1. Introduction
The atmosphere supports a wide range of dynamics that lead to instabilities and turbulence extending from the stable boundary layer (SBL) into the mesosphere and lower thermosphere (MLT) under conditions of stable mean stratification. In nearly all cases, these dynamics occur in multiscale (MS) flows comprising superpositions of mean, large-scale, and GW motions, various flow instabilities, and turbulence (e.g., Fritts and Alexander 2003). Wavenumber and frequency spectra at GW scales reveal characteristic variations with altitude that indicate 1) slower growth of variances with altitude than would accompany conservative GW propagation, 2) constraints on vertical gradients of horizontal velocities and temperatures consistent with tendencies for instability, and 3) dominance of the variances by low-frequency inertia–GWs at altitudes from ~0 to 100 km (e.g., VanZandt 1982; Fritts and VanZandt 1993).
MS flows, and GW amplitude growth with increasing altitude, have a number of implications for atmospheric structure and the evolution of GW spectra and instability dynamics with increasing altitude. These include the following:
wave–wave interactions that occur at all GW amplitudes but play more dominant roles as amplitudes increase (e.g., Klostermeyer 1991; Vanneste 1995; Sonmor and Klaassen 1997),
local instabilities that constrain GW amplitudes at small and large vertical scales (e.g., Lombard and Riley 1996; Andreassen et al. 1998; Fritts et al. 1998, 2009a,b, 2013, hereafter F13; Achatz 2005, 2007),
approximate universality of the GW wavenumber and frequency spectra with altitude (e.g., Dewan and Good 1986; Smith et al. 1987; Tsuda et al. 1989; Fritts and VanZandt 1987, 1993; Cot and Barat 1990; Eckermann 1999), and
characteristic “sheet and layer” (S&L) structures that are ubiquitous from the SBL to the MLT (e.g., Coulman 1973; Gossard et al. 1985; Dalaudier et al. 1994; Coulman et al. 1995; Luce et al. 1995, 1999; Balsley et al. 1998, 2003, 2013; Muschinski and Wode 1998; Nastrom and Eaton 2001; Chuda et al. 2007; Fritts et al. 2004, 2013).
Instabilities and turbulence at small spatial scales are of considerable interest throughout the atmosphere because of their local effects, their roles in weather and climate, the risks they pose for aircraft, and their implications for atmospheric and astronomical observations (e.g., Reitar 1969; Coulman 1969; McIntyre 1990; Garratt 1994; Mahrt 1999; Stull 2003; and references therein). The most recognized instabilities in stably stratified flows are Kelvin–Helmholtz instability (KHI) and GW breaking [e.g., Fritts and Rastogi (1985) and references therein], though other sources also contribute, especially in the SBL and accompanying MS flows extending into the thermosphere (e.g., Sun et al. 2002, 2004; Fritts et al. 2004, 2013).
Spatial scales of MS flows and their associated instabilities increase with altitude because of the exponential increase of kinematic viscosity ν and the suppression of instabilities at small Reynolds numbers, Re = UL/ν, where U and L are characteristic velocity and length scales (Fritts et al. 2009a, 2013, 2014). Spatial localization of instabilities also emphasizes the MS nature of the underlying flows throughout the atmosphere (e.g., Fritts and Rastogi 1985; Eaton et al. 1995; Yamada et al. 2001; Luce et al. 2002; Hecht et al. 2005; Lehmacher et al. 2007; Pfrommer et al. 2009; Baumgarten and Fritts 2014). The turbulence arising from such events contributes mixing and transport but is localized and inhomogeneous in nature (e.g., Fritts and Wang 2013, hereafter FW13). The efficiency of these processes is expected to depend strongly on event character (Fritts and Dunkerton 1985; Coy and Fritts 1988; McIntyre 1989), hence cannot be described by a mean turbulent or “eddy” diffusion (McIntyre 1990). Background turbulence intensities are often significant, however, and appear to impose a turbulent viscosity νturb ≫ ν that reduces the effective Re for successive instabilities in the MLT, where the transitional instability scales can be directly assessed using imaging of these dynamics (e.g., Lübken et al. 1993, 2002; Lübken 1997; Rapp et al. 2004; Fritts et al. 2014; Hecht et al. 2014).
