beyond which a forecast probability distribution of a system state variable becomes indistinguishable from its climatological probability distribution (DelSole, 2004; DelSole & Tippett, 2007; Lorenz, 1969; Schneider & Griffies, 1999; Shukla, 1981). From this formal definition, it is apparent that the predictability limit is not a universal metric but depends on the variable and its climatology. For example, the same variable can have a different predictability if a different time scale or observation is used to construct its climatology. Using a two‐dimensional (2D) model of statistically stationary, isotropic homogeneous turbulence and kinetic energy as the metric of error growth, Lorenz (1969, hereinafter L69) established the important result that fully developed 2D turbulence with a power‐law energy spectrum

How to apply Lorenz's predictability framework to T‐C intensity is, however, a nontrivial question for several reasons. First, the evolution of a TC is by definition time‐dependent. Unlike the classical predictability framework in which a fully developed stationary state of turbulence is well‐defined, TCs evolve with time, starting first as a tropical depression followed by subsequent development into a mature TC at peak intensity and eventual dissipation. While TCs may possess a stable equilibrium regime during which the quasi‐stationary property applies (Kieu, 2015; Kieu and Wang, 2017; Hakim, 2011, 2013; Rotunno & Emanuel, 1987), in practice a T‐C intensity forecast requires predicting T‐C intensity from its very early stage to its end. The question of practical T‐C intensity predictability is therefore fundamentally different from the traditional framework for intrinsic predictability in which a statistically stationary energy spectrum is required. That is, T‐C intrinsic intensity predictability cannot be defined for the entire T‐C evolution, simply because it is meaningless to have a climatological distribution that is a function of both forecast lead‐time and time. The T‐C intensity problem in this sense can be pictured as a Lorenz (1963)‐type model with its butterfly‐wing attractor expanding over time; for this case, predictability is not well‐defined.

Second, T‐C dynamics are not homogeneous and isotropic, even at the quasi‐stationary stage. Unlike homogenous isotropic turbulence in which all points and directions in space are indistinguishable, TCs possess a unique center in physical space where the dynamics and thermodynamics are different from elsewhere. Furthermore, T‐C turbulence possesses coherent structures ranging from convective‐to meso‐scale, as illustrated in Figure 1. These coherent structures prevent one from applying the traditional spectral analysis for homogenous isotropic turbulence to study T‐C error growth. Hence, the question of estimating intrinsic T‐C‐intensity predictability is still open, despite the strong suggestion of limited practical intensity predictability from real‐time forecasts and numerical simulations (Kieu and Moon, 2016; Kieu et al., 2018; Emanuel & Zhang, 2016; Hakim, 2011, 2013; Judt et al., 2015).

Given the above issues, which are very specific to T‐C intensity, the objectives of this study are to (a) present a system‐appropriate definition of the T‐C turbulence spectrum so that the scales of T‐C turbulence can be quantified, and (b) propose estimates of intrinsic T‐C intensity predictability within the present multiscale spectral analyses.

In this study, a cloud‐resolving idealized configuration of the Hurricane Weather Research and Forecasting (HWRF) model was used to examine T‐C turbulence. Similar to the setting used in Kieu et al. (2018, hereinafter K18), the HWRF simulations were carried out with a nested resolution of ∼8.1, 2.7 km, and 900 m to best capture the convective‐scale turbulence (see Section S1 of the Supporting Information S1 for more details of all model configurations used in this study).

With this HWRF idealized configuration and a control simulation that can capture the simulated T‐C quasi‐stationary stage as in K18, an external axisymmetric error is then added to the model vortex at the mature stage to examine the evolution and related upscale growth of the error. Here, we give the initial error ^{−1} and is added every 6 hr during the mature stage of the model vortex, resulting in a total of 12 perturbed integrations between 72 and 120 hr. Perturbing other variables such as temperature or relative humidity gave the same results and will not be presented here.

With the main objective of studying how high‐radial‐wavenumber noise could grow upscale given T‐C background dynamics,

Given the nearly circular homogeneity of the TC for each radius, we employ in this study a combination of Fourier and Bessel decompositions to analyze T‐C turbulence in polar coordinates. Specifically, the Fourier transform is first applied to any variable

The Fourier components

Technically, this decomposition sequence is equivalent to the Fourier‐Bessel transform, which is a natural extension of the 2D Fourier transform (Piessens, 2010). In fact, it can be shown that a particular class of the Fourier‐Bessel transform with the zeroth‐order Bessel function is equivalent to the 2D Fourier transform of circularly symmetric functions (see, e.g., Piessens, 2010). In all Bessel decompositions presented in this study, the discrete version of the Hankel transform 3 developed by Leutenegger (2007) is applied for a finite domain range

Because of the lack of homogeneity of T‐C turbulence, a direct 2D Fourier‐transform of

Given the Fourier coefficients

One can therefore define the energy spectral density in

The integration of Equation 8 over both the

Thus

Several remarks are in order here on the above spectral decompositions when applying them to T‐C turbulence. First, unlike homogeneous isotropic turbulence in which there is circular (or spherical in 3D) symmetry of the energy spectrum in the Cartesian wavenumber space (i.e., the 2D energy spectrum can be considered a function of the single wavenumber

Second, the Fourier‐Bessel spectral decomposition is applied only to the 2‐D wind field at

