## 1. Introduction

Ogura and Phillips (1962) derived the original anelastic model through systematic formal asymptotics using the flow Mach number as the expansion parameter. Their goal was to derive a set of model equations that would simultaneously represent internal gravity waves and the effects of advection while suppressing any sound modes. Such a model would not only cast the notion that compressibility plays only a subordinate role in the majority of atmospheric flow phenomena in systematic mathematical terms, but it would also lend itself to numerical integration without the necessity of handling the fast yet unimportant acoustic modes through computationally cumbersome numerical means. Such “soundproof” models may also be considered as conceptually important building blocks in a model hierarchy for analyses of fundamental processes of weather and climate (Held 2005).

By design, the soundproof models should be able to address deep atmospheres vertically covering a typical pressure scale height *h*_{sc} ∼ 10 km or more, and nonhydrostatic flow regimes corresponding to horizontal scales down to 10 km or less (cf. Bannon 1996). Thus, the characteristic vertical and horizontal length scales for the design regime of these models are comparable to *h*_{sc}. As can be seen in Table 1, the characteristic acoustic time scale *t*_{ac} is small, on the order of *O*(ε), relative to the advection time *t*_{adv}, whereas the time scale for internal waves, *t*_{int} (i.e., the inverse of the Brunt–Väisälä frequency *N*), is on the order of *O*{ε[(*h*_{sc}/*θ*)*dθ*/*dz*]^{−1/2}}, where *θ* is the potential temperature. As a consequence, if we follow Ogura and Phillips (1962) and construct an asymptotic single time scale model that resolves the advection time scale and includes internal waves at the same time, we would have to adopt a weak stratification so that (*h*_{sc}/*θ*)*dθ*/*dz* = *O*(ε^{2}).

For typical flow Mach numbers of

The presence of multiple scales in the soundproof models is, nevertheless, an issue because both the spatial structures and frequencies of internal waves featured by the soundproof models only approximate those represented by the full compressible flow equations. As a consequence, there are two necessary conditions for the validity of the soundproof models over the targeted advective time scales:

- (a) the spatial structures of corresponding internal wave eigenmodes of the soundproof and compressible systems should be asymptotically close as ε → 0, and
- (b) the accumulation of phase differences between such soundproof and compressible internal waves should remain asymptotically small at least over the advective time scale.

- (i) compare the internal wave eigenmode structures of the compressible Euler equations and selected soundproof models;
- (ii) assess the approximation errors due to “soundproofing” for both the spatial eigenmodes and the associated frequencies in terms of the Mach number; and
- (iii) demonstrate, as our main result, that internal wave solutions of the soundproof and compressible models remain asymptotically close for
*t*=*O*(*t*_{adv}) for sufficiently weak stratification. Specifically, for both Lipps and Hemler’s and Durran’s soundproof models, the corresponding bound on the stratification isThis corresponds to realistic stratifications withover 10–15 km.

The rest of the paper is organized as follows. In section 2 we summarize the model equations to be studied. In section 3 we introduce a new set of variables that explicitly reveal the multiscale nature of fully compressible flows within the regime of stratifications from (1). In section 4, using formal asymptotic analysis and vertical mode decompositions, we compare the vertical internal wave eigenmodes and eigenfrequencies for the pseudoincompressible and the Lipps and Hemler anelastic models with those of the compressible equations and show that they are asymptotically close as long as (*h*_{sc}/*θ*) *dθ*/*dz* = *O*(ε^{μ}) for any *μ* > 0. In that section we also assess the time it takes compressible and soundproof internal waves to accumulate leading-order deviations of their phases because of these differences in the dispersion relations, and this will lead to the abovementioned principal result in (3). In section 5 we draw conclusions and provide an outlook for future work.

## 2. Compressible and soundproof model equations

The exposition in this section of the three sets of model equations to be analyzed subsequently closely follows Klein (2009). Here, we restrict our considerations to flows under gravity, but without Coriolis effects and nonresolved-scale closures, and present consistent dimensionless forms of the compressible Euler equations and of two soundproof models.

### a. Compressible Euler equations

*ρ*,

**v**) are the density and flow velocity,

*P*=

*p*

^{1/γ}=

*ρθ*is a modified thermodynamic pressure variable,

*θ*is potential temperature, and

*π*=

*p*/

^{κ}*κ*, where

*κ*= (

*γ*− 1)/

*γ*, and

*γ*=

*c*/

_{p}*c*is the ratio of the specific heat capacities. Let an asterisk, for the moment, denote dimensional variables; then the dimensionless quantities appearing in (4) are defined aswhere

_{υ}*c*

_{ref}=

*h*

_{sc}=

*p*

_{ref}/

*ρ*

_{ref}

*g*, and where

*p*

_{ref},

*ρ*

_{ref}, and

*g*respectively denote the sea level pressure, the corresponding density at a temperature of, say, 300 K, and the acceleration of gravity.

### b. Pseudoincompressible model

*P*to match a prescribed background distribution

*P*≡

*P*

*z*). Thus, we find

### c. Anelastic model

*ρ*≡

*ρ*

*z*), and by slightly modifying the pressure gradient and gravity terms. With these modifications, we obtainIn all three cases,

*θ*

*z*) is the mean background potential temperature distribution that defines the background pressure variable,

*P*

*z*), and the background density,

*ρ*

*z*), via

*d*

*p*

*dz*= −

*ρ*

*g*,

*p*

*ρ*

*θ*

*P*

*P*

*p*

^{1/γ}. For later reference we note the exact solution,We also note that in the anelastic model (7) the pressure-related quantity

*π̂*is defined asthat is, it is a density-scaled perturbation of the pressure

*p*but not of the Exner pressure

*π*.

## 3. Scaled variables

*θ*′ is the potential temperature perturbation away from a static background distribution,

**v**is the velocity with vertical component

*w*, and

*π*′ is the perturbation Exner pressure. The hydrostatic background variables satisfyThis yields the equivalent advective form of the compressible Euler equations:Now the following transformation of variables will explicitly reveal the asymptotic scalings to be discussed in the sequel. First we introduce a time coordinate nondimensionalized by the characteristic advection timeand then we letThe velocity

**v**was nondimensionalized by

*ν*determines the scaling of the dynamic potential temperature perturbations. Its specific value as given in (14a) implies the correct scaling for internal gravity waves, as we will see shortly. Furthermore,

*π*

*z*) denotes the background Exner pressure distribution given the stratification from (1). We assume a pressure perturbation amplitude on the order of the Mach number,

*O*(ε), so as to not preclude leading-order acoustic modes at this stage.

*θ̃*,

*π̃*,

**ṽ**satisfyThese equations are obtained from a straightforward

*equivalent*transformation of the compressible flow equations in (4) without any asymptotic simplifications.

Besides the tendencies of temporal change, there are three groups of terms in (16): the terms multiplied by ε^{−ν} induce internal waves, the terms multiplied by ε^{−1} represent the acoustic modes, and the terms on the right-hand side cover all nonlinearities. In fact, all terms on the left-hand sides are linear in the unknowns. Notice that all terms on the right are nonsingular as ε → 0; that is, they are *O*(ε^{α}) with *α* ≥ 0. This clean Mach number scaling of acoustic, internal wave, and nonlinear (advective) terms justifies in hindsight the choice *ν* = 1 − *μ*/2 introduced earlier.

*π̂*in the unscaled anelastic model from (7) already denotes a deviation from the background pressure according to (9), whereas we had retained the full dimensionless Exner pressure in writing down the compressible and pseudoincompressible models. Thus, we have replaced (14b) with

*π̂*= ε

*π̃*(

*τ*,

**x**,

*z*; ε) in deriving (18b).

We observe that the potential temperature transport equations are in agreement between all three models. This was to be expected since in the present adiabatic setting this equation reduces to a simple advection equation. The momentum equations of the compressible and pseudoincompressible models are in complete agreement, whereas the anelastic model’s momentum equation lacks the respective last terms on the left and right from (16b) or (17b) that combine to yield ε^{μ−1}(* ^{ν}θ̃*)

**∇**

*π̃*.

^{1}This reduces baroclinic vorticity production in the anelastic model in essence to the effects of horizontal gradients of buoyancy (cf. Smolarkiewicz and Dörnbrack 2008).

The only difference between the compressible Euler equations from (16) and the pseudoincompressible model is found in the Exner pressure evolution equation (16c), which becomes the pseudoincompressible divergence constraint in (17c). The anelastic divergence constraint in (18c) again differs from the pseudoincompressible one through an additional term involving the background potential temperature stratification.

## 4. Internal gravity waves

### a. Gravity wave scaling

*τ*=

*O*(ε); for internal waves,

*τ*=

*O*(ε

^{ν}); and for advection,

*τ*=

*O*(1). In this section we consider solutions that do not feature sound waves but evolve on time scales comparable to the internal wave time scale. The only “sound term”

*O*(ε

^{−1}) in the momentum equation is the one involving the pressure gradient. This term will reduce to

*O*(ε

^{−ν}) and thus induce changes on the internal wave time scale only, provided that the pressure perturbations satisfy

*π̃*= ε

^{1−ν}

*π** with

*π** =

*O*(1). By introducing this additional rescaling of the pressure fluctuations and by adopting an internal wave time coordinate

*ϑ*= ε

^{−ν}

*τ*, the compressible, pseudoincompressible, and anelastic systems can be represented aswith the choices of switching parameters summarized in Table 2.

We observe that in the gravity wave scaling all differences between the compressible model on the one hand and both of the soundproof models on the other hand are *O*(ε^{μ}) or smaller (i.e., at least on the order of the stratification strength). At leading order in ε, all models agree from a formal scaling perspective, although switching off the pressure tendency by letting *A* = 0 fundamentally changes the mathematical type of the equations from strictly hyperbolic to mixed hyperbolic–elliptic. We will demonstrate below through formal asymptotics that this, nevertheless, affects the internal gravity wave solutions only weakly. Between the pseudoincompressible and anelastic systems there is no such singular switch, however, so that their solutions will differ only by *O*(ε^{μ}) at least on internal wave time scales with *ϑ* = *O*(1).

### b. The constraint on the stratification

*O*(ε

^{μ}) in the linearized part on the left and terms

*O*(ε

^{ν}) in the nonlinear part of the equations on the right. This suggests that for

*μ*<

*ν*(i.e., for ε

^{μ}≫ ε

^{ν}), the linearized internal wave eigenmodes and eigenvalues of the three systems differ by

*O*(ε

^{μ}) only, and the nonlinearities represent even higher-order effects. In this setting, we may expect solutions of the three models that start from comparable internal wave initial data to remain close with differences

*O*(ε

^{μ}) over the internal wave time scale with

*ϑ*=

*O*(1). However, we are really interested in flow evolutions over advective time scales with

*τ*= ε

*=*

^{ν}ϑ*O*(1). Over such longer time scales, the expected differences in the internal wave eigenfrequencies

*O*(ε

^{μ}) will accumulate to phase shifts on the order of ε

*=*

^{μ}ϑ*O*(

*τ*ε

^{μ−ν}) =

*O*(ε

^{μ−ν}). As a consequence, the linearized internal wave solutions of the three models should remain asymptotically close even over advective time scales provided that

*d*

*θ*

*dz*=

*O*(ε

^{2/3}), the internal wave dynamics of the compressible, pseudoincompressible, and anelastic models should remain asymptotically close in terms of the flow Mach number

*over advective time scales*. This is a considerable improvement over the Ogura and Phillips’ original condition for the validity of their anelastic model, which requires that

*d*

*θ*

*dz*=

*O*(ε

^{2}). For

Another indication that at the threshold of *μ* = ⅔, the dynamics change nontrivially arises as follows. When *μ* = ⅔ we have *ν* = 1 − *μ*/2 = *μ*, so that the leading nonlinearities on the rhs of (19), which are *O*(ε^{ν}), become comparable to the perturbation terms of the linearized system on the lhs of (19), which are *O*(ε^{μ}). Thus, for *μ* ≤ ⅔ any perturbation analysis of internal waves in compressible flows that go beyond the leading-order solution must necessarily account for nonlinear effects.

