1. Introduction
The importance of gravity wave–fine structure (GW–FS) interactions throughout the atmosphere is discussed in a companion paper by Fritts et al. (2013, hereafter Part I). As defined by Part I, GW–FS interactions are a subset of more general multiscale interactions among GWs, planetary waves, tides at higher altitudes, and mean wind and stability fields that encompass the mutual influences and evolutions of motions spanning a wide range of spatial and temporal scales. GW–FS interactions arguably occur to varying degrees at essentially all altitudes all of the time. Importantly, studies addressing GW–FS interactions to date suggest significant differences from those accompanying individual GWs in terms of GW amplitudes, spectral evolution, instabilities, and turbulence occurrence, intensities, and statistics. Thus, we expect GW–FS interactions to play central roles wherever GW energy and momentum transport, and deposition accompanying GW dissipation, yield significant forcing, induced mean motions, and/or mixing.
Examples of apparent GW–FS interactions (or more general multiscale) dynamics, and their consequences, that have eluded a clear understanding to date include 1) intermittent, layered small-scale turbulence in the stable boundary layer (SBL) under relatively quiescent conditions; 2) the dynamics leading to “sheet and layer” structures in winds, stability, and various turbulence parameters at smaller vertical scales extending from the SBL into the thermosphere; and 3) clear examples of larger- and smaller-scale GW superpositions and interactions accompanying increases in mean stability at the tropopause and the mesopause. Factors that appear to contribute to intermittent small-scale turbulence and layering in the SBL include GWs, density currents, intrusions, and meandering motions that constitute local and “nonlocal” influences on SBL turbulence (Mahrt 1998, 1999; Chimonas 1999; Monti et al. 2002; Sun et al. 2002, 2004; Balsley et al. 2003, 2008, 2013). Initial analytic studies of GW–turbulence interactions assumed local shear instability or “overturning” accompanying small or negative Richardson numbers in order to assess the mutual interactions of a GW and turbulence (Chimonas 1972; Finnigan and Einaudi 1981; Fua et al. 1982). Subsequent direct numerical simulations (DNSs) of GW–FS interactions by Fritts et al. (2009a, hereafter F09a) and in Part I confirmed both the roles of small-scale shears and overturning in driving turbulence generation and the strong departures of such evolutions from GW instability dynamics occurring without FS influences (Fritts et al. 2009b,c, hereafter F09b and F09b). Part I further identified intrusions comprising superposed GW and FS flows as a principle driver of turbulence events.
There is now extensive observational evidence of strong layering of winds and stability, and of their dynamical origins, at vertical scales ranging from meters to tens or hundreds of meters in the SBL and hundreds of meters to several kilometers extending from the middle troposphere into the mesosphere and lower thermosphere (MLT). Evidence in the SBL comes from high-resolution vertical profiles that suggest local instabilities in sufficiently strong, small-scale layers to yield sharp sheets of very high stability, often multiple sheets separated by weakly stratified layers, large temperature structure parameter (
Evidence of sheet and layer structures, and indications of associated turbulence variability, in the troposphere, stratosphere, and the MLT come from a variety of remote sensing and in situ measurements. These include 1) high-resolution balloon and unmanned aerial vehicle (UAV) measurements of winds, temperatures, 
Observational evidence of similar FS in the oceans and of the internal wave, instability, and turbulence dynamics that drive their generation and evolution preceded observations in the atmosphere. Useful references to, and reviews of, these contributions include Woods (1968, 1969), Osborn and Cox (1972), Woods and Wiley (1972), Garrett and Munk (1972, 1979), Gregg and Briscoe (1979), VanZandt (1982), Müller et al. (1986), Gregg (1987), Thorpe (1987, 1999), and VanZandt and Fritts (1989).
In the atmosphere, multiscale interactions are typically enhanced near strong GW sources and where GWs are refracted to smaller vertical scales and larger amplitudes in a common volume. Examples include strong jet streams or frontal systems (Koch et al. 2005), the lower stratosphere during strong mountain wave responses (Lilly 1978; Smith et al. 2008), and the lower thermosphere above the polar summer mesopause (Fritts et al. 2004). The tropopause and the polar summer mesopause, in particular, impose a large increase in buoyancy frequency N as GWs propagate upward, and this implies significant increases in GW amplitudes, stronger GW interactions, and an increase in local instabilities accompanying larger and superposed GWs (VanZandt and Fritts 1989). This is because a single GW encountering an increased N will experience increasing amplitude [a ~ (N/N0)1/2], assuming no GW dissipation and neglecting amplitude increases accompanying decreases in mean density. Such dynamics appear to account for the strong statistical increases in radar backscatter at each of these sites.
Measurements of ɛ and the thermal energy dissipation rate χ, and related quantities (e.g., winds, temperatures, 
New measurement capabilities can characterize turbulence events and their evolution, structure, and spectral character in space and time. As an example, high-resolution aircraft measurements of turbulence spectra and structure functions (Wroblewski et al. 2003) confirmed earlier predictions of structure functions in DNS of Kelvin–Helmholtz instability (KHI) (Werne and Fritts 2000) that differed from previous theoretical estimates. Measurements during the 1999 Cooperative Atmosphere–Surface Exchange Study (CASES-99) and DNS of idealized GW breaking confirmed the occurrence of nearly lognormal distributions of ɛ and χ for turbulence events that span sufficiently long times or large volumes (Frehlich et al. 2004; F09b). Similar measurements of ɛ distributions in the SBL, in contrast, suggest multiple, superposed quasi-lognormal distributions (B. B. Balsley 2013, personal communication).
Thus, there are compelling reasons to address GW–FS interactions in a systematic fashion employing DNSs that fully describe the larger-scale GW field, the instability structures that arise, and the turbulence that results. Part I described four idealized GW–FS superpositions in order to identify the multiscale dynamics that contribute most to the larger-scale flow evolutions and the turbulence sources, characteristics, and intermittency at smaller scales. Our purpose in this paper is to examine the characteristics and statistics of ɛ and χ within the turbulence fields arising in the DNS described in Part I. Specifically, we will define the dependence of turbulence dissipation morphologies on the turbulence sources and evolutions, quantify the dissipation statistics and their evolutions and correlations in space and time, and discuss the implications of these dissipation fields for measurements of turbulence parameters.
