## 1. Introduction

High-resolution probing of the middle atmosphere reveals characteristic vertical wavenumber spectra of gravity wave perturbations of horizontal velocities, temperatures, and densities. In particular, the power spectral densities at vertical scales less than a few kilometers or so are well fitted by a power law of the form *ab*(*N*) *M*^{−3}, where *M* is the Fourier wavenumber, and *b*(*N*) depends on the Brunt–Väisälä frequency *N* and is derived from elementary gravity wave formulas (e.g., see Smith et al. 1987; Tsuda et al. 1991). The scaling factor *a* is nearly constant, typically around 1/10–1/6, although lower values are sometimes encountered, particularly in the stratosphere (e.g., Hines 1993a; Eckermann 1995).

A number of gravity-wave-based theories have been developed to explain these spectral shapes. Early efforts invoked wave amplitude saturation due to static instabilities as the primary cause (Dewan and Good 1986; Smith et al. 1987). Weinstock (1990) argued that the random fluctuations of saturated small-scale waves produced a diffusive damping of larger-scale waves, and his theoretical development yielded *M*^{−3} model spectra [see also subsequent models based on these ideas by Gardner (1994), Zhu (1994), and Medvedev and Klaassen (1995)]. Dewan (1991) derived gravity wave spectra that resembled observations by assuming a cascade of saturated wave energy from larger to smaller spatiotemporal scales. All of these theories arrived at *M*^{−3} spectra by applying dissipative processes of some kind to the wave field.

An altogether different theory was developed in a series of papers by Hines (1991a,b,c; 1993a,b; 1996; 1997a,b,c), which he has called Doppler-spread theory. While dissipation of wave energy and pseudomomentum are incorporated (e.g., Hines 1997a,b), such processes play no significant role in producing observed spectral shapes within this theory. The Doppler-spread model is most easily introduced by considering a short wave of initial vertical wavenumber *m*_{0} that propagates through a preexisting long wave of vertical wavenumber *m*_{lw}, such that |*m*_{lw}| ≪ |*m*_{0}|. In modeling this situation, Doppler-spread theory assumes that the velocity oscillation of the long wave acts like a time-invariant, vertically varying mean-flow profile that refracts the short wave, changing its vertical wavenumber and amplitude as it propagates and thus “spreading” its spectral signature through *M* space. When the long wave is replaced by a spectrum of many randomly phased long waves, their superposed velocity oscillations refract the short wave more randomly. When all waves (both longer and shorter) are considered, Hines (1991a,b) argued that, over time, a characteristic statistical spread of a given gravity wave’s vertical wavenumber results, which is centered about *m*_{0} and has a width dependent mainly on the horizontal velocity variance of the other waves in the spectrum.

Hines (1991a,b) further argued that this “Doppler spreading” of gravity wave *m* values was appreciable in the middle atmosphere. Advective interactions among oceanic gravity waves are also substantial (e.g., Holloway 1980; Desaubies and Smith 1982; Sherman and Pinkel 1991), and Munk (1981) speculated that spreading effects might explain the formation of spectral shapes ∼*M*^{−3}, which are often observed at large *M* within the ocean (e.g., D’Asaro and Morehead 1991). Since then, a number of sophisticated modeling approaches have been developed to study nonlinear interactions among oceanic gravity waves (see the review of Müller et al. 1986). Some of these models, which have sought to incorporate the major advective interactions among the waves, have generated oceanic wave spectra in agreement with observations (e.g., Orlanski and Cerasoli 1981; Flatté et al. 1985; Shen and Holloway 1986; Allen and Joseph 1989; Hibiya et al. 1996).

Hines (1991a) proposed that Doppler-spreading effects could also explain the formation of *M*^{−3} spectra within the atmosphere. Rather than attempting to adapt the complex models of oceanic wave dynamics to the atmosphere, Hines (1991b, 1993b) pursued an atmospheric theory by first developing a statistical model of Doppler-spread interactions among atmospheric gravity waves. This Doppler-spreading model was then combined with models of incident (unspread) wave-field spectra to derive Doppler spread fluctuation spectra (Hines 1991b, 1993b).

These Doppler-spread spectra contained features that resembled observations. For example, the theory predicted a “cutoff wavenumber” *M*_{C}, which roughly marked a transition from essentially unspread spectra at *M* ≪ *M*_{C} to Doppler spread tail spectra at *M* ≫ *M*_{C} that attenuated with increasing *M.* Predicted values of *M*_{C} agreed well with the locations of shape transitions in observed spectra from various regions of the atmosphere (Hines 1991b). This cutoff wavenumber assumed primary importance in a subsequent Doppler-spread parameterization of gravity-wave momentum deposition (Hines 1997a,b), which, despite including a number of untested assumptions and uncertain parameter values, performed well when used in global middle atmosphere models (Mengel et al. 1995; Lawrence 1997; Manzini et al. 1997).

Despite these successes, Doppler-spread theory could not self-consistently yield the required *M*^{−3} spectral shape at large *M.* This shortcoming was investigated in some depth by Hines (1993b). He identified the following three key approximations within the theory that could cause disagreement among predicted and observed spectra.

*Approximation 1.*In quantifying how the velocity oscillations of all other waves affect any given gravity wave, these velocities were simplified to be(a) purely horizontal

(b) invariant with time

(c) invariant with horizontal displacement.

*Approximation 2.*Doppler spreading was modeled using WKB wave formulas for single linear plane waves.*Approximation 3.*Gravity waves experienced negligible damping up to some maximum wavenumber*m*_{M}beyond which waves were completely dissipated into turbulence.

Hines (1993b) considered Approximation 1 to be the most potentially serious and, citing the oceanic model of Allen and Joseph (1989), argued that eliminating it should lead to *M*^{−3} Doppler-spread spectra at *M* ≫ *M*_{C}. Here, we investigate this by progressively relaxing (and ultimately eliminating) Approximation 1. Our formulation attempts to maintain and build upon the analytical and illustrative approach to the Doppler-spreading problem initiated by Hines. To this end, three simple models are considered initially. They are depicted schematically in Figs. 1a–c and are addressed theoretically in sections 2–4. More detailed simulations of wave spreading follow in later sections.

## 2. Refraction by a stationary background wind *U*(*z*)

We begin by reconsidering the standard problem of a linear, small amplitude, plane gravity wave, with initial parameters *m*_{0} (vertical wavenumber), *ω*_{0} (intrinsic frequency), and *K*_{0} (horizontal wavenumber), which propagates upward through a larger-scale height-varying background zonal velocity field *U*(*z*), as depicted schematically in Fig. 1a. We refer to this wave hereafter as the short wave, for reasons that become clearer in section 3. For simplicity, we assume that the background Brunt–Väisälä frequency *N,* Coriolis parameter *f,* and density scale height *H*_{ρ} are constants.

Since ∂*U*/∂*t* = *U*_{t} = 0, then Ω, the frequency of the short wave as measured from the ground, remains constant (e.g., Jones 1969). Since *U*_{x} = *U*_{y} = 0 and *f* is assumed to be constant, then the short wave’s horizontal wavenumber *K* remains constant so that *K* = (*k*^{2} + *l*^{2})^{1/2} = *K*_{0} (e.g., Eckermann 1992). Thus the ground-based horizontal phase speed *c*_{x} = Ω/*K* also remains constant. The short-wave velocity perturbation is plotted in Fig. 1a oscillating about its constant *c*_{x} value.

*z*is given by the conventional Doppler relation

*ω*

*z*

*KU*

*z*

*φ,*

*φ*is the azimuth subtended by

**K**= (

*k, l*) =

*K*(cos

*φ,*sin

*φ*). On propagating from height

*z*

_{0}to

*z*

_{1},

*ω*varies as

*x*–

*z*plane so that

*φ*= 0° or 180° and cos

*φ*= ±1, whereupon (2a) simplifies to

*ω*

*z*

_{1}

*ω*

*z*

_{0}

*k*

*U,*

*k*=

*K*cos

*φ*= ±

*K*

_{0}.

*m*(

*z*) responds to the shifts in

*ω*(

*z*) through the dispersion relation

*α*

^{2}= 1/(4

*H*

^{2}

_{ρ}

*N̂*=

*N*/|

*f*| and

*ω̂*

*ω*/|

*f*| (so that 1 <

*ω̂*

^{2}

*N̂*

^{2}) and dimensionless wavenumbers,

*μ*(

*z*) =

*m*/

*m*

_{c}and

*κ*=

*k*/

*m*

_{c}. Here

*m*

_{c}is an arbitrary constant wavenumber, which we use to nondimensionalize the problem: it assumes greater significance when these formulas are applied in section 3 to the Doppler-spreading problem, where it corresponds approximately to the cutoff wavenumber derived by Hines (1991b). Thus, (3a) now becomes

*α̂*

*α*/

*m*

_{c}.

Since *k* is constant, the refraction of wave parameters is completely specified by the frequency variations given by (2) and the ensuing *m* variations implied by (3). Thus refraction effects are conveniently summarized by simultaneously plotting variations of the normalized variables (*μ,* *ω̂*

*μ*|,

*ω̂*|

*m*

_{c}= 2

*π*(2.5 km)

^{−1},

*N̂*= 203, and

*f*evaluated at 45°N. Logarithmic axis scales are chosen for |

*μ*| and

*ω̂*|

*μ*|,

*ω̂*|

*κ.*Hence, hereafter we refer to these dotted curves synonymously as either dispersion curves or

*κ*curves. The |

*κ*| values are labeled on selected curves in Fig. 2. Note that these dispersion curves tend to flatten at |

*μ*| ≲ 0.04 and also at the frequency limits

*ω̂*

^{2}

*N̂*

^{2}and

*ω̂*

^{2}

*ω̂*

^{2}

*N̂*

^{2}is accurate and (3b) can be simplified to

*β*encapsulates both the sense and magnitude of the short wave’s refraction. This same

*β*term arises in spectral studies of shifting of gravity wave parameters by a steady background wind (e.g., Fritts and VanZandt 1987; Gardner et al. 1993; Fritts and Lu 1993; Eckermann 1995). Substituting

*β*back into (2b) yields

*ω̂*(

*z*

_{1})

*ω̂*(

*z*

_{0})

*κN̂β.*

Note from (5) that, in the midfrequency range, the amount of refraction *β* can be quantified solely through the change in the wave’s vertical wavenumber *μ,* an approach used to quantify refraction effects in some previous spectral studies (Hines 1991b, 1993b; Fritts and Lu 1993). This does not, however, diminish the importance of the accompanying changes in *ω̂,*

To illustrate these refraction effects, we consider five different short waves, each launched separately from a starting height *z*_{0} and represented in Fig. 2 by a symbol at their initial (Δ*U* = 0) parameter values (|*μ*_{0}|, *ω̂*_{0}|*U*_{max} ≤ *U*(*z*) ≤ *U*_{max} over a height interval, Δ*z,* as depicted in Fig. 1a. We choose *U*(*z*_{0}) = 0 and *U*_{max} = 8.3 m s^{−1}, so that *β*_{max} = *U*_{max}*m*_{c}/*N* = 1. Solid curves in Fig. 2 show the variations in |*μ*| and *ω̂*|

We see in all cases that the (|*μ*|, *ω̂*|*μ*_{0}| values experience greater excursions through (|*μ*|, *ω̂*|*ω̂*^{2}*U*(*z*) ≳ 1 m s^{−1}. At heights where *U*(*z*) ≲ −1 m s^{−1}, this same wave is also refracted to a turning (reflection) level *ω̂*|*ω̂*_{c}*ω̂*_{c}*ω̂*_{c}*N̂* for |*μ*| ≳ 0.04: see Marks and Eckermann (1995)]. Whether a critical level or turning level occurs first depends on the shear profile *U*_{z}(*z*), but either way this wave does not emerge at *z* = *z*_{0} + Δ*z* when background wind variations of this magnitude are present.

## 3. Doppler spreading

We now ascribe the velocity variations −*U*_{max} ≤ *U*(*z*) ≤ *U*_{max} to one full vertical cycle of a “long” (small |*m*|) gravity wave oscillation of peak horizontal-velocity amplitude *U*_{max} and negligible vertical-velocity amplitude. This situation is depicted in Fig. 1b. Since the formulas in section 2 assumed *U*_{t} = *U*_{x} = *U*_{y} = 0, this interpretation assumes that the velocity structure of the long wave acts like a stationary background flow velocity *U*(*z*) in refracting any given short wave that propagates through it, and thus follows Approximations 1 and 2 from section 1 (Hines 1993b).

*μ*

_{lw},

*κ*

_{lw}, and

*ω̂*

_{lw}

*U*

_{t}=

*U*

_{x}=

*U*

_{y}= 0 imply that the long wave’s velocity oscillation can be approximated by

*m*

_{lw}=

*m*

_{c}

*μ*

_{lw}and

*ϑ*

_{lw}is the phase at

*z*= 0 (taken to be zero here; see Fig 1b). We also stipulate that

*μ*

^{2}

_{lw}

*α̂*

^{2}

The resulting refraction of short-wave parameters (*m, ω*) induced by the velocity structure (7) of the long wave is a two-wave deterministic analog of the statistical Doppler spreading among a random spectrum of gravity waves modeled by Hines (1991b, 1993b). Since the governing formulas do not differ from those presented in section 2, the solid curves in Fig. 2 can be reinterpreted here as specifying the (deterministic) Doppler spreading of these five short waves by one cycle of the long-wave oscillation in Fig. 1b.

To study this Doppler spreading of short waves in more detail, Fig. 3 extends Fig. 2 by summarizing results for a large number of short waves spanning a wide range of initial pairings (|*μ*_{0}|, *ω̂*_{0}|*U*_{max} = 8.3 m s^{−1}, so that −1 ≤ *β* ≤ 1, and the symbol plotted at each (|*μ*_{0}|, |*ω̂*_{0}|*z,* while the other symbols denote waves that cannot. The triangular symbols clustered near the top left of Fig. 3 indicate waves that are refracted to turning levels (*ω̂*^{2}*ω̂*^{2}_{c}*ω̂*^{2}

### a. Cutoff vertical wavenumber *m*_{c}

Note that all short waves for which |*μ*_{0}| ≥ 1 (|*m*| ≥ *m*_{c}) are refracted to critical levels by the long wave (a dot-dashed line at |*μ*| = 1 is plotted in Fig. 3 for reference). Given that *β*_{max} = 1, this follows directly from (5). This is a simplified replication of a more general statistical result derived by Hines (1991a,b; 1993b), who predicted a cutoff in unshifted wavenumbers at |*μ*_{0}| ∼ 1 when *β*_{max} ∼ 1 due to the high probability of Doppler spreading of |*μ*_{0}| > 1 waves beyond a dissipative high-wavenumber limit *μ*_{M} = *m*_{M}/*m*_{c}. Hines also proposed that waves at |*μ*| > *μ*_{M} were obliterated (approximation 3), which, he argued, acted to relax any interacting wave spectrum for which *β*_{max} > 1 to an equilibrium value of *β*_{max} ∼ 1. This in turn produced a Doppler-spread vertical wavenumber spectrum that peaked at |*m*| ≈ *m*_{c} (|*μ*| ∼ 1) (Hines 1991b).

### b. Cutoff horizontal wavenumber *k*_{c} ≈ *m*_{c}*/2*

Figure 3 also reveals a cutoff horizontal wavenumber, *κ*_{c}, beyond which all waves are either absorbed at critical levels or vertically reflected at turning levels. This can be seen by starting at the bottom left of Fig. 3 and moving upward toward the |*μ*| = 1 line, so that successively encountered dispersion curves increase in their |*κ*| value. Eventually a *κ* curve is encountered on which there are no filled circles (i.e., no waves which propagate freely through the interval Δ*z*). Closer inspection reveals that this occurs at |*κ*| = *κ*_{c} ≈ 0.5.

*μ*| < ∞) must satisfy the inequality

*ω̂*

_{0}

*κN̂β*

^{2}

*N̂*

^{2}

*ω̂*

^{2}

_{c}

*N*

^{2}for sufficiently large |

*κ*| (Marks and Eckermann 1995). If

*N̂*

^{2}≫ 1, as is usually the case in the middle atmosphere, then it follows from (8) that, once

*κN̂β*

_{max}

*N̂*

*ω̂*

_{0}

*β*over the range −

*β*

_{max}≤

*β*≤

*β*

_{max}. Since

*β*

_{max}= 1 in Fig. 3, then (9) yields no freely propagating waves at |

*κ*| ≥

*κ*

_{c}≈ 0.5, as evident in Fig. 3. Given the Hines (1991b) arguments that Doppler-spread spectra converge to an equilibrium of

*β*

_{max}∼ 1, then our finding of

*κ*

_{c}≈ 0.5 for

*β*

_{max}= 1 illustrates that a cutoff horizontal wavenumber

*k*

_{c}≈

*m*

_{c}/2 is also predicted by the Doppler-spread model of Hines (1991b, 1993b), as extended here.

The “case III” model of oceanic gravity waves of Allen and Joseph (1989) predicted *M*^{−3} and ^{−3} spectra beyond hypothesized Eulerian cutoff wavenumbers *M*_{c} and _{c}, respectively, where *M*_{c} in the atmosphere using Doppler-spread theory (i.e., *M*_{c} ∼ *m*_{c}), Hines (1991b, 1993b) cited their findings as evidence that a fully generalized Doppler-spread theory should yield *M*^{−3} spectra at *M* ≫ *m*_{c} as observed. Spectral analysis of high-resolution velocity and temperature data from stratospheric aircraft by Bacmeister et al. (1996) revealed a clear transition from shallower power-law spectra at smaller horizontal wavenumbers to ^{−3} spectra at large *π* (5 km)^{−1}, which is of the order of *k*_{c} = *m*_{c}/2 at these heights (e.g., Tsuda et al. 1991; Allen and Vincent 1995). Bacmeister et al. (1996) noted that separable linear spectral models also predict the observed spectral shapes and transition point quite well (*k*_{c} = *m*_{c} in these models).

### c. Approximation 1 and the “longness” criterion

Hines (1991b, 1993b) employed Approximation 1 to model Doppler spreading among waves of all wavelengths and arbitrary amplitudes, without testing this approximation in any formal way. As a necessary first condition for Approximation 1 to hold, we imposed scale separation between the long and short wave through a longness criterion |*μ*_{lw}| ≪ |*μ*_{0}| and thereafter assumed that the approximation was valid. Yet |*μ*_{lw}| ≪ |*μ*_{0}| is well satisfied when |*κ*_{lw}| ≫ |*κ*| and/or *ω̂*_{lw}|*ω̂*_{0}|*κ*_{lw}| ≪ |*κ*| and *ω̂*_{lw}|*ω̂*_{0}|*κ*_{lw}| ≪ |*κ*|, since this usually also ensures that both |*μ*_{lw}| ≪ |*μ*_{0}| and *ω̂*_{lw}|*ω̂*_{0}|*μ*_{0}|, *ω̂*_{0}|*μ*_{lw}|, *ω̂*_{lw}|

However, even in these cases Approximation 1 may not be valid. For example, in sections 2 and 3 our initial longness criterion |*μ*_{lw}| ≪ |*μ*_{0}| in no way justified neglecting the *U*_{z} variations of the long wave that refracted the short wave. In the same way, |*κ*_{lw}| ≪ |*κ*| and *ω̂*_{lw}|*ω̂*_{0}|*U*_{x} and *U*_{t} terms can be ignored in computing the propagation of the short wave (Approximations 1c and 1b, respectively). However, since Approximation 2 is usually assured by our longness criterion (e.g., Eckermann and Marks 1996; see also section 9), we can use WKB ray theory to trace any given short wave through a long wave without discarding the horizontal and temporal oscillations of the long wave, thereby assessing directly the validity of Approximation 1. We pursue this in the following sections.

## 4. Short-wave ray solutions in the presence of a propagating long wave

*m*

_{lw},

*ω*

_{lw}, and

*K*

_{lw}, which satisfy the dispersion relation (3a). We set

*m*

_{lw}< 0 and

*ω*

_{lw}> 0 so that the vertical phase velocity is downward ((

*c*

_{z})

_{lw}< 0) and vertical group velocity is upward ((

*c*

_{gz})

_{lw}> 0). We also confine long-wave propagation along the

*x*axis so that

*K*

_{lw}=

*k*

_{lw}and the phase of the long wave is given by

_{lw}=

*ω*

_{lw}here.

We launch the short wave from a position (*x*_{0}, *z*_{0}), at a time *t*_{0} when *θ*_{lw}(*x*_{0}, *z*_{0}, *t*_{0}) delivers a node in the long wave’s velocity oscillation (see section 4b). This situation is profiled vertically in Fig. 1c, where the propagation of the long wave is depicted by arrowing its downward vertical phase velocity (*c*_{z})_{lw}.

### a. Wave action equation

*A*=

*E*/

*ω*is the short wave’s action density,

*E*is its total energy density,

**U**is the background velocity, and

**c**

_{g}= (

*c*

_{gx},

*c*

_{gy},

*c*

_{gz}) is the intrinsic group velocity of the short wave (Andrews and McIntyre 1978).

### b. Velocity oscillation of the long wave

*B*

_{z}

_{lw}

*c*

_{gz}

_{lw}

*A*

_{lw}

*B*

_{z})

_{lw}is the vertical flux of long-wave action density (e.g., Schoeberl 1985). As in section 3, we assume

*μ*

^{2}

_{lw}

*α̂*

^{2}

*ρ*is approximately constant across the interval Δ

*z*in Fig. 1c. This in turn yields, through (12), an approximately constant peak horizontal velocity amplitude

*U*

_{max}, and a long-wave velocity oscillation,

**U**

_{lw}(

*x, z, t*) = (

*U*

_{lw}(

*x, z, t*),

*V*

_{lw}(

*x, z, t*),

*W*

_{lw}(

*x, z, t*)), given by

The velocity oscillation of the long wave along the *x*–*z* plane is depicted in Fig. 4, extending the simple one-dimensional depiction of the problem in Fig. 1c. The phase fronts of the long wave lie along the *x*′ axis and move downward with time along the *z*′ axis at the phase speed *C*_{lw} = −|**c**_{lw}|, where **c**_{lw} is the phase velocity.

### c. Ray equations for the short wave

*d*/

*dt*is time differentiation following the group motion of the short wave. The wavenumber vector (

*k, l, m*) is again taken to be zonally aligned (

*l*= 0) and remains so since ∂

**U**

_{lw}/∂

*y*= 0 and, thus,

*dl*/

*dt*= 0. The time variation of the long-wave background yields an additional short-wave ray equation (e.g., Jones 1969; Eckermann and Marks 1996)

### d. Transformation to the “long-wave frame”

*x*′,

*z*′) in Fig. 4, where

*ζ*= arctan(−

*k*

_{lw}/

*m*

_{lw}) is the elevation of the long wave’s phase fronts above the horizontal plane. Since there are no long-wave phase variations along

*x*′, then, when viewed from this frame, the phase (10) transforms to

*m*

^{′}

_{lw}

*z*′, which equals the total wavenumber

*K*

^{tot}

_{lw}

*k*

^{2}

_{lw}

*m*

^{2}

_{lw}

^{1/2}in magnitude but is negative (

*m*

^{′}

_{lw}

*K*

^{tot}

_{lw})

*z*′ at the phase velocity

*C*

_{lw}= −

*ω*

_{lw}/

*K*

^{tot}

_{lw}

*ω*

_{lw}/

*m*

^{′}

_{lw}

*θ*

^{′}

_{lw}

*z*

*t*

*m*

^{′}

_{lw}

*z*

*C*

_{lw}

*t*

*x*

*y*

*z*

*x*

*y*

*z*

**V**

*t,*

**V**= (0, 0,

*C*

_{lw}) in (18) transforms to a new frame (

*x*",

*z*") that has the same orientation as (

*x*′,

*z*′) in Fig. 4 but moves downward along

*z*′ at the long-wave phase velocity

*C*

_{lw}.

*x*",

*z*") for

**V**= (0, 0,

*C*

_{lw}) as the long-wave frame since it is oriented with the phase fronts of the long wave and follows their movement. Since

*z*" =

*z*′ −

*C*

_{lw}

*t,*then from (17b) the long-wave phase is given by

*θ*

^{"}

_{lw}

*z*

*m*

^{′}

_{lw}

*z*

*K*

^{tot}

_{lw}

*z*

*x*–

*z*plane, its phase varies only along

*z*", while its velocity oscillates along the

*x*" (=

*x*′) axis (see Fig. 4), and thus (13a) and (13c) transform and then combine to yield

*U*

^{"}

_{lw}

*z*

_{max}

*θ*

^{"}

_{lw}

*z*

*U*

^{"}

_{lw}

*z*") is the velocity along the

*x*" axis, and

_{max}is the peak

*total*velocity amplitude of the long wave in the

*x*–

*z*plane and equals

*U*

_{max}[1 + (

*k*

_{lw}/

*m*

_{lw})

^{2}]

^{1/2}=

*U*

_{max}|sec

*ζ*|. The remaining velocity component (13b), which does not influence short-wave propagation, oscillates orthogonal to the page in Fig. 4, producing elliptical orbits in the

*x*–

*y*plane.

### e. Ray solutions for the short wave

*x*",

*z*"). Since

*A*is invariant under Galilean transformations (e.g., Grimshaw 1984), then, using (20), the continuity equation for action density (11) takes the form

*c*

^{"}

_{gx"}

*c*

^{"}

_{gz"}

*x*" and

*z*", respectively. Since the long wave produces gradients along

*z*" alone, according to (19) and (20), then the first two terms in (21) vanish, giving the solution (Thorpe 1989; Eckermann and Marks 1996)

*z*",

*B*

^{"}

_{z"}

*k*" and

*m*" are the wavenumber components of the short wave along

*x*" and

*z*" respectively, which relate to those along

*x*and

*z*according to

*C*

_{lw}

*m*

*k*" and Ω" are both constants during propagation through the long wave. The remaining ray equation (23b) is redundant since

*m*" follows from the dispersion relation (3), so that

### f. Simplifications for |*k*_{lw}| ≪ |*m*_{lw}| (*ζ* → 0) as a test of Approximation 1b

Our amended longness criterion of section 3c was |*k*_{lw}| ≪ |*k*|, so that |*ω*_{lw}| ≪ |*ω*| and |*m*_{lw}| ≪ |*m*|. The implied low frequency of the long wave also implies (*k*_{lw}/*m*_{lw})^{2} = tan^{2}*ζ* ≪ 1 through the dispersion relation (3a), and thus *ζ* ≈ 0.

Thus, to simplify our analysis of these short-wave solutions, we consider first the inertial limit *ζ* = 0, which eliminates horizontal variations of the long-wave oscillation (*k*_{lw} → 0) but retains its temporal oscillation. The velocity oscillations of the long wave now become purely horizontal, and the long-wave phase moves downward along *z* at the vertical phase velocity (*c*_{z})_{lw}, consistent with the simplified depiction in Fig. 1c. This invokes approximations 1a and 1c but not approximation 1b, and thus allows us to isolate and test approximation 1b.

*c*

_{gz}

*c*

_{z}

_{lw}

*A*

*k*is constant, and

*c*

_{z}

_{lw}

*m*

*m*is determined by the dispersion relation

*m*yields a quartic polynomial with two real and two imaginary roots for free-propagating waves, and the relevant real root for

*m*is easily identified. The general form of these solutions for

*m*can be illustrated by adopting the midfrequency approximation (4) so that (29) simplifies to

*m*and positive

*ω.*Solving for

*m*in (30) yields a quadratic with roots (Broutman and Young 1986).

*c*

_{z})

_{lw}< 0, the discriminant in (31) is always positive, so solutions always exist. This is not always so in the full quartic solution, where solutions are imaginary when the wave is refracted to a turning level (

*ω*

^{2}>

*ω*

^{2}

_{c}

*c*

_{z})

_{lw}also ensures one positive and one negative root for

*m*in (31), and we choose the latter since we set

*m*< 0 initially.

## 5. Limiting behavior of the *ζ* = 0 short-wave solutions

*ζ*= 0) short-wave solutions of section 4f. In defining these limits, it proves useful to introduce the ratio

*B*

_{z}=

*Ac*

_{gz}is the vertical flux of short-wave action density.

### a. The “fast” limit: *r* ≫ 1, *r* → ∞

For *r* ≫ 1, then *c*_{gz} ≫ −(*c*_{z})_{lw} and so the short-wave packet propagates upward much faster than the downward moving long-wave phase fronts. Thus, the time that the short wave takes to propagate through one vertical wavelength of the long wave is very much less than the period of the long wave.

“Fast limit” solutions follow by letting *r* → ∞, whereupon the solutions of section 4f reduce to those of section 3. Equation (33) simplifies to the familiar conservation with height of *B*_{z} = *c*_{gz}*A* (e.g., Schoeberl 1985), the same solution that governs the amplitude of the long wave in section 4b and of waves in Doppler-spread theory (Hines 1991b). Similarly, (*c*_{z})_{lw}*m* becomes negligibly small in this limit, whereupon (28) and (29) reduce to the original equations (1) and (3a) that we used in section 3 to model spreading effects initially.

Thus the longness criterion of section 3c is a necessary but insufficient condition for Approximation 1b to hold. We see here that *r* must also remain sufficiently larger than unity in order for the short-wave solutions of section 4f to be approximated acceptably by the *r* → ∞ solutions of Approximation 1.

### b. The “slow” limit: 0 < *r* ≪ 1, *r* → 0

For *c*_{gz} ≪ −(*c*_{z})_{lw} and thus 0 < *r* ≪ 1, the vertical group propagation of the short wave is much slower than the speed of the long-wave phase fronts. Thus, the vertical distance travelled by the short wave during one period of the long wave is very much less than the vertical wavelength of the long wave. This is easily seen from Fig. 1c: the long wave propagates its phase rapidly through the short wave for *c*_{gz} ≪ −(*c*_{z})_{lw}, so that many long-wave cycles pass through the short wave as it propagates slowly upward through the interval Δ*z.*

*r*→ 0, the ray equations (14a) and (14b) vanish so that

*m, k,*and hence

*ω*remain constant for the short wave, as originally discussed by Jones (1969) (see also Zhong et al. 1995; Eckermann and Marks 1996). From (33), (

*c*

_{z})

_{lw}

*A*remains constant; since (

*c*

_{z})

_{lw}is also constant in this problem, then

*A*

*A*(see also Broutman et al. 1997).

For upward propagating waves, it can be shown that the refraction of any given short wave by a long wave always lies somewhere between (but never equals) these fast and slow limits. Of course, most short-wave solutions differ significantly from either limit, as shown in Fig. 5, since changes in *m,* Ω, and *ω* are all significant during propagation (see also Eckermann and Marks 1996). This is investigated in more depth in sections 6–8.

### c. The “antipropagating” solutions: *r* < 0

When *r* < 0 and *c*_{gz} and (*c*_{z})_{lw} have the same sign, the long wave and short wave propagate in opposing vertical directions. In this case, the range of refraction of short-wave parameters by the long wave can exceed the fast limit described above, as discussed by Eckermann and Marks (1996).

This situation is more relevant to gravity waves in the ocean, where upward propagating and downward propagating gravity waves coexist with approximately equal energies (e.g., Munk 1981; D’Asaro and Morehead 1991). Here, the “phase–group condition” *c*_{gz} = (*c*_{z})_{lw} (*r* = −1) in particular has received considerable attention, initially because it was where certain resonant wave–wave interactions were thought to interchange wave action most efficiently (e.g., McComas and Bretherton 1977). The phase–group condition produces *A* → ∞ according to (33) and is a caustic where ray theory fails (Broutman 1986). Using a midfrequency approximation, *r* = −1 occurs when Ω" − *kU*_{lw}(*z, t*) → 0 (Broutman and Young 1986), so the mathematical condition for the singularity resembles a critical-level-type condition as viewed within the long-wave frame: indeed, the conventional critical level follows from these solutions as (*c*_{z})_{lw} → 0. However, phase–group caustics differ physically from critical levels in that *m* remains finite [see (31)], and short waves can survive these encounters. Accurate analysis of these events requires an extended ray theory that deals properly with the caustic singularity (e.g., Broutman 1986; Broutman et al. 1997).

Since gravity wave variance is dominated by upward propagating (*r* > 0) waves (e.g., Hirota and Niki 1985; Eckermann et al. 1995), antipropagating solutions are less immediately relevant to the atmosphere, and so we confine attention in the remainder of the paper to *r* > 0 solutions. Nevertheless, short waves undergoing successive reflections at turning levels may experience some of these antipropagating interactions with long waves during the downward portions of their group propagation. For example, the “separatrix crossings” at phase–group caustics (Broutman and Young 1986; Bruhwiler and Kaper 1995) resemble aspects of the Eckart’s resonances for ducted modes considered by Fritts and Yuan (1989). We also show in appendix A that upward propagating gravity waves may undergo similar sorts of caustic interactions in environments where the mean flow has a downward component.

### d. Phase–group ratios of refracted short waves

Figure 6 reproduces the refraction of the five short waves in Fig. 2, which used the solutions of section 3. A sample long wave is depicted in Fig. 6 with a shaded circle and is positioned to the left and below the initial position of the three largest-|*μ*_{0}| short waves, thereby satisfying longness criteria for these waves initially. This long wave has a vertical phase speed (*c*_{z})_{lw} = −1 m s^{−1}, and we assume that the short waves in Fig. 6 were refracted by it.

The labeled broken curves in Fig. 6 are contours of constant vertical group velocity *c*_{gz}. As the short waves are refracted to larger |*μ*| values, they intersect contours of progressively smaller *c*_{gz} values. Since (*c*_{z})_{lw} = −1 m s^{−1}, then from (32) these contours can be interpreted here as *r* values for these short waves. Clearly, as short waves are refracted by the long wave to |*μ*| ≫ 1, then 0 < *r* ≪ 1, and so, if anything, they tend more towards the slow limit of section 5b rather than the fast limit *r* ≫ 1 that was implicitly assumed when computing their refraction in section 3.

This result, together with the findings of section 5a, reveal that approximation 1b fails for the short waves at both large and intermediate |*μ*| values. In light of this, we now recompute the refraction of short waves using the *ζ* = 0 shortwave solutions of section 4f that do not use approximation 1b.

## 6. Shortwave refraction by a vertically propagating long wave

*c*

_{z})

_{lw}= −1 m s

^{−1}, as in Fig. 6. Given initial short-wave parameters (

*m*

_{0},

*ω*

_{0}) at the node of the long-wave velocity oscillation, then, from (29),

*ω*varies as

*m*varies so as to satisfy the dispersion relation (3a).

We note immediately from Fig. 7 that the range of short-wave refraction is reduced substantially compared to Fig. 2. This is consistent with the inferences from Fig. 6, where waves at |*μ*| ≫ 1 tended toward 0 < *r* ≪ 1, and so their refraction characteristics should more closely resemble the slow limit solutions of section 5b, in which no variation of short-wave parameters occurs.

This suppression of short-wave refraction can also be understood on inspecting (35). As short waves are refracted to |*m*| > |*m*_{0}|, then −*kU*_{lw} is negative (since *ω* must decrease) but (*m* − *m*_{0})(*c*_{z})_{lw} is positive [recall that *m, m*_{0}, and (*c*_{z})_{lw} are all negative]. Thus, the feedback effect of the latter term in (35) reduces the refraction compared to (2b). Similarly, when −*kU*_{lw} is positive and |*m*| < |*m*_{0}| in (35), then (*m* − *m*_{0})(*c*_{z})_{lw} is negative, which again reduces the refraction in (35) compared to (2b).

The feedback term (*m* − *m*_{0})(*c*_{z})_{lw} also produces changes in the ground-based frequency Ω, according to (28) (since Ω" is constant). These changes oppose the changes in *ω* due to the long-wave velocity *U*_{lw}. Note that the feedback term is limited in magnitude as *m* → 0 and so cannot ultimately prevent turning levels. However, as *m* → ∞, then (*m* − *m*_{0})(*c*_{z})_{lw} → ∞, which from (29) ensures that critical levels do not occur: the accelerations of the long-wave oscillation always refract short waves away from a Jones critical level (*ω* = *f*) (Eckermann and Marks 1996).

Thus, in repeating the presentation of Fig. 3 using these short-wave ray solutions, a critical-level criterion is no longer relevant. We replace it by following Approximation 3 of Hines (1993b) and introducing a maximum wavenumber *μ*_{M}, beyond which waves are dissipated completely into turbulence. A value of *μ*_{M} = 200 is adopted from section 4 of Hines (1991b).

Figure 8 shows four plots, each depicting the end result of subjecting short waves to a long-wave velocity oscillation *U*_{lw} in the range ±8.3 m s^{−1} (*β*_{max} = 1), for which the theory of section 3 predicted a sharp cutoff at |*μ*_{0}| ∼ 1. In each plot, a different value of (*c*_{z})_{lw} was used: (a) −10 m s^{−1}, (b) −1 m s^{−1} (as in Figs. 6–7), (c) −0.1 m s^{−1}, and (d) −10^{−3} m s^{−1}. The |(*c*_{z})_{lw}| isoline is shown with a broken curve on each plot, and the long wave must lie somewhere along it in each case. The line |*μ*| = *μ*_{M} is also shown.

Clearly the results in Figs. 8a–c differ substantially from those in Fig. 3. In Fig. 8a, the characteristics at |*μ*| < *μ*_{M} are nearly independent of |*μ*|, with free propagating short waves occurring everywhere except at *ω̂*_{0}|*c*_{z})_{lw} = −1 m s^{−1}], a collection of waves appear at 1 ≳ |*μ*_{0}| ≳ 100 that are now reflected at turning levels by the long wave. For (*c*_{z})_{lw} = −0.1 m s^{−1} in Fig. 8c, we now see that waves for which |*κ*| ≳ 10^{0.3} can no longer propagate freely. Note, however, that there is still no evidence of a short-wave cutoff at |*μ*_{0}| ∼ 1—many free-propagating short waves still exist at |*μ*_{0}| ≫ 1. Only as (*c*_{z})_{lw} → 0 do characteristics resembling those in Fig. 3 return in Fig. 8d (since *r* ≫ 1). However, the |(*c*_{z})_{lw}| isoline in Fig. 8d, on which the long wave must lie, yields |*μ*_{lw}| ≫ 1, whereupon the longness criterion is violated for most short waves on this diagram.

Thus we conclude that incorporating the vertical propagation of the long wave removes any tendency for a distinct cutoff in free-propagating short waves at |*μ*_{0}| ∼ 1 when maximum normalized long-wave wind variations of *β*_{max} = 1 are present, in contrast to the findings of Doppler-spread theory. Note, however, that the Doppler-spread analysis of Hines (1991b, 1993b) did not impose any longness criteria on interacting waves, and so it could still simulate a cutoff since it would not reject the results in Fig. 8d (even though Approximations 1 and 2 mostly fail in this case).

We shall now look to incorporate additional effects that we have omitted up to this point, to see whether they affect these findings in any significant way.

## 7. Effect of the vertical velocity and horizontal wavelength of the long wave (Approximations 1a and 1c)

Approximation 1 also neglects the effects of long-wave vertical velocities (Approximation 1a) and horizontal wavelengths (Approximation 1c), as has our analysis up to this point. Hines (1991b, 1993b) extended his analysis to investigate how larger-scale vertical velocities might modify his Doppler-spread theory (see also appendix A). We can gauge the effects in our simplified solutions by now considering the general (*ζ* ≠ 0) ray solutions of section 4e, in which the effect of both the vertical velocity and the horizontal wavelength of the long wave on the short wave are included.

Figure 9 considers the same five short waves depicted in earlier figures. Here, these waves were refracted by a long wave with (*c*_{z})_{lw} = −1 m s^{−1} and *ζ* = −1.5° (panel a) and +1.5° (panel b). These small *ζ* values yield only small *W*_{lw} values, but |*ζ*| values any larger than these yield high-frequency long-wave oscillations and violations of longness criteria for the short waves.

Since *k*_{lw} ≠ 0 here, (14a) no longer vanishes, so some spreading of short-wave *k* values now occurs. Thus, the refraction of short waves in Fig. 9 is no longer confined along a *κ* curve. While we note some differences in the refraction characteristics from Fig. 7, it is clear that the horizontal wavelength and vertical velocity of the long-wave oscillation do not produce major changes in the refraction characteristics of these short waves. Note, however, that short-wave refraction is greater for *ζ* < 0 than for *ζ* > 0. In the former case, increases and decreases in *k* accentuate similar changes in *ω,* whereas for *ζ* > 0, *k* tends to increase as *ω* decreases (and vice versa), which acts to reduce the range of short-wave refraction.

## 8. Effect of more than one long-wave oscillation

*m*and

*ω*values. However, Doppler-spread theory considers the statistical properties of waves refracted by a randomly phased spectrum of many other waves. To study this to some extent, we consider a background field of linearly superposed long waves of the form

*i*denotes the

*i*th long-wave oscillation. The superposition is linear in that we have ignored any Doppler spreading of individual long-wave parameters or WKB violations that might arise as these long waves interact with each other. This is similar to the approach taken by Henyey and Pomphrey (1983) in their modeling of short-wave refraction in the ocean.

*J*= 8 long waves and gave each one the same peak amplitude of (

*U*

_{max})

_{i}= 4 m s

^{−1}, which from (36a) and using the same

*m*

_{c}and

*N*values as before, yields an rms

*β*value of

*x*

_{0}=

*z*

_{0}=

*t*

_{0}= 0 through nine separate realizations of the superposed long-wave velocity field (36a)–(36c), changing the initial phases (

*ϑ*

_{lw})

_{i}in each simulation. To match our earlier results as much as possible, we set

**U**

_{lw}(0, 0, 0) = 0 so that the short wave always commenced propagation within a stationary atmosphere. This required choosing (

*ϑ*

_{lw})

_{i}=

*n*180°, where

*n*is a random integer. The short-wave ray equations were integrated using a 1 s time step.

Results for the five short waves considered earlier are shown in Fig. 10, for cases where (*k*_{lw})_{i} < 0 (panel a) and (*k*_{lw})_{i} > 0 (panel b). Nine curves are plotted for each short wave (i.e., 45 curves in each panel), summarizing short-wave refraction within each of the nine realizations of the long-wave velocity field (36a)–(36c). In addition to turning levels, ray integrations were also terminated if the short wave propagated either to a height of 100 km or for a time exceeding ∼2.66 days (∼3.75 inertial periods). These limits allowed free-propagating short waves to propagate through a minimum of eight cycles of each long-wave oscillation, thus allowing a more statistical picture of the short-wave refraction to emerge.

The range of refraction in Fig. 10 is greater than in Figs. 7 and 9. This occurs because *β* values greater than unity can arise here as the individual long waves superpose (*β*_{max} ≈ 4 in these experiments). Despite this, there were no occurences of either critical levels or |*μ*| values exceeding *μ*_{M} = 200, although two of the five waves were refracted to turning levels in some integrations (see Fig. 11). As in Fig. 9, short-wave refraction was greater for (*k*_{lw})_{i} < 0 (*ζ*_{i} < 0) than for (*k*_{lw})_{i} > 0. Indeed, the basic characteristics of the refraction in Fig. 10 do not differ greatly from those in Figs. 7 and 9. In particular, there is again no tendency for short waves at |*μ*_{0}| ≳ 1 to be removed via refraction to small vertical scales (|*μ*| > *μ*_{M}).

*μ*

_{0}| ≈ 0.9 and

*ω̂*

_{0}|

*k*

_{lw})

_{i}< 0 simulations (Fig. 11a), the short-wave group sometimes moves downward for a time. That the wave remains free propagating in these instances can be understood from the ray equation

*c*

^{gnd}

_{gz}

*x, z, t*) is the ground-based vertical group velocity of the wave. Instances of downward group motion in Fig. 11a occur when

*W*

_{lw}(

*x, z, t*) <

*c*

^{gnd}

_{gz}

*x, z, t*) < 0 so that the intrinsic frame is moving downward and the intrinsic group velocity remains upward (

*c*

_{gz}(

*x, z, t*) > 0).

## 9. Validity of WKB ray theory (Approximation 2)

We have made extensive use of ray theory in sections 4–8, which requires that WKB plane-wave solutions for the short waves remain valid (i.e., that Approximation 2 always holds). We ensured this initially for the short waves using the longness criterion of section 3c. Here we briefly address the applicability of WKB ray theory for short waves within propagating long-wave fields.

For ray theory to remain valid within arbitrary environments, changes in wavenumber and frequency must always be much slower than the phase variations of the wave motion itself. Dimensionless parameters have been derived that check whether these conditions hold and can be evaluated at each time step during ray integrations (e.g., Einaudi and Hines 1970; Broutman 1984; Eckermann and Marks 1996). The results show that, typically, ray theory only breaks down at caustics (e.g., Broutman and Young 1986) or when the long-wave shear is so strong that it produces gradient Richardson numbers ≲1 (Broutman 1984). In the scale-separated ray simulations reported here, WKB requirements were violated and ray theory failed only as waves approached critical levels and turning levels, at which point we terminated further tracing of the ray. Elsewhere, approximation 2 remained valid. Thus, while we have demonstrated limited validity for approximation 1 in this study, approximation 2 is much more robust, giving us confidence that WKB ray methods can provide further insights into aspects of wave–wave interactions and wave spectra (e.g., Flatté et al. 1985; Henyey et al. 1986).

## 10. Discussion

The idealized problems in this paper were devised to provide insights into the range of validity of Approximations 1–3 of Doppler-spread theory, as well as interactions among atmospheric gravity waves in general. Critical scrutiny of these approximations was anticipated in two recent review articles by Hines (1996, 1997c). In these articles, Doppler-spread theory was cast within the broad general framework of advective nonlinearity, with mathematical development minimized to deflect criticisms of the theory based on the approximations used in earlier studies. Thus, potential ramifications of the present results for Doppler-spread theory have been anticipated to some extent and preemptively rebutted by Hines (1996, 1997c).

For example, Doppler-spread theory used Approximations 1–3 to derive model spectra of relative potential-temperature fluctuations at large Fourier wavenumbers *M* that asymptoted to *M*^{−1} shapes for *f* = 0 (see appendix B), as opposed to the observed *M*^{−3} shape. Hines (1993b, 1997c) argued that this breakdown in the theory at large *M* arose due to the onset of nonnegligible spreading by wave-induced vertical-velocity fluctuations (i.e., Approximation 1a broke down) but that Doppler-spread theory gave acceptable results for waves of intermediate |*m*| values up to ∼10 *m*_{c}. Derived spectra in this range had shapes close to *M*^{−3} at *M* > *M*_{c} ∼ *m*_{c}.

Our scale-separated simulations indicated that it was Approximation 1b—the neglect of time variations of the background wave field—that broke down most notably at large |*m*|. Furthermore, it failed not just for large |*m*| waves, but for all but the smallest |*m*| (fastest *c*_{gz}) waves (see Fig. 6). When Approximation 1b was removed for the specific case of a short wave propagating through a long-wave background, any tendency for a sharp increase in the probability of short-wave obliteration at |*μ*_{0}| ≳ 1 (|*m*| ∼ *m*_{c}) was eliminated. Extensions of this model to include horizontal wavelengths (i.e., removal of approximation 1c), vertical-velocity fluctuations (removal of approximation 1a), and a background containing more than one long wave did not lead to any major changes in these basic conclusions.

The imposition of scale separation was critiqued by Hines (1997c) because spreading is produced only by long waves and so additional spreading by waves of similar or shorter spatial scales is omitted. Spreading by waves of all scales is permitted within Doppler-spread theory. While this of course is a valid criticism, scale separation was imposed here to ensure consistency with Approximations 1–2 of the original theory of Hines (1991b, 1993b). The view taken here is that these scale-separated simulations represent a proper application and extension of the original equations of Hines (1991b, 1993b) to a restricted class of interactions to which they are applicable. These simulations supported the basic contention of Doppler-spread theory that gravity wave *m* values are spread more as one progresses to larger |*m*_{0}| values (e.g., Figs. 7, 9, and 10), and culminated in section 8 with numerical simulations that resembled the approach taken by Henyey and Pomphrey (1983) to model oceanic wave interactions.

Thus, while Doppler-spread theory is conceived to be a comprehensive model of broadband nonlinear advective processes, we view its current mathematical formulations as a simplified version of the “eikonal” (ray tracing) spectral models of oceanic gravity waves (e.g., Flatté et al. 1985; Henyey et al. 1986; Broutman et al. 1997). Hines (1991b, 1996, 1997c), on the other hand, argued that Doppler-spread theory corresponded best to “case III” of a general statistical mechanical theory of oceanic gravity waves developed by Allen and Joseph (1989). This model is discussed extensively by Hines (1991b, 1993b, 1996, 1997c), and an atmospheric version of it was presented recently by Chunchuzov (1996). Hines (1991b, 1997c) argued that the Allen and Joseph (AJ) model represented a full formal embodiment of Doppler-spread interactions, and he deferred to its findings at *M* ≫ *M*_{c}, where it produced the required *M*^{−3} spectrum. Hines (1996, 1997c) further argued that the Doppler-spread theory of Hines (1991b, 1993b) complemented the AJ model by explaining physically both the formation of wavenumber cutoffs and the initial transition to decaying “tail spectra” at scales smaller than the cutoff wavenumbers.

However, the AJ model differs from eikonal models. It does not include refraction, but instead tracks the velocities and displacements of air parcels under the influence of a prescribed canonical distribution of wave modes. These Lagrangian displacements and velocities are then transformed to Eulerian measurement frames, and spectra are computed. While no modes exist at wavenumbers |*m*| > *m̃*_{c} (where *m̃*_{c} is the Lagrangian cutoff wavenumber), advective nonlinearities within the Eulerian frame produce variance at *M* ≫ *M*_{c} ∼ *m̃*_{c} with an *M*^{−3} spectrum. This variance departs from the gravity-wave dispersion surface and so is considered to be nonwavelike and essentially devoid of physical significance when viewed from the Eulerian frame (Allen and Joseph 1989): Hines (1996, 1997c) has coined the term “wavulence” to describe it.

Conversely, no violations of the dispersion relation occur for waves in an eikonal formulation within the Eulerian frame: the system always remains entirely wavelike (Approximation 2), even as waves are refracted to larger |*m*| values to produce tail spectra. Allen and Joseph (1989) discuss other possible differences between their theory and eikonal theories on p. 216 of their paper (while noting that precise relationships between them are unclear). Given that the Doppler-spread formulation of Hines (1991b, 1993b) is viewed here as being a simplified eikonal formulation, then the degree to which Doppler-spread theory corresponds to the AJ model is unclear at present.

While Doppler-spread theory now defers to the AJ model in many areas, the original formulation of Hines (1991b, 1993b) is still used to predict the cutoff wavenumber *m*_{c}. This cutoff is a primary parameter in recent Doppler-spread parameterizations of gravity-wave momentum deposition (Hines 1997a, 1997b), yet no clear cutoff occurred in our scale-separated simulations once Approximation 1b was eliminated (see Fig. 8). If Doppler-spreading processes do produce a cutoff at *M* ∼ *M*_{c} and *M*^{−3} spectra thereafter, as argued by Hines (1991b, 1993b), then it seems that further extensions and refinements of the theory are needed to explain this result self-consistently.

Many important effects were omitted from our simulations. For example, we did not incorporate vertical profiles of mean wind velocities *U**z*), which refract waves appreciably as they propagate through the middle atmosphere in a manner consistent with the theory of sections 2 and 3 (e.g., Lindzen 1981; Eckermann 1995). Other important omissions include wave-induced thermal gradients, different horizontal propagation azimuths, damping, and saturation. While these processes can be incorporated within ray-based simulations (e.g., Marks and Eckermann 1995; Eckermann and Marks 1996), others cannot, such as strongly nonlinear non-scale-separated wave–wave and wave–turbulence interactions (e.g. Frederiksen and Bell 1984; Müller et al. 1986; Allen and Joseph 1989; Weinstock 1990; Sonmor and Klaassen 1996).

The experience gained from oceanic and laboratory studies may aid further understanding of atmospheric gravity wave interactions and spectra. The oceanic review of Müller et al. (1986) specifically recommended direct nonlinear numerical simulations of the oceanic wave field, and models of this kind have produced spectra in good agreement with observations (e.g., Orlanski and Cerasoli 1981; Shen and Holloway 1986; Hibiya et al. 1996). Recent numerical and laboratory experiments have also generated *M*^{−3} spectra as gravity waves become unstable and break (Benielli and Sommeria 1996; Bouruet-Aubertot et al. 1996), and wave overturning also observed by Shen and Holloway (1986) within their numerically simulated ^{−3} spectral regime. These issues relate to Approximation 3 of Doppler-spread theory, which we have said little about here, and merit further investigation in the atmosphere where wave dissipation is large. Similar models of atmospheric gravity wave spectra are still very limited (Frederiksen and Bell 1984; Huang et al. 1992) but are now feasible using existing codes and computing capabilities (e.g., Fritts et al. 1996).

## 11. Conclusions

In this study, we assessed key approximations used in Doppler-spread models of atmospheric gravity waves. Our results confirmed that the velocity oscillations of long waves produce important refractive spreading of short-wave vertical wavenumbers and frequencies in the atmosphere. However, we found that Doppler-spread models of these effects, which ignore the propagation of the long-wave velocity field, did not adequately describe either the amplitudes or group trajectories of short waves as they were refracted to shorter vertical wavelengths. We employed ray methods to describe more accurately the refraction and amplitude characteristics of short waves propagating through long waves. These solutions revealed a considerable reduction in the amount of wavenumber spreading compared to Doppler-spread predictions. Most notably, a cutoff in unspread vertical wavenumbers at |*m*_{0}| ∼ *m*_{c} (|*μ*_{0}| ∼ 1), which is a key aspect of Doppler-spread models of gravity-wave fluctuation spectra and momentum deposition, no longer occurred.

The simulations reported here dealt with a restricted and idealized class of advective wave–wave interactions. Nevertheless, the results indicated that more work is needed to understand the manner by and extent to which Doppler-spread-like processes influence the spectra and momentum dissipation of gravity waves within the atmosphere.

## Acknowledgments

I would particularly like to thank Dave Broutman of the University of New South Wales, Sydney, Australia, for valuable discussions and advice throughout this study. Thanks also to Joan Alexander, Colin Hines, and Bryan Lawrence for comments on an earlier draft, and to Mary Anderson and Jane Ford at Computational Physics, Inc., and Mary-Ann Lindsey at the Naval Research Laboratory for their help with the administration and public release of this paper. This research was supported by Contract N00014-95-C-2152 of the Naval Research Laboratory, Washington, D.C.

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## APPENDIX A

### Gravity Wave Propagation through a Nonhorizontal Shear Flow

*U*(

*z*), 0,

*W*(

*z*)), In the midfrequency approximation, a gravity wave propagating through such a region must satisfy the dispersion relation

*N*and small curvature in the flow profile. Equation (A1) was used by Hines (1991b, 1993b) to gauge the effect that random

*U*and

*W*oscillations of larger-scale waves might have on the Doppler spreading of shorter waves. Hines (1991b) noted that (A1) produced a quadratic equation for

*m.*Our purpose here is to highlight some interesting physical consequences of these solutions.

*W*(

*z*) appears in place of (

*c*

_{z})

_{lw}and Ω appears in place of Ω" (note also that Ω and Ω" are both constants). Hence, from (31), the solution of (A1) is

*c*

_{gz}

*W*

*A*

*c*

_{z})

_{lw}replaced by −

*W*(

*z*). Thus, the effects discussed in section 5 for the short-wave solutions (27) and (31) can also be applied to this problem.

*W*(

*z*) > 0, the solutions have the properties discussed in sections 5a and 5b. Thus critical levels vanish, and the amount of wavenumber–frequency refraction is reduced. Note in particular that when

*c*

_{x}−

*U*vanishes (the standard critical-level criterion for

*W*= 0), then, from (A1) and (A2), the wave survives and has a vertical wavenumber

*m*=

*m*

_{r}and frequency

*ω*= −

*m*

_{r}

*W,*where

*W*(

*z*) < 0 has much in common with the antipropagating solutions discussed in section 5c. No solutions for

*m*are possible when the discriminant in (A2) is negative. This also occurs at

*m*=

*m*

_{r}, which in this case corresponds to the condition

*c*

_{gz}

*W.*

*A*→ ∞ according to (A3), which is a caustic singularity equivalent to the one studied by Broutman and Young (1986). The explosive buildup of wave action density near these caustic points implied by (A5) and (A3) suggests that wave saturation may be possible, depending on the initial amplitude of the wave (see also Broutman 1986; Broutman et al. 1997). WKB conditions break down in these regions, and Broutman and Young (1986) argued that “separatrix crossings” occur at the caustic point, which correspond to a transfer from one root to the other in (A2), an assertion supported in later studies by Broutman (1986) and Bruhwiler and Kaper (1995). The caustic (

*m*=

*m*

_{r}) occurs at an intrinsic horizontal phase speed of

*W*≠ 0, the “critical level” criterion |

*c*

_{x}−

*U*| = 0 can be replaced by a more general “caustic level” criterion

*c*

_{gz}+

*W*= 0 implied by (A3) and (A5). The critical level then follows as a limiting form of the caustic-level condition as

*c*

_{gz}and

*W*both approach zero.

Figure A1 shows the variation of |*m*| as a function of *c*_{x} − *U* for various values of *W* and *k.* The turning points on the *W* < 0 curves correspond to the caustic point, as defined by (A4) and (A6), and demark where the solutions cross from one of the roots in (A2) to the other. Note too that the *W* > 0 curves yield *m* = *m*_{r} at *c*_{x} − *U* = 0.

## APPENDIX B

### Asymptotic “Doppler Spread” Spectra at Large *M*

^{2}(

*k, m*). Approximations 1 and 2, which use “fast limit” solutions (see section 5a), yielded a spectral conservation relation (e.g. Hines 1991b)

*B*

_{z}with height for short-wave solutions in the fast limit. Hines used midfrequency approximations, which, from (5), yield

*dm*

_{0}/

*dm*= (

*m*

_{0}/

*m*)

^{2}, so (B1) becomes

^{2}

*k, m*

^{2}

_{0}

*k, m*

_{0}

*m*

_{0}

*m*

^{−1}

*m*values by larger-scale waves using probability density functions, which he integrated through the right hand side of (B2) using models of

^{2}

_{0}

*k, m*

_{0}) to yield a one-dimensional model Doppler-spread spectrum Θ

^{2}(

*M*). The

*m*

^{−1}factor in (29) carried through this integration and ultimately yielded an asymptotic

*M*

^{−1}dependence of the spectrum at

*M*≫

*M*

_{C}(where

*M*

_{C}=

*N*/2

*σ*

_{T}in his derivation and

*σ*

_{T}is the total horizontal velocity variance of the wave field).

*ζ*= 0 short-wave solutions in the midfrequency approximation. Since in this particular problem nondissipating short waves conserve −(

*c*

_{z})

_{lw}

*A*(

*r*+ 1) from (33), where

*A*=

*E*/

*ω*≈ −

*Em*/

*kN,*and

*k, N,*and (

*c*

_{z})

_{lw}are constants, then this implies

*r*

_{0}is the phase/group ratio at

*m*

_{0}. Combining (30) and (35) to eliminate frequency terms, and then differentiating, yields

*M*

^{−1}spectral shape at large Fourier wavenumbers

*M.*Note that this derivation holds only for a short wave propagating though a single low-frequency (

*ζ*≈ 0) long wave of constant amplitude in an unsheared mean flow.