Heating and Kinetic Energy Dissipation in the NCAR Community Atmosphere Model

1DECEMBER 2003 BOVILLE AND BRETHERTON 3877

BYRON A. BOVILLE National Center for Atmospheric Research,* Boulder, Colorado

CHRISTOPHER S. BRETHERTON Department of Atmospheric Sciences, University of Washington, Seattle, Washington

(Manuscript received 2 August 2002, in ®nal form 24 March 2003)

ABSTRACT Conservation of energy and the incorporation of parameterized heating in an atmospheric model are discussed. Energy conservation is used to unify the treatment of heating and kinetic energy dissipation within the Community Atmosphere Model, version 2 (CAM2). Dry static energy is predicted within the individual physical parameterizations and updated following each parameterization. Hydrostatic balance leads to an ef®cient method for determining the temperature and geopotential from the updated dry static energy. A consistent formulation for the heating due to kinetic energy dissipation associated with the vertical diffusion of momentum is also derived. Both continuous and discrete forms are presented. Tests of the new formulation verify that the impact on the simulated climate is very small.

1. Introduction izations in CAM2 are clearly separated from the solution of the resolved adiabatic dynamics. In fact, CAM2 sup- The solution of the equations of motion in an atmo- ports three ``dynamical cores'': an Eulerian spectral spheric model requires the parameterization of processes core, as in CCM0 through CCM3 (CCM0±3); a semi- that occur on scales smaller than those explicitly repre- Lagrangian spectral core based on Williamson and Ol- sented by the model. Parameterizations are typically in- son (1994); and a ®nite volume core based on Lin and cluded for several processes (e.g., radiation, convection, Rood (1997). Horizontal diffusion is the only parame- vertical diffusion), which are treated as physically dis- terized process that acts across columns in CAM2 and tinct, even though they are intimately coupled. Because it is treated within the dynamical cores. Conservation of the difference between horizontal and vertical length of energy within the resolved dynamics is the respon- scales, the parameterized processes typically operate en- sibility of the dynamical core and the latter two (which tirely in the vertical direction, treating the atmosphere as contain implicit diffusion) include ad hoc ®xers to en- a set of distinct columns, which can be treated indepen- force conservation. dently. This paper discusses conservation of energy in Energy conservation was not a signi®cant consider- the Community Atmosphere Model, version 2 (CAM2; ation in developing the original CCM0, which contained more information available online at http://www.ccsm. an energy imbalance of ϳ10 W mϪ2 (Williamson 1988), ucar.edu/models/atm-cam/), particularly within the pa- primarily due to inconsistencies in the vertical numerical rameterization suite, and a uni®cation of the application approximations. Energy conservation was taken more of parameterized thermodynamic processes in terms of seriously in CCM1±3, which conserved energy to ϳ0.4 dry static energy. WmϪ2 (e.g., Boville and Gent 1998). This level of CAM2 is the successor to the National Center for conservation was partly due to a cancellation of errors Atmospheric Research (NCAR) Community Climate since individual processes conserved energy only to ϳ1 Model, Version 3 (CCM3; Kiehl et al. 1998). In common WmϪ2. Energy conservation is now a serious issue with many atmospheric models, the column parameter- because of the application of CAM2 in multicentury coupled ocean±atmosphere simulations in which even * The National Center for Atmospheric Research is sponsored by imbalances of 0.4 W mϪ2 can cause spurious long-term the National Science Foundation. trends. For historical reasons, the parameterizations of ther- Corresponding author address: Dr. Byron Boville, NCAR, P. O. modynamic processes in CCM0±3 have involved eval- Box 3000, Boulder, CO 80307-3000. uating tendencies for a mixture of temperature, potential E-mail: [email protected] temperature and dry static energy. Both computational

᭧ 2003 American Meteorological Society

Unauthenticated | Downloaded 09/30/21 02:12 PM UTC 3878 JOURNAL OF CLIMATE VOLUME 16 constraints and the evolution of the conceptual frame- is dI/dt ϩ dW/dt ϭ Q, where I is the internal energy work for the individual parameterizations have contrib- per unit mass, W is the work done by the ¯uid through uted to this diversity. In addition, the forcing terms have expansion, Q is the heating rate, and d/dt represents a been applied using a mixture of time and process split- substantive derivative. For a variable mixture of ideal ting (see below). Recently, Williamson (2002) has re- gases, each with a ®xed speci®c heat at constant volume formulated the CCM3 parameterization suite entirely in c␷ , p␣ ϭ RT, and I ϵ c␷ T, where R and c␷ are the time-split form. In this formalism, the parameterizations apparent (mass weighted) constants for the mixture, ␣ operate sequentially, on pro®les modi®ed by the pre- is speci®c volume, p is pressure, and T is temperature. vious parameterization. Noting that the work term is dW/dt ϵ p(d␣/dt) ϭ (d/

