{"Bibliographic":{"Title":"The LFM model - 1976 : a documentation","Authors":"","Publication date":"1977","Publisher":""},"Administrative":{"Date created":"08-20-2023","Language":"English","Rights":"CC 0","Size":"0000101995"},"Pages":["P82\nA\nQC\n851\nOF\nCOMMUNITY\nU6N5\nno.60\nNOAA Technical Memorandum NWS NMC 60\n*\n*\nwater\nSTATES\nOF\nTHE LFM MODEL - 1976: A DOCUMENTATION\nNational Meteorological Center\nWashington, D. C.\nDecember 1977\nnoaa\nNATIONAL OCEANIC AND\nNational Weather\nATMOSPHERIC ADMINISTRATION\nService","National Weather Service, National Meterological Center Series\nThe National Meteorological Center (NMC) of the National Weather Service (NWS) produces weather anal-\nyses and forecasts for the Northern Hemisphere. Areal coverage is being expanded to include the entire\nglobe. The Center conducts research and development to improve the accuracy of forecasts, to provide\ninformation in the most useful form, and to present data as automatically as practicable.\nNOAA Technical Memoranda in the NWS NMC series facilitate rapid dissemination of material of general\ninterest which may be preliminary in nature and which may be published formally elsewhere at a later\ndate. Publications 34 through 37 are in the former series, Weather Bureau Technical Notes (TN), Na-\ntional Meterological Center Technical Memoranda; publications 38 through 48 are in the former series\nESSA Technical Memoranda, Weather Bureau Technical Memoranda (WBTM). Beginning with 49, publications\nare now part of the series, NOAA Technical Memoranda NWS.\nPublications listed below are available from the National Technical Information Service (NTIS), U.S.\nDepartment of Commerce, Sills Bldg., 5285 Port Royal Road, Springfield, Va. 22161. Prices vary for\npaper copies; $3.00 microfiche. Order by accession number, when given, in parentheses.\nWeather Bureau Technical Notes\nTN 22 NMC 34 Tropospheric Heating and Cooling for Selected Days and Locations over the United States\nDuring Winter 1960 and Spring 1962. Philip F. Clapp and Francis J. Winninghoff, 1965.\n(PB-170-584)\nTN\n30\nNMC 35 Saturation Thickness Tables for the Dry Adiabatic, Pseudo-adiabatic, and Standard Atmo-\nspheres. Jerrold A. LaRue and Russell J. Younkin, January 1966. (PB-169-382)\nTN 37 NMC 36\nSummary of Verification of Numerical Operational Tropical Cyclone Forecast Tracks for\n1965. March 1966. (PB-170-410)\nTN 40 NMC 37\nCatalog of 5-Day Mean 700-mb. Height Anomaly Centers 1947-1963 and Suggested Applica-\ntions. J. F. O'Connor, April 1966. (PB-170-376)\nESSA Technical Memoranda\nWBTM\nNMC\n38 A Summary of the First-Guess Fields Used for Operational Analyses. J. E. McDonell, Feb-\nruary 1967. (AD-810-279)\nWBTM NMC 39 Objective Numerical Prediction Out to Six Days Using the Primitive Equation Model--A Test\nCase. A. J. Wagner, May 1967. (PB-174-920)\nWBTM NMC 40 A Snow Index. R. J. Younkin, June 1967. (PB-175-641)\nWBTM NMC 41 Detailed Sounding Analysis and Computer Forecasts of the Lifted Index. John D. Stackpole,\nAugust 1967. (PB-175-928)\nWBTM NMC 42 On Analysis and Initialization for the Primitive Forecast Equations. Takashi Nitta and\nJohn B. Hovermale, October 1967. (PB-176-510)\nWBTM NMC 43 The Air Pollution Potential Forecast Program. John D. Stackpole, November 1967. (PB-176-\n949)\nNMC 44 Northern Hemisphere Cloud Cover for Selected Late Fall Seasons Using TIROS Nephanalyses.\nWBTM\nPhilip F. Clapp, December 1968. (PB-186-392)\n45 On a Certain Type of Integration Error in Numerical Weather Prediction Models. Hans\nWBTM\nNMC\nOkland, September 1969. (PB-187-795)\n46 Noise Analysis of a Limited-Area Fine-Mesh Prediction Model. Joseph P. Gerrity, Jr., and\nWBTM\nNMC\nRonald D. McPherson, February 1970. (PB-191-188)\nNMC 47 The National Air Pollution Potential Forecast Program. Edward Gross, May 1970. (PB-192-\nWBTM\n324)\nWBTM NMC 48 Recent Studies of Computational Stability. Joseph P. Gerrity, Jr., and Ronald D. McPher-\nson, May 1970. (PB-192-979)\n(Continued on inside back cover)","A\nQC\n851\nU6N5\nmo.60\nNOAA Technical Memorandum NWS NMC 60\nTHE LFM MODEL - 1976: A DOCUMENTATION\nJoseph F. Gerrity, Jr.\nNational Meteorological Center\nWashington, D. C.\nDecember 1977\nATMOSPHERIC SCIENCES\nLIBRARY\nFEB 9 1978\nN.O.A.A.\nU.S. Dept. of Commerce\nAND NOAA ATMOSPHERIC\nUNITED STATES\nNATIONAL OCEANIC AND\nNational Weather\nAMOUNT\nDEPARTMENT OF COMMERCE\nATMOSPHERIC ADMINISTRATION\nService\nJuanita M. Kreps, Secretary\nRichard A. Frank, Administrator\nGeorge P. Cressman, Director\nUS SOFFARMENT OF\n78\n0435","Foreword\nDuring the past year, the Limited-Area, Fine-Mesh Model (LFM) has become a\nmajor contributor to the National Weather Service's numerical guidance.\nThis paper provides a relatively complete documentation of that model, a\nprimer for those who might need to use the code.\nThe selected material answers recurring questions, highlights those aspects\nmost amenable to future improvements, and functions as a handy reference to\nthe model's approximations. Some of the equations are incomplete; sufficient\ndetail is provided, however, to reconstruct them fully.\nThis paper defers model performance and evaluation to future studies.\nAcknowledgments\nThe originators of the LFM model are J. G. Howcroft and A. J. Desmarais.\nThey were encouraged in the task by G. P. Cressman, F. G. Shuman, and J. A.\nBrown. In the construction of the current version of the model, contributions\nwere made by J. G. Howcroft, W. Carlton, K. A. Campana, P. Polger, J. E.\nMcDonell, A. Nierow, E. Costello, R. Hollern, J. Irwin, and J. E. Newe11.\nSince the conversion of the model to the latest NOAA computer system, J.E.\nNewell has made major contributions to the improvement of the model's\nreliability and timeliness. Special acknowledgment is also due to Harry\nBrown and Harlan Saylor who generously shared their expertise in the inter-\npretation of numerical forecasts with the meteorologist-programmers involved\nin the development work. Finally, acknowledgment is made of the assistance\nprovided by J. D. Stackpole who was largely responsible for the development\nof the physical parameterizations that were adapted to the LFM model from\nthe National Meteorological Center's (NMC) hemispheric 6-layer model.\nAbstract\nThe Limited-Area, Fine-Mesh model (LFM), used operationally at the National\nMeteorological Center (NMC), is documented. Emphasis is placed upon the\nmodel's mathematical and physical approximations, and upon the methods used\nto initialize the model's o-coordinated variables from the analyzed fields.\nThe information presented was current at the end of 1976.\nii","CONTENTS\nForeword\nii\nAcknowledgments\nii\nAbstract\nii\n1.\nModel equations--hydrodynamic system\n1\n2.\nVertical coordinate\n4\n3.\nDiabatic and subgrid scale physical processes\n8\n3.1\nAtmospheric water content\n9\n3.2\nRadiative heat transfer\n12\n3.3\nSensible and latent heat transfer from the sea\n12\n3.4\nSurface frictional drag\n14\n3.5\nConvection\n15\n4.\nNumerical approximations\n15\n4.1\nTime integration and space smoothing\n17\n4.2\nLateral boundary conditions\n17\n4.3\nHorizontal approximations\n18\n4.4\nVertical approximations\n20\n5.\nCalculation of large-scale precipitation\n22\n6.\nCalculation of convective precipitation\n25\n6.1\nRemarks on the convective routine\n31\n7. Dry adiabatic convection\n32\nAppendices:\nA. Method of initialization of the o coordinate geopotential\nheights and potential temperature\n33\nB.\nRadiative heat transfer details\n42\nB.1\nAstronomical computations\n42\nB.2\nLong-wave cooling\n44\nB.3\nShort-wave warming\n44\nB.3.1\nDirect absorption\n45\nB.3.2\nIndirect heating\n46\niii","CONTENTS--Continued\n48\nC. Wind initialization\n48\nGeneral methodology\nc.1\n48\nNondivergent wind\nC.2\n54\nIrrotational component of the wind\nC.3\n54\nC.3.1\nComputation\n57\nC.4\nRemarks\n64\nAnalysis of water vapor\nD.\n64\nD.1 Relative humidity\n65\nD.2 Conversion to precipitable water\n67\nLiterature cited\niv","1.0 Model Equations--hydrodynamic system\nThe hydrodynamic equations use quasi-static approximations\nbut with sigma as the vertical coordinate. 2 Four domains are dis-\ncriminated in the vertical. The vertical coordinate, sigma, is\ndistinguished by a different subscript to identify each domain as\nfollows:\nboundary layer domain,\nOB\ntroposphere domain,\no\nT\nstratosphere domain,\no\nS\ncap layer domain.\no C\nThe absence of a sigma-subscript implies that the equation\napplies for all sigma domains upon specialization of the sigma\ncoordinate.\nThe quasi-static equations of motion are written for reference to\na set of plane Cartesian coordinates, X and y, on a polar stereo-\ngraphic map projection that is fixed to the rotating Earth. The\norigin of the Cartesian system is taken as the projection of the\nEarth's North Pole upon the map; the projection's scale factor,\nm(0), is defined by the equation:\nm(0) = (1 + sin 60°)/(1 + sin 0)\n(1.1)\nimplying that the map is true at a latitude (0) of 60° north.\n- (1.2)\n(1.3)\n1Lorenz, E. N. (1967).\nShuman, F. G. , and Hovermale, J. B. (1968).\n-1-","-2-\nThe variables u and V are defined in terms of the horizontal\nwind velocity (relative to the rotating Earth) in the zonal (us) and\nmeridional (vs) directions, positive \"contrasolum\" and toward the\nNorth Pole of the Earth respectively.\nThe defining equations may be expressed as\nu = = us sin(A+ 11) - vs - - cos (X + 11),\n(1.4a)\nv=usc(1)-vs sin (1+ 11) .\n(1.4b)\nIn equations (1.4), l stands for the longitude of a point measured\ncontrasolum from the Greenwich meridian. The 11 is a necessary com-\npensation because the positive sense of the X Cartesian coordinate\nruns from the North Pole toward the Equator along the 15° West\nlongitude meridian. (The NMC hemispheric model uses the 10° East\nlongitude meridian. ) One sets 11 = (15t/180) radians in equations (1.4).\nThe velocity components us and VS are calculated from observations\nusing the constant radius of the equivalent spherical earth, 6371 km.\nAll the partial derivatives, with respect to time t, and the\nhorizontal coordinates X, y, hold the sigma coordinate fixed.\nThe other parameters in equations (1.2) and (1.3) are:\nthe absolute vorticity,\nn\nthe geopotential,\n$\ncp the specific heat of pure dry air at constant pressure,\nthe potential temperature,\n0\nthe Exner function,\nTT\ndo\n& the O coordinate vertical velocity, dt\nThe absolute vorticity n is defined as the sum of the Coriolis\nparameter f and the relative vorticity 5. The value of f is defined\nin terms of the horizontal coordinates by:\n(1.5)\nf=25","-3-\nwith R(x,y) = (x2 + y2) / [a(1 + sin 60°),12,\n(1.6)\nwith \"a\" standing for the mean earth radius, 6.371x 10° 6 m, and So, the\naverage value of the angular velocity of the earth's rotation about\nits axis,\nR=7.292 x 10-5 sec-1. *\nThe relative vorticity 5 is given by the equation:\n(1.7)\nThe hydrostatic equation is written:\n(1.8)\nThe conservation of mass is expressed by the continuity equation\nin which pressure p is the mass variable,\n(1.9)\n-\nThe form of equation (1.9) follows from the hydrostatic equation (1.8)\nand the ideal gas law:\n(1.10)\np = p R T O\nin which p is the air density and R the gas constant for pure dry air.\nOne also uses the definitions of TT and 0,\n==(R)R\n(1.11)\nOETN-1\n(1.12)\nwith P a constant reference pressure (1000 mb) and T the air temperature.\n*Time is calculated in sidereal units. It may be useful to note that if\n(I,J) are the indices for the LFM grid points with (1,1) at the lower\nleft side of the grid, then the North Pole would have the indices (27,49).\nAll points along the column I=27 lie on the 105°W meridian. Thus,\n(x2+y2) = (190.5)2 ((I-27)2+(J-49)2) is in units of square km.","-4-\nThe hydrodynamic equations, stated above, are applicable to\ninviscid, compressible but quasi-static, motions of pure dry air.\nTo\ncomplete the system of equations, one must add a physical equation.\nThe air motions are assumed to be isentropic. The specific entropy\nmay be related to the logarithm of potential temperature. Thus, in\nthe case of no \"diabatic\" heating, one has the physical equation,\n(1.13)\n-\n.\nThe atmospheric model modifies the basic hydrodynamic equations\nfor a variety of nonadiabatic and irreversible physical processes,\nespecially those related to atmospheric water content. Subsequent\nsections will elaborate on these modifications.