Our paper is organized as follows. Section 2 reviews our numerical formulation and the two GW–FS DNS described in Part I that will be employed here. Section 3 describes the evolutions of the dissipation fields, their dependence on turbulence sources, the probability distribution functions (PDFs) arising from these dynamics, the correlations among these fields, and their implications for measurements. Section 4 discusses the relationship of these results to measurements throughout the atmosphere and their implications for understanding the character of more general multiscale interactions. Section 5 presents a summary and our conclusions.
2. Model formulation and case studies
a. Model formulation
The computational domain is chosen to accommodate both an initial GW and mean FS motions having a smaller vertical scale. This is accomplished with a periodic domain oriented along the phase lines (or group velocity) of the GW (referred to as the streamwise direction) and having the minimum length required to enable the specified FS to also be periodic in the domain. Any periodic domain places constraints on other motions that can arise owing to nonlinear interactions, but a tilted domain enables much greater diversity in the modes that can be excited at lower frequencies. Because of the periodic constraints, we assumed an initial state comprising a superposed monochromatic GW and undisturbed FS, which differs from the superposition that would arise for a GW packet propagating into such an FS field. Despite these idealizations, we expect the resulting dynamics to be representative of such GW–FS interactions, at least after initial transient responses, because instabilities and turbulence require several Tb to arise.
The computational domain is designed to accommodate an initial monochromatic GW having an intrinsic frequency of ω ~ N/10, such that the domain is inclined at an angle φ = 5.73° from horizontal and has dimensions (X′, Y′, Z′). The GW phase-normal wavelength is λ = Z′ such that λz = Z′/cosφ. The FS vertical wavelength is specified to be λ cosφ/5, corresponding to five wavelengths in the vertical projection of the end of the domain. This is approximately 2 times larger than considered by F09a to allow instabilities and turbulence that are not artificially constrained by viscosity and thermal diffusivity at small scales. With these choices, the streamwise domain length is X′ = Z′/(5 tanφ) = 1.993Z′, and the spanwise dimension is chosen to be Y′ = 0.5X′ to ensure an ability to define any instability structures that might arise for the cases considered. Additional details on the code architecture and the numerical methods are provided by Part I.
b. GW–FS cases
As in Part I, we assume an initial GW having a nondimensional amplitude 
Initial FS velocities, U and V, are assumed sinusoidal in either the plane of GW propagation or as a rotary field. These correspond to cases L0 and R considered by Part I, with clockwise FS wind rotation with increasing altitude in case R. The maximum FS shear is assumed to be [d(U, V)/dz]max = 2N so that the minimum FS Ri = N2/[(dU/dz)2 + (dV/dz)2] = 0.25.
3. Dissipation field evolutions and statistics
a. Overview of domain-averaged and vertical profile evolutions
Part I provides a detailed assessment of instability and turbulence character and influences on the 2D and 3D motion fields for cases L0 and R. Here we extend the overview to include the domain-averaged dissipation evolutions and statistics for these two cases. The mechanical and thermal energy dissipation terms appear in the energy equation as ɛ and Riχ (where they have comparable magnitudes), hence we will employ these forms in our discussion here. Time series of the domain-averaged ɛ and Riχ, denoted [ɛ] and [Riχ], respectively, for cases L0 and R are compared with the evolutions of the primary GW amplitudes and the 3D (l ≠ 0) “turbulent” kinetic and potential energies (TKE and TPE, respectively, excluding only motions having l = 0) in the top panels of Fig. 1. Full-domain PDFs for ɛ and Riχ are shown from t = 4Tb to 24Tb at intervals of 4Tb for the two cases in the bottom four panels of Fig. 1. Vertical profiles of 〈N2〉, log〈ɛ〉, and log〈Riχ〉, where the angle brackets denote a spanwise average, are displayed at the center of the streamwise domain from t = 2Tb–24Tb at intervals of 2Tb in Fig. 2.
(top) Time series of (left) GW streamwise velocity (black) and θ (red) amplitudes, (middle) mean 3D (l ≠ 0) TKE (black) and TPE (red), and (right) mean ɛ (black) and Riχ (red) from t = 0 to 25Tb. Results for cases L0 and R are shown with solid and dashed lines, respectively. (bottom) Full-domain PDFs of (first column),(third column) ɛ and (second column),(fourth column) Riχ for cases L0 and R. Times are colored as shown in the bottom left panel. For reference, those PDFs at t = 8Tb and 16Tb in the bottom left panel are approximately lognormal.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-059.1
(top) Vertical profiles of nondimensional spanwise-averaged 〈N2〉 for cases L0 (red) and R (black) from t = 0 to 24Tb at intervals of 2Tb. Nondimensional spanwise-averaged log10〈ɛ〉 (black) and log10〈Riχ〉 (red) for (middle) case L0 and (bottom) case R from t = 0 to 24Tb at intervals of 2Tb. The mean nondimensional 〈N2〉 = 4π2, and profile offsets are 300. Scales for log10〈ɛ〉 and log10〈Riχ〉 are linear and offset by 4 decades of 〈ɛ〉 and 〈Riχ〉. The initial magnitudes of 10−4 correspond to the initial profiles at t = 0 in each case.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-059.1
GW amplitude and 2D KE and PE evolutions for cases L0 and R are shown by Part I to be essentially identical out to t ~ 7Tb (Fig. 1), after which relatively small differences occur. The 3D TKEs and TPEs differ significantly, however; are driven by the sharp differences in instability character; and arise about 1Tb–2Tb later in case R because spanwise FS shears suppress instabilities accompanying strictly streamwise FS shears in case L0 (Fig. 1, and Figs. 5, 8, and 11 in Part I). Corresponding domain-averaged [ɛ] and [Riχ] are seen to correlate closely with the 3D TKEs and TPEs, apart from the large initial values of [ɛ] and the rapidly increasing [Riχ] in each case, both of which occur prior to instability and turbulence. The higher [ɛ] in case R at early times is due to the doubled FS shear variance, while the rapidly increasing [Riχ] in both cases is due to the evolution of θ FS accompanying FS advection of GW θ′ gradients. Once instabilities and turbulence arise, [ɛ] and [Riχ] track 3D TKE and TPE well, except that earlier maxima in TKE and TPE in both cases exhibit larger dissipation relative to the TKE and TPE maxima—consistent with the more intense turbulence and smaller turbulence scales accompanying the initial turbulence events.
PDFs of ɛ and Riχ in the bottom panels of Fig. 1 exhibit evolutions that differ from those seen for the idealized GW breaking DNS (F09b) in several respects. Whereas those for idealized GW breaking exhibit significant skewness prior to vigorous instability and turbulence generation (especially at the larger GW amplitude considered by F09b), both PDFs approach lognormal distributions for both GW amplitudes at later stages of the evolutions. PDFs for cases L0 and R, in contrast, exhibit less overall skewness but much greater structure prior to initial instabilities, especially in case R. Following instabilities and turbulence generation, both cases exhibit PDFs of ɛ that become quasi lognormal, while PDFs of Riχ retain high skewness to late times. We will see below that PDFs of ɛ and Riχ in subsets of the full domain exhibit very significant structure, large variations in median and mean values, and more nearly multiple-lognormal PDFs that reflect the local instability and turbulence dynamics within the larger-scale flow. Thus, domain-averaged dissipation statistics do not provide a complete picture of the complexity of these fields for multiscale flows.
Profiles of 〈N2〉, log〈ɛ〉, and log〈Riχ〉 in Fig. 2 reveal strong similarities and significant differences between the two cases and between the two dissipation fields, depending on the stage of instability and turbulence evolution and location. Profiles of 〈N2〉 clearly reveal similarities in the large- and small-scale features where cases L0 and R have each experienced instability and turbulence generation, or the lack thereof. They also exhibit clear differences where cases L0 and R are at different stages in the transition to instability and turbulence. Examples include regions where cases L0 and R exhibit initial instabilities at different times (see profiles at t = 4Tb and 8Tb) and where earlier instabilities and turbulence have led to differing sheet and layer structures (see t = 8Tb–12Tb).
Profiles of log〈ɛ〉 and log〈Riχ〉 for case L0 in the middle panel of Fig. 2 reveal various features and correlations, depending on the stage and spatial scale of instability and turbulence. A close correspondence is seen at locations of relative maxima accompanying strong, small-scale θ gradients and vorticity sheets and subsequent instability and turbulence events (see the narrow maxima at t = 8Tb–12Tb). Similar correlations are seen where turbulence layers are much deeper, and in which log〈ɛ〉 and log〈Riχ〉 maxima exhibit similar variations in altitude (see the more prominent events from t = 12Tb–16Tb). Other deep turbulence events exhibit very different correlations, however, having log〈Riχ〉 relatively weaker in the center of the layer where log〈ɛ〉 is large, but exhibiting maxima at one or both edges of the turbulence layer where log〈ɛ〉 is small (see the stronger events at t = 8Tb). This latter behavior suggests efficient mixing accompanying strong turbulence that reduces or largely eradicates strong θ gradients within the turbulence layer and reestablishes them in the edge regions, as noted in multiple observations cited above.
Profiles of log〈ɛ〉 and log〈Riχ〉 in case R closely resemble those in case L0, exhibiting similar close correspondence accompanying strong, small-scale θ gradients and vorticity sheets until instabilities and turbulence arise. These strong sheets are often stronger than seen in case L0 owing to suppression of initial instabilities in case R by spanwise shears, as discussed in Part I. Case R also exhibits reasonable correlations accompanying deeper turbulence layers at later times (see the middle and upper layers at t = 12Tb and 14Tb). Not seen in these profiles at these locations and times are maxima in log〈Riχ〉 at the edges of strong turbulence layers seen in case L0. Such correlations do occur, however, but in different regions than shown in the vertical profiles in Fig. 2.
b. Case L0: Aligned GW and FS velocities
1) ɛ-field evolution
The evolution of the ɛ field for case L0 is shown with streamwise vertical (hereafter streamwise) cross sections of the spanwise mean over the computational domain from t = 5Tb to 13Tb in the left columns of Figs. 3 and 4 (the right columns show the corresponding fields for case R to be discussed below for easy comparison). These images reveal the same instability and turbulence dynamics shown in the vorticity field in the left columns in Figs. 4 and 5 of Part I. Because the streamwise ɛ images highlight the sites of turbulence generation and ɛ maxima, however, they illustrate both the larger-scale GWs and smaller-scale instability and turbulence features more clearly than the corresponding vorticity fields.
Streamwise vertical cross sections of spanwise-averaged log10〈ɛ 〉 at t = 5, 6, 7, 8, 9, and 10Tb for cases (left) L0 and (right) R. The color scale is the same for all panels and spans 5 decades of intensities.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-059.1
As in Fig. 3, but at t = 10.5, 11, 11.5, 12, 12.5, and 13Tb.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-059.1
Prior to the dominant initial instabilities, ɛ is relatively small and accompanies the laminar vorticity sheets. Maximum values of ɛ quickly surpass their initial magnitudes during the early stages of shear and overturning instabilities at t ~ 5Tb, however (Fig. 3), and continue to achieve their maximum values during instability events and turbulence generation thereafter. Only at later times (t > 20Tb) do ɛ values in the vorticity sheets again become comparable to those accompanying decaying turbulence. Median, mean, and maximum values of ɛ are seen to largely decrease with time after the second maximum in energy dissipation extending from t ~ 11Tb to 15Tb, apart from weaker, less frequent intrusions occurring at later times.
The image at t = 6Tb in Fig. 3 (see also Fig. 3 of Part I) reveals initial small-scale fluctuations in ɛ accompanying initial instabilities in the relatively uniform and horizontally extended vorticity sheets (also having N2 < 0 owing to FS advection) in the upper-middle portion of the domain. Similar fluctuations are also seen in the more distorted flow in the lower portion of the domain beginning at t = 6Tb. The deeper overturning event in the lower portion of the domain exhibits a transition to turbulence within about 1Tb, and the ɛ fields confirm much enhanced dissipation in this and adjacent layers thereafter. Prior to instability, nondimensional ɛ maxima are about 10−2; after the transition to turbulence, ɛ maxima are about 10 times larger.
In the initial overturning event and thereafter, the ɛ fields reveal the complex transitional dynamics and their subsequent evolutions more clearly than the vorticity magnitudes shown by Part I. Maximum values of ɛ largely accompany the more conspicuous, larger-scale events. These events include 1) initial and subsequent shear instabilities occurring at smaller vertical scales (which are relatively weak), 2) the merging of regions of strong instability and turbulence sharing a common advecting layer (see Figs. 3 and 4 at left at t = 6Tb–8Tb and 10Tb–12Tb), 3) the evolutions of intrusions after t ~ 8Tb and 12Tb in Figs. 3 and 4, and that emerging beginning at t ~ 18Tb in the left column of Fig. 5, and 4) KHI having larger scales. The ɛ fields also nicely illustrate the effects of large horizontal gradients of horizontal velocity leading to fluid “plunging” reminiscent of GW breaking at the onset of intrusions seen to occur at t ~ 7Tb and 10Tb–12Tb in Figs. 3 and 4. These events are particularly conspicuous at t = 7Tb and 11.5Tb and will be evaluated in greater detail below.
As in Fig. 3, but at t = 18, 18.5, 19, 19.5, 20, and 20.5Tb for the upper half of the domain.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-059.1
2) ɛ and Riχ correlations
We employ streamwise cross sections of both spanwise-averaged and local ɛ and Riχ distributions to explore their character within the instability and turbulence fields, their statistical responses to instability and turbulence dynamics, and their implications for atmospheric observations at two representative times. In Fig. 6 spanwise averages (left column) and local values at the spanwise center of the domain (right column) are illustrated for both ɛ and Riχ fields (top and bottom panels) at t = 8Tb and 11.5Tb (top and bottom four-panel sets).
(left) (top to bottom) Spanwise averaged log10〈e〉 at t = 8Tb, log10〈Riχ〉 at 8Tb, log10〈e〉 at t = 11.5Tb, and log10〈Riχ〉 at 11.5Tb. (right) (top to bottom) Local log10〈e〉 at t = 8Tb, log10〈Riχ〉 at 8Tb, log10〈e〉 at t = 11.5Tb, and log10〈Riχ〉 at 11.5Tb.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-059.1
Comparing first the 〈ɛ〉 and 〈Riχ〉 fields at t = 8Tb (Fig. 6, two top left panels), we see fairly strong correlations between 〈ɛ〉 and 〈Riχ〉 maxima on nearly laminar (and coincident) vorticity and θ sheets where strong gradients are preserved prior to significant instability. But there are relatively few of these, given the significant occurrence of streamwise-aligned vortices and associated turbulence due to initial shear instability at the stronger vorticity sheets at this time (Fig. 5 in Part I). The consequence of instability and turbulence at these sites is mixing of the velocity gradients and dramatic reductions in the initial vorticity. Large θ gradients are preserved, however, because they are mixed to the edges of the turbulent layers. There are even fewer quasi-coherent sheets having high 〈ɛ〉 and 〈Riχ〉 at t = 11.5Tb, the sheets appear to be broader, and 〈ɛ〉 and 〈Riχ〉 are less well correlated because shear instabilities have become quite pervasive by this time. The strongest cases of coherence between the 〈ɛ〉 and 〈Riχ〉 fields accompany the strong and deep instability and turbulence events throughout the evolution, whether they are due to shear instability, initial GW breaking, or the more general intrusions occurring at later times. However, approximate spatial coherence does not imply that the dissipation fields exhibit similar responses.
Turbulence enhances both dissipation fields, but to very different degrees and at different times, depending on the event character, stage of evolution, and degree of mixing. The range of values in each case spans about 4 decades. Because strong turbulence drives θ gradients to the edges of a mixing region, however, those regions having the most active recent or current mixing exhibit large 〈ɛ〉 but often much smaller relative 〈Riχ〉. Clear examples can be seen at both times in Fig. 6. At t = 8Tb, the most turbulent portions of the flow seen to have the highest 〈ɛ〉 (top left panel) have lower relative 〈Riχ〉 (second row, left panel) by about 1–2 decades. These regions include the plunging turbulent fluid right of center at the top of the domain and the more diffuse ɛ enhancements throughout the streamwise extent in approximately the lower one-third of the domain.
At t = 11.5Tb (Fig. 6, two bottom left panels), large 〈ɛ〉 enhancements again accompany the plunging turbulent fluid to the right of center, the strong, but more diffuse, turbulence along the intrusion in its wake, and the KHI at the upper right. As seen at t = 8Tb, large reductions are seen in 〈Riχ〉 compared to 〈ɛ〉 accompanying mixing due to an intrusion. In each case, obviously turbulent regions in 〈ɛ〉 are much less conspicuous in 〈Riχ〉. Similar effects were noted previously accompanying turbulence and mixing due to KHI by Fritts et al. (2003, 2011, 2012). These apparently robust results imply possible challenges for measurements of motions or turbulence intensities that rely on refractive-index variations for radar backscatter having sufficient signal to noise. These implications are discussed further in section 4.
We now examine the differences in perspective on case L0 dynamics provided by the spanwise-averaged and local dissipation fields (left and right columns in Fig. 6). The most obvious benefit of the local dissipation fields is a far superior definition of individual instability and turbulence structures. These exhibit much finer sheets having large ɛ and Riχ and filamentary rather than more diffuse enhancements in 〈ɛ〉 and 〈Riχ〉, and these features remain highly correlated at the finer scales. Sheets having large ɛ and Riχ are especially sharp prior to becoming fully turbulent and at the edges of the strongest turbulence in Riχ at both times. Those with large Riχ also appear more confined spatially than those with large ɛ (also see discussion of PDFs below). The regions of strongest turbulence at t = 8Tb exhibit large ɛ throughout. Large Riχ, however, are confined to the edges of these regions to a greater degree, because strong turbulence rapidly mixes θ gradients to the edge regions. The same effect is seen at t = 11.5Tb, but to a lesser degree because it is an earlier stage in the turbulence evolution of this event and the θ gradients in their interiors have not yet been eradicated. Implications of these results for various measurements will be discussed in section 4.
The large-scale KH billows seen at the upper right at t = 11.5Tb are defined somewhat more clearly in the left than in the right panels in Fig. 6, owing to significant 3D character and spanwise modulation of the shape, strength, and location of these dynamics. Large ɛ and Riχ in these KH billows exhibit strong correlations, because the billows are strongly turbulent but not fully mixed.
Large ɛ and Riχ in regions without obvious strong turbulence also exhibit correlations at both times, particularly in the remaining strong vorticity and θ sheets. But these sheets are neither as highly correlated as they are at earlier times, nor as they are in the mean fields. Sheets of large 〈ɛ〉 and 〈Riχ〉 exhibiting high coherence at t = 8Tb are thin, because they are among the few sheets that have not yet succumbed to shear instability. This is because of the tendency noted in Part I for only those vorticity sheets that have their vorticity enhanced by the GW to undergo initial shear instability. Sheets of large ɛ and Riχ at t = 8Tb remain reasonably correlated at larger spatial scales but are less strongly correlated at smaller scales. Inspection of the ɛ and Riχ fields at intermediate altitudes reveal sheets of large ɛ or Riχ that do not have a counterpart in the other field and sheets that do correlate at larger scales, but that have small FS variations within the sheets due to local instability and turbulence.
At t = 11.5Tb most sheets exhibit clear evidence of instability and turbulence and are much more apparent in Riχ than in ɛ in both the mean and local fields. This is because all sheets having either positive or negative spanwise vorticity have been subjected to GW vorticity enhancements enabling instability owing to GW phase progression spanning more than a GW period. Not all sheets have undergone instability, however, because local vorticity enhancements depend on GW amplitudes and superposition.
3) ɛ and χ PDF variations
The PDFs of ɛ and Riχ throughout the full computational domain reveal significant differences between the two and significant changes in each with time. These differences are even greater if we consider localized regions of the overall flow exhibiting differing instability dynamics and/or different stages of these individual event evolutions. To examine these differences, we compute the same distributions, but for small subsets of the computational domain, with each representing only 1% of the full domain (each 10% of the depth and 10% of the streamwise extent). These PDFs for ɛ and Riχ in case L0 are shown in the first and third columns of Figs. 7 and 8 at t = 8Tb and 11.5Tb, respectively, corresponding to the images displayed in Fig. 6. The panels show increasing heights (bottom to top) and successive streamwise locations (left to right) with the different line colors (blue to red).
As in Fig. 1 (bottom) for (top to bottom) PDFs in one-tenth domain intervals in the vertical at t = 8Tb. (first column),(second column) Log10(ɛ) PDFs for cases L0 and R and (third column),(fourth column) log10(Riχ) PDFs for cases L0 and R. Line colors in each panel show PDFs for (blue to red) left to right one-tenth domain-width intervals in each altitude interval.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-059.1
As in Fig. 7, but for t = 11.5Tb.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-059.1
Considering first the ɛ PDFs at t = 8Tb (Fig. 7, left), we note more highly structured PDFs with significant skewness and kurtosis than seen at t = 11.5Tb (Fig. 8, left). This is likely due to the evolution of initial instabilities at the earlier time. Maximum values of ɛ accompany the intrusion at the left and center near the top of the domain at t = 8Tb (blue, green, and orange PDFs, top two panels at left). We also see large variability in the character of individual PDFs, with a number having approximately lognormal character and significant continuity in the streamwise direction, while others exhibit several distinct maxima and high skewness and/or kurtosis. PDFs exhibiting multiple maxima typically include peaks at small ɛ owing to regions that have not yet experienced significant instability (panels 4, 7, 8, and 9, bottom to top, at the domain edges). PDFs exhibiting high skewness and/or kurtosis are seen in regions of incipient instability prior to strong turbulence where large values of ɛ are less pronounced (panels 4–8 from bottom at streamwise locations right of center, and in panels 6–9 at streamwise locations left of center) and where large values of ɛ are predominant owing to active, strong turbulence generation (top panel, left of center).
Turning now to the ɛ PDFs at t = 11.5Tb (Fig. 8, left), we note much greater variability in the streamwise direction and a significantly larger fraction of large values (by about 10 times) than at t = 8Tb, owing to simultaneous, strong turbulence in different regions. As at t = 8Tb, regions having quasi-lognormal character are seen at several streamwise locations at most vertical positions. These include the regions of most intense turbulence seen in Fig. 6 accompanying the intrusion extending from the left center through the streamwise domain boundary to the center right and the KHI at the upper right. Regions exhibiting multiple maxima again include quite small ɛ values, indicative of regions that have either eluded significant instability and turbulence generation or experienced strong turbulence decay following earlier instability events. Of these possibilities, strong turbulence decay appears less likely than an absence of turbulence generation given the long turbulence decay times accompanying GW breaking (F09b).
PDFs of Riχ differ markedly from those of ɛ throughout the case L0 evolution. At the large majority of locations at both t = 8Tb and 11.5Tb (cf. first and third columns in Figs. 7 and 8), Riχ distributions have much smaller full widths at half maxima, but often much stronger tails (hence large kurtosis), than the corresponding ɛ PDFs. The Riχ PDFs also differ from those for ɛ in having much smaller variations in the median values, which are nearly constant in altitude and very similar at the two times displayed, unlike the median ɛ values, which vary by about 2 decades. Two types of features of the Riχ PDFs appear to contribute to these differences from the corresponding ɛ PDFs. The first is characterized by very sharp decreases in Riχ somewhat above the median values at both times in regions at which median ɛ values are very small. These mirror the same tendency seen in Fig. 2 but are often much sharper in the much-smaller sampling regions shown in Figs. 7 and 8. The second includes tendencies for relatively small numbers of large values of Riχ that make only small contributions to the PDFs within each subset of the domain, but contribute to high skewness and kurtosis.
c. Case R: GW and rotary FS velocities
1) ɛ-field evolution
The evolution of the 〈ɛ〉 field for case R is shown from t = 5Tb to 13Tb in the same format as for case L0 in the right columns of Figs. 3 and 4. Relative to case L0, larger 〈ɛ〉 values accompanying initial instabilities are slower to arise, owing to delayed instability onset, but become comparable to those occurring in case L0 where instability has occurred by t ~ 7Tb. The morphology is very different between the two cases, however, because case R FS enables stronger vorticity sheets to develop prior to strong 3D instability. Once instabilities arise, case R 〈ɛ〉 attains values comparable to those seen in case L0 (t = 7Tb–9Tb in Fig. 3) in regions where turbulence is well developed. In regions where initial instability has been suppressed by case R spanwise FS shears, however, 〈ɛ〉 is significantly higher in case R owing to the persistence of strong initial vorticity sheets. Thereafter, 〈ɛ〉 values are comparable in the two cases, but case R is distinguished from case L0 in having somewhat more highly structured instability and 〈ɛ〉 fields accompanying the turbulence transitions. Larger-scale turbulence structures are nevertheless remarkably similar in the two cases. Very similar features are also seen to be common to the two cases extending to later times (Fig. 5), at which 〈ɛ〉 values are comparable accompanying the intrusion seen in each case.
2) ɛ and χ correlations
Streamwise cross sections of 〈ɛ〉 and 〈Riχ〉 and ɛ and Riχ distributions for case R at t = 8Tb and 11.5Tb are displayed in Fig. 9 in the same format as for case L0 in Fig. 6. Comparing first the 〈ɛ〉 and 〈Riχ〉 fields at t = 8Tb (two top left panels), we see stronger correlations between 〈ɛ〉 and 〈Riχ〉 maxima on more nearly laminar vorticity and θ sheets than seen in case L0 because of the delayed occurrence of instability in case R. Where strong instabilities have occurred accompanying the initial overturning feature in the lower and upper portions of the domain, correlations depend on location and stage of instability and turbulence generation. Regions exhibiting no instability, or only initial instabilities at small scales that have not yet generated strong turbulence and mixing, display strong correlations—that is, on thin vorticity and θ sheets where dissipation is strong (see the region at x′ ~ 0.8–1.8 and z′ ~ 0–0.2). Regions in which instabilities have yielded strong turbulence are well correlated where turbulence has not yet caused strong mixing (vorticity and θ sheets at the top of the unstable region from x′ ~ 0.4 to 0.9 at z′ ~ 0.1–0.2 and from x′ ~ 1.0 to 1.7 at z′ ~ 0.4). In contrast, regions where strong mixing has occurred exhibit large 〈ɛ〉 but much weaker 〈Riχ〉 (by a decade or two) and weak correlations. Examples include the regions of large 〈ɛ〉 at x′ ~ 0.1 and z′ ~ 0.1 and at x′ ~ 1.1–1.9 and z′ ~ 0.2–0.3.
As in Fig. 6, but for case R.
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-13-059.1
Dissipation fields shown at t = 11.5Tb in Fig. 9 (bottom left) reveal very similar correlations (or lack thereof) to those seen at t = 8Tb. The sole-remaining strong vorticity and θ sheets exhibiting KHI at the upper right are strongly correlated (with and without KHI), as are the regions of strong, developing turbulence at the leading edge of the plunging fluid plume near the center of the domain. Other regions that have already experienced turbulence and mixing are less well correlated because large 〈ɛ〉 causes mixing and destruction of thermal gradients, as discussed above. Other more diffused vorticity and θ sheets remain approximately correlated spatially because mixing accompanying their prior instabilities is highly confined in the vertical.
Local fields of ɛ and Riχ at t = 8Tb and 11.5Tb (Fig. 9, right) exhibit the same tendencies noted for the spanwise-averaged fields but also reveal the FS within these fields far better than the spanwise-averaged fields (Fig. 9, left). As for case L0, local FS sheets having enhanced ɛ and Riχ reveal the turbulence character, intermittency, and dominant spatial scales clearly. Highly variable ɛ and Riχ intensities are seen in regions where mean values are large and small. Also seen are features indicative of local instability dynamics that are not apparent in the averaged fields. See, as examples, the filamentary vorticity and θ sheets that appear relatively laminar in the averaged fields at lower altitudes but exhibit clear FS indicating instabilities and strong spanwise variations in the local fields. Vorticity sheets are particularly fine in regions that have not yet become fully turbulent.
Comparing the dissipation fields for case R with those for case L0, we see several clear differences that must have their roots in the different instability dynamics in the two cases. In our discussion of case L0, we noted that initial instabilities arising at earlier times on the more intense vorticity sheets seen at middle levels at t = 8Tb and at lower levels at t = 11.5Tb yield strong local mixing that strongly reduces the vorticity maxima but retains strong θ sheets. This leads to the lack of correlations between ɛ and Riχ seen at these sites in Fig. 6. Referring to Fig. 9, however, we see that correlations between ɛ and Riχ at larger scales accompanying these vorticity and θ sheets are preserved to a higher degree at both times. We attribute these differences to the initial instabilities and mixing at these sites in case L0 and to the suppression of these initial instabilities and the occurrence of subsequent instabilities on vorticity sheets at smaller vertical scales in case R. There are also large differences between the two cases in both the ɛ and Riχ and the 〈ɛ〉 and 〈Riχ〉 fields at t = 8Tb that are attributed above to the very different initial instabilities accompanying overturning and the times at which these occur. At t = 11.5Tb, in contrast, we see strong similarities between the larger-scale turbulence events in cases L0 and R, though there remain larger-scale differences in the details because of their very different instability evolutions. Similarities at t = 11.5Tb include nearly equal TKE and TPE (Fig. 1); very similar large-scale GW structures accounting for the occurrence of similar instabilities at nearly identical locations; comparable regions in each case where ɛ is large, local mixing is strong, and Riχ is small as a result; and very similar θ sheets in each case.
3) ɛ and Riχ PDF variations
PDFs of ɛ and Riχ for case R in subdomains constituting 10% of the depth and 10% of the streamwise extent are shown in the second and fourth columns of Figs. 7 and 8 at t = 8Tb and 11.5Tb, respectively, for comparison with those for case L0 discussed in section 3b.
Referring to subdomain PDFs of ɛ and Riχ for case R at t = 8Tb (Fig. 7, second and fourth columns), we see for both ɛ and Riχ significant structure at a majority of locations. Comparing these PDFs with the ɛ and Riχ fields shown in Fig. 9, we see that the few PDFs exhibiting smooth distributions at this time correspond to subdomains in which there is strong turbulence and large ɛ. Those subdomains exhibiting PDFs having multiple, sharp maxima occur where there are strong vorticity and θ sheets but little or no instability and turbulence. PDFs having ɛ and Riχ maxima at widely spaced magnitudes occur in subdomains spanning several vorticity or θ sheets (see the top panels of Fig. 9 and the PDFs between z′ = 0.4 and 0.8). Other regions exhibit multiple, closely spaced maxima that likely occur because of variability along individual vorticity and θ sheets. Several subdomains at this time also yield PDFs having a narrow peak, high skewness, sharp reductions above the peak, and a long tail having small amplitudes at higher magnitudes. This PDF character is suggestive of sheets that have not yet become unstable (as seen for GW breaking by F09b) or regions of active mixing and reductions of strong gradients in Riχ. The former is clearly the case here, however. The large differences in the PDFs for cases L0 and R are due to their very different stages in the transition to turbulence due to their different initial instability dynamics.
PDFs of ɛ and Riχ for case R at t = 11.5Tb are displayed in the second and fourth columns of Fig. 8. These differ dramatically from those at t = 8Tb because instabilities and turbulence have occurred on the large majority of vorticity sheets by t = 11.5Tb. However, they bear a close resemblance to those in case L0 at this time. This is likely because the differing instability dynamics in the two cases have led to similar turbulence generation owing to control of these events by the similar larger-scale, quasi-2D motions in the two cases noted above. As seen in case L0, PDFs of Riχ are typically much narrower (having high kurtosis) than those of ɛ at depths of z′ ~ 0.4–0.8 where ɛ is large, turbulence is strong, and there has been significant mixing. In these cases, the maxima in the PDFs of Riχ have very nearly the same magnitude, independent of the dynamics leading to turbulence. Also as in case L0, many of the ɛ PDFs, and some of the Riχ PDFs, exhibit approximately lognormal character or evidence of more than one nearly lognormal PDF having different median values and widths. The former suggest dominance of the subdomain statistics by a single large turbulence event; the latter suggest contributions to the subdomain statistics from two or more events having different median values and widths.
Also seen in case R, as in case L0, are PDFs of Riχ having sharp decreases above sharp maxima (as seen for case R at t = 8Tb) with corresponding PDFs of ɛ that are more nearly lognormal. These differ from the PDFs characterizing strong turbulence that has not yet mixed fully. They also differ from the PDFs accompanying monochromatic GW breaking in which both ɛ and Riχ PDFs remain largely lognormal following instability and turbulence generation (see F09b and section 3a). Indeed, these Riχ PDFs closely resemble those prior to strong instabilities and turbulence in the GW breaking DNS of F09b. However, this cannot be the explanation because those displayed in Fig. 8 are following strong turbulence. Instead, this PDF character in both cases L0 and R appears to arise as a result of strong mixing that destroys the strong initial gradients within active turbulence and reestablishes weaker gradients at the edges of the mixed regions. These dynamics may account for the relatively uniform values of Riχ at the PDF peaks following strong mixing in both cases L0 and R.
4. Discussion
Our idealized DNSs of GW–FS interactions have yielded a number of insights into the evolution and behavior of turbulent dissipation in such flows in the atmosphere and oceans. Here we discuss these results in relation to previous modeling studies and measurements.
Turbulence events in the atmosphere are seen to have different character, depending on the dynamics contributing to their generation (Mahrt 1998, 1999; Sun et al. 2002, 2004). Many in situ measurements in the SBL, the troposphere, and the stratosphere have shown turbulence events to exhibit strong layering of various quantities. These include θ and humidity (Dalaudier et al. 1994; Muschinski and Wode 1998; Chimonas 1999; Balsley et al. 2003); 
Despite the simplicity of our GW–FS DNS initial conditions, the resulting turbulence events capture many of the characteristics of similar observed events noted above. Preferred turbulence generation mechanisms include KHI, GW breaking, and intrusions that closely resemble the structure of density currents. In our DNS, however, there is not a dominant low-frequency GW that has the potential to yield sustained KHI extending over many hours. Close similarities between observations and our GW–FS DNS results include the following:
turbulent dissipation events that are strongly layered in the vertical and lead to sheet-and-layer θ structures, owing to stable stratification and velocity FS;
turbulent dissipation events that occur where the larger-scale GW fields impose favorable conditions, such as strong shear layers (vorticity sheets) and intrusions;
multilognormal PDFs of ɛ suggestive of several contributing turbulence events; and
ɛ and Riχ distributions that indicate eradication of large θ gradients and small Riχ where ɛ is large, suggesting relative insensitivity of measurements in such regions with radar or in situ instrumentation where
Indeed, our GW–FS DNS exhibit many similarities to multiscale dynamics throughout the atmosphere, where these dynamics can be quantified to some degree. Specific examples include KHI observed by various radars from the SBL into the MLT (Fritts and Rastogi 1985; Eaton et al. 1995; Lehmacher et al. 2007) and by airglow and noctilucent cloud-imaging and resonance lidars near the mesopause (Witt 1962; Hecht 2004; Hecht et al. 2005; Pfrommer et al. 2009). The PDFs of ɛ in Figs. 7 and 8 indicate typical nondimensional values of a local (spanwise)-mean 〈ɛ〉 ~ 10−4–3 × 10−2, depending on the local dynamics. These imply dimensional spanwise-mean values (denoted with a subscript D) in the SBL of 〈ɛD〉 ~ 3 × 10−7–10−4 m2 s−3 (assuming a physical domain depth of about 300 m, FS structure having an approximately 60-m vertical scale, and Tb ~ 300 s), which agree extremely well with in situ measurements over approximately 100-m flight paths for SBL dynamics at these scales (B. B. Balsley 2013, personal communication).