In CCM0±3, heating was considered equivalent to dt)p␣ Ϫ ␣(dp/dt), ␻ ϵ dp/dt and cp ϵ c␷ ϩ R, the ®rst temperature tendency within the parameterization suite, law is so that energy was not conserved for the intermediate d(cT) d␣ states which follow time split parameterizations. The ␷ ϩ p ϭ Q, (1) total heating due to all parameterizations was correctly dt dt reported to the dynamics and applied in a nearly con- or, equivalently, servative fashion. This manuscript describes a method for applying heating to the dry static energy so that the d(cT) RT␻ p ϪϭQ. (2) energy is locally conserved at all stages within the pa- dt p rameterization suite, regardless of the splitting method. The methods are equivalent when height is the vertical Here Q may have contributions from the divergence of coordinate but the distinction becomes important for radiative ¯uxes, molecular conduction, phase change pressure-based vertical coordinates, as used in most and precipitation processes, chemical reactions, and vis- global models. The meaning of energy conservation is cous dissipation. In the lower atmosphere, molecular stated precisely, including the effects of variable gas conduction is usually replaced by turbulent transport. mixtures, although CAM2 is still somewhat inconsistent For a variable mixture of gases, c␷ , R, and cp are in its treatment of variations in water vapor. determined by their mass-weighted average over the The vertical diffusion in CAM2 has been reformu- gases in the mixture. In CAM2, we make the common lated using dry static energy and both the kinetic energy approximation that dry air is well mixed and that water ¯ux and the heating due to kinetic energy dissipation vapor is the only additional substance present in suf®- are derived. These terms depend correctly on the stress cient abundance to affect the total pressure. Let super- tensor as discussed by Fiedler (2000) and Becker (2001). scripts ␻ and d denote water vapor and dry air, respec- In CCM1±3, the kinetic energy ¯uxes were ignored. tively. Then the apparent constants are We begin by reviewing the total energy equation in dwd section 2 and its application to the physical parameter- cppϭ c ϩ (c pϪ c p)q, (3) izations, together with the hydrostatic approximation in R ϭ Rdwdϩ (R Ϫ R )q, (4) section 3. The splitting procedures used for parameterizations are summarized in section 4, the procedure for where q is speci®c humidity. Water vapor is suf®ciently deriving temperature and geopotential from dry static abundant in the Tropics so that vertical variations in R energy is discussed in section 5, and vertical diffusion may change the sign of the static stability, while hori- and dissipation of kinetic energy are discussed in section zontal variations may produce pressure gradients even

6. Energy conservation considerations for the dynamical with uniform temperatures. Variations in cp are often cores are commented on brie¯y in section 7, the impact considered to be less important, since they affect the of the changes on the CAM2 simulation is discussed in response to heating, but do not directly affect the density d section 8, and conclusions appear in section 9. and pressure gradients. CAM2 uses cp ϭ cp, so that (2) dd becomes dT/dt Ϫ RT␻/pcppϭ Q/ c . This is the thermodynamic equation solved by both the spectral (Eu- 2. Total energy conservation lerian and semi-Lagrangian) cores, although cp is used Energy conservation is a important property of the in the second term due to an oversight. The ®nite volume equations of motion, which should be maintained when core converts (2) to a virtual potential temperature equa- approximations are made in order to solve them. Energy tion. We retain a variable cp in the derivations below, conservation can be guaranteed by forming an evolution but properly accounting for variations in cp in a model equation for total energy and verifying that it is satis®ed will probably require predicting an enthalpy (cpT) re- by the approximate equations. The total energy equation lated variable. This will become increasingly important is obtained by adding the kinetic energy equation to the as the thermosphere is incorporated into CAM2, since ®rst law of thermodynamics. The derivations of these diffusive separation of constituents results in large var- equations may be found in Gill (1982), among other iations of cp and R above ϳ120 km. sources, and will only be summarized here. The kinetic energy equation is derived from the mo- A fundamental statement of the ®rst law for a ¯uid mentum equation,

Unauthenticated | Downloaded 09/30/21 02:12 PM UTC 1DECEMBER 2003 BOVILLE AND BRETHERTON 3879 pץ dV d (F ϩ Q , (9 ´ ١␣ ␶, (5) (s ϩ K) ϭ ␣ Ϫ ´ ١␣ ١p Ϫ g Ϫ␣ϩ 2⍀ ϫ V ϭϪ t K 0ץ dt dt where ␶ is the stress tensor due to molecular or turbulent where s ϵ cpT ϩ⌽is the dry static energy. The dry viscosity and other notation is standard. De®ning kinetic static energy includes the internal energy (c␷ T), the po- energy K ϵ V ´ V/2, and taking the scalar product of tential energy (⌽), and the work done (RT ϭ p␣)to V with (5) we obtain raise the volume of a ¯uid element from 0 to ␣ at constant p. dK (FK Ϫ D, (6 ´ ١␣ ١p Ϫ wg Ϫ ´ ϭϪ␣V dt 3. Hydrostatic approximation where the last two terms are the divergence of the dif- Up to this point, we have not made the hydrostatic fusive ¯ux (FK ϭ V ´ ␶) and the dissipation (D ϭ approximation: .(١V ´ ␶) of kinetic energy, as in Fiedler (2000␣Ϫ pgץ We obtain the total energy equation by adding (6) and ϭϪ␳g ϭϪ . (10) ␣ zץ V ϭ ␣Ϫ1d␣/dt), and noting ´ ١) using mass continuity ,(1) that wg ϵ d⌽/dt, where ⌽ϵgz is the geopotential: However, CAM2 is a hydrostatic model and we require d that the state be in hydrostatic balance in addition to (pV ϩ F ) ϩ Q . (7)´ ١␣cTϩ⌽ϩK) ϭϪ) dt ␷ K 0 conserving energy. Furthermore, each atmospheric column is treated independently within the physical parameterization suite and the parameterizations are time The heating Q 0 ϭ Q Ϫ D in (7) excludes kinetic energy dissipation, which merely converts one form of energy split. A sequence of intermediate states is constructed to another. Energy conservation requires that Q in (2) and we require both energy conservation and hydrostatic include all kinetic energy dissipation, whether it arises balance for each state. In fact, within the CAM2 pa- explicitly from viscous processes in (5), or implicitly rameterization suite, we require local conservation for from imperfections in the numerical approximations of both energy and constituents within each layer of every the dynamical core. The local value of the implicit dis- column. sipation is usually dif®cult to determine, but the globally Satisfying these requirements begins by forming a dry averaged value may be determined by integrating (7). static energy equation from (2). Expanding the sub- Converting (7) to ®xed volume elements and assum- stantive derivative in the second term of (2) and applying (10), ing that pV and FK vanish on the top and bottom bound- aries, the global integral of (7) is pץ ds zt Q V ´ p , (11) d ϭ ϩ ␣ ϩ hh١ tץcTϩ⌽ϩK)␳ dz dA dt ΂΃) dt ͵͵ ␷ Azϭ0 where Vh is the horizontal velocity. We emphasize that

zt (11), which is adopted in CAM2, is an exact statement ϭ Q ␳ dz dA, (8) of the ®rst law of thermodynamics in a hydrostatic sys- ͵͵ 0 Azϭ0 tem. A similar dry static energy equation, in which the

