Description and Preliminary Results of the 9-level UCLA General Circulation Model

Transcription

1 July 6, 003 :08 am Description and Preliminary Results of the 9-level UCLA General Circulation Model Max J. Suarez, Akio Arakawa This manuscript has been re-typed from: Suarez, M. J. and A. Arakawa, 979: Description and preliminary results of the 9-level UCLA general circulation model. Fourth Conference on Numerical Weather Prediction of the American Meteorological Society, October 9 - November, 979, Silver Spring, MD. Minor typos in the original manuscript have been corrected here. Michelle Beckman of Colorado State University re-typed the manuscript. The embedded PBL and vertical differencing scheme described here have been used in the Colorado State University General Circulation Model for many years, and are still be ing used as of Fall 00.. Department of Atmospheric Sciences, UCLA, 405 Hilgard Ave., Los Angeles, California Current affiliation: NASA Goddard Space Flight Center, Greenbelt, MD 077

2 . Introduction During the last five years, 6 and -level versions of the general circulation model (GCM) described in Arakawa and Lamb (977) have been used at UCLA. Some results from the analysis of a recent July simulation using the 6-level version are presented by Mechoso et al in this volume. Such simulations have shown that, although the model compares quite favorably with the state of the art in general circulation modeling, it has certain deficiencies. Some of these deficiencies are particular to it; others are common to most GCMs. In an effort to improve the model, both for general circulation studies and for numerical weather prediction, two major design changes have been made. () A potential enstrophy conserving advection scheme in the equation of motion has replaced the previous scheme, which only conserved enstrophy for non-divergent flow. It is hoped that this scheme will improve the simulation of topographically forced motions. A description of the scheme and its application to the shallow-water system of equations is presented by Arakawa and Lamb in this volume. () The treatment of planetary boundary layer (PBL) has been dramatically modified by making the predicted PBL top a coordinate surface. This change is discussed in detail in the following sections.. The vertical coordinate A deficiency noted in the UCLA GCM was the rather poor simulation of fields strongly influenced by PBL processes. Precipitation in the tropics, for example, was less sensitive to sea surface temperature than in either nature of earlier versions of the model. This was thought to be due to the indirect way in which GCM layers and the parameterized PBL were coupled. In this model, GCM layers were defined by conventional σ-coordinate surfaces, (a modification of Phillips, 957), as indicated schematically by the heavy lines in Fig. a. The well-mixed PBL, whose depth was predicted, was then allowed to occupy any fraction of the lower atmosphere. Typically, it was confined to a part of the lowest GCM layer, as shown on the right of the figure, but it could also penetrate to higher GCM layers, as shown on the left. Coupling between this PBL and GCM layers was handled by writing prognostic equations for the magnitude of the discontinuities at the PBL top.

3 We felt that the difficulties arising from this rather indirect coupling between PBL and GCM layers might be overcome by making the PBL the lowest layer of the GCM. Such an arrangement of GCM layers is shown in Fig..b. Figure : (a) Old Model. (b) New model with arrangement of the GCM with PBL as the lowest layer The vertical σ -coordinate adopted for the new model is defined formally as follows: let p be the pressure; p T, the pressure at the top of the model atmosphere, taken as constant; p B, the pressure at the PBL top; p S, the pressure at the Earth s surface; and p I a constant pressure between p T and a realistic lower bound of p B (see Fig. ). Then σ is given by Figure : The modified σ-coordinate. σ corresponds to the top of the PBL, and σ to the surface.

4 σ p p I for p p I p T p p I, T p p I for p p B p I p p B, I p p B + for p p S p T p p I, B () With π defined by π π K p I p T for σ < 0, π L p B p I for 0 < σ <, π M p S p B for < σ, () the pressure p may be written as p p I + σπ for σ <, p B + ( σ )π for σ <, (3) Then, if we let d denote the differential under constant horizontal coordinates and time dp πdσ (4) for all σ except at σ 0 and σ where π is not uniquely defined. From (3), the individual time derivative of pressure, ω Dp Dt, is given by ω πσ σ V t π for σ <, + ( σ ) V π V + pb for < σ, t t (5) where σ Dσ Dt and V is the horizontal velocity. Recall that π π K const. for - σ < 0. Gradients in the p - and σ -coordinate systems are related by 3