\nThe LFM is a limited-area numerical model. The equations out-\nlined above are replaced by numerical approximations based on finite-\ndifferencing. Subsequent sections will discuss the numerical\nintegration methods together with the treatment of boundary and\ninitial conditions.\n2.0 Vertical Coordinate\nThe vertical coordinate, sigma, is an elaboration of the system\nintroduced by Phillips (1957) it incorporates some aspects of a\nquasi-\nLagrangian system (Starr, 19452 and Shuman and Hovermale, 1968).\nThe explicit use of a cap layer to top the hodel-atmosphere\nrepresents a strong solution to two problems. First, the equations\nused in the model are clearly invalid at very low pressure: second, there\nis no satisfactory way either to provide realistic input data or to\ninterpret forecast output data at the very low pressures associated\nwith the top of the model atmosphere.\nThe definition of the oc is:\noc=p/pc\n(2.1)\nin which Pc is a function of horizontal position and time. This cap-\nlayer domain is not partitioned into sublayers, as are other sigma\nPhillips, N. A. (1957).\nStarr, V. P. (1945).","-5-\ndomains: It is treated as a homogeneous, isentropic fluid. The layer\nis initialized with an original horizontal wind of zero, although the\nwind can change, thereafter, in accordance with the equations of motion.\nThe potential temperature is constant everywhere within the layer at\nthe initial time and, because the motions are assumed to be isentropic,\nremains constant. The base of the layer, where the pressure has the\nvalue PC' is taken to be a material surface so that &C vanishes\nidentically there and also at the top of the cap where the pressure\nalso vanishes.\nAppendix A describes the methodology used for the initialization\nof the cap layer. The equations of motion (with no vertical advection\nterm), govern this autobarotropic layer, in conjunction with the\ncontinuity equation, i.e.,\n(2.2a)\n(2.2b)\n(2.3)\nwith\np\n(2.4)\nThe geopotential is determined by the upward integration of\nthe hydrostatic equation that is applicable in the lower o domains.\nThe stratospheric domain is contained between the tropopause and\nthe cap base. The coordinate 's is defined by:\n(2.5)\nwith PT the \"tropopause\" pressure, a function of horizontal position\nand time, and Pc the previously defined \"cap base\" pressure. The\ntropopause and cap base are both treated as material surfaces so that\n: vanishes at both interfaces.","-6-\nThe use of a stratospheric domain was motivated by several con-\nsiderations. First, it provides a reasonable way to partition the\nvertical resolutions of the numerical model. Second, the horizontal\nvariability of potential temperature is minimized in any o surface in\ncomparison to what it would be along an. isobaric surface that inter-\nsects the tropopause. Third, the standard sigma system permits oro-\ngraphically induced undulations in the coordinate surfaces to extend\nto great heights, but the introduction of the tropopause (as done here)\nrestricts the vertical extent of this kinematic influence of orographv.\nFinally, to the extent that the actual tropopause behaves as a material\nsurface, the method for interpolation of forecast output to isobaric\nsurfaces can be made to utilize the predicted \"tropopause\" surface to\nsimulate the observed vertical shear of the wind and temperature.\nThe principal simplification of the hydrodynamic equations that\nfollows from the use of the stratospheric sigma domain is that 20/20\ndoes not vary with 's' Thus, one has the continuity equation:\nPP s a -\n(2.6)\nin which\n(2.7)\nThe equation (2.6) may be differentiated with respect to 's to\nobtain a diagnostic2 equation for os:\n(2.8)\nAppended to equation (2.8) are the boundary conditions:\n's = at 's = 0 and 10s=1.\n(2.9)\nWithin the LFM model, the stratospheric sigma-domain is partitioned\ninto two equal mass subdomains for purposes of calculating the vertical\nstructure of the several variables.\n1F. G. Shuman, National Meteorological Center, NWS/NOAA, Camp Springs, Md. ,\npersonal communication.\n2A diagnostic equation is one that does not contain a time derivative, in\ncontrast to a prognostic equation.","-7-\nIt may be noted at this point that the use of material surfaces to\ndefine the interfaces of the several o domains does not give rise to an\nexplicit equation for \"following the material surfaces.\" The only\ndifficulties in this regard are related to the implicit assumption that\nthe flow and the material surfaces will remain well-behaved. In fact,\ncomputational noise does arise in the numerical integration and requires\nthe use of somewhat ad hoc procedures to maintain the good behavior of\nthe subdomains and their interfaces. These procedures may be rational-\nized on the basis that they constitute a redefinition of the material\nsurfaces that are to serve as the O domain interfaces.\nThe remaining two 0 coordinate domains are the troposphere and the\nboundary layer. The tropospheric o coordinate is defined by:\nTHE\n(2.10)\nin which PT is again the pressure at the \"tropopause,\" PG is the pressure\nat the level of the surface geopotential, it varies horizontally and in\ntime, and PB is a constant pressure (50 mb) assigned to the pressure\ndepth of the boundary layer. The tropospheric domain is therefore seen\nto be contained between the material surface tropopause and the top of\nthe boundary layer. The interface between the troposphere and the\nboundary layer is not treated as a material surface. Consequently, a\nset of continuity or compatibility conditions define the interconnections\nbetween the troposphere and the boundary layer. These interface con-\nditions are based on the continuity of vertical flux through the non-\nmaterial interface.\nFor the troposphere, as for the stratosphere, the continuity\nequation takes a form that permits one to derive, by its vertical\ndifferentiation, a diagnostic equation for the \"vertical velocity\" 'ST.\nThe continuity equation is (because PB is constant):\na P\n(2.11)\n-\nThe equation","-8-\nThe boundary conditions to which 'T are subject are :\n(2.13a)\n(2.13b)\n1.\nEquation (2.13b) reflects continuity of vertical mass transport through the\ninterface between the troposphere and boundary layer domains. The parameters\nUB and VB in (2.13b) are the vertically integrated, horizontal wind com-\nponents of the boundary layer domain.\nThe o coordinate in the boundary layer domain is defined by:\n(2.14)\nin which PG is the pressure at the ground and PB is a constant (50 mb).\nThe continuity equation takes the form:\n(2.15)\nThis equation may be considered to be a diagnostic equation for 'B'\nthe\n\"vertical\" velocity. Only one boundary condition is needed. The boundary\ncondition follows from the treatment of the ground as a material surface,\n(2.16)\nDiabatic and Sub-grid Scale Physical Processes\n3.0\nThe hydrodynamic system of equations is generally sufficient to\ndescribe the predominant aspects of the meteorological behavior of the\natmosphere for phenomena with horizontal length scales greater than\n1000 km and time periods of a few days. For smaller scales of motion or\nfor longer period forecasts, it is generally believed that diabatic physical\nprocesses must be taken into account in some explicit fashion.\nFor operational weather prediction, one must formulate a procedure\nfor the prediction of the evolution of the water vapor content of the\natmosphere. This is necessary to facilitate the forecasting of precipitation","-9-\nand cloudiness. This requirement exists quite independently of the\nsignificance of the atmospheric water content as a factor in the hydro-\ndynamic behavior of the atmospheric circulation.\nIn view of the disclaimers given above, we must state the ration-\nale for incorporating into the LFM model the physical processes discussed\nin this section. There are basically two reasons: first, the LFM model\nis based upon the NMC 6-1ayer PE model and second, the mean sea level\npressure field is sensitive to small changes in the thermal structure of\nthe lower atmosphere.\nExpanding on these two points, we note that the NMC hemispheric\nmodel has been used to produce forecasts up to 5 days in duration. The\nevolution of the atmosphere over such periods may include episodes in which\ndiabatic processes play a significant role. For this reason, the NMC\n6-layer PE model was designed to include computationally simple parameteri-\nzations of the physical processes generally included in general circulation\nmodels. Most of these processes have been carried over into the LFM model.\nThe second factor is most evident in the summer season in which the\nmean sea level pressure field displays features that are largely the result\nof low-level heating or cooling of the atmosphere. The thermal low of the\nsouthwestern United States is a primary example of this effect. A more\nsignificant effect of diabatic processes is found in the production of a\nlow-level trough in proximity with the warm ocean current near the east\ncoast of the United States. This trough is considered to owe its genesis\nto sensible heat transfer to the atmosphere from the sea and to the\nvariation of surface roughness across the coastal zone.\nIn the following, we shall present the general outline of the methods\nused to represent the phase changes and precipitation of atmospheric water,\nthe eddy, convective and radiative transfer of heat, and finally the parame-\nterization of eddy momentum transfer between the ground and the air. The\ndetailed computational system will be presented subsequent to the intro-\nduction of the finite difference system of equations in section 4 of this\npaper.\n3.1\nAtmospheric water content\nThe atmospheric model is designed to recognize the presence of water\neither as vapor or liquid. The liquid phase is admitted only briefly,\nhowever, because it is forced to either evaporate or to precipitate. Suspen-\nsion of the liquid or solid phases of water in clouds is explicitly avoided,\nbut the relative humidity is used implicitly to parameterize the presence of\nclouds in the computation of heat transfer by either radiation or convection.","-10-\nThe concentration of water vapor may be represented by the specific\nhumidity q, defined as the ratio of the mass of water vapor to the mass of\nmoist air. In the absence of water phase changes, the specific humidity\nof an air parcel remains unchanged. It follows that:\nJ\nq\n(3.1)\nThe maximum amount of water vapor that can be contained in a unit\nvolume is mainly a function of the temperature of the vapor, with larger\nconcentrations admitted with higher vapor temperatures. Because the vapor\nis mixed with the dry gases of the atmosphere, the vapor temperature is\ngenerally equal to that of the mixture. One notes further that the con-\ncentration of atmospheric water vapor is usually only a few parts per\nthousand, which means that the temperature of the mixture is dominated by\nthat of the dry gas. Thus, to predict the saturation value of specific\nhumidity qs, it suffices as a first approximation to predict the temperature\nof the dry gas components of the mixture.\nThe water vapor plays only a passive dynamic role in the model.\nWater vapor transport is assumed to be determined by the velocity field of\nthe dry gas. The saturation value of specific humidity is determined\nfrom the temperature of the dry gas.\nConversely, the hydrodynamic equations are treated as being applicable\nto pure dry air only. The influence of atmospheric water is restricted to\nthe production of diabatic heating effects which result from phase changes\nor indirect influences upon radiative heat transport.\nThe parameter actually used to represent water vapor concentration\nis defined in terms of the specific humidity q by the equation:\n(3.2)\nin which g is the acceleration of gravity and O, ,On refer to the limits\non o in the integral. Because o and q are dimensionless quantities, the\nvariable W has dimensions of mass per unit area. W may be converted to\nthe parameter \"precipitable water\" by dividing W by the density of water.\nBecause g is a constant, one may derive an equation for W by using\nthe continuity equation (1.9) and equation (3.1),\n(3.3)\nO 1\n1 The code computes W in units of gm/cm2, , but outputs W in units of kgm/m2.","-11-\nin which the last term's symbolism means that the difference should be\ntaken between the value of the bracketed quantity at the two values of O.\nA saturation value of W (Wg) is defined by analogy to (3.2),\ndp\ndo\n(3.4)\nin which as is the saturation value of the specific humidity.\nThis is an appropriate place to introduce a device that has found\nwide use in numerical weather prediction and in operational forecasting.\nExperience indicates that fractional cloud cover and even rainfall may\noccur when the relative humidity measured by a balloon-borne hygristor or\npredicted by a numerical model is appreciably less than 100 percent. This\ndoes not belie the validity of cloud physics, but indicates a problem inherent\nin the discrete sampling of a field with large variability. To account for\nthis fact, a reduced saturation value of W is used in the LFM model. The\nreduced value may be referred to as WM' and defined by:\n(3.5)\nWM = (SATRH)W\nin which SATRH is a constant less than unity. The parameter SATRH is\npresently assigned a seasonally variable value ranging between 0.90 and\n0.96. Whenever it is found necessary to estimate the degree of saturation\nof the water vapor, we use the W in lieu of Ws.\nIn the course of solving the system of hydrodynamic equations and\nthe water vapor conveyance equation, it can happen that W is predicted to\nexceed WM. The implication is that the water vapor has reached its\nsaturation value and that condensation has occurred.\nThe LFM model contains an algorithm by which the condensed water is\nprecipitated, and in some cases evaporated into lower-level unsaturated\nlayers. In conjunction with this redistribution of the condensed water\n(W-WM) , the algorithm modifies the temperature of the dry air to reflect\nthe addition of the latent heat of condensation that is associated with\nthe implied phase change.\nBecause the numerical integration of the model equations is carried\nout with a small time step (6 min), the algorithm is rather simple and\nis applied only at discrete moments and it therefore appears to be in the\nnature of an adjustment. Because the initiation of the algorithm follows\nfrom the prediction made by the hydrodynamic and water vapor transport","-12-\nequations, the adjustment is considered, somewhat imprecisely, to model\nthe larger-scale precipitation process with the attendant large-scale\nheat release.\nAnother adjustment is also made, however, on the basis of the\npredicted mass and water vapor distribution, but this adjustment is\ndesigned to estimate the contribution of moist (cumulus) convection to\nboth the production of precipitation and to the static stabilization of\nthe model atmosphere. It will be seen, when this algorithm is detailed\nsubsequently, that the requirements for this adjustment are related\nsolely to the prediction of the existence of a conditionally unstable air\nmass with a water vapor content exceeding an empirically arbitrary thresh-\nold value. At most, an indirect relationship exists between the diagnosis\nthat this moist convective adjustment should be applied and the kinematic\nstructure of the velocity field.\n3.2\nRadiative heat transfer\nThe transfer of heat via radiation is treated by the LFM model in\na fashion that is very elementary in comparison to procedures used in many\ngeneral circulation and boundary-layer models. A detailed exposition of\nthe methods used is given in appendix B.\nThe short wave radiation received from the sun is modulated by\nastronomical factors so that diurnal and seasonal variations are considered.\nDirect absorption of the solar beam by water vapor is parameterized, and\nindirect heating of the boundary layer is simulated in response to the\namount of incident short wave radiation. This entire process is sensitive\nto the estimated distribution of water vapor and cloudiness.\nThe specification of long-wave radiative cooling at a constant rate\nsubject only to the diagnosis of cloud cover is probably too elementary to\npermit an adequate simulation of the diurnal temperature cycle. This\nlimitation has been found to result in different, systematic errors in the\nboundary layer forecasts initiated at the two synoptic times 00 and 12 GMT.\nThe current formulation of this physical process is subject to change and\nwe leave the description of details to the cited appendix.\nSensible and latent heat transfer from the sea\n3.3\nThe model's boundary-layer potential temperature is modified to\naccount for sensible heat transfer from a warmer sea surface. Additionally,\nthe boundary-layer water vapor is not permitted to become less than an\narbitrary fraction (30%) of the saturation value associated with the sea\nsurface temperature.\n1At all levels above diagnosed clouds (if any), the long-wave cooling\nproceeds at the rate of 1.44°C/day.","-13-\nWe note that the sea surface temperature is introduced as a fixed\nparameter into the model from an analysis that is updated daily through\nthe use of satellite, buoy, and ship observations.\nThe formulation of the sensible heat received from the sea surface\nfollows the simple estimation of the heat flux solely as a function of\nthe air-sea temperature difference. If H is the heat flux from the sea\nin ergs/ (cm2 sec), , and if this heat is absorbed within the 50 mb (400m)\ndeep boundary layer, then the rate of change of boundary layer potential\ntemperature may be given by:\n(3.6)\nA typical formula¹ for H is:\n(3.7)\nwhich we may estimate using 0 B' the boundary layer potential temperature,\nand Ts, the sea surface temperature,\nH\n(3.8)\nor\n(3.9)\n=\n8.0 x 10\n.\nThe model uses :\n(3.10)\n8.0\nor\nKH = cm2/sec\n(3.11)\nwhich gives the final formula:\n(3.12)\n1 cf. (eq. 2.5), Priestley, C. H. B. (1959).","-14-\nThis value is set at zero when I's < OB. If the air sea temperature\ndifference is ten (10°C) degrees, equation (3.12) gives a heating rate of\n3.6°C per hour.\nThe LFM model does not parameterize turbulent transfer above the\nboundary layer. The heat added to the boundary layer will be transported\nupward primarily through the parameterization of dry or moist convection.\n3.4 Surface frictional drag\nWithin the boundary layer of the LFM model, the eddy momentum transfer\nbetween the air and surface is parameterized using a bulk formula.\nThe surface stress T is estimated using a dimensionless drag coefficient\nCD,\nTHE\n(3.13)\nin which p is the air density and VB is the average wind velocity in the\nboundary layer and VB is the wind speed. The stress is assumed to\ndiminish to zero at the top of the boundary layer so that the frictional\nacceleration of the boundary layer wind velocity is:\n(3.14)\nFor a 50 mb deep boundary layer, one may use AZ = 4.0 X 102 m.\nIntroduction of this value into equation (3.14) together with equation (3.13),\nin which VB is given in m sec-1 yields:\n(3.15)\nThe drag coefficient CD is taken from Cressman. 1 CD varies from 15 X 10-4\nto 95 X 10-4, with smallest values over the ocean and largest values over\nthe Rocky Mountains.\nTo illustrate the magnitude of this effect, suppose that /\nis\n10 m sec-1 and that the CD is 95 X 10-4, then, in 1 hr (3600 sec), the\nfractional change in VB, is:\n(3600) 10-4\n(3.16)\n(10)(3600)=+85.\nOver the ocean, where CD is 15 X 10-4, this change would be about 17 percent.\n1Cressman, G. P. (1960).","-15-\nThis numerical estimate, taken in conjunction with other questionable\naspects of friction's parameterization, 1 suggest that a reevaluation should\nbe made of the significance of this process in the LFM.\n3.5 Convection\nThe LFM model incorporates parameterizations of both \"moist\" and \"dry\"\nconvective mixing that take effect through the adjustment of the temperature\nand horizontal winds.\nThe adjustments associated with dry convection are made prior to every\ntime step of the model and insure that the tendencies are calculated from\ntemperature data which are nowhere superadiabatically stratified.\nThe moist convection is invoked at every time step, during the first\nforecast hour and, subsequently, at only hourly intervals. This convection\nroutine comes into play if the relative humidity in a layer exceeds 75 percent\nand a moist air parcel with the thermodynamic characteristics of the layer\nwould reach saturation, upon pseudo-adiabatic expansion, at a pressure higher\nthan that associated with the air in the model layer immediately above it.\nThis moist convective adjustment produces an adjustment in the temperature\nand horizontal winds of the two contiguous layers; in addition, an estimate\nof subgrid scale convective rainfall is made, but no changes are permitted\nin the large-scale water vapor field.\nThe details of these two convective adjustment algorithms will be\npresented subsequently.\n4.0 Numerical Approximations\nThe differential equations governing the model atmosphere are solved\nby means of numerical integration. The technique replaces derivatives by\ndifferences calculated on a discrete set of spatial points and temporal\ninstants. This set of points is termed a finite difference grid or lattice.\nTo denote the value of a model parameter F at a particular location\non the lattice, a set of indices (i, j, k, and n) are used; for example,\n(x=iAx, y=jAy, o=kAo, t=nAt)\n(4.1)\nin which Ax, Ay, so, and At are the differences between the grid points\nin the relevant coordinate directions.\n1Gerrity, J. P. (1976 and 1977).","-16-\nA convenient symbolic convention for use in expressing finite difference\noperations was introduced by Shuman1, and is used here. The notation involves\nthe use of subscripts to denote differences and superscripts with an attendant\noverstroke, or bar, on the parameter to denote averages. It becomes necessary,\nhowever, to admit fractional values of the indices and thereby imply the\nformation of intermediate quantities at locations that are not coincident with\npoints of the grid-point lattice. These are always intermediate functions,\nhowever, and the totality of operations always results in a value applicable\nat a point which is a member of the set of grid points.\nAn example might clarify this notion. Consider the centered finite\ndifference approximation to a first derivative in the X coordinate:\n- Fi-1,j,k] -\n(4.2)\nIn Shuman's notation, this is expressed in two steps. First, a difference\noperation is defined as:\n(4.3)\nIf i is an integer, Fx would have to involve values of F that are not\ndefined on the grid lattice; or if i were an odd multiple of 12, F\nX\nitself would have to be defined between grid points. The notation,\nhowever, always introduces a second step involving the application of a\ncorresponding averaging operation. In this example, one uses a \"bar x\"\naverage defined by\n(4.4)\nShuman, F. G. (1962).\nThe following papers also elaborate on aspects of this notation:\n2Gerrity, J. P. (1973).\n3Gerrity, J. P., McPherson, R. D., , and Polger, P. D. (1972).\n4Robert, A. J. , Shuman, F. G., and Gerrity, J. P. Jr. (1970).\n5Grammeltvedt, A. (1969).","-17-\nThe same comments apply here as did for the difference operation.\nWhen these two operations are both applied, one gets the approximation\ngiven in (4.2), viz.\n(4.5)\n4.1 Time integration and space smoothing\nThe time integration is carried out using the leap-frog method in\nconjunction with a spatial filter on the (n-1) time level field. The\nvery first time step is made using a forward difference. The interested\nreader is referred to NMC Office Note No. 1291 for more details on this\ntime integration technique.\nIn connection with the treatment of the diabatic processes, complete-\nness requires the recognition that some processes (e.g., moist convection/\nradiation) are calculated only at hourly intervals and are treated either\nas adjustments to the solution or as steady-valued quantities during the\ninterim between reevaluations. Most of these ad-hoc techniques have only\nheuristic and/or empirical justifications.\n4.2 Lateral boundary conditions\nBecause the boundary of the LFM model's area of integration is located\nin geographical areas where the atmospheric behavior is unsteady, a method\nfor specifying time-dependent lateral boundary conditions is required.\nThe actual behavior of the atmosphere on the boundaries is unknown, but a\nset of forecast values is generally available from previous integrations\nof hemispheric-scale models.\nThese hemispheric forecasts are used to specify the time derivatives,\nor tendencies, of the \"marched\" variables of the LFM model. 2 The tendencies\nare constructed so that they are uniform over 6-hour intervals: they are\nformed through biquadratic interpolation from the hemispheric model's grid\npoints to those of the LFM model.