## 1. Introduction

The increasing power of digital computers and interest in mesoscale meteorology have resulted in anew type of forecast or research model: the nonhydrostatic regional model. These models use one oftwo sets of governing equations: anelastic equations,which filter out sound waves, or elastic equations,which retain them. Because the elastic equations include sound waves, the use of an explicit time-differencing scheme is very costly; numerical stabilityrequires a very small time step. For this reason modelers have made considerable efforts to improve thenumerics. For time differencing two methods areused. The semi-implicit scheme (Golding 1992; Tappand White 1976; Tanguay et al. 1990) integrates thoseterms responsible for the acousticâ€“gravity modes implicitly. The time-splitting scheme (Dudhia 1993;Chen 1991; Klemp and Wilhelmson 1978; Skamarockand Klemp 1992; Tao and Simpson 1993; Tripoli andCotton 1982) isolates terms responsible for the acoustic modes and integrates them with a smaller explicittime step than that used for the remaining terms.

The purpose of this note is to examine what effectdo various numerical schemes have on the representation of acousticâ€“gravity waves and their generationby hydrostatic adjustment, the generation of acousticâ€“gravity waves by nonhydrostatic disturbances (Bannon 1995). In section 2 we find the small-amplitudewave motion generated by impulsively adding massor heat to a nonhydrostatic, compressible, adiabatic,and isothermal atmosphere. Given that mesoscalemodels always generate sound waves, the remainderof this paper deals with how the present temporal-and spatial-differencing schemes handle these waves.In section 3 we solve the hydrostatic adjustment problem using a semi-implicit scheme and compare thoseresults with the exact solution found in section 2. Werepeat our analysis with the split time-differencingscheme in section 4. Finally, we give our conclusionsin section 5.

## 2. Governing equations

*T*

_{0}. Theperturbation momentum equations are

*u*and

*w*denote the velocity components in the

*x*and

*z*direction, respectively, at time

*t,*and

*Î¸*is thedeviation of the potential temperature from the reference temperature

*Î¸*

_{0}(

*z*) =

*T*

_{0}exp(

*Îºz*/

*H*). The variable

*Ï€*denotes the Exner pressure function. In addition,

*g*is the gravitational acceleration,

*H*is the scale height,

*C*

_{p}is the specific heat at constant pressure, and

*Îº*=(

*Î³*âˆ’ 1)/

*Î³,*where

*Î³*is the ratio of the specific heatsof air at constant pressure and constant volume. Because we are focusing on the interplay of gravity andsound waves, the Coriolis force is neglected.

*N*= [(

*Î³*âˆ’ 1)

*g*/

*H*]

^{1/2}is the constant Bruntâ€“VÃ¤isÃ¤lÃ¤frequency and

*Q*denotes a source of heating. Finally,the perturbation pressure (continuity) equation is

*M*denotes a source of mass.

*M*

*M*

_{0}

*Î´*

*x*

*Î´*

*z*

*h*

*te*

^{âˆ’Î±t}

*U*

*t*

*z*= 0, where

*Î´*isthe Dirac delta function,

*U*is Heavisideâ€™s step function,

*Î±*> 0, and

*h*>

*a*> 0. Our choice of

*M*introduces apoint source of mass that eventually decays. Becausewe are finding the Greenâ€™s function in the spatial dimensions, we could find the solution for any other spatial forcing via convolution. Similar considerations apply to

*Q*; however, the vertical structure has beenchanged so that it is consistent with the one given byBannon (1995).

*x,*

*z*= 0, where

*k*and

*s*are the transform variables of the Fourierand Laplace transform, respectively.

_{M}, theparticular solution arising from the mass source term,and

_{Q}, the particular solution arising from the theheat source term. The solution

_{M}is

*z*=

*h,*while the second term gives awave that appears to emanate from a source located at

*z*= âˆ’

*h.*In reality it is the reflection of the direct wavefrom the surface at

*z*= 0. The last term gives the Lambsurface wave.

_{Q},

_{1}= Î“ + 1/(2

*H*) and Î“

_{2}= Î“ âˆ’ 1/(2

*H*). Thefirst term in (2.17) arises from the discontinuities in theheating function at

*z*=

*h*Â±

*a.*The second term represents the direct waves generated by the heating function. The most interesting aspect of this term is the factor1/(Î“

_{1}Î“

_{2}), which will yield the steady-state solution afterhydrostatic balance is established. Finally the third termis the reflection of the direct wave due to the presenceof the surface.

