Experiments with a Spectral Convection Model

A new spectral moist convection model that employs both the least as­ sumptions in moist physics and a very accurate solution method is pre­ sented. The temperature and pressure in the model are diagnostically de­ termined from thermodynamics. There is no need to predict water vapor and condensate separately; rather, they are diagnostically separated from the predicted total airborne water. The model allows a modular separation of dynamics and thermodynamics; the link between dynamics and thermo­ dynamics is through the pressure gradient force. The modular separation allows the possibility of having a detailed, fine resolution, nonhydrostatic cloud model and a coarse resolution, hydrostatic model which can be run side by side with the identical moist thermodynamics. The height coordi­ nate of the nonhydrostatic model can also extend into the hydrostatic re­ gime. The only differences between the hydrostatic and nonhydrostatic models are spatial resolution and the way vertical motion is computed. We have performed numerical experiments in the nonhydrostatic model for acoustic adjustment and moist convection. The discontinuity in thermody­ namics due to phase change is modified in the model by the "gradual satu­ ration" technique.


INTRODUCTION
Global models have become important tools for weather and climate simulations.How ever, these models have simplified hydrostatic dynamics and coarse vertical and horizontal resolution, so they are unable to explicitly simulate most of the cloud patterns that are crucial to climate dynamics.Because of their important effects on radiative transfer, hydrological cycles and apparent heat sources, moist convections must be more accurately treated in cli mate models.The real atmosphere contains a myriad of cloud structures which modulate ra diative fluxes and which modify atmospheric structure by condensing water at one level and 1Department of Atmospheric Sciences, National Taiwan University, Taipei, Taiwan, ROC 652 TAO, Vol.JO, No. 4, December 1999 reevaporating it at another level.This deep and shallow moist convection can be modeled more accurately with a nonhydrostatic model.Nonhydrostatic convection models can be based on unfiltered or filtered systems.The filtered systems include the anelastic and Boussinesq equations.Ogura and Phillips ( 1962) derived the anelastic system under the assumptions that the percentage range of potential temperature is small and the time scale is set by the Brunt Vaislilii a frequency.The anelastic equations reduce to the Boussinesq equations under the additional assumption that the vertical scale of motion is small compared with the depth of an isentropic atmosphere.In an unfiltered system, one has to cope with the propagation of acous tic waves.Often the pressure is used as a prognostic variable of the model.Since pressure is not a conservative property, the prognostic equation for pressure has been derived from other conservation laws under some approximation of moist thermodynamics, which usually in volves a definition of equivalent potential temperature.This is also a problem with existing general circulation models, since they are based on the quasi-static primitive equations in sigma coordinates and have a host of thermodynamic approximations associated with the use of equivalent potential temperature, moist static energy, etc, With progress in computers and computational techniques, we have often experienced that the return to the first principles of physics enables a model to cope more easily with the complexity of the atmosphere.Ooyama (1990) proposed a "primitive" form of moist thermo dynamics.Instead of using pressure as a prognostic variable, Ooyama's model uses the con servation of the entropy density, the momentum density and the total moisture density as predicted variables.With an accurate definition of entropy density in the moist atmosphere, Ooyama's proposal involves the least assumptions in reversible moisture physics.Since the pressure field no longer is a prognostic variable, Ooyama's proposal also allows the extension of a nonhydrostatic height coordinate model into the hydrostatic regime.This extension may be useful in simulations of weather systems that involve a wide range of horizontal scales.
We have constructed a new spectral moist convection model that employs both the least assumptions in moist physics and a very accurate solution method.Our modeling effort in volves the Fourier-Chebyshev spectral discretization similar to that in Kuo and Schubert (1988) and the moist thermodynamics of the "primitive" form in Ooyama (1990).We believe a sound basis for moist thermodynamics and an accurate treatment of discretization are important for the improvement of cloud modeling.The governing equations are presented in section 2. Sec tion 3 describes the solution method.Numerical results are covered in section 4. Section 5 contains the concluding remarks.