5 p σ + p σ (6) σ p σ may be obtained by operating p on (3). Then p σ -- π----- for σ <, π σ σ ( σ ) π + p B for σ <. π σ (7) 3. The model s governing equations 3. The continuity equation The continuity equation in pressure coordinates is ω p V p 0. (8) Using (7) for p V, and (4) and (5) for ω p, (8) yields 3. The equation of state π (9) t σ ( πv) ( πσ ) σ The model atmosphere is assumed to be a perfect gas, so that α RT , (0) p where α is the specific volume, T is the temperature, and R is the gas constant. 3.3 The hydrostatic equation With (4), the hydrostatic equation dφ αdp becomes dφ παdσ, () where φ is the geopotential, gz, and z is height. The following alternative forms of the hydrostatic equation can be derived from () and will be useful: 4

6 dφ RT d ln p () c p θdp (3) dlnθ c p dp, (4) d θ -- where c p is the specific heat at constant pressure, θ is the potential temperature T P, P ( p p 0 ) κ, κ R c p and p 0 is a standard pressure. 3.4 The first law of thermodynamics The specific entropy is c p lnθ, and the first law of thermodynamics is D lnθ Dt ---- c p T --Q (5) or D -----θ Dt ----, (6) c p P -- Q where Q is the diabatic heating rate, and D Dt is the material time derivative D Dt V. (7) t σ + σ σ σ Combining (6) with the continuity equation (9) we obtain the flux form corresponding to (6): --- π ( πθ) t σ ( πvθ) ( πσ θ) Q σ σ c --- p P (8) 3.5 The water vapor and ozone continuity equations Let r be the mixing ratio of either water vapor or ozone. The continuity equation for either variable is 5

7 Dr S Dt (9) where S is the source term. The corresponding flux form is 3.6 The momentum equation --- ( πr). (0) t σ ( πvr) ( πσ r) + πs σ σ The horizontal component of the momentum equation is DV fk V, () Dt p φ + F where DV Dt is the horizontal acceleration, f is the Coriolis parameter, k is the vertical unit vector, and F is the horizontal frictional force. Using (7) in the left hand side of () may be rewritten as --- t V σ V + σ qk ( πv) +, () σ σ --V where q ( f+ ζ) π and ζ k σ V. We shall refer to q as the potential vorticity. Using (7) and replacing σ φ σ by ( φσ) σ φ and ( σ ) φ σ by ( ( σ )φ) σ φ, the pressure gradient force may be written as -- ( σφ) φ π for σ, π σ p φ σ φ -- [ ( σ )φ] φ φ π p π σ π σ B for < σ. (3) Substitution of () and (3) in () gives the form of the momentum equation used in the model. 3.7 The vertical boundary conditions and their applications to the continuity equation It is assumed that the top of the model atmosphere and the Earth s surface are material surfaces, thus 6

8 ( πσ ) σ 0, (4) and ( πσ ) σ 0. (5) Since σ is the PBL top, ( πσ ) σ represents the rate of net addition of mass, per unit horizontal area, to the PBL from the layer above (the free atmosphere). Thus ( πσ ) σ ge ( M B ), (6) where E is the entrainment of mass into the PBL from above and M B is the upward cumulus mass flux through the PBL top. Combining these with the continuity equation we obtain equations for πσ and the PBL top pressures: and the surface p B ( πv) dσ ge ( M, (7) t B ) p S ( πv) dσ, (8) t πσ σ σ ( πv) dσ for σ 0, (9) πσ σ σ ( πv) σ σ p B d for 0 σ. t (30) 4. The vertical difference scheme 4. The vertical index The model atmosphere above the PBL top is divided into L layers by L levels of constant σ. These layers and the PBL carry the three-dimensional prognostic variables and are identified with integer values of l. The levels which divide the layers carry πσ and are identified with half-integer values of l. The upper boundary p p ( σ T ), the level p p I ( σ 0 ), the PBL top p p B ( σ ), and the lower boundary p p S ( σ )are identified with 7

9 l, l K+, l L+ M and l M+, respectively (see Fig. 3). In the following description of the model, superscripts will be used for the vertical index. The superscript l, however, will be omitted when there is no danger of confusion. Define, for an arbitrary variable A, ( δa) l  l +  l. (3) Here the ˆ is a reminder that the variable is at a half-integer level. Figure 3: The vertical index of the model. 4. The continuity equation The continuity equation is written in the form π δπσ ( ) ( πv) for l.,,.., M. (3) t δσ Here the superscripts l for π, πv δπσ ( ), and δσ have been omitted. πσ at l, l L+ M, and l M+ are given by (4), (6), and (5). 8

