2.1 Scaling the Euler equations for finite-amplitude internal waves

We consider gravity waves in a two-dimensional domain, neglecting the Coriolis force. Viscous effects as well as external heating sources are also ignored. With respect to these basic assumptions, the time evolution of the flow is governed by the compressible Euler equations for an ideal gas, which may be written in non-dimensionalized form as

with $(x,z)$ the horizontal and vertical coordinates and $\boldsymbol{e}_{x}$ and $\boldsymbol{e}_{z}$ the corresponding unit vectors. Here, $D_{t}=\unicode[STIX]{x2202}_{t}+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ is the material derivative. The velocity vector field is denoted by $\boldsymbol{v}$ . The Exner pressure is related to the usual pressure by

where $\unicode[STIX]{x1D705}=2/7$ is the ratio of the ideal gas constant $R$ to the specific heat capacity at constant pressure $c_{p}$ . From the temperature $T$ we define the potential temperature to be

The independent and dependent variables are non-dimensionalized according to table 1. The Mach and the Froude numbers are given by $Ma=v_{ref}/C_{s}$ and $Fr=v_{ref}/\sqrt{gL_{ref}}$ where

is the speed of sound and $g$ the gravitational acceleration. Let $H_{p},H_{\unicode[STIX]{x1D703}}$ , $L_{ref}$ and $N_{ref}$ denote the pressure and potential temperature scale height, a typical wavelength and the reference Brunt–Väisälä frequency, respectively. The distinguished asymptotic limit adopted by Grimshaw (Reference Grimshaw1974) and Achatz et al. (Reference Achatz, Klein and Senf2010) is reproduced by assuming:

1. (i) low Mach number,

   (2.5) $$\begin{eqnarray}\sqrt{\frac{\unicode[STIX]{x1D705}}{1-\unicode[STIX]{x1D705}}}Ma=\unicode[STIX]{x1D700}\ll 1;\end{eqnarray}$$

2. (ii) leading-order stable stratification,

   (2.6) $$\begin{eqnarray}\frac{H_{p}}{\overline{\unicode[STIX]{x1D6E9}}}\frac{\text{d}\overline{\unicode[STIX]{x1D6E9}}}{\text{d}z}=O(1)\;(\unicode[STIX]{x1D700}\rightarrow 0)\quad \text{where }H_{p}=\frac{RT_{ref}}{g}\approx 10~\text{km};\end{eqnarray}$$

3. (iii) non-hydrostatic motions,

   (2.7) $$\begin{eqnarray}x_{ref}=z_{ref}\equiv L_{ref}\approx 4~\text{km};\quad \text{and}\end{eqnarray}$$

4. (iv) large internal wave Froude number

   (2.8) $$\begin{eqnarray}Fr_{int}=\frac{v_{ref}}{N_{ref}L_{ref}}=O(1)\;(\text{as }\unicode[STIX]{x1D700}\rightarrow 0)\quad \text{where }N_{ref}=\sqrt{\frac{g}{\overline{\unicode[STIX]{x1D6E9}}}\frac{\text{d}\overline{\unicode[STIX]{x1D6E9}}}{\text{d}z}}\approx 0.02~\text{Hz}.\end{eqnarray}$$

On account of (2.5), (2.6) and (2.8) we find that $L_{ref}/H_{p}=O(\unicode[STIX]{x1D700})$ , and we let

for specificity. Combining the scaling assumptions (i)–(iv), we obtain the distinguished limit $Fr=\unicode[STIX]{x1D700}^{1/2}$ which links the Froude number to the Mach number, so that $\unicode[STIX]{x1D700}$ is the only parameter appearing in (2.1).

Table 1.

Non-dimensionalization of variables.

2.2 Asymptotic scalings

Here we prepare the ground for the subsequent summary of the WKB analysis in § 3 by anticipating the essential spatio-temporal and amplitude scalings of the dependent variables from Achatz et al. (Reference Achatz, Klein and Senf2010). Let

denote compressed coordinates that resolve spatial scales comparable to the pressure scale height and the associated advection time scale. Furthermore, for any variable $V$ , let

denote its slowly varying leading-order horizontal background and the collection of its $i$ th and higher-order contributions in $\unicode[STIX]{x1D700}$ with wave-induced variations on the fast and slow scales. Then the asymptotic scaling of the WKB solutions to be discussed below can be summarized as

In the velocity field the wave appears at $O(1)$ , whereas potential temperature and Exner pressure decompose into leading-order hydrostatic background variations $\unicode[STIX]{x1D703}_{0}$ and $\unicode[STIX]{x1D70B}_{0}$ satisfying

and higher-order fluctuations. Potential temperature perturbations are of order $O(\unicode[STIX]{x1D700})$ , whereas Exner pressure deviations appear first at $O(\unicode[STIX]{x1D700}^{2})$ . See Achatz et al. (Reference Achatz, Klein and Senf2010) and § 3 below for more detail.