\nAvailable from Office of Director, NMC, Washington, D.C.\n2An exception is the precipitable water which is kept constant on the\nboundary.","-18-\nOnly the tendencies on three grid rows, surrounding the LFM area, are\nactually used. These hemispheric model tendencies are combined with the\ntendencies calculated by the LFM model's equations. At the outermost row\nof grid points, the hemispheric model's tendencies are used alone because\nthe LFM's equations do not admit construction of a tendency at those points.\nAt the next inner row, the hemispheric model's tendencies are given a\nweight of 2/3 and combined with 1/3 of the LFM's calculated tendency. The\nweights are reversed at the next inner row.\nThis \"sponge-type\" of boundary condition was adapted from the work of\nthe Fleet Numerical Weather Facility in Monterey.\nIn the rare event that hemispheric tendencies are unavailable, the LFM\ntendencies are combined, as above, with zero replacing the hemispheric\nmodel's contribution.\n4.3 Horizontal approximations\nThere are four basic horizontal finite difference approximations used\nin the numerical equations. They involve: the advection of a scalar, the\ndivergence of a flux, the convective term in the equations of motion, and the\npressure gradient and Coriolis term. Stating these terms here should suffice\nto reconstruct most of the difference equations.\nFirst, consider an advection term such as:\n(4.6)\nThe map scale factor appears as a divisor of the horizontal wind components\nbecause the numerical calculation carries the quotients, us rather than\nu and V. The finite difference form of (4.6) is expressed by writing two\nsteps. First,\n+ ADV(0) = + m2 xy xy\n(4.7)\nADV is seen to be an intermediate quantity, one that is defined at the\ncenter of a \"grid box.\" \" The value of ADV (0) is returned to the grid point\nwhich ao is wanted as follows:\nat\nat\nA\nxy\n(4.8)\n.\nat\nA","-19-\nThe operation in (4.8) amounts to taking the average of ADV in the four\nboxes adjacent to the relevant grid point. Somewhat greater accuracy of\nthis advection term can be achieved by using a higher order formula than\nis expressed in (4.8), but this refinement will require some significant\nmodifications to the current computer code.\nAn example of the approximation of the divergence of a flux can be\ngiven by reference to the precipitable water equation:\n(4.9)\n-\nAgain, a two-step procedure is used. First, one calculates in the grid boxes\na quantity WST defined by:\nWST = ADV( xy\n(4.10)\nin which ADV (W) is of the form (4.7) with W in lieu of 0. The final step\nis:\n(4.11)\nWST\nat\nFD\nIn equations (1.2) and (1.3) in section 1, the equations of motion\nwere written in the invariant form. We shall consider the approximation\nof the terms involving horizontal derivatives in the equation for u.\nFirst, consider\n(4.12)\nwhich combines the convective term and the Coriolis term. We again have a\ntwo-step procedure. First, calculate\n(4.13)\nand then return the quantity to the relevant grid point by\n(4.14)\nFinally, we consider the pressure gradient terms,","-20-\n(4.15)\n0\ncp\nWe note first that ₫ and TT are defined on sigma-surfaces, whereas 0 and\nare defined as sigma-layer average values. This point will be expanded\nupon in the next section. It is mentioned here to justify a \"bar o\"\noperator that appears on and T in the following. The now familiar two-\nstep method again applies. First, calculate\n(4.16)\ny\n= o + 0 xy -o TT X\nand then\nxy\n(4.17)\n.\np\n4.4 Vertical approximations\nThe treatment of the vertical finite differences in the NMC 6-1ayer\nmodels is unique. In fact, the LFM (and 6L PE) both have seven sigma\nlayers within which the variables u,v are calculated. The acronym 6-1ayer\nmodel derives from the conviction that the uppermost, or seventh, layer\nhas no meteorological significance; it is regarded solely as a computa-\ntional device for bounding the model atmosphere from above. In addition\nto u and V within the seventh layer, the variable is also calculated\nfor this layer. It is primarily this quantity's variation that impacts\nupon the solutions obtained elsewhere in the model atmosphere.\nBesides ap/doc, the model also calculates ap/do and ap/dos from the\nappropriate forms of the continuity equation. The value of B is\ntreated as a constant. These several variables do not vary in the vertical\ndirection. We need simply note that once they have been calculated, they\nserve to specify the pressure p at the upper and lower o-coordinate surfaces\nthat bound the seven o-layers. With the pressure specified, one gets the\nvalues of the Exner function IT by direct evaluation.\nThe potential temperature 0 is carried as a variable in the lower six\nlayers. The approximation of the vertical advection of 0 is discussed\nsubsequently; for the present we shall assume that a set of O's is given.\nThe calculation of the geopotential at the eight sigma-surfaces bounding\nthe seven o layers is made from the hydrostatic equation, using the specified\nvalue of at the bottom of the model atmosphere. If we let k be an integer\ndenoting the sigma-level or sigma-layer counting from 1 upward, the hydro-\nstatic equation is approximated by:","-21-\n(4.18)\nThe vertical velocity o is carried at the sigma-levels bounding the\nsigma-layers. There are only four nonzero values of : permitted in the\nLFM model, because the other four sigma-levels correspond to material sur-\nfaces (top and bottom of the computational layer, the tropopause and the\nground). In spite of the vanishing of specific o's, it is convenient\nto assume that all eight may be referenced by a subscript k running from\none to eight, from the ground to the top of the model. For completeness,\nit should be noted that & 2 is not unique. Reference to equations (2.13b)\nand (2.15) shows that : at this level must be scaled appropriately for use\nin either the tropospheric layer above or the boundary layer below the o\nlevel.\nWe may next consider the two forms of vertical difference approxima-\ntions to differential terms. First, vertical advection of quantity (say 0)\nin layer k is estimated by:\n(4.19)\nthe choices for Do follow from the assumption that the layer values Ok are\napplicable at the midpoint (with respect to o) of the kth o layer. We may\nnote that within the model stratosphere, the scheme (4.19) will return identi-\ncal estimates of vertical advection in both stratospheric o layers.\nThe final vertical approximation is used only in the layer precipitable\nwater W equation. The variable W is zero except within the lower three\nlayers of the model. The form to be approximated is the divergence of the\nvertical flux, ow.\nOne uses:\nfrom :\n(4.20)\nThe appropriate value of W for introduction in (4.20) is found by assuming\nthat the specific humidity q varies linearly with pressure. So in general\none has:\n=\nwhere F1,2 are appropriate functions for carrying out the interpolation.","-22-\n5.0 Calculation of \"Large-scale\" Precipitation\nThe calculation of the large-scale precipitation is made using the\npredicted layer precipitable water quantities W(k) (where k is an index:\n1,2 or 3), the flux of water vapor into the top tropospheric layer (within\nwhich no vapor is permitted), and the value of SATW(k) last available for\nthe layers. The calculation begins with the top tropospheric layer and\nworks downward through layers 3, 2 and 1. Condensation heating or evapora-\ntive cooling is calculated and applied to the temperature in the layer.\nIf AM is the mass of vapor condensed (positive value) or evaporated (nega-\ntive value) in the layer, the change in the layer's potential temperature\nis calculated by the formula:\nT X Md 1\nin which M is the mass of dry air per unit area in the layer, L is the\nd\nlatent heat of vaporization or condensation, 0 is the potential temperature,\nT is the thermodynamic temperature, and cp is the specific heat.\nIf the layer is \"super-saturated,\" say\nSUPER = W - WSAT > 0,\nthe mass of condensed vapor per unit area, AM, is\nAM - 191400 (q-qs) Po do\n1\n= SUPER.\nMd is C Po\nwith g the acceleration of gravity and C is a constant to convert pressure\nto the appropriate cgs units.\nOne also uses the relation,\nT=0\nwith i - the mean value of the Exner function in the layer.","-23-\nOne then has :\nSUPER\ncp TT g .\nC Do Po\nWithin the code, the constants L, CP' g, and C are combined by writing\nL g cp c = 4409 2\nwhere C is the conversion factor from mb to cgs units (i.e., , Appo is\ncarried in mb in the code). The factor 2 in the denominator is needed\nto account for the use of TT (2 i°) rather than it in the code.","-24-\nThe algorithm used in the code is given as a flow diagram:\nWas water vapor introduced into layer 4,\nYES\nNO\nthe topmost tropospheric layer?\nRAIN = 0\nRAIN = Excess Vapor\n4409 * RAIN\n0 (4) = 0 (4) +\n2\n*\n1/3 (Po) T\nSet k=3\nSUPER = Wk - SATW k\nB = RAIN\nRAIN = RAIN + SUPER\nLESS\nGREATER\nIS RAIN\nTHAN 0\nTHAN 0\nEQUAL\nTO 0\nSUPER = 0\nB = 0\n0\nRAIN\n=\nSUPER = 0\nB = 0\nSUPER = SUPER - B\nWk = W1k - SUPER\nOk = Ok + 4409 * SUPER\no\n2\n(Do Po)k\n*\nk= k-1\nNO\nIs k=0\nYES\nAdd RAIN to Accumulated\nPrecipitation\nFINIS","-25-\nThe preceding algorithm starts at the top of the water-bearing layers\nby forming precipitation from the vapor transported up into the top tropo-\nsphere layer of the model. This layer is assumed to be dry in the mean -\nbut the vapor transported into it is permitted to heat the layer as shown,\nwhen the vapor condenses. The precipitate thus formed falls into the next\nlower layer where it is added to the difference between W and the saturation\nvalue of W in the layer.\nIf the result is negative, all of the precipitate (i.e., rain) is\nevaporated: increasing W and lowering the temperature. No rain remains to\nfall into the next lower layer.\nIf the result is positive, we have one of two cases:\nCase 1 is that in which W is less than SATW by an amount less than the\nrain falling into the layer from above. This case requires the evaporation\nof a part of the rain sufficient to saturate the layer; and the cooling of\nthe layer temperature to account for the evaporation of part of the rain and\nfinally a reduction in the value of the rain falling into the next layer.\nCase 2 is that in which W is greater than SATW. In this case, we require\nthe reduction of W to SATW, the warming of the temperature to account for the\ncondensation of the excess of W above SATW, and the augmentation of the rain\nfalling into the next lower layer.\nThis exposes the physical interpretation of the central algorithm. One\nproceeds to the next lower layer. When all layers have been treated, the\nresidual rain, if any, is added to a location which accumulates the rainfall\namount attributed to the model.\nIt should be noted that large-scale rain cannot reach the ground from\nan upper layer unless the intervening layers are first brought to \"saturation\"\nas specified by SATRH and the most recent calculation of SATW. The value of\nSATW is calculated every even time step, and kept at that value for the sub-\nsequent odd time step.\n6.0 Calculation of Convective Precipitation\nAfter the \"large-scale\" precipitation and its associated changes in\nW and temperature have been calculated, the model turns to the estimation of\nconvective rain and to the adjustments in temperature and wind that may be\nappropriate in association with the convection. Within the present LFM code\nthis consideration of convection is done only once each hour of forecast time,\nexcept for the very first hour in which it is considered at each time step.\nDuring the first hour, neither the convective rainfall nor the large scale\n1\nrain is accumulated into the total rainfall.\n1 This omission of the precipitation formed in the first hour was implemented\nbecause of the occasional large amounts produced by the initial convective\nadjustments, especially at low latitudes.","-26-\nThe algorithm employed is schematically presented in the following flow\ndiagram. The method proceeds through the layers k=1,2, and 3, i.e., , upward\nfrom the boundary layer. The process is iterated permitting mixing of two\nlayers to occur up to twenty times or until no mixing is required in a full\npass through the three layers.