## 1. Introduction

Internal gravity waves are one prominent feature of atmospheric flows on length scales from ∼10 to 100 km and are responsible for a number of important effects. As Bretherton and Smolarkiewicz (1989) and Lane and Reeder (2001) show, convecting clouds emit gravity waves that alter their environment, rendering it favorable for further convection by reducing convective inhibition (CIN). Chimonas et al. (1980) investigated a feedback mechanism between saturated regions and gravity waves that can trigger convection. They hypothesize that gravity waves contribute to the organization of individual convective events into larger-scale structures like squall lines. Vertically propagating gravity waves are associated with vertical transport of horizontal momentum. The dissipation of these waves in the stratosphere exerts a net force on middle atmospheric flows known as gravity wave drag (GWD) (see, e.g., Sawyer 1959; Lindzen 1981). McLandress (1998) demonstrates the necessity of including the effects of GWD in general circulation models (GCMs) to obtain realistic flows. Joos et al. (2008) find that gravity waves are also important for the parameterization of cirrus clouds. Because GCMs have a spatial resolution of 100–200 km, gravity waves cannot be resolved in these models and their effects have to be parameterized. Kim et al. (2003) provide an overview of concepts of GWD parameterizations in GCMs.

Moisture in the atmosphere significantly affects the propagation of internal waves. Barcilon et al. (1979) propose a model for steady, hydrostatic flow over a mountain with reversible moist dynamics. This model distinguishes between saturated and nonsaturated regions by a switching function that depends on the vertical displacement of a parcel: if the parcel is displaced beyond the lifting condensation level (LCL), it is treated as saturated and the dry stability frequency is replaced by the reduced moist stability frequency. Barcilon et al. (1980) extend the model to nonhydrostatic flows with irreversible condensation, and Barcilon and Fitzjarrald (1985) to nonlinear, steady flow. These authors find that moisture can significantly reduce the mountain drag, which is closely related to the wave drag. Jusem and Barcilon (1985) employ a nonlinear, nonsteady, nonhydrostatic anelastic model that explicitly includes the mixing ratios of liquid water and vapor to define heating source terms for the potential temperature. Besides finding again that moisture can reduce drag, they also find that moisture does reduce the wave intensity and increases the vertical wavelength. While the first result is also found in the present paper, instead of an increased vertical wavelength we observe an increase of the vertical wavenumber by moisture, corresponding to a reduced vertical wavelength.

Durran and Klemp (1983) employ a fully compressible model combined with prognostic equations for water vapor, rainwater, and cloud water to simulate moist mountain waves. They also find that moisture reduces the vertical flux of momentum and the amplitude of the generated wave patterns. Further, they observe an increase in vertical wavelength for nearly hydrostatic waves. Attenuation of gravity waves by moisture and an increase of vertical wavelength are also found in the analysis of wave propagation in a fully saturated atmosphere in Einaudi and Lalas (1973).

Although there is extensive literature dealing with the parameterization of drag from convectively generated waves, there are very few attempts to include the effect of moisture in parameterizations of orographic waves. In their review, Kim et al. (2003) mention only the work of Surgi (1989), investigating the introduction of a stability frequency modified by moisture into the orographic GWD parameterization.

Klein and Majda (2006) derive a multiscale model for the interaction of nonhydrostatic internal gravity waves with moist deep convective towers from the conservation laws of mass, momentum, and energy combined with a classical bulk microphysics scheme. In agreement with the regime of nonrotating, nonhydrostatic gravity waves described by Gill (1982, chapter 8), the characteristic horizontal and vertical scales for the gravity waves are assumed comparable to the pressure scale height, *h*_{sc} ∼ 10 km. LeMone and Zipser (1980) provide an indication for the characteristic horizontal scales of the narrow deep convective towers: they analyze data obtained during the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) and find that the median diameter of “convective events” related to tropical cumulonimbus clouds is about 900 m (see also Stevens 2005). Thus, a horizontal “microscale” of 1 km is used as the second horizontal length scale in the multiscale ansatz to describe the tower-scale dynamics. The assumed time scale of 100 s is compatible with the typical value of 0.01 s^{−1} for the stability frequency in the troposphere.

By using an asymptotic ansatz representing these scales, this paper presents the derivation of a reduced model for modulation of internal waves by moisture. For a first reading, one can study the summary of the model below in section 1, skip the technical derivation in section 2, and go immediately to the new phenomena and application of the model in sections 3 and 4. The reduced model allows for an analytical investigation, concisely revealing a number of mechanisms by which moisture affects internal waves. It confirms already known facts but also allows one to hypothesize new effects that might be important for improved parameterizations of internal waves. Further, the model equations themselves might serve as a starting point for the development of such parameterizations.

The asymptotic ansatz used here is the one introduced by Klein and Majda (2006), with the slight modification of adding a constant horizontal background flow. Using weighted averages, the obtained leading order multiscale equations are converted into a closed system of equations for the gravity wave scale only. In this system, the effective vertical mass fluxes are obtained analytically so that no additional physical closure assumptions beyond those made in adopting the asymptotic scaling regime are required. The resulting equations are an extension of the anelastic equations, linearized around a moist adiabatic, constant wind background flow. Mathematical analysis of these equations reveals, among other effects, that moisture introduces a lower horizontal cutoff wavenumber, the existence of which is, to the authors’ knowledge, a new hypothesis. The essential moisture-related parameter in the model is the area fraction of saturated regions on the microscale, reminiscent of a smaller-scale version of the “cloud cover fraction,” a parameter routinely computed and used in GCMs. Jakob and Klein (1999) discuss this parameter in the context of microphysical parameterizations in the ECMWF model. They find that a uniform value for cloud cover over one cell is not sufficient and divide the cell into a number of subcolumns to approximately represent a spatially inhomogeneous distribution of cloud cover inside a cell. Trying to link the saturated area fraction arising in the present model to such decompositions of cloud cover might be a promising ansatz to include moisture effects in GWD parameterizations in a systematic way.

### a. Summary of the model

The new model for gravity wave–convective tower interactions consists of equations describing linearized, anelastic moist dynamics plus two equations for the conditionally averaged tower-scale dynamics. Here *u* is the horizontal velocity, *w**θ**w*′ and *θ*′ are conditional averages of deviations from *w**θ**π* is the Exner function, *ρ*^{(0)} the leading order background density, and *θ _{z}*

^{(2)}the background stratification, while

*u*

^{∞}is a constant horizontal background velocity. The source term

*C*

_{−}is a constant cooling term related to evaporating rain in nonsaturated areas, and

*σ*is the saturated area fraction mentioned earlier. See section 2 for details.

#### 1) Linearized anelastic moist dynamics

#### 2) Averaged tower-scale dynamics

*σ*, representing the effect that, if a parcel rises and starts condensing water, the release of latent heat will effectively reduce the restoring force that the parcel experiences. Because of the short time scale in this model, the only conversion mechanism between moist quantities that has a leading order effect is evaporation of cloud water into vapor and condensation of vapor into cloud water in fully saturated regions. As a consequence,

*σ*itself does not change with time in the present model except for being advected by the mean flow. In fact, for the scalings assumed in the present derivation, the physical effect of nonsaturated rising parcels eventually becoming saturated when lifted sufficiently is not present. An extension of the model to capture this effect is the subject of current work and beyond the scope of the present paper. See also our comments in section 3e.

The release and consumption of latent heat by averaged small-scale up- and downdrafts in saturated areas is described by the source term *θ _{z}*

^{(2)}

*w*′ in (1c). A positive

*w*′ results in a positive contribution to

*θ*

*w*′ models latent heat consumption. However, the microscale model not only provides the source term for the large-scale dynamics, it is also affected by them in return through the

*w*

*σ*= 0,

*C*

_{−}= 0 and

*w*′(

*τ*= 0) =

*θ*′(

*τ*= 0) = 0—the system in (1) reduces to the anelastic equations linearized around a constant-wind background flow with stable stratification,

*θ*

_{z}^{(2)}> 0 (see, e.g., Davies et al. 2003):

## 2. Derivation of the model

This section provides the derivation of the model given by (1a–d) and (2). The present modification of the original asymptotic regime considered by Klein and Majda (2006) is developed in section 2a together with a justification for the particular scaling of the constant horizontal background wind velocity. Section 2b describes the closure of the leading order equations by weighted averages.

### a. Multiple scales ansatz

*M*, describing the ratio of a typical flow velocity

*u*

_{ref}to the speed of sound waves; the barotropic Froude number

_{B}, providing a measure of the importance of rotational effects for flows on the bulk microscale. These parameters are defined as

*l*

_{bulk}is the length scale of the bulk microphysics,

*p*

_{ref}and

*ρ*

_{ref}are typical values for pressure and density, Ω is the rate of earth’s rotation, and

*g*the gravity acceleration. Following Majda and Klein (2003), these parameters are related to a universal expansion parameter ε in the following distinguished limit:

*baroclinic*Froude number, will be discussed shortly in the context of (10).

*u*, vertical velocity

*w*, density

*ρ*, pressure

*p*, potential temperature

*θ*, and the mixing ratios of water vapor

*q*, cloud water

_{υ}*q*, and rainwater

_{c}*q*. The scales considered in the derivation are a time scale of

_{r}*t*≈ 100 s, a vertical length scale equal to

_{τ}*h*

_{sc}≈ 10 km, and two horizontal scales

*l*≈ 10 km and

*l*

_{bulk}≈ 1000 m. As discussed in the introduction, these scales correspond to a combination of the regimes of nonhydrostatic gravity waves and deep convective towers. To resolve them, new coordinates are introduced by rescaling the “universal” coordinates

*x*and

*t*, which resolve the reference length scale of

*l*

_{ref}≈ 10 km and time scale of

*t*

_{ref}≈ 1000 s, by powers of ε. The new coordinates resolving the short scales are

*H*, defined as

_{qυ}*q*

_{υ}^{(0)}is the leading-order water vapor mixing ratio and

*q*

_{vs}.] For the warm microphysics considered here saturated regions and clouds coincide and

*H*is the leading-order characteristic function for cloudy patches of air, which equals unity inside clouds and zero between them.

_{qυ}*u*

^{∞}. To avoid inconsistencies in the derivation, we also add a second coordinate

*τ*′ corresponding to the time scale set by advection of flows with

*u*

^{∞}velocity over the short, tower-scale distances resolved by the

*η*coordinate. The terms related to

*τ*′ will eventually drop out by sublinear growth conditions and do not appear in the final model. In terms of ε the new coordinate is

*η*also depend on

*τ*′. The horizontal velocity is assumed to be independent of the small horizontal coordinate

*η*, so we use an ansatz

*u*

^{∞}of ∼100 m s

^{−1}, a value of

*u*

^{∞}= 0.1, corresponding to 10 m s

^{−1}, will be used throughout this paper. The reason for this apparent inconsistency between the asymptotic scaling of

*u*

^{∞}and the actual value used for it is that, as shown by Klein (2009), the inverse time scales of advection and internal waves for flows on a length scale

