The Vadas-Fritts ray-tracing model for convectively generated gravity waves is analyzed using the stationary phase approximation and is interpreted in terms of a ray Jacobian approximated by the density of rays. The Vadas-Fritts model launches rays from the convective source region, with initial conditions for the ray-tracing deduced from a near-field integral representation. In the far-field the rays are binned in space-time grid cells. The contribution of each ray to the spatial wave amplitude is determined by its spectral amplitude and by the local density of rays within the grid cells. The present analysis accomplishes two things. First, the stationary phase analysis gives the formal initial conditions for the ray-tracing, which mostly agree with the Vadas-Fritts initialization but also suggest some refinements. Secondly, the Jacobian and ray-density analysis shows how the Vadas-Fritts model can be generalized to follow a beam of rays with a single moving grid cell.
 Gravity waves generated by tropospheric convection can propagate high into the atmosphere, reaching altitudes of 200 kilometers or more. While high resolution simulations using fully nonlinear mesoscale atmospheric models can capture the processes near the tropospheric source, extending these models with sufficient resolution to such high altitudes is computationally impractical. Thus, alternative approaches are required to model the deep penetration of convectively generated gravity waves into the middle and upper atmosphere.
 The Vadas-Fritts model consists of a Fourier-Laplace integral representation for the near-field around the convective forcing, and a ray-tracing for the propagation of the gravity waves into the far-field. The initial conditions for the far-field ray-tracing are deduced from the near-field integral representation.
 In this paper, we examine the Vadas-Fritts ray-tracing model. Our main analysis tool is the stationary phase approximation for the near-field integral representation, which we present based on the theory of Lighthill . The stationary phase approximation, like the Vadas-Fritts theory, takes spectral wave amplitudes from the near-field integral representation and produces a ray solution for the gravity waves emerging from the forcing. It thus provides all the information necessary to initialize a far-field ray-tracing. Mostly, the stationary phase theory supports the Vadas-Fritts ray-tracing initialization, but there are some small differences that suggest potential refinements to the specification of the initial conditions.
 We also examine the Vadas-Fritts method of computing the spatial wave amplitudes in the ray-tracing. Their method is new, at least in the context of gravity waves, in the way that it accounts for the three dimensional geometrical spreading of the rays. In principle, the effects of geometrical spreading on wave amplitudes can be obtained by advecting a ray-tube volume element along each ray, but this is notoriously difficult in practice, as discussed by Hasha et al. . The Vadas-Fritts method is a practical alternative to the ray-tube method and is potentially useful for other gravity wave applications. We analyze the Vadas-Fritts method in a different but equivalent way to their formulation, making explicit use of a ray Jacobian and relating the ray Jacobian to the density of the rays. The advantages of this approach are summarized at the conclusion of the paper.
 The paper is organized as follows. In section 2 the Vadas-Fritts model equations are presented, along with an integral representation for the solution. This is followed in section 3 by a presentation of the stationary phase approximation for the integral representation. In section 4 the ray Jacobian and density of rays are discussed in relation to the Vadas-Fritts method of wave amplitude calculation. In section 5 the specific forms for the convective forcing terms of the Vadas-Fritts model are introduced, and examples of the solution are presented based on the stationary phase analysis. In section 6 the stationary phase and Vadas-Fritts ray initializations are compared. In section 7, the extension to a non-uniform background is discussed. In section 8 our main results are summarized, with suggestions for further analysis.
 We assume a uniform background throughout this paper, except to indicate in section 7 how the method can be extended to a non-uniform background. We also make the Boussinesq approximation and ignore dissipation. While dissipative effects are an important component of the Vadas-Fritts ray-tracing model, they are not needed for our purposes and so are omitted. The presentation is fairly general until section 5, where the specific Vadas-Fritts forcing terms are introduced.
 We give a modified derivation of the Vadas-Fritts integral representation using a Fourier transform in time, rather than the Laplace transform of the Vadas-Fritts theory. The modified derivation is based on work by Lighthill , and its equivalence to the Vadas-Fritts solution is shown in Appendix A.
2. The Vadas-Fritts Model
 We consider a simplified form of the Vadas-Fritts model, as used in section 2.1.1 of Vadas and Fritts . It is Boussinesq, incompressible, inviscid, and without the influence of the earth's rotation. The background is uniform and at rest. The linearized governing equations are [see Vadas and Fritts, 2001, equation (2)]
Here t is time, (u, v, w) are the perturbation velocity components in x = (x, y, z), with z positive upwards, N is the unperturbed buoyancy frequency, g = 9.8 m s−2 is gravitational acceleration, p and θ are the perturbation pressure and potential temperature, and and are the mean potential temperature and density.