\nThe amount of rain attributed to the convection CNRAIN is calculated from\nthe amount of water vapor that would have to condense in order to release\nsufficient heat to raise the temperature of the upper of the two layers\nby the full difference ADJ| Several physical constants have been combined\nto yield the constant 4.1 X 10-4, viz.,\n4.1 X\nCD is . .24 cal gm-1deg-1 ; L is 600 cal gm-1; 103 is the conversion factor to\nchange the units of DPPk+1 from mb to gm ; g is 980 cm sec-2.","-27-\nCONVECTIVE PRECIPITATION AND MIXING ALGORITHM FLOW DIAGRAM\nNWET = 0\nKUO = 0\nk = 1\nRH = Wk SATWK\nNote:\n>\nIS RH = 1. - YES Go To B\nIn both A and B\npaths\nKUO =\nNO\nKUO + 1,\nif adjustment is\ndone.\nIS RH < . .75 - NO\nGo To A\nYES\nC\nEVENTUAL\nRETURN\nk = k+1\nNO\nIS k= 4\nYES\nYES\nIS KUO = 0\nNO\nYES\nIS NWET > 20\nNO\nFINIS\nNWET = NWET + 1","-28-\n75 RH VI 1.\nMoist but unsaturated layer:\n.\nA\nCalculate the thermodynamic\ntemperature of the layer in degrees C\nCalculate the saturation vapor\npressure of the layer k\nCalculate the saturation mixing\nratio of the layer k\nCalculate the mixing ratio of the\nlayer k\nCalculate the pressure, PCL, and the\ntemperature, TCL, at the lifting\ncondensation level of the layer k\nReturn to C\nIs the PCL less than or equal to\nYES\nthe mean pressure of layer k+1?\nNO\nCalculate the saturation vapor\npressure at the temperature, TCL.\nCalculate the pseudo-equivalent\npotential temperature OD E\nof\na\nparcel in layer k lifted to the\nLCL dry adiabatically\nCalculate the potential temperature\n§ k+1 corresponding to O E at the mean\npressure of layer k+1\nADJ = 0 k+1 - AD k+1\nReturn to C\nIs ADJ < -0.1\nNO\nYES\nCalculate CNRAIN =\n-ADJ * 4.1*10-1 4 *DPP k+1\n( i° )\nk+1","-29-\nAdd CNRAIN to total\nprecipitation\nDo mixing\ncf.\nnote on mixing\nKUO = KUO+1\nReturn to C","-30-\nSaturated layer: RH > 1.\nB\nCalculate the thermodynamic\ntemperature of the layer k\nin degrees C\nCalculate the saturation\nvapor pressure of the layer k\nCalculate the pseudoequivalent\npotential temperature of layer k\n8 Ek\nCalculate the potential temperature\n8\nk+1 corresponding to 0 E k at the\nmean pressure of layer k+1\nCalculate ADJ = 0 k+1 - k+1\nIS ADJ\n- .1\nNO\nReturn to C\n<\nYES\nDo mixing cf.\nKUO = KUO+1\nnote on mixing","-31-\nThe following note clarifies the mixing process:\nLet DPP (k) be the difference in pressure across layer k and DPP (k+1)\nbe that across layer k+1.\nThe potential temperatures are adjusted or mixed according to (note\nthat ADJ is negative) :\nADJ * DPP (k+1)\n(k) = (k) +\n,\nDPP (k) + DPP (k+1)\n*\nDPP(k)\n0 (k+1) = 0 (k+1)\nDPP(k) + DPP (k+1)\nThe wind component mixing is done by setting the component in each\nlayer equal to the mass weighted averages of the two layers:\nu(k) DPP(k) + u(k+1) DPP (k+1)\n,\nDPP(k) + DPP (k+1)\nv(k) DPP(k) + v(k+1) PP(k+1)\nDPP(k) + DPP (k+1)\n6.1 Remarks on convection routine\nThe routine outlined has the effect of reducing the conditional\ninstability of the model atmosphere whenever the relative humidity exceeds\n75 percent. The reduction of the conditional instability is brought about\nby warming the upper layer and cooling the lower layer. Above the boundary\nlayer, the temperature changes are equal in the two layers leaving the\naverage potential temperature of the two layers unmodified. However, when\nthe boundary layer and the layer above it are adjusted, the boundary layer\npotential temperature is diminished approximately five times more than the\nupper layer's potential temperature is increased. This preserves the mass-\nweighted average of the two layers' potential temperature. It seems unreason-\nable, however, to simulate convection by cooling the boundary layer so signi-\nficantly. The possibility exists that the implied increase in boundary-layer\nrelative humidity will lead to the \"large-scale\" precipitation algorithm\nreleasing latent heat within the boundary layer and producing the associated\n\"large-scale\" rain at the next time step. It is important, therefore, to\nrecognize that the influence of convection on rainfall is not confined\nexclusively to the CNRAIN computation.","-32-\n7.0 Dry Adiabatic Convection Adjustment\nAt the end of each forecast step, the potential temperature 0 in each\ncontiguous pair of layers (beginning with the boundary layer) is checked\nto insure that the stratification is not \"super adiabatic.\" If a super- -\nadiabatic stratification is found, an adjustment to uniform potential tem-\nperature is made. The uniform temperature O is given in terms of the\ninitial values Ok by the formula:\nkPk - -\nShould more than two contiguous layers be found to be unstably stratified,\nthe adjusted temperature § is set to\nThis examination and adjustment is repeated, beginning once more with\nthe boundary layer, until the entire column is adjusted to a stable (or\nneutral) stratification.\nIn conjunction with this dry adiabatic adjustment of the potential\ntemperature, the horizontal wind components are also mixed to uniform\nvalues û,v. The u,v values are determined from the (u,v)'s in the layers\nby the same weighting scheme used for 8.","-33-\nAppendix A\nThe Method for Initialization of the o-coordinate Geopotential\nHeight and Potential Temperature\nInput from objective analysis programs consists of geopotential height\nat the ten mandatory pressure levels, temperature at the surface of the\nearth T* and at the mandatory pressure levels with the exception of 1000 mb,\nthe pressure and temperature of the analyzed \"tropopause\" surface.\nThe temperature at 1000 mb is constructed in subroutine \"T1000\" If\nthe analyzed geopotential of the 1000 mb surface ZZ is smaller than the\n10\ngeopotential of the model's orography Z* the temperature at 1000 mb, T10'\nis calculated from the formula:\n+ .006 (2* - 210)\n(A-1)\nIf the value of Z*-Z10 is nonpositive, then T 10 is calculated from the 850\n850 mb temperature T8 and geopotential Z8, by the formula:\n(A-1a)\nThe interpolation of the pressure level data to the O levels and layers\nis done in subroutine \"PTOSIG,\" which operates upon one vertical strip of\ndata at a time.\nThe first computation obtains the surface pressure at the model's\norographic surface, Z*. This is done by finding the two consecutive pressure\nlevels which embrace the model orography. If, however, the height of the\n1000 mb surface is above Z*, the 850 and 1000 mb levels are used.\nIf we denote the pressure and geopotential of the isobaric surface data\nlying above (below) Z* by a subscript k (k-1), the calculation of p*, the\nsurface pressure, follows from application of an assumption that the geo-\npotential is a quadratic function of the logarithm of pressure. This for-\nmulation is given by Shuman and Hovermale (op. cit, P. 530) in somewhat\ndifferent fashion, so it will be explained here in what appears to be a more\ndirect way.","-34-\nBy the use of the truncated Taylor series expansion one may write,\n(A-2)\np\n(1np\n1np)\n+\nwhere is taken to be a function of Inp and the expansion is about the\npoint Inp; and its derivatives are to be evaluated at 1np for insertion\ninto (A-2). In order to find the second derivative, one resorts to the\nhydrostatic equation:\ndinp\n(A-3)\nso that\na2g\nalnp2\nThe approximation (A-2) may be rewritten,\n(A-4)\n(1np 1np)2\nnow, if one selects 1np to satisfy\n(A-5)\n1np = 1/2 (1n Pk + In Pk-1) :\nand assumes that T varies linearly with 1np in the layer so that\n(A-6)\none may determine 10 in (A-4) by evaluating the expression at 1n Pk and ln\nPk-1' and then averaging the two resulting expressions. In this way, one\nfinds that of must be obtained from the formula:","-35-\n-\n= 1/2 +\n(A-7)\nTo determine the proper approximation of the first derivative [to the second\norder implicit in equation (2) ], one may difference the two expressions derived\nby evaluating (A-4) at In Pk and In Pk-1' to get\n=\n(A-8)\nThus, if one uses (A-5), (A-6), (A-7), and (A-8), the constants in (A-4) are\ndetermined. It is then necessary only to evaluate (A-4) given 0* to obtain\na quadratic equation for ln p*:\n(A-9)\nEquation (A-9) may be compared with the formula given in Shuman and\nHovermale (op. cit.). Comparison will be aided if we introduce the notation:\nh* = Inp*\n(A-10)\nb\n= gZ*\nwhence equation (A-9) becomes\n(A-11)\ngZ* = gZ3 = - (h* h3) - (1/2)b (h* - h3)2\nwhich is identical to the formula given by Shuman and Hovermale. Having\nexposed the details, we may now consider the solution of the quadratic\nequation for h*.","-36-\nShuman notes that the coefficient b may be vanishingly small which\nimplies an isothermal lapse rate, a common enough meteorological event.\nBy reference to equation (A-4), one notes that in the isothermal case the\nvariation of 0 with lnp becomes linear. In that event, the solution of\n(A-11) is\n(A-12)\nFrom this observation, we learn that the general solution must be written,\n2 (gZ3 - gz#)\n(A-13)\nEquation (A-13) has selected the positive sign for the radical and, following\nShuman, has expressed the solution in such a way that as b 0, the solution\n(A-13) approaches solution (A-12).\nIn order for the solution of (A-13) to be real valued, one must have\na2 + 2b\n(A-14)\nAn investigation of this inequality demonstrates that it cannot be\nviolated unless highly erroneous numerical values find their way into the\ninput data for height and temperature.\nThe surface pressure, p*, may therefore be calculated from the value,\nh*, given by equation (A-13) by means of the formula,\n(A-15)\np* = exp (h*).\nFrom the value of p* and the analyzed value of the tropopause pressure,\np**, the pressure at each a level within the troposphere (including the\nboundary layer) may be determined using the definitions:\nfor the boundary layer (BTHICK is 50 mb) and\nBTHICK\n-","-37-\nfor the troposphere. This calculation is carried out in subroutine \"PATSIG.\"\nConsider now a particular o-level, and assume that we find that it is\nlocated between the two mandatory pressure levels Pk and Pk-1 where\nPk-1 > Pk\nThe geopotential 0 is once more assumed to be a quadratic function of the\nInp, [cf. eq. (A-2) ] above. The solution for 0 at the pressure of the O\nsurface is straight forward.\nAs an example of the computation, we may consider the calculation of\nthe geopotential at the top of the model boundary layer where and\np=PB = p*-50 mb. It will be assumed that PB is greater than 850 mb. The\nformula for °B obtained from equation (A-12) is :\nT\n2\n.85)\n(ln\n1000\n(A-16)\n+\nR\nHaving calculated all the heights within the troposphere, the\ninitialization turns to the construction of the model stratosphere and\nthe computational cap. The initialization code observes that an \"error\nstop\" must occur if one of the tropospheric o levels lies above (at lower\npressure than) the 100-mb level. This stop does not abort the run if the\ntropopause pressure is equal to that at the sigma-level in question. None-\ntheless, this potential error stop points out a difficulty that must be\naddressed when the tropopause is analyzed to be near 100 mb; viz., there will\nbe essentially no stratospheric analysis data available to the initialization\ncode.\nAn examination of Gustafson's (1965) 1 explanation of his tropopause\nanalysis system suggests that the subtropical leaf of the tropopause will\nusually be located between 200 and 150 mb at a potential temperature of\napproximately 355°K. There is not, however, any constraint in his analysis\nsystem to insist that the tropopause pressure be greater than 100 mb.\nGustafson, A. F. (1965).","-38-\nThe first step in the calculation of the stratospheric o levels is\nthe determination of Po' the pressure at the top of the model stratosphere.\nThe method used is basically that described by Shuman and Hovermale (op. cit.,\np. 530), although there is some change in notation in the code which obscures\na direct comparison.\nTwo input parameters into this calculation, ZTOP and 0 -.5' the geo-\npotential height of the \"top\" of the atmosphere and the potential temperature\nin the computational cap layer, are largely arbitrary quantities. The princi-\npal constraints upon them are that 0 5 be sufficiently large to insure\nthat,\nit exceeds the stratospheric potential temperatures, and that ZTOP be suffi-\nciently large so that there can be a real valued solution to the equation de-\nrived by Shuman and Hovermale (loc. cit.).\nThe solution for PO is constrained to be at least 15 mb less than the\n\"tropopause\" pressure.\nHaving determined acceptable values ZTOP, 0\nand Po' the model\n5\ncalculates the geopotential height at the top of the stratosphere by in-\ntegrating down from the top of the atmosphere via the equation,\nThe code then calculates the height of the mid-level - of the stratosphere\nwhere the pressure is\nPs = 12 po+ p**).