Note that there is no noticeable transition or change in the structure of the linear eigenmodes and eigenvalues considered in the next section as *μ* decreases below the threshold of *μ* = ⅔. The importance of this threshold is associated entirely with the more subtle effects just explained.

The present estimates rely on the linearized equations. However, since all three models considered feature the same leading nonlinearities represented by the nonlinear advection of potential temperature and velocity in (19a) and (19b) [see the terms *O*(ε^{ν})], we expect asymptotic agreement of the solutions over advective time scales as long as the fast linearized dynamics do not already lead to leading-order deviations between the model results (i.e., as long as *μ* > ⅔). A mathematically rigorous proof of the validity of the fully nonlinear pseudoincompressible, and possibly the anelastic, models over advective time scales is a work in progress.

### c. Vertical mode decomposition and the Sturm–Liouville eigenvalue problem

*z*= 0 and

*z*=

*H*=

*O*(1), respectively, and seek horizontally traveling waves described byInserting this ansatz into (19), neglecting the nonlinearities, and eliminating

*W*(

*z*),with boundary conditionsHere we have used the following abbreviations:andSee the appendix for details of the derivation, and note that

*θ*

^{B},

*θ*

^{C}are to be read as

*θ*

*B*and

*C*, respectively.

*A*= 0 (i.e., for either the anelastic or the pseudoincompressible model), and for any fixed horizontal wavenumber vector

**, (23) and (24) represent a classical Sturm–Liouville eigenvalue problem, about which the following facts are well known (Zettl 2005):**

*λ*- (i) There is a sequence of eigenvalues and associated eigenfunctions,
, with 0 < Λ _{0}^{0}< Λ_{1}^{0}… , and Λ_{k}^{0}→ ∞ as*k*→ ∞. - (ii) The
form an orthonormal basis of a Hilbert space of functions *f*:[0,*H*] → ℝ with scalar product. Note that the scalar product and, thus, the Hilbert space are independent of the horizontal wavenumber .*λ* - (iii) The vertical mode number
*k*equals the number of zeroes of the associated eigenmodes on the open interval 0 <*z*<*H*(i.e., excluding the boundary points). Thus,*k*= 0 represents the leading, vertically nonoscillatory mode.

**. The only differences in the linearized eigenmodes between the pseudoincompressible and the present anelastic model consist of the scaling factor of**

*λ**θ*

*P*

*ρ*

*W*(

*z*) in (26) and the slightly different way in which the background potential temperature distribution enters the Sturm–Liouville equation. Specifically,Notice that the compressible and pseudoincompressible models share the definition of

*W*as well as that of

*ϕ*

_{BC}.

### d. Asymptotics for the compressible internal wave modes

*A*= 1] is

*nonlinear*in the eigenvalue Λ. Here we construct first-order accurate approximations to the

*weakly*compressible eigenvalues and eigenfunctions, for which

*λ*

^{2}≫ ε

^{μ}/Λ

*c*

^{2}, so that the compressibility term in the denominator of the first term in (23) remains a small perturbation, and we may expand the solution aswhere the

*W*

_{k}^{0}are taken to be the eigenfunctions corresponding to the pseudoincompressible model.

Notice that there is a set of eigenvalues with Λ = 1/*ω*^{2} = *O*(ε^{μ}) that correspond to the system’s high-frequency acoustic modes. Those will not be considered further in this paper.

*W*

_{k}^{1}(

*z*) are then expanded in terms of the leading-order eigenfunction basis,

*O*(1) cancel identically because (Λ

_{k}

^{0},

*W*

_{k}^{0}) already solve the eigenvalue problem for

*A*= 0 and

*B*= 1. At

*O*(ε

^{μ}) we have, letting

*ϕ*

_{BC}≡

*ϕ*for simplicity of notation,whereMultiplying by

*W*

_{k}^{0}, integrating from

*z*= 0 to

*z*=

*H*, and using the orthonormality from item (ii) above as well as the fact that

*W*

_{k}^{0}is the leading-order eigenfunction with eigenvalue Λ

_{k}

^{0}, we find that the left-hand side and the first term on the right cancel each other, whereas the remaining two terms yieldSimilarly we find, after multiplication with