*h*

_{sc}with a typical velocity of

*u*/

*u*

_{ref}=

*O*(1) and a background stratification

*θ*

*O*(ε

^{4}) stratifications, the advection time scale and the time scale of internal waves are asymptotically separated in the limit ε → 0. To retain both effects, that is, advection and internal gravity waves, in the leading order equations for an

*O*(ε

^{2}) stratification as will be used here according to (15), the inverse advection time scale has to be on the order of ε

^{−1}. For the

*O*(ε

^{−1}) scaling of

*u*

^{∞}in (9), (10a) becomes

*t*, an

_{τ}*O*(ε

^{−1}) scaling of

*u*is necessary to address the nonhydrostatic regime by analyzing the scaling of the baroclinic Froude number (11). It denotes the ratio between advection and wave speed and indicates the importance of nonhydrostatic effects. For an

*O*(ε

^{2}) stratification,

*t*

_{τ}^{−1}∼ ε

^{−1}, while

*t*

_{ref}∼ 1 according to (10). Thus, the scaling of Fr

_{baroclinic}reads

*u*

^{∞}= 0, then

*u*/

*u*

_{ref}=

*O*(1) and Fr

_{baroclinic}is

*O*(ε) so that nonhydrostatic effects would not be contained in the leading order equations.

*O*(ε

^{2}) stratifications and advection based on the reference velocity

*u*

_{ref}in the limit ε → 0 is, however, obscured for finite values of ε ≈ 0.1, which are typical for realistic atmospheric flows: let

*k*and

*m*denote the horizontal and vertical wavenumber of an internal wave. For nonhydrostatic waves, these are of comparable magnitude. The length scale

*h*

_{sc}provides a reference value for both wavenumbers of

*N*= 0.01 s

^{−1}, we obtain a dimensional value for the phase velocity of

*O*(ε

^{−1}) scaling of

*u*

^{∞}is required to retain advection in the limit ε → 0, a value of

*u*

^{∞}= 0.1, corresponding to dimensional values of about 10 m s

^{−1}, agrees very well with the time scale of internal waves actually obtained with realistic values of ε

_{actual}= 0.1. The reason is that the factor (

*π*)

^{−1}in (14) is

*O*(1) in the limit ε → 0 but is comparable to ε

_{actual}= 0.1.

For a reference velocity *u**_{ref} = *h*_{sc}*N* = 100 m s^{−1}, no separation of the internal wave time scale and the advection time scale occurs. In principle, an equivalent derivation can be conducted if the governing equations are nondimensionalized using *u**_{ref}. This changes the distinguished limit (5) and the expansions of the horizontal and vertical velocity, avoiding an ε^{−1} scaling of the leading order *u*^{∞}. The small parameter then is the amplitude of wave-induced perturbations of the velocity field. The justification for setting *u*^{∞} = 0.1 is required in this derivation, too.

*θ*

*z*) = 1 +

*ϵ*

^{2}

*θ*

^{(2)}(

*z*) as

*θ*from a moist adiabat, showing that

*θ*

^{(2)}should satisfy the moist adiabatic equation

*p*

^{(0)}is the leading order of the pressure and Γ**,

*L***, and

*O*(1) scaling factors arising from the nondimensionalization in Klein and Majda (2006).

The expansions of all other dependent variables are adopted from Klein and Majda (2006), except that variables depending on *η* in their derivation now also depend on *τ*′.

*ϕ*∈ {

*u*,

*w*,

*θ*,

*π*,

*q*,

_{υ}*q*,

_{c}*q*} are split below as

_{r}*η*coordinate, while

*ϕ̃*denotes deviations from this average.

The descriptions of the scalings and the ansatz are basically a repetition of what is done in Klein and Majda (2006), so the reader is referred to the original work for a detailed discussion. The focus here is the derivation of a closed model from the resulting leading order equations and an analysis of the model’s properties. The derivation is presented in an *x*–*z* plane here. This simplifies the notation and numerical examples presented below will be of this type, too. However, this is not an essential restriction.

*π*

^{(3)}=

*p*

^{(3)}/

*ρ*

^{(0)}and

*θ*

_{z}^{(2)}(z) is the potential temperature gradient of the background,

*C*

_{d}^{(0)}the leading order source term from vapor condensing to cloud water or cloud water evaporating, and

*H*= 1) and the nonsaturated (

_{qυ}*H*= 0) cases:

_{qυ}#### 1) Saturated

#### 2) Nonsaturated

*O*(1) scaling factor from the nondimensionalization, and

*q*

_{υ}^{(0)},

*q*

_{c}^{(0)}, and

*q*

_{r}^{(0)}are the leading order mixing ratios of the saturation water vapor, water vapor, cloud water, and rainwater. Key steps of the derivation can be found in the appendix.

### b. Computing the mass flux closure

To obtain a closed set of equations, an equation for

*H*and average over

_{qυ}*η*to get

*η*, we get

*σ*. Considering the definition (7) of the switching function

*H*, we see that

_{qυ}*x*,

*z*,

*τ*),

*σ*is the area fraction of saturated regions on the

*η*scale. Using (25) and (16) we can write (23) as

*H*and average to get

_{qυ}*H*(

_{qυ}*H*− 1) = 0,

_{qυ}*q*

_{r}^{(0)}is only advected with the background flow on the chosen short time scale. The same holds for

*q*

_{υ}^{(0)}so that the evaporation source term

*C*

_{−}is also only advected and can thus be computed once at the beginning of a simulation and then be obtained by suitable horizontal translations. Combining (28), (32), and (30) with (19) yields the final model, (1) and (2).