GOVERNING EQUATIONS
We consider the two-dimensional (x-z) case described below.The "primitive" form of moist thermodynamics makes model predictions strictly in terms of conservative properties, in partiCular the density of dry air �.density of total airborne moisture 1J, entropy density a, the momentum densities U=pu, W=pw, where p;=�+7J. (2.6,2.7,2.8) The above constitute eight equations for the five prognostic variables �. 1], <1, U, W, and the four diagnostic variables p, u, w and p.The system is closed by the thermodynamic diagnosis, the input of which is �.f/,<1, and the output of which is temperature, pressure and the partition of 1 7 into its vapor and condensate parts.This requires writing two formulas (depending on whether the total airborne moisture 11 is entirely in the vapor phase or is partially condensed) for the entropy a(,�,17,T), iteratively solving for two temperatures (T1 and T2) and then using Here 77v and rte are the densities of vapor and condensate respectively, Pa and Pv the partial pressures of dry air and water vapor respectively, E(T) the saturation vapor pressure and 1J.(T)=E(T)l(RvT) the mass density of saturated vapor.
When the hydrostatic approximation is made, as described by Ooyama (1990) and DeMaria (1995), (2.2) is replaced by a simple diagnostic equation (2.9) where (2.10) is the Laplace adiabatic sound speed.Derivations of the vertical motion diagnostic equations can be found in appendix A. The replacement of (2.2) with (2.9) is the only change necessary to convert the nonhydrostatic equations to their hydrostatic form.This diagnostic equation is a one-dimensional (height) second-order elliptic equation that can be solved efficiently using a direct method.DeMaria (1995) found that hydrostatic solutions of (2.9) are very sensitive to the method used to solve the diagnostic vertical velocity equation.The sensitivity can be elimi nated, as described by DeMaria (1995), by adding an extra term to the diagnostic equation that ensures the solution does not drift away from the hydrostatic balance due to numerical ap proximation.With the diagnostic vertical velocity equation, Ooyama's formulation allows us to design a numerical model in height coordinates that can be used in hydrostatic and nonhydrostatic regimes . .

SOLUTION METHODS
The simulation of moist convection places great demands on the spatial discretization schemes used in numerical models.We will use a scheme which is spectral in both directions.
We shall solve the above system of equations on the domain 0 $ x $ L, 0 $ z $ H, with the assumption that all variables are periodic in x and W = 0 on z = O,H.In the x direction, Fourier basis functions are used so that the periodicity is built into each basis function.In the z direc tion, Chebyshev polynomial basis functions are used; the top and bottom boundary conditions are not satisfied by each basis function, but rather by the series as a whole.Details of the Chebyshev tau method can be found in Fulton and Schubert (1987) and Kuo and Schubert (1988).In the following we discuss the spectral method for solving the system.

a. Fourier-Chebyshev method
The dependent variables (e.g., c;) are approximated by the series expansions

b. Pressure gradient across a cloud edge
The pressure field is diagnosed from the equation of state in our model.The fields of temperature and liquid water density can have discontinuities in the first derivative across a cloud edge due to phase change.With a sufficiently smooth density field, the first derivative discontinuity in the temperature field causes the first derivative discontinuity in the pressure field across a cloud edge.To avoid the Gibb's phenomena in a spectral model, we need to adapt the "graduation saturation" technique.Since the temperature and the liquid water den sity fields do not explicitly appear in (3.5),only treatment for the pressure field is required.The basic idea is presented in equations (3.10) and (3.11) in this section.The numerical results are presented in section 4 (i.e., Figs.l, 2, and 3).
At any spatial point, p=P(.;,17,cr), thus we have (3.10) The P-coefficients are known functions of (x,h,s) so that (3.10) for V p could now be used for the pressure gradient force in the momentum equations in the spectral model.However, Ooyama discusses how this can cause Gibbs' phenomena near cloud edges.As a solution he proposes weighted averages of the ?-coefficients for saturated and unsaturated conditions.The overlap of the weighting coefficients is adjusted to the model spatial resolution.The weighting coeffi cients are If the �o> and �(Z) are computed from T1 (temperature in unsaturated region) and T2 (tem perature in saturated region) respectively, then the weighted average of P� across cloud edge is given by (3.llc) Equation (3.11) can also apply to the calculations of P11 and Pa• The fonnula for P 1 /1l, P 11(2), Pt» p�<2>, P001 and P0l2J can be found in Ooyama (1990).The derivation of the fonnula is given in appendix B.