 The forcing term in (3) has the separable form q(x)r(t) and is localized in space and time. As in the Vadas-Fritts calculations, we find solutions that are valid for times after the forcing has finished. The forcing represents a single convective plume. We ignore the contribution from waves that initially propagate downward from the source and are reflected back upward by the ground. Ground reflected waves can be included by adding an image source below the ground. Cases with multiple convective plumes and ground reflection are studied by Vadas and Fritts . Throughout our paper, reference to the Vadas-Fritts model means its simplest version, with the above specifications.
The frequency is ω, and the wave number is k = (k, l, m). The spatial integrals above and the wave number integrals below are all three-dimensional, and all integration limits are ±∞. The inverse Fourier transform of (9) is
Our Fourier transform notation is the same as in equation (A4) of Lighthill  and differs from that of equation (2) of Vadas and Fritts  in that they place the 1/2π factors in the inverse transform rather than the forward transform.
 Taking the Fourier transform of (6) in time and space, and solving for W gives
with kh2 = k2 + l2. The dispersion relation for gravity waves is defined by the relation B = 0. We let σ denote a frequency that satisfies the dispersion relation, i.e.
We take σ to be positive, so the zeros of B are at ω = ±σ(k).
This is the stationary phase condition. It is equivalent to
since the group velocity vector cg has magnitude N∣ sin θ ∣/K.
 In the stationary phase solution (22), the position, time, and wave number cannot all be set independently. They are related by the stationary phase condition (27) or (28).
 For example, if K and θ are fixed, then according to (28) the position r moves at speed ∣cg∣t. This is equivalent to following a ray. The stationary phase solution (22) then shows that the wave amplitude decreases along the ray as t−3/2 or as r−3/2. This decrease is the result of geometrical spreading.
 The reason that the wave number argument of G in (22) has a negative sign is to match the negative sign in the associated wave phase, with frequency −σ. The combination −k, −σ has the same group velocity and corresponds to the same ray as the combination k, σ. Nevertheless, these are two linearly independent terms, and both are needed to satisfy arbitrary initial conditions.
4. Jacobian Approximation and Ray Density
 The stationary phase solution (22) contains the factor
Here J is the Jacobian determinant
of the ray transformation between wave number and spatial coordinates. The ray transformation is the stationary phase condition (27) or (28).
 The stationary phase solution (22) is expressed in terms of the Jacobian as
We refer to W1 as the spectral stationary phase solution, or the spectral ray solution.
 The aim here is to find a practical approximation for the Jacobian J that can be used in a numerical ray-tracing algorithm to convert spectral wave amplitudes, obtained from the near-field integral representation, to spatial wave amplitudes along the ray. This is done here by considering the density of rays.
 We rewrite J as the ratio of infinitesimal volume elements J = dx/dk. As noted by Lighthill [1996, equation (A11)], the spatial volume element of size dx = Jdk is occupied by waves with wave number k lying in a spectral volume element of size dk. The spatial volume element dx is carried along the ray and expands in size as t3, a result of the geometrical spreading of the waves away from the source. The factor of t3 is obtained from J ∼ r6/t3 in (29) with r = ∣cg∣t along the ray. For a uniform background, the size of the spectral volume element dk remains constant following the ray.
 Let wave number space be discretized with grid cells of finite volume δk. Suppose we launch one ray from each wave number grid cell. Then the density of rays in the wave number domain is 1/δk. The associated density of rays in the spatial domain is 1/δx, where δx ≃ Jδk. The last expression is approximate because J is approximated by its value at the center of the wave number grid cell. The finite-sized volume element δx, like the infinitesimal volume element dx, is carried along the ray and expands in size (approximately as t3) due to geometrical spreading. To summarize,
 Next we introduce a fixed spatial grid with grid cells of fixed size Δx. Let the number of rays in a particular spatial grid cell at time t be n. Then in that grid cell,
 In ray-tracing, it is usually the squared wave amplitude in some form (e.g. wave action, wave momentum flux), that is advected along the ray. Working with the vertical velocity (until section 7), the squared wave amplitude for w1 is, from (31) and (36),
where 〈 〉 is an average over the wave phase, the asterisk indicates a complex conjugate, and the value for 〈W1W1*〉 refers to some central ray in the spatial grid cell. Alternatively, the solution can be averaged over each grid cell:
Here j is an index for each ray.
 The steps for the ray-tracing algorithm are then:
 1. Discretize wave number space with elements of size δk.
 2. Launch one ray for each wave number element.
 3. Assign each ray its spectral wave amplitude 〈W1W1*〉.
 4. Ray-trace from position x0 at time t = 0.
 5. Bin the rays spatially, in grid cells of size Δx.
 6. Use (39) to compute the spatial wave amplitude. This is close to the Vadas-Fritts algorithm. They use somewhat different initial conditions, as we shall discuss in section 6. They also bin the rays in four-dimensional grid cells that are discretized in time as well as in space.
 The ray Jacobian appears implicitly in the Vadas-Fritts theory through a normalization factor that is the quotient of the wave number volume element Δk and the spatial volume element Δx [Vadas and Fritts, 2009, equation (62)].
 Note the distinction between a ray-tube method [Hasha et al., 2008] and the ray-density method (39). The ray-tube method computes geometrical spreading rates explicitly and has to keep track of the varying size of δx along the ray. Each ray individually provides the spatial wave amplitude on that ray. The ray-density method computes geometrical spreading rates implicitly from the density of rays in a spatial volume element of predetermined size Δx. Each ray individually provides the spectral wave amplitude along that ray, but many ray-tracings are needed to estimate the spatial wave amplitude from the ray density.
5. Vadas-Fritts Forcing
 We now set the forcing terms q(x) and r(t) in (3), using the same functional form and parameter settings following Vadas and Fritts . The spatial dependence q(x) takes the Gaussian form
where q0, L, D, and z0 are constants.
 The temporal dependence r(t) is
where τ is a constant, and a = 2π/τ. The Fourier transform (41) of r(t) is
 The buoyancy frequency N = 0.02 s−1. The forcing scales are, spatially, L = 20/4.5 km, D = 10/4.5 km, and temporally, τ = 12 min. The forcing is centered at z0 = 7 km. We set the arbitrary forcing magnitude q0 = 103ms−1 because this value conveniently gives O(1) perturbation velocities (in ms−1) over much of the spatial domain of interest. The value for q0 may seem large, but see the comments of Vadas and Fritts [2004, p. 788] about how the forcing is largely balanced by pressure and potential temperature perturbations rather than by large vertical motions.
 Since the forcing functions r(t) and q(x) are real, their Fourier transforms satisfy R(ω) = R*(−ω) and Q(k) = Q*(−k), where the asterisk denotes a complex conjugate. Hence from (19) and (21),
We write F = ∣F∣exp(ıβ). From (19), the complex argument β of F is equal to the sum of the complex arguments of Q and R. These arguments are, respectively,
Since the term in square parentheses in (45) is real, the argument function in (45) is either zero or π, depending on whether the sign of that term is positive or negative, respectively. Since a and σ are positive, we can write a − σ instead of a2 − σ2 in (45).
 The spectral ray solution W1 is then, starting with (32) and taking α from (25),
The spectral amplitude F(k), defined generally in (19), becomes
Figure 1 shows solutions calculated from the exact integral (16). Figure 1 (top) plots w at z = 70 km and t = 45 min. This corresponds to the third row of Figure 6 of Vadas and Fritts , but without ground reflection. Figure 1 (bottom) is a vertical cross section of w for the case without ground reflection in Figure 1c of Vadas and Fritts , and is plotted here using their same contour values. These calculations were made with a discrete Fourier transform approximation to the Fourier integral (16), on a spatial grid of (128, 128, 192) points in x, y, z, respectively. The grid spacing is 4 km in the horizontal directions, and 2.33 km in the vertical.
Figure 2 compares the exact integral solution (16) for w and its stationary phase approximation w1 in (49). The comparison is at t = 30 min and at three heights of 30, 40, and 50 km. The solutions are symmetric about the x axis. The stationary phase solution w1 is zero along the vertical axis at x = 0 because no rays reach these points. The rays that are directed vertically upward from the source have σ = N and hence zero group velocity.