## 3. Analytical properties of the model

In this section we point out some analytical properties of the model. The dispersion relation and group velocity is computed, and we find that moisture reduces the absolute value of the group velocity and changes its direction. For solutions with a plane-wave structure in the horizontal and in time, a Taylor–Goldstein equation for the vertical profiles is derived, revealing that moisture introduces a lower cutoff horizontal wavenumber and may cause critical layers. A way to assess the amount of released condensate is sketched, and a possible extension of the presented model to include nonlinear effects from dynamically changing area fractions *σ* will be explained briefly.

### a. Dispersion relation

*γ*→ 1) =

*O*(ε) (see Klein and Majda 2006) reads

*ϕ*∈ {

*u*

*w*

*θ*

*π*,

*w*′,

*θ*′}, into (1) and (2) and assume, for the purpose of this section,

*C*

_{−}= 0, that is, the absence of source terms from evaporating rain, and that

*σ*is uniform in

*x*. By successive elimination of

*ϕ̂*, we are left with roots

*ω*

_{intr}= 0 corresponds to a vortical mode while the nonzero solutions are gravity waves. Choosing

*σ*= 0, this is equal to the dispersion relation for the pseudoincompressible equations derived in Durran (1989). Equation (39) can be rewritten as

*ω*

_{intr}is the so-called intrinsic frequency that would be seen by an observer moving with the background flow. Interestingly, for the incompressible case with

*ρ*

_{0}= const, in which the ¼ term vanishes, the formula in (40) is equal to the dispersion relation for internal gravity waves in a rotating fluid (see, e.g., Gill 1982) but with the Coriolis parameter

*f*

^{2}replaced by

*σ*Θ

_{z}

^{(2)}.

*α*between the direction of the wavenumber vector (

*k*,

*m*) of a wave and the horizontal:

### b. Group velocity

*k*and

*m*yields the group velocity:

*σ*= 0), incompressible (

*μ*= 0, so no ¼ term) atmosphere,

**c**

_{g}simplifies to the well-known expression for the group velocity of internal waves in a stratified fluid (see, e.g., Lighthill 1978):

**c**

_{g,dry,inc}⊥ (

*k*,

*m*); that is, the direction in which these waves transport energy is perpendicular to their phase direction. Because of the ¼ term, this no longer holds for (42), but waves with upward directed phase—that is, either positive

*m*and positive branch in (40) and (42) or negative

*m*and negative branch in (40) and (42)—still have a downward-directed group velocity and vice versa.

With increasing *σ*, the coefficient in (42) decreases and eventually, for *σ* = 1, vanishes. Thus, moisture reduces the transport of energy by waves and in completely saturated large-scale regions there is no energy transport by waves at all, only advection of energy by the background flow.

_{g}and the horizontal depending on

*σ*for a flow with

*u*

^{∞}= 0.1 or, dimensionally,

*N*= 0.01 s

^{−1}and

*u*

_{∞}= 10 m s

^{−1}. For all modes, moisture decreases the angle of the group velocity, so we expect the angle between the propagation direction of wave packets and the horizontal to decrease with increasing

*σ*. This is demonstrated in the stationary solutions shown in section 4a(2).

### c. Taylor–Goldstein equation

*x*and

*τ*. Apply an ansatz,

*c*=

*ω*/

*k*being the horizontal phase speed observed at a fixed height

*z*and

*ϕ*∈ {

*u*

*w*

*θ*

*π*,

*w*′,

*θ*′}. The additional factor with parameter

*μ*, as in the derivation of the dispersion relation, describes the amplitude growth caused by the decreasing density in the anelastic model. Inserting (45) into (1) and (2) and eliminating all

*ϕ*

^{(k)}except for

*w*

^{(k)}yields

*μ*= ½ as in section 3a so that the final equation reads

*σ*= 0 and without the ¼ term) becomes the well-known equation for dry internal gravity waves (see, e.g., Etling 1996). The coefficient in (47) is the square of the local vertical wavenumber

*λ*(

*k*) = 2

*π*/

*m*(

*k*) depends on

*σ*for

*k*= 1, … , 4, constant

*θ*

_{z}^{(2)}= 1, and

*u*

^{∞}= 0.1. Obviously, moisture reduces the vertical wavelength.

#### 1) Critical layers

*z*for which

_{c}*m*(

*z*,

*k*) → ∞ as

*z*→

*z*, indicating a critical layer (see, e.g., Bühler 2009). In the dry case without shear, this only happens if at some height

_{c}*c*is equal to

*u*

^{∞}. In the moist case, critical layers also arise from the vertical profile of

*σ*so that noncritical dry flows can develop critical layers if moisture is added. Also,

*z*depends on

_{c}*k*in that case. A detailed investigation of the local structure of solutions in the presence of critical layers will not be presented here but will be the subject of future work.

### d. Cutoff wavenumbers

*ω*

_{intr}is zero and the dispersion relation (39) can be rewritten to express the vertical wavenumber

*m*as a function of the horizontal wavenumber

*k*only:

*m*becomes imaginary. Thus, there is an upper limit of the horizontal wavenumber up to which internal waves actually propagate. Different from the dry case, moisture also introduces a lower cutoff; as for

*m*also becomes imaginary. So, only horizontal wavenumbers

*k*with

*σ*gets closer to unity and the range of propagating wavenumbers narrows. For

*σ*= 1, the only propagating mode left is

*k*=

*u*

^{∞}.

A typical value for the stability frequency in dimensional terms is 0.01 s^{−1}, corresponding to ^{−1}, that is, *u*^{∞} = 0.1, and a not very moist atmosphere with *σ* = 0.1. Then the upper cutoff wavenumber is *k*_{up} = 10 and the lower one is *k*_{low} = 10*σ* corresponding to small amounts of moisture can already filter a significant range of wavelengths. For *σ* = 0.2, the maximum wavelength is 14 km and is further reduced to about 8 km for *σ* = 0.5. This low-wavenumber cutoff is especially interesting in the context of GWD parameterizations, as it primarily affects near-hydrostatic modes with long horizontal wavelengths, which are the most important ones in terms of GWD.

### e. Release of condensate

*ξ*(

*x*,

*z*,

*τ*) the displacement of the parcel at (

*x*,

*z*) at time

*τ*. For a given vertical velocity field

*w*

*ξ*can be computed for a given

*w*

*z*

_{0}at time

*τ*= 0. This parcel has a

*η*-scale distribution of water vapor, given by

*q*(

_{υ}*η*,

*x*,

*z*, 0). The air is saturated wherever

*q*(

_{υ}*η*,

*x*,

*z*

_{0},

*τ*) ≥

*q*

_{vs}(

*z*

_{0}) and condensation will take place if the parcel is displaced upward, so the amount of water vapor in the parcel is reduced according to the decrease of saturation water vapor mixing ratio. Denote by

*δq*(

_{υ}*ξ*;

*x*,

*z*

_{0}) the condensate released by a parcel, initially located at (

*x*,

*z*

_{0}), if it is displaced upward from

*z*

_{0}to

*z*

_{0}+

*ξ*. For a parcel with

*q*(

_{υ}*η*,

*x*,

*z*

_{0}) ≥

*q*

_{vs}(

*z*

_{0}) for every

*η*, this amount can be approximated by

*σ*(

*x*,

*z*) is the horizontal area fraction of saturated small-scale columns and the condensate release can be approximated as

*z*

_{0}to

*z*

_{0}+

*ξ*, so

*δq*is more like a lower bound for the condensate release. However, as our linear model is only valid for small displacements anyhow, (56) will be a decent approximation for the actual condensate release except for peculiar distributions of

_{υ}*q*(

_{υ}*η*) with large nonsaturated regions that are very close to saturation.

There is an interesting possible extension of the model emerging from this derivation: if we assume leading order saturation everywhere from the start, that is, *σ* according to the first-order water vapor distribution *q _{υ}*

^{(1)},

*σ*is no longer passively advected by the background flow. Instead, the equation for

*σ*then contains

*w*

## 4. Stationary and nonstationary solutions

A projection method is used to solve the full system (1), (2) numerically. It consists of a predictor step, advancing the equations in time ignoring the divergence constraint and the pressure gradient. In a second step, the predicted velocity field is projected onto the space of vector fields satisfying the anelastic constrain by applying the “correct” pressure gradient, obtained by solving a Poisson problem at each time step. The predictor step uses a third-order Adams–Bashforth scheme in time together with a fourth-order central difference scheme for the advective terms. The application of this scheme to advection problems was investigated in Durran (1991) and found to be a viable alternative to the commonly used leapfrog scheme. To solve the Poisson problem occurring in the projection step, we use the discretization described in Vater (2005) and Vater and Klein (2009) with slight modifications to account for the density stratification *ρ*_{0}(*z*).

In section 4a, visualizations of analytical stationary solutions for the case with constant coefficients are shown. The effect of the lower cutoff wavenumber is demonstrated as well as the damping of wave amplitudes by moisture and the reduced angle of propagation. In sections 4b and 4c, nonstationary, numerically computed, approximate solutions are shown. First, the effects of a cloud envelope being advected through a mountain wave pattern are discussed, and then waves initiated by a perturbation in potential temperature and traveling through a pair of clouds are investigated. A reduction of momentum flux by moisture is demonstrated for stationary and nonstationary mountain waves.

The code solves the nondimensionalized equations, but for a more descriptive presentation all quantities have been converted back into dimensional units in the figures.

### a. Steady-state solutions for a uniform atmosphere

*θ*

_{z}^{(2)},

*σ*, and

*u*

^{∞}are constant and periodic boundary conditions are assumed in the horizontal direction, analytical solutions of the form

*k*= 2

_{n}*πn*/

*l*, in which

*l*equals the length of the domain, can be derived, whereas every

*w*

^{(n)}is a solution of (47) with

*k*=

*k*. For constant coefficients, these solutions simply read

_{n}**c**

_{g}, we choose the negative branch in (48) and set

*B*

^{(n)}= 0. The coefficient

*A*

^{(n)}is determined according to the lower boundary condition:

*h*describes the topography.

#### 1) Sine-shaped topography

*π*] × [0, 1] or [0, 62.8] km × [0, 10] km, where only a single mode with

*k*= 2 is excited. Values for

*σ*are

*σ*= 0,

*σ*= 0.02, and

*σ*= 0.05. The stratification is

^{−1}and

*u*

^{∞}= 0.1 or 10 m s

^{−1}. The lower cutoff wavenumbers are

*k*

_{low}=

*k*

_{low}=

*H*= 0.04 or 400 m. Figure 3 shows contour lines of the vertical velocity

*w*

*σ*. The first figure shows the dry solution. The second shows the solution for

*σ*= 0.02 where still

*k*

_{low}<

*k*and the excited wave is propagating. Compared to the dry case, the direction of propagation is slightly tilted to the vertical, corresponding to the increase of the vertical wavenumber

*m*(

*k*) with increasing

*σ*. In the last figure with

*σ*= 0.05, the solution has completely changed. Now

*k*<

*k*

_{low}, so the excited wave no longer propagates but is evanescent and its amplitude decays exponentially with height.

#### 2) Witch of Agnesi

*H*is the height of the hill,

*L*a measure of its length, and

*x*the center of the domain so that the top of the hill is in the middle of it. Figure 4 shows solutions with

_{c}*N*= 201 for

_{x}*σ*= 0,

*σ*= 0.1, and

*σ*= 0.5 for

*u*

^{∞}= 0.1 or 0.01 s

^{−1}and 10 m s

^{−1}. The computational domain is [0, 8] × [0, 1] or [0, 80] km × [0, 10] km but the solution is plotted only on [2, 6] × [0, 1]. Set

*H*= 0.04 or 400 m and

*L*= 0.1 or 1000 m.

The dashed lines visualize the average over the slopes (44) of all propagating modes, where each mode is weighted by its amplitude. With increasing *σ*, the slope of the wave pattern is reduced according to the reduced slope of the group velocity pointed out in 3b. This is compatible with the following consideration: by decreasing the stability *N* of the atmosphere, moisture increases Fr_{baroclinic} [cf. (11)], which is small for waves close to hydrostatic balance. As hydrostatic waves propagate only in the vertical and the horizontal component increases as waves become more nonhydrostatic, it seems reasonable that an increase of Fr_{baroclinic} by moisture leads to a reduction of the propagation angle.

^{1}

*σ*.

### b. Mountain waves disturbed by a moving cloud envelope

In this subsection, we demonstrate the effect of a cloud packet being advected through an established mountain wave pattern, so the fully time-dependent problem, (1) and (2), is solved numerically. The domain is [−3, 3] × [0, 1.5] or [−30, 30] km × [0, 15] km in dimensional terms. To realize a transparent upper boundary condition, a Rayleigh damping layer, as described in Klemp and Lilly (1978), reaching from *z* = 1 to *z* = 1.5 is used. Subsequent figures show the solution on the subset [−2, 2] × [0, 1]. The topography is a Witch of Agnesi hill with *H* = 0.04 or 400 m and *L* = 0.1 or 1000 m and the maximum is located at *x* = 0. The background flow is linearly increased from *τ* = 0 to *τ* = 0.25 up to its maximum value *u*^{∞} = 0.1 or 10 m s^{−1}.

*τ*= 3 with

_{σ}*σ*≡ 0. Then, a cloud packet described by a Gaussian-distributed

*σ*, defining its local intensity and envelope, is introduced and advected with velocity

*u*

^{∞}, crossing the domain and finally exiting at the right boundary. For

*τ*≥

*τ*, set

_{σ}*s*= 0.25,

_{x}*s*= 0.25, and

_{z}*z*= 0.4. To avoid a sudden introduction of moisture at

_{c}*τ*=

*τ*, set

_{σ}*τ*the maximum of

_{σ}*σ*is outside the actual domain of computation at

*x*= −4. The cloud is then advected with velocity

*u*

^{∞}= 0.1 into the domain so that its maximum enters at

The simulation uses 300 nodes in the horizontal direction and 75 in the vertical, resulting in a resolution of Δ*x* = Δ*z* = 0.02 or 200 m. The time step is Δ*τ* = 0.05 or 5 s.

The top four panels in Fig. 5 show contour lines of *w**σ*_{max} = 0.5. In the first panel, the cloud pattern has entered the domain from the left. The middle three panels show how the cloud envelope travels through the mountain waves, while the last shows a dry reference solution at *τ* = 50 for comparison. Considering the third and fourth panels and comparing them with the dry solution in the last panel, we see how the cloud packet has damped the waves in the upper region. Also, in the fourth panel a decreased propagation angle is observed.

Figure 6 shows the net vertical flux of horizontal momentum (62) over time for the dry case, the case with *σ*_{max} = 0.5, and a third simulation where only the value of *σ*_{max} has been changed to 0.2. The negative momentum flux increases toward zero as the cloud envelope travels through the wave pattern; that is, its magnitude is reduced as in the analysis of the stationary solutions.

### c. Waves traveling through clouds

The domain is [−5, 5] × [0, 1.25] or [−50, 50] km × [0, 12.5] km and there is no background flow here; that is, *u*^{∞} = 0. The stability frequency is ^{−1}. Between −30 to −10 km and 10 to 30 km, two cloud packets are located.

All initial values are zero except for a concentrated Gaussian peak of negative *θ**x*, *z*) = (0, 0.5). Figure 7 visualizes the distribution of *σ* as well as the initial *θ**z* = 10 km, and 10 more nodes to realize the sponge layer between *z* = 10 km and *z* = 12.5 km. The resulting resolution is Δ*x* = Δ*z* = 250 m. The time step is Δ*τ* = 0.1 or 10 s. For comparison, a reference solution is computed with identical parameters but *σ* ≡ 0.

The initial potential temperature perturbation starts to excite waves, which form a typical X-shaped pattern (not shown, see, e.g., Clark and Farley 1984). Figure 8 shows a cross section through the vertical velocity at 5 km of the cloudy case (continuous line) as well as the noncloudy reference simulation (dashed line). In the first figure, waves have formed and started traveling outward. Inside the cloud, the updraft from the entering wave is amplified by latent heat release. Because the wave is also slowed down inside the cloud, there is some steepening before the cloud. In the next figure the steepening before and the amplification inside the cloud are even more pronounced. In the last figure one can see the newly formed extrema and a generally strongly distorted distribution of vertical velocity inside the cloud. In the region behind the cloud, the amplitude of the wave is noticeably reduced compared to the noncloudy case.

## 5. Conclusions

The paper presents the derivation and analysis of a model for modulation of internal waves by deep convection. In the analysis, the dispersion relation, group velocity, and Taylor–Goldstein equation of the extended model are computed. Moisture, represented by the saturated area fraction *σ*, introduces multiple effects compared to the dry dynamics: by altering the group velocity, it inhibits wave propagation and changes the propagation direction of wave packets. It introduces a lower cutoff horizontal wavenumber below which modes turn from propagating into evanescent. This is in contrast to the dispersion properties of waves in a nonrotating dry atmosphere for which only an upper cutoff wavenumber exists.

The lower cutoff leads to a moisture-related reduction of the vertical flux of horizontal momentum. As gravity wave drag (GWD) is closely related to momentum flux, including this effect in parameterizations of GWD could improve simulations because near-hydrostatic modes with small horizontal wavenumber, which are primarily affected by the cutoff, significantly contribute to wave drag. We also note that moisture can cause critical layers for flows that would be noncritical under dry conditions.

Examples of stationary solutions obtained analytically demonstrate the cutoff, the reduced angle of propagation of wave packets, and the reduced momentum flux. The examples include stationary solutions for mountain waves excited from sine-shaped and Witch of Agnesi topographies. The nonstationary results show how a cloud packet, represented as a Gaussian-distributed *σ*, is advected through Witch of Agnesi mountain waves. A significant damping of the waves by the cloud pattern is observed, and the reduction of momentum flux is documented. The second example begins with a small perturbation of potential temperature between two clouds so that the excited waves travel through them. Inside the clouds, we observe an amplification of the amplitudes of wave-induced up- and downdrafts, while beyond the clouds wave amplitudes are damped.

The derivation starts from the results in Klein and Majda (2006) and yields a model for the interaction of nonhydrostatic internal gravity waves with a time scale of 100 s and a length scale of 10 km and convective hot towers with horizontal variations on a 1-km scale. An analytically computed closure of the model is achieved without requiring additional approximations by applying weighted averages over the small 1-km horizontal scale. In the final model, moisture is present only as a parameter *σ* that describes the area fraction of saturated regions on the tower scale.

The resulting model involves anelastic, moist, large-scale dynamics described by an extension of the linearized dry anelastic equations. The equations include a source term for the potential temperature determined by two additional equations for the averaged dynamics on the small tower scale. The averaged equations for the small scales, in turn, include terms from the large-scale dynamics so that there is bidirectional coupling between large- and tower-scale flow components in the model.

The presented model provides some interesting possibilities for future research. The similarity of the saturated area fraction *σ* to the cloud cover fraction in GCMs might make the model a good starting point for the development of GWD parameterizations that include moist effects. Also of interest with respect to GWD parameterizations would be an attempt to validate the hypothesized lower horizontal cutoff wavenumber in a model employing a full bulk microphysics scheme. An extension of the model to the case of weak undersaturation, in which *σ* will turn into a prognostic variable and the model will become nonlinear, is work in progress.

## Acknowledgments

This research is partially funded by Deutsche Forschungsgemeinschaft, Project KL 611/14, and by the DFG Priority Research Programs “PQP” (SSP 1167) and “MetStroem” (SPP 1276). We thank Oliver Bühler, Peter Spichtinger, and Stefan Vater for helpful discussions and suggestions. We also thank the three anonymous reviewers for their very helpful comments on the initial version of the manuscript.

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## APPENDIX

### Key Steps in the Derivation of the Model

*S̃*is a given external source of energy (e.g., radiation). Inserting (9) and (15), plus the expansions of

_{θ}^{ϵ}*ρ*,

*w*, and

*p*in Klein and Majda (2006), yields the following leading-order equations:

#### a. Horizontal momentum

*τ*′ +

*u*

^{∞}

*η*= const and employing a sublinear growth condition for the higher-order quantities

*u*

^{(1)}and

*p*

^{(4)}, we conclude that the right-hand side must be zero and the equation simplifies to

*u*

^{∞}is assumed to be constant, there is no term

*w*

^{(0)}

*u*

_{z}^{∞}.

#### b. Vertical momentum

*ρ*

^{(1)}= 0 here and employ again the sublinear growth condition. The last equation then becomes

*c*and

_{υ}*c*are constants and employing the Newtonian limit, expanding the equation of state yields

_{p}*π*

^{(3)}: =

*p*

^{(3)}/

*ρ*

^{(0)}.

#### c. Mass

#### d. Potential temperature

*S̃*= 0. Again, the advective derivative of

_{θ}^{ϵ}*θ*

^{(4)}along

*τ*′ −

*η*characteristics vanishes by sublinear growth condition

*δq*: =

*q*

_{vs}−

*q*. So we can distinguish the regime of saturated air, where the saturation deficit

_{υ}*δq*

_{vs}is nonzero only at higher orders, and the nonsaturated regime with

*q*

_{c}^{(0)}= 0 (i.e., zero leading order cloud water mixing ratio).