10 Corresponding to (7), (8), (9), and (30), L p B [ πvδσ] l ge ( M, (33) t B ) l M p S [ ( πv)δσ] l t l, (34) ( πσ ) l + l [ ( πv )δσ] l l 0 for l K, + p, (35) B for K< l< M. t σ l 4.3 The pressure gradient force Corresponding to (3), the pressure gradient force may first be written in the form: ( p σ) δφσ ( ) -- πσ ( ) π for l.,,.., L; π δσ δ{ ( σ )φ} δφ -- πσ ( ) π π δσ δσ p B for l M. (36) Here δ is defined by (3). The value of φ at half integer levels, φˆ, will be specified later. It follows that M M π ( φ)δσ. p πφδσ φˆ S p S (37) l l Here π p B for l K+, K +,, L and π p S p B for l M have been used. The first term on the right hand side of (37) is a gradient vector, and a line integral of its tangential component taken along an arbitrary closed curve on the sphere always vanishes. As in the continuous case, therefore, only the second term contributes to such a line integral, and therefore, only when φ S is variable (i.e., a non-horizontal bottom surface) can there be any generation of the circulation of vertically integrated momentum. The finite difference for of (3) that is consistent with (36) may be written as 9

11 -- δφσ ( ) φ π for l,,, L; π δσ ( p σ) φ -- {( φ φˆ π S ) π+ ( φˆ B φˆ S ) p B } for l M. (38) 4.4 The kinetic energy generation by the pressure gradient force The kinetic energy generation by the pressure gradient force is obtained through multiplication of (38) by πv. After some manipulations including use of the continuity equation (9), we obtain for πv p φ ( πvφ) δ{ ( πσ + p t)φ} δσ φ δφσ ( ) V π δσ t {( πσ ) l + ( δσ φl φˆ l + ) + ( πσ ) l ( φˆ l φ l )} for l,,, L ; * (39) ( πvφ) δ{ ( πσ + p t)φ} δσ [( φ φˆ S ) V π t + ( φˆ B φˆ S ) V pb ] πσ t M ( φˆ B φ) for l M. The area integral of (39) over the entire globe makes the first term vanish. The vertical sum of (39) xδσ over all layers makes the second term become φ S p S t. This represents the conversion from potential energy to kinetic energy through redistribution of vertically integrated mass over the Earth s topography. The terms with the underlines must then represent πωα, which is the rate of conversion from πc P T to --π V. 4.5 The first law of thermodynamics Corresponding to (8), the first law of thermodynamics may be written in vertically discrete form: --- δπσ ( θ) π ( πθ) ( πvθ) Q --- for l,,, M. (40) t δσ c p P 0

12 The value of θ at half integer levels, θˆ, will be specified later; whatever its specification is, however, the form (40) guarantees that an analog of the global integral of θ with respect to mass is conserved when Q 0. Making use of θ T P in (40) and then adding and subtracting δπσ ( c p T) ( δσ) to the right hand side we obtain δπσ c --- ( πc t p T) ( πvc p T) ( p T) δσ c θπ p t + V P c p ( πσ ) l + Tˆ l + P l θˆ l + + ( πσ ) l P δσ l θˆ l Tˆ l + πq. (4) The area integral over the entire globe and the vertical sum, after multiplication by δσ, make the first and second terms of the right hand side of (4) vanish, respectively. The terms with the underlines must then represent πωα, as can be confirmed by taking the limit as δσ 0 and using (5) and (0). 4.6 Energy conversion and the hydrostatic equation We now require that the terms identified with the energy conversion, πωα, on the right hand sides of (39) and (4) be identical. First, by comparing the terms that involve πσ, we obtain φ l φˆ l + l + ( Tˆ P l θˆ l + )c p for l,,, L; (4) φ l φ l ( P l θˆ l Tl )c p for l,,, M. (43) We further note that P l is constant for l,,, K ; a function of π for l K+, K +,, L ; and a function of π and p B for l M. Thus φ δφσ ( ) c δσ p θπ dp for l K+,,, L ; dπ (44) φˆ B φˆ S c p θπ P M p, (45) B