Because turbulence events play important, but unquantified, roles from ~0 to 100 km, they have been the subject of many studies using a wide range of instrumentation. Tower-mounted instruments sample winds and temperatures at ~5–20 Hz or higher (e.g., Sun et al. 2002, 2004), yielding spatial resolution that depends on SBL wind speed. Research aircraft and UAVs measure turbulence fluctuations along quasi-horizontal paths at altitudes from ~50 m to 15 km and spatial scales above ~0.3–20 m, depending on the platform and airspeed (e.g., Muschinski et al. 2000; Whiteway et al. 2003; Wroblewski et al. 2007; Balsley et al. 2013; Lawrence and Balsley 2013). Recent balloonborne instrumentation has achieved exceptional spatial resolution, approximately a few millimeters, that characterizes velocity fluctuations extending into the viscous range of turbulence at stratospheric altitudes (e.g., Haack et al. 2014; Schneider et al. 2015). At higher altitudes, rocketborne instruments measure turbulence at scales down to a few meters in the MLT, thus also sampling into the viscous range at these altitudes (e.g., Rapp et al. 2004).
Because turbulence plays key roles in many applications, and because high-resolution in situ measurements are costly and not routine or global, there have been efforts to define a surrogate for turbulence intensity in the ocean (e.g., Osborn 1980; Dillon 1982; Wesson and Gregg 1994; Moum 1996; Ferron et al. 1998; Gargett 1999) and more recently in the atmosphere (e.g., Clayson and Kantha 2008; Love and Geller 2012). The method involves relating the Ozmidov scale, LO = (ε/N3)1/2, and the Thorpe scale, LT = 〈d′2〉1/2, where the angle brackets denote a suitable spatial average and ε, N, and d′ are the mechanical energy dissipation rate, the buoyancy frequency, and the Thorpe displacement, respectively (Thorpe 1977). The ratio of these scales, C = LO/LT, is an attractive choice for estimating LO in the atmosphere because LT can be easily assessed with high-resolution radiosonde profiling of θ, which yields d′ at approximately 5-m resolution.
The challenge has been to demonstrate that there is a quasi-universal value of C, which has been suggested by various researchers (e.g., Dillon 1982; Clayson and Kantha 2008), though it is acknowledged that there is considerable scatter in these estimates. As examples, numerical simulations of KHI by Smyth and Moum (2000) suggest an increase of C with time, while a recent study using radiosonde and high-resolution in situ measurements by Schneider et al. (2015) found LO and LT to be largely uncorrelated. Hence, a major goal of this paper is to use a direct numerical simulation (DNS) of instabilities and turbulence in a MS flow to assess the relations between LT, LO, C, ε, N, and other measures of flow structure and turbulence intensity. These include potential temperature, θ; the local Richardson number, Ri = N2/(dU/dz)2 ~ 0.1 (where dU/dz is the local wind shear); the thermal energy dissipation rate χ; and the buoyancy Reynolds number, Reb = ε/νN2 = (LO/LK)4/3, where LK = (ν3/ε)1/4 is the Kolmogorov microscale.
From an atmospheric perspective, we note that a significant fraction of ocean turbulence is expected to accompany KHI, whereas additional sources, especially GW breaking, are known to contribute to instabilities and turbulence in the atmosphere. Importantly, C has not been extensively evaluated in the atmosphere, nor has its dependence on the character of turbulence events, which is essentially unknown at this time.
The above quantities were assessed using high-vertical-resolution wind and temperature measurements collected aboard a DataHawk unmanned aerial vehicle (UAV) at Dugway Proving Ground (DPG) in Utah at altitudes from ~50 to 400 m under SBL conditions during October 2012 (B. B. Balsley et al. 2015, unpublished manuscript, hereafter BB). These measurements reveal a number of correlations that suggest specific dynamics and evolutions, many of which bear close resemblance to structures and correlations seen in our MS DNS to be discussed below. These include S&L structures in the wind and potential temperature (θ) fields, with sheets comprising strong θ and velocity gradients separated by layers having much weaker gradients, various instabilities types, and turbulence patches exhibiting structures and characteristics determined largely by the instability events. Also noted were large variations in estimates of LO, LT, and C due to the large variations in ε. Example profiles from one DataHawk flight will be used to demonstrate the similarities in the measured and DNS profiles as motivations for our discussion of MS DNS results below.
DNS is now able to address atmospheric dynamics at realistic spatial scales and Re spanning a wide range of altitudes. As an example, KHI in the SBL with N ~ 0.03 s−1, an initial minimum Ri ~ 0.1, and Re ~ 5,000, with velocity and length scales, U and L, based on the initial mean shear and ν ~ 1.5 × 10−5 m2 s−1, can be described fully by DNS at a KH wavelength, λKH ~ 4πh ~ 10 m. At ~80 km in the MLT, N ~ 0.014 s−1, Ri = N2/(dU/dz)2 ~ 0.1, ν ~ 1 m2 s−1, and Re ~ 5000 imply λKH ~ 4 km. These capabilities mean that we can now address instability and turbulence dynamics spanning many spatial scales and altitudes at realistic Re (for either ν or νturb). Recent DNSs performed of the same dynamics for Re varying by 10 times confirm that a lower Remin DNS can capture the larger-scale instability and turbulence characteristics occurring at higher Re, provided that Remin exceeds the threshold for strong instability and turbulence (e.g., Dimotakis 2000; Monin and Yaglom 2007; L. Wang and D. C. Fritts 2015, unpublished manuscript).
Our goals in this study include the following: 1) characterize the evolutions of instability and turbulence events for a higher-Re MS flow than previously performed, 2) explore the roles of these events and larger-scale wave–wave interactions in the formation and evolution of S&L structures, and 3) perform an assessment of C = LO/LT for various stages of several of the instabilities yielding significant turbulence.
The paper is organized as follows. The MS DNS is described in section 2. Section 3 provides a comparison of measured and modeled profiles and discusses the character and evolution of the overall flow, the roles of instabilities in creating and maintaining S&L structures, and the detailed evolutions of several instability and turbulence events occurring within the larger-scale MS DNS. Section 4 examines the relationships between ε, χ, LT, LO, Reb, and the variations of C = LO/LT with event character and evolution. Section 5 provides a discussion of these results in relation to previous studies and our conclusions.
2. Model formulation
a. Problem specification
For a MS DNS in a tilted domain, the bulk Ri relating the velocity and stability scalings is 
Our solution algorithm is pseudospectral and employs Fourier series in each direction, the time integration method of Spalart et al. (1991), a variable time step (due to varying velocities and resolution) with a CFL limit of 0.68, and a minimum spatial resolution of Δx/LK < 1.8LK (typically ~1.6) throughout the DNS. For reference, Δx/LK ~ 1.5–2.1 are suggested thresholds for DNSs sufficient to resolve essentially all turbulence dissipation (Moin and Mahesh 1998; Pope 2000; Zikanov 2011; Chung and Matheou 2012). For this assessment, LKmin = (ν3/εmax)1/4 is evaluated every 30–50 time steps to guide required resolution changes, where εmax is the largest ε obtained from the two-dimensional (2D) velocity spectra in the normal plane for each x′, y′, and z′ (or the equivalent average of ε over the same plane). Because turbulence events are far more variable in z′ (εmax varies by over 100 times in z′ at 11.5Tb), LKmin is usually defined on a horizontal plane where a single, strong event makes the dominant contribution. A “two-thirds rule” spectral truncation is also applied at each time step to avoid backscatter to larger spatial scales. Further details on the DNS employed for these applications are provided by F13 and Werne et al. (2005).
b. Initial and environmental conditions
As in F13, we assume an initial monochromatic GW having an amplitude of 
Computational domain aligned along the GW phase at an inclination of ϕ = sin−1(ω/N) from the horizontal. The domain and geophysical coordinates are (x′, y′, z′) and (x, y, z), respectively; the GW velocity and θ′ fields are initially uniform along x′ (blue velocity profile and dashed phase inclination); streamwise FS velocities are uniform along x (red velocity profