## 1. Introduction

Large-scale, midlatitude, atmospheric flows exhibit a tendency to be in hydrostatic and geostrophic balance. However, rapid, localized heat sources embedded in the synoptic-scale flow (e.g., mesoscale convective systems) can force the synoptic-scale flow away from hydrostatic and geostrophic balance. This imbalance is characterized by the creation of potential vorticity anomalies and the generation of a spectrum of buoyancy and acoustic waves. The unbalanced atmosphere adjusts toward a new hydrostatic and geostrophic state, determined by the magnitude and geometry of the potential vorticity anomaly. Blumen (1972) and Gill (1982) have reviewed the fundamental problem of the adjustment of a geophysical flow after an initial perturbation from a balanced state. Typically the initial perturbation is an addition of momentum or of mass. Most of the literature emphasizes the subsequent evolution of the flow toward a state of geostrophic balance, the time- and length scales of the adjustment, and the partitioning of the energy between the final steady state and the transients. In contrast, recent studies (Raymond 1986; Bretherton 1988, 1993; Bretherton and Smolarkiewicz 1989; Shutts and Gray 1994) indicate that considerable insight into the workings of moist convection can be gained by the examination of the response to prescribed heat sources.

This paper describes the hydrostatic and geostrophic adjustment to a prescribed, rapid, localized heating in a compressible atmosphere. The approach is analytic and holds for linearized dynamics. We focus on the initial response to the heating and the final steady-state equilibrium. From these solutions we infer the wave energetics during the process of adjustment and the net parcel displacements. We also assess the ability of several common approximations of the fully compressible equations to model the adjustment problem correctly. For example, the need to filter acoustic modes from model governing equations has led to the anelastic approximation [see Bannon (1996b) for a review and synthesis of these theories]. An attempt to relax the assumption of local incompressibility in the anelastic approximation has led to the pseudo-incompressible theory of Durran (1989). Another approximation commonly used in mesoscale modeling that retains both compressibility and numerical efficiency is the modified-compressible approximation (Klemp and Wilhelmson 1978). The present study also serves as an extension of the work of Bannon (1995a,b, 1996a) on hydrostatic adjustment to the fully three-dimensional problem.

Section 2 describes the compressible model of the three-dimensional hydrostatic and geostrophic adjustment process. Section 3 presents the compressible solutions. Section 4 presents an intercomparison of the anelastic, pseudo-incompressible, and modified-compressible approximations. Section 5 discusses the salient features of the adjustment process and the ability of these approximations to simulate the adjustment adequately.

## 2. The compressible model

### a. Governing equations

*f*plane in Cartesian coordinates linearized about an isothermal, resting base state arewhere

*Q*is the heating rate per unit volume,

*g*is the vertical acceleration due to gravity,

*f*is the constant Coriolis parameter, and

*γ*=

*c*

_{p}/

*c*

_{υ}= 1.4 is the ratio of the specific heats. A subscript

*s*denotes a static, base-state quantity, and a superscript prime denotes a perturbation from the base state. The base-state quantities arewhere

*T*∗,

*p*∗,

*θ*∗ =

*T*∗, and

*ρ*∗ are constants satisfying the ideal gas law and Poisson's relation. Here,

*κ*=

*R*/

*c*

_{p}, and

*H*

_{s}=

*RT*

_{s}/

*g*is the density scale height in an isothermal atmosphere. The isothermal base state is an analytically convenient representation of a statically stable atmosphere. Note that the height

*z*= 0 corresponds to a location in the midtroposphere, not the surface of the earth. The values of the parameters chosen for this study are representative of the

*U.S. Standard Atmosphere*in the midtroposphere:Equations (2.1a,b,c) are the momentum equations. Equations (2.1d,e,f) are the entropy equation, the linearized Poisson's relation, and the mass conservation equation, respectively.

