## 1. Introduction

Atmospheric gravity waves generated in the lee of mountains extend over scales across which the background may change significantly. The wave field can persist throughout the layers from the troposphere to the deep atmosphere, the mesosphere and beyond (Fritts et al. 2016, 2018). On this range background temperature and therefore stratification and background density may undergo several orders of magnitude in variation. Also, dynamic viscosity and thermal conductivity cannot be considered constant on such a domain (Pitteway and Hines 1963; Zhou 1995).

Two predominant regimes for the waves can be identified: the homosphere and the heterosphere (Nicolet 1954). These two are separated by the turbopause, usually somewhere in the mesopause region. Below the turbopause molecular viscosity is negligible. Hence, if diffusion occurs, it is caused by turbulence. Due to the missing damping and the thinning background air, mountain waves amplify exponentially when extending upward. This phenomenon is also called anelastic amplification.

At certain heights the amplitudes cannot grow any further due to limiting processes. Those may be static or dynamic instabilities that act on the small scale comparable to the wavelength. For instance, Klostermeyer (1991) showed that all inviscid nonlinear Boussinesq waves are prone to parametric instabilities. The waves do not immediately disappear by the small-scale instabilities, rather the perturbations grow comparably slowly such that the waves persist in their overall structure over several more wavelengths. However, turbulence is produced. Lindzen (1970) modeled the effect of turbulence on the wave by harmonic damping with a constant kinematic eddy viscosity. The eddy viscosity is exactly such that it saturates the wave, meaning that viscous damping and anelastic amplification are balanced (Lindzen 1981; Fritts 1984; Dunkerton 1989; Becker 2012). Pitteway and Hines (1963) referred to this instance as amplitude-balanced wave.

Above the turbopause molecular viscosity takes over. It is usually modeled by a constant dynamic viscosity. In combination with the thinning background density, the kinematic viscosity increases exponentially with height resulting in a rapid decrease in amplitude.

In the process of becoming saturated the amplitude becomes considerably large such that the waves cannot be considered linear. Pioneering work on the mathematical description of nonlinear gravity waves was accomplished by Grimshaw (1972, 1974). This paper aims to extend Lindzen’s linear wave saturation theory with the aid of Grimshaw’s nonlinear wave description.

## 2. How the modulation equations solve the compressible Navier–Stokes equations—A brief review

*L*

_{r}≈ 1...10 km denotes the reference wavelength (hence the subscript

*r*) and

*υ*

_{r}is the reference velocity. Note that variables labeled with an asterisk denote dimensional quantities throughout the paper. To separate the hydrostatic background from the flow field associated with the wave the first ingredient necessary is a small scale separation parameter

*H*

_{θ}≈ 10...100 km is the potential temperature scale height. This choice for the scale separation parameter was introduced by Achatz et al. (2010).

The authors considered inviscid flows. To take viscous damping into account, we need to compare inviscid and viscous terms. The (eddy) viscosity is not a constant throughout the atmosphere and among others depends on temperature. Midgley and Liemohn (1966, their Fig. 9) gave a realistic vertical profile of the effective kinematic viscosity combining eddy and molecular effects. Hodges (1969) computed the eddy diffusion by gravity waves near the mesopause to be *K*_{r(eddy)} ≈ 10^{6...7} cm^{2} s^{−1}, which compares well with Midgley and Liemohn (1966) and Lindzen (1981). In the scaling regime of Achatz et al. (2010) as well as Schlutow et al. (2017) these values correspond to Reynolds numbers of Re ≈ 10...100 when the Mach number is Ma ≈ 0.1...0.01. These numbers are also supported by a review of Fritts (1984).

*ρ*

_{r},

*p*

_{r},

*κ*

_{r}, and

*μ*

_{r}represent the reference density, pressure, thermal conductivity, and dynamic viscosity, respectively. The constant

*κ*= 2/7 is the ratio of the ideal gas constant

*R*to the specific heat capacity at constant pressure

*c*

_{p}for diatomic gases. Equations (3) provide a distinguished limit for multiple-scale asymptotic analysis. Introducing compressed coordinates for the horizontal, vertical, and time axis,

*T*(

*z*). Two dimensionless background variables, which vary on the large scale, will appear in the final modulation equations, the Brunt–Väisälä frequency or stratification measure

*N*and the background density

*ρ*. They have to be calculated from the temperature by solving

**v**, buoyancy

*Nb*, and kinematic pressure

*p*, where

**e**

_{z}the unit vector pointing in the vertical direction;

**U**= (

**v**,

*b*,

*p*)

^{T}contains the prognostic variables and

*ϕ*is the wave’s phase.

We want to give a remark at this point. By construction, the WKB ansatz is a nonlinear approach. In the limit *ε* → 0 the amplitudes are finite. In fact, the ansatz converges to the nonlinear plane wave of Boussinesq theory, which is known to be an analytical solution. Mean-flow interaction and Doppler shift are leading-order effects, which is in contrast to weakly nonlinear theory where they appear as higher-order corrections. In the *ε* limit, the weakly nonlinear approach converges to the linear plane wave.

The nonlinear WKB ansatz is inserted into the compressible NSE and terms are ordered with respect to powers of *ε* and harmonics.

## 3. Governing equations: Grimshaw’s dissipative modulation equations

*k*

_{x}= const) and only modulated in the

*z*direction,

*k*

_{z}, wave action density

*ρa*, and mean-flow horizontal wind

*u*. Equations (9a)–(9c) are closed by

**k**,

*ω*, and

*ω*′ represent the wavenumber vector, extrinsic frequency, and vertical linear group velocity, respectively. Note that primes denote derivative with respect to the vertical wavenumber throughout this paper. Extrinsic frequency is defined by the sum of intrinsic frequency and Doppler shift. It is linked to the wavenumber vector by the dispersion relation for nonhydrostatic gravity waves. It was shown in Schlutow et al. (2017) that the modulation equations equally hold for hydrostatic waves, where the horizontal wavelength is much larger than the vertical, if the dispersion relation is replaced by

*ω*=

*Nk*

_{x}/|

*k*

_{z}| +

*k*

_{x}

*u*. The prognostic variables determine the asymptotic solution as described in the previous section and explained in Schlutow et al. (2017).

*ϕ*≡

**k**and −∂

*ϕ*/∂

*t*≡

*ω*. Cross differentiating these expressions, they add up to zero. The second equation, (9b), governs the conservation of wave action density being the ratio of wave energy density and the intrinsic frequency. In terms of the polarization relation the leading-order first harmonics are computable via

*p*corresponding to the mean-flow horizontal kinematic pressure is unknown and needs additional investigation to close the system (9).

**F**and an inhomogeneity

**G**where

**y**= (

*k*

_{z},

*a*,

*u*)

^{T}is the prognostic vector.

## 4. General stationary solutions of the modulation equations

In this section we will explore general stationary solutions before we focus on particular solutions for which we will present stability analysis in the upcoming sections.

*p*/∂

*x*= 0, which can be computed analytically by a formula of Schlutow et al. [2017, their (5.20)]. When we multiply (9b) by

*k*

_{x}and subtract (9c), we obtain

**Y**(

*z*) = (

*K*

_{z},

*A*,

*U*)

^{T}depicting a mountain lee wave fulfills

*ω*′(

*K*

_{z}). Note that we label the stationary solution by capital letters. Given any well-behaved, slowly varying mean-flow horizontal wind

*U*(

*z*), the remaining two variables of the general stationary solution are given explicitly by

*z*= 0.

## 5. Lindzen-type mountain lee wave and the wave-Reynolds number

So far, we considered all background variables of our governing PDE as functions of *z*. In this section we will prescribe these functions in a piecewise fashion in order to construct a typical mountain lee wave that gets saturated by some small-scale instability process comparable to Lindzen (1981). The complete solution is divided into an unsaturated as well as a saturated middle-atmospheric part and a deep-atmosphere part.

### a. The unsaturated middle-atmospheric solution

*N*= const, some typical value for the middle atmosphere. Then (6) gives

*H*=

*κN*

^{−2}= const denotes the dimensionless (local) pressure scale height. Second, we assume that the mean-flow horizontal wind is piecewise constant as well, so

*U*= const. These assumptions can be weakened, which we will discuss in the concluding section 7. It follows immediately by (18) and horizontal periodicity that

*K*

_{z}= const making it a plane wave.

*z*

_{break}the integral in (19) vanishes and the amplitude of the wave grows exponentially with the inverse density,

### b. The saturated middle-atmospheric solution

*z*

_{break}, the wave saturates by some small-scale instability process producing turbulence, which balances the anelastic amplification. The exact kinematic eddy viscosity that keeps the amplitude leveled in (19), such that

*A*= const, is then given by

*z*only by the background density and can be computed inserting (22) into (14),

### c. The deep-atmosphere solution

*z*

_{turbo}as in the heterosphere the molecular viscosity dominates, which is modeled by a constant dynamic viscosity

*μ*

_{mol}implying

*A*, which decays quickly for

*z*→ +∞. Here, typical values

*N*and

*U*for the deep atmosphere are used.

### d. The wave-Reynolds number

*C*

_{gz}and represents the velocity scale for this type of Reynolds number. The term

## 6. Stability of the saturated wave

_{wave}=

*O*(1) as here the amplitude is at its maximum and one can expect most likely nonlinear behavior. We assess stability by analyzing the evolution of small perturbations. Due to their smallness, we can linearize the governing equations in vector form (12) around the stationary solution

**Y**. Applying the ansatz

**y**. The Jacobian matrices of the flux and inhomogeneity evaluated at

**Y**are given by

**y**= (

*k*

_{z},

*a*)

^{T}and

_{wave}=

*O*(1) (in Fig. 1 at 70 and 110 km), in other words, that the edges have no influence on the perturbation, then one can extend the domain to the infinities. Consequently, we consider the differential operator associated with the EVP as closed and densely defined on

*L*

^{2}, the space of vector valued square integrable functions on the real line equipped with the norm

*M*) = 0], which happens if

*μ*, which can also be interpreted as the vertical wavenumber of the perturbation. The real part of

*λ*represents the instability growth rate, if it is positive, and the imaginary part is a frequency. Thus, (39) provides also a dispersion relation linking the perturbation’s wavenumber with its frequency.

One can readily show that either *λ*_{1} or *λ*_{2} of (39) has positive real part for reasonable wave solutions implying that they are unconditionally unstable. We want to point out that when *H* → ∞ and Λ → 0, (39) reduces to the spectrum of the inviscid nonlinear Boussinesq plane waves (Schlutow et al. 2019) having the classical modulational stability criterion Ω″ > 0 (Grimshaw 1977).

### a. Transient (in)stability

In this subsection we want to investigate the characteristics of the instability that is presented in the previous section. We put the “in” of the section title into parentheses because there is the possibility that the primary wave “survives” the instability. One particular type of those harmless instabilities is called transient instability. Despite the fact that the instability’s norm grows exponentially in time, the instability vanishes at each given point in space if one waits long enough.

To show such characteristics a prerequisite for mathematical rigor must be fulfilled: the linear differential operator of the EVP must be well-posed, which means its spectrum has a maximum real part or, in other words, the instability growth rate is bounded. The reader finds this tedious verification in the appendix. We want to give an interesting remark at this point. The “well-posedness” depends on a criterion, Ω″ > 0, which turns out to be equivalent to the modulational stability criterion of plane waves in Boussinesq theory.

*iμ*→

*iμ*−

*α*, so

*L*

^{2}must therefore decay exponentially at −∞ for all times. But simultaneously, its

*L*

^{2}norm grows exponentially in time. This seeming paradox resolves when the perturbation propagates sufficiently fast toward +∞ (i.e., upward). Perturbations of this behavior are called transient instabilities (Sandstede and Scheel 2000). For illustrative purpose, the unweighted and weighted spectrum of an example wave, which we will discuss in more detail in the following section, are plotted in Fig. 2.

### b. Numerical investigation of the transient instabilities

We use the finite-volume numerical solver presented in Schlutow et al. (2019) for the governing equations, (9), to compute the evolution of a tiny Gaussian initial perturbation of the saturated wave in the region where Re_{wave} = *O*(1). The results are shown in Fig. 3. The simulation is set up by *N* = 1 and *K*_{x} = 1. We discretize the equations on 2000 grid points in *z* and integrate in *t* over 6000 time steps. As can be seen in the figure, the perturbation amplifies exponentially and propagates to the right (i.e., upward), as theory suggests. We can also observe that the perturbation’s wavelength compares with the scale height. Or in other words the instability varies on the large scale. The corresponding spectra to this case as computed by (39) and (41) are plotted in Fig. 2. The amplitude and the vertical wavenumber undergo strong modulations due to the exponentially growing perturbation. Furthermore, the initially constant mean-flow horizontal wind experiences acceleration. The analytic maximum instability growth rate according to (A6) is 2.2 for this particular case. In terms of an approximated *L*_{2} norm that we compute numerically, the actual growth rate of the Gaussian perturbation is found to be 1.6. The difference between theoretical and observed rate occurs because the perturbation is not optimal.

## 7. Summary and discussion

In this paper we investigated nonlinear mountain waves, which are governed by Grimshaw’s dissipative modulation equations being asymptotically consistent with the compressible Navier–Stokes equations and the dissipative pseudoincompressible equations likewise.

We introduced a wave-Reynolds number characterizing the stationary solution. When this dimensionless quantity is of order unity, the wave amplitude saturates by small-scale instabilities and assumes a maximum as anelastic amplification by the thinning background air is exactly balanced by the turbulent damping. We analyzed this regime with respect to modulational stability as nonlinearities dominate for large-amplitude waves. It turned out that transient instabilities emerge that propagate upward. We tested this analysis solving the modulation equations numerically and found excellent agreement with the theory.

In the framework of saturated nonlinear wave theory, which we presented in this paper, the saturated mountain wave does initially not accelerate the mean-flow horizontal wind. Instead, a mean-flow horizontal kinematic pressure gradient emerges that keeps the horizontal wind constant by balancing the viscous forces. The wave persists structurally and loses energy directly to turbulence, which in turn damps altitudinal amplification. Eventually, the mean flow is accelerated by a transient instability in the saturation zone propagating upward while growing and transferring kinetic energy to the mean flow.

Our investigations have two implications being of interest for gravity wave parameterizations in numerical weather prediction and climate modeling. First, the induced mean flow behaves wavelike. Its evolution is governed by a dispersion relation for the linear perturbation. In conclusion, it may be interpreted as an upward-traveling secondary wave of larger scale than the primary wave. Secondary waves with wavelengths comparable to the primary modulation scale were investigated by Vadas and Fritts (2002), Vadas et al. (2003), and Becker and Vadas (2018). The authors propose a generating mechanism based on body forces produced by the dissipating primary wave. In contrast to this model, our secondary wave is generated by direct wave–mean-flow interaction that compares to Wilhelm et al. (2018).

Second, this novel picture of saturated mountain waves may explain the bias between the onset of small-scale instability and the actually observed mean-flow acceleration (Achatz 2007). We give an extension to the established picture where waves become unstable at the breaking height and deposit their momentum and energy onto the mean flow at this level. As it turns out, only in combination with modulational instability does an initially saturated nonlinear wave induce a mean flow. This modulational instability has its own transient velocity and growth rate, which we can compute explicitly. Then the mean-flow acceleration depends on these two quantities.

In our derivations we assumed piecewise analytic solutions being matched. This assumption can be weakened to almost arbitrary background, fulfilling hydrostatics and the ideal gas law, as well as almost any mean-flow horizontal wind. However, these functions of height must be restricted to converge to constant values at the infinities. The resulting spectrum describing the temporal evolution of the perturbation would be much more complicated but still analytically assessable by Fredholm operator theory (Schlutow et al. 2019). In conclusion, our results are valid in a much more realistic atmosphere.

## Acknowledgments

The author thanks Prof. Ulrich Achatz, Prof. Rupert Klein, and Prof. Erik Wahlén for many helpful discussions during the preparation of this article. The research was supported by the German Research Foundation (DFG) through Grants KL 611/25-2 of the Research Unit FOR1898 and Research Fellowship SCHL 2195/1-1.

## APPENDIX

### Well-Posedness: Are the Instability Growth Rates Finite?

*λ*

_{1,2}of (39) by

*D*

_{Y}(

*μ*). The real part being the instability growth rate may be expressed by

*λ*

_{1,2}(

*μ*)) has no poles. Thus, we are only concerned with its behavior at the infinities. We have

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