*s*= 0and branch points at

*s*= Â±

*a*

_{1}

*i*and

*s*= Â±

*a*

_{2}

*i*where

*c*

^{2}=

*Î³gH.*Cole and Greifinger (1969) encountereda similar situation in their study of acousticâ€“gravitywaves generated by a nuclear explosion. Our method ofinverting (2.14) is as follows: invert the Laplace transform using Talbotâ€™s (1979) method, as programmed byMurli and Rizzardi (1990), for fixed

*k.*Invert the Fouriertransform using a fast Fourier transform (FFT) with2048 wavenumbers and Î”

*k*= 0.0306796 km

^{âˆ’1}. Thismethod is so accurate that the solution essentially vanishes in those domains ahead of the wave front, as itshould.

To clarify the role of the mass and heat sources, weconsider *Ïˆ*_{M}(*x, z, t*) and *Ïˆ*_{Q}(*x, z, t*) separately. Figure 1illustrates *Ïˆ*_{M}(*x, z, t*) for different values of *Î±* and timewhile Fig. 2 gives *Ïˆ*_{Q}(*x, z, t*). The case of *Î±* = 0.1 s^{âˆ’1}gives a nonzero forcing for approximately 70 s, while*Î±* = 1 s^{âˆ’1} gives a nonzero forcing for approximately 5s. These figures show the following: when we impulsively force the pressure field, most of the energy excitesacoustic waves, which propagate rapidly outward witha very sharp wave front. On the other hand, if we impulsively force the thermodynamics equation, gravitywaves are primarily excited, which radiate away fromthe source much more slowly.

Having obtained the manner in which our model atmosphere adjusts to a nonhydrostatic disturbance, wenext examine how well the semi-implicit integrationscheme captures hydrostatic adjustment.

## 3. The semi-implicit scheme

In this section we solve (2.1)â€“(2.4) using the numerical techniques of semi-implicit time differencingand staggered spatial finite differencing. Because weonly have variations in *x* (and not in *y*), the so-calledB and C grids become identical in this two-dimensionalproblems, with *u* at the horizontal midpoint and *w, Î¸,*and *Ï€* at the horizontal grid point. Our goal is to discoverthe effect that both spatial and temporal finite differencing has on our solution.

*Ï€*=

*e*

^{kHz}

*Ïˆ*

*z.*

An important consideration for any numerical schemeis its stability. Instability occurs if the Laplace transformhas a singularity in the right half of the *s* plane. Fromthe branch points of Î“, we find that stability occurs forany time step. If we applied the semi-implicit schemeto those terms describing acoustic waves, as Tapp andWhite (1976) did, then stability would only occur if *N*Î”*t* < 1.

The actual inversion of the Laplace transform in thisparticular problem is rather difficult. The origin of thisdifficulty lies with the sinh(*s*Î”*t*) and cosh(*s*Î”*t*) terms inthe solution, which introduce an infinite number of singularities and branch points along the imaginary *s* axis.Even if we could invert the Laplace transform exactly,the inverse would consist of an infinite series from theevaluations of the residues associated with singularitiesplus an infinite number of branch cut integrals. Consequently, we can only keep a finite number of theseterms and integrals in our numerical calculations; theneglected terms are several orders of magnitude smallerthan the inverse.

Following Evans (1993) we used a scheme by Alaylioglu et al. (1975) to evaluate the inversion integral.By using this scheme we avoid an unacceptably largenumber of function evaluations of the Laplace transformduring the inversion. We used the product formula forthe hyperbolic cosine and sine so that we could controlthe exact number and location of the singularities. Weincreased the number of terms until the solution converged. Because of computational costs for this newscheme, we could only use 1024 wavenumbers in theinversion of the Fourier transform and Î”*k* = 0.0306796km^{âˆ’1}.

We begin by repeating the calculations from the previous section for small Î”*t* and Î”*x* = 1 km. For Î”*t* =0.1 s (not shown) we recover the continuous results,except that there is a hint of a small-scale wave in theacoustic wave field. Figures 3 and 4 show the wave fieldat *t* = 120 s and *t* = 240 s, for Î”*t* = 2.5 s and Î”*t* =10 s. If the forcing occurs in the pressure equation, thereis considerable deviation from the exact solution (seeFig. 3). In particular, the disturbances remain close tothe point of excitation while the large amplitudes at thewavefront, associated with the outwardly propagatingacoustic waves, are greatly reduced. This is not surprising in light of the well-known property that semi-implicit schemes slow down the faster propagatingwaves.

When the thermodynamics equation is forced (seeFig. 4), there is considerable agreement between thefinite-differenced solutions with Î”*t* = 2.5 s and Î”*t* =10 s and the corresponding exact solution (Fig. 2a,b).Because this forcing excites primarly gravity rather thanacoustic waves, the semi-implicit scheme is better atcapturing the adjustment process than when the pressureequation was forced.

In summary, the semi-implicit scheme reproduces hydrostatic adjustment well for small time steps. However,for time steps used in operational models, the semi-implicit scheme gives a poor representation of hydrostatic adjustment if the source of excitation involvesmass as the properties of the scheme trap shorter acoustic waves near the point of excitation.

## 4. The time-splitting scheme

Unlike the semi-implicit scheme, time splitting divides the various terms of the dynamical equations intotwo groups according to whether they are necessary todescribe acoustic waves or not. Each group is then integrated separately with its own particular time step.The time step used to integrate the acoustic terms istypically taken to be a fifth to a third smaller than thatused for the remaining terms.

After taking the Laplace and Fourier transform of thissystem of differenced equations, it is necessary to solvea large set of ordinary differential equations. Even whenthe ratio of the acoustic time step to the conventionaltime step is *O*(1), the corresponding algebra becomesintractable and we gain little insight. Therefore, we investigate the time-splitting scheme numerically using amodel described by Chen (1991). We eliminated theacoustic damper and eddy turbulence flux parameterization that were added to the model after publicationof Chenâ€™s (1991) paper while retaining the fourth-orderShapiro spatial filter. The acouticâ€“gravity terms wereintegrated with a Matsuno time-differencing scheme; thesimulations were sufficiently short that the well knowndamping of the Matsuno scheme was not a problem.

We conducted two groups of experiments. In the first,the acoustic and gravity time steps equal each other. Inthe second group, the gravity time step was 10 timeslarger than the acoustic time step.

*x*= Î”

*z*= 200 m and Î”

*t*= 0.1 s) version of themodel; this corresponds to our version of the exact solution. The forcing function is given by (2.9) and (2.10)with

*Î±*= 0.1 s

^{âˆ’1}. We approximated the delta functionsby the formula

*Ïµ*

_{x}= 0.1 km

^{2}. A similar expression approximatedthe delta function in the vertical direction. These solutions are shown in Fig. 5a,b and 6a,b and correspondto the times

*t*= 120 s and

*t*= 240 s, respectively.

For the next numerical experiments we ran the modelwith an advective time step of Î”*t* = 2 s and an acouticâ€“gravity time step Î”*t* = 0.2 s. Figures 5a,b and 6a,b givethese results. In general, the effect of time splitting isvery small, introducing some smoothing of the wavefront in Fig. 5. In Fig. 6 the largest differences occurat *t* = 240 s. These differences arise because of the poortemporal resolution of the forcing functions, because itis evaluated in the advective time-step portion of thecode.

## 5. Conclusions

In this note we examined how hydrostatic adjustment,the elimination of nonhydrostatic disturbances throughthe generation of acousticâ€“gravity waves, occurs in aprototypical compressible, nonhydrostatic model. In order to determine whether current numerical techniquesfaithfully simulate this process, it was necessary to findan exact solution to the hydrostatic adjustment problem.This was done by examining the wave motions arisingin an isothermal, compressible, nonrotating atmosphereover a flat earth subject to impulse forcing. If the pressure equation is forced, these wave motions consist ofa direct acousticâ€“gravity wave, a reflected acousticâ€“gravity wave, and a Lamb wave. If the thermodynamicsequation is forced, most of the disturbances are excitedas gravity waves.

Having explored the exact solution to the problem,we reworked the same problem using the semi-implicitscheme. If the pressure equation was forced, primarilyacoustic waves were generated. For time steps equal to2.5 or 10 s, these acoustic waves were trapped near thepoint of excitation. On the other hand, when the thermodynamics equation was forced, most of the disturbances were excited as gravity waves and the semi-implicit scheme did considerably better.

We examined the time-splitting scheme by numericalexperimentation using a numerical model by Chen(1991). Generally the time-splitting scheme performedwell with some smoothing of the acousticâ€“gravity wavefront. When the thermodynamics equation was forced,the evaluation of the forcing function at the larger, advective time step introduced errors.

## Acknowledgments

The author would like to thank Dr.R. Wayne Higgins for his careful reading of the manuscript, Dr. Peter R. Bannon for several insightful discussions, Dr. Chiang Chen for the use of his model and assistance in running that model, and the helpful commentsof Dr. Ming Xue and another anomyous reviewer.

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Same as Fig. 1 except that the waves are solely generated by heating. The *Ïˆ* field is made nondimensional by dividing it by*Îº**cQ*_{0}*e*^{h/(2H)}/(*Î³H*). The fields in (c) and (d) have been multiplied by 100 and *a* = 0.5 km.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 1 except that the waves are solely generated by heating. The *Ïˆ* field is made nondimensional by dividing it by*Îº**cQ*_{0}*e*^{h/(2H)}/(*Î³H*). The fields in (c) and (d) have been multiplied by 100 and *a* = 0.5 km.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 1 except that the waves are solely generated by heating. The *Ïˆ* field is made nondimensional by dividing it by*Îº**cQ*_{0}*e*^{h/(2H)}/(*Î³H*). The fields in (c) and (d) have been multiplied by 100 and *a* = 0.5 km.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 1 for the case *Î±* = 0.1 s^{âˆ’1}, except that we finite differenced the governing equations using a semi-implicit scheme.For (a) Î”*t* = 2.5 s and *t* = 120 s; (b) Î”*t* = 2.5 s and *t* = 240 s; (c) Î”*t* = 10 s and *t* = 120 s; and (d) Î”*t* = 10 s and *t* = 240 s.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 1 for the case *Î±* = 0.1 s^{âˆ’1}, except that we finite differenced the governing equations using a semi-implicit scheme.For (a) Î”*t* = 2.5 s and *t* = 120 s; (b) Î”*t* = 2.5 s and *t* = 240 s; (c) Î”*t* = 10 s and *t* = 120 s; and (d) Î”*t* = 10 s and *t* = 240 s.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 1 for the case *Î±* = 0.1 s^{âˆ’1}, except that we finite differenced the governing equations using a semi-implicit scheme.For (a) Î”*t* = 2.5 s and *t* = 120 s; (b) Î”*t* = 2.5 s and *t* = 240 s; (c) Î”*t* = 10 s and *t* = 120 s; and (d) Î”*t* = 10 s and *t* = 240 s.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 2 for the case *Î±* = 0.1 s^{âˆ’1} except that the governing equations were finite differenced using a semi-implicit scheme.For (a) Î”*t* = 1 s and *t* = 120 s; (b) Î”*t* = 1 s and *t* = 240 s; (c) Î”*t* = 10 s and *t* = 120 s; and (d) Î”*t* = 10 s and *t* = 240 s.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 2 for the case *Î±* = 0.1 s^{âˆ’1} except that the governing equations were finite differenced using a semi-implicit scheme.For (a) Î”*t* = 1 s and *t* = 120 s; (b) Î”*t* = 1 s and *t* = 240 s; (c) Î”*t* = 10 s and *t* = 120 s; and (d) Î”*t* = 10 s and *t* = 240 s.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 2 for the case *Î±* = 0.1 s^{âˆ’1} except that the governing equations were finite differenced using a semi-implicit scheme.For (a) Î”*t* = 1 s and *t* = 120 s; (b) Î”*t* = 1 s and *t* = 240 s; (c) Î”*t* = 10 s and *t* = 120 s; and (d) Î”*t* = 10 s and *t* = 240 s.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Numerical simulation of hydrostatic adjustment with the time-splitting scheme when the pressureequation has been impulsively forced at *x* = 0 and *z* = 2 km. In (a) and (b) we show the solutions whenboth the acousticâ€“gravity and advective time steps are equal (Î”*t* = 0.1 s). (a) is the solution at *t* = 120 s,while (b) gives the solution at *t* = 240 s. (c) and (d) are identical to (a) and (b) except that the advectivetime step (Î”*t* = 2 s) is 10 times larger than the acousticâ€“gravity time step (Î”*t* = 0.2 s). Other parametersare Î”*x* = Î”*z* = 200 m.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Numerical simulation of hydrostatic adjustment with the time-splitting scheme when the pressureequation has been impulsively forced at *x* = 0 and *z* = 2 km. In (a) and (b) we show the solutions whenboth the acousticâ€“gravity and advective time steps are equal (Î”*t* = 0.1 s). (a) is the solution at *t* = 120 s,while (b) gives the solution at *t* = 240 s. (c) and (d) are identical to (a) and (b) except that the advectivetime step (Î”*t* = 2 s) is 10 times larger than the acousticâ€“gravity time step (Î”*t* = 0.2 s). Other parametersare Î”*x* = Î”*z* = 200 m.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Numerical simulation of hydrostatic adjustment with the time-splitting scheme when the pressureequation has been impulsively forced at *x* = 0 and *z* = 2 km. In (a) and (b) we show the solutions whenboth the acousticâ€“gravity and advective time steps are equal (Î”*t* = 0.1 s). (a) is the solution at *t* = 120 s,while (b) gives the solution at *t* = 240 s. (c) and (d) are identical to (a) and (b) except that the advectivetime step (Î”*t* = 2 s) is 10 times larger than the acousticâ€“gravity time step (Î”*t* = 0.2 s). Other parametersare Î”*x* = Î”*z* = 200 m.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 5 except that we have forced the thermodynamics equation with *a* = 0.5 km.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 5 except that we have forced the thermodynamics equation with *a* = 0.5 km.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2

Same as Fig. 5 except that we have forced the thermodynamics equation with *a* = 0.5 km.

Citation: Monthly Weather Review 125, 12; 10.1175/1520-0493(1997)125<3357:HAINCM>2.0.CO;2