a. One-dimensional experiment
Our first experiment (EXPl) is the calculation of the pressure gradient across a cloud edge in one dimension.We consider a domain of

X (m) (b)
Dry air density e Waler density T)   The basic state satisfies hydrostatic balance.Superimposed on the basic state is a temperature anomaly defined by We have also set the e' equal to zero in our initial condition.Thus only the p ' and u' anomaly exist along with the T' anomaly.Since ;' = 0, the anomaly (bubble) has no buoyancy and will not rise.Moreover, hydrostatic balance is violated because we have a p ' but not a �' superimposed on the hydrostatic basic state.
Figure 4 shows the U, W, a', f, p' and T" in physical domain at time 0.3s for the calculation with .6.T= 2.5K in (4.3).The perturbation temperature in Fig. 4d is given by T" = T -T -T', the difference between T and the initial T. To see how fast the acoustic wave can make the hydrostatic adjustment, we have plotted the time series at the center of the domain for the variables of divergence, T', ;1 and p ' in Fig. 7. Figure 7 indicates that it takes about 3 to 4 seconds for these variables to reach a steady state.Figure 8 is similar to Fig. 7 except for an experiment with .6.T = 7 .5K in (4.3).Interest ingly, the time series in both cases are very similar.This indicates the atmosphere reaches the "anelastic balance" ( CJp' I dt :;: :: : 0 and au I CJx + aw I ()z :;: :: : 0 ) or converts a zero ;' to a finite value of ; ' in 3 to 4 seconds, regardless of the size of .6.T. In other words, anelastic models are just as good as compressible models if the transient acoustic waves are not the focus of modeling.Discussions on one-dimensional acoustic adjustment with an isothermal basic state can be found in Bannon (1995).Duffy (1997) examined hydrostatic adjustment through the generation of acoustic-gravity waves.

2.
3. Namely, we have the initial condition for our third experiment The experiment is designed so that p' <= 0 in the initial condition.Thus, contrary to the second experiment, we do not experience significant acoustic wave radiation in this experi-ment.Figures 9 and 10 are the U, W, a', T', p' and ;1 in physical space for t=150s and t=300s respectively.We observe strong updraft in the center of the warm bubble and relatively weak downward motion in a broad area adjacent to the rising bubble.On top of the rising bubble there is high pressure while below the bubble top is a slightly broad area of low pres sure.
Figure 11 shows the time series in the first stage of EXP3 (t <7 .5s)at the point (x = 1250m, z = 780m) for divergence of momentum density, the difference between T and the initial T (i.e., T" ), the difference between ; and the initial � (i.e., �11 ), and the perturbation pressure p'.The point (x=l 250m, z=780m) is above the rising warm bubble where we expect a region of high pressure.Figure 11 indicates the presence of acoustic waves in that ;11 is out of phase with the momentum density divergence.The formation of a high pressure region above a warm rising bubble is associated with a series of transient acoustic waves.
The fourth experiment (EXP4) is a rising moist bubble experiment in a hydrostatic atmo sphere.W � consider a basic state of where " is the mixing ratio of total water density with respect to the dry air density g.The basic state dry air density in (4.7c) is computed from the equation of state.Now we consider a moist bubble with temperature perturbation T' given by (4.5) T(x,z) = T(z) + T ' (x, z) ,  ,.--. .X (M) . (Fig.   T" , the difference between T and the initial T (unit K), (c) �,,,the dif ference between � and th e initial � (unit kg m-3) and (d) perturbation pressure p' (unit hPa).
Equation (4.8) gives positive values of a', T ' and r ( and a negative value of �, (positive buoyancy) for the bubble.Figures 12 and 13 are the results at t=150s and t=240s respectively for EXP4.With the initial maximum T of 2.SK, the results indicate that th e perturbation tem perature T' decreases with time before the condensation takes place (i.e., t=150s).Fr om Fig. 13 we see that the maximum perturbation temperatilre T' is about 4K which is higher than the initial value of 2.5K.This is due to the release of the latent heat.Figure 13 also indicates a very small warming (0.0lK) and drying ( r ( = -0.0128kgm•3) outs ide the cloud as a result of forced do wnward motion in a constant 8 atmosphere.The perturbation pressure distribution is mo re complicated than the pressure distribution in Figs. 9 and 10.This is probably also due to the  L -. 128 -.010 -.12111  �in the initial conditions, there are positive anomalies of fl and er which are brought up by the updraft.In addition, the positive anomaly of T' and the negative anomaly of �' are associated with the condensation of liquid water.Rotunno et al. (1988) viewed the no tilted updraft situ ation as the "optimal state" for the squall line.Moreover, they argued that the tilted updraft may be stemmed from the imbalance of the vorticity across the low level gust front.On the other hand, Seitter and Kuo (1983) argued the tilted updraft may be stemmed from the liquid water loading effect in the updraft/downdraft interface.The study of tilted updraft is of funda mental importance in understanding long-lived mesoscale convection.EXP5 suggests that our .0 025 r-r-r-r-r--,-,r-r-.. .--;r-r-. ..,. ..-,. .. ..,--.-, , -.0 06 .0020 model is capable of simulating condensation associated with a tilted updraft.

CONCLUDING REMARKS
Some points of the model worth noting are as follows.
l.The temperature and pressure are diagnostically determined from thermodynamics.
2. There is no need to predict water vapor and condensate separately; rather, they are diagnostically separated from the predicted total airborne water.�T -) =--(v;-+ pC -)--g+-(-Q a-+-!4r)+ g!4rz = 1 is the transformation from physical space to Fouriern > 0 Chebyshev spectral space and (3.1) is the transformation back.The evaluation of (3.2) can be done by the fast Fourier transform and the fast Chebyshev transform.The total Nz collocation points for the fast Chebyshev transform in the vertical are determined asWith the nonlinear terms defined by, computed by the transform method.To eliminate aliasing error in the quadratic nonlinear terms in the transform method, 3M points in the x direction and 3N/2 points in the z direction are needed in the physical domain.The tau equations for (2.1 )-(2.5) are dU mn + A ( 1 ,0) + fJCO.I) + p "(l,0) mn + ('0,0) + fJ<O•l) + (� + ry" )g + p "--1! !! !. .+ E c1,o> + p<o.1) = o the x derivative of the spectral coefficients is denoted by the superscript (1,0) and the z derivative of the spectral coefficients is denoted by the superscript (0,1).The time integration of (3.5) is done with the fourth-order Runge-Kutta scheme {or all the mo " des -M '5.m '5, -Mand 0 '5, n ::::; N with the exception of the spectr� coefficients w m,N -1 and w m ,N.According to the 'r method, the last two vertical modes of w m n are t� be obtaineq from the vertical boundary conditions (W (0) = W(H) = 0).Namely, we solve w m , N -1 and w m, N by 6b)The relation between A ��oi and Amn (the spectral coefficient of A) is Hung-Chi Kuo & Chao-Tzuen Cheng A_c1.o) = • (2mn)A.m n l L mn' while the relation between B�� I ) and B mn (the spectral coefficient of B) is B(O,I ) = ___±_ __ � pB Although the spectral evaluation of z derivatives by (3.8) looks at first sight more difficult than the spectral evaluation of x derivatives by (3.7), such is not the case.Equation (3.8) yields the (backward) recurrence formula "(0,I) "(0,1) _ 4 " cn_I B m,n-1 -Bm,n+I -H nBm,n (n = 1, 2, . . . ,N -1) x0 = lOOOm and A. x = 250m.The thermodynamic variables are specified according to (4.1) with different constants a and b.We take b = 65 m2s•2K1 and a= 220 m2s-2K1 for the entropy density a.Similarly, we take b = 0.01 kg m-3, a= 0.0078 kg m•3 for total water density1J and take b=0.0095 kg m3, a=l.1135 kg m-3 for dry air density �.The profiles of a, � and 1J are shown in Fig. 1.The profiles of the diagnosed temperature T, pressure p and condensed water density 11c are shown in Fig. 2.There are discontinuities in the first derivative of the T, p and 1J,.fields across the cloud edge.Figure 3 shows the pressure gradient calculated from (a) direct differentiation of the p profile of Fig. 2b by 48 grid points Fourier spectral method and from (b) Fourier spectral method by (3.10) and (3.11) with 48 grid points.The A. T12 used here is 10-1 K.The Chebyshev method yields results similar to Fig. 3, and thus is not shown here.It is concluded from Fig. 3 that the pressure gradient calculation by (3 .10)and (3 .11)greatly reduces the Gibbs phenomenon in the Fourier-Chebyshev spectral method.b.Two-dimensional experiments For the two-dimensional experiments, we have used L = H = 2500m, M = 16, N = 32 andA.t = 0.075s in our calculation.To improve time integration efficiency, the subgrid diffusion process in the model is handled in spectral space by applying the Lanczos filter to the tendency