heating Q 0 does not include D, may be obtained from where dA is an area element and zt is the height of the (9) by neglecting K, as discussed in Gill (1982). This model top. The globally integrated total energy may approximation is not made in CAM2. change only due to the globally integrated nondissipa- From the hydrostatic form of (5), we obtain the hy- tive heating, which can be expressed as the net boundary drostatic kinetic energy equation, ¯uxes of sensible, latent, and radiative energy. Implicit dissipation due to the numerical approximations will dK (F Ϫ D, (12 ´ ١␣ p Ϫ ١ ´ ϭϪ␣V result in an imbalance in (8), with the left-hand side dt hh K usually being less than the right-hand side. In order to maintain energy balance, the globally averaged implicit and the total energy equation is still (9). Note that in dissipative heating, determined by this imbalance, can the hydrostatic approximation, the kinetic energy is re- be added to (2) as either a globally uniform heating, or de®ned to include only the horizontal wind, K ϭ Vh ´ V /2. The pressure tendency and all advective and hor- other ad hoc function. The implicit heating for the h CAM2 dynamical cores is discussed in section 7. izontal gradient terms are treated within the dynamical The form of the total energy equation used to con- core, so that within the parameterizations (11) and (12) struct conservative parameterizations is a variation of are just

(t)ccϭ Q , (13ץ/sץ) Applying the product rule for a divergence and .(7) ,tץ/pץ␣ pV) ϭ d(␣p)/dt Ϫ)´١␣ ,using mass continuity (z Ϫ D , (14ץ/ Fץ␣t)cKcϭϪץ/Kץ) which is substituted into (7) to obtain

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where the subscript c refers to the column parameteri- most applications, Fs(zt) is just the net radiative ¯ux, zations and FK is the vertical ¯ux of kinetic energy due while Fs(zs) also includes the sensible heat ¯ux. In the to parameterized stresses. We distinguish between Qc ϭ thermosphere, emission of longwave radiation is very Q 0 ϩ Dc and Q ϭ Q 0 ϩ D, because kinetic energy inef®cient and absorption of solar radiation is largely dissipation within the dynamical core, whether implicit balanced by downward molecular diffusion of heat. or explicit, is treated separately. Within the parameter- Therefore, in upper-atmosphere applications Fs(zt) may izations, the total energy is de®ned as s ϩ K and its include a downward diffusive ¯ux of heat due to solar tendency is given by (13) ϩ (14). Total energy is locally heating above zt. conserved by using (13) to add all parameterized heat- The hydrostatic approximation is used to pose the ing, including Dc. Hydrostatic balance is maintained by parameterizations in pressure coordinates in CAM2. using (10) to obtain T and ⌽ from s. Since the pressure tendency is treated within the dy- In the CAM2 column parameterizations, only vertical namical core, the pressure is assumed to be invariant diffusion and orographic gravity wave drag dissipate over the parameterization step. This procedure is not kinetic energy by acting on momentum. The dissipation strictly correct in the presence of parameterized water was computed in CCM2±3 by neglecting FK in (14) so ¯uxes, which should change the local pressure. Omitting t for each process. This is this effect implies a compensating ¯ux of dry air. Theץ/Vץ ´ t ϭϪVץ/KץD ϭϪ actually the correct heating for orographic (zero phase net surface ¯ux of water (precipitation minus evapo- speed) gravity waves, which have no vertical energy ration) should change the surface pressure, but instead ¯ux. The correct vertical integral is still obtained for implies a surface ¯ux of dry air. For the ®nite volume molecular and turbulent diffusion processes, despite ne- core, the dry pressure in each layer is corrected after glecting FK. The correct forms of FK and D for the the parameterizations, while conserving the masses of vertical diffusion parameterization in CAM2 are derived all constituents. An energy and momentum conserving below. form has now been tested, but is not in CAM2. The Latent energy is conserved by conserving constitu- Eulerian and semi-Lagrangian cores require a ®xed re- ents individually and computing the energy associated lationship between the layer pressures and the surface with phase changes. As in previous versions of the pressure, so a simple correction to the dry mass does CCM, the latent heat of fusion is neglected in CAM2, not work and only a global correction to the dry air but not in the land and sea ice models that determine mass is applied. the surface ¯uxes. This inconsistency results in a globally and annually averaged conservation error of ϳ0.2 4. Process and time splitting WmϪ2, as mentioned in Boville and Gent (1998). Of course, the error is systematic and is much greater over The set of parameterizations in a model may be log- ice surfaces. The latent heat of fusion has been imple- ically conceived of as operating simultaneously (process mented in CAM2 and will be discussed in a separate splitting) or sequentially (time splitting) within a time manuscript. step. In process splitting, each parameterization com- A statement of vertically integrated energy conser- putes a heating based on the same input state, vation within the physical parameterizations acting on Q ϭ Q (s, T, ⌽, . . .), (16) each column can be obtained by adding the vertical ͸ i i integrals of (13) and (14) and writing the heating Qc as the net latent heating due to moist processes plus the where Qi is the heating due to a particular parameter- convergence of an upward total sensible heat ¯ux Fs ization. Although the same states enter each parame- due to radiation and vertical transport: terization, the parameterizations are not independent. For example, radiative heating depends on cloud prop- -zt erties, which depend on convection and large-scale con ץ zt (s ϩ K)␳ dz ϭ Fsz| s ϩ LC,␷ (15) densation. Therefore, the solution may depend on the t ͵ cץ []zs order of the parameterizations. In time splitting, the or- where L␷ is the latent heat of vaporization, zs and zt are der dependence of the parameterizations is made ex- the surface and model top heights, Fs is the heat ¯ux, plicit. The partial time derivative in (13) is evaluated and C is the vertically integrated net condensation rate. after each parameterization and the state is updated se- In CCM0±3, C was just the precipitation rate. CAM2 quentially: includes a prognostic cloud water that acts as a reservoir s ϭ s ϩ ␦tQ (s , T , ⌽ , . . .). (17) of liquid water, so that C is not balanced instantaneously iiϪ1 iiϪ1 iϪ1 iϪ1 by precipitation. We have assumed that the stress van- After the application of all I parameterizations, the net ishes at zt and that V(zs) ϭ 0, so FK(zs) ϭ FK(zt) ϭ 0. heating is calculated as Even if the motion of the sea and ice surfaces is ac- Q ϭ (s Ϫ s)/␦t. (18) counted for when computing the surface stress, one must I still assume that FK(zs) ϭ 0 unless the kinetic energy In CCM3, the ®rst three parameterizations (penetra- ¯ux is also accounted for in the ocean and sea ice. For tive convection, shallow convection, large-scale con-

