Turbulence Evolution, Structure, and Anisotropy - AMS Journals

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Gravity Wave Instability Dynamics at High Reynolds Numbers. Part II: Turbulence Evolution, Structure, and Anisotropy DAVID C. FRITTS, LING WANG, JOE WERNE, TOM LUND,

AND

KAM WAN

NorthWest Research Associates, Colorado Research Associates Division, Boulder, Colorado (Manuscript received 31 December 2007, in final form 22 October 2008) ABSTRACT This paper examines the character, intermittency, and anisotropy of turbulence accompanying wave instability, breaking, and turbulence evolution and decay for gravity waves (GW) having a high intrinsic frequency, amplitudes above and below nominal convective instability, and a high Reynolds number. Wave breaking at both amplitudes leads to an extended inertial range of turbulence, with turbulence energies that maximize within ;1 wave period of the onset of breaking. Turbulence sources include both shear and buoyancy, with shear being the major contributor. Turbulence displays considerable intermittency both within and across the phase of the breaking gravity wave and exhibits clear anisotropy throughout the evolution. Turbulence anisotropy is found at all spatial scales and all times but is most pronounced in the most statically stable phase of the GW and at late times as the turbulent flow restratifies.

1. Introduction The companion paper by Fritts et al. (2009, hereafter Part I) describes two examples of gravity wave (GW) instability and breaking in idealized direct numerical simulations (DNS) at large GW amplitudes, a relatively high GW frequency v ; N/3.2, where N is the buoyancy frequency, and a Reynolds number Re 5 104 sufficient to allow vigorous instability, strong wave–wave interactions and energy transfers among modes, and the development of a broad spectrum of inertial range turbulence. Part I focused on the initial GW instability character, the impacts of instability and breaking on the GW amplitude and fluxes, the character of the 2D motion field that survives to late times, and the evolution of bulk turbulence properties and their relation to the phase structure of the initial GW. Our purpose in this paper is to explore in detail the evolution, morphology, and anisotropy of the turbulence fields arising in these same simulations. Our earlier numerical studies of gravity wave instability and breaking employed either a compressible code (Andreassen et al. 1994, 1998; Fritts et al. 1994, 1996,

Corresponding author address: David C. Fritts, NorthWest Research Associates, Colorado Research Associates Division, 3380 Mitchell Lane, Boulder, CO 80301. E-mail: [email protected] DOI: 10.1175/2008JAS2727.1 2009 American Meteorological Society

1998) or our current Boussinesq code, but at Reynolds numbers and spatial resolution that we considered insufficient to characterize the structure, dynamics, and anisotropy of the resulting inertial range of turbulence (Fritts et al. 2003, 2006). These studies nevertheless captured limited inertial range turbulence dynamics and enabled an assessment of the vorticity dynamics driving the turbulence cascade (Arendt et al. 1997, 1998; Andreassen et al. 1998; Fritts et al. 1998) that appears to be sufficiently robust to also describe vortex interactions within turbulence fields at much higher Re (Fritts et al. 2003, 2006). These dynamics manifest as waves propagating on vortices having varying axial, radial, and azimuthal structures, which we have termed Kelvin ‘‘twist’’ waves on account of their discovery by Lord Kelvin more than a century ago (Kelvin 1880). As suggested by their name, twist-wave dynamics are driven by distortions, or twisting, of the vortex (due to nonzero axial or radial vorticity) that induce differential advection at nearby locations and result in the propagation of these twist waves along the vortex with a speed that depends on the twist-wave parameters [see the analytic and turbulence assessments of these dynamics by Arendt et al. (1997, 1998) and Fritts et al. (1998)]. Twist waves are easily excited by distortions of the vortex or proximity of adjacent vortex structures, and they readily achieve large amplitudes and lead to vortex stretching, unraveling, or

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fragmentation, which we have suggested may be key elements in the cascade of enstrophy to smaller spatial scales. Indeed, the same vorticity dynamics are observed to occur accompanying the transition to, and evolution of, turbulence due to Kelvin–Helmholtz (KH) instability at sufficiently high resolution and Reynolds numbers (Werne and Fritts 1999; Smyth 1999; Fritts and Werne 2000). No study until this paper series, however, has explicitly allowed for both wave–wave interactions and instability leading to a broad 3D turbulence spectrum in order to examine the competition between the two processes and the implications for their respective evolutions. An interesting aspect of both GW breaking and KH instability simulations is the indication of anisotropy within the turbulence field arising from background shear and/or stratification. Indeed, the isotropy (or anisotropy) of turbulence arising in geophysical flows is of considerable interest to a much broader community and has been the subject of numerous experimental, numerical, and theoretical studies for more than six decades. According to classical theory (Kolmogorov 1941, 1962; Obukhov 1941), the smallest scales of turbulence should achieve an isotropic state, regardless of the structure or complexity of the large-scale flow. Since these initial studies, numerous papers have addressed the validity of these ideas, but there has been little consensus regarding the degree to which small-scale motions achieve an isotropic state. Several studies suggested that local isotropy in either sheared or stratified flows is unlikely and should not be expected at even the smallest spatial scales (e.g., Uberoi 1957; Townsend 1959). Excellent reviews of this work were provided by Champagne (1978), Mestayer (1982), Brown et al. (1987), Hunt et al. (1991), Van Atta (1991), and Sreenivasan (1991). In addition to experiments, both analytical and numerical studies have also shown direct influences of the large-scale on small-scale turbulence structures, suggesting a communication of anisotropy to scales extending into the dissipation range (George and Hussain 1991; Brasseur and Wei 1994; Yeung et al. 1995; Shen and Warhaft 2000). These influences were demonstrated formally for sheared and stratified flows within the framework of the Navier–Stokes equations by Durbin and Speziale (1991) and Pettersson-Reif and Andreassen (2003). Similar conclusions were reached in direct numerical simulation of turbulence arising from KH instability, but only recently have studies begun to address Reynolds numbers sufficient to assess anisotropy, and its causes and effects, with confidence (Smyth and Moum 2000; Werne and Fritts 1999, 2000, 2001; Fritts et al. 2003). Although all of the studies referenced above seem to cast doubt on Kolmogorov’s notion of small-scale isotropy, none of them considered Reynolds numbers high

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enough to produce the required broad separation between the inertial and dissipation ranges. High Reynolds number implies very small-scale dissipative motions, which are difficult to measure in the laboratory and require unattainable computational resources to simulate. Recent advances in experimental techniques, however, have enabled high-fidelity measurements of smallscale motions in fairly high Reynolds number flows. Praskovsky (1992) and Praskovsky and Oncley (1994) used both wind tunnel and atmospheric boundary layer data to investigate Kolmogorov’s (1962) refined similarity hypothesis at Taylor microscale Reynolds numbers in the range Rl 5 (20/3)1/2K/(ne)1/2 ; 2–12 3 103 (Pope 2000), where K is the turbulence kinetic energy, n is kinematic viscosity, and e is the mechanical energy dissipation rate. These measurements spanned almost two decades of the inertial range and confirmed universal scaling at scales deep within the inertial range. Thoroddsen and Van Atta (1992a) reached a similar conclusion in analyzing cylinder wake data at Rl ; 550. Saddoughi and Veeravalli (1994) and Saddoughi (1997) performed very careful measurements in the National Aeronautics and Space Administration (NASA) Ames 80–120-ft wind tunnel at Rl as high as 2000. These authors provided convincing evidence that the existence of small-scale isotropy is governed by the ratio of the Kolmogorov to mean-shear time scales, Sc* 5 S(n/e)1/2, where S is the strain rate magnitude. In particular, small-scale isotropy is found only for Sc* , 0.01, which increases with increasing shear and decreases with increasing Reynolds number. Thus, one can only expect to see local isotropy in sheared flows if the Reynolds number is sufficiently high. This fact can be used to reconcile many of the prior contradictory measurements that did not satisfy this criterion. In addition to mean shear, stable stratification provides a second source for anisotropy. This source is usually important in atmospheric flows and is thought to provide a significant contribution to the observed smallscale anisotropy (Thoroddsen and Van Atta 1992b; Kaneda and Yoshida 2004). Unlike anisotropy induced by mean shear, stratification-induced small-scale anisotropy has not been studied as broadly. From theory we know that the mechanisms for stratification and shear-based anisotropy are similar. Both mean shear and buoyancy produce accelerations that act in clearly defined directions. The accelerations due to stratification are proportional to the potential temperature fluctuations, which decrease in magnitude with decreasing scale. Thus, as in the case with mean shear, one would expect the small-scale anisotropy induced by stratification to become negligible at sufficiently high Reynolds number. This limiting Reynolds number does not appear to have

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been quantified, nor has a more exacting criterion (such as Sc* , 0.01 for shear flow) been proposed to date. Departures from stationarity, homogeneity, and isotropy also have important implications for measurements and inferences of turbulence and larger-scale dynamics in the atmosphere, the oceans, and the laboratory because most measurement techniques depend on sensitivity to gradients of quantities induced by turbulence motions. DNS of KH instability by Werne and Fritts (2000, 2001) made predictions of structure functions and turbulence variances that departed significantly from expectations of stationary, homogeneous, isotropic turbulence theory and that were later confirmed by very high-resolution direct measurements (Wroblewski et al. 2003). Fritts et al. (2003) have highlighted a potential for biases in atmospheric measurements by radars due both to insensitivity to turbulence where mixing has largely eradicated refractive index variations and to anisotropic structures leading to specular sheets in regions of very high stratification at the edges of well-mixed layers. These imply a potential for biases in radar and other estimates of spectral moments (Muschinski 1996). Here, we employ the simulations described in Part I for detailed studies of the turbulence fields arising from GW breaking. The simulations were performed for a monochromatic GW having an intrinsic frequency v ; N/3.2, nondimensional GW amplitudes a 5 u9/c 5 0.9 and 1.1, no mean wind, constant stability N2, a bulk Richardson number Ri 5 4p2, and Re 5 lz2/nTb 5 104. Here, Tb 5 2p/N is the buoyancy period, u9 and c are the horizontal perturbation velocity amplitude and the horizontal trace speed of the GW, a 5 1 corresponds to a zero vertical potential temperature gradient at the least statically stable phase of the GW, lz is the vertical wavelength of the primary GW, and n is kinematic viscosity. As described in Part I, the computational domain was inclined along the GW phase to allow easy computation of turbulence statistics at constant GW phase; the simulations were triply periodic employing a Fourier series representation in each dimension; domain dimensions were (X9, Y9, Z9) 5 (3.4, 2.22, 1); and a maximum spectral resolution of (Nx, Ny, Nz) 5 (2400, 1600, 800) Fourier modes was required to describe the turbulence field at its most intense phase. This allowed an inertial range of turbulence extending well over a decade of scales and Rl ; 150–200 during its most intense phase. The domain width was chosen to allow adequate resolution of the dominant instability modes having spanwise wavenumbers. Sections 2 and 3 describe the turbulence dynamics, the correlations with the primary GW, the spectra, and the anisotropy arising for the two

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primary GW amplitudes. Our summary and conclusions are presented in section 4.

2. Turbulence evolution at a 5 1.1 a. Turbulence dynamics Following our earlier studies of the vorticity dynamics within the turbulence spectrum arising from wave breaking (Andreassen et al. 1998; Fritts et al. 1998), we employ the negative eigenvalue (l2) of the tensor defined as L 5 V2 1 S2, where V and S are the rotation and strain tensors, with antisymmetric and symmetric components Vij 5 (›iuj 2 ›jui)/2 and Sij 5 (›iuj 1 ›jui)/2, to identify flow features having strong rotational character (Jeong and Hussain 1995). Negative eigenvalues correspond to flow features having the greatest rotational (as opposed to shearing) character, and the magnitude is a measure of rotational intensity. Pure shearing motions, in contrast, make no contribution to l2. Thus, l2 represents sensitivity to a subset of total vorticity that is more relevant to turbulence dynamics and that provides a clear ability to distinguish between the two contributions to total vorticity. These features comprise the majority of the turbulence field (Jeong and Hussain 1995). They also allow us to follow the transition from initial instability structure, through vortex interactions and instabilities, to fully developed turbulence and its subsequent decay. Figure 1 displays 3D volumes of l2, viewed from the side and from below at those times depicted with 2D cross sections of vorticity magnitude in Fig. 2 of Part I. In both views, GW propagation is to the left. Volumetric animations of the l2 field evolutions are available online (see http://www.cora.nwra.com/ dave/GWBmovies.html). These provide a continuous and much clearer view of the vortex evolution and dynamics than can be conveyed in the still images presented here. Examining the volumetric views of the l2 fields depicted in Fig. 1, we note (consistent with Fig. 2 of Part I) that instability structures at t 5 7 occur at small spatial scales. They also have typically oblique alignments and small amplitudes. Between t 5 7 and 10, however, the weak initial optimal perturbations are replaced by structures having largely streamwise alignments (with spanwise wavenumbers l ; 2–5 relative to the gravest spanwise mode l 5 1; see Part I) that have already achieved significant amplitudes. As seen more clearly in our flow animations noted above, the instability progression is from 1) initial counterrotating horizontal vortices (aligned along the GW propagation direction) that achieve significant fluid displacements and rotational character to 2) horseshoe vortices closed in the

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FIG. 1. 3D views of l2 viewed from the side, normal to (x9, z9), and from below, normal to (x9, y9), at t 5 7, 10, 12, 15, and 30 Tb for a 5 1.1. Red and yellow regions denote the most intense rotation; small l2 magnitudes are transparent; images are shown relative to a fixed GW phase structure, with the least statically stable phase (having an upward and leftward GW velocity) 1/4 of the domain depth from the top boundary; and GW propagation is to the left in all panels. Optimal perturbations with largely oblique alignments are seen to occur at early times; finiteamplitude instabilities occur at dominant spanwise wavenumbers of l ; 2–5, consistent with linear instability and the discussion in Part I. Instability progression is rapid thereafter and quickly leads to locally intense turbulence. See text for further details. Color tables are scaled for each time because there is large variability in l2 magnitudes in time.

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high-shear region above the phase of lowest GW static stability (upper 1/4 of the domain), having negative spanwise vorticity, and self-advecting downward with the GW phase motion, to 3) ringlike vortices arising where fluid advected downward within the horseshoes encounters the oppositely sheared phase of the GW (having strong positive spanwise vorticity at their lowest extents). These regions prove to be the sites of most intense turbulence for several Tb thereafter, although they are advected by the larger-scale, quasi-2D flow throughout the evolution. The progression of instability toward turbulence summarized above exhibits all of the features that were diagnosed in much greater detail by Andreassen et al. (1998) and Fritts et al. (1998) in numerical results at a far lower (and more easily understood) Re and in analytic studies by Arendt et al. (1997, 1998). These involve strong vortex stretching by orthogonally aligned vortex structures, intertwining and leapfrogging of adjacent, coaligned vortices, and all of the twist-wave dynamics identified in our previous studies. The dominant interactions are driven by mutual vortex stretching (or compression) and include 1) mode-0 Kelvin twist waves (with axial vorticity variations) excited by axial vortex stretching, 2) mode-1 twist waves (with radial vorticity variations) due to radial vortex displacements, and 3) mode-2 twist waves (with intertwined, helical corotating vorticity variations) resulting from vortex flattening due to divergent radial advection (Arendt et al. 1997, 1998; Fritts et al. 1998). As at much lower Re, the excitation, propagation, and evolution of these twist waves appear to largely account for the character and evolution of the turbulence field and the cascade of energy to smaller scales of motion. Indeed, turbulence accounts for the majority of the velocity and potential temperature variances during active wave breaking and the reduction of the primary GW amplitude (Part I). The choice to display l2, which highlights flow features having strong rotational character (Jeong and Hussain 1995), enables the flow animations to display the small-scale vorticity structures and their temporal evolutions more clearly than can be seen in Fig. 1. Because of the much larger current Re, however, this simulation exhibits a turbulence cascade to much smaller scales very quickly. This occurs between the images at t 5 10 and 12 in Fig. 1. Similar vorticity dynamics also extend to much later times but are more sporadic and occur at larger spatial scales as energy dissipation rates decrease.

b. Turbulence statistics and correlations with the 2D GW field To provide insights into the turbulence evolution, as well as the sources and sinks of turbulence energy, we

derive the evolution equations for the velocity and potential temperature variances for all motions except those having the wavenumber of the initial GW (denoted eddies); the GW, having wavenumber (0, 0, 21), is given by U9(x, z) 5 (U9, V9, W9) and Q(x, z), P(x, y, z) is pressure, and the eddy motions are described by u 5 (u, y, w) and u in geophysical coordinates. Denoting averages over GW phase planes by angle brackets, these equations may be written as ›t Æu2 æ 5 22[Æuwæ›z U9 1 Æu2 æ›x U9] 1 Re2 1 ›n2 Æu2 æ 2 Æu›x Pæ 2 Re2 1 Æ=u =uæ,

(1)

›t Æy2 æ 5 22[Æuyæ›x V9 1 Æywæ›z V9] 1 Re2 1 ›n2 Æy 2 æ 2 Æy›y Pæ 2 Re2 1 Æ=y =yæ,

(2)

›t Æw2 æ 5 22[Æuwæ›x W9 1 Æw2 æ›z W9] 1 Re2 1 ›n2 Æw2 æ 2 Æw›z Pæ 2 Re2 1 Æ=w =wæ 1 RiÆuwæ,

and (3)

›t Æu2 æ 5 22Æuwæ›z Q9 1 (Re Pr)2 1 ›n2 Æu2 æ 2 (RePr)2 1 Æ=u =uæ,

(4)

where subscripts denote directional partial derivatives and n denotes the direction normal to the GW phase surfaces. The first terms on the right-hand side (rhs) of Eqs. (1)–(4) are the shear sources of eddy kinetic energy and the buoyancy source of eddy potential energy, where ‘‘eddy’’ includes turbulence as we have defined it with l 6¼ 0 as well as other 2D wave motions having l 5 0, but different k and m than the initial GW; the second terms on the rhs of Eqs. (1)–(4) represent diffusion of variances normal to the GW phase; the third terms on the rhs of Eqs. (1)–(3) are the pressure-strain terms that couple the component velocity variances; and the last terms on the rhs of Eqs. (1)–(4) are the dissipation terms, which may be rewritten as ei 5 Re2 1 Æ=ui =ui æ

and

x 5 ›i u›i u 5 2(RePr )2 1 Æ=u =uæ,

(5) (6)

where a subscript i denotes the x, y, or z direction, ›i denotes differentiation in the ith direction, and repeated indices imply summation in Eq. (6). From these equations, we see that the only sources of eddy velocity and potential temperature variances within the primary GW field are shear and buoyancy terms, respectively, via Reynolds stresses acting on the GW velocity and potential temperature gradients. Shear sources fuel horizontal and vertical velocity variances, with relative contributions that depend on the phase slope

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FIG. 2. (top left) Domain-averaged ‘‘turbulence’’ energies [having wavenumbers (k, l 6¼ 0, m)]. Streamwise, spanwise, and vertical kinetic energies and potential energy are shown with heavy solid, light solid, light dashed, and heavy dashed lines, respectively. (top right) (middle right) Shear and buoyancy production of turbulence energy, (middle left) mechanical (solid) and thermal dissipation (dashed) rates, Ree and x, and (bottom) energy component ratios for a 5 1.1. (bottom left) Energy ratios of spanwise-to-streamwise and vertical-to-streamwise kinetic energy are shown with heavy and light solid lines, respectively. (bottom right) The potential-to-kinetic energy ratio. Units are nondimensional, with shear sources and mechanical energy dissipation Ree scaled by lz2/Tb3 and buoyancy sources and thermal dissipation x scaled by Rib2lz2/Tb. All quantities are shown only to t 5 40 because they change little after that time. Energy ratios likely reflect some low-frequency GW motions having l 6¼ 0 at late times, as well as spanwise motions having zero frequency and no potential energy.