Fig. 3 .
Fig. 3.The corresponding pres sure gradient V p of Fig. 2b from (a) direct differ entiation of the p profile of Fig. 2b by 48 points Fourier spectral method and from (b) Fourier spec tral method with 48 points by (3.10) and (3.11).
Figures 5 and 6 are similar to Fig. 4 except at time 1.Ss and 30s respectively.Figure 6 reveals the motion and density fields associ ated with a rising bubble.This rising bubble (now e' < 0 ) can be viewed as the result of the hydrostatic adjustment by the acoustic waves.In contrast, Figs. 4 and and 5 indicate the mo tion and density fields associated with the transient acoustic waves.

CONTOUR
Figure 14 presents the time series of the maximum value of �', W, 1Jc and T' for the ¢ = 0 and ¢ = 1C 16 cases.The results from Fig. 14 are expected in that the tilted updraft ( </> = 1C I 6) produced less condensed water, T' , and Win the later stage.The numerical results of the perturbation fields in the x-z domain at t=240s with the initial momentum density condition computed with ¢ = 1C 16 are presented in Fig. 15.Because of the vertical gradient of 17, er and
HHIE•H TO 1.nlHE•U CONTOUR INfER•AL or l.IHHE•n LABEL$ SCAL[O 81 l.UH9(c I c )p in (A.4) can be written as (c I c )'P = (c I c )p R T = p C2 where C2 is the .6) is the Richardson vertical motion equation in differential form.We now derive th e vertical equation based onOoyama (1990) moist thermodynamics in a-z vertical coordinate.The derivation is similar to th at in the height coordinate by DeMaria (1995).The a-z coordinate is defined byz ' = H( Z -Zs J .• , H -z swhere z, is the topographic height.The conservation equations after the coordinate transform can be written as aa' + ac a'vi ) ac a ' w ' ) = Q at ax� .+ Ck' er' ' r For the saturated air we with the equation of state p =(� a + 77* R,, ) T = e R.a T + E(T ) and obtainThe entropy density for the saturated air is a = 17cv)n I__ �)n ; + 17C(T ) + D (T ) .