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Ϫ1 t, (solid), (left) forץ/Tץ FIG. 1. Heating rate Q/cp (dash, K day ) and temperature tendency

an idealized heating Q/cp ϭ 1 and (right) for the actual net tropical convective heating from .t use the average tropical pro®le of T and q from CAM2ץ/Tץ CAM2. Both solutions for densation), which involve condensation and latent heat given below, allowing (17) to be used at the same cost release, were time split while the last three parameter- as (19). The difference is shown in Fig. 1 for a uniform Ϫ1 izations (radiation, vertical diffusion, and gravity wave heating of Q/cp ϭ 1 K day and for the net convective drag) were process split. Time splitting the moist pro- heating from CAM2 over the western equatorial Paci®c cesses allows precise enforcement of the constraint that Ocean. Applying a uniform heating in the vertical results relative humidity Յ100%. In CAM2, all parameteri- in a temperature tendency that decreases with height. zations are time split. The surface turbulent ¯uxes are Heating increases both the temperature and the thickness computed after radiation and are applied as boundary of lower layers, therefore increasing ⌽ in layers above. conditions in vertical diffusion. In the absence of heating at upper levels, conservation t Ͻ 0as⌽ increases. This canץ/Tץ Application of (17) is straightforward if z is the ver- of s would require tical coordinate, since ⌽ is invariant. However, CAM2 be clearly seen for the convective heating, which is Ͼ0 actually solves the parameterizations in pressure coor- below 200 mb (except in the surface layer) and ϳ0 dinates and the sequence de®ned by (17) requires that above. The resulting temperature tendency is smaller

Ti and ⌽i be derived from si after each parameterization, than Q/cp at all levels and is negative above 200 mb. using (10). The time-split parameterizations in CCM3 We note that the time-splitting method does not allow (and earlier versions) were not actually posed as in (17). all properties of the original equations to be preserved Instead, heating was equated with temperature tendency, for intermediate states. We have chosen energy conser- so that vation and hydrostatic balance as the properties to pre- T ϭ T ϩ ␦tQ /c , (19) serve, while potential temperature conservation has not iiϪ1 ip been preserved within the parameterization suite. and hydrostatic balance was imposed by integrating (10) to determine ⌽i. While this procedure is apparently more straightforward, it does not conserve energy within the parameterization suite. 5. Discrete equations for s, T, and ⌽ Energy was conserved outside of the parameterizations by reporting the net heating from the time-split We de®ne a compact notation for use in the discrete parameterization as equations below. For an arbitrary variable ␺, let a sub- QQTI Ϫ T script denote a discrete time level, with current step ␺n ϭϩiI , (20) and next step ␺ . The model has L layers in the ver- ͸ nϩ1 cciϭ1 ␦t pp tical, with indices running from top to bottom. Let ␺ k kϪ with Qi ϭ 0 for time-split parameterizations. Williamson denote a layer midpoint quantity and let ␺ denote the (2002) time split all parameterizations in CCM3 using value on the upper interface of layer k while ␺ kϩ denotes this method. the value on the lower interface. The relevant quantities

An ef®cient method to solve for Ti and ⌽i from si is are then

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␺ kϩ ϭ (␺ kkϩ ␺ ϩ1)/2, k ∈ (1,2,3,...,L Ϫ 1) algorithms in use today. In fact, for ``well-behaved'' algorithms (including those in CAM2), H is an upper- ␺ kϪ ϭ (␺ kϪ1 ϩ ␺ k)/2, k ∈ (2,3,4,...,L) triangular matrix with the special property that all ␦␺kkϭ ␺ ϩ Ϫ ␺ kϪ, ␦␺kϩ ϭ ␺ kϩ1 Ϫ ␺ k, above-diagonal entries in each column are identical. That is, the thickness of a layer l in (23) is independent ␦␺kϪ ϭ ␺ kkϪ ␺ Ϫ1, ␺ ϭ (␺ ϩ ␺ )/2, nϩ nnϩ1 of the level k to which the sum is taken. Therefore, the solution of (27) can proceed from the bottom up and is ␦␺nnϭ ␺ ϩ1 Ϫ ␺nn, ␦t ϭ t ϩ1 Ϫ tn. much cheaper than the matrix notation might lead one Using the above notation, the dry static energy at step to believe. n and level k is

kkkk snpnϭ cT ϩ⌽, (21) 6. Vertical diffusion A general vertical diffusion parameterization for a which can be calculated from Tn by integrating (10): hydrostatic model can be written in terms of the diver- p gence of diffusive ¯uxes: ⌽ϭ⌽ Ϫ RT d lnpЈ, (22) ץ 1 ץ ͵ s ps (u, ␷, q) ϭϪ (F , F , F ) (28) z u ␷ qץ t ␳ץ where ⌽s is the geopotential at the earth's surface and ץ 1 ץ -p is the surface pressure. A fairly arbitrary discreti s s ϭϪ F ϩ D, (29) z sץ t ␳ץ -zation of (22) can be represented using a triangular hy drostatic matrix Hkl: where D is the heating rate due to the dissipation of k resolved kinetic energy in the diffusion process. In the kllkl ⌽ϭ⌽ϩs RTH . (23) ͸ notation of section 2, (Fu, F␷ ) are the stress tensor com- lϭL ponents ␶31, ␶32 corresponding to horizontal stresses on Note that (23) is often written in terms of the virtual interfaces normal to the vertical direction. These are the d temperature T␷ ϭ TR/R . Using (23) in (21), only stresses relevant for hydrostatic column processes.