(or intrinsic frequency) of, and the position within, the GW field. GW potential temperature gradients likewise fuel eddy potential temperature variance. The pressurestrain terms drive conversions between horizontal and vertical velocity variances but no net sources. In particular, we see that horizontal and vertical gradients within the primary GW fields can act as eddy variance sources, with magnitudes that depend on the Reynolds stresses averaged over that phase of the GW. In contrast, the vertical potential temperature flux, seen to be negative during wave breaking (Fig. 7 of Part I), acts as both a buoyancy source of eddy potential temperature variance and a conversion between eddy vertical velocity and potential temperature variances. Although this decomposition is not as general as in our assessment of 2D and 3D motions in Part I, the above equations nevertheless indicate clearly the expected eddy variance sources, sinks, and exchanges among different motions.

1) DOMAIN-AVERAGED

QUANTITIES

Temporal evolutions of domain-averaged turbulence quantities as defined above, but only for motions with l 6¼ 0, are shown in Fig. 2 for a 5 1.1. Velocity and potential temperature variances (top left) are seen to exhibit sharp increases beginning at t ; 9 (with all times in units of the buoyancy period Tb), with maxima at t ; 12 in response to corresponding spikes in shear and buoyancy production at slightly earlier times (top right). Shear production is seen to dominate at all times, peaking at t ; 11, with buoyancy production (middle right panel) peaking at the same time, but having a significantly smaller domain-averaged magnitude. Thus, shear production accounts for the majority of net turbulence energy generation, even for a GW that is initially convectively overturning. This is because an opposite heat flux also accounts for the conversion of kinetic to

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potential energy accompanying shear production and because the transition to strong turbulence accompanies the penetration of fluid advected downward within the horseshoe vortices (and the formation and breakdown of the ringlike vortices between t 5 11 and 12). Both production terms decay quickly as the GW amplitude decays, however, and buoyancy production exhibits a significant oscillation with a dominant period of ;14 thereafter, reflecting energy exchanges between kinetic and potential energies of the 6(1, 0, 1) motion field (Part I). At the peak turbulence intensity, t ; 12, we obtain an estimate of a mean Rl ; 400 based on domain-averaged quantities and a peak value averaged over the phase of the GW having the most intense turbulence of Rl ; 670. Because instability growth and energy extraction from the primary GW is highly efficient at a 5 1.1, turbulence energy production is large for only ;3Tb. Energy transfers from larger to smaller scales, and mechanical and thermal energy dissipation at smaller scales, are likewise very efficient when turbulence is strong and extended in spatial scales, yielding turbulence durations [full width, half maximum (FWHM)] of only t ; 5Tb, with significant mechanical and thermal energy dissipation (second left panel in Fig. 2) occupying even less time (;3Tb FWHM). Velocity variance ratios (lower left) are ;0.5 prior to wave breaking and increase to ;0.8–1.2 during active wave breaking and turbulence, suggesting an evolution toward equipartition and isotropy accompanying strong turbulence. By comparison, we anticipate velocity variance ratios of 1.0 for stationary homogeneous isotropic turbulence, for which there are no preferred directions. The ratio of potential temperature variance to total velocity variance (lower right), however, is ;0.4 during active wave breaking and increases gradually toward ;0.9 thereafter. Based on domain-averaged quantities, the simulated turbulence appears to depart systematically from homogeneity and isotropy, in line with previous expectations cited above. We perform below other assessments of turbulence isotropy, as a function of wavenumber and locally within the GW phase, to more fully quantify the observed anisotropy. The systematic departures of the variance ratios from unity during strong turbulence and thereafter clearly suggest the influences of preferred directions, sources, sinks, and energy conversions. Indeed, these departures from homogeneity and isotropy exhibit similar tendencies to those noted in previous numerical studies of KH instability (Smyth and Moum 2000; Werne and Fritts 2000, 2001) and anticipated in theoretical efforts addressing the influences of shear and stratification (Durbin and Speziale 1991; Pettersson-Reif and Andreassen 2003).

Collectively, our DNS results provide additional evidence that turbulence in the presence of large-scale shear and stratification is neither homogeneous nor isotropic at any scale.

2) VARIATIONS WITH GW

PHASE

We now examine the distribution of turbulence velocity and potential temperature variances and sources within the phase of the breaking GW. The primary GW structure, turbulence variances, and energy sources, averaged over planes along the GW phase, are shown throughout wave breaking in Fig. 3. The upper panel shows the primary GW streamwise velocity (dashed) and potential temperature (solid) perturbations over a vertical wavelength at seven times during and after breaking, with the GW phase having the maximum potential temperature perturbation centered in the domain at each time (this is the same phase displayed in our Fig. 1 and also in Fig. 2 of Part I). Turbulence velocity variances (middle) and shear and buoyancy production (bottom) are shown relative to the primary GW phase at the same times. Turbulence variances and sources indicate that instability growth and turbulence generation are relatively localized within the GW phase structure during active wave breaking. Velocity variances (with horizontal, spanwise, and vertical components shown with long-dashed, light solid, and dashed–dotted lines, respectively) achieve their maximum values between the maximum streamwise shear and the minimum static stability within the GW phase. Horizontal velocity variance, in particular, exhibits a sharp maximum immediately above the (descending) site of maximum shear generation at t ; 11–12 (consistent with GW shear being the only direct source of horizontal turbulence kinetic energy) and is somewhat more downstream and more diffused vertically as turbulence decays. The spanwise and vertical velocity variances mirror this behavior to some degree, but are smaller in magnitude and more distributed at the outset (and throughout the evolution), suggesting a finite equilibration time for kinetic energy redistribution among the three components. Potential temperature variance is somewhat more uniform across the GW phase and maximizes within the most statically stable phase of the GW because of advection of turbulence out of active source regions and its generation of larger potential temperature gradients in this phase of the GW. Buoyancy production of potential temperature variance is more variable spatially than shear production of velocity variance because it is dependent on both the distribution of velocity variances and the variable stratification within the GW field. We have seen that separations between locations of maximum velocity

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FIG. 3. (top) Streamwise- and spanwise-averaged horizontal (x) velocity (dashed) and potential temperature (solid) defining the primary GW structure. (middle) Turbulence potential temperature and component velocity variances, with potential energy and horizontal (x), spanwise, and vertical velocity variances shown as heavy solid, long-dashed, light solid, and dashed–dotted, respectively, averaged along the phase of the initial GW after removal of the phase-plane means at t 5 10, 11, 12, 13, 14, 15, and 20 for a 5 1.1. (bottom) Shear (dashed) and buoyancy (solid) sources of turbulence energy. Profiles are shown as functions of the vertical domain coordinate relative to a fixed initial GW phase, with the maximum GW streamwise velocity gradient and maximum potential temperature perturbation, located in the domain center at each time (same phase as in Fig. 1). Successive profiles are displaced horizontally by increments of (top) 2, (middle) 0.4, and (bottom) 1 unit.

variances (and mechanical energy dissipation rates) and those of maximum potential temperature variances (and refractive index fluctuations) can have significant implications for measurements of large-scale dynamics and inferences of mean structures, turbulence quantities, measurement biases, and inferred mixing in applications to shear instability (Gibson-Wilde et al. 2000; Fritts et al. 2003), and we expect that similar tendencies will accompany GW breaking.

Referring to Fig. 3, we see tendencies during energetic wave breaking (t ; 10–13) both for 1) phases of maximum streamwise velocity and maximum potential temperature to become closer and for 2) positive gradients of both streamwise velocity and potential temperature to exceed the negative gradients in magnitude, thus providing an explanation of the transient, positive correlation noted in the discussion in Part I. The presence of the most intense turbulence within the statically

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most unstable GW phase also appears to support the earlier theoretical arguments for a large turbulence Prandtl number accompanying GW breaking (Fritts and Dunkerton 1985; Coy and Fritts 1988; McIntyre 1989).

3) DISTRIBUTIONS OF MECHANICAL

AND

THERMAL ENERGY DISSIPATION

We saw above that both turbulence energy sources and velocity and potential temperature variances exhibit significant variability within the GW phase structure and with time. Thus, we should expect the mechanical and thermal energy dissipation rates to exhibit similar variability. These fields are displayed together in volumetric views from the side and below in Fig. 4, with the same format and times as Fig. 1 for easy comparison. These two fields are separated clearly only at the earliest times (T ; 7 and 10) because this is when largescale GW and instability structures create distinct largescale thermal gradients in the adjacent fluid. Because differential advection (stretching or shearing) is the cause of enhanced thermal gradients throughout the flow evolution, it is challenging to separate the dissipation fields spatially at later times because of the smaller spatial scales on which these dynamics operate in highRe turbulence. Indeed, the dissipation fields begin to entwine as initial instabilities achieve finite amplitudes and become even more closely entwined within the regions of active turbulence (t ; 12–15). In regions where turbulence has been advected out of the least stable GW phase, however, and at later times when turbulence energy sources have abated, the dissipation fields become increasingly distinct, and thermal dissipation occurs largely in quasi-horizontally stratified layers within the more stratified regions of the flow (t ; 15–30). Indeed, these layers are initially partially inclined along the GW phase, but they become more horizontally aligned as turbulence intensities subside. This is a distinct difference from the initial evolutions of these fields accompanying KH instability (Werne and Fritts 2001; Fritts et al. 2003). The reason is that turbulence accompanying KH instability remains confined to the shear layer at which instability arose, rapidly mixes the fluid within the billows, and drives the thermal gradients to the edges of the turbulence layer where they evolve largely separately to late times. In contrast to KH instability, turbulence due to GW breaking remains most active near the least stable phase of the GW, and turbulence that has already formed is advected into the increasingly stratified fluid as the least stable GW phase progresses downward. Nevertheless, turbulence due to wave breaking also results in regions of more intense local turbulence (which mix gradients to the edge regions) interleaved with regions of weaker tur-

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bulence having significantly greater thermal gradients and stratification, both during active turbulence and at late stages of decay. We examine the statistics of turbulence energy dissipation rates with histograms of log (Re e) and log x in the upper panels of Fig. 5 at the times for which 3D views are shown in Figs. 1 and 4. These reveal highly non-Gaussian distributions of log (Ree) and log x at t 5 7 and a still clearly non-Gaussian distribution of log x at t 5 10. This is because initial instability structures and vortex dynamics have only begun to drive a turbulence energy cascade, the dominant source of turbulence energy is shear (Fig. 2), and responses at small scales occur first in the velocity or vorticity fields [see the discussion of Eqs. (1)–(4) above]. Although there are large gradients yielding strong dissipation at t 5 10, the instability structures, and hence the peak dissipation values, are still relatively uniform. The turbulence spectra are also highly nonstationary at this time (see below) and require finite time to transfer enstrophy to the smallest spatial scales. Turbulence energy and dissipation achieve their maximum values at t ; 12, but a clear inertial range persists for several additional GW periods. At these times, the dissipation histograms appear to have very nearly lognormal distributions [Gaussian log (Ree) and log x distributions], to achieve maximum mean and median values at t ; 12, and to achieve maximum distribution widths at t ; 12, which decrease with time thereafter. In particular, we note the following: 1) mean dissipations and widths increase from the onset of GW breaking until the time of maximum turbulence energy generation and dissipation, 2) mean dissipations and widths decrease following maximum turbulence energy generation and dissipation, and 3) the dissipation distributions remain approximately lognormal to late times in the turbulence decay.

c. Turbulence spectra and anisotropy We now discuss the turbulence velocity and potential temperature spectra, their evolution, and their anisotropy throughout wave breaking and restratification of the flow. Total velocity variance spectra (denoted ui9ui9) along each direction in the computational domain (x9, y9, z9), averaged over the normal plane, are shown from t 5 1–79 in intervals of 3Tb in the upper panels of Fig. 6. Early spectra exhibit obvious and significant anisotropy, with smaller spatial scales along y9 and z9 than along x9. This reflects the near-streamwise orientations of the initial instability structures discussed above. Indeed, we see clear evidence of the preferred instability scales discussed in Part I in the y9 spectra at wavenumbers ;1–5 at the earliest times, shifting to smaller scales and then

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FIG. 4. As in Fig. 1, but for the mechanical (yellow/red) and thermal (blue) dissipation fields at a 5 1.1. Regions of intense dissipation are red and light blue, respectively. Light areas in these panels indicate interleaved or entwined regions of mechanical and thermal dissipation. Color scales are different at each time because of large variations in dissipation with time.

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FIG. 5. Distributions of (left) mechanical energy dissipation e and (right) thermal energy dissipation x magnitudes throughout the model domain for a 5 (top) 1.1 and (bottom) 0.9 for the five times displayed in Figs. 4 and 12, respectively. During and after strong turbulence, these distributions suggest that energy dissipation rates are approximately lognormal to a reasonable approximation.

disappearing as maximum spectral amplitudes are achieved. The z9 spectra exhibit a large maximum at wavenumber 1 at early times because of the large primary GW. Spectra become more nearly isotropic by t ; 10 at higher wavenumbers with elevation of all spectral amplitudes, and they remain nearly indistinguishable on the expanded scales in Fig. 6 (top). Component velocity variance spectra (denoted u92, 2 y9 , and w92) are shown in the lower panels of Fig. 6 along each direction in the computational domain at t 5 12, 15, 20, and 30 (shown in black, red, orange, and

green, respectively). Here, the velocity variance spectra along the direction of the spectral computation have been elevated by 4/3 for comparison with all other components. This factor is a consequence of velocities and gradients at specific scales being smaller along than they are normal to the direction of spectral computation. These spectra suggest that there are systematic differences in spectral amplitudes that are small at low wavenumbers and at early times but that increase toward higher wavenumbers at early times and at all wavenumbers at late times (and small e). As discussed

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FIG. 6. (top) Total velocity variance spectra in the streamwise, spanwise, and vertical directions [in domain coordinates (x9, y9, z9), left to right] for a 5 1.1 from t 5 1 to 79 in increments of 3Tb. Earlier times are shown in dark blue, followed by light blue, green, yellow, orange, and red, and the spectra are labeled from t 5 1 to 16 on the left-hand side. Nyquist scales were increased as the turbulence dissipation rate e decreased to conserve computational resources. Potential temperature variance spectra are almost indistinguishable and are not shown. (bottom) Component velocity variance spectra (without removal of streamwise GW motions) along the (left) streamwise, (middle) spanwise, and (right) vertical directions in the computational domain coordinates (x9, y9, z9) are shown for a 5 1.1 at t 5 12, 15, 20, and 30 (black, red, orange, and green, respectively). Component velocity variance spectra in the streamwise, spanwise, and vertical directions and potential temperature variance spectra are shown with solid, dotted, dashed–dotted, and long-dashed lines, respectively. In each case, the component velocity variance along the direction of the spectral computation (i.e., u92 in the case of x9 spectra) has been elevated by 4/3 to account for the ratio of component variances expected in isotropic turbulence. The reference line in each panel shows a slope of 25/3.

further below, however, the large apparent differences in spectral amplitudes are not by themselves indications of significant turbulence anisotropy. Ratios of velocity variance spectra (again denoted u92, 2 y9 , and w92) and three spectral orientations (x9, y9, z9) are shown at three times in Fig. 7, again with the spectral amplitude along the direction of spectral computation elevated by 4/3 for comparison. These help to quantify the different spectral amplitudes more precisely than is possible from the spectra in Fig. 6 alone. In most cases, they also suggest that departures from isotropy increase with both wavenumber and time (with the most intense turbulence at the earliest times). The spectral amplitude ratios are closest to unity when turbulence is most intense (t ; 12–15) and at the larger scales within the inertial range (typically at wavenumbers of ;5–10, with the inertial range extending from ;2 to 40 when tur-

bulence is strong). Apparent departures from isotropy are often large; however, they are also partially an artifact of the continuity equation. This implies departures from unity at high wavenumbers in the ratios of velocity variance spectra when one component is along the direction of spectral computation, and unity when neither variance is in this direction—for instance, u92(m9)/y92(m9) (Werne and Fritts 2001). The continuity equation requires (Werne and Fritts 2001) ð 9‘ u9i 2 (k9) 5 2k9 [u9j 2 (k)/k2 ]dk (7) k9

for component wavenumber k and i 6¼ j, and we can employ this relation to check the true departures from isotropy by comparing the measured spectral amplitudes with those computed from the other fields through Eq. (7). Thus, to assess true departures from isotropy, we

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FIG. 7. Ratios of spectral amplitudes as functions of component wavenumbers for times of t 5 12 (heavy solid), 15 (light solid), and 20 (dashed) for the simulation at a 5 1.1. Spectral amplitude ratios are (top) w92/u92, (middle) y92/u92, and (bottom) u92/y92 as functions of (left) streamwise, (middle) spanwise, and (right) vertical wavenumber (k9, l9, m9). All wavenumber magnitudes here are relative to m9 5 1 in order to compare the same scales for different domain dimensions.

compute the ratio of the velocity variance spectrum in one direction to its spectrum implied by Eq. (7) computed in a different direction, denoted u92/u92(y92) for measured u92 compared to u92 computed from y92. These ratios are displayed in Fig. 8 and are seen to be much nearer unity than those in Fig. 7. They are also nearer unity for all wavenumbers spanning inertial and dissipation range scales, and for the stronger turbulence during active wave breaking than at later times. However, there remain

systematic departures from unity of ;20% that indicate clear and persistent departures from isotropy even during the most active turbulence, with departures increasing to ;50% or larger at late stages of restratification, consistent with values noted in our previous assessment for stratified shear flows (Werne and Fritts 2001). We have also computed these ratios as functions of GW phase and found that departures decrease (increase) by factors of ;2 at the phase of most (least) intense turbulence. These departures

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FIG. 8. Ratios of spectral amplitudes computed from the DNS data and predicted from other spectra employing Eq. (7) streamwise, spanwise, and vertical velocity perturbations and spectral indices (top to bottom) for times of t 5 12 (heavy solid), 15 (light solid), and 20 (dashed) for the simulation at a 5 1.1. Departures of these ratios from unity are indications of true anisotropy in the turbulence fields. Notation is as in Fig. 7.

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from isotropy occur despite Sc* 0.01, which is the requirement for approach to isotropy at the smaller scales of motion in sheared flows (Saddoughi and Veeravalli 1994; Saddoughi 1997). Thus, we conclude that stratification likely makes a significant, perhaps preferential, contribution to the anisotropy observed in this simulation.

3. Turbulence evolution at a 5 0.9 a. Turbulence dynamics We noted in Part I that the simulations for a 5 1.1 and 0.9 exhibit dissimilar initial scales but similar instability orientations leading to wave breaking. Compared to the initial vorticity structures shown for a 5 1.1 in Fig. 1, those for a 5 0.9 (see Fig. 9) appear to be approximately twice as large, consistent with the Floquet analysis discussed in Part I. The cascade to turbulence and the associated vorticity dynamics that follow are qualitatively very similar in the two simulations, despite requiring approximately twice as long for a 5 0.9. As for a 5 1.1, the sites of most rapid and intense turbulence generation are those where horseshoe vortices, followed by ringlike vortices, form at the upstream edges (left edges, and lower GW phase) of initial counterrotating streamwise vortices within the most unstable GW phase. However, these structures are actually somewhat more complex than described above for both GW amplitudes. Both evolutions show the left edges of the ‘‘vortex rings’’ to evolve from the horseshoe vortices, with the right (downstream) edges of the rings having the opposite slope to the horseshoe vortices in the streamwisephase normal plane, arising because of enhanced positive spanwise vorticity accompanying downward advection within the horseshoe vortices into a region of existing positive spanwise vorticity, and extending to lower altitudes into the opposite phase of the GW field. Because turbulence remains correlated with the most unstable GW phase, and because the GW phase progresses downward with time, the more active turbulence sources are advected leftward and upward in Fig. 9 (along the GW group velocity), while regions of decaying turbulence in the opposite GW phase are advected rightward and downward. A major difference between the evolutions displayed in Figs. 1 and 9, apart from those noted above, is that patches of intense turbulence appear to be more variable and discrete at a 5 0.9 than at a 5 1.1.

b. Turbulence statistics and correlations with the primary GW 1) DOMAIN-AVERAGED

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Temporal evolutions of domain-averaged turbulence quantities for a 5 0.9 are shown in Fig. 10. As seen for

a 5 1.1, velocity and potential temperature variances (top left) are seen to exhibit sharp increases accompanying wave breaking, with maxima at t ; 23. Relative to a 5 1.1, however, maximum variances are smaller by ;2, as is the duration of strong turbulence, and the subsequent decay of turbulence variances is much slower for a 5 0.9. Shear and buoyancy sources contribute for a 5 0.9 in approximately the same ratio as for a 5 1.1, but with amplitudes smaller by ;3 because of weaker instabilities and slower wave breaking occurring for a 5 0.9. Shear production also contributes for approximately twice the time for a 5 0.9 for the same reason, but buoyancy production is small and brief, likely because the GW was not initially overturning in the potential temperature field. Whereas for a 5 1.1 buoyancy production peaked during turbulence generation, for a 5 0.9 buoyancy production is small at early stages but becomes more significant and oscillatory at a range of periods thereafter, suggesting that GWs (with l 5 0 and/ or l 6¼ 0) are contributing to conversions between turbulence kinetic and potential energies. Velocity variance ratios for a 5 0.9 (Fig. 10, lower left) largely mirror their behavior for a 5 1.1. These are ;0.5 prior to wave breaking and increase to ;0.8–1.2 during active wave breaking and turbulence. The ratio of turbulence potential temperature to velocity variance (Fig. 10, lower right) is likewise ;0.5 during active wave breaking and increases gradually toward ;0.9 thereafter, perhaps due to the presence of stable stratification at nearly all locations in the flow following (and accompanying) turbulence decay. As for a 5 1.1, there are several factors that may cause departures of domainaveraged quantities from expectations of isotropy, among them the exclusion of streamwise (l 5 0) modes from the computations. For a 5 0.9, we estimate a mean Rl ; 320 based on domain-averaged quantities and a peak value averaged over the GW phase having the most intense turbulence of ;500, each ;20% smaller than at a 5 1.1.

2) VARIATIONS WITH GW

PHASE

We now examine the distribution of turbulence velocity and potential temperature variances and sources within the phase of the breaking GW for a 5 0.9. As above, primary GW structure, component velocity variances, and sources, averaged on planes along the GW phase, are shown throughout wave breaking in Fig. 11. Variances and sources for a 5 0.9 again indicate that instability growth and turbulence generation are relatively localized within the GW phase structure during active breaking. Velocity variances again achieve their maximum values between the maximum streamwise shear and the minimum static stability within the GW

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FIG. 9. As in Fig. 1, but for a 5 0.9. Note color scales are different at each time.

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FIG. 10. As in Fig. 2, but for a 5 0.9.

phase, whereas the horizontal variance (along x), in particular, exhibits a sharp maximum immediately above the site of maximum shear generation at t ; 22–24 and somewhat further downstream as turbulence decays. The spanwise and vertical velocity variances again mirror this behavior but do not exhibit the clear maximum displayed by the horizontal (x) component, which suggests a finite equilibration time for kinetic energy redistribution among the three components. As for a 5 1.1, turbulence potential energy is more uniform across the GW phase and maximizes within the most statically stable phase. A more significant difference between the results for a 5 0.9 and 1.1 is the behavior of buoyancy production across the GW phase following wave breaking. Whereas buoyancy production was seen to be more variable spatially than shear production for a 5 1.1 at early times, the magnitude of buoyancy production remains much larger for a 5 0.9 at much later times following wave breaking. For a 5 1.1, this term decayed by a factor of ;4 within ;3Tb, but for a 5 0.9, this term remains large and variable more than ;20Tb thereafter. This is likely a consequence of exchanges between kinetic and potential energy driven either by GWs with l 5 0 influencing turbulence buoyancy production or by GWs with l 6¼ 0 contributing exchanges within our defined turbulence field.

As for a 5 1.1, we see a tendency during wave breaking for a 5 0.9 for the phases of maximum streamwise velocity and maximum potential temperature to become closer, suggesting a transient, positive vertical potential temperature flux, consistent with that noted in Part I. Similarly, the presence of the most intense mechanical turbulence at the least statically stable GW phase supports previous arguments for a large turbulent Prandtl number, even for a 5 0.9.

3) DISTRIBUTIONS OF MECHANICAL AND THERMAL ENERGY DISSIPATION

The structures and evolutions of the mechanical and thermal energy dissipation fields for a 5 0.9 are displayed in Fig. 12 at the same times the turbulence fields are shown in Fig. 9. As seen for a 5 1.1 and discussed above, there are several phases in the evolutions of these fields. The first accompanies instability growth and turbulence generation and leads to dissipation fields that occur at larger scales and are spatially separated (t ; 20–22). A second phase accompanies the transition to small turbulence scales, during which the dissipation fields are closely entwined (t ; 25–30). A third accompanies turbulence decay within the more stable GW phase and exhibits regions, especially in thermal dissipation, that evolve from being more aligned with the

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FIG. 11. As in Fig. 3, but for a 5 0.9 at t 5 20, 22, 24, 26, 28, 30, and 40. Note the similarities between the two simulations in both the GW structure and amplitude evolution and in the distribution of turbulence within the GW field. The major difference is the slower evolution of wave breaking at a 5 0.9 (;2 times as long). Successive profiles are displaced horizontally by increments of (top) 2, (middle) 0.2, and (bottom) 0.2 units.

GW phase at early times to being more horizontal at later times (t ; 30–40). Apart from the delayed and longer turbulence evolution, and the larger initial instability and turbulence scales, these dissipation field evolutions closely parallel those discussed for a 5 1.1 above. Histograms of log (Ree) and log x are shown for a 5 0.9 in the lower panels of Fig. 5 at the times for which 3D views are shown in Figs. 9 and 12. As for a 5 1.1, these exhibit significant departures from lognormal distributions at t 5 20 and 22, with distributions appearing more nearly lognormal thereafter. The initial distributions are not as sharply peaked for a 5 0.9 as for a 5 1.1, however, suggesting a less sudden transition to small scales for a 5 0.9, consistent with the longer duration of wave breaking and the weaker turbulence arising in this

case. At later stages, the distributions follow the general trend noted for a 5 1.1, with maximum mean and median dissipation values at the termination of large GW amplitude reductions and decreases in these values and the distribution widths at later times. A major distinction is that the temporal evolutions of the dissipation statistics are much slower for a 5 0.9.

c. Turbulence spectra and anisotropy Turbulence kinetic and potential energy spectra, their evolutions, and their anisotropy throughout wave breaking and the subsequent restratification of the flow for a 5 0.9 resemble closely those displayed above for a 5 1.1. The most apparent differences between the spectra for a 5 0.9 and 1.1 are 1) the more distinct peaks at smaller

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FIG. 12. As in Fig. 4, but for a 5 0.9. Note that color scales are different at each time.

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FIG. 13. As in Fig. 8, but for a 5 0.9 at t 5 24 (heavy solid), 30 (light solid), and 40 (dashed).

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spanwise wavenumbers during initial instability, 2) the slightly smaller maximum spectral amplitudes at a given wavenumber, and 3) the slower reduction in spectral amplitudes at late times for a 5 0.9. The first highlights the larger initial instability scales occurring for a 5 0.9; the second and third reflect the smaller initial turbulence energies and their persistence to late times for a 5 0.9. Thus, we will not display the results for a 5 0.9 corresponding to Figs. 6 and 7 but will summarize the indications of turbulence anisotropy (analogous to Fig. 8 above) in Fig. 13. As for a 5 1.1, the ratios of measured and predicted spectral amplitudes reveal departures from unity (and from isotropy) at virtually all spatial scales and times, again with departures as large as ;20% during the most energetic turbulence and increasing with time, and with departures that are smaller (larger) by ;2 in the most (least) turbulent phase of the GW. Indeed, the variations of each ratio with spatial scale and time bear a very close resemblance to the results for a 5 1.1, suggesting similar influences of stratification and/or shear on turbulence structure throughout both wave breaking evolutions. In this simulation Sc* ; 0.01 during most intense turbulence, again suggesting that shear plays a small role. As for a 5 1.1, this suggests that stratification plays a greater role than shear in accounting for anisotropy occurring for a 5 0.9.

4. Summary and conclusions We employed high-resolution direct numerical simulations of GW breaking for GWs having an intrinsic frequency v ; N/3.2 and initial amplitudes of a 5 0.9 and 1.1 to study the transition to turbulence and the evolution, character, and anisotropy of the resulting turbulence spectrum at a Reynolds number of Re 5 lz2/ nTb 5 104. Corresponding Taylor microscale Reynolds numbers were Rl ; 670 and 500 for a 5 1.1 and 0.9, respectively, at the most energetic phase of the GW. The high Re enabled a vigorous and extended inertial range of turbulence sufficiently broad to assess anisotropy as a function of wavenumber. Turbulence transitions and evolutions for the two GW amplitudes exhibit both surprising similarities and significant differences. Similarities among the two simulations include 1) initial instabilities having similar structures and orientations, comprising streamwise-aligned, counter-rotating vortices linked as horseshoe vortices at the upstream ends; 2) a high degree of intermittency and spatial variability of strong turbulence originating with initial instability structures but also persisting to late times; 3) an inertial range of turbulence extending well over a decade of spatial scales in regions where, and at times

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when, turbulence is sufficiently strong; 4) evolutions of the variance spectra with time, and correlations of turbulence variances and shear and buoyancy production terms with the phase of the initial GW; and 5) significant and systematic departures of the resulting turbulence from expectations of stationarity, homogeneity, and isotropy, and similar evolutions of these departures as functions of wavenumber and time. Significant differences between the wave breaking simulations for a 5 0.9 and 1.1 include 1) larger coherent structures in the initial turbulence fields for a 5 0.9 and 2) less well defined and less transient shear and buoyancy production for a 5 0.9. Our major results include 1) verification of the predictions of linear Floquet and optimal perturbation theory for the instability modes triggering GW breaking; 2) definition of the dynamics of the transition from laminar to turbulent flow; 3) quantification of the competition between 2D energy exchanges and 3D instabilities in driving GW amplitude reductions and postbreaking dynamics; 4) apparent confirmation of the role of twist-wave turbulence dynamics in driving the cascade to smaller spatial scales; 5) confirmation of departures of turbulence structure and statistics from those predicted for stationary, homogeneous, isotropic turbulence noted, or predicted, in previous studies; and 6) definition of the relations between turbulence energies (and character) and GW phase that is key to anticipating other effects, including transport efficiency and implications for observations. We anticipate that these results, and similar results for other GW amplitudes, frequencies, environments, and superpositions, will play an important role in crafting more accurate parameterizations of GW transport, mixing, and spectral interactions important to their accurate description in large-scale numerical weather prediction, climate, and general circulation models of relevance to a much broader community. Acknowledgments. Support for this research was provided by AFOSR contracts F49620-03-C-0045 and FA9550-06-C-0129, NASA contracts NAS5-02036 and NAS5-02069, and NSF Grants ATM-0314060 and ATM0435789. We also acknowledge the DoD High Performance Computing Modernization Office for access to large computational resources and Randall Hand and Paul Adams of ERDC for their assistance in making the accompanying flow animations.

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