\nThis is done by finding the first mandatory pressure level which has lower\npressure than Ps or, if none are found, by taking the smallest mandatory\npressure level (100 mb). Analysis data at this level are denoted by a sub-\nscript k and that at the next lower mandatory layer are denoted by a sub-\nscript, k-1. The height at Ps is then calculated using the quadratic equation\n(A-2).\nAt this point the geopotential height of each o-coordinate level has\nbeen specified and the code turns to the calculation of potential temperatures\nfor each o layer.\nWith the exception of the boundary layer, the potential temperatures for\na o-coordinate layer are calculated using the hydrostatic equation applied to\nthe o layer.","-39-\nIn the case of the boundary layer, if the model topography is at or\nabove the 850 mb isobaric surface, the calculation proceeds as above;\notherwise, the boundary layer temperature, T1, is obtained by interpolation\nfrom the surface temperature T* and pressure p*, and that at the first\nmandatory level above the boundary layer (say Pk):\nT is then converted into potential temperature via\n1\nThis computation of 10 from the surface temperature, without reference to\nthe hypsometric equation, will produce a modification of all the o level\nheights in a subsequent step of the code.\nThe code next checks to determine if any portion of the initialized\nair column is superadiabatically stratified. The check is carried out by\nscanning the potential temperatures in contiguous layers from the boundary\nlayer up to the uppermost stratospheric layer. It is required that the\npotential temperature be nondecreasing. If two contiguous layers k and\nk+1, where Pk Pk+1' are found to possess potential temperatures Ok and\n0k+1' such that:\nthen the potential temperatures are modified. The modification consists in\nsetting both potential temperatures equal to a weighted average of their\nprevious values; viz.,\n(\n(TkPk - kt2Pkt2)","-40-\nAfter this modification has been effected, it is determined if the\npotential temperature in layer k+2 is such that:\nIf it is, then that potential temperature is worked into a new weighted\naverage with the previously modified values OK and 'k+1' and all three\nlayer values of potential temperature are set equal to the new weighted\naverage, viz.,\n(TkPk - \"kk2Pkt2)\nwhere we have used\nThis process is continued (involving additional layers as necessary)\nuntil the uppermost stratospheric layer has been checked with respect to\nthe layer beneath it. If any layer was found to be unstable, the code\nwill at this point repeat the check starting anew at the bottom of the\ncolumn.\nFinally, the potential temperature in the computational cap layer is\ncompared with the potential temperature of the uppermost stratospheric\nlayer. If the process just completed has resulted in a higher potential\ntemperature in the top stratospheric layer than in the cap layer, the\nentire initialization is started anew. If instead, the cap potential\ntemperature exceeds that of the upper stratospheric layer, the dry con-\nvective temperature is completed.\nAT the end of the \"PTOSIG\" subroutine, the geopotential heights of\nthe o levels are recalculated from the possibly modified potential temper-\natures by integrating the hypsometric equation up from the surface. It is\nat this point that the use of surface temperature in calculating 1 1 will\neffect a change in all the heights above the ground.\nBefore concluding this summary of the PTOSIG calculations, a question\nmay be raised with respect to the dry convective adjustment method. The\nmethod used is designed to conserve the sum of potential and internal energy\nin the entire air column. The calculation is based on a particular scheme for\ncalculating the change in the enthalpy in a layer due to a change in potential","-41-\ntemperature. That method involves an approximation for the layer mean value\nof TT (the Exner function) which is not that used elsewhere in the model. The\napproximation may be written:\nwhereas the usual approximation of the model is\nIf the customary approximation for the layer mean value of TT was used\none would have\n\"k+1)(Pk- Pk+1' + (1 \"k+2) (Pk+1 Pk+2)\nin place of the expression previously cited.","-42-\nAppendix B\nRadiative Heat Transfer Details\nB.1 Astronomical computations\nThe solar ray is depleted by water vapor absorption as it passes through\nthe model atmosphere. The absorbed radiation warms the atmosphere. To calcu-\nlate this effect one must know the angle of incidence of the solar ray. This\ncomputation requires the use of formulas based upon astronomical considerations.\nFirst, one must determine the solar declination S which is a function of\nthe day of the year. The formula used in the model is :\nsin S = sin(23° 261 38 ') * sin (SIG)\n(B-1)\nwith\nTT\nSIG = D + 180 * [279.9348 + 1.914827 * sin (D)\n- 0.07952 COS D + 0.019938 * sin (2D)\n(B-2)\n.00162 cos (2D) ]\n-\nin which\n(B-3)\nD 365.242 [DD]\n=\n.\nThe factor 365.242 is the length of the mean tropical year in apparent solar\ndays and DD is the day of the year after January 1. The justification for\nequation (B-1) is its approximate fit to data cited in the Smithsonian\nMeteorological Tables-1951 (cf. P. 495) which were extracted from the\nAmerican Ephemeris and Nautical Almanac for the year 1950.\nThe equation of time for the sun is also calculated from an empirical formula.\nSince this correction to Greenwich Mean Time is trivial in this current\napplication, we will not cite the formula employed.\nThe second astronomical quantity required is the sun's local hour angle.\nThis quantity depends upon the Greenwich Mean Time and the longitude of the\nplace for which the computation is to be done. If the local solar hour\nangle is known for any one location at a specified meridian, it may be con-\nverted to the value appropriate for any other meridian location.","-43-\nIn practice, the local hour angle is calculated for the meridian 75°E\nlongitude in the LFM (for 100°E longitude in the 6L PE). This meridian\nis\nthat extending from the North Pole parallel to the positive y-axis of the\npolar stereographic map when the origin of the map's coordinate system is\ntaken to be the North Pole. The local hour angle is reckoned positively\ntoward the west from the intersection of the meridian with the semicircle\nof the equator which is above the horizon (cf. P. 19, Woolard and Clemence,\n1966)¹.\nThe conversion of the reference meridian's angle h to that h())\nfor the meridian (a) of concern is carried out using the formula:\nh(2)2T-1+ho =\n(B-4)\ncos h(1) = cos(ho-1),\ncos h()) = cos h cos l + sin ho sini.\nThis is shown graphically looking at a polar stereographic map in the\nfigure below.\nl=0\nSun's meridian\nh(1)\nno\n1=0 reference meridian\nh hour angle at\nreference meridian\nNP\nh(a) hour angle at l\nmeridian\nNP North Pole\nMeridian of interest\nThe hour angle at the Greenwich meridian hG is given in terms of\nGreenwich mean time TT by the relation,\nhc = 15(TT + 12)\nwhere h is in degrees and TT in hours. Since the Greenwich meridian is 75°\nwestward from the LFM's reference meridian, the Sun's local hour angle ho at\nthe reference meridian measured in degrees is\nho = 15(5) = + hG\nho = 15(TT + 12 + 5)\n.\n1Woolard, E. W. , and Clemence, G. M. (1966).","-44-\nThis formula is used to write:\nSOLHR = (The)\n(B-5)\nin the LFM code's calculation. The sine and cosine of SOLHR are subsequently\nused in equation (B-4) to calculate the cosine of the local hour angle at any\npoint of interest. In the latter computation, the geographic symmetry of the\ngrid array is used to minimize storage and computation requirements. The\nformula for the cosine of the zenith distance of the Sun is:\n(B-6)\nCOSZEN = sino sind + coso cos8 cos h\nwith cos h given by (B-4). The COSZEN value is set at zero if it is less than\n0.173659, that is when the Sun's altitude above the horizon is less than 10\ndegrees.\nB.2 Long-Wave cooling\nFor purposes of calculating the cooling associated with terrestrial\n(long-wave) radiative heat flux divergence, the model assumes that clouds\nexist in a layer if the relative humidity is at least 60 percent. A check is\ndone to determine the uppermost of the three water vapor bearing layers within\nwhich the relative humidity indicates cloudiness. All layers of the model\n(but not the isentropic cap layer) above the highest cloudy layer are then\ncooled at the rate 1.6667 X 10-5 degrees sec-1. This cooling rate is\napplied at each time step. A check is made each hour to revise the estimated\ncloudiness and the associated long-wave cooling. Note that the three upper- -\nmost layers of the model are always subjected to this cooling because there is\nnever any water vapor in or above them.\nAn additional factor, related to long-wave radiative cooling within the\nmodel's boundary layer, is accounted for. If the relative humidity is less\nthan 60 percent in all layers, and if the ground is covered with snow or ice,\nand if the Sun's altitude is less than 10 percent, then the model's boundary\nlayer is cooled at an augmented rate. The total rate of cooling in the\nboundary layer (including the regular effect and that just outlined) is\n4.4445 X 10- 5 degrees sec-1 (i.e., 0.16 degrees per hour).\nB.3 Short-Wave warming\nThe short-wave (i.e., solar) radiation is permitted to warm the model\natmosphere in two ways. Direct absorption of radiative energy by water vapor\nis estimated in the lowest three layers. Indirect heating of the boundary\nlayer is calculated under conditions designed to parameterize the sensible\nheat transfer from the ground which has been warmed by the radiation reaching\nit.","-45-\nB.3.1 Direct absorption\nThe solar radiative energy absorbed by the water vapor in layer k\n(k=1,2,3) Ek is related to the quantity AH(LEV) by the relation:\nEk = cos Z [AH(k) - AH(k+1)]:\n(B-7)\nEk is in cal cm-2 min-1.\nAH (LEV) is the energy absorbed by water vapor down to the bottom of layer\nk=LEV. The value of AH(LEV) is calculated from the path length of water\nvapor above the level.\nFor the LFM model, no water vapor is present above layer k=3. Thus,\nheating by water vapor absorption occurs only in layers k=1, 2, and 3. The\ncode calculates AH(LEV) for LEV=1,...,7, but\n(B-8)\nAH(LEV) E 0 for LEV > .\nTo illustrate the computation, let's define:\n(B-9)\nin which Wk is the precipitable water (cm) and (10-3 Pk) is the nondimen-\nsionalized mean pressure of layer k. The calculation is done only for times\nwhen the Sun's altitude is greater than 10 degrees, so the cosine of the\nSun's zenith distance (cos Z) is always appreciably greater than zero.\nOne may further define:\nPT (1) = PT*,\nPT (2)\n(B-10)\n(3)\nPT*\nPT\n=\n(4)\nPT*\nPT\nThe calculation of AH(LEV) is made from PT (LEV) using the set of\nrelations:\nif PT 1,\nAH = .20 + .15 10g10\nif 10-1 PT < 1,\nAH = .20+.10 log10 PT\nif 10-2 SPT < 10-1\nAH = .15 .05 log10 PT\nif 10-3 PT < 10-2,\nAH = .13 + .04 logio PT\nif 10T4 S PT < 10-3,\nAH = .025 + .005 PT\nif PT < 10-4\nAH = 0","-46-\nThe heating rate H1 in degrees sec-1 of the moist air within layers\nk=1,2,3 is calculated using the relations\nE1,\n(B-11)\nwhere (Po)B and 1/3(Po)T are the differences in pressure (mb) across the\nrelevant layers of the model. The constant is necessary to account for the\nconversion to an intrinsic heating rate. It does appear from the omission\nof a division by IT that the assumption has been made that O 12 T. This\nomission results in an underestimate of the warming, which is larger in\nthe upper layers where O > T.\n.3.2 Indirect heating\nTo parameterize the warming of the lower atmosphere through sensible\nheat transfer from the ground which is itself warmed by absorption of in-\ncoming solar radiation, the following quantity is calculated:\nGNDWRM =\n{1.86\nAH(1) }\nZ\ncos\n-\n(B-12)\n{ 1 - TOPCLD } { 1 - GNDALB }.\nThe first factor was discussed above in connection with the conversion of\nunits; 1.86 is the value adopted for the solar radiative heat flux reaching\nthe ground if the atmosphere is relatively dry. AH(1) is the depletion of\nthe solar beam by water vapor. The cosine of the Sun's zenith distance cos\nZ accounts for the obliquity of the solar beam by water vapor. The factor\n(1 - TOPCLD) has the following set of values:\n(1 - TOPCLD) = 0 if RH1 60% ,\n= .25 if RH1 < 60%, RH2 60% ,\n.50 if RH1 and RH2 < 60%, RH3 60%,\n=\n= 1.0 if RHK < 60% for k=1,2 and 3.