To illustrate first the properties of T‐C turbulence, Figure 1 shows the structure of the relative‐vorticity perturbation, which is obtained from the 900‐m resolution domain during the mature stage (80–120 hr into the integration). One notices in Figure 1 a range of features whose spatial scales are from

Physically, the existence of such organized rings of turbulence as seen in Figure 1 is due to the T‐C dynamics, which has strong vertical motion in the eyewall/rainband regions that distinguish a T‐C vortex from pure 2D vortices. Note that if very high resolution is used to capture much finer spatial scales (<

Given the properties of T‐C turbulence, we next examine the saturated‐error energy spectrum associated with the T‐C convective‐scale structures. This spectrum provides the predictability limit as a function of scale for time‐evolving error‐energy spectra, which is essential for determining TC‐intensity predictability. Figure 2 shows the error kinetic energy (EKE) spectrum

If one applies the classical predictability interpretation as in Lorenz (1969)'s study that all power spectra with slopes less than −3 imply indefinite predictability, it is tempting to infer that the EKE spectral density obtained seen in Figure 2 indicates that T‐C intensity is more predictable for the low azimuthal wavenumbers than that of high wavenumber

To provide further insight into the EKE spectral distribution in the radial direction, Figure 2b shows the spectral density *k* wavenumbers, which have very different power laws for high wavenumbers in the spectral space

Physically, the spectral difference between the

With the error‐energy background shown in Figure 2, we now examine how an initial small‐scale error grows in this T‐C‐background environment. This is a central step in quantifying T‐C intensity predictability because the timescale for an initial error to grow and approach its stationary background spectrum in a multiscale system is defined as the predictability limit. This definition of predictability limit is general for any multiscale turbulent fluid flow that has no multiple equilibria, and it has been applied over a wide range of scales and problems (see, e.g., L69; Aurell et al., 1996, 1997; Boffetta et al., 1998; Durran et al., 2013; Judt, 2018; Leith, 1971; Métais & Lesieur, 1986; Rotunno & Snyder, 2007; Tribbia & Baumhefner, 2004). Thus, the upscale growth of an initial error in a given background spectrum is an inherent property of multiscale systems, which is more relevant to the predictability limit than the simple linear error growth in a low‐order model (e.g., Palmer et al., 2014; Vallis, 2017).

Figure 3 shows the evolution of EKE spectrum in the *k* direction shows a slower error saturation, starting first with high‐

Second, unlike classical 3D turbulence models in which an initial error grows upscale from the smallest spatial scale to a larger scale progressively, one notices in Figure 3 that the spectrum tends to progress differently for different wavenumbers, with a faster growth rate for the high wavenumbers in both the

We note that the behavior of the spectral growth for the very high wavenumber regimes (

In this study, the error‐energy spectrum of fully developed turbulence and related error growth in a simulated TC were examined. Using the Fourier‐Bessel transform for high‐resolution HWRF model simulations, it was shown that T‐C convective‐scale turbulence does not possess properties typical of isotropic homogeneous turbulence. Instead, T‐C turbulence tends to organize strongly around the eyewall and spiral‐band regions, with a well‐defined center similar to the background vortex. These organized patterns of T‐C convective‐scale turbulence are persistent during the mature stage of the model‐simulated T‐C vortex, regardless of model configurations, physical parametrizations, or variables used to quantify T‐C turbulence.

From a practical standpoint, the consequence of such distinctive patterns of T‐C turbulence is significant, because it suggests that the classical interpretation of homogenous turbulence likely does not be apply to T‐C intensity predictability. That is, it is not sufficient to use a single wavenumber to represent the spectrum of T‐C error energy in the two‐dimensional plane. Instead, one must consider the azimuthal and the radial directions jointly to properly define the error‐energy spectral density. Our analyses show that area‐averaged TC turbulence can be approximately treated as homogeneous only in the azimuthal direction, which has an energy spectrum with a “∼−3” slope for low‐

Further analyses of the spectral growth of T‐C EKE revealed several noteworthy features that are very specific to T‐C turbulence. Looking along the azimuthal‐wavenumber direction, it is found that the EKE spectrum grows most rapidly for the high

We should emphasize that any intensity predictability implication derived from the simulations of T‐C turbulence must be viewed with caution. This is because our approach of using the Fourier‐Bessel transform in the 2D plane at each level is just one among several possible approaches. One could, for example, choose the radial direction of wind profile for energy spectra as in Vonich and Hakim (2018) or use an axisymmetric model for spectral analyses along the radial‐vertical directions (Hakim, 2013), which could provide different estimation for T‐C intensity predictability. Further development using the spectral approach requires a derivation of an error‐energy spectral equation that can account for how the error energy grows upscale in radial‐azimuthal wavenumber space. Such an equation does not exist currently but will be needed before one can understand the intrinsic predictability limit of T‐C intensity.

This research was partially supported by the Office of Naval Research (ONR) Young Investigator Award (N000141812588) and ONR/Tropical Cyclone Rapid Intensification program (N000142012411). The first author also wishes to thank the NCAR/Advanced Study Program visiting program for their summer support and hospitality during the preparation of this work. We also thank Falko Judt and two anonymous reviewers for their helpful comments and suggestions. Richard Rotunno is supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under cooperative agreement 1852977.

All HWRF simulations in this study employ an input sounding taken from Jordan (1958). Model output data used to generate spectral analyses in this study is available at: