Modulation of Internal Gravity Waves in a Multiscale Model for Deep Convection on Mesoscales

Daniel Ruprecht FB Mathematik, Freie Universität Berlin, Berlin, Germany

Search for other papers by Daniel Ruprecht in
Current site
Google Scholar
PubMed
Close
,
Rupert Klein FB Mathematik, Freie Universität Berlin, Berlin, Germany

Search for other papers by Rupert Klein in
Current site
Google Scholar
PubMed
Close
, and
Andrew J. Majda Courant Institute of Mathematical Sciences, New York University, New York, New York

Search for other papers by Andrew J. Majda in
Current site
Google Scholar
PubMed
Close
Full access

Abstract

Starting from the conservation laws for mass, momentum, and energy together with a three-species bulk microphysics model, a model for the interaction of internal gravity waves and deep convective hot towers is derived using multiscale asymptotic techniques. From the leading-order equations, a closed model for the large-scale flow is obtained analytically by applying horizontal averages conditioned on the small-scale hot towers. No closure approximations are required besides adopting the asymptotic limit regime on which the analysis is based. The resulting model is an extension of the anelastic equations linearized about a constant background flow. Moist processes enter through the area fraction of saturated regions and through two additional dynamic equations describing the coupled evolution of the conditionally averaged small-scale vertical velocity and buoyancy. A two-way coupling between the large-scale dynamics and these small-scale quantities is obtained: moisture reduces the effective stability for the large-scale flow, and microscale up- and downdrafts define a large-scale averaged potential temperature source term. In turn, large-scale vertical velocities induce small-scale potential temperature fluctuations due to the discrepancy in effective stability between saturated and nonsaturated regions.

The dispersion relation and group velocity of the system are analyzed and moisture is found to have several effects: (i) it reduces vertical energy transport by waves, (ii) it increases vertical wavenumbers but decreases the slope at which wave packets travel, (iii) it introduces a new lower horizontal cutoff wavenumber in addition to the well-known high wavenumber cutoff, and (iv) moisture can cause critical layers. Numerical examples reveal the effects of moisture on steady-state and time-dependent mountain waves in the present hot-tower regime.

Corresponding author address: Daniel Ruprecht, FB Mathematik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany. Email: ruprecht@zib.de

Abstract

Starting from the conservation laws for mass, momentum, and energy together with a three-species bulk microphysics model, a model for the interaction of internal gravity waves and deep convective hot towers is derived using multiscale asymptotic techniques. From the leading-order equations, a closed model for the large-scale flow is obtained analytically by applying horizontal averages conditioned on the small-scale hot towers. No closure approximations are required besides adopting the asymptotic limit regime on which the analysis is based. The resulting model is an extension of the anelastic equations linearized about a constant background flow. Moist processes enter through the area fraction of saturated regions and through two additional dynamic equations describing the coupled evolution of the conditionally averaged small-scale vertical velocity and buoyancy. A two-way coupling between the large-scale dynamics and these small-scale quantities is obtained: moisture reduces the effective stability for the large-scale flow, and microscale up- and downdrafts define a large-scale averaged potential temperature source term. In turn, large-scale vertical velocities induce small-scale potential temperature fluctuations due to the discrepancy in effective stability between saturated and nonsaturated regions.

The dispersion relation and group velocity of the system are analyzed and moisture is found to have several effects: (i) it reduces vertical energy transport by waves, (ii) it increases vertical wavenumbers but decreases the slope at which wave packets travel, (iii) it introduces a new lower horizontal cutoff wavenumber in addition to the well-known high wavenumber cutoff, and (iv) moisture can cause critical layers. Numerical examples reveal the effects of moisture on steady-state and time-dependent mountain waves in the present hot-tower regime.

Corresponding author address: Daniel Ruprecht, FB Mathematik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany. Email: ruprecht@zib.de

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, hsc ∼ 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 the averaged vertical velocity, and θ the average potential temperature, w′ and θ′ are conditional averages of deviations from w and θ within the deep convective clouds; π is the Exner function, ρ(0) the leading order background density, and θz(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

i1520-0469-67-8-2504-eq1
and
i1520-0469-67-8-2504-e1

2) Averaged tower-scale dynamics

i1520-0469-67-8-2504-eq2
and
i1520-0469-67-8-2504-e2
Moisture affects the large-scale dynamics given by (1) in two ways. It reduces the effective stability of the atmosphere by a factor of 1 − 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 θz(2)w′ in (1c). A positive w′ results in a positive contribution to θ, modeling latent heat release, whereas a negative 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 source term on the rhs of (2b). Finally, for the chosen time and length scales, the mass conservation equation reduces to the anelastic divergence constraint (1d).

Note that, if all moisture related terms vanish—that is, for 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, θz(2) > 0 (see, e.g., Davies et al. 2003):
i1520-0469-67-8-2504-eq3
and
i1520-0469-67-8-2504-e3

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

In the derivation by Klein and Majda (2006), three primary dimensionless parameters occur: the Mach number M, describing the ratio of a typical flow velocity uref to the speed of sound waves; the barotropic Froude number Fr, describing the ratio of a typical flow velocity to the speed of external gravity waves; and the bulk microscale Rossby radius RoB, providing a measure of the importance of rotational effects for flows on the bulk microscale. These parameters are defined as
i1520-0469-67-8-2504-e4
in which lbulk is the length scale of the bulk microphysics, pref and ρref 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:
i1520-0469-67-8-2504-e5
The scaling of a fourth dimensionless parameter, the baroclinic Froude number, will be discussed shortly in the context of (10).
The starting point of the model development is the conservation laws for mass, momentum, and energy combined with a slightly modified version of the bulk microphysics model from Grabowski (1998). The prognostic quantities in the original equations are the horizontal velocity u, vertical velocity w, density ρ, pressure p, potential temperature θ, and the mixing ratios of water vapor qυ, cloud water qc, and rainwater qr. The scales considered in the derivation are a time scale of tτ ≈ 100 s, a vertical length scale equal to hsc ≈ 10 km, and two horizontal scales l ≈ 10 km and lbulk ≈ 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 lref ≈ 10 km and time scale of tref ≈ 1000 s, by powers of ε. The new coordinates resolving the short scales are
i1520-0469-67-8-2504-e6
The model distinguishes between saturated and nonsaturated regions by a switching function Hqυ, defined as
i1520-0469-67-8-2504-e7
in which qυ(0) is the leading-order water vapor mixing ratio and is the leading-order saturation water vapor mixing ratio, computed from Bolton’s formula for the saturation vapor pressure. [See Emanuel (1994) for the original formula and Klein and Majda (2006) for the derivation of an asymptotic expression for qvs.] For the warm microphysics considered here saturated regions and clouds coincide and Hqυ is the leading-order characteristic function for cloudy patches of air, which equals unity inside clouds and zero between them.
We modify the ansatz for the horizontal velocity by introducing a constant background velocity 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
i1520-0469-67-8-2504-e8
All quantities depending on η also depend on τ′. The horizontal velocity is assumed to be independent of the small horizontal coordinate η, so we use an ansatz
i1520-0469-67-8-2504-e9
Although this scaling would formally suggest dimensional values for 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 hsc with a typical velocity of u/uref = O(1) and a background stratification θ, using the distinguished limit (5), are
i1520-0469-67-8-2504-e10
Thus, except for very weak O4) 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 O2) 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−1) scaling of u in (9), (10a) becomes as required, and both time scales are of the same order.
One can also see that if the employed time scale is tτ, an O−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 O2) stratification, tτ−1 ∼ ε−1, while tref ∼ 1 according to (10). Thus, the scaling of Frbaroclinic reads
i1520-0469-67-8-2504-e11
If u = 0, then u/uref = O(1) and Frbaroclinic is O(ε) so that nonhydrostatic effects would not be contained in the leading order equations.
The scale separation revealed in (10) between internal waves for O2) stratifications and advection based on the reference velocity uref 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 hsc provides a reference value for both wavenumbers of
i1520-0469-67-8-2504-e12
Now, according to the dispersion relation for internal waves (see e.g., Bühler 2009), the horizontal phase velocity is
i1520-0469-67-8-2504-e13
Using a typical value for the stability frequency of N = 0.01 s−1, we obtain a dimensional value for the phase velocity of
i1520-0469-67-8-2504-e14
Thus, while an O−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 εactual = 0.1. The reason is that the factor (2π)−1 in (14) is O(1) in the limit ε → 0 but is comparable to εactual = 0.1.

For a reference velocity u*ref = hscN = 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.

The potential temperature is expanded about a background stratification θ(z) = 1 + ϵ2θ(2)(z) as
i1520-0469-67-8-2504-e15
As discussed in Klein and Majda (2006), considering realistic values of convectively available potential energy (CAPE) constrains deviations of θ from a moist adiabat, showing that θ(2) should satisfy the moist adiabatic equation
i1520-0469-67-8-2504-e16
Expansions about a moist adiabatic background are also considered, for example, in Lipps and Hemler (1982). In (16) p(0) is the leading order of the pressure and Γ**, L**, and are 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′.

All quantities ϕ ∈ {u, w, θ, π, qυ, qc, qr} are split below as
i1520-0469-67-8-2504-e17
where
i1520-0469-67-8-2504-e18
denotes the small-scale average with respect to the 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 xz plane here. This simplifies the notation and numerical examples presented below will be of this type, too. However, this is not an essential restriction.

The leading order equations for the averages resulting from this ansatz are
i1520-0469-67-8-2504-eq4
and
i1520-0469-67-8-2504-e19
with π(3) = p(3)/ρ(0) and θz(2)(z) is the potential temperature gradient of the background, Cd(0) the leading order source term from vapor condensing to cloud water or cloud water evaporating, and the leading order source term representing cooling by evaporation of rain in nonsaturated regions. The equations for the perturbations read
i1520-0469-67-8-2504-e20
The leading order equations for the moist species are split into separate equations for the saturated (Hqυ = 1) and the nonsaturated (Hqυ = 0) cases:

1) Saturated

i1520-0469-67-8-2504-e21

2) Nonsaturated

i1520-0469-67-8-2504-e22
Here is again an O(1) scaling factor from the nondimensionalization, and , qυ(0), qc(0), and qr(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 in (19c) will be derived from the perturbation equations (20) and the leading order equations (21) and (22) obtained from the bulk microscale model. We stress that the closure is computed analytically and does not require the introduction of additional physical closure assumptions.

Multiply (21a) by Hqυ and average over η to get
i1520-0469-67-8-2504-e23
Considering (22b) and noting that in saturated regions trivially satisfies the same transport equation, we find
i1520-0469-67-8-2504-e24
Define
i1520-0469-67-8-2504-e25
As is independent of η, we get
i1520-0469-67-8-2504-e26
It will turn out that the dynamics induced by moisture in this model can all be tied to this variable σ. Considering the definition (7) of the switching function Hqυ, we see that
i1520-0469-67-8-2504-e27
So, for a fixed point (x, z, τ), σ is the area fraction of saturated regions on the η scale. Using (25) and (16) we can write (23) as
i1520-0469-67-8-2504-e28
Now, an expression for
i1520-0469-67-8-2504-e29
is required. Multiply (20) by Hqυ and average to get
i1520-0469-67-8-2504-eq5
and
i1520-0469-67-8-2504-e30
where
i1520-0469-67-8-2504-e31
and, using (22a) and Hqυ(Hqυ − 1) = 0,
i1520-0469-67-8-2504-e32
From (21) and (22) one can see that qr(0) is only advected with the background flow on the chosen short time scale. The same holds for qυ(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

The leading order density for a near-homentropic atmosphere in the Newtonian limit (γ → 1) = O(ε) (see Klein and Majda 2006) reads
i1520-0469-67-8-2504-e33
in nondimensional terms. Thus, the anelastic constraint in (1) can be rewritten as
i1520-0469-67-8-2504-e34
Applying a standard plane wave ansatz would lead to a complex-valued dispersion relation because in an atmosphere with decreasing density the amplitude of gravity waves grows with height. However, we do obtain a real valued expression by allowing for a vertically growing amplitude readily in the solution ansatz. Insert
i1520-0469-67-8-2504-e35
with 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
i1520-0469-67-8-2504-e36
and
i1520-0469-67-8-2504-e37
The solution with ωintr = 0 corresponds to a vortical mode while the nonzero solutions are gravity waves. Choosing
i1520-0469-67-8-2504-e38
results in the real-valued dispersion relation
i1520-0469-67-8-2504-e39
For σ = 0, this is equal to the dispersion relation for the pseudoincompressible equations derived in Durran (1989). Equation (39) can be rewritten as
i1520-0469-67-8-2504-e40
Here ωintr is the so-called intrinsic frequency that would be seen by an observer moving with the background flow. Interestingly, for the incompressible case with ρ0 = 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 f2 replaced by σΘz(2).
For the incompressible case, the dispersion relation can be written as a function of the angle α between the direction of the wavenumber vector (k, m) of a wave and the horizontal:
i1520-0469-67-8-2504-e41

b. Group velocity

Taking the derivative of (40) with respect to k and m yields the group velocity:
i1520-0469-67-8-2504-e42
The group velocity is the traveling velocity of packets of waves with close-by wavenumbers. It is identified with the transport of energy. In a dry (σ = 0), incompressible (μ = 0, so no ¼ term) atmosphere, cg simplifies to the well-known expression for the group velocity of internal waves in a stratified fluid (see, e.g., Lighthill 1978):
i1520-0469-67-8-2504-e43
One essential feature of these waves is cg,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.

The ratio of the vertical and horizontal component of the group velocity determines the slope at which a wave packet propagates:
i1520-0469-67-8-2504-e44
Figure 1 shows the angle between a line with slope Δg and the horizontal depending on σ for a flow with = 1 and 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

A simplified but very elucidating class of solutions is those with height-dependent profile but plane wave structure in x and τ. Apply an ansatz,
i1520-0469-67-8-2504-e45
with 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
i1520-0469-67-8-2504-e46
Set μ = ½ as in section 3a so that the final equation reads
i1520-0469-67-8-2504-e47
This is a Taylor–Goldstein equation, which in the incompressible dry case (i.e., with 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
i1520-0469-67-8-2504-e48
Figure 2 shows how the steady-state vertical wavelength λ(k) = 2π/m(k) depends on σ for k = 1, … , 4, constant θz(2) = 1, and u = 0.1. Obviously, moisture reduces the vertical wavelength.

1) Critical layers

Note that, if there is a height zc for which
i1520-0469-67-8-2504-e49
then m(z, k) → ∞ as zzc, indicating a critical layer (see, e.g., Bühler 2009). In the dry case without shear, this only happens if at some height 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, zc depends on 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

For steady-state solutions, the intrinsic frequency ωintr 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:
i1520-0469-67-8-2504-e50
We neglect the ¼ term as this simplifies the following derivation without qualitatively affecting the result. From (50) one can see that for
i1520-0469-67-8-2504-e51
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
i1520-0469-67-8-2504-e52
m also becomes imaginary. So, only horizontal wavenumbers k with
i1520-0469-67-8-2504-e53
are propagating while waves with horizontal wavenumbers outside this range are evanescent. For increasing moisture, 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. Assume a background flow of 10 m s−1, that is, u = 0.1, and a not very moist atmosphere with σ = 0.1. Then the upper cutoff wavenumber is kup = 10 and the lower one is klow = 10 ≈ 3.162. Expressed in dimensional zonal wavelengths, this means that only wavelengths between roughly 6 and 20 km propagate, while larger or smaller wavelengths are evanescent. The maximum wavelength decreases as 1/ so that small values of 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

To assess the amount of condensate released by condensation in a parcel of air, the vertical displacement from its initial position has to be computed. Denote by ξ(x, z, τ) the displacement of the parcel at (x, z) at time τ. For a given vertical velocity field w we have
i1520-0469-67-8-2504-e54
so ξ can be computed for a given w by solving (54).
Consider a parcel at height z0 at time τ = 0. This parcel has a η-scale distribution of water vapor, given by qυ(η, x, z, 0). The air is saturated wherever qυ(η, x, z0, τ) ≥ qvs(z0) 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 δqυ(ξ; x, z0) the condensate released by a parcel, initially located at (x, z0), if it is displaced upward from z0 to z0 + ξ. For a parcel with qυ(η, x, z0) ≥ qvs(z0) for every η, this amount can be approximated by
i1520-0469-67-8-2504-e55
If the parcel is not saturated everywhere, according to (27), σ(x, z) is the horizontal area fraction of saturated small-scale columns and the condensate release can be approximated as
i1520-0469-67-8-2504-e56
This approximation fails to account for small-scale areas that are initially not saturated but reach saturation somewhere on the parcel’s rise from z0 to z0 + ξ, so δqυ 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 qυ(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, , and define σ according to the first-order water vapor distribution qυ(1), σ is no longer passively advected by the background flow. Instead, the equation for σ then contains w, making this modified model nonlinear. The discussion of this extension is the subject of future work.

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 ρ0(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

If θz(2), σ, and u are constant and periodic boundary conditions are assumed in the horizontal direction, analytical solutions of the form
i1520-0469-67-8-2504-e57
with kn = 2πn/l, in which l equals the length of the domain, can be derived, whereas every w(n) is a solution of (47) with k = kn. For constant coefficients, these solutions simply read
i1520-0469-67-8-2504-e58
To avoid energy propagating downward, that is, a negative vertical component of cg, we choose the negative branch in (48) and set B(n) = 0. The coefficient A(n) is determined according to the lower boundary condition:
i1520-0469-67-8-2504-e59
where h describes the topography.

1) Sine-shaped topography

We illustrate the change of the vertical wavenumber and the lower cutoff for a simple sine-shaped topography
i1520-0469-67-8-2504-e60
on a domain [0, 2π] × [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 or 0.01 s−1 and u = 0.1 or 10 m s−1. The lower cutoff wavenumbers are klow = × (1/0.1) ≈ 1.41 and klow = × (1/0.1) ≈ 2.24, respectively. The height of the topography is H = 0.04 or 400 m. Figure 3 shows contour lines of the vertical velocity w for the three different values of σ. The first figure shows the dry solution. The second shows the solution for σ = 0.02 where still klow < 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 < klow, so the excited wave no longer propagates but is evanescent and its amplitude decays exponentially with height.

2) Witch of Agnesi

A more complex case is the Witch of Agnesi topography, exciting modes of all wavenumbers
i1520-0469-67-8-2504-e61
in which H is the height of the hill, L a measure of its length, and xc the center of the domain so that the top of the hill is in the middle of it. Figure 4 shows solutions with Nx = 201 for σ = 0, σ = 0.1, and σ = 0.5 for = 1 and 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 Frbaroclinic [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 Frbaroclinic by moisture leads to a reduction of the propagation angle.

An important mechanism caused by gravity waves is the vertical transfer of horizontal momentum. While for propagating steady-state modes the horizontally averaged vertical flux of momentum1
i1520-0469-67-8-2504-e62
is constant, it vanishes for evanescent modes. Thus, by turning propagating modes into evanescent ones, moisture inhibits the momentum transfer by gravity waves. Table 1 lists the values of the horizontally averaged vertical flux of momentum at the top of the domain for different values of 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.

The simulation is run until τσ = 3 with 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 ττσ, set
i1520-0469-67-8-2504-e63
with sx = 0.25, sz = 0.25, and zc = 0.4. To avoid a sudden introduction of moisture at τ = τσ, set
i1520-0469-67-8-2504-e64
so that at τσ the maximum of 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
i1520-0469-67-8-2504-eq6
at the left boundary and would leave the domain at
i1520-0469-67-8-2504-eq7
The maximum is always located at a height of 4 km.

The simulation uses 300 nodes in the horizontal direction and 75 in the vertical, resulting in a resolution of Δx = Δz = 0.02 or 200 m. The time step is Δτ = 0.05 or 5 s.

The top four panels in Fig. 5 show contour lines of w at different times for a cloud envelope with σmax = 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 σmax = 0.5, and a third simulation where only the value of σmax 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 or 0.01 s−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, placed at the center of the domain with its maximum at (x, z) = (0, 0.5). Figure 7 visualizes the distribution of σ as well as the initial θ. The simulation uses 400 nodes in the horizontal direction, 40 nodes up to 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 Δx = Δz = 250 m. The time step is Δτ = 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.

REFERENCES

  • Barcilon, A., and D. Fitzjarrald, 1985: A nonlinear steady model for moist hydrostatic mountain waves. J. Atmos. Sci., 42 , 5867.

  • Barcilon, A., J. C. Jusem, and P. G. Drazin, 1979: On the two-dimensional, hydrostatic flow of a stream of moist air over a mountain ridge. Geophys. Astrophys. Fluid Dyn., 13 , 125140.

    • Search Google Scholar
    • Export Citation
  • Barcilon, A., J. C. Jusem, and S. Blumsack, 1980: Pseudo-adiabatic flow over a two-dimensional ridge. Geophys. Astrophys. Fluid Dyn., 16 , 1933.

    • Search Google Scholar
    • Export Citation
  • Bretherton, C. S., and P. K. Smolarkiewicz, 1989: Gravity waves, compensating subsidence, and detrainment around cumulus clouds. J. Atmos. Sci., 46 , 740759.

    • Search Google Scholar
    • Export Citation
  • Bühler, O., 2009: Waves and Mean Flows. Cambridge University Press, 341 pp.

  • Chimonas, G., F. Einaudi, and D. P. Lalas, 1980: A wave theory for the onset and initial growth of condensation in the atmosphere. J. Atmos. Sci., 37 , 827845.

    • Search Google Scholar
    • Export Citation
  • Clark, T. L., and R. D. Farley, 1984: Severe downslope windstorm calculations in two and three spatial dimensions using anelastic interactive grid nesting: A possible mechanism for gustiness. J. Atmos. Sci., 41 , 329350.

    • Search Google Scholar
    • Export Citation
  • Davies, T., A. Staniforth, N. Wood, and J. Thuburn, 2003: Validity of anelastic and other equation sets as inferred from normal-mode analysis. Quart. J. Roy. Meteor. Soc., 129 , 27612775.

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., 1989: Improving the anelastic approximation. J. Atmos. Sci., 46 , 14531461.

  • Durran, D. R., 1991: The third-order Adams–Bashforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev., 119 , 702720.

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., and J. B. Klemp, 1983: A compressible model for the simulation of moist mountain waves. Mon. Wea. Rev., 111 , 23412361.

    • Search Google Scholar
    • Export Citation
  • Einaudi, F., and D. P. Lalas, 1973: The propagation of acoustic–gravity waves in a moist atmosphere. J. Atmos. Sci., 30 , 365376.

  • Emanuel, K. A., 1994: Atmospheric Convection. Oxford University Press, 592 pp.

  • Etling, D., 1996: Theoretische Meteorologie. Springer, 354 pp.

  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Grabowski, W. W., 1998: Toward cloud-resolving modeling of large-scale tropical circulations: A simple cloud microphysics parameterization. J. Atmos. Sci., 55 , 32833298.

    • Search Google Scholar
    • Export Citation
  • Jakob, C., and S. A. Klein, 1999: The role of vertically varying cloud fraction in the parameterization of microphysical processes in the ECMWF model. Quart. J. Roy. Meteor. Soc., 125 , 941965.

    • Search Google Scholar
    • Export Citation
  • Joos, H., P. Spichtinger, U. Lohmann, J-F. Gayet, and A. Minikin, 2008: Orographic cirrus in the global climate model ECHAM5. J. Geophys. Res., 113 , D18205. doi:10.1029/2007JD009605.

    • Search Google Scholar
    • Export Citation
  • Jusem, J. C., and A. Barcilon, 1985: Simulation of moist mountain waves with an anelastic model. Geophys. Astrophys. Fluid Dyn., 33 , 259276.

    • Search Google Scholar
    • Export Citation
  • Kim, Y-J., S. D. Eckermann, and H-Y. Chun, 2003: An overview of the past, present and future of gravity-wave drag parameterization for numerical climate and weather prediction models. Atmos.–Ocean, 41 , 6598.

    • Search Google Scholar
    • Export Citation
  • Klein, R., 2009: Asymptotics, structure and integration of sound-proof atmospheric flow equations. Theor. Comput. Fluid Dyn., 23 , 161195.

    • Search Google Scholar
    • Export Citation
  • Klein, R., and A. Majda, 2006: Systematic multiscale models for deep convection on mesoscales. Theor. Comput. Fluid Dyn., 20 , 525551.

    • Search Google Scholar
    • Export Citation
  • Klemp, J. B., and D. K. Lilly, 1978: Numerical simulation of hydrostatic mountain waves. J. Atmos. Sci., 35 , 78107.

  • Lane, T. P., and M. J. Reeder, 2001: Convectively generated gravity waves and their effect on the cloud environment. J. Atmos. Sci., 58 , 24272440.

    • Search Google Scholar
    • Export Citation
  • LeMone, M. A., and E. J. Zipser, 1980: Cumulonimbus vertical velocity events in GATE. Part I: Diameter, intensity, and mass flux. J. Atmos. Sci., 37 , 24442457.

    • Search Google Scholar
    • Export Citation
  • Lighthill, J., 1978: Waves in Fluids. Cambridge University Press, 504 pp.

  • Lindzen, R. S., 1981: Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res., 86 , 97079714.

  • Lipps, F. B., and R. S. Hemler, 1982: A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci., 39 , 21922210.

    • Search Google Scholar
    • Export Citation
  • Majda, A., and R. Klein, 2003: Systematic multiscale models for the tropics. J. Atmos. Sci., 60 , 393408.

  • McLandress, C., 1998: On the importance of gravity waves in the middle atmosphere and their parameterization in general circulation models. J. Atmos. Sol.–Terr. Phys., 60 , 13571383.

    • Search Google Scholar
    • Export Citation
  • Sawyer, J. S., 1959: The introduction of the effects of topography into methods of numerical forecasting. Quart. J. Roy. Meteor. Soc., 85 , 3143.

    • Search Google Scholar
    • Export Citation
  • Stevens, B., 2005: Atmospheric moist convection. Annu. Rev. Earth Planet. Sci., 33 , 605643.

  • Surgi, N., 1989: Systematic errors of the FSU global spectral model. Mon. Wea. Rev., 117 , 17511766.

  • Vater, S., 2005: A new projection method for the zero Froude number shallow water equations. PIK Report 97, Potsdam Institute for Climate Impact Research, 114 pp. [Available online at http://www.pik-potsdam.de/research/publications/pikreports/.files/pr97.pdf].

    • Search Google Scholar
    • Export Citation
  • Vater, S., and R. Klein, 2009: Stability of a Cartesian grid projection method for zero Froude number shallow water flows. Numer. Math., 113 , 123161.

    • Search Google Scholar
    • Export Citation

APPENDIX

Key Steps in the Derivation of the Model

The nondimensional conservation laws for mass, momentum, and energy (expressed as potential temperature) in Klein and Majda (2006) read
i1520-0469-67-8-2504-ea1
where
i1520-0469-67-8-2504-ea2
is the source term related to evaporation and condensation, while θϵ is a given external source of energy (e.g., radiation). Inserting (9) and (15), plus the expansions of ρ, w, and p in Klein and Majda (2006), yields the following leading-order equations:

a. Horizontal momentum

i1520-0469-67-8-2504-ea3
The second equation can be rewritten as
i1520-0469-67-8-2504-ea4
By integrating this equation along a characteristic τ′ + uη = 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
i1520-0469-67-8-2504-ea5
Note that as u is assumed to be constant, there is no term w(0)uz.

b. Vertical momentum

i1520-0469-67-8-2504-ea6
We assume ρ(1) = 0 here and employ again the sublinear growth condition. The last equation then becomes
i1520-0469-67-8-2504-ea7
Assuming that the specific heats cυ and cp are constants and employing the Newtonian limit, expanding the equation of state yields
i1520-0469-67-8-2504-ea8
Using this and the hydrostatic balance for the leading-order density and pressure, one obtains
i1520-0469-67-8-2504-ea9
with π(3): = p(3)/ρ(0).

c. Mass

i1520-0469-67-8-2504-ea10
Sublinear growth yields
i1520-0469-67-8-2504-ea11

d. Potential temperature

i1520-0469-67-8-2504-ea12
Assume that there are no external sources of heat; that is, θϵ = 0. Again, the advective derivative of θ(4) along τ′ − η characteristics vanishes by sublinear growth condition
i1520-0469-67-8-2504-ea13
From the water vapor transport equation in Klein and Majda (2006) we get
i1520-0469-67-8-2504-ea14
with δq: = qvsqυ. So we can distinguish the regime of saturated air, where the saturation deficit δqvs is nonzero only at higher orders, and the nonsaturated regime with and qc(0) = 0 (i.e., zero leading order cloud water mixing ratio).

e. Saturated air

i1520-0469-67-8-2504-ea15
Again, by using sublinear growth condition, the equations simplify to
i1520-0469-67-8-2504-ea16

f. Nonsaturated air

i1520-0469-67-8-2504-ea17
Sublinear growth yields
i1520-0469-67-8-2504-ea18

Fig. 1.
Fig. 1.

Angle between the direction of group velocity and the horizontal for wavenumbers k = 1, … , 4 depending on σ in a steady-state flow with = 1 or 0.01 s−1 and u = 0.1 or 10 m s−1.

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

Fig. 2.
Fig. 2.

Vertical wavelength for modes k = 1, … , 4 depending on σ for = 1 or 0.01 s−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

Fig. 3.
Fig. 3.

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

Fig. 4.
Fig. 4.

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

Fig. 5.
Fig. 5.

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

Fig. 6.
Fig. 6.

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

Fig. 7.
Fig. 7.

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; the contour interval is −0.025, with the outer line corresponding to θ = −0.025.

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

Fig. 8.
Fig. 8.

Cross section of w at different times for the cloudy and noncloudy case. The dotted line is the cross section through 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

Table 1.

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

Table 1.

1

For once here, the bar over uw denotes the horizontal average in x and not in η.

Save
  • Barcilon, A., and D. Fitzjarrald, 1985: A nonlinear steady model for moist hydrostatic mountain waves. J. Atmos. Sci., 42 , 5867.

  • Barcilon, A., J. C. Jusem, and P. G. Drazin, 1979: On the two-dimensional, hydrostatic flow of a stream of moist air over a mountain ridge. Geophys. Astrophys. Fluid Dyn., 13 , 125140.

    • Search Google Scholar
    • Export Citation
  • Barcilon, A., J. C. Jusem, and S. Blumsack, 1980: Pseudo-adiabatic flow over a two-dimensional ridge. Geophys. Astrophys. Fluid Dyn., 16 , 1933.

    • Search Google Scholar
    • Export Citation
  • Bretherton, C. S., and P. K. Smolarkiewicz, 1989: Gravity waves, compensating subsidence, and detrainment around cumulus clouds. J. Atmos. Sci., 46 , 740759.

    • Search Google Scholar
    • Export Citation
  • Bühler, O., 2009: Waves and Mean Flows. Cambridge University Press, 341 pp.

  • Chimonas, G., F. Einaudi, and D. P. Lalas, 1980: A wave theory for the onset and initial growth of condensation in the atmosphere. J. Atmos. Sci., 37 , 827845.

    • Search Google Scholar
    • Export Citation
  • Clark, T. L., and R. D. Farley, 1984: Severe downslope windstorm calculations in two and three spatial dimensions using anelastic interactive grid nesting: A possible mechanism for gustiness. J. Atmos. Sci., 41 , 329350.

    • Search Google Scholar
    • Export Citation
  • Davies, T., A. Staniforth, N. Wood, and J. Thuburn, 2003: Validity of anelastic and other equation sets as inferred from normal-mode analysis. Quart. J. Roy. Meteor. Soc., 129 , 27612775.

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., 1989: Improving the anelastic approximation. J. Atmos. Sci., 46 , 14531461.

  • Durran, D. R., 1991: The third-order Adams–Bashforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev., 119 , 702720.

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., and J. B. Klemp, 1983: A compressible model for the simulation of moist mountain waves. Mon. Wea. Rev., 111 , 23412361.

    • Search Google Scholar
    • Export Citation
  • Einaudi, F., and D. P. Lalas, 1973: The propagation of acoustic–gravity waves in a moist atmosphere. J. Atmos. Sci., 30 , 365376.

  • Emanuel, K. A., 1994: Atmospheric Convection. Oxford University Press, 592 pp.

  • Etling, D., 1996: Theoretische Meteorologie. Springer, 354 pp.

  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Grabowski, W. W., 1998: Toward cloud-resolving modeling of large-scale tropical circulations: A simple cloud microphysics parameterization. J. Atmos. Sci., 55 , 32833298.

    • Search Google Scholar
    • Export Citation
  • Jakob, C., and S. A. Klein, 1999: The role of vertically varying cloud fraction in the parameterization of microphysical processes in the ECMWF model. Quart. J. Roy. Meteor. Soc., 125 , 941965.

    • Search Google Scholar
    • Export Citation
  • Joos, H., P. Spichtinger, U. Lohmann, J-F. Gayet, and A. Minikin, 2008: Orographic cirrus in the global climate model ECHAM5. J. Geophys. Res., 113 , D18205. doi:10.1029/2007JD009605.

    • Search Google Scholar
    • Export Citation
  • Jusem, J. C., and A. Barcilon, 1985: Simulation of moist mountain waves with an anelastic model. Geophys. Astrophys. Fluid Dyn., 33 , 259276.

    • Search Google Scholar
    • Export Citation
  • Kim, Y-J., S. D. Eckermann, and H-Y. Chun, 2003: An overview of the past, present and future of gravity-wave drag parameterization for numerical climate and weather prediction models. Atmos.–Ocean, 41 , 6598.

    • Search Google Scholar
    • Export Citation
  • Klein, R., 2009: Asymptotics, structure and integration of sound-proof atmospheric flow equations. Theor. Comput. Fluid Dyn., 23 , 161195.

    • Search Google Scholar
    • Export Citation
  • Klein, R., and A. Majda, 2006: Systematic multiscale models for deep convection on mesoscales. Theor. Comput. Fluid Dyn., 20 , 525551.

    • Search Google Scholar
    • Export Citation
  • Klemp, J. B., and D. K. Lilly, 1978: Numerical simulation of hydrostatic mountain waves. J. Atmos. Sci., 35 , 78107.

  • Lane, T. P., and M. J. Reeder, 2001: Convectively generated gravity waves and their effect on the cloud environment. J. Atmos. Sci., 58 , 24272440.

    • Search Google Scholar
    • Export Citation
  • LeMone, M. A., and E. J. Zipser, 1980: Cumulonimbus vertical velocity events in GATE. Part I: Diameter, intensity, and mass flux. J. Atmos. Sci., 37 , 24442457.

    • Search Google Scholar
    • Export Citation
  • Lighthill, J., 1978: Waves in Fluids. Cambridge University Press, 504 pp.

  • Lindzen, R. S., 1981: Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res., 86 , 97079714.

  • Lipps, F. B., and R. S. Hemler, 1982: A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci., 39 , 21922210.

    • Search Google Scholar
    • Export Citation
  • Majda, A., and R. Klein, 2003: Systematic multiscale models for the tropics. J. Atmos. Sci., 60 , 393408.

  • McLandress, C., 1998: On the importance of gravity waves in the middle atmosphere and their parameterization in general circulation models. J. Atmos. Sol.–Terr. Phys., 60 , 13571383.

    • Search Google Scholar
    • Export Citation
  • Sawyer, J. S., 1959: The introduction of the effects of topography into methods of numerical forecasting. Quart. J. Roy. Meteor. Soc., 85 , 3143.

    • Search Google Scholar
    • Export Citation
  • Stevens, B., 2005: Atmospheric moist convection. Annu. Rev. Earth Planet. Sci., 33 , 605643.

  • Surgi, N., 1989: Systematic errors of the FSU global spectral model. Mon. Wea. Rev., 117 , 17511766.

  • Vater, S., 2005: A new projection method for the zero Froude number shallow water equations. PIK Report 97, Potsdam Institute for Climate Impact Research, 114 pp. [Available online at http://www.pik-potsdam.de/research/publications/pikreports/.files/pr97.pdf].

    • Search Google Scholar
    • Export Citation
  • Vater, S., and R. Klein, 2009: Stability of a Cartesian grid projection method for zero Froude number shallow water flows. Numer. Math., 113 , 123161.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Angle between the direction of group velocity and the horizontal for wavenumbers k = 1, … , 4 depending on σ in a steady-state flow with = 1 or 0.01 s−1 and u = 0.1 or 10 m s−1.

  • Fig. 2.

    Vertical wavelength for modes k = 1, … , 4 depending on σ for = 1 or 0.01 s−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.

  • Fig. 3.

    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.

  • Fig. 4.

    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.

  • Fig. 5.

    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.

  • Fig. 6.

    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.

  • Fig. 7.

    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; the contour interval is −0.025, with the outer line corresponding to θ = −0.025.

  • Fig. 8.

    Cross section of w at different times for the cloudy and noncloudy case. The dotted line is the cross section through 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.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 180 56 3
PDF Downloads 108 46 9