*H*is the Heaviside step function,

*d*is the heating half depth,

*δ*is the Dirac delta function, and

*s*is an arbitrary function describing the horizontal structure of the heating. Here

*H*and

*s*are dimensionless functions, and

*δ*has dimensions inverse to those of its argument. The magnitude of the heating is specified by Δ

*p,*that is, the amplitude of the initial pressure perturbation. Before the heating is applied, all fields satisfy the base state. The initial conditions are thus

*z*= −

*D*= −6 km where the vertical velocity is zero. The finite atmosphere is bounded both above and below the heated layer by rigid boundaries at

*z*= ±

*D*where the vertical velocity is zero. The rigid boundary condition on the pressure field isEquation (2.5) is derived by setting

*w*′ = 0 in (2.1c,d) and by applying the condition that, because there is no heating on the boundary,

*θ*′ = 0 at the boundaries.

### b. Potential vorticity conservation

*q*′, isand

*ζ*′ is the vertical component of the relative vorticity. The steady-state potential vorticity can be found by integrating (2.6) with respect to time. Using (2.3) we findwhere the subscript

*f*denotes a final, steady-state quantity.

*x*and

*y,*and by applying the vertical coordinate transformation defined by

*P̂*

_{f}

*k,*

*l,*

*z*

*P*

^{′}

_{f}

*k,*

*l,*

*z*

*e*

^{z/2Hs}

*S*(

*k,*

*l*) is the Fourier transform of

*s.*The compressible Rossby height,

*H*

_{R}, is the

*e*-folding distance for the solutions to (2.11).

The Green's function procedure by which (2.11) is solved is presented in appendix A. We apply a numerical inverse fast Fourier transform to the solutions of (2.11) to obtain solutions in physical space. A Lanczos smoothing factor (Arfken 1970) is applied to the wind field in order to avoid convergence problems with the derivative of a Fourier series.

### c. Energetics

*w*′

*ρ*′

*g,*and between available elastic and available potential energy by the work done against the pressure,

*w*′

*p*′/(

*γH*

_{s}). Figure 1 presents a schematic of the energetics.

## 3. Compressible solution

*a*determines the horizontal half-width of the heating function. The perturbation fields of density, potential temperature, and velocity are normalized byrespectively. For an initial pressure perturbation Δ

*p*of 7.38 hPa, these scales are 1 × 10

^{−2}kg m

^{−3}, 3.5 K, and 100 m s

^{−1}, respectively.

### a. Initial response

*t*= 0, with (2.4), implies that the density perturbation immediately after the heating (denoted by a subscript +) is zero:

*ρ*

*t*

_{+}

*ρ*

^{′}

_{+}

*u*

^{′}

_{+}

*υ*

^{′}

_{+}

*w*

^{′}

_{+}

*p*

^{′}

_{+}

*p*

*H*

*z*

*d*

*H*

*z*

*d*

*s*

*x,*

*y*

*a*= 100 km. The horizontal axis is scaled by the Rossby radius of deformation,For the parameters prescribed in this study with

*d*= 5 km, the Rossby radius is 1890 km. The initial potential temperature and pressure fields are confined vertically to the heated layer, and decay horizontally according to (3.1). The vertical structure of the initial potential temperature is the same as that of the warming rate,

*Q,*which is constant.

### b. Steady-state fields

Figure 3 presents the steady-state perturbation pressure, density, potential temperature, and velocity fields for the infinite atmosphere. The initial potential vorticity anomaly (2.8) is created by the heating function (2.3) whose horizontal structure is given by (3.1) with *a* = 100 km. This vertical potential vorticity dipole determines the structure of the steady state. The dipole is characterized by a cyclone/anticyclone located below/above the initially warmed layer. A vertical asymmetry exists in this structure due to the exponential decay of the base-state density. The magnitude of the pressure perturbation associated with this feature decays with horizontal distance from the heating center. The existence of the low-level cyclone implies a net loss of mass in the vertical column near the heating center because the steady state is in hydrostatic balance. Such a feature does not exist in the horizontally homogeneous adjustment problem (Bannon 1995a).

*x,*is derived by integration of the

*y*momentum equation (2.1b) from

*t*= 0

_{+}to

*t*→ ∞. The net vertical displacement, Δ

*z,*is derived by integration of the adiabatic form of the heat equation (2.1d) from

*t*= 0

_{+}to

*t*→ ∞. We findThe total displacement vector is defined as Δ

**s**= (Δ

*x,*Δ

*z*). The

*x*and

*z*components are normalized byNote that the

*x*normalization increases with height like exp(

*z*/

*Hs*). At

*z*= 0 with Δ

*p*= 7.38 hPa the displacements are Δ

*d*

_{x}= 5.05 km and Δ

*d*

_{z}= 348 m.

*x*displacement normalization is inversely proportional to the base-state density, the amplitude of the

*x*component of the normalized displacement vectors is less dependent on

*z*than the scaled velocity field. The horizontal displacement vectors indicate flow away from the top of the heated column and flow toward the bottom of the heated column. Upward vectors exist in the heated column, with weaker compensating subsidence outside of this region, and a weak low-level return flow directed toward the heated column.

The structure of the steady-state fields reflects the net displacements. The positive density anomaly immediately above the heated layer reflects the net upward expansion of the heated layer into the adjacent atmosphere. A corresponding feature is apparent in the potential temperature field. A negative potential temperature perturbation exists immediately above the heated layer, reflecting the net upward advection of lower, base-state values of potential temperature. The positive density anomaly immediately below the heated layer is the result of the return flow at low levels. The geostrophically balanced velocity anomaly at low levels is another indication of this return flow.

### c. Effects of vertical boundaries

Figures 5 and 6 illustrate the effect of vertical boundaries on the steady-state pressure field and net displacements, respectively. In the semi-infinite atmosphere, the negative pressure perturbation at low levels is much stronger and broader than that in the infinite atmosphere (Fig. 5a). This feature corresponds to the enhancement of the low-level inflow in the semi-infinite atmosphere (Fig. 6a). In contrast, the vertical displacement field is diminished.

In the finite atmosphere, the positive pressure perturbation at high levels is much larger and broader than in either the semi-infinite or infinite atmosphere (Fig. 5b). The rigid upper boundary suppresses the net upward expansion of the heated layer but the horizontal displacement vectors are enhanced in the finite atmosphere, relative to the semi-infinite atmosphere. This effect is most pronounced for the lowest horizontal wavenumber modes whose Rossby heights, *H*_{R}, are large.

### d. Effects of heating geometry

The effect of the heating function's aspect ratio on the structure of the steady-state pressure field is presented in Fig. 7 for an infinite atmosphere. (We note that the Rossby radius is a function of heating half-depth and thus the scaling of the horizontal axis is variable for each panel.) We define the aspect ratio as *δ* = *d*/*a,* where *a* is the horizontal decay half-width of the heating function *s* and *d* is its vertical half-depth [defined in Eq. (2.3)]. We present results for aspect ratios *δ* = 50^{−1}, 20^{−1}, 5^{−1}, and 2^{−1}. The locations of the pressure perturbation maxima and minima coincide with the location of maximum and minimum potential vorticity anomalies. A continuous transition in the geometry of the steady-state solution is not observed as the aspect ratio is varied smoothly. Rather, it is more appropriate to consider the transition of the solution geometry as the depth scale is held constant and the horizontal scale is varied, and vice versa. For example, decreasing the heating half-depth while holding *a* constant produces wider and flatter structures with larger amplitude (compare Fig. 7a with Fig. 7c and Fig. 7b with Fig. 7d).

### e. Energetics

*κ*=

*R*/

*c*

_{p}= 28% of the total energy; the remainder is available potential energy. This distribution holds for all wavelengths.