For later reference, we use the ideal gas equation of state to express the leading-order density and temperature as

and

respectively. In addition we also define the ‘logarithmic background derivative’ of any $f_{0}\in \{\unicode[STIX]{x1D703}_{0},\unicode[STIX]{x1D70B}_{0},\unicode[STIX]{x1D70C}_{0},T_{0}\}$ by

Taking the derivative with respect to $Z$ of (2.14) and substituting with (2.16) we find that

We can identify $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D703}}=N^{2}$ as the local Brunt–Väisälä frequency squared, and $-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70C}}^{-1}$ may be interpreted as the local density scale height. Combining (2.13)–(2.17), it turns out that all background variables can be expressed by the background temperature,

So once the vertical background temperature profile is known, all profiles of the remaining background variables are determined. If we now insert the distinguished limit (2.9) as well as the multiple-scale representation from (2.12) into the Euler equations (2.1) and substitute by (2.17), we arrive at a system of evolution equations

for the new set of prognostic variables

where $B$ and $P$ relate to the buoyancy and the kinetic pressure, respectively.

The expression

in (2.19) represents higher-order terms which do not contribute to the leading-order WKB solutions because either they are literally of higher order and just do not appear, or they do appear in the expanded equations up to the order considered but cancel each other. We note in passing that (2.19) without the $C$ -terms constitutes Durran’s pseudo-incompressible model equations, and this corroborates the conclusion of Achatz et al. (Reference Achatz, Klein and Senf2010) that the full compressible and pseudo-incompressible models share the WKB wave packet solutions in the considered flow regime. We will refer to system (2.19) henceforth as the scaled Euler equations.

3.1 A spectral WKB expansion

The scaled Euler equations are a set of nonlinear, first-order partial differential equations (PDEs) whose coefficients depend only on the stretched coordinate $Z$ due to the background stratification. To account for the nonlinearities we exploit an asymptotic spectral expansion approach,

for the vector (2.20). The subscript $n$ keeps track of the order in $\unicode[STIX]{x1D700}$ while the $m$ -subscript carries the order of the harmonics. So, resonant coupling of the wave with the mean flow, i.e. zeroth harmonics, occurs already at leading order (Chu & Mei Reference Chu and Mei1970; Miura & Kruskal Reference Miura and Kruskal1974). We restrict $\hat{U} _{n,-m}=\hat{U} _{n,m}^{\ast }$ to maintain real values, where $^{\ast }$ indicates the complex conjugate of a variable. To account for the slowly varying background state we apply the WKB assumption that the coefficients and the phase are functions of only the slow variables $(X,Z,T)$ , such that

The differential operators become

The phase should be also expanded in $\unicode[STIX]{x1D700}$ , like

We need only the even gauge functions because effects of any $\unicode[STIX]{x1D6F7}^{(1)}$ are already captured by $\arg (\hat{U} _{0,1})$ . Similarly, higher-order odd phases are also included in the corresponding coefficients. Since we will only analyse the system up to $M=1$ , these higher-order effects do not matter in this work, so we keep simply $\unicode[STIX]{x1D6F7}\equiv \unicode[STIX]{x1D6F7}^{(0)}$ . By means of (3.3) we can define the local wave vector and the frequency,

By inserting the ansatz (3.1) into the governing equations (2.19), we obtain an equation of the form

The coefficients ${\mathcal{C}}_{k,l}$ represent the terms which contain the ansatz functions $\hat{U} _{n,m},\unicode[STIX]{x1D6F7}$ , their derivatives and the background variables. One can now establish that in the limit $\unicode[STIX]{x1D700}\rightarrow 0$ the coefficients must vanish independently of the order in $\unicode[STIX]{x1D700}$ and the order of the harmonics, i.e.  ${\mathcal{C}}_{k,l}=0$ for all possible $k$ and $l$ . The result is an equation hierarchy providing constraints on the ansatz functions which will give us the modulation equations. In order to achieve independence with respect to the harmonics Whitham (Reference Whitham1965), Grimshaw (Reference Grimshaw1974), Miura & Kruskal (Reference Miura and Kruskal1974) and Tabaei & Akylas (Reference Tabaei and Akylas2007) assumed a priori that the solution vector $\boldsymbol{U}$ is, in addition to its spatio-temporal dependence, also a periodic function of the fast phase $\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x1D6F7}$ , so

If one now averages (3.7) with respect to the fast phase over a period $2\unicode[STIX]{x03C0}$ , then one can exploit the fact that the harmonics are mutually orthogonal. In terms of this argument it was justified that the claim of independently vanishing coefficients must hold with regard to the average.

We encounter two issues when we want to apply this technique to our approach in order to establish independently vanishing coefficients: (i) the fast phase depends on $\unicode[STIX]{x1D700}$ and (ii) it also depends itself on space and time. Due to these properties the assumption of periodicity must be justified in terms of the asymptotic series expansion of $\boldsymbol{U}$ . Miura & Kruskal (Reference Miura and Kruskal1974), who applied the nonlinear WKB method to the Korteweg–DeVries equation, wrote in this regard: ‘it remains unsettled whether or in what sense the series $\boldsymbol{U}\,[\ldots ]$ approximates the solution of the original problem’. However, we show in appendix A an alternative approach where periodicity is not needed by introducing the notion of weak asymptotic solutions based on Danilov, Omelyanov & Shelkovich (Reference Danilov, Omelyanov, Shelkovich and Karasev2003). It requires only that the ansatz functions be twice differentiable and is hence more general than the averaging method. The key ideas for overcoming the aforementioned issues are (i) to formulate (3.7) in a weak sense and (ii) to replace the Cartesian coordinates by a map onto some phase-following coordinates.

3.2 Leading-order analysis

In the following sections we present the equation hierarchy obtained by the weak asymptotic analysis. From ${\mathcal{C}}_{0,0}=0$ we find solenoidality of the velocity field of the primary harmonics,

and constraints for some mean-flow components (the zeroth harmonics),

From ${\mathcal{C}}_{0,1}=0$ we get a linear homogeneous system of equations for the primary harmonics,

with the anti-Hermitian coefficient matrix

Here, $\hat{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}-\hat{u} _{0,0}k_{x}$ represents the intrinsic frequency, which is the frequency one observes in a reference frame moving with the leading-order mean flow. It differs from $\unicode[STIX]{x1D714}$ only by a Doppler shift term. Equations ${\mathcal{C}}_{0,2}=0$ hold by the same solenoidality (3.9).

3.3 First-order analysis

From ${\mathcal{C}}_{1,0}=0$ we obtain next-order constraints for the mean flow:

where $\text{c.c.}$ stands for the complex conjugate of the previous terms. Primary harmonics are fixed by ${\mathcal{C}}_{1,1}=0$ : it follows a linear inhomogeneous system for $\hat{U} _{1,1}$ ,

with $\unicode[STIX]{x1D64D}$ being a matrix

and $\boldsymbol{L}$ being a linear differential operator

where $\unicode[STIX]{x1D6FF}_{T}=\unicode[STIX]{x2202}_{T}+\hat{u} _{0,0}\unicode[STIX]{x2202}_{X}+\text{i}k_{x}\hat{u} _{1,0}+\text{i}k_{z}{\hat{w}}_{1,0}$ . Equations ${\mathcal{C}}_{1,2}={\mathcal{C}}_{1,3}=0$ yield the solenoidality,

for the velocity field of the secondary harmonics.

3.4 Second-order analysis

From ${\mathcal{C}}_{2,0}=0$ we obtain, among others,

Here, we have already used the solenoidality (3.9). At this point we can stop the iteration as we have found enough constraints to construct leading-order solutions.