*W*

_{j}^{0}for

*j*≠

*k*and integration,Because of normalization of the eigenfunctions, it turns out that

This determines the first-order perturbations in terms of ε^{μ} from (28) and (29). For a forthcoming companion paper, two of the authors are currently working on a rigorous proof that the remainders are actually *O*(ε^{2μ}) as indicated in (28). A *remark* is in order. If Λ_{j}(*p*, *q*, *r*) is a simple eigenvalue of a Sturm–Liouville operator *L*(*p*, *q*, *r*) on [0, 1]—that is, if there exists a unique eigenfunction *W _{j}* such that

*L*(

*p*,

*q*,

*r*)

*W*= −(

_{j}*pW*′

_{j})′ +

*qW*=

_{j}*r*Λ

*with*

_{j}W_{j}*W*(0) =

_{j}*W*(1) = 0—then Λ

_{j}_{j}(

*p*,

*q*,

*r*) depends analytically on the functions

*p*and

*q*in a neighborhood of the coefficients. The derivative of the eigenvalue Λ

_{j}and eigenvector

*W*are given by the expressions in (32) and (33) (see Kato 1995; Kong and Zettl 1996).

_{j}### e. Examples

*h*

_{sc}=

*p*

_{ref}/

*ρ*

_{ref}

*g*∼ 8.8 km and

*T*

_{ref}= 300 K, the potential temperature distribution from Fig. 1 results, showing a vertical variation of about 40 K over ∼13 km. The maximum relative deviation between

*P*

*p*

^{1/γ}and

*ρ*

*λ*= 0.5, 2.0, 8.0, corresponding to horizontal wavelengths of 110.6, 27.6, and 6.9 km, respectively, the eigenvalues for the compressible and soundproof systems deviate from each other by less than two percent. Figure 2 shows the leading-order relative difference between the Sturm–Liouville eigenvalues for the pseudoincompressible and anelastic models on the one hand and the first-order approximations of the eigenvalues for the compressible model on the other hand. The approximate eigenvalues for the compressible case have been computed here from the first iterate of a Picard iteration in terms of Λ in (23)—that is, from the perturbed regular Sturm–Liouville equationThe resulting Λ

_{k}

^{1}equals the compressible eigenvalue of mode number

*k*up to errors

*O*(ε

^{2μ}) as shown rigorously by two of the authors in a forthcoming paper. The Λ

_{j}

^{1}(

*k*) for

*j*≠

*k*resulting from (35) have no physical meaning.

We observe that the relative deviation of the eigenvalues between the soundproof and compressible cases is surprisingly small in practice. According to our previous analysis, we would expect deviations of the same order of magnitude as the relative vertical variation of the potential temperature, which in the present case is ε^{μ} ∼ 0.1. Yet, the maximum relative deviation between the eigenvalues is less than 0.02 (in modulus) for the cases documented in Fig. 2 for mode number *k* = 0, and it decreases rapidly for larger *k*. The situation is very similar for other horizontal wavenumbers (not shown).

The deviations in the vertical structure functions are similarly small as demonstrated in an exemplary fashion by the differences in the vertical velocity structure functions, ** λ**| = 0.5 and

**| = 8.0 in Fig. 3. We have observed similarly small differences in the structure functions for a range of vertical mode numbers. Note, however, that we have assumed**

*λ**H*= 1.5, so that deep internal modes with characteristic scales much larger than the pressure scale height are excluded. A systematic study of such deep modes as well as much larger horizontal scales is left for future work.

### f. The long-wave limit

**| ≪ 1, as in this case the two terms in the denominator,**

*λ**λ*

^{2}− ε

^{μ}/Λ

*c*

^{2}, could become comparable. That this is not the case becomes clear after multiplication of the entire equation (23) by

*λ*

^{2}and considering the rescaled eigenvalue Λ*(

*λ*) =

*λ*^{2}Λ. The Sturm–Liouville equation for this variable then readsAs

*λ*

^{2}vanishes, the equation approaches a well-defined limit in which second term on the left vanishes asymptotically, and the term ε

^{μ}/Λ*

*c*

^{2}remains a small perturbation in the denominator of the second-derivative term. As a consequence, the long-wave limiting behavior of the original eigenvalues will bewhere Λ*

_{k}(0) is an eigenvalue of the limit problemwith the same rigid-wall boundary conditions. Of course, to correctly capture the behavior of internal wave modes at large horizontal scales we will have to include the Coriolis effect. This is left for future work.