\nRHk is the relative humidity in layer k.","-47-\n(1 - GNDALB) is 0.10 over normal land surfaces, and zero over the\nsea or landpoints covered by snow or ice.\nFor dry air over land with the Sun in the zenith, the heating rate\nis about 0.7 degrees per hour.","-48-\nAppendix C\nWind Initialization\nC.1 General methodology\nThe LFM initialization of the horizontal wind operates on two data\nsources: the hemispheric 6-layer model forecast (12 hr) winds in sigma-\nlayers and the isobaric analysis of the wind. Both data sources are\nprovided on the LFM grid points. The analyzed winds are provided for\nthe mandatory isobaric levels between 850 and 100 mb. To these is appended\na 1000-mb wind-field calculated geostrophically from the 1000-mb geo-\npotential height analysis. The isobaric-levels winds are interpolated to\nthe sigma-layers of the LFM model.\nThe initialization code calculates for each sigma-layer a nondivergent\nwind which has the same relative vorticity as the interpolated analysis\nwinds. The code also calculates for each sigma-layer an irrotational wind\nwhich has the same divergence as the interpolated coarse-mesh forecast winds.\nThe \"initialized wind\" is produced as the sum of the nondivergent and irrota-\ntional components.\nThis note concentrates on the mathematical methods used to process the\ninput winds between the point at which they have been constructed in the\nsigma-layers of the LFM grid and the point at which the initialized wind is\nconstructed.\nC.2 Nondivergent wind\nThe method used to process the analyzed winds into nondivergent winds\nis contained in subroutine HANS* of the \"Fine Mesh Initialization\" (FMINI)\ncode.\nFrom the wind components u and V, one forms the relative vorticity 5\nby the finite difference formula:\n(C-1)\n.\nThe polar stereographic map factor is represented by m. The field of 5\nis calculated for each square (formed by grid points) within the fine-mesh\ngrid array.\nOne next implicitly constructs an additional set of grid points\nsurrounding the actual fine-mesh grid. The wind components on these\npoints are assumed to satisfy the relations;\n*The name of this subroutine recalls Hans Okland's contribution.","-49-\n+\n(C-2)\n,\nm\nIN\nm\nOUT\n(C-3)\nm\nIN\nm\nOUT\nwhere n\nV\nstands for the wind component normal to the boundary and\nV\nstands for the wind component parallel (tangent) to the boundary.\nUsing these implicit winds the value of 5 may be computed for the grid\nsquares surrounding the actual fine-mesh grid. We observe that the method\nof extrapolating the wind components implies that the tangential wind\nvanishes midway between the \"in-points\" and \"out-points.\" This midway locus\ncoincides with the centers of the grid squares constructed around the actual\nfine-mesh grid.\nFigures la and 1b may be helpful in visualization of what has been said\nto this points.\n+\nO\nO\nO\no\no\no\nO\n+\nO\nO\nO\nO\nO\n+\nFigure 1a. The actual fine mesh grid points \"+\" and the centers of the fine-\nmesh grid squares \"o.\" The u and V winds are given on the \"+\" points. The\nrelative vorticity is calculated at the \"o\" points. The line connecting the\noutermost grid points denotes the fine-mesh boundary.","-50-\nX\nX\nX\nX\nX\nX\nx\nX\nX\nx\nX\nX\nx\nX\nFigure 1b. - - Repeat of figure la, but with an additional set of grid points \"x\"\non which implicit wind components are constructed by extrapolation and an\nadditional set of grid squares \"A\" on which 5 is calculated using the newly\nconstructed \"x\" wind components. The line connecting the outermost \"A\" points\ndenotes the locus along which the tangential wind is implied to be identically\nzero by reason of the formulae used to extrapolate winds to the \"x\" points.","-51-\nIn order to insure the nondivergence of the wind field (to be constructed),\nit will be calculated from a scalar streamfunction 4 by the finite difference\nformulas:\n(C-4a)\nv=\n(C-4b)\nThe divergence of û and v is given by:\n(C-5)\n;\nsubstitution yields\n(C-6)\nThe relative vorticity of u, V is given by the formula:\n(C-7)\nSubstitution gives\n(C-8)\n.\nyy\nThe computational problem reduces therefore to solving the finite\ndifference equation:\n(C-9)","-52-\nwhere 5 is the relative vorticity computed from the analyzed winds at the\n\"0\" points of the fine-mesh grid (cf. figure 1a). Because the relative\nvorticity 5 is located at \"o\" points, the streamfunction 4 will also be\nlocated at \"o\" points. The problem associated with solving the equation\n(C-9) relates to the fact that the finite difference operator on 4 cannot\nbe formed on the outer row of \"o\" points. This gives rise to a require-\nment for \"boundary conditions\" on 4.\nThe several steps outlined above, that involve setting up an additional\nstrip of winds by extrapolation outward and the computation of an additional\nstrip of 5's, are understood to be artifices by which the equation for 4 may\nbe extended and supplied with boundary conditions, sufficient to permit a\nsolution of the equation to be calculated by a standard numerical method.\n+\nFigure 2. -- -Repeat of figure 1b, but containing an additional set of points \" \"\nlocated at the center of an imagined additional strip of grid squares. These\n\".\" points are useful to illustrate the boundary condition imposed on the\nstream function 4 in the Poisson equation.\nFigure 2 shows the structure of a grid that includes all the artificial\n(and implicit, as far as coding is concerned) points and grid centers. We\nnoted above that the winds extrapolated, to the \"x\" points implied that the\ntangential winds vanish midway between these points and the outermost set of\nfine-mesh grid points. We denote the tangential wind by UT and assume that\nit is given by the streamfunction 4 in accordance with equation (C-4). For\nspecificity, we will consider the boundary curve parallel to the X coordinate,\nso\n.\n(C-10)","-53-\nFrom the streamfunction, defined only at the center of grid boxes,\nUT may be constructed only at grid points. It is assumed therefore that\nthe wind in the center of grid boxes can be constructed from grid point\nvalues by averaging. Denoting a grid box value of the wind u, we have\n(C-11)\nThe vanishing of the tangential wind at the grid box centers denoted by\n\"A\" points in figure 2 requires that:\nxy\n(C-12)\n= 0\nat those points.\nThus, equation (C-12) provides a connection between the streamfunction\non the \" \" box center points and the \"A\" and \"o\" streamfunction points.\nHence, the boundary condition on 4 in the Poisson equation (C-9) is\n(C-13)\nThe relation (C-13) is satisfied if:\n(C-14)\nThis (eq. C-14) is the boundary condition used in the numerical solution of\nthe Poisson equation.\nThe exposition given above is obscured in the present coding of sub-\nroutine HANS by the effort to minimize storage requirements and by the\ncomplexity of the simultaneous indexing of grid-point and grid-square lo-\ncations. It is possible to carry through the exposition only through the use\nof diagrams (cf. figs. 1 and 2) and a knowledge of the mathematical\nprinciples involved.\nThe solution of the Poisson equation is carried out using successive\nover-relaxation with a convergence criterion that implies an error of 10-7\nsec-1 in the residual. The streamfunction that results is used to construct\nthe nondivergent wind on the fine-mesh grid points, \"+.\"","-54-\nC.3 Irrotational component of the wind\nThe basic input to this computation is the sigma-layer wind field predicted\nby the hemispheric, coarse mesh version of the 6L PE model from the initial data\nvalid 12 hours prior to the time for which the LFM model is being initialized.\nThese winds have been interpolated onto the grid points of the LFM model.\nThe objective of the computation to be described below is to determine\na sigma layer irrotational wind field with similar sigma layer divergence\npattern to that possessed by the 6L PE forecast wind field. The irrotational\nwind is constrained to vanish on the outermost set of F.M. grid points. It\nonly approximately possess the divergence of the 6L PE forecast winds. The\napproximation is best on the largest resolvable scales.\nC.3.1 Computation\nThere are five steps in the computation:\n1. A divergence field is calculated at the fine-mesh grid points.\n2.\nA velocity potential is calculated having a value of zero for its\nnormal derivative on the boundary and satisfying a Poisson equation\nat the interior grid points.\n3. A wind field is calculated in the grid boxes from the velocity\npotential.\n4. A smooth-desmooth filter is applied to the derived box winds to\nobtain interior grid point winds.\n5. A zero wind value is inserted on the boundary grid points.\nLet be the wind at the F.M. grid points provided by the\nhemispheric model. We form the divergence D of these winds, at the interior\ngrid points by the formula:\n(C-15)\nxyx\nD = m2 xyy + y\nThe map factor, defined at the grid points is denoted by m.\nThe irrotational wind (u,v) to be calculated at the center of grid boxes\nis related to a scalar velocity potential X 1 by:","-55-\n(C-16)\nThus, the finite difference Poisson equation,\n(C-17)\nis established at the interior grid points.\nIn the LFM initialization code scaled versions of X1 and D are used, viz.,\nx = (Ax) - 1 x 1\n(C-18)\nDiv = [(Ax) /m2] D\nThe Poisson equation is therefore expressed:\n(C-19)\nThis equation is solved by an iterative sequential relaxation method. The\niteration is assumed to have converged to an acceptable level when:\n(C-20)\nTo interpret this condition, we may introduce (u,v) and (UC'Vc) recalling\nDIV (Ax) the\n(C-21)\nand\n(C-22)","-56-\nSo convergence occurs when:\nVc\n(C-23)\n-\nm\ny\nX\nwith Ax = 190.5 X 10 3 (= 2 x 105) in the units employed throughout the code.\nConsequently, convergence occurs when the difference, between the diver-\ngence of the winds obtained from the hemispheric model and those implied by\nthe solution velocity potential, is less than /2 X 10-5sec-1. This is not a\nparticularly stringent condition.\nIn the course of solving the Poisson equation, the boundary condition of\na vanishing normal derivative of X is employed. After the solution is obtained\nfor X, the grid box-winds (u,v) are set by the equation (C-16).\nWinds (u,v) are constructed at the grid points, from the grid-box winds\n(u,v), by averaging\n,\nm\n(C-24)\nxy\n3/2\nis\nV\nThe averaging is sufficient to derive grid-point winds only at interior\ngrid points. The winds (u,v) on the boundary grid points are derived by use\nof a wall condition.\nThe final set of grid-point, irrotational winds is calculated by applying\na filter to the (u,v) field. Note that the averaging operator 11 xy'\nhas\nbeen used twice in the preceding steps: first, to obtain box winds from\nand, second, to obtain the grid-point winds (u,v). The filter used is denoted\nto imply an inverse to \"-xxyy\". We will remark upon these\nby\nfilters in section C.4.\nThe final irrotational winds provided by the process outlined above are\nobtained at interior grid points via:\n-x-x-y-y\n(C-25)\n-x-x-y-y\nTo complete the field of (uD ,VD), the value zero is inserted at boundary\ngrid points.","-57-\nC.4 Remarks\nThe completely initialized wind field is obtained by adding the non-\ndivergent wind components (cf. section C.2) and the irrotational wind com-\nponents (cf. section C.3). No explicit balancing of these winds to the mass\nfield is performed in the initialization, but during the first hour of the\nLFM numerical integration a version of the Euler-backward time integration\nmethod is employed. Experience suggests, however, that this method has little\neffectiveness in suppressing gravity-wave noise unless it is applied over a\nmuch longer time interval.