k We note that the vertical diffusion of heat is written here skkkϭ cT ϩ⌽ ϩ RT llHkl, (24) in terms of s, which is the variable diffused in CAM2. npns͸ n lϭL The vertical diffusion in CCM3 operated on ␪, which is very similar to operating on s, although it is easier ϭ (ckkϩ R Hkk)T kϩ⌽ kϩ. (25) pnn to write formally conservative operators for s. The interface geopotential in (25) is de®ned as Following Holtslag and Boville (1993), the diffusive ¯uxes are de®ned as kϩ1 ץ (⌽ϭ⌽ϩkϩ RTllHkl, (26 s ͸ F ϭϪ␳K (u, ␷), (30) zץ lϭL u,␷ m and c and R are evaluated from (3) and (4), using q . ץ p n t The de®nition of the hydrostatic matrix H depends Fq,sϭϪ␳K q,s(q, s) ϩ ␳K q,s␥ q,s. (31) zץ on the numerical method used in the dynamics and is subject to constraints from energy and mass conserva- The viscosity Km and diffusivities Kq,s are the sums of t tion (see, e.g., Williamson and Olson 1994). The de®- turbulent componentsK m,q,s , which dominate below the m nitions of H for the three dynamical methods used in mesopause; and molecular componentsK m,q,s , which CAM2 are given in appendix A. dominate above ϳ120 km. The nonlocal transport terms

If sn is modi®ed by diabatic heating in a time-split ␥ q,s apply in the boundary layer and represent the effects process, then the new snϩ1 ϭ sn ϩ Qn␦t can be converted of horizontally subgrid-scale eddies with vertical scale into Tnϩ1 and ⌽nϩ1 using (25): comparable to the boundary layer depth. We form the equation for total energy, equivalent to T kkk(s ϩ )(ckkR Hkk),Ϫ1 (27) nϩ1 ϭ nϩ1 Ϫ⌽nϩ1 p ϩ (9), from (28)±(29): kkk sץ ␷ץ uץ Eץ withcqpn and R evaluated from (3) and (4) using ϩ1 (recall that cckdϵ in CAM2). Once H is de®ned, (26) ϭ u ϩ ␷ ϩ (32) tץ tץ tץ tץ pp and (27) can be solved for Tnϩ1 and ⌽nϩ1. Calculating Fץ Fץ Fץ T and ⌽ from s involves the same amount of compu- 1 tation as calculating ⌽ and s from T. ϭϪ u u ϩ ␷ ␷ ϩϩs D (33) zץ zץ zץThe only signi®cant constraint imposed on the nu- ␳΂΃ Fץ Fץ merical algorithm to obtain (27) is that H be triangular, 1 which implies that ⌽(p) depends only on T(pЈ Ն p). ϭϪKE ϩ s . (34) zץ zץThis constraint is a property of the continuous system ␳΂΃ represented by (22) and is obeyed by most numerical The diffusive kinetic energy ¯ux in (34) is

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FKE ϵ uFu ϩ ␷F␷ , (35) where we have assumed zero boundary ¯uxes for kinetic energy. This leads to and the kinetic energy dissipation is g Dk ϭ (dkϩ ϩ dkϪ ϩ dkϩ ϩ dkϪ) (43) ␷ 2␦kp uu␷␷ץ uץ 1 D ϵϪ Fu ϩ F␷ . (36) z kϩ kϩ kϩץ zץ␳΂΃ du,␷ ϭ ␦ (u, ␷)nϩF u,␷ ,1Յ k Յ L Ϫ 1 (44)

These de®nitions correspond precisely to the more gen- Lϩ LLϩ du,␷ ϭϪ2(u, ␷)nϩF u,␷ . (45) eral expressions of FKE and D in terms of the stress tensor introduced after (5). To show that D is positive According to (43), the internal dissipation of kinetic k de®nite, we use (30) to expand for Fu and F␷ : energy in each layer D is the average of of the dissipation on the bounding interfacesdkϮ , given by (44) 22 u,␷ ␷ and (45). Expanding (44) using (40) and recalling thatץ uץ D ϭ (K tmϩ K ) ϩ Ն 0. (37) mm u ϭ (u ϩ u )/2, z nϩ nϩ1 nץz ΂΃ץ΂΃[] (g␳2K )kϩ In CAM2, (28)±(31) are converted to pressure co- dkϩ ϭ m [(␦kϩu )2 ϩ ␦kϩu ␦kϩu ], (46) ordinates using (10) and solved using a Euler backward un2␦kϩp ϩ1 nϩ1 n time step. Like the continuous equations, the discrete for 1 Յ k Յ L 1 and similarly fordkϩ . The discrete equations are required to conserve momentum, total en- Ϫ ␷ ergy, and constituents. The discrete forms of (28)±(29) kinetic energy dissipation is not positive de®nite, be- are cause the last term in (46) is the product of the vertical difference of momentum at two time levels. The surface ␦ (u, ␷, q)k ␦kF layer is heated by the frictional dissipation associated n ϭ g u,␷,q (38) ␦t ␦kp with generating the surface stress, since the surface stress is opposed to the bottom-level wind. However, if kk Lϩ Lϩ ␦nss ␦ F |FF | or | | is large enough to change the sign of ϭ g ϩ Dk. (39) u ␷ k LL ␦t ␦ p unϩ or␷ nϩ in (45), the dissipative heating may be negative. For interior interfaces, 1 Յ k Յ L Ϫ 1, A series of time-split operators is actually de®ned by ␦kϩ(u, ␷) (38)±(41) and (43±45). First, ␥ q,s are used to update the F kϩ ϭ (g␳2K )kϩ nϩ1 (40) input pro®les. Although q Ͼ 0, the updated pro®le for u,␷ mn kϩ n ␦ p a constituent may occasionally contain negative values, kϩ ␦ (q, s) in which case ␥ q is discarded for that pro®le. As men- F kϩ ϭ (g␳2K )kϩ nϩ1 ϩ (␳K tk␥ )ϩ . (41) q,s q,s n␦kϩp q,s q,s n tioned in Holtslag and Boville (1993), this problem usually arises under rapidly changing conditions for which Lϩ Surface ¯uxesF u,␷,q,s are provided explicitly at time n the boundary layer formulation in the turbulence model Lϩ 1Ϫ by separate surface models for land, ocean, and sea ice is not strictly appropriate. Second,FFu,␷,s,q and u,␷,s,q are 1Ϫ while the top boundary ¯uxes are usually F u,␷,q,s ϭ 0. used to update the bottom and top model layers, re- t The turbulent diffusion coef®cientsK m,q,s and nonlocal spectively. Third, (38) is inverted for (u, ␷)nϩ1, using transport terms ␥ q,s are calculated for time n by the the method given in appendix B. Fourth, the diffusive k turbulence model, which is identical to CCM3. The mo- heating, D is added to sn before solving (39) for snϩ1. lecular diffusion coef®cients are only included if the Finally, (38) is solved for qnϩ1. model top is above ϳ90 km, in which case nonzero top boundary ¯uxes may be included for heat and some 7. Energy conservation in the dynamical cores constituents. The formulation of thermospheric processes in a vertically extended version of CAM2 will The conservation properties of the numerical ap- be described elsewhere. proximations in the dynamical cores are beyond the Similarly to the continuous form (36), Dk is deter- scope of this manuscript and we only comment on issues mined by separating the kinetic energy change over a that are closely related to conservation within the phys- time step into the kinetic energy ¯ux divergence and ical parameterizations. Following Williamson (2002), the kinetic energy dissipation. The discrete system is the parameterization suite may be either time or process required to conserve energy exactly: split from the dynamics, regardless of the method used