For the steady state, each of the energy terms (AEE, APE, and KE) is calculated directly. The total energy in the final state is less than that in the initial. The difference is energy lost to the acoustic and gravity waves. Figure 8 presents the energetics as a function of horizontal wavelength, *λ* = 2*π*/*k.* We refer to this as the “white noise” spectrum, because each of the steady-state, vertically integrated energy terms in (2.12) is calculated at a particular wavenumber k and is normalized by the total energy residing in that wavenumber initially. This method removes the total energy dependence on heating half-width *a* and is hence identical to the response from a white noise forcing. We note that scales larger than the circumference of the earth in Fig. 8 are an artifact of the analytic method and the use of an *f* plane.

The large-wavelength limit in Fig. 8a gives the same partitioning as that in the 1D, acoustic adjustment problem (Bannon 1995a). This limit represents the case in which the heating is horizontally homogeneous and the adjustment occurs entirely in the vertical. There is no kinetic energy; a fraction *κ* = 28% of the total energy generates waves and the remaining energy is partitioned between the available elastic and potential energies. This large-wavelength limit corresponds to an initial heating with aspect ratio *δ* approaching zero. As *λ* decreases (i.e., as *δ* increases) this fraction of wave energy monotonically increases, but the fraction of energy attributed to elastic and potential energy decreases. The fraction of kinetic energy increases as *λ* decreases and reaches a maximum near *λ* ≈ 7200 km. At smaller length scales, the fraction of elastic energy is negligible. Available elastic energy is only significant at very large (synoptic) scales. For smaller scales, the wave energetics dominate and all the steady-state perturbations rapidly asymptote to zero.

This wavenumber dependence of the energetics has the following interpretation. For a small horizontal-scale heating, the mass adjustment occurs relatively quickly in the horizontal and the effects of Coriolis deflection are negligible. Consequently, all of the steady-state perturbation fields and energies are small. As the horizontal scale of the heating increases into the synoptic range, Coriolis deflection becomes increasingly significant. At these synoptic scales, the steady state is characterized by larger horizontal pressure perturbations, geostrophic velocity, and kinetic energy. Larger quantities of available potential and elastic energy also exist in the steady state due to the restriction of the horizontal mass adjustment by geostrophic motions. As the horizontal scale increases beyond the synoptic range, horizontal accelerations are reduced by the decreasing horizontal pressure gradients of the initial state. At these very large scales, the steady state is characterized by smaller horizontal pressure perturbations, geostrophic velocity, and kinetic energy. The degree to which horizontal mass adjustment occurs continues to decrease, and the steady-state available potential and elastic energy continues to increase.

Figure 8 also describes the effect of vertical boundary conditions on the energetics. At large wavelengths the energetics of the semi-infinite atmosphere (Fig. 8b) is exactly equal to that for the infinite atmosphere. This behavior reflects the fact that the energetics in Lamb's hydrostatic adjustment problem are unaffected by a rigid lower boundary. At smaller wavelengths, the major effect of the boundary is to shift the peak in the kinetic energy to a wavelength of about 5300 km.

The finite-atmosphere energetics are characterized by suppressed wave generation at large wavelengths. The existence of rigid boundaries above and below the heated layer suppresses the vertical expansion of the heated layer and the generation of acoustic waves. The available potential and elastic energies become larger at larger wavelengths, but the kinetic energy maximum is increased and shifted to smaller scales (*λ* ∼ 4600 km).

Table 1 presents the effect of the vertical boundaries and the heating half-depth, *d,* on the amplitude and wavelength of the kinetic energy maximum. The amplitude increases with increasing *d.* Although this increase is monotonic in the infinite atmosphere, it is not monotonic when the heating occurs very close to a boundary. The finite atmosphere produces the largest maximum, independent of *d.* The wavelength of this maximum increases monotonically with increasing *d.* However, the rate of increase is not the same in each atmosphere. The wavelength of the maximum is larger when boundaries are present for small heating half-depth (*d* < 5 km) and is smaller for large heating half-depth (*d* ≥ 5 km).

