• Andreassen, O., C. E. Wasberg, D. C. Fritts, and J. R. Isler, 1994: Gravity wave breaking in two and three dimensions. Part I: Model description and comparison of two-dimensional evolutions. J. Geophys. Res.,99, 8095–8108.

  • Barat, J., 1983: The fine structure of the stratospheric flow revealed by differential sounding. J. Geophys. Res.,88, 5219–5228.

  • Dunkerton, T. J., 1984: Inertia–gravity waves in the stratosphere. J. Atmos. Sci.,41, 3396–3404.

  • ——, 1987: Effect of nonlinear instability on gravity wave momentum transport. J. Atmos. Sci.,44, 3188–3209.

  • ——, 1997: Shear instability of internal inertia–gravity waves. J. Atmos. Sci.,54, 1628–1641.

  • Durran, D. R., 1991: The third-order Adams–Bashforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev.,119, 702–720.

  • Fritts, D. C., and P. K. Rastogi, 1985: Convective and dynamical instabilities due to gravity wave motions in the lower and middle atmosphere: Theory and observations. Radio Sci.,20, 1247–1277.

  • ——, and L. Yuan, 1989: Stability analysis of inertio–gravity wave structure in the middle atmosphere. J. Atmos. Sci.,46, 1738–1745.

  • ——, J. F. Garten, and O. Andreassen, 1996: Wave breaking and transition to turbulence in stratified shear flows. J. Atmos. Sci.,53, 1057–1085.

  • Hazel, P., 1972: Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech.,51, 39–61.

  • Howard, L. N., 1961: Note on a paper of John W. Miles. J. Fluid Mech.,10, 509–512.

  • Klostermeyer, J., 1991: Two- and three-dimensional parametric instabilities in finite-amplitude internal gravity waves. Geophys. Astrophys. Fluid Dyn.,61, 1–25.

  • Lelong, M.-P., and T. J. Dunkerton, 1998: Inertia–gravity wavebreaking in three dimensions. Part II: Convectively unstable waves. J. Atmos. Sci.,55, 2489–2501.

  • Lombard, P. N., and J. J. Riley, 1996: On the breakdown into turbulence of propagating internal waves. Dyn. Atmos. Oceans,23, 345–355.

  • Miles, J. W., 1961: On the stability of heterogeneous shear flows. J. Fluid Mech.,10, 509.

  • Munk, W. H., 1981: Internal waves and small-scale processes. Evolution of Physical Oceanography, B. A. Warren and C. Wunsch, Eds., The MIT Press, 264–291.

  • O’Sullivan, D., and T. J. Dunkerton, 1995: Generation of inertia–gravity waves in a simulated lifecycle of baroclinic instability. J. Atmos. Sci.,52, 3695–3716.

  • Patterson, G. S., and S. A. Orszag, 1971: Spectral calculations of isotropic turbulence: Efficient removal of aliasing interactions. Phys. Fluids,14, 2538–2541.

  • Winters, K. B., and E. A. D’Asaro, 1994: Three-dimensional wave instability near a critical level. J. Fluid Mech.,272, 255–284.

  • Yamanaka, M. D., and H. Tanaka, 1984: Multiple “gust layers” observed in the middle stratosphere. Dynamics of the Middle Atmosphere, J. R. Holton and T. Matsuno, Eds., Terra Scientific, 117–140.

  • Yuan, L., and D. C. Fritts, 1989: Influence of a mean shear on the dynamical instability of an inertio-gravity wave. J. Atmos. Sci.,46, 2562–2568.

  • View in gallery

    Density field, with velocity vectors superposed, of an inertia–gravity wave with leftward and downward phase propagation at t = 0 in (a) the vertical plane of wave propagation and (b) the vertical plane transverse to the propagation.

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    Time evolution of the density field in a numerical simulation with IGW parameters (R, a) = (0.95, 0.7): (a, b, c) xz plane at y = 23.4 km; (d, e, f) y–z plane at x = 11.7 km. Three times are shown: at (a, d) 0.42Tw, (b, e) 0.63Tw, and (c, f) 0.70Tw, where Tw is the IGW period.

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    (Continued)

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    Vertically integrated kinetic energy spectrum for the simulation of Fig. 2 at t = 0.70Tw. Contours are 0.1 and 0.5 of the maximum value, and the primary wave is excluded. The horizontal resolution is 256 × 256; only the significant part of the spectrum is shown.

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    Volume-integrated Reynolds stress as a function of time for the simulation of Fig. 2.

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    Filtered vertical velocity field for the simulation of Fig. 2 at t = 0.63Tw: (a) xz plane at y = 23.4 km and (b) yz plane at x = 11.7 km.

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    Contours of local Richardson number J, at 16 equally spaced intervals from min(J) to J = 0.25, for a convectively stable inertia–gravity wave with (R, a) = (0.95, 0.7).

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    Cross section of density in the transverse vertical plane at x = 1.45 km in a simulation with IGW parameters (R, a) = (0.6, 0.95).

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    Hovmöller plot of filtered density for the simulation of Fig. 7, at y = 4.9 km, z = 0.75 km. IGW propagation is from right to left as a function of time. The filtered data were normalized by a temporally smoothed version of the maximum value at each time step in the plot.

  • View in gallery

    Vertically integrated kinetic energy spectrum as in Fig. 3, with amplitude a = 0.95, for (a) R = 0.6, (b) R = 0.7, (c) R = 0.8, and (d) R = 0.9. The horizontal resolution is 128 × 128; only the significant part of the spectrum is shown.

  • View in gallery

    Energy of the most unstable mode for 0.6 ⩽ R ⩽ 0.95, with amplitude a = 0.95, as a function of time. The energy has been normalized by the energy of the primary wave at t = 0.

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    Predicted and observed shear instability thresholds as a function of R.

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    Vertically integrated kinetic energy spectrum in a series of runs with IGW parameters (R, a) = (0.95, 0.8) for (a) N/f = 10, (b) N/f = 20, and (c) N/f = 30. The horizontal resolution is 256 × 256; only the significant part of the spectrum is shown.

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Inertia–Gravity Wave Breaking in Three Dimensions. Part I: Convectively Stable Waves

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Abstract

The three-dimensional breakdown of a large-amplitude, convectively stable inertia–gravity wave is examined numerically as a function of primary-wave frequency and amplitude. The results confirm that inertia–gravity waves in this region of parameter space break down preferentially via shear instability. In low-frequency waves the instability is ubiquitous, occurring simultaneously throughout the wave field, and the spectrum of instability energy is approximately, but not exactly, isotropic in azimuthal orientation. In higher-frequency waves, shear instability develops adjacent to the region of reduced static stability, and displays a preference for intermediate azimuths (e.g., near 45°). Near-inertial waves experience the fastest growing instabilities. The growth rate of shear instability drops off rapidly as the wave frequency is increased and, for all frequencies, increases with increasing wave amplitude. At most frequencies, the onset of modal shear instability occurs at a wave amplitude slightly above the theoretical stability boundary determined from a local Richardson number argument.

Corresponding author address: Dr. Timothy J. Dunkerton, Northwest Research Associates, Inc., P.O. Box 3027, Bellevue, WA 98009-3027.

Email: tim@nwra.com

Abstract

The three-dimensional breakdown of a large-amplitude, convectively stable inertia–gravity wave is examined numerically as a function of primary-wave frequency and amplitude. The results confirm that inertia–gravity waves in this region of parameter space break down preferentially via shear instability. In low-frequency waves the instability is ubiquitous, occurring simultaneously throughout the wave field, and the spectrum of instability energy is approximately, but not exactly, isotropic in azimuthal orientation. In higher-frequency waves, shear instability develops adjacent to the region of reduced static stability, and displays a preference for intermediate azimuths (e.g., near 45°). Near-inertial waves experience the fastest growing instabilities. The growth rate of shear instability drops off rapidly as the wave frequency is increased and, for all frequencies, increases with increasing wave amplitude. At most frequencies, the onset of modal shear instability occurs at a wave amplitude slightly above the theoretical stability boundary determined from a local Richardson number argument.

Corresponding author address: Dr. Timothy J. Dunkerton, Northwest Research Associates, Inc., P.O. Box 3027, Bellevue, WA 98009-3027.

Email: tim@nwra.com

1. Introduction

Inertia–gravity waves (IGWs)—internal gravity waves influenced by the Coriolis effect—are a common feature of geophysical flows. In the earth’s atmosphere, IGWs dominate the gravity wave signal in the upper troposphere and lower stratosphere and, as such, may be instrumental in vertical mixing and stratosphere/troposphere exchange (O’Sullivan and Dunkerton 1995). Horizontal wavelengths of midlatitude IGWs range from ∼200 to 1000 km. Vertical wavelengths are much smaller, on the order of ∼1 to 5 km, and characteristic periods are several hours or more. IGWs are also important in the ocean since they account, on average, for nearly half of the energy found in the oceanic internal waveband (Munk 1981).

High-resolution measurements of wind and temperature in the lower stratosphere, obtained by the differential sounding technique, have provided unique information on the structure of atmospheric inertia–gravity waves. These observations reveal “gust layers” and turbulent microstructures evidently associated with the breakdown of IGWs (Barat 1983; Yamanaka and Tanaka 1984). Turbulent layers—possibly associated with Kelvin–Helmholtz billows—are found primarily in regions of low Richardson number, supporting the conjecture (based on a local Richardson number argument) that IGWs break down via shear instability (Dunkerton 1984; Fritts and Rastogi 1985). Fritts and Yuan (1989) further substantiated this prediction by examining the stability of a single IGW as a function of frequency and azimuthal angle for a fixed value of local Richardson number. They found that the growth rate of instability is largest in a direction transverse to the IGW propagation and drops off rapidly as the frequency is increased away from the inertial frequency. When a mean shear is added, the instability tends to align with the shear; growth rates are also enhanced (Yuan and Fritts 1989). It should be noted that Fritts and Yuan (1989) used a hyperbolic tangent profile to approximate the IGW velocity field in the most unstable region and assumed a steady, parallel flow instead of a propagating IGW. A somewhat more realistic stability analysis performed by Dunkerton (1997), using velocity profiles similar to those of an inertia–gravity wave packet, showed that the growth rate of shear instability in a near-inertial IGW is almost independent of azimuth. Like Fritts and Yuan, Dunkerton (1997) assumed a steady, parallel flow. The instability of propagating large-amplitude IGW, over most of the parameter space, remains to be examined.

Direct numerical simulations of gravity wave breakdown in three dimensions have, until now, focused mainly on nonrotating internal waves with overturned isopycnals (Winters and d’Asaro 1994; Andreassen et al. 1994; Fritts et al. 1996). In convectively unstable waves, breakdown occurs via a combination of parallel shear instability and transverse convection, in agreement with Dunkerton’s (1997) analysis. In nonrotating, convectively stable waves without mean shear, local shear instability cannot occur immediately, but parametric instabilities are possible (Klostermeyer 1991; Lombard and Riley 1996), albeit on a longer timescale (Dunkerton 1987).

Rotation introduces a horizontal velocity component transverse to the direction of IGW propagation. The vertical shear of this component has a maximum at the point of minimum static stability so that transverse shear instability is possible in convectively stable waves (Dunkerton 1984; Fritts and Rastogi 1985). The vertical wavelength of IGW is reduced by rotation so that for convectively stable waves, shear instability can also occur in a direction parallel to IGW propagation (Dunkerton 1997).

Our numerical problem differs in several respects from its nonrotating counterpart, namely in the broad range of temporal and spatial scales that must be resolved simultaneously and the inherent three-dimensionality of the wave field. This renders the computation very memory and time intensive. The calculation can be economized, as done here, by considering a dynamically similar problem in which the ratio of buoyancy and Coriolis frequency (N/f) is artificially reduced.

The objectives of this study are to simulate the onset of instability in a monochromatic, large-amplitude, propagating inertia–gravity wave; to identify the dominant modes of instability; and to assess the nature and growth rate of instability as a function of primary-wave amplitude and frequency. Part II of this series examines the convectively unstable case (Lelong and Dunkerton 1998).

The paper is organized as follows. Section 2 introduces the numerical model and reviews the basic properties of inertia–gravity waves. The concept of a local Richardson number for a single IGW is also discussed and a brief description of the various terms that contribute to the energy budget is given. Section 3 is devoted to the numerical simulations. We first examine in detail the results of a high-resolution simulation of a low-frequency wave (section 3a). The nature and growth rates of instability are then examined as a function of primary-wave frequency (section 3b) and amplitude (sections 3c,d). Simulations in sections 3a–d were performed with N/f, the ratio of buoyancy to Coriolis frequency, set equal to 10. In section 3e, we examine the effect of increasing N/f to geophysically more realistic values. Conclusions are drawn in section 4.

2. Background

a. Numerical model

The model’s governing equations are the three-dimensional, nonlinear, Boussinesq equations of motion with f-plane rotation, a thermodynamic or density equation, and a continuity equation:
i1520-0469-55-15-2473-e2-1a
where u = (u, υ, w) is the velocity vector, p and ρ are pressure and density respectively, ρ0 is a constant reference density, and ρ(z) is the (assumed linear) background density profile.

The numerical model is an efficient pseudospectral code for (2.1a–c) in a triply periodic domain of dimension Lx, Ly, Lz. The code is pseudospectral in that the computation of nonlinear terms is performed in physical space while the evaluation of derivatives and time stepping are done in spectral space. Optimized FFTs are used to transform fields back and forth between physical and spectral space.

We implemented a third-order Adams–Bashforth (AB3) time stepping algorithm, which has been shown to remain very stable over long time integrations (Durran 1991). Aliasing errors are minimized with the 8/9 rule (Patterson and Orszag 1971). This allows significantly more modes to be retained than the exact two-thirds rule, nonetheless maintaining reasonable accuracy.

A sixth-order hyperviscous operator is used to confine dissipative effects to a narrow range at the high-wavenumber end of the spectrum. In spectral space it is written:
ν6khkmax2mmmax23
where kmax and mmax denote the maximum horizontal and vertical wavenumbers, that is, wavenumbers corresponding to the smallest resolved spatial scales. An advantage of this formulation is that the energy dissipation rate at the smallest resolved scales is constant (for a given choice of coefficient ν6), independent of the resolution. It should be noted that the hyperdiffusion is anisotropic in physical space when kmaxmmax, as generally true in these simulations. Although the scale selection of instability is unaffected by hyperdiffusion in most of the experiments reported here, the secondary breakdown and decay of unstable structures certainly is; consequently, their evolution at late times cannot be viewed as entirely realistic.

b. IGW structure

In all cases, the initial condition consists of randomly phased white noise superimposed on a single, linear inertia–gravity wave with leftward and downward phase propagation, having dimensional wavenumber (k, l, m) = (kmin, 0, mmin) in a computational domain of dimension Lx = Ly = 2π/kmin and Lz = 2π/mmin. The inertia–gravity wave satisfies the polarization relations:
i1520-0469-55-15-2473-e2-3a
We have chosen, without loss of generality, to orient the coordinate system such that the wave propagation is in the xz plane. Here k (=kmin) and m (=mmin) denote the horizontal and vertical wavenumbers respectively, and Rf/ω, where ω is the primary-wave frequency:
i1520-0469-55-15-2473-e2-4
In this paper, we restrict attention to the range 0.6 ⩽ R ⩽ 0.95. The buoyancy frequency is
i1520-0469-55-15-2473-e2-5
The wave phase is defined as
ϕkxmzωt.
IGW amplitude a has been normalized by the intrinsic horizontal phase speed ĉ = ω/k, such that a = 1 corresponds to the critical amplitude for the onset of static (convective) instability. In all cases considered here, a < 1; that is, the primary wave is convectively stable. From the polarization relations, the horizontal velocity components u and υ are elliptically polarized. They approach circular polarization as ω approaches the inertial frequency f. Note also that as ω approaches f (at fixed k and a), the density ρ and vertical velocity w tend toward 0. The various phase relationships are illustrated in Fig. 1, showing the density field and velocity vectors associated with an inertia–gravity wave in the vertical plane of wave propagation (Fig. 1a) and transverse to the propagation (Fig. 1b).

c. Local Richardson number

The stability of a steady, parallel shear flow in a stratified fluid is determined, in part, by the Richardson number, defined as the ratio of the vertical buoyancy gradient to the square of the vertical shear. A necessary but nonsufficient condition for instability of a stratified shear flow is that the Richardson number drop below 1/4 somewhere in the flow (Miles 1961; Howard 1961). While the flow associated with an inertia–gravity wave, strictly speaking, is neither steady nor parallel, its long horizontal wavelength and period suggest that locally in space and time, it may be approximated as such. We therefore define a local Richardson number:
i1520-0469-55-15-2473-e2-7
where ρ̃ = ρ + ρ is the sum of ambient and perturbation density fields. The orientation of the horizontal velocity vector is a function of position along the wave phase. In order to invoke parallel flow arguments, we substitute in (2.7) the profile of ũ, the projection of horizontal velocity (u, υ) in the azimuthal direction α:
ũuαυα.
In terms of R, a, ϕ, and α, J becomes
i1520-0469-55-15-2473-e2-9
Here J attains its absolute minimum for values of ϕ = π and α = π/2 (Dunkerton 1984; Fritts and Rastogi 1985) corresponding to the position on the wave phase where static stability is reduced and vertical shear of the transverse velocity component is maximum (refer to Fig. 1b). However, for adequately large a < 1 (convectively stable waves) and relatively large values of R (low frequencies), the range of (α, ϕ) values for which J < 0.25 is quite broad. As demonstrated in a later section, this has important implications for the nature and likelihood of shear instability in IGWs.

We refer to instabilities with α = 0 as “parallel” to the plane of IGW propagation, and instabilities with α = π/2 as “transverse” to this plane.

d. Energetics

Equations for the kinetic energy and density variance are readily obtained from (2.1a,b):
i1520-0469-55-15-2473-e2-10a
where 〈· · ·〉 denotes a volume-integrated average. The potential energy is related to the density variance through a factor of N2.

The first term on the rhs of (2.10a) is the Reynolds stress. It is zero when summed over all wavenumbers and represents the exchange of kinetic energy between various spectral modes. The second term is the buoyancy flux. It appears with opposite sign (up to a factor of N2) in the density variance equation and represents the transfer between kinetic and potential energies. The first term on the rhs of (2.10b) is the equivalent of the Reynolds stress for potential energy. Final terms represent the dissipation of kinetic and potential energies through hyperdiffusion. Examination of the spectra of the various terms that govern the energy of the flow will enable us to identify the physical mechanisms that are responsible for the instability. For example, a shear-driven instability will be characterized by a flow of kinetic energy from the primary wave to the unstable mode, whereas a flow of potential energy from the wave to the unstable mode indicates that convection is responsible for the onset of instability. In general, as shown by several authors, modes of instability in the convectively supercritical regime may have mixed character such that both buoyancy and shear terms are important.

e. IGW modeling strategy

Previous studies of shear instability (e.g., Hazel 1972) showed that the horizontal scale of instability is approximately six to seven times the depth of the unstable shear layer. Noting the large disparity between horizontal and vertical wavelengths of inertia–gravity waves, it is apparently necessary to resolve simultaneously length scales spanning several orders of magnitude—an impossible task, given the limited resolution attainable with modern computers. This problem can be avoided by performing simulations with a reduced value of N/f. Whereas typical geophysical values of this ratio are on the order of 100, we used a value of 10 for the majority of runs presented here. This modification effectively brings together the two timescales 1/N and 1/f and, by the same token, increases the aspect ratio of vertical to horizontal wavelength without a qualitative change in the underlying physics of the problem. To be sure, there are measurable changes when N/f is reduced:for example, the horizontal extent of unstable flow is diminished and unstable perturbations are directly affected by rotation (which may or may not be realistic). However, the fundamental nature of the problem is unchanged, and as will be shown, similar behavior is found over a wide range of N/f values.

We performed many simulations with a relatively low resolution of 128 × 128 × 32 grid points in the x, y, and z directions, respectively. This proved to be sufficient to resolve adequately the scale of instability and to calculate a growth rate. In order to resolve the evolution of overturning Kelvin–Helmholtz billows resulting from shear instability, higher resolution is necessary. Several runs at 256 × 256 × 64 were also performed. At this resolution, it became possible to simulate the overturning and rollup of individual KH billows in low-frequency IGWs although, as noted above, their further evolution and demise could not be accurately simulated. Isotropic resolution was chosen in the horizontal so as not to bias the scale selection process. In the last subsection, we demonstrate the effect of varying N/f in three additional high-resolution runs performed with N/f = 10, 20, 30.

In all cases, the vertical wavelength of the primary wave was held fixed and the horizontal wavelength was adjusted to yield the desired frequency. Simulation parameters are summarized below:
i1520-0469-55-15-2473-e2-11a
To ensure numerical stability, the buoyancy period Tb = 2π/N must be resolved adequately. This was achieved in most cases with a time step of Δt ≈ 0.01Tb.

3. Results

a. Breakdown of low-frequency IGW

Figures 2a–f show the time evolution of xz and yz cross sections of the density field for a low-frequency (R = 0.95) wave of amplitude a = 0.7, obtained in a higher-resolution run (256 × 256 × 64 grid). At t = 0, isopycnals in the yz plane are flat since the primary wave field is independent of y. As the instability develops, slight ripples appear at the position of maximum vertical shear. In the parallel plane, this occurs at ϕ = π/2 and 3π/2, and in the transverse plane at ϕ = 0 and π. The ripples intensify and take on the familiar appearance of Kelvin–Helmholtz billows by the time t = 0.63Tw, where Tw is the wave period. In the parallel plane, billows above and below the region of reduced static stability are out of phase and oppositely tilted, indicating a “sinuous” mode of instability. The timescale of instability is fast compared to a wave period; therefore, the instability is well developed before the wave propagates a significant distance. The instability appears to have developed simultaneously at all positions of extreme vertical shear.

A plot of the vertically integrated two-dimensional kinetic energy spectrum (Fig. 3) reveals that the unstable modes lie on a ring of nondimensional radius k2 + l2/kmin ≈ 20–25. This figure (and others like it, to be shown later) indicates the cumulative growth of instability energy up to the time shown. A ring or “halo” structure in Fig. 3 is representative of spectra in the latter part of the simulation’s linear growth stage (after the decay of initial transients but prior to IGW breakdown). The primary wave’s horizontal wavelength in this case is 30 km. Thus wavenumber 20 corresponds to a scale of 1500 m. This is consistent with the linear theory of shear instability, which predicts that the horizontal scale of instability is approximately 6–7 times the depth of the unstable shear layer (e.g., Hazel 1972). The instability is clearly shear driven, as confirmed by the behavior of the Reynolds stress term, which transfers kinetic energy from the primary wave to the unstable modes (Fig. 4). In contrast, the density gradient term, which transfers potential energy from the primary wave to the instability, remains several orders of magnitude smaller until the instability reaches finite amplitude. The filtered vertical velocity field (corresponding to the cross sections in Figs. 2a and 2d) offers further evidence that the primary wave shear is supplying energy to the instability (Fig. 5). The perturbed fields appear tilted against the wave shear, indicating a flow of energy from the wave to the instability. For parallel modes, a three-lobed structure centered about the position of maximum zonal velocity (e.g., at x ≈ 6–8) was also found by Dunkerton (1997).

The approximately isotropic nature of the instability spectrum can be understood, in part, by examining the local Richardson number. Contours of J(α, ϕ) < 0.25 relevant to this simulation (with R = 0.95, a = 0.7) are shown in Fig. 6. While the minimum of J occurs at (α, ϕ) = (π/2, π), the criterion for shear instability is met at most positions along the wave phase, and in all azimuths. At any phase an angle α exists for which the vertical shear of the horizontal velocity attains an extremum. In physical space, any vertical cross section will reveal streets of oppositely tilted billows at the heights corresponding to the extrema of the velocity shear in that plane. The three-dimensional picture that emerges is of billow structures with azimuthal orientation changing as a function of height (i.e., as a function of wave phase). This is a quite different picture from what had been anticipated based on the earlier stability analysis of Fritts and Yuan (1989). Instead of being confined to the transverse direction, the instability occurs simultaneously in all directions with nearly equal growth rate, in agreement with Dunkerton (1997).

Closer inspection of the spectrum in Fig. 3 reveals that the halo is not exactly uniform in azimuth, nor is it exactly symmetric about the two axes. This behavior is related to the propagation of the primary IGW and the effect of rotation on the instability, as discussed in section 3e.

b. Effect of primary-wave frequency on the instability

To assess the effect of the primary-wave frequency on the instability, we performed a series of experiments at fixed amplitude a = 0.95 for the range of frequencies 0.6 ⩽ R ⩽ 0.95.

In contrast to the low-frequency case discussed in the previous subsection, the instability of an IGW with R = 0.6, a = 0.95 takes several wave periods to develop. Another difference is that the preferred modes of instability are neither in a parallel nor transverse direction, but somewhere in between. After about two wave periods, a single mode dominates having an azimuthal orientation of about 45°. Figure 7 shows a vertical cross section of density in the transverse direction at t = 2.1Tw. Roll-up occurs reminiscent of the KH billows seen in the low-frequency simulation, but in this case the instabilities are vertically asymmetric about their centerline, being somewhat taller underneath where static stability is lower and somewhat shallower above where static stability is higher. The center of maximum isopycnal displacement due to instability is located above the static stability minimum, in a region where the IGW shear vector is rotated clockwise approximately 45° from a transverse orientation. This agrees qualitatively with the observed orientation of instability. The diagonal instability therefore has its locus of origin in a region where the shear vector determines the azimuthal orientation of instability. The “locus of origin” can be identified with the region of maximum isopycnal displacement due to instability, insofar as this region contains the critical level of the unstable perturbation, and the critical level occurs inside a region of Ri < 1/4, in agreement with theory (see Dunkerton 1997, and references therein).

A Hovmöller plot of the filtered density perturbation (Fig. 8) lends further insight on the nature of the most unstable mode. Early times are characterized by a reorganization of the randomly phased noise; gradually, a signal aligned with the primary wave emerges. In this plot, the perturbation was normalized by a temporally smoothed version of the maximum perturbation at each time step in order that the same contours would appear throughout the figure. (The smoothed time series, not shown, displays nearly constant exponential growth during the linear instability growth phase.) We see that instability energy tends to track the motion of the primary wave but is not exactly constant in time when viewed with respect to a smoothed growth rate. Rather, it vacillates somewhat about a constant exponential growth. The interpretation of this behavior is suggestive, to some extent, of a “local” or absolute instability in the IGW frame of reference. However, the vacillation of disturbance energy complicates this interpretation, and further study is warranted. We have seen evidence in a yt Hovmöller plot (not shown) that the instability packets wobble slightly in the transverse direction, which could account for the vacillation observed at constant y.

Diagonal modes are favored over purely transverse modes apparently because the latter cannot keep step with the zonally propagating IGW. Propagation of the primary inertia–gravity wave is relevant in cases, such as this one, where instabilities require several wave periods to develop. Another effect that may be important is vertical advection within the unstable layer, due to the upward IGW velocity component in the region of minimum stability.

Figure 9a shows the corresponding vertically averaged spectrum of instability energy. The isotropy that we find to be characteristic of low-frequency IGW has completely disappeared and is replaced by modes at intermediate angles as noted above. The dominant mode is centered at (k, l) ≈ (18, 16). Since the wave fields are real, there is an identical peak on the opposite side of the origin; that is, Ek,l = Ek,l. Interestingly we do not observe symmetry about the k and l axes in this simulation; there is less symmetry about these axes than in the low-frequency run. Evidently something causes the symmetry of the horizontal wavenumber spectrum to be broken (e.g., rotation or vertical advection). Without such effects, and with vertically symmetric (or periodic) boundary conditions, the instability spectrum is expected to be symmetric about both axes, so only a single quadrant need be considered. The importance of rotation, for example, must be measured against the timescale of instability; for modes with relatively small growth rate, it is likely to have a more significant effect. The same can be said of vertical advection.

Vertically averaged spectra for R = 0.7, 0.8, 0.9 are shown in Figs. 9b,c,d. At R = 0.9, the spectrum is still approximately isotropic, and the instability has shifted to smaller wavenumber, in agreement with the contraction of IGW horizontal wavelength. (Recall that the vertical wavelength of IGW is invariant in these simulations.) The actual physical scale is 1300 m compared to 1500 m at R = 0.95. By R = 0.8, a faint halo is still visible, but modes at α = 0 and at α = π/4 have begun to dominate. The halo disappears entirely by R = 0.7. Instead, four distinct diagonal lobes are apparent.

The kinetic energy of the most unstable modes at each R is plotted versus time in Fig. 10. A linear slope indicates exponential growth. An estimate of the growth rate can be obtained with a least squares fit during the period of exponential growth (the actual growth rate is equal to one-half the slope of the straight line since energy corresponds to the squared velocity amplitude). Growth rates ωi obtained with this method are shown in Table 1. The lowest frequency wave exhibits the fastest-growing instability. The growth rate drops off rapidly as we decrease R. This is in agreement with the predictions of linear stability theory (Fritts and Yuan 1989; Dunkerton 1997).

c. Effect of primary-wave amplitude on the instability

Simulations were performed with primary-wave amplitudes ranging from just above the (theoretical) shear instability threshold to just below the convective instability threshold for 0.6 ⩽ R ⩽ 0.95. In all cases, the overall nature and scale of the instability proved insensitive to the primary-wave amplitude; however, growth rates were found to increase with increasing amplitude (Table 1).

d. Threshold region

In order to test the utility of the local Richardson number criterion in determining a critical amplitude for shear instability, we performed a series of simulations with 0.6 ⩽ R ⩽ 0.95, increasing the amplitude gradually until shear instability was detected. The criterion used to assess whether shear instability is present was based on the behavior of the Reynolds stress in the primary-wave energy balance. When the instability is shear driven, the Reynolds stress associated with the primary wave becomes negative after an initial “reorganizing” of the white noise. This signals a flow of kinetic energy from the wave to the unstable modes. Below the critical amplitude, the Reynolds stress oscillates in time with the primary-wave period and remains negligibly small. Since small amplitude waves generally exhibit weaker growth rates, the simulations were run over five wave periods. If the Reynolds stress behavior did not deviate from its oscillatory behavior in that time, the flow was deemed stable to shear instability.

Figure 11 displays the computed instability threshold and the corresponding threshold determined theoretically from a local Richardson number argument; that is,
i1520-0469-55-15-2473-e3-12
(Dunkerton 1984; Fritts and Rastogi 1985). At R = 0.95, the two thresholds agree. Here, the instability develops quickly on a faster timescale than the wave period; therefore, the Richardson number criterion based on the approximation of steady, parallel flow provides a good measure of the actual threshold. As R decreases, the thresholds begin to diverge. The growth rate of instability is now slow compared to the IGW period and the actual threshold amplitude is larger than predicted by the local Richardson number criterion. The steady, parallel-flow approximation breaks down when the growth rate of the instability becomes weak. This effect is most pronounced for smaller amplitude waves since they exhibit the slowest growth rates.

According to the Richardson number criterion, the first modes to become unstable should be transverse modes. We did not see evidence for this at any R. Instead, the asymptotically dominant modes appear to be those with wavenumbers oriented at ∼π/4. Evidently transverse modes are at a competitive disadvantage with respect to nontransverse modes as the theoretical stability curve is approached, except at large R where growth rates in all azimuthal directions are rapid compared to the timescale of IGW propagation.

Below the shear-instability threshold, there is a relatively very slow cascade of energy to smaller scales, possibly attributable to resonant interactions (not shown).

e. Variation of N/f

To demonstrate that the above results apply to geophysical flows with larger values of N/f, additional runs were performed at (R, a) = (0.95, 0.8) with a 256 × 256 × 32 grid and N/f = 10, 20, 30. Two-dimensional spectra for R = 0.95, shown in Figs. 12a–c, demonstrate that the dynamical behavior of instability for low-frequency waves was correctly simulated at a reduced value of N/f. The halo structure is a robust feature of the instability, and its radius is consistent with theoretical expectations. Consider a wave with frequency ω, horizontal wavelength λh, and horizontal wavenumber kmin = 2π/λh at N/f = 10. As N/f is increased by a factor of x, so is the horizontal wavelength of the wave in order to preserve the original frequency ω. Therefore, at N/f = 10x, the horizontal wavelength λh of the primary wave is λh = h and the corresponding horizontal wavenumber kmin = kmin/x. The horizontal scale of the instability, on the other hand, depends on the depth of the shear layer (held constant in these simulations) and is unaffected by this change. In the new configuration, the halo should have radius 2r ≈ 40 for N/f = 20 and radius 3r ≈ 60 for N/f = 30. Figure 12 confirms that this is indeed the case. Growth rates for the three cases do not differ significantly (Table 1).

Closer inspection of these spectra and comparison to that of Fig. 3 reveals that the halo is not exactly uniform in azimuth, nor is it exactly symmetric about the k and l axes. If rotation were unimportant for the instability, the spectra would be symmetric about the k axis (and therefore symmetric about the l axis) as noted in section 3a. Nonuniformity of amplitude in the halo (at R = 0.95) indicates that the effective growth rate, taking into account the zonal propagation of IGWs, remains a weak function of azimuth even in low-frequency IGWs. When rotation is important on the timescale of instability, the spectrum of instability energy appears as if it has been rotated in the (k, l) plane so that the halo’s maximum and minimum amplitudes are displaced from a purely parallel or transverse orientation.

Simulations were also performed at intermediate R, varying N/f in the same range. These results (not shown) suggest that diagonal modes exist in each case, although their horizontal wavenumber is approximately constant and greater symmetry was observed in each of the four quadrants with increasing N/f, as might be expected as the direct influence of rotation on unstable perturbations diminishes.

4. Conclusions

The three-dimensional breakdown of a large-amplitude, convectively stable inertia–gravity wave was examined numerically as a function of primary-wave frequency and amplitude. The results confirm that inertia–gravity waves in this region of parameter space break down preferentially via shear instability. In low-frequency waves the instability is ubiquitous, occurring simultaneously throughout the wave field, and the spectrum of instability energy is approximately, but not exactly, isotropic in azimuthal orientation. It is expected that this “global” breakdown will lead to a greater mixing efficiency due to breaking IGW, relative to that of convectively unstable (nonrotating) gravity waves. Further studies are under way to verify this conjecture. In higher-frequency waves, shear instability develops adjacent to the region of reduced static stability and displays a preference for intermediate azimuths (e.g., near 45°). Near-inertial waves experience the fastest growing instabilities. The growth rate of shear instability drops off rapidly as the wave frequency is increased and, for all frequencies, increases with increasing wave amplitude. At most frequencies, the onset of modal shear instability occurs at a wave amplitude slightly above the theoretical stability boundary determined from a local Richardson number argument.

The fact that instability growth rates in low-frequency IGW depend only weakly on azimuth is consistent with the behavior of the most unstable modes (along the real-k axis) in Dunkerton’s (1997) stability analysis at large R. Instabilities with wave vector orientation parallel to the direction of IGW propagation develop in sinuous fashion on the flanks of a zonal “jet” coincident with the region of minimum static stability. Although parallel instabilities do not arise at the point of minimum local Richardson number, as do transverse modes, their growth is aided by the presence of twin instability critical levels on either side of the jet maximum (Dunkerton 1997).

As R = f/ω is reduced, growth rates weaken and the halo radius contracts in wavenumber space, while new modes emerge with approximately diagonal azimuthal orientation. Interestingly, we do not find that modes with purely transverse orientation dominate anywhere adjacent to the threshold amplitude for shear instability. Such modes are unfavored apparently due to their slow growth rate, lack of zonal group propagation, and inability to keep step with the zonal phase propagation of the primary IGW.

In a companion paper, we explore the instability of convectively unstable inertia–gravity waves (Lelong and Dunkerton 1998). It is shown that the properties of shear instability in low-frequency IGW are similar whether the waves are subcritical or supercritical to convection. Shear instability remains the dominant mechanism of breakdown in near-inertial IGW. On the other hand, instabilities at intermediate R are sensitive to IGW amplitude in the vicinity of the convective instability threshold a = 1.

Acknowledgments

This research was supported by the National Science Foundation, Grants ATM-9123797, ATM-9500613, and OCE-9521275, and by the Air Force Office of Scientific Research, Contract F49620-92-C-0033. The computations were performed on the Cray C90 at the Pittsburgh Supercomputing Center and on the Cray Y-MP and Cray J916 at the National Center for Atmospheric Research. Discussions with Donal O’Sullivan, Jim Riley, and Pete Lombard during the course of this work are gratefully acknowledged.

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Fig. 1.
Fig. 1.

Density field, with velocity vectors superposed, of an inertia–gravity wave with leftward and downward phase propagation at t = 0 in (a) the vertical plane of wave propagation and (b) the vertical plane transverse to the propagation.

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 2.
Fig. 2.

Time evolution of the density field in a numerical simulation with IGW parameters (R, a) = (0.95, 0.7): (a, b, c) xz plane at y = 23.4 km; (d, e, f) y–z plane at x = 11.7 km. Three times are shown: at (a, d) 0.42Tw, (b, e) 0.63Tw, and (c, f) 0.70Tw, where Tw is the IGW period.

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 2.
Fig. 2.

(Continued)

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 3.
Fig. 3.

Vertically integrated kinetic energy spectrum for the simulation of Fig. 2 at t = 0.70Tw. Contours are 0.1 and 0.5 of the maximum value, and the primary wave is excluded. The horizontal resolution is 256 × 256; only the significant part of the spectrum is shown.

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 4.
Fig. 4.

Volume-integrated Reynolds stress as a function of time for the simulation of Fig. 2.

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 5.
Fig. 5.

Filtered vertical velocity field for the simulation of Fig. 2 at t = 0.63Tw: (a) xz plane at y = 23.4 km and (b) yz plane at x = 11.7 km.

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 6.
Fig. 6.

Contours of local Richardson number J, at 16 equally spaced intervals from min(J) to J = 0.25, for a convectively stable inertia–gravity wave with (R, a) = (0.95, 0.7).

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 7.
Fig. 7.

Cross section of density in the transverse vertical plane at x = 1.45 km in a simulation with IGW parameters (R, a) = (0.6, 0.95).

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 8.
Fig. 8.

Hovmöller plot of filtered density for the simulation of Fig. 7, at y = 4.9 km, z = 0.75 km. IGW propagation is from right to left as a function of time. The filtered data were normalized by a temporally smoothed version of the maximum value at each time step in the plot.

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 9.
Fig. 9.

Vertically integrated kinetic energy spectrum as in Fig. 3, with amplitude a = 0.95, for (a) R = 0.6, (b) R = 0.7, (c) R = 0.8, and (d) R = 0.9. The horizontal resolution is 128 × 128; only the significant part of the spectrum is shown.

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 10.
Fig. 10.

Energy of the most unstable mode for 0.6 ⩽ R ⩽ 0.95, with amplitude a = 0.95, as a function of time. The energy has been normalized by the energy of the primary wave at t = 0.

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 11.
Fig. 11.

Predicted and observed shear instability thresholds as a function of R.

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Fig. 12.
Fig. 12.

Vertically integrated kinetic energy spectrum in a series of runs with IGW parameters (R, a) = (0.95, 0.8) for (a) N/f = 10, (b) N/f = 20, and (c) N/f = 30. The horizontal resolution is 256 × 256; only the significant part of the spectrum is shown.

Citation: Journal of the Atmospheric Sciences 55, 15; 10.1175/1520-0469(1998)055<2473:IGWBIT>2.0.CO;2

Table 1.

Horizontal wavenumbers and growth rate of most unstable modes in convectively stable IGWs.

Table 1.
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