L within the parameterization suite. The Eulerian and [(uk )2 ϩ (␷ k )2 ϩ skk]␦ p semi-Lagrangian dynamical cores are both process split ͸ nϩ1 nϩ1 nϩ1 kϭ1 from the parameterizations, with heating and speci®c

L forces being applied within the dynamics. The ®nite ϭ [(uk )2 ϩ (␷ k )2 ϩ skk]␦ p ϩ ␦t(F Lϩ ϩ F 1Ϫ), volume (fv) core is entirely adiabatic and is time split ͸ nnn ss kϭ1 from the parameterizations, taking updated T, u, ␷ as (42) input. The current CAM2 implementation of the fv core

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takes Tnϩ1 ϭ Tn ϩ Q/cp as the input state, leading to a CCM3 are the inclusion of: the prognostic cloud water small energy imbalance. This issue will be addressed in variable of Rasch and KristjaÂnsson (1998), the gener- a future model revision. alized cloud overlap treatment in the radiative transfer The dissipation of kinetic energy into heat D must be of Collins (2001), the updated treatment of the infrared calculated explicitly and included in the heating Q in radiative effect of water vapor of Collins et al. (2002), the ®rst law of thermodynamics (2), in order to conserve and a simple parameterization for evaporation of con- energy. In Newtonian ¯uids, D is a positive de®nite vective rainfall. quantity given by the product of the stress tensor and Changing between updating T and updating s had no the velocity gradient, as discussed by Fiedler (2000) and signi®cant impact on the simulation. The change does Becker (2001). The spectral Eulerian core in CAM2 alter the conceptual framework for incorporating param- includes a biharmonic horizontal diffusion operator that eterizations and energy conservation within the param- cannot be represented by a symmetric stress tensor and eterization suite can be veri®ed. In fact, a few minor therefore the kinetic energy dissipation cannot be cor- conservation errors in CAM2 were identi®ed during this rectly de®ned, as noted by Becker (2001). Instead, FK work that will be ®xed in the next version. Most notably, -V/ the radiative heating rates (¯ux divergences) are comץ) t)d, whereץ/Vץ)´ t ϭϪVץ/Kץis ignored and D ϭϪ .t)d is the speci®c force from the diffusion process. Note puted hourly, saved, and applied to multiple time stepsץ t)d Ͼ 0. The ¯uxes should be saved instead to allow for pressureץ/| V|)ץ that D Ͻ 0 (cooling) if Both the ®nite volume and semi-Lagrangian dynam- changes. On an annual average basis, the standard con- ical cores in CAM2 have some diffusion implicit in their ®guration of CAM2 conserves energy to Ͻ0.3 W mϪ2, numerical approximations, for which D is unknown. In of which ϳ0.2WmϪ2 is due to the neglect of the latent principle, D could be determined as a residual in (6), heat of fusion within the atmosphere. The remainder is by predicting K in addition to V. In practice, (8) is used due to lack of conservation within the spectral dynamical d to de®ne the global integral of D, which is then applied core, including the use of cp from (3) instead ofcp . in (2) as a uniform heating. For all three dynamical For a model that does not extend at least into the cores, the globally averaged value of D is ϳ2WmϪ2 mesosphere, such as the standard version of CAM2, (D. Williamson 2002, personal communication). almost all of the vertical diffusion takes place in the Conservation of water vapor within the transport boundary layer. Figure 2 shows the annually averaged component of the dynamical core is also required in heating from vertical diffusion over the bottom 100 mb order to conserve the latent energy included in Q. The of the atmosphere. Note that 1 m W kgϪ1 corresponds ®nite-volume core conserves constituents explicitly to a temperature tendency of 0.0864 K dayϪ1. The dif- while the semi-Lagrangian transport algorithm, used by fusion heats throughout the boundary layer on average, both the spectral Eulerian and semi-Lagrangian cores, except in polar regions. Turbulent mixing of s will cool contains an ad hoc mass ®xer to ensure global conser- the upper part of the boundary layer, since s increases vation, as discussed in Rasch and Williamson (1990). upward in a stably strati®ed atmosphere. However, the Sources in the constituent equations are always time input of sensible heat at the surface dominates the tur- split in CAM2. bulent cooling over most of the globe. The mixing in polar regions is primarily mechanically driven and the surface sensible heat ¯ux is small, resulting in cooling 8. CAM2 results in the upper part of the boundary layer. The formalism discussed above is implemented in The difference in vertical diffusion heating between CAM2. The impact of the changes compared to CCM3 the CAM2 and CCM3 forms is very small, as shown have been evaluated with a series of 5-yr simulations in Fig. 2, where the contour interval for the difference using the standard CAM2 con®guration of spectral Eu- is 0.1 of that for the heating itself. The difference due lerian dynamics with a triangular truncation at total to diffusing s rather than ␪ is negligible. Most of the wavenumber 42 (T42) and 26 layers in the vertical, change comes from using the correct de®nition (36) for extending from the surface to a top interface at ϳ2 mb. kinetic energy dissipation. The dissipative heating for (The tests actually used a prerelease version of CAM2 the two de®nitions is shown in Fig. 3. Kinetic energy that had slightly different adjustable constants for some dissipation is distributed through the boundary layer in parameterizations.) All of the simulations used clima- the CCM3 form, while it is mostly con®ned in the sur- tological sea surface temperatures. The comparison to face layer in the CAM2 form. The turbulent mixing of ``CCM3'' has been made by returning speci®c processes momentum in the boundary layer results in a relatively to the CCM3 form while retaining the CAM2 form else- smooth decrease of kinetic energy in the vertical. How- where. The CCM3 form requires updating T instead of ever, most of the kinetic energy change results from the t in- kinetic energy ¯ux in (34), rather than dissipation. Theץ/Kץ s after parameterizations, computing D from stead of (37), mixing ␪ instead of s, and all of the above CCM3 form ignores this distinction. The vertical inte- changes together. In addition to the time-splitting meth- gral of the heating is identical for the two de®nitions, od and vertical diffusion parameterization discussed given the same input pro®le, and is indistinguishable in above, the primary differences between CAM2 and climate simulations.

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FIG. 2. Annual and zonal average of the heating from vertical diffusion (mW kg Ϫ1) for (left, contour interval 10) the CAM2 diffusion of s and (right, contour interval 1) the difference between CAM2 and CCM3 diffusion. Negative contours are dashed.

The global and annual average of the explicit dissi- energy. Vertical diffusion and gravity wave drag, which pative heating due to surface stress and vertical diffusion alter kinetic energy, conserve total energy by including is 2.0 W mϪ2 in CAM2. The surface layer accounts for heating due to kinetic energy dissipation. This heating 1.65 W mϪ2 of the global mean, while the rest of the was included in an ad hoc form in previous versions atmosphere accounts for only 0.35 W mϪ2. Although (CCM1±3), rather than depending correctly on the stress Lϩ LϪ the contributions ofddu,␷ andu,␷ were not separated in tensor and velocity gradient. the CAM2 output, the dissipative heating in the surface Within the parameterization suite, heating is always layer results almost entirely from the surface stress applied to the dry static energy s in CAM2, rather than terms. In fact, the heating is mostly due to surface stress being applied to T, as in CCM3 and earlier versions. in the oceanic storm tracks with somewhat more dis- Therefore, CAM2 conserves energy at each stage within sipation in the Southern Hemisphere than in the North- a time step, except for any nonconservation within the ern Hemisphere (Fig. 3). There is a distinct seasonal dynamical core. Earlier versions approximately con- cycle, maximizing in winter in each hemisphere, when served energy in aggregate, but not within the param- the dissipation in the storm track exceeds 8 W mϪ2 (not eterization suite. shown). For time-split parameterizations, applying heating to s requires inverting the hydrostatic equation to determine T and ⌽ from the modi®ed s. The inversion method 9. Conclusions developed for CAM2 is actually as ef®cient as evalu- The parameterized heating in CAM2 has been refor- ating the hydrostatic equation to determine ⌽ and s from mulated around the unifying concept of conservation of T. We note that time splitting is not necessarily the best

FIG. 3. Annual and zonal average of the heating from kinetic energy dissipation (mW kg Ϫ1) for (left) the CAM2 diffusion of s and (right) the CCM3 diffusion. The contour interval is 1 mW kgϪ1 and negative contours are dashed.

Unauthenticated | Downloaded 09/30/21 02:12 PM UTC 3886 JOURNAL OF CLIMATE VOLUME 16 way of treating the physical parameterizations. As part The rather unusual looking diagonal (l ϭ k) term comes of the uni®cation of the formulation discussed here, the from deriving the layer average value of ⌽ CAM2 code has been structured so that the parameter- pk 1 k ization suite can easily be converted to any combination ⌽ϭ k dp ⌽ dp (A4) p ͵ of time and process splitting. pkϩ1

Incorporating the ``new'' formalism in CAM2 did not ͵ kϩ1 p change the simulation signi®cantly. The largest system- k atic change is that most of the heating from kinetic 1 p ϭ ␦k(p⌽) ϩ RT dp , (A5) energy dissipation is applied in the surface layer, rather k ␦ p ͵ kϩ1 than being distributed through the boundary layer. The []p importance of the formalism lies in the precise de®nition where we have integrated by parts and applied the hy- of conservation of energy, which can be monitored at drostatic equation between (A4) and (A5). Expanding each stage within a time step and enforced within the ␦k(p⌽) and assuming that T is constant within layers, numerical approximations. 1 The net energy conservation in CAM2 is very similar ⌽ϭk [(pkϩ⌽Ϫkϩ pkϪ⌽kϪ) ϩ RTkkk␦ p]. (A6) to that in CCM3 but does not arise from cancellation ␦kp of errors. The remaining conservation errors in the pa- Substituting ⌽kϪ ϭ⌽kϩ ϩ RkT k␦k lnp from (A3), rameterization suite arise from the omission of the latent heat of fusion and from the application of a ®xed ra- pkϪ␦k lnp ⌽ϭ⌽kkϩ ϩRTkk1 ϩ , (A7) diative heating rate over multiple time steps. In the next ΂΃␦kp generation of CAM, the parameterizations will conserve precisely and the net conservation will depend only on and the second term on the rhs of (A7) gives the k ϭ the properties of the dynamical core. l element of (A3).

Acknowledgments. The authors thank Prof. David APPENDIX B Randall and two anonymous reviewers for their com- ments, which led to signi®cant improvements on the Solution of Implicit Vertical Diffusion Equations original version of the manuscript. CFB also thanks Equations (38)±(41) constitute a set of four tridi- NCAR for supporting his sabbatical, during which much agonal systems of the form of this work was completed. kkϩ1 kk kkϪ1 k ϪA ␺nϩ1 ϩ B ␺nϩ1 Ϫ C ␺nϩ1 ϭ ␺nЈ, (B1) APPENDIX A where ␺nЈ indicates u, ␷,q,or s after updating from time n values with the nonlocal and boundary ¯uxes. The Hydrostatic Matrix superdiagonal (Ak), diagonal (Bk), and subdiagonal (C k) CAM2 supports three separate numerical methods for elements of (B1) are solving the dynamical equations: spectral Eulerian, 1 ␦t semi-Lagrangian, and ®nite volume dynamics. In both k 22 kϩ A ϭ (g ␳ K)n , (B2) the Eulerian and semi-Lagrangian dynamics the hydro- ␦kkp ␦ ϩp static matrix H in (23)±(27) is de®ned in terms of an Bkkkϭ 1 ϩ A ϩ C , (B3) ``energy conversion'' matrix C as 1 ␦t Hklϭ C lk␦lp, (A1) C k ϭ (g22␳ K)kϪ . (B4) ␦kkp ␦ Ϫp n 1/pll Ͼ k,  The solution of (B1) has the form Clk ϭ 1/2pll ϭ k, (A2)  ␺ kkkϭ E ␺ Ϫ1 ϩ F k, or (B5) 0 l Ͻ k. nϩ1 nϩ1 kϩ1 kϩ1 kkϩ1 The matrix C is used in the evaluation of ␻ ϭ dp/dt in ␺nϩ1 ϭ E ␺nϩ1 ϩ F . (B6) (2). Here H and C are required to obey (A2) in order Substituting (B6) into (B1) gives to conserve energy in the conversion of potential into C kkkk␺ ϩ AFϩ1 kinetic energy. k kϪ1 nЈ ␺nϩ1 ϭ ␺nϩ1 ϩ . (B7) For the ®nite volume dynamics, the hydrostatic matrix BkkkϪ AEϩ1 BkkkϪ AEϩ1 H is de®ned as Comparing (B5) and (B7), we ®nd  l ␦ lnplϾ k, C k  E k ϭ L Ͼ k Ͼ 1, (B8) pkkϪ ␦ lnp BkkkϪ AEϩ1 Hkl ϭ 1 Ϫ l ϭ k, (A3) k ␦ p ␺ kkkϩ AFϩ1  F k ϭ nЈ , L Ͼ k Ͼ 1. (B9) 0 l Ͻ k. BkkkϪ AEϩ1

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The terms E k and F k can be determined upward from k ÐÐ, J. K. Hackney, and D. P. Edwards, 2002: An updated param- ϭ L, using the boundary conditions: eterization for infrared emission and absorption by water vapor in the National Center for Atmospheric Research Community E Lϩ1 ϭ F Lϩ1 ϭ AL ϭ 0. (B10) Atmosphere Model. J. Geophys. Res., 107, 4664, doi:10.1029/ 2001J0001365. k Finally, (B7) can be solved downward for␺nϩ1 , using Fiedler, B. H., 2000: Dissipative heating in climate models. Quart. the boundary condition: J. Roy. Meteor. Soc., 126, 925±939. Gill, A. E., 1982: Atmosphere±Ocean Dynamics. Academic Press, C 11ϭ 0 ⇒ E ϭ 0. (B11) 662 pp. Holtslag, A. A. M., and B. A. Boville, 1993: Local versus nonlocal CCM1±3 used the same solution method, but with boundary-layer diffusion in a global climate model. J. Climate, the order of the solution reversed, which merely requires 6, 1825±1842. kϪ1 kϩ1 writing (B6) for␺␺nϩ1 instead ofnϩ1 . The order used Kiehl, J. T., J. J. Hack, G. B. Bonan, B. A. Boville, D. L. Williamson, here is particularly convenient because the turbulent dif- and P. J. Rasch, 1998: The National Center for Atmospheric fusivities for heat and all constituents are the same but Research Community Climate Model: CCM3. J. Climate, 11, 1131±1149. their molecular diffusivities are not. Since the terms in Lin, S.-J., and R. B. Rood, 1997: An explicit ¯ux-form semi-La- (B8)±(B9) are determined from the bottom upward, it grangian shallow-water on the sphere. Quart. J. Roy. Meteor. is only necessary to recalculate Ak, C k, E k, and 1/(Bk Soc., 123, 2477±2498. Ϫ AkE kϩ1) for each constituent within the region where Rasch, P. J., and D. L. Williamson, 1990: Computational aspects of molecular diffusion is important. moisture transport in global models of the atmosphere. Quart. J. Roy. Meteor. Soc., 116, 1071±1090. ÐÐ, and J. E. KristjaÂnsson, 1998: A comparison of the CCM3 model REFERENCES climate using diagnosed and predicted condensate parameterizations. J. Climate, 11, 1587±1614. Becker, E., 2001: Symmetric stress tensor formulation of horizontal Williamson, D. L., 1988: The effect of vertical ®nite difference ap- momentum diffusion in global models of atmospheric circula- proximations on simulations with the NCAR community climate tion. J. Atmos. Sci., 58, 269±282. model. J. Climate, 1, 40±58. Boville, B. A., and P. R. Gent, 1998: The NCAR Climate System ÐÐ, 2002: Time-split versus process-split coupling of parameteri- Model, version one. J. Climate, 11, 1115±1130. zations and dynamical core. Mon. Wea. Rev., 130, 2024±2041. Collins, W. D., 2001: Parameterization of generalized cloud overlap ÐÐ, and J. G. Olson, 1994: Climate simulations with a semi-La- for radiative calculations in general circulation models. J. Atmos. grangian version of the NCAR Community Climate Model. Mon. Sci., 58, 3224±3242. Wea. Rev., 122, 1594±1610.

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