## 4. Comparison of anelastic, pseudo-incompressible, and modified-compressible approximations

### a. Governing equations

*δ*

_{1},

*δ*

_{2}, and

*γ** as follows:A similar notation was introduced by Durran (1989). Note that in (4.1a),

*γ** is treated as a flag for the anelastic approximation and may not satisfy the typical relationships for the ratio of specific heat capacities. We still denote

*γ*=

*c*

_{p}/

*c*

_{υ}= 1.4 as the ratio of specific heat capacities. For all but the anelastic approximation,

*γ** =

*γ.*In this section, we consider the response of the anelastic, pseudo-incompressible, and modified-compressible atmospheres to the instantaneous heating, (2.3), with horizontal structure (3.1). We apply the infinite atmosphere boundary condition and the initial conditions (2.4).

### b. Initial response

*N*

^{2}

_{s}

*f*

^{2}associated with the vertical decay scale for a particular Fourier mode. This difference represents the effects of the buoyancy adjustment process.

Figure 9b presents a plot of the initial anelastic pressure. Its amplitude is approximately 70% smaller than the compressible one and its structure exhibits a positive pressure perturbation above the heated layer and a small, negative perturbation below the heated layer. These differences reflect the effects of the acoustic adjustment.

Figure 9d presents a plot of the initial anelastic density field, found from (4.1a). Unlike the compressible atmosphere, the anelastic atmosphere produces a nonzero initial density perturbation that is negative in the heated layer, and positive above and below the heated layer. Again this structure is achieved entirely by the acoustic adjustment.

*t*= 0,

*p*

^{′}

_{+}

The pseudo-incompressible initial response is more complicated. In the other cases, we made the implicit assumption that the fields are bounded at *t* = 0. This assumption allows the elimination of terms upon temporal integration about infinitesimal regions. However, this assumption is invalid for the pseudo-incompressible atmosphere because (4.1b) implies that there must be a net divergence during the heating. The solution of the pseudo-incompressible initial response is outlined in Appendix B.

### c. Potential vorticity conservation and steady-state fields

*H**(

*z,*

*d*) =

*H*(

*z*+

*d*) −

*H*(

*z*−

*d*) and

*δ**(

*z,*

*d*) =

*δ*(

*z*+

*d*) −

*δ*(

*z*−

*d*). The final state potential vorticity is identical in the anelastic, pseudo-incompressible, and compressible atmospheres. There is a potential vorticity dipole that conserves potential vorticity globally. For the modified-compressible atmosphere [

*δ*

_{1}= 0,

*δ*

_{2}= 1,

*γ** =

*γ*] there is, in addition to the dipole, a potential vorticity monopole, the

*H** term, that violates the requirement that the potential vorticity be globally conserved (Obukhov 1963).

Figure 10 presents the steady-state pressure perturbation fields for two of the compressibility approximations. The anelastic steady-state pressure perturbation field is not shown, because it is identical to the compressible solution (see Fig. 3a). In addition, the structure of the anelastic steady-state density and velocity perturbation fields is also identical to the compressible solution. However, the anelastic steady-state potential temperature field is, generally, larger than that of the compressible solution (Bannon 1995b). The modified-compressible steady-state pressure perturbation field is characterized by an anomalously weak anticyclone aloft and an anomalously strong, broad, low-level cyclone. Since this solution is equal to the sum of the compressible solution plus a contribution from the potential vorticity monopole, these errors are due to the monopole. The pseudo-incompressible approximation produces a steady-state pressure perturbation field characterized by an anomalously narrow and weak low-level cyclone, and a slightly stronger upper-level anticyclone. Since these errors exhibit very little structure in the horizontal direction, it suggests that they are most significant at very low horizontal wavenumber. Appendix A indicates that this approximation decreases the vertical decay scale associated with any particular horizontal Fourier mode and that this shift is maximized at zero wavenumber. Thus the errors associated with the pseudo-incompressible approximation are confined to low wavenumbers.

### d. Energetics

*δ*

_{1}= 0,

*δ*

_{2}= 0,

*γ** = 1] removes elastic energy from the energy balance and contains no projection of the heating onto AEE. In spite of removing AEE from the balance, the resulting anelastic energy equation is consistent, because (4.11b) reduces to the anelastic continuity equation,The pseudo-incompressible balance [

*δ*

_{1}= 1,

*δ*

_{2}= 0,

*γ** =

*γ*] retains the generation of available elastic energy by the heating but removes its storage. The remaining two terms in (4.11b) must instantly balance the AEE generation. For

*t*> 0, the balance involves only conversion between available potential and kinetic energy, and wave flux divergence. The available elastic energy plays no part in the pseudo-incompressible balance for

*t*> 0. In contrast, the modified-compressible balance [

*δ*

_{1}= 0,

*δ*

_{2}= 1,

*γ** =

*γ*] retains the storage of available elastic energy for

*t*> 0. However, there is no initial generation by the heating of available elastic energy, by (4.5), because there is no initial pressure perturbation in this approximation.

## 5. Discussion

We have presented analytical, initial, and steady-state solutions to a linear hydrostatic and geostrophic adjustment problem in a compressible, stratified atmosphere. An instantaneous heating function with top-hat vertical structure and horizontally decaying spatial structure creates the initial imbalance. The heating produces a pressure, entropy, and potential vorticity disturbance confined to the heated layer, to which the flow fields in turn adjust. Such heating can be used to model a synoptic-scale atmospheric response to an embedded, mesoscale convective cluster, or the linear mesoscale response to an isolated cloud.

The steady-state solutions are obtained from the initial conditions by consideration of potential vorticity conservation. The heating function produces a vertical potential vorticity dipole that uniquely determines the steady state for prescribed vertical boundary conditions. For a warming, the steady state is characterized by an upper-level anticyclone and a low-level cyclone. This low-level, low pressure perturbation in the hydrostatic and geostrophic steady state is a result of a net mass loss in the heated column due to horizontal transport. It is absent from the one-dimensional, hydrostatic adjustment problem (Bannon 1995a).

The amplitudes of the steady-state perturbation fields are dependent on the length scale of the heating. Consider an initial heating with *a* = 100 km, which generates an initial potential temperature anomaly of 10 K. In this case the pressure scale, Δ*p,* is 29.65 hPa. Then the steady-state fields have maximum values of 1.42 hPa, 6.37 K, 0.0086 kg m^{−3}, and 5.8 m s^{−1} for pressure, potential temperature, density, and velocity, respectively. The maximum displacements are 295 m and 58.7 km in the *z* and *x* directions, respectively. These values do not change significantly when the heating half-width is on the synoptic scale (e.g., *a* = 1000 km). However, the steady-state perturbations are significantly smaller when the heating half-width is *a* = 10 km. In this case, the fields have maximum values of 0.22 hPa, 1.0 K, 0.0014 kg m^{−3}, and 0.92 m s^{−1} for pressure, potential temperature, density, and velocity, respectively. The maximum displacements in this case are 47 m and 9.4 km in the *z* and *x* directions, respectively. It is clear that heating produced by a typical mesoscale convective system produces a significant synoptic-scale balanced response. The adjustment to a smaller-scale heating, such as an isolated thunderstorm, is dominated by a wave response.