 We now consider the ray-density method for computing wave amplitudes. The idea is to compute the spatial solution w1 in (49) from a ray-density approximation for the Jacobian J. From (48),
The ray-density approximation for w1 in (49) is then
 In the Vadas-Fritts calculation the spatial grid cells of size Δx are fixed in space, but they can also move to follow a group of rays. We show two examples, one with a moving spatial grid cell and one with a fixed spatial grid cell. Both cases have the same spatial grid cell volume Δx and dimensions 4 km by 4 km by 2.33 km, in x, y, and z, respectively. This is the same spatial grid size used in the calculation of Figure 1. The solutions are calculated every 0.75 min from t = 25 min to t = 59.5 min and plotted as a function of t.
 The wave number grid cell is centered at (k0, 0, m0) and has 21 wave numbers in each of the three components for a total of 213 = 9261 launched rays. The central wave number corresponds to a horizontal wavelength of 24.3 km, a vertical wavelength of 19.3 km, an intrinsic frequency of σ = 0.6N, and a group velocity that reaches x = (80, 0, 70) km at t ≃ 45 min. The wave number grid cell size is δk = δk δl δm with δk = 1.72, δl = 1.83, and δm = 3.28, each in units of 10−6m−1.
 In Figure 3, we compare solutions for the case of a moving spatial grid cell. The solid curves denote the stationary phase solution (49) for w1 (Figure 3, top) and its peak amplitude (Figure 3, bottom). The circles denote the corresponding values obtained from the ray-density method using (54) for Figure 3 (top) and (53) for Figure 3 (bottom). The wave phase in (54) is evaluated along the ray using the central wave number noted above. The center of the spatial grid cell moves from about x = (44, 0, 42) km at t = 25 min to about x = (105, 0, 91) km by t = 60 min. Recall that the source is centered at x0 = (0, 0, 7) km.
 The theoretical stationary-phase prediction for the number of rays contained in the moving spatial grid cell is
Here we have used (36) and the expression for J in (29). Relative to the source, the center of the spatial grid cell is located at a fixed angle θ = cos−1(σ/N) of about 53° from the vertical.
Figure 4 shows the number of rays in the grid cell predicted theoretically (solid curve) by (56) and the number of rays counted in the ray-tracing calculation (circles).
Figure 5 shows the solutions when the grid cell remains at a fixed location centered around the point x = 80 km, y = 0, z = 70 km. The solid curves denote the stationary phase solution (49) for w1 (Figure 5, top) and its peak amplitude (Figure 5, bottom). The stationary phase amplitude grows until about 42 min and then decreases. The circles denote the corresponding solution from the ray-density method using (54) in Figure 5 (top) and (53) in Figure 5 (bottom). Because we are considering a limited wave number range in the ray-tracing, with a bandwidth in wave number magnitude δK/K ≃ 0.17, the grid cell contains rays for only a short time. No rays propagate into the grid cell until t ≃ 41 min, and none are left in the grid cell after t ≃ 49 min. This time range, and the associated bandwidth for K, are consistent with the stationary phase condition (27).
6. Initialization of the Ray-Tracing
 We now compare the ray initialization obtained from stationary phase with the ray initialization implemented in the Vadas-Fritts model. Vadas-Fritts advect along each ray a spectral wave-momentum flux. We can most readily explain the differences between the Vadas-Fritts and stationary phase initializations by working instead with the spectral vertical velocity, as in previous sections. We consider spectral amplitudes more generally in section 7.
6.1. Stationary Phase Initialization
 The stationary phase condition (28) shows that all rays are launched from x0 = (0, 0, z0) at t = 0. The initial wave phase is π/4 + βR, as indicated by (48). The mean square spectral wave amplitude for the ray-tracing is 2∣F∣2, as also indicated by (48). As noted in the discussion preceding (19), our solution for F is valid for times t > τ, after the forcing vanishes. We can still use this F to initialize the ray-tracing, but the solution is not valid until t > τ.
 In summary, the ray-tracing initial conditions from stationary phase are:
 Vadas-Fritts initialize their ray-tracing with a spectral amplitude derived from an integral representation involving a Laplace transform in time. In Appendix A, we demonstrate that their result can be reproduced using the Fourier transform approach developed in earlier sections, giving an equivalent in (A7). The spectral amplitude for times t > τ, after the forcing stops, is
This is equivalent to the spectral amplitude 2∣F∣ in the stationary phase solution (49), with ∣F∣ given by (51). Vadas-Fritts account for any additional constant factors, such as the (2π)3/2 in the stationary phase solution, through their normalization, which is done by comparing the ray solution with the exact integral solution [Vadas and Fritts, 2009, p. 162].
The rays are launched at time ti from position x0 with initial phase ϕ0. The sign before the integral in (63) is negative in the Vadas-Fritts notation but positive here. The integral is taken along the ray and reduces to σ(t − ti) for a uniform background.
 Vadas-Fritts choose ϕ0 to be the wave phase at t = τ. This is a natural choice, given that the wave phase contains the phase of in (A7), and this solution for is not valid until t = τ. Vadas-Fritts then adjust the launch time of the rays to find the best ray approximation to the exact integral solution, settling on ti = τ/2 [Vadas and Fritts, 2009, p. 161].
 For the Vadas-Fritts choice of ϕ0, the stationary phase solution indicates that the rays should be launched at ti = τ from the ray-dependent position x0 + cgτ, where cg is the group velocity vector of the ray. This is consistent with the stationary phase prediction that all rays originate from x0 at t = 0. Alternatively, one can simply launch all rays from x0 at t = 0 with ϕ0 set to the wave phase at t = 0, as in section 6.1. Although ϕ0 contains the phase of in (A7), which is not correct until t = τ, this initialization will give the correct stationary phase solution for times t > τ.
 The Vadas-Fritts initialization of the wave phase omits the π/4 phase shift in the stationary phase solution (49). Our calculations indicate that this has a relatively small effect on the solution for the Vadas-Fritts parameters.
Here s(x, t) is the ray solution for any field variable, and S(k, t) is the corresponding ray variable in Fourier space. For the previous results s = w1 of (22) and (31), and S = W1 of (32). The regions and are related point-wise by the group velocity condition, which for a non-uniform background is x = ∫ cgdt, with the integral taken along the ray.
 For sufficiently small volumes, (64) approximates to
where the finite volume elements δx and δk introduced in section 4 correspond to and , respectively.
 For a uniform background, 〈SS*〉 and δk are constant along the ray, and δx can be estimated by the ray density method, as in (35).
 For a non-uniform background, 〈SS*〉 and δk both vary following the ray, but if 〈SS*〉 represents a conserved quantity, such as the wave-action density, then by definition the product of 〈SS*〉 and δk is constant following the ray during conservative propagation. Using the ray-density approximation δx = Δx/n, as in (35), we obtain
The zero subscript above indicates an initial value, for example at t = τ in (41) when the forcing has just finished. We have assumed that one ray is launched from each wave number volume element of size δk0. Equation (66) is the generalization of (38) for a non-uniform background.
 Another way of deriving (66) is through the conservation equation for wave action in the form given by Hayes :
Here A is the wave-action density per unit volume, and JH is the Jacobian of the Hayes formulation:
The vector a = (a, b, c) is a label for each ray. To conform with Vadas-Fritts, we choose a = k0, the initial wave number of the ray. Then
where Ã0 is the spectral wave-action density at x0.
 Using, as in (35), the ray-density approximation δx ≃ Δx/n with (69) and (71) gives
This is equivalent to (66) with A = 〈ss*〉 and Ã0 = (2π)3〈SS*〉0.
 The integral ray formulation (64) and the Hayes ray formulation (67) are completely general in that the background can vary in all directions and in time, and the wavefield can be transient or steady state. Thus the corresponding ray-density approximations (66) and (72) are also general. It is only necessary to assume a uniform background for the near-field Fourier analysis, which determines the initial condition for (64) and the constant in (67).
8. Summary and Outlook
 We used the stationary phase method to analyze the Vadas-Fritts ray-tracing model of gravity waves generated by a convective plume. Like Vadas-Fritts, stationary phase takes spectral wave amplitudes from a near-field integral representation and converts them to spatial wave amplitudes along raypaths. It provides all quantities necessary to initialize a ray-tracing, as we discussed in section 6. It also expresses the conversion factor from spectral to spatial wave amplitudes in terms of a ray Jacobian. As discussed in section 4, the Vadas-Fritts method in essence uses the density of rays to approximate the ray Jacobian, and the ray Jacobian to convert spectral wave amplitudes to spatial wave amplitudes.
 Introducing the ray Jacobian explicitly is beneficial in three ways. First, the ray Jacobian is useful in itself for estimating geometrical spreading rates, which affect local wave amplitudes. While the analytic expression for the ray Jacobian in (29) is valid for a uniform background, this is sometimes sufficient for rough estimates of geometrical spreading rates in more general backgrounds, as in Table 1 of Fritts and Vadas . The present analysis suggests that this Fritts-Vadas estimate can be improved. They assume that the gravity wave momentum flux is proportional to r−2 following a ray, where r is the distance from the source. This takes into account geometrical spreading in the horizontal directions, but the geometrical spreading is three dimensional for a transient source of gravity waves. The wave amplitude is then proportional to r−3/2, as noted at the end of section 3, and the momentum flux is proportional to r−3.
 Secondly, the accuracy of the Vadas-Fritts conversion of spectral to spatial wave amplitudes along the ray is equivalent to the accuracy of the ray-density approximation for the ray Jacobian. Thus the ray Jacobian approximation alone can be analyzed and tested in order to estimate the number of rays needed for accurate wave amplitude calculation in a specific application.
 Third, the ray Jacobian identifies the conversion of spectral to spatial wave amplitudes as a local transformation. Thus the Vadas-Fritts method can be applied locally to follow a beam of rays, with a single moving grid cell rather than a three dimensional spatial grid of many stationary cells. We showed an example in Figure 3. The moving grid cell may be a useful generalization of the Vadas-Fritts method.
 We limited our analysis to the simplest case of a uniform windless background, except for the derivation in section 7 that indicates how to extend the method to a non-uniform background. For further tests and analysis, the Vadas-Fritts method could be compared with the exact integral solution and stationary phase approximation of Shutts  for mountain waves in wind shear, and of Dupont and Voisin  for gravity waves generated by a translating oscillating source in a uniform background. Both of these models have regions of strong ray focusing where standard ray methods break down, including caustics. The ray density method has the potential to smooth out caustics and related focusing regions and produce more accurate estimates of the wave amplitudes than obtained from standard spatial ray-tracing methods. The ability of the ray-density method to smooth out caustics has been demonstrated in a case of electromagnetic waves by Didascalou et al. .
 We used a Fourier transform in time for the near-field integral representation in (9) and (10) rather than the Laplace transform in time of the Vadas-Fritts theory. We check that our result matches Vadas-Fritts, i.e. that the inverse Fourier transform
gives the same as Vadas and Fritts [2009, equation (12)] obtained by inverse Laplace transform. Here W(k, ω) is defined in (9) as the space-time Fourier transform of the vertical velocity w.
 We substitute for W(k, ω) from (11) and displace the integration path in (A1) by −ıϵ to obtain
The small positive ϵ will ultimately tend to zero. As noted by Lighthill [1996, 1978, p. 267], the introduction of ϵ ensures that the system is completely undisturbed in the limit of large negative t. See also the causality comments by Voisin [2003, equation (2.16)].
 From the integrand of (A2) and R(ω) in (42) we have
The poles of this integrand are at B = 0 (not ω = ± a) and correspond to the frequencies that satisfy the dispersion relation ω = ± σ(k).
 For t < 0, before the forcing in (41) is turned on, the exponentials exp(ıωt) and exp[ıω(t − τ)] vanish as ωi → −∞, where ωi is the imaginary part of ω. The integration path is thus closed in the lower half plane with a semicircle of infinite radius. Since the poles at B = 0 are on the real ω axis, exterior to the integration path, , as expected for times that precede the forcing.
 For t > τ, after the forcing is turned off, the exponentials exp(ıωt) and exp[ıω(t − τ)] vanish as ωi → +∞. The integration path is thus closed in the upper half plane. The poles B = 0 on the real ω axis then reside within the integration path. The residue at each pole is obtained by replacing B in (A5) with its derivative Bω = 2ω(kh2 + m2) and evaluating the integrand at the respective pole, ω = ±σ(k). The integral (A5) is 2πı times the sum of the two residues, resulting in
after taking the limit ϵ → 0. Using the dispersion relation and, from below (41), 2π = aτ, (A6) reduces to the solution given by Vadas and Fritts [2009, equation (12)]:
To convert from our notation to that of Vadas-Fritts, let , τ → σt, σ → ω, , and .
 We thank Sharon Vadas for discussions of her model and the reviewers for their comments. This work was supported by the Office of Naval Research through NRL's base 6.1 work units 9440 (Subgrid-scale Dynamics of Middle and Upper Atmospheres) and 4461 (The Boundary Paradox), and by NASA through its Global Modeling and Analysis Program, contract NNTG06HM191.