13 φ M φˆ S c p θπ----- P M. (46) π Elimination of φˆ from (4) and (43) yields φ l φ l + c p ( P l + P l )θ l + for l,,, L. (47) Eqs. (46) and (47) are the finite difference hydrostatic equation of the model. Eqs. (44) and (45), however, must also be satisfied by properly specifying P and θˆ. 4.7 Specification of P and θˆ The vertical sum of (44) δσ gives L L ( φδσ ) l φˆ B + c p θπ dp δσ l. (48) dπ l K+ l K+ L Using (45), (46), (47) and the identity ( φδσ ) l φ M L + ( φ l φ l + )σˆ l +, l K+ l K+ (48) may be rewritten as L l K+ c( p l + p l )θˆ l + σˆ l + p c p θπ dp. (49) p M c θπ dp M L p π + c B p θπ dp δσ l dπ l K+ Let θˆ l + be a linear combination of θ l and θ l +. Thus θˆ l + a l + θ l + b l + θ l + for l K+, K +,, L, (50) where a + l + b l +. (5) Substituting (50) into the left hand side of (49) and comparing the coefficients of θ l with those in the right hand side, we obtain

14 π dp δσ l [( P dπ l + P l )σˆ l + a l + + ( P l P l )σˆ l b l ] for l K+, K +,, L, (5) and π P. (53) p π P M P B π ( M P L)b L + Using (5) and the definition of δσ, (5) may be rewritten as ( πp) δσ l σˆ l + ( b π l + P l + a l + P l + ) σˆ l ( b l P l + a l P l ) for l K+, K +,, L. (54) If we make the choice b l + P l + a l + P l + l + l + Pˆ ( pˆ p 0 ) κ for l K+, K +,, L, (55) where l + pˆ p I + σˆ l + π, (54) can readily be integrated to give P l ( Pˆ pˆ ) l + ( Pˆ pˆ ) l , (56) + κ l + l pˆ pˆ π dp l l + ( Pˆ P l )σˆ l + P l l + + ( Pˆ )σˆ l dπ ( δσ) l (57) and l + Pˆ P a l l P l + P l P b l + l + Pˆ l P l + P l (58) for l K+, K +,, L. The form of P given by (56) was proposed by Phillips (974). 3

15 We can show that applied to l M, with b given by (58), satisfies (53). Thus L + P M PSpS P p B B + κ p S p B (59) From (59) with P S P B + π M, π P, (60) p M P P S B B π P. (6) π M P P M S When a b, we have a θ -conserving vertical difference scheme (Arakawa, 97). The choice given by (58), however, does not guarantee that an analog of the global integral of θ with respect of mass is exactly conserved when Q 0. With a proper choice of the values of σ for half-integer levels, however a and b can be made sufficiently close to, except at the PBL top, under normal values of p B. Numerical experiments using the model with Q 0 and ( πσ ) σ 0 in fact showed that the global integral of θ was practically conserved. At the PBL top, a and b are not necessarily close to. For the case of a very thin PBL, for example, (58) gives a L +, b L + 0 and therefore θ L + θ L from (50). Under such situations the global integral of θ is not conserved even approximately, when ( πσ ) σ ge ( M B ) is non-zero; but the conservation is not expected physically as will be shown in the next section. For l,,, K, θˆ l + is chosen to conserve an analog of the global integral of lnθ with respect to mass. It is given by θˆ l + lnθ l lnθ l +, with. (6) θ l + θ l P l κ l l + pˆ pˆ p0 The motivation for these choices is given by Arakawa and Lamb (977) and Tokioka (978). 4

16 4.8 The water vapor and ozone continuity equation form Corresponding to (0), the water vapor and ozone continuity equation is written in the --- δπσ ( r) ( πr) ( πvr) πs. (63) t δσ Whatever the specification of r at half-integer levels, rˆ, the form (63) guarantees that an analog of the global integral of r with respect to mass is conserved. + The choice of rˆl for l,,, L is based on conservation of an analog of the global integral of lnr with respect to mass. The form is identical to (6). This choice prevents generation of negative values of r, as far as the vertical differencing is concerned. At the PBL top, we choose + rˆl r L. (64) This also prevents generation of negative values of r at r L when ( πσ ) L + > 0. At present we are ignoring the possibility of conditional instability of the computational kind (CICK) discussed by Arakawa and Lamb (977), because the new model has a relatively high vertical resolution for the lower troposphere. 4.9 The vertical advocation of momentum The momentum equation of the model is based on (). This form, which is not the momentum flux form, is used to implement the potential enstrophy conserving horizontal difference scheme. The vertical advection of momentum, σ V σ, for layer l is written as where , (65) ( πδσ) l ( πσ ) l V l l ( Vˆ ) ( πσ ) l + l + [ + ( Vˆ V l )] l + Vˆ ( V l + V l + ) for l,,, L (66) and 5

17 L + Vˆ V L. (67) The form (66) is chosen to conserve kinetic energy under advection processes. The motivation for choosing the form (67) will be given in the next section. 5. Discontinuity assumption at the PBL top The layer below σ represents the PBL of the model, where turbulent transfer plays a dominant role. In the atmosphere, the vertical extent of the PBL is frequently well-defined by a transition layer where vertical gradients of the thermodynamic variables and wind are large. Since this transition layer is usually thin and the vertical structure within it depends on the details of the turbulence, we have little hope of simulating it explicitly in the GCM. Nevertheless, when such a sharp transition exists, we wish to include its effects on the PBL properties. We have chosen, therefore, to model these rapid transitions as discontinuities in the temperature, moisture and wind fields. In this section we will denote the level immediately above the PBL top σ by B+ and the level immediately below by B-. The relationship between the values of the prognostic variables for the GCM layers and their values at these two levels will be specified below. For an arbitrary variable, A, the size of the discontinuity of the PBL top is given by A ( A) B+ ( A) B-. (68) In the presence of such a discontinuity the grid-scale vertical flux of A, πσ A g, is discontinuous, though πσ itself is continuous. We must require, however, that the total vertical flux of A must be continuous. If we denote by F A the sub-grid scale vertical flux of A by turbulence and cumulus convection, plus, when A θ, the radiative flux, we have -- ( πσ ) g σ A B+ ( F A ) B+ -- ( πσ ). (69) g σ A B- ( F A ) B- We may now distinguish between turbulent, cumulus, and radiative fluxes. Assuming that, at σ, A A B- inside cumulus clouds and A A B+ in the cloud environment, the upward flux of A due to cumulus clouds at B+ is M B A, where M B is the cumulus mass flux through the PBL top. At level B- there is no vertical flux of A due to cumulus clouds. Since we are assuming there is no turbulence at level B+, (3) becomes 6

18 -- ( πσ ) g σ A B+ + M B A ( R A ) B+ -- ( πσ ), (70) g σ A B- ( F A ) B- where R A is, when relevant, the radiative flux. Returning now to the finite difference form of the equation for A, we see that if we associate  L + with A B+ (as we have done implicitly in choosing θˆ L + +, rˆl and L + Vˆ in (50) with l L, (64), and (67)), we must include the fluxes M B A and ( R A ) in B+ the budgets of the GCM layer l M. There remains only to specify A B- (necessary to determine A ). We do this by assuming the PBL is well-mixed in the vertical, and so velocity and the appropriate thermodynamic variables are taken independent of height in the PBL. The total downward flux of A is given by ( πσ g)a F A A. For the budget of A, however, we must consider the conversion to the subgrid scale part of A (and the destruction due to radiation, when relevant), given by ( g π) ( A σ)f A per unit mass. This conversion is unbounded at the PBL top due to the discontinuity of A there; but integration with respect to mass through the infinitesimally thin transition layer should give a finite amount of the conversion, C A. Thus the budget of A for the transition layer may be written as Eliminating C A -- ( πσ ). (7) g σ -- ( A A B+ ) ( F B- A ) A B+ B+ + ( F A ) A B- B- ( F A ) B- between (7) and (69) and using (68) we obtain C A -- ( πσ ). (7) g σ -- ( A) ( F A ) A B+ Returning again to the finite difference form of the equation for following quantity as C A for the discrete system: A, we can identify the C A -- ( πσ ) L ( A L A M ) L -- ( AL) ( A M ) g ga ( L A M )FL + A, (73) which becomes identical to (7) if we define 7

19 A  L + A M (74) and  L + A L. (75) The choice of rˆ and Vˆ given by (64) and (67), respectively, has the form of (75). The choice of θ given by (50) and (58) has been made from the point of view of the energy conversion, but approaches the form (75) when the PBL becomes thin. ) 6. Outline of the horizontal difference scheme The horizontal grid of the model is based on the spherical coordinates and corresponds to the C-grid shown by Fig. of Arakawa and Lamb in this volume. All scalar variables, such as πθ, and r are defined at the h-point of the figure. At present, the entire horizontal differencing is designed to maintain second order accuracy, though we plan to replace the horizontal advection terms by fourth order schemes in the near future. The horizontal differencing for the first law of thermodynamics is based on the form (40), rather than the form (4). This was done to increase the accuracy for horizontal advection of θ near steep mountains, where p changes rapidly along a σ-surface. The horizontal flux of θ is formulated to conserve the global integral of θ and lnθ, with respect to mass, for l K, and of θ and θ for l> K. These choices do not formally guarantee conservation of the total energy for adiabatic and frictionless processes, but numerical experiments with the model under such conditions showed that the total energy was practically conserved. The horizontal flux of r is formulated to conserve the global integral of r and lnr in both the vertical and horizontal differencing prevents generation of negative values of r, as far as the space differencing is concerned. The horizontal differencing of the continuity equation and the momentum equation is based on the potential enstrophy and energy conserving scheme for the shallow water equations, the Cartesian grid version of which is presented by Arakawa and Lamb in this volume. The scheme for the model is obtained by replacing h of the shallow water equation scheme by π. When πσ 0 and when there is no generation of vorticity by pressure gradient and frictional forces, the scheme guarantees conservation of the potential enstrophy, defined by the global integral of --q with respect to mass, where q is defined in Section 3.6. The horizontal differencing for the pressure gradient force (38) is the simplest second order scheme for the C- grid. 8

20 7. Outline of the time differencing The time differencing is the leapfrog scheme for the basic dynamical processes, with a Matsuno step inserted periodically to avoid the separation of the solution produced by the leapfrog scheme. Contributions from the remaining processes are computed after every fifth step and added to the prognostic variables evenly over the following five steps. To avoid the use of an extremely short time interval necessary to maintain computational stability, a revised version of the scheme presented by Arakawa and Lamb (977, Section VI, C, ) is used near the poles. 8. Other aspects and the present status of the model The model uses the PBL entrainment formulation recently proposed by Randall (979). This formulations accounts for both unstable and stable conditions, for both deepening and shallowing (positive and negative entrainment) and for the major effects of stratocumulus and cumulus clouds. the cumulus parameterization, which includes the calculation of M B, is based on the theory of Arakawa and Schubert (974), as implemented by Lord (978). The changes described in this paper have been implemented in a 9-level version of the model, with the upper boundary at 50 mb. Experiments with idealized steep mountains have been made to test each of the major modifications and the model is now read for general circulation simulation experiments. Preliminary results will be presented at the Conference. Acknowledgements Special thanks are due to Dr. D. Randall who suggested using the PBL top as a coordinate surface. The authors wish to express their gratitude to all members of the UCLA General Circulation Research Project, in particular Mrs. C. H. Moeng and Mr. D. Sinton, who implemented many of the changes. Thanks are also due to Mrs. S. Lovell for typing the manuscript and to Mrs. B. Gladstone for drafting the figures. This research was supported by the Climate Dynamics program of NSF under ATM 78-09, by ONR through NEPRF under N C-003 under NGR

21 References Arakawa, A., 97: Design of the UCLA general circulation model. Numerical Simulation of Weather and Climate, Tech. Rep. No. 7, Dept. Meteorol., UCLA. Arakawa, A., and W. H. Schubert, 974: Interaction of a cumulus cloud ensemble with the largescale environment. Part I. J. Atmos. Sci., 3, Arakawa, A., and V. R. Lamb, 977: computational design of the basic dynamical processes of the UCLA general circulation model. Methods in Computational Physics, 7, Academic Press, New York. Lord, S. J., 978: Development and observational verification of a cumulus cloud parameterization, Ph. C. Thesis, Dept. Atmos., Sci., UCLA. Phillips, N. A., 957: A coordinate system having some special advantages for numerical forecasting. J. Meteorol., 4, Phillips, N. A., 974: Application of Arakawa s energy-conserving layer model to operational numerical weather prediction. U. S. Dept. of Commerce, NMC, Office Note 04. Randall, D. A., 979: The entraining moist boundary layer. Fourth Symposium on Turbulence, Diffusion, and Air Pollution, Jan. 5-8, 979. Reno, NV. American Meteorological Society. Tokioka, T., 978, some consideration on vertical differencing. J. Meteorol. Soc. Japan, 56,