The effect of vertical boundary conditions has also been considered. The semi-infinite and the finite atmospheres demonstrate several departures from the infinite model. In the infinite atmosphere, the low-level cyclone is narrower and weaker than in the semi-infinite atmosphere. The lower boundary inhibits the vertical expansion of the heated layer while concentrating the strength of the horizontal low-level flow near the boundary. In the finite atmosphere, a rigid lid suppresses the net upward expansion of the heated layer and concentrates the horizontal upper-level flow near the boundary to an even larger extent. Consequently, a smaller fraction of energy is projected onto wave energy in the finite atmosphere. The relative closeness of a boundary can be measured by the ratio of geometric distance to the boundary divided by the Rossby height. Thus the impact of the boundaries is most pronounced at large horizontal wavelengths or when the heating takes place closer to the boundaries. At small horizontal wavelengths, or when the heating takes place sufficiently far away from the boundaries, the solutions are less affected by the choice of boundary condition.

The compressible solution to the hydrostatic and geostrophic adjustment problem has been used to evaluate the anelastic, modified-compressible, and pseudo-incompressible approximations. In the anelastic atmosphere, the initial response to the heating includes an instantaneous acoustic adjustment whose effect is to broaden, weaken, and deepen the initial pressure field relative to the compressible case. In addition, the adjustment produces a rarefaction at the levels of the heating with weak compressions above and below. The steady-state pressure field in the anelastic atmosphere is the same as that for the compressible atmosphere. Then by the hydrostatic and geostrophic relations and the ideal gas law, the density, velocity, and temperature fields are also identical. The anelastic energy conservation equation contains no storage or generation of available elastic energy. In this sense the anelastic approximation is energetically consistent. The steady-state anelastic energetics best model the compressible energetics. Its kinetic energy is identical to the compressible kinetic energy, and its available potential energy contains smaller errors than either the modified-compressible or pseudo-incompressible atmospheres.

The modified-compressible approximation exhibits several shortcomings. Among these is the failure to conserve potential vorticity globally. In this approximation, the heating produces a potential vorticity monopole as well as a dipole. This monopole violates the principle of global potential vorticity conservation and produces an anomalous low pressure field, with an anomalously weak upper-level anticyclone and strong low-level cyclone. In the modified-compressible atmosphere the initial response to the heating occurs at constant pressure. As a consequence, the entropy is manifested as a negative density anomaly. This strong buoyancy forcing would most likely overemphasize the convective nature of the flow evolution. Furthermore, the modified-compressible energetics inconsistently allows for the storage but not the diabatic generation of available elastic energy. The steady-state energetics are characterized by anomalously small kinetic and potential energy. These errors are largest at large wavelengths, where the acoustic adjustment process is most important and where levels of available elastic energy are largest.

The pseudo-incompressible approximation also exhibits several shortcomings. In the pseudo-incompressible atmosphere the initial response to the heating has a complicated structure including a net flow divergence accomplished by a pulse in the velocity field. The pseudo-incompressible steady-state pressure field contains an anomalously weak and narrow low-level cyclone. These errors are primarily confined to large horizontal wavelength structures. The pseudo-incompressible energetics inconsistently allow for the diabatic generation but not the storage of available elastic energy. By removing elastic energy from the time-dependent energetics, some of the elastic energy initially generated is ambiguously projected onto available potential energy. As is the case with the other approximations, these errors are largest at large horizontal wavelengths.

The modification (Almgren 2000) of the pseudo-incompressible equations to include a time-dependent base state with the assumption that the heating occurs at constant pressure does provide the correct solution for the wavenumber zero case, but it is inapplicable to the more general problem of three-dimensional flow presented here.

The scope of the present investigation has focused on the initial and steady-state response to a prescribed heating in a compressible atmosphere. It provides insight into the problem of hydrostatic and geostrophic adjustment in a realistic compressible atmosphere as well as in several approximate model formulations. However, the transient solution is required to complete the analysis. Buoyancy and acoustic waves account for a considerable fraction of the total energy generated by the heating. Solving the time-dependent problem and partitioning the energetics between the various wave modes requires an analytical technique that solves for each individual wave mode explicitly. Such an investigation is in progress.

## Acknowledgments

Partial financial support for this work was provided by the National Science Foundation under NSF Grants ATM-9521299 and ATM-9820233.