3.5 Derivation of the modulation equations

The objective of this section is to exploit solvability constraints from the asymptotic analysis (3.9)–(3.21) in order to derive a closed set of equations which we will refer to as modulation equations. Equation (3.11) has non-trivial solution only if $\det (\unicode[STIX]{x1D648})=0$ , yielding the dispersion relation

Note that $\text{sgn}(\hat{\unicode[STIX]{x1D714}})=\pm 1$ indicates two different branches of the solution. By definition (3.5), the dispersion relation is essentially a prognostic equation for $\unicode[STIX]{x1D6F7}$ , which is by definition (3.6) in the form of a Hamilton–Jacobi equation. We find a particular vector ${\mathcal{U}}\in \ker (\unicode[STIX]{x1D648})$ providing the polarization relation

which is exactly the same polarization one would obtain in Boussinesq theory. For a detailed discussion on polarization relations in different soundproof approximations we refer to Davies et al. (Reference Davies, Staniforth, Wood and Thuburn2003). We deduce that $\operatorname{rank}(\unicode[STIX]{x1D648})=3$ . Since $\dim (\ker (\unicode[STIX]{x1D648}))=1$ , we get a general solution for (3.11) by

with a complex-valued scalar $A$ that we later refer to as the wave amplitude. Note in particular that $\hat{B}_{0,1}\equiv A$ . By means of solenoidality of the several harmonics in the velocity, (3.9) and (3.20), we can show that $\unicode[STIX]{x1D64D}(\hat{U} _{1,2},\boldsymbol{k})\hat{U} _{0,-1}=0$ , which simplifies (3.17). A necessary condition for the solvability of (3.17) is that any vector $\boldsymbol{L}\hat{U} _{0,1}\in \operatorname{img}(\unicode[STIX]{x1D648})$ has to be orthogonal to every ${\mathcal{U}}\in \ker (\unicode[STIX]{x1D648})$ by the rank–nullity theorem:

If equation (3.25) holds, then

must be true independently, if $A\neq 0$ , which will serve to fix the degrees of freedom in

This argument essentially means that the real and imaginary parts of the right-hand side of (3.25) must vanish. Equation (3.26) is true if

which is derived using (3.16). This equation of motion yields essentially the evolution of the wave energy $\widetilde{E}\equiv 2\unicode[STIX]{x1D70C}_{0}|A|^{2}$ , which will be extensively discussed in § 4. With the help of the consistency relations, $\unicode[STIX]{x2202}_{T}\boldsymbol{k}+\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D714}=0$ and $\unicode[STIX]{x2202}_{X}k_{z}=\unicode[STIX]{x2202}_{Z}k_{x}$ , that we obtain by cross-differentiating the definitions (3.5) and (3.6), we find

with the group velocity

Combining $\unicode[STIX]{x1D70C}_{0}\times$  (3.29) with (3.30) and exploiting that $\unicode[STIX]{x1D70C}_{0}(\unicode[STIX]{x2202}_{Z}+\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70C}})\,\cdot \,=\unicode[STIX]{x2202}_{Z}(\unicode[STIX]{x1D70C}_{0}\,\cdot \,)$ results in a conservation law

for the wave action density ${\mathcal{A}}=\widetilde{E}/\hat{\unicode[STIX]{x1D714}}$ .

Let us return to the orthogonality of the kernel vector and the image vector (3.25). Exploiting (3.27) gives an equation of motion for the argument of $A$ being basically some slow phase in addition to the fast phase $\unicode[STIX]{x1D700}^{-1}\unicode[STIX]{x1D6F7}$ , i.e.

which contains the first-order mean-flow horizontal wind. With the aid of the derivations from above, we can handle the mean-flow equations (3.13)–(3.16) and (3.21). Using once more solenoidality from (3.9) and (3.20), it is possible to settle all the terms depending on $\hat{U} _{1,1}$ , so

If we plug (3.24) into (3.34), we obtain an equation of motion for the leading-order mean-flow horizontal wind,

The six equations (3.22), (3.32), (3.33), (3.36), (3.37) and (3.38) that we derived in this section constitute a coupled system of equations for the six unknowns

These are the aforementioned modulation equations. Once they are solved, the leading-order, weak asymptotic solution of the scaled Euler equations has been found. It is worth mentioning that further solvability constraints for the first-order mean flow will show up when we keep iterating the asymptotic scheme. The equations are in line with the calculations of Achatz et al. (Reference Achatz, Klein and Senf2010). However, in addition, we provide explicitly a prognostic equation for the slow phase $\arg (A)$ and a constraint for the first-order mean-flow horizontal wind since they contribute to the leading-order solution. Equation (3.14) containing $\hat{B}_{1,0}$ can be considered to be uncoupled as it does not affect the leading-order solution and it does not appear in one of the other modulation equations.

5.1 The horizontally travelling wave solution

In this section we want to investigate the horizontally travelling wave solution (C1), because it is possible to find analytic, explicit solutions in this case, even for arbitrary stratification. These solutions already possess some interesting properties such as nonlinear phase and amplitude, as well as arbitrary mean-flow horizontal wind.

If $C_{z}=0$ , then $\hat{u} _{0,0}$ disappears in (5.7), and hence (5.8) is solved by any real function

Although we did not show the equation, an evolution equation for $\hat{u} _{1,0}$ emerges at higher order in the asymptotic iteration. But fortunately the same argument holds due to (5.9), so it turns out that $\hat{u} _{1,0}(\unicode[STIX]{x1D701})=u_{f1}(\unicode[STIX]{x1D701})$ is also an arbitrary function. Equation (5.12) reduces to a quadratic polynomial that is solved by

It is worth pointing out that a negative discriminant results in exponential growth or decay of the wave in the vertical direction corresponding to a reflective layer. To prevent this, we constrain $N/K_{x}>|C_{z}-u_{f0}|$ at any point. Also, we must not have $u_{f0}=C_{x}$ at any point as it possibly causes singularities, which corresponds to a critical layer.

To solve (5.5), we drop the time derivative and rewrite it as

which is an equation for the vertical wave action flux $G=\unicode[STIX]{x1D70C}_{0}c_{gz}a$ . Its validity requires that $c_{gz}\neq 0$ , which is fulfilled if $k_{z}\neq 0$ and bounded. This manipulation is feasible as $\boldsymbol{c}_{g}$ depends only on $\unicode[STIX]{x1D701}$ due to (5.13) and (5.14). As soon as we find a solution for $G$ , the amplitude can then be computed by $|A|=\sqrt{G\hat{\unicode[STIX]{x1D714}}/(2\unicode[STIX]{x1D70C}_{0}c_{gz})}$ . By the method of characteristics a solution is given by

with

The function $G_{0}$ can be determined by a boundary condition at $\unicode[STIX]{x1D701}_{0}$ . Equation (5.17) can be further simplified if we plug in the definition (3.31) and use the dispersion relation (5.4), yielding

Note that the characteristic curves given through $\unicode[STIX]{x1D6F9}$ are locally orthogonal on every line with constant phase $\unicode[STIX]{x1D6F7}$ . This remarkable property will be investigated in detail in appendix B and used to construct the substitution that was postulated in order to obtain weak asymptotic solutions in § 3 and appendix A. Taking advantage of (5.9), we can compute a solution to (5.6) by separation of variables, so

with $L_{\unicode[STIX]{x1D6F9}}$ some constant which may be interpreted as a slow wavenumber. A general solution to (5.7) is found upon inserting (5.5):

where $P_{f}$ is some function.

Before we proceed to derive an explicit, analytic solution, let us consider the energy budget. With the aid of (5.16) we can compute the energy exchange rate presented in § 4 as

If $G$ is locally confined, the energy exchange between wave and mean flow happens on the characteristic curves defined by $\unicode[STIX]{x1D6F9}$ , and is strongest where the characteristic curve of the peak of $G$ meets the level of strongest vertical shear.

We were able to derive a complete general solution in the case of a horizontally travelling wave. However, the solution is still not explicit since integrals emerged for $\unicode[STIX]{x1D6F7}$ , $\unicode[STIX]{x1D6F9}$ and $\arg (A)$ . They may be computed numerically, although for some particular background stratification profiles and horizontal mean-flow winds these integrals may have analytic solutions.

Consider a background with constant mean flow, such that

and a stratification profile

which represents a bell function with a peak of $N_{max}^{2}$ centred at $\unicode[STIX]{x1D701}_{0}$ and width $\unicode[STIX]{x0394}\unicode[STIX]{x1D701}$ . For $\unicode[STIX]{x1D701}\rightarrow \pm \infty$ it converges to $N_{min}^{2}$ . When we insert these assumptions into (5.14), we obtain

Here, we chose $C_{x}=N_{min}/K_{x}+U_{f}$ . This can be integrated by means of standard antiderivatives, so

and

The slow phase becomes simply $\arg (A)=L_{\unicode[STIX]{x1D6F9}}\unicode[STIX]{x1D6F9}$ .

Figure 1.

Illustration of the horizontally travelling wave solution without mean flow experiencing non-uniform stratification. (a) Squared Brunt–Väisälä frequency profile. (b) Horizontal wind component multiplied by the square root of the background density in the stationary coordinate system $(x,z)$ ; also, the translational coordinate system $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701})$ that moves with the constant velocity $C_{x}$ to the right. (c) Profile that one would observe at $x_{obs}$ .

In conclusion, for this particular stratification we found an analytic travelling wave solution. It is plotted in figure 1 for some choice of $O(1)$ parameters and $G_{0}$ a bell function. This travelling wave is characterized by spatial modulation of phase as well as amplitude which is completely controlled by the stratification (see figure 1 a). The structure is that of a locally confined wave packet. For $\unicode[STIX]{x0394}\unicode[STIX]{x1D701}\rightarrow \infty$ the background tends to uniformity and the travelling wave merges with a plane wave. As the wave travels to the right, an observer at position $x_{obs}$ (see figure 1 b) would experience an upward-propagating wave packet with downward travelling wave troughs and crests. As we decided on a constant mean-flow horizontal wind, there is no shear. So, there is no energy exchange between the wave and the mean flow.

If $C_{x}=0$ , the result resembles a stationary wave in the original $(X,Z)$ coordinates. Such a wave can be interpreted as a mountain lee wave. The orography then may fix the boundary condition for $G_{0}$ . It is also possible to construct analytic travelling wave solutions in the same fashion, where $N=\text{const.}$ , corresponding to an isothermal background, and $u_{f0}(\unicode[STIX]{x1D701})$ is a prescribed profile. For example, choosing $u_{f0}(\unicode[STIX]{x1D701})=u_{0}+\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D701}$ linear also provides standard antiderivatives. Here, the energy exchange rate $q$ is controlled by the constant shear $\unicode[STIX]{x1D6EC}$ . This could be used to study the energy transport due to orography in circulation models.

5.2 The isothermal, horizontally periodic travelling wave

The previous section was dedicated to the horizontally travelling wave solutions (C1) where we found exact analytic solutions to the modulation equations. This section examines case (C2), that is, the isothermal travelling wave solutions, where the local Brunt–Väisälä frequency, $N$ , and the logarithmic background derivative of the density, $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70C}}$ , are constant. In this case a complete leading-order solution of the scaled Euler equations cannot be achieved as we did not provide the explicit evolution equation for $\hat{u} _{1,0}$ , which is mandatory for computing $\arg (A)$ in (5.6). This is because the argument in (5.13), where $\hat{u} _{1,0}$ disappeared in the evolution equation, does not hold since in general we allow for a non-zero vertical relative velocity, $C_{z}\neq 0$ . However, the remaining modulation equations are uncoupled, so they may possess some insightful solution.

Even though the fourth-order polynomial in (5.12) has an analytic solution, its calculation does not provide any further enlightenment. We differentiate (5.4) instead with respect to $\unicode[STIX]{x1D701}$ , which avoids lengthy terms and gives us an evolution equation for the vertical wavenumber. Closure of the system as well as substantial simplification can be gained if we restrict ourselves to horizontally periodic travelling waves, such that the horizontal derivatives in the modulation equations drop out. So, we may rewrite the modulation equations as

The prime indicates the derivative with respect to $k_{z}$ , such that $\hat{\unicode[STIX]{x1D714}}^{\prime }(k_{z})=c_{gz}(k_{z})$ .

This system has been studied numerically by Rieper et al. (Reference Rieper, Achatz and Klein2013a ). A similar system has been investigated by Dosser & Sutherland (Reference Dosser and Sutherland2011). If we combine (5.28) with (5.29) and integrate, we obtain

with $u_{0}=u_{0}(\unicode[STIX]{x1D701}+C_{z}\unicode[STIX]{x1D70F})$ being fixed by some boundary conditions. Substituting into (5.27) gives

reducing the number of equations. As we seek stationary solutions in the translational reference frame, we drop the time derivatives and constrain $u_{0}=\text{const.}$ , making it some mean-flow horizontal wind offset, to get rid of its $\unicode[STIX]{x1D70F}$ -dependency. By that we can integrate (5.31) and solve for

The constant $a_{0}$ has absorbed $u_{0}$ . Hereinafter, we treat $a=a(k_{z})$ as function of $k_{z}$ , so it becomes a diagnostic quantity. Inserting it into (5.32) yields a single, nonlinear, autonomous ordinary differential equation (ODE) for $k_{z}$ , which can be written after some basic manipulation as

The denominator of $f$ arises after differentiating (5.33) and using the identity

The ODE (5.34) has equilibrium solutions if the numerator of $f$ vanishes, which happens either for $\hat{\unicode[STIX]{x1D714}}^{\prime }(k_{z}^{-})=0$ or for $a(k_{z}^{+})=0$ . By definition (3.31) it turns out that $k_{z}^{-}=0$ . Instead of solving (5.33) for $k_{z}^{+}$ , which means again finding the root of a quartic function, we can use it to fix its constant

Both equilibria seem uninteresting at first glance: for $k_{z}^{+}$ the amplitude vanishes and for $k_{z}^{-}$ there is no oscillation in the vertical direction, such that wave vector is purely horizontal. But maybe there exists some solution connecting asymptotically the two equilibria, which can then be considered as asymptotic rest states, such that

Such a solution may be called a wave front. Consider, without loss of generality, $k_{z}^{+}<0$ and an initial condition $k_{z}(0)=k_{z}^{0}\in I=(k_{z}^{+},k_{z}^{-})$ . We want to settle two questions: first, for which initial condition and set of parameters does a solution exist? And second, to which of the two asymptotic rest states does the solution converge? That is, to determine the sign of $f$ in (5.34). As we seek upward-propagating wave packets, we shall set $\text{sgn}(\hat{\unicode[STIX]{x1D714}})=+1$ , such that $\hat{\unicode[STIX]{x1D714}}^{\prime }>0$ on $I$ . To guarantee that $|A|\in \mathbb{R}^{+}$ , we must constrain $a>0$ on $I$ . In particular, we require $a(k_{z}^{-})=a_{-}>0$ , which we can use to determine

It emerges that $C_{z}>0$ , because we can estimate $\hat{\unicode[STIX]{x1D714}}(k_{z}^{+})<N$ for all $k_{z}^{+}<0$ . Our wave therefore always travels upwards. Note that by (2.18) and $\unicode[STIX]{x1D702}_{T}=0$ (isothermal background), we get $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70C}}=-\unicode[STIX]{x1D705}^{-1}N^{2}$ . So, the numerator of $f$ is negative on $I$ . In order to obtain the sign of the denominator, we must consider (5.35) where

If we evaluate (5.35) at $k_{z}^{-}=0$ , we find that

As the denominator must not assume a root, $a$ cannot have a saddle point on $I$ . Neither is $a$ allowed to exhibit a local maximum, because ${a^{2}}^{\prime \prime }$ would switch sign and would in particular assume a root as it is continuous. The continuity of $a$ prohibits also a local minimum, since it would imply a local maximum, which we have already ruled out. In conclusion, $a$ must not have any critical points, which yields the restriction $a^{\prime }>0$ on $I$ , i.e.  $a$ is strictly increasing in $k_{z}$ . This is fulfilled regarding (5.39) if $\hat{\unicode[STIX]{x1D714}}^{\prime }<C_{z}$ on $I$ . It turns out that

This gives us a critical relative velocity for the existence of bounded travelling wave solutions

where

Demanding $C_{z}>C_{z}^{crit}(k_{z}^{+})$ generates by (5.38) a lower bound on

The critical asymptotic mass-specific wave action density $a_{-}^{crit}$ is illustrated in figure 2. Given that the previous constraints hold, by continuity ${a^{2}}^{\prime \prime }$ is positive on $I$ and the sign of $f$ is therefore negative. This means that for any $k_{z}^{0}\in I$ the solution converges to the asymptotic rest state $k_{z}^{+}$ as $\unicode[STIX]{x1D701}\rightarrow +\infty$ . Vice versa, for $\unicode[STIX]{x1D701}\rightarrow -\infty$ the solution tends to $k_{z}^{-}=0$ . So, as long as for some $k_{z}^{+}$ we choose $a_{-}$ such that it is in the white zone in figure 2, the solution converges. As soon as $a_{-}<a_{-}^{crit}$ (grey zone), the solution will run into a singularity and grow without bound.

Figure 2.

Critical asymptotic mass-specific wave action density. The isothermal travelling wave solutions cannot exist if the asymptotic rest states $k_{z}^{+}$ and $a_{-}$ belong to the grey zone.

We want to point out that $C_{z}$ is also the vertical component of the phase velocity. Multiplying by $k_{z}$ therefore gives us the frequency in the stationary coordinate system. We may interpret (5.42) in terms of a critical frequency for a front-like travelling wave. In particular, for a lower boundary condition on $k_{z}$ , i.e. an initial condition in the stationary system, this leads to a minimal frequency for the existence of travelling wave solutions.

The ODE (5.34) can be solved with the explicit Euler method. For some $O(1)$ parameters (table 2) that fulfil the requirements, a particular solution illustrating the travelling wave front’s properties is plotted in figure 4. Panel (a) shows the directly computed solution of the vertical wavenumber $k_{z}$ , panel (b) has the mass-specific wave action density $a$ diagnosed from (5.33), and panel (c) displays the leading-order horizontal wind

which is also shown in figure 3 from a two-dimensional perspective. We chose $u_{0}=-K_{x}a_{-}$ for (5.30), so the mean-flow horizontal wind drops to zero after the front has propagated through. To compute $\unicode[STIX]{x1D6F7}$ from (5.11), we applied the very same Euler method. To compute $\hat{u} _{0,1}$ according to (3.24), we set $\arg (A)=\text{const.}$ because we lack an equation for it. The error produced by this oversimplification occurs in the wake of the front when $k_{z}$ tends to zero, such that the slow phase $\arg (A)$ becomes vertically dominant and maybe produces some long-wavelength vertical oscillation. From the two-dimensional point of view, the leading term in $\unicode[STIX]{x1D6F7}$ is then still $K_{x}\unicode[STIX]{x1D709}$ . However, this will not change the overall picture as the induced mean-flow regarding (5.30) is not affected by the slow phase. Again this travelling wave is characterized by spatial modulation of amplitude as well as phase.

Figure 3.

Illustration of the isothermal travelling wave front in two dimensions propagating with velocity $\boldsymbol{C}$ . Horizontal wind $u_{\ast }$ is as given by (5.45); the potential temperature scale height $H_{\unicode[STIX]{x1D703}}$ is shown for comparison.

Figure 4.

Illustration of the isothermal travelling wave front in one dimension. (a) Vertical wavenumber. (b) Mass-specific wave action density. (c) Horizontal wind as given by (5.45) and the potential temperature scale height $H_{\unicode[STIX]{x1D703}}$ for comparison.

Table 2.

$O(1)$ parameters defining an isothermal, horizontally periodic travelling wave.

The mass-specific wave action density and therefore the amplitude remain constant in the wake of the front. So, at first glance, it seems that the wave induces wave energy into the mean flow. Because one would usually expect increasing amplitude with height due to the thinning air when the wave propagates upwards and due to the fact that the wave energy scales with the background density, this ‘extra’ energy should then accelerate the mean flow. This interpretation turns out to be incorrect, which becomes clear if we compute the wave–mean-flow energy exchange rate from § 4 in the stationary ( $X,Z$ ) coordinate system,

Here, we used (5.30) and the estimates from this section. We can already conclude that energy is exchanged directly in the front where the gradient is located. The rate is always negative. On the other hand, the wave gains energy from the mean flow and the mean-flow horizontal wind decelerates. This surprising behaviour is demystified if we consider also the wave energy flux from (4.1). It tells us that the wave energy is transported with the group velocity $\hat{\unicode[STIX]{x1D714}}^{\prime }$ . But the front moves with $C_{z}$ , which is always greater than $\hat{\unicode[STIX]{x1D714}}^{\prime }$ as we suppressed any critical points for $a$ .

In summary, the fast travelling wave front can only be sustained if it obtains energy from the mean flow. One may ask if inverted travelling wave fronts could exist as solutions to the isothermal, horizontally periodic modulation equations. These would be solutions where the asymptotic rest states switched. So, $k_{z}^{-}\neq 0$ and $a_{-}=0$ but $k_{z}^{+}=0$ and $a_{+}\neq 0$ . These solutions would be referred to as travelling wave backs. It turns out that backs cannot be solutions under any circumstances. We state this claim here unproved as the argumentation goes beyond the scope of this survey.