Assuming typical GW vertical wavelengths of about 5 and 15 km at 75- and 90-km altitudes (with Tb ~ 300 s), respectively, the corresponding values of 〈ɛD〉 are about 3 × 10−5–10−2 m2 s−3 near 75 km and about 10−3–3 × 10−1 m2 s−3 near 90 km, which also agree very well with in situ measurements spanning about 1 km in the MLT (Lübken et al. 2002; Rapp et al. 2004). Estimations of 〈ɛD〉 employing radars, on the other hand, are more likely best compared with DNS values averaged over suitable volumes that would potentially include entire, or multiple, instability and turbulence events and hence more closely approximate our domain-averaged results. We caution, however, that such large-volume estimates can also be biased toward larger values by contributions to variances that might be attributed to turbulence, but which may also be due to small-scale and/or high-frequency GWs.
Given the above comparisons to measurements in the SBL and MLT, we believe that our DNS can provide useful, quantitative, and robust insights into shallow and deep GW–FS instability dynamics (the latter where the Boussinesq approximation is not valid), for which the instability and turbulence scales are themselves of limited vertical extent. The reason is that shallow instability dynamics and intensities must depend more strongly on the local GW and FS environment than on its evolution on longer time scales. Turbulence scales and intensities must nevertheless depend on energy inputs at larger scales. In the MLT where GW scales, amplitudes, and vertical group velocities may be quite large, we expect instability and turbulence dynamics to be dictated by the dominant contributors to energy fluxes in each case.
Despite our idealized initial conditions, Boussinesq code, and compact DNS domain imposing highly discretized large-scale motion fields, our GW–FS DNS have allowed a degree of quantification of various instability and turbulence dynamics in 3D and time that is far beyond current measurement capabilities in the atmosphere and oceans. They have also yielded estimates of 〈ɛD〉 that agree very well with available measurements in the SBL and MLT. Other turbulence parameters can also be easily assessed, such as the Kolmogorov and Ozmidov scales, the buoyancy Reynolds number, structure functions, influences of anisotropy on turbulent dissipation, and correlations among these fields, as needs warrant.
5. Summary and conclusions
We examine the dissipation fields arising in two DNSs of GW–FS interactions in this paper. The two DNSs employ a common GW and either a planar or rotary oscillatory FS (denoted cases L0 and R) having a vertical wavelength about 5 times smaller than the imposed GW. They were chosen for analysis because of the very different initial instabilities and turbulence transitions caused by the absence or presence of FS in a plane orthogonal to the streamwise plane of GW propagation. Strong 2D wave–wave interactions precede local instabilities and turbulence and are nearly identical in the two cases because these dynamics are determined by the mutual deformations of the GW and FS flows confined largely to the streamwise plane. The 2D GW fields largely dictate the occurrence and character of instabilities thereafter.
Significant findings include the following:
several types of instabilities accompanying multiscale dynamics lead to episodic bursts of strong turbulent energy dissipation that last several Tb;
layered mixing leads to “sheet and layer” features in the θ fields that persist to late times;
characteristics of the dissipation fields depend strongly on local instability dynamics, but the larger-scale turbulence events are dictated by quasi-2D GW dynamics;
PDFs of ɛ and Riχ exhibit highly nonlognormal distributions prior to turbulence generation that depend on the character of the vorticity and θ fields;
following turbulence generation, local PDFs of ɛ and Riχ often exhibit several lognormal distributions, suggesting more than a single turbulence source;
strong mixing can yield regions having large 〈ɛ〉 and smaller 〈Riχ〉 that may cause measurements of turbulence quantities in these regions to be very challenging;
strong mixing can also result in much less variable 〈Riχ〉 than 〈ɛ〉, strong log10Riχ PDF maxima with sharp decreases at larger log10Riχ, and strong positive skewness of log10Riχ PDFs at large 〈ɛ〉; and
measurements of θ, u = (u′, υ′, w′), ɛ, and Riχ in the plane of larger-scale motions can provide a highly detailed view of the character and evolution of instability and turbulence dynamics, transport, and mixing.
The apparent correspondence of small-scale dynamics, instabilities, and dissipation fields and statistics seen in our GW–FS DNSs with similar features in the atmosphere suggests that such dynamics are relatively robust in general multiscale flows. Clearly, local instability and turbulence dynamics will depend on the character of the contributing fields, and these dependencies remain to be investigated, both observationally and numerically. However, the initial comparisons discussed here suggest that new, less idealized GW–FS modeling studies will prove valuable in interpreting existing and new observations, predicting structures and correlations that remain to be assessed observationally, planning for future measurement programs, and perhaps guiding new or improved parameterizations of these dynamics from the SBL into the MLT.
Acknowledgments
Support for this research was provided by the Army Research Office under Contract W911NF-12-C-0097, NASA under Contract NNH09CF40C, and the National Science Foundation under Grants AGS-1242943, AGS-1242949, and AGS-1250454. We gratefully acknowledge access to large computational resources provided by the DoD High Performance Computing Modernization Program for our various studies.
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