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## APPENDIX A

### Green's Functions Solution of the Steady-State Pressure Equation

*B*= (Δ

*p*/

*g*)

*H*

_{s}

*S*(

*k,*

*l*). We apply the principle of superposition such that

*P̂*

_{f}

*g*

_{+d}

*g*

_{−d}

*g*

_{+d}is the solution to (A.1) corresponding to the delta function forcing at

*z*= +

*d,*and

*g*

_{−d}is the solution corresponding to the forcing at

*z*= −

*d.*The solution for the forcing at

*z*= −

*d*satisfiesand has the general formwhere

*a,*

*b,*

*c,*and

*d*are constants with respect to

*z.*Four conditions are required to determine these constants. The first two conditions are provided by vertical boundary conditions in the upper and lower regions of (A.3). The third condition is that the pressure must be continuous at

*z*= −

*d.*The fourth condition follows from an integration of (A.2) about an infinitesimally small region containing

*z*= −

*d.*This fourth condition isA similar approach is used to solve for

*g*

_{+d}.

Table A1 presents the solutions for the various approximations and boundary conditions considered in this paper. In some cases, only *g*_{+d} is given since *g*_{−d} can be retrieved from *g*_{+d} by setting *d* → −*d* and changing the sign of the function.

## APPENDIX B

### Initial Response of a Pseudo-incompressible Atmosphere to an Instantaneous Heating

*y*direction, with initial condition (2.4), and a heating function of the form

_{0}

*δ*

*t*

*δ*

_{1}= 1,

*δ*

_{2}= 0,

*γ** =

*γ*], isEquation (B.2), with (B.1), implies that there must be some delta-function temporal structure in the velocity field. The temporal structure of the fields are generalized as, for example,where

*H*is the Heaviside step function, and

*δ*′ is the derivative of the delta function. The higher-order terms (not shown) correspond to higher-order derivatives of the delta function. This generalization no longer allows us to claim that the initial velocity field is zero using (2.1a)–(2.1c), nor can we immediately retrieve the initial potential temperature perturbation from (2.1d). We derive a diagnostic equation for the pressure field, using (B.2), (2.1a)–(2.1c), and (4.1a). The result isThe nonhomogeneous solution to (B.4) requires the nonhomogeneous solutions for

*υ*and

*θ.*The nonhomogeneous pressure solution includes a delta-prime temporal dependence that satisfiesThe pressure is related to the

*y*momentum through the equationWith the initial condition (2.4),

*p*

_{δ′}implies a step function temporal dependence of

*υ*in (B.6):This step function temporal dependence of

*υ*′ and (2.1b) implies a delta-function temporal dependence of

*u*′,

*υ*

_{H}

*fu*

_{δ}

*θ*. One findsThe step function temporal dependence of

*p*′ may now be found via the diagnostic pressure equation (B.4) with

*θ*

_{H}and

*υ*

_{H}:The step function temporal dependence of

*ρ*′ is found using Poisson's equation (4.1a) with

*γ** =

*γ,*By solving the equations (B.7)–(B.12) sequentially, we can find the initial response corresponding to the instantaneous heating source. An instant after the heating, the particular solution isThus the pseudo-incompressible atmosphere's initial response to an instantaneous heat source involves the generation of instantaneous vertical and horizontal displacements, accomplished by delta-function vertical and horizontal velocity pulses. Corresponding to these velocity pulses is a pressure field with a temporal dependence given by the derivative of a delta function. The result of the instantaneous displacement is the existence of meridional velocity and density perturbations an instant after the heating.

The amplitude and wavelength of the kinetic energy maximum as a function of the heating half-depth, *d,* in the infinite, semi-infinite, and finite compressible atmospheres. The amplitude is expressed as a fraction of the initial energy residing in that wavelength. The wavelength is given in parentheses in units of 10^{3} km. Here *D* = 6 km for the semi-infinite and finite atmospheres