## 5. Conclusions

In this paper we have addressed the formal asymptotics of weakly compressible atmospheric flows involving three asymptotically different time scales for sound, internal waves, and advection. Both the pseudoincompressible and a particular anelastic model yield very good approximations to the linearized internal wave dynamics in a compressible flow for realistic background stratifications and on length scales comparable to the pressure and density scale heights. These soundproof models should be applicable for stratification strengths (*h*_{sc}/*θ**d**θ**dz*) < *O*(ε^{2/3}), where ε is the flow Mach number. This constraint guarantees that the soundproof and compressible internal waves evolve asymptotically closely even over advective time scales. For typical flow Mach numbers *θ**h*_{sc} ∼ 8.8 km for *T*_{ref} = 300 K and that typical tropospheric heights are about 10–15 km, the estimate for the validity of the soundproof models yields realistic potential temperature variations of *δ**θ**h*_{sc}/*θ**d**θ**dz*) = *O*(ε^{2}) and implied unrealistically weak background stratifications.

A number of important open questions remain to be addressed, such as (i) Could either of the soundproof models be also justified even for (*h*_{sc}/*θ**d**θ**dz*) = *O*(1), and if so, what are the pertinent flow regimes when linear as well as nonlinear effects are taken into account? (ii) Is there a mathematically rigorous justification of the present formal asymptotic results? (iii) How does inclusion of Coriolis effects influence the regime of validity of these soundproof models, especially with regard to horizontal scales comparable to synoptic or even planetary distances, and vertical extensions much larger than the pressure scale height? See also the discussions in Davies et al. (2003) and Almgren et al. (2006) in this context.

R. K. thanks the Johns Hopkins University and the U.S. National Center for Atmospheric Research for hosting him during his 2009 sabbatical leave, the Wolfgang Pauli Institute at Wirtschaftsuniversität Wien for their generous hospitality during an intense week of joint research with D. B., Dr. Veerle Ledoux from Gent University for providing the open source Sturm–Liouville eigen-problem solver “MATSLISE,” Deutsche Forschungsgemeinschaft for partial support through the MetStröm Priority Research Program (SPP 1276), and through Grant KL 611/14. U. A. and R. K. both thank the Leibniz-Gemeinschaft (WGL) for partial support within their PAKT program. D. B. acknowledges support from the ANR-08-BLAN-0301-01 project. O. K. and R. K. thank Alexander-von-Humboldt Stiftung for partial support of this work through their Friedrich-Wilhelm-Bessel prize program. P. K. S. acknowledges partial support by the DOE Award DE-FG02-08ER64535.

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# APPENDIX

## Derivation of the Sturm–Liouville Eq. (23)

*B*ε

^{μ}

*θ*

^{B}. Then we haveIntroduce the vertical mode expansion from (22). Then the first two equations in (A1) yieldEliminate

*z*derivative and eliminate

*ω*,where we have usedthe definition of

*π*

*θ*

^{B}≡

*θ*

*A*≠ 0. Realize thatwhich, given

*P*

*κ*

*π*

^{1/γκ}from (8), is obvious for

*C*= 0 and follows from

*θ*

*z*) = 1 + ε

^{μ}

*z*) for

*C*= 1. Letting

*ρ*

_{C}=

*P*

*θ*

^{C},

*ϕ*

_{BC}=

*θ*

^{C}/

*θ*

^{B}

*P*

*ω*

^{2}, we collect (A5)–(A7) to obtain the Sturm–Liouville equation from (23):where

Characteristic inverse time scales.

Switching parameters in Eq. (19).

^{1}

Notice that *μ* − 1 + *ν* = 2(1 − *ν*) − 1 + *ν* = 1 − *ν*.

^{}

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.