#### e. Saturated air

#### f. Nonsaturated air

Vertical wavelength for modes *k* = 1, … , 4 depending on *σ* for ^{−1} and *u*^{∞} = 0.1 or 10 m s^{−1}. The dimensional horizontal wavelengths corresponding to *k* = 1, … , 4 are approximately 63, 31, 21, and 16 km. Values of *σ* where *λ*(*k*, *σ*) = 0 indicate critical layers.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Vertical wavelength for modes *k* = 1, … , 4 depending on *σ* for ^{−1} and *u*^{∞} = 0.1 or 10 m s^{−1}. The dimensional horizontal wavelengths corresponding to *k* = 1, … , 4 are approximately 63, 31, 21, and 16 km. Values of *σ* where *λ*(*k*, *σ*) = 0 indicate critical layers.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Vertical wavelength for modes *k* = 1, … , 4 depending on *σ* for ^{−1} and *u*^{∞} = 0.1 or 10 m s^{−1}. The dimensional horizontal wavelengths corresponding to *k* = 1, … , 4 are approximately 63, 31, 21, and 16 km. Values of *σ* where *λ*(*k*, *σ*) = 0 indicate critical layers.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Contour lines of the steady-state vertical velocity for a sine-shaped topography with *k* = 2 for (top) *σ* = 0, (middle) *σ* = 0.02, and (bottom) *σ* = 0.05. The interval between contours is 0.2 m s^{−1}; dotted contours represent negative values.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Contour lines of the steady-state vertical velocity for a sine-shaped topography with *k* = 2 for (top) *σ* = 0, (middle) *σ* = 0.02, and (bottom) *σ* = 0.05. The interval between contours is 0.2 m s^{−1}; dotted contours represent negative values.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Contour lines of the steady-state vertical velocity for a sine-shaped topography with *k* = 2 for (top) *σ* = 0, (middle) *σ* = 0.02, and (bottom) *σ* = 0.05. The interval between contours is 0.2 m s^{−1}; dotted contours represent negative values.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Contour lines of steady-state vertical velocity for a Witch of Agnesi topography with *H* = 400 m and *L* = 1000 m for (top) *σ* = 0, (middle) *σ* = 0.1, and (bottom) *σ* = 0.5. The interval between contours is 0.25 m s^{−1}; dotted contours represent negative values. The dashed lines visualize the averaged slope of the group velocity of all propagating modes.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Contour lines of steady-state vertical velocity for a Witch of Agnesi topography with *H* = 400 m and *L* = 1000 m for (top) *σ* = 0, (middle) *σ* = 0.1, and (bottom) *σ* = 0.5. The interval between contours is 0.25 m s^{−1}; dotted contours represent negative values. The dashed lines visualize the averaged slope of the group velocity of all propagating modes.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Contour lines of steady-state vertical velocity for a Witch of Agnesi topography with *H* = 400 m and *L* = 1000 m for (top) *σ* = 0, (middle) *σ* = 0.1, and (bottom) *σ* = 0.5. The interval between contours is 0.25 m s^{−1}; dotted contours represent negative values. The dashed lines visualize the averaged slope of the group velocity of all propagating modes.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Contour lines of vertical velocity at different times for a cloud packet advected through the waves excited by a Witch of Agnesi. Interval between contours is 0.25 m s^{−1} in dimensional terms; dotted contours represent negative values. The two thin circles are the *σ* = 0.05 and *σ* = 0.25 isolines. The last figure shows a reference solution with *σ* ≡ 0 at *τ* = 50 for comparison.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Contour lines of vertical velocity at different times for a cloud packet advected through the waves excited by a Witch of Agnesi. Interval between contours is 0.25 m s^{−1} in dimensional terms; dotted contours represent negative values. The two thin circles are the *σ* = 0.05 and *σ* = 0.25 isolines. The last figure shows a reference solution with *σ* ≡ 0 at *τ* = 50 for comparison.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Contour lines of vertical velocity at different times for a cloud packet advected through the waves excited by a Witch of Agnesi. Interval between contours is 0.25 m s^{−1} in dimensional terms; dotted contours represent negative values. The two thin circles are the *σ* = 0.05 and *σ* = 0.25 isolines. The last figure shows a reference solution with *σ* ≡ 0 at *τ* = 50 for comparison.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Net vertical flux of horizontal momentum across *z* = 10 km over time. The solid line is the dry reference simulation, the dash line is a moving cloud envelope with max(*σ*) = 0.2, and the dash–dotted line represents a cloud envelope with max(*σ*) = 0.5.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Net vertical flux of horizontal momentum across *z* = 10 km over time. The solid line is the dry reference simulation, the dash line is a moving cloud envelope with max(*σ*) = 0.2, and the dash–dotted line represents a cloud envelope with max(*σ*) = 0.5.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Net vertical flux of horizontal momentum across *z* = 10 km over time. The solid line is the dry reference simulation, the dash line is a moving cloud envelope with max(*σ*) = 0.2, and the dash–dotted line represents a cloud envelope with max(*σ*) = 0.5.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

The saturation fraction modeling two clouds. The dashed line is the cross section along which the vertical velocity is plotted in Fig. 8. The solid lines are isolines of *σ* with an interval of 0.1 and the outer isoline corresponding to *σ* = 0.1. The dotted line shows the initial distribution of *θ**θ*

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

The saturation fraction modeling two clouds. The dashed line is the cross section along which the vertical velocity is plotted in Fig. 8. The solid lines are isolines of *σ* with an interval of 0.1 and the outer isoline corresponding to *σ* = 0.1. The dotted line shows the initial distribution of *θ**θ*

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

The saturation fraction modeling two clouds. The dashed line is the cross section along which the vertical velocity is plotted in Fig. 8. The solid lines are isolines of *σ* with an interval of 0.1 and the outer isoline corresponding to *σ* = 0.1. The dotted line shows the initial distribution of *θ**θ*

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Cross section of *w**σ* at the same height, but multiplied by a factor 0.05 so that the shape is reasonably visible in the given scaling of the *y* axis.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Cross section of *w**σ* at the same height, but multiplied by a factor 0.05 so that the shape is reasonably visible in the given scaling of the *y* axis.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Cross section of *w**σ* at the same height, but multiplied by a factor 0.05 so that the shape is reasonably visible in the given scaling of the *y* axis.

Citation: Journal of the Atmospheric Sciences 67, 8; 10.1175/2010JAS3269.1

Vertical flux of horizontal momentum for different values of *σ* in the constant coefficient, steady-state solution.

^{1}

For once here, the bar over *u**w**x* and not in *η*.