\nAt several points, the use of averaging and \"inverse-averaging\" operators\nhas been indicated. It is considered desirable to illuminate the response\ncharacteristics of these filters here although they have been discussed else-\nwhere by Shuman (1957).","-58-\nThe smoother \"--xyxy\" has the 9-point stencil:\nx1/subscript(1)\nx1\n16\n16\n8\nx1/4\nx1/18\nx1/20\nx1/11\n1\nx1/2\n16\nIf we denote this operator by S, it is easy to show that it can be\ndecomposed into a linear combination of the identity operator I and two\n5-point operators A and B with the stencils associated with alternative\nLaplace operators:\nA:\nx 1\n0\nx 1\nX o\n(-4)\nx 0\n1\nx 0\nX 0\nB:\nX\nX 0\nX 1\n(-4)\nX 1\nX 0\nx 1\nOne has:\nS\n=\n(C-26)\nIf S is applied to a functional form:\n(C-27)\nm,n\nits spectral response Rk,l is:\n= .25 (1+a) (1+b))\n(C-28)\nwith a = COS kAx and b = coslAx.","-59-\nThe operator\n-x-x-y-y\" has the stencil:\nx 1/ 20\nx(-6)\nx 1\n16\n16\nx(-6)\nx36\n36\n(-6)\n16\n(6)\nx 1/6\nX\nIf we denote the operator by S-1, one may show that it can be\nexpressed in terms of the operators I, A, and B by:\n(C-29)\nThe spectral response of s-1 may be found to be:\n- .5a) (1.5 - ,5b)\n(C-30)\nwhere a = cos kAx and b = cos lAx.\nThe combined operator SS is representable by:\ns =\n(C-31)\nThe spectral response K of the product of S and -1 is:\n= (1+a) (1+b) (1.5-.5a (1. 5-.5b)\n(C-32)\nor\nRx, = + a - .5a2) (1.5 +b - .5b2).\n(C-33)\nThe range of kAx (and lAx) is from zero to TT. The range of a (and b)\nis from +1 to -1. In both cases, we are passing from long waves to short\nwaves.","for S operator; the table is symmetric across the diagonal.\n.000\n1.0\n.000\n.000\n0.9\n.009\n.002\n.000\nThe responses are tabulated below for kAx (lx) at intervals of 0.1 TT.\n0.8\n.042\n.020\n.005\n.000\n0.7\n.119\n.071\n.032\n.008\n.000\n0.6\n.250\n.173\n.103\n.048\n.012\n.000\n0.5\n(kAx/)\n.428\n.327\n.226\n.135\n.062\n.016\n.000\n0.4\n.520\n.630\n.397\n.274\n.164\n.076\n.019\n.000\n0.3\n.818\n.718\n.592\n.452\n.312\n.186\n.086\n.022\n.000\n0.2\nRk,l\n.952\n.882\n.774\n.638\n.488\n.337\n.201\n.093\n.024\n.000\n0.1\nResponse function\n.794\n.654\n.500\n.345\n.206\n.095\n.024\n.000\n.976\n.904\n0\n1.00\n(l/)\n0.7\n0.8\n0.9\n1.0\no\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6","1.0\n4.00\n3.90\n3.95\n0.9\nRk,l of the desmooth operator; the table is symmetric across the diagonal.\n3.63\n3.76\n3.81\n0.8\n3.22\n3.42\n3.54\n3.59\n0.7\n2.74\n2.97\n3,15\n3.26\n3.31\n0.6\n2.25\n2.48\n2.69\n2.86\n2.96\n3.00\n0.5\n1.81\n2.02\n2.23\n2.41\n2.56\n2.66\n2.69\n0.4\nkAx/TT\n2.16\n1.45\n1.62\n1.81\n2.00\n2.30\n2.38\n2.41\n0.3\n1.20\n1.32\n1.47\n1.64\n1.81\n1.96\n2.09\n2.16\n2.19\n0.2\n1.83\n1.05\n1.12\n1.24\n1.38\n1.54\n1.69\n1.95\n2.02\n2.04\n0.1\n1.00\n1.02\n1.10\n1.21\n1.34\n1.50\n1.65\n1.79\n1.90\n1.98\n2.00\n0\nResponse\nlsx/TT\n0\n0.7\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.8\n0.9\n1.0","1.0\nof the combined smooth-desmooth operator si the table is symmetric across\n.000\n0.9\n0.\n.033\n.008\n0.8\n0.\n.135\n.068\n.018\n0.7\n0.\n.326\n.211\n.101\n.026\n0.6\n0.\n.562\n.429\n.277\n.137\n.037\n0.5\n0.\n.775\n.660\n.504\n.325\n.159\n.042\nkAx/T\n0.4\n0.\n.914\n.842\n.718\n.548\n.354\n.175\n.045\n0.3\n0.\n.982\n.948\n.870\n.741\n.565\n.364\n.180\n.048\n0.2\n0.\n1.000\n.988\n.960\n.880\n.752\n.570\n.368\n.181\n.048\n0.1\n0.\nthe main diagonal.\nResponse Rkel\n.996\n.994\n.961\n.876\n.750\n.569\n.369\n.181\n.047\no\n1.00\n0.\nlAx/T\n0.4\n0.7\n0.8\n0.9\n1.0\no\n0.1\n0.2\n0.3\n0.5\n0.6","-63-\nIn view of the response function R it is reasonably clear that the\ndivergence of the initialized wind field will approximate that of the\nhemispheric forecast winds only in the larger scales. The value kAx = .2TT\nis associated with a scale (wavelength) of about 2000 km, and about 98\npercent of this scale in the divergence ought to be retained. Conversely,\nonly 20 percent of the divergence with a scale of 500 km is retained.","-64-\nAppendix D\nThe Analysis of Water Vapor\nD.1 Relative humidity\nThe analysis of the initial water vapor distribution is carried out in\nterms of the relative humidity within each of the model's three sigma-\ncoordinate layers that are closest to the Earth's surface. The lowest\nlayer, the model's boundary layer, has a fixed pressure-depth of fifty milli-\nbars (5 kpa). The pressure depth Dp of the remaining two layers is given by:\n= = 1/3(pg-p*-50)\nin which Ps is the air pressure at the Earth's surface and p* is the air\npressure at the level of the tropopause. The data processing of radiosonde\nobservations is carried out using predicted values of the two values, Ps\nand p*. Other observations and the initial guess are processed on the\nassumption that normal pressure-depths of the sigma-coordinate layers are\nvalid.\nBeside radiosonde observations of the relative humidity, the analysis\ncode employs pseudo-observations constructed from the surface weather and\ncloud observations, reported by land and ship stations, following the method\ndeveloped by Desmarais.\nThe initial guess or background field for\nthe analysis is derived from a combination of the layer relative humidities,\npredicted during the previous synoptic cycle by the hemispheric model, and\nthe climatological values of the fields.\nAt each analysis grid point, the background field, B, is constructed from\nthe predicted first guess, G, and climatology, C, according to the following\nalgorithm.\nNorth of 35°N latitude\nB=G\nBetween 20°N and 35°N\nB = W,G + (1-Wq)\nwith W1 III (sino - sin 20°)/(sin 35° - sin 20°)\nBetween 15°N and 20°N\nB=W2C + (1-W2)D\nwith W2 III (sind - sin 15°)/(sin 20° - sin 15°)\nand D = 40%\nSouth of 15°N\nB=D=40%\nDesmarais, A. J. (1971).","-65-\nThe resulting background field is then reduced by a factor SATRH which\nvaries seasonally between 0.90 to 0.96 according to the relations:\nconstant .96,\nNov. 21 to Mar. 20\nlinear decrease from .96 to .90,\nMar. 20 to May 20\nconstant .90,\nMay 21 to Sep. 20\nlinear increase from .90 to .96.\nSept. 21 to Nov. 20\nThe relative humidity observation obtained from radiosonde observations\nare assigned to the grid points within 2 Ax of the observation point. The\nreports derived from surface weather and cloud observations are similarly\nassigned to the nearest grid points. Radiosonde data take precedence over\nother observations. If more than one observation is assigned to a grid point,\na distance weighted average is used. These \"observations\" are adjusted at\nall grid points located south of 30°N according to the formula:\nR=R = * 2.* sino\nin which R is the original observation and R is the finally assigned value.\nThe background field is then assigned to fill in all grid points which\nhave not been assigned an observational value. The resulting field is then\nsmoothed to eliminate very small-scale irregularities. The result is accepted\nas the initial relative humidity analysis.\nD.2 Conversion to precipitable water\nThe precipitable water content of a sigma-coordinate layer is given by :\ndo\nq\nin which o > Superscript(0); g is the acceleration of gravity; q is the specific humidity;\n2\nand ap is the partial derivative of the air pressure with respect to the vertical\ncoordinate O.","-66-\nGiven the relative humidity, RH, of the layer, and the temperature and\npressure in the layer, W for the layer is constructed as follows:\nFirst, the \"Saturation specific humidity\" as in the layer is calculated\nusing the known temperature and pressure. The saturation precipitable water\nis estimated by approximation of the integral\nSAT W = as do dp do\n.\nFinally the layer precipitable water W is obtained from :\nW = RH* SATW , .\nThis initialized value of W is constrained to be no greater than Wmax--\nmax\nWMAX = SATRH*SATW\nwhere SATRH is the parameter that varies seasonally between .90 and 96. 1\n1More information on the NMC humidity analysis methods is contained in\nChu, R. (1977) and Chu, R., , and Parrish, D. (1977).","LITERATURE CITED\nChu, R., , 1977: Objective humidity analysis on isentropic surfaces for a\nlimited area. NMC Office Note 135, National Meteorological Center,\nNWS/NOAA, Washington, D.C.\nChu, R., and Parrish, D., 1977: Humidity analyses for operational predic-\ntion models at the National Meteorological Center. NMC Office Note 140,\nNational Meteorological Center, NWS/NOAA, Washington, D.C.\nCressman, G. P., 1960: Improved terrain effects in barotropic forecasts.\nMonthly Weather Review, 88, 327-342.\nDesmarais, A. J., (National Meteorological Center, NWS/NOAA, Washington,\nD.C.) 1971: Notes on mean relative humidity analyses in sigma layers for\nthe PE model. 10 pp. (unpublished manuscript).\nGerrity, J. P. , 1973: Numerical advection experiments with higher order\naccurate, semimomentum approximations. Monthly Weather Review, 101,\n231-234.\n, 1976: Modeling the planetary boundary layer: frictional influ-\nence. NMC Office Note 131, National Meteorological Center, NWS/NOAA,\nWashington, D.C., 14 pp.\n1977: On the parameterization of surface frictional drag and\n,\nthe existence of an inertial oscillation. NMC Office Note 138, National\nMeteorological Center, NWS/NOAA, Washington, D.C., 16 pp.\nGerrity, J. P., McPherson, R. D., and Polger, P. D., 1972: On the efficient\nreduction of truncation error in numerical weather prediction models.\nMonthly Weather Review, 100, 637-643.\nGrammeltvedt, A., 1969: A survey of finite-difference schemes for the\nprimitive equations of a barotropic fluid. Monthly Weather Review, 97,\n384-404.\nGustafson, A. F., 1965: Objective analysis of the tropopause. NOAA\nTechnical Memorandum NMC 33, National Meteorological Center, NWS/NOAA,\nWashington, D.C.\nLorenz, E. N., 1967: The nature and theory of the general circulation of\nthe atmosphere. World Meteorological Organization, Unipub, New York,\np. 6-19.\nPhillips, N. A., 1957: A coordinating system having some special advantages\nfor numerical forecasting. Journal of Meteorology, 14, 184-185.\nPriestley, C. H. B. , 1959: Turbulent transfer in the lower atmosphere.\nUniversity of Chicago Press, Chicago, 130 pp.\n67","68\nRobert, A. J., Shuman, F. G., , and Gerrity, J. P. , Jr., 1970: On partial\ndifference equations in mathematical physics. Monthly Weather Review,\n98, 1-6.\nShuman, F. G., 1957: Numerical methods in weather prediction. II.\nSmoothing and filtering. Monthly Weather Review, 85, 357-361.\n1962: Numerical experiments with the primitive equations.\n,\nProceedings of the International Symposium on Numerical Weather Prediction,\nTokyo, Japan (1960), Meteorological Society of Japan, Tokyo, 85-107.\nShuman, F. G., and Hovermale, J. B., 1968: An operational six-layer primi-\ntive equation model. Journal of Applied Meteorology, 7, 525-547.\nStarr, V. P. , 1945: A quasi-Lagrangian system of hydrodynamical equations.\nJournal of Meteorology, 2, 227-237.\nWoolard, E. W., and Clemence, G. M., 1966: Spherical astronomy. Academic\nPress, New York, 453 pp.","(Continued from inside front cover)\nNOAA Technical Memoranda\nNWS NMC 49\nA Study of Non-Linear Computational Instability for a Two-Dimensional Model. Paul D.\nPolger, February 1971. (COM-71-00246)\nNWS NMC 50\nRecent Research in Numerical Methods at the National Meteorological Center. Ronald D.\nMcPherson, April 1971.\nNWS NMC 51\nUpdating Asynoptic Data for Use in Objective Analysis. Armand J. Desmarais, December\n1972. (COM-73-10078)\nNWS NMC 52\nToward Developing a Quality Control System for Rawinsonde Reports. Frederick G. Finger\nand Arthur R. Thomas, February 1973. (COM-73-10673)\nNWS NMC 53\nA Semi-Implicit Version of the Shuman-Hovermale Model. Joseph P. Gerrity, Jr. , Ronald D.\nMcPherson, and Stephen Scolnik. July 1973. (COM-73-11323)\nNWS NMC 54\nStatus Report on a Semi-Implicit Version of the Shuman-Hovermale Model. Kenneth Campana,\nMarch 1974. (COM-74-11096/AS)\nNWS NMC 55\nAn Evaluation of the National Meteorological Center's Experimental Boundary Layer model.\nPaul D. Polger, December 1974. (COM-75-10267/AS)\nNWS NMC 56\nTheoretical and Experimental Comparison of Selected Time Integration Methods Applied to\nFour-Dimensional Data Assimilation. Ronald D. McPherson and Robert E. Kistler, April\n1975. (COM-75-10882/AS)\nNWS NMC 57\nA Test of the Impact of NOAA-2 VTPR Soundings on Operational Analyses and Forecasts.\nWilliam D. Bonner, Paul L. Lemar, Robert J. Van Haaren, Armand J. Desmarais, and Hugh M.\nO'Neil, February 1976. (PB-256075)\nNWS NMC 58\nOperational-Type Analyses Derived Without Radiosonde Data from NIMBUS 5 and NOAA 2 Temp-\nerature Soundings. William D. Bonner, Robert van Haaren, and Christopher M. Hayden, March\n1976. (PB-256099)\nNWS NMC 59\nDecomposition of a Wind Field on the Sphere Clifford H. Dey and John A. Brown, Jr.\nApril 1976. (PB-265-422)","NOAA CENTRAL LIBRARY\nCIRC QC851.U6N5 no.60\nGerrity, Jos The LFM model - 1976 : a\nd\n3 8398 0002 1503 2\nNOAA SCIENTIFIC AND TECHNICAL PUBLICATIONS\nNOAA, the National Oceanic and Atmospheric Administration, was established as part of the Depart-\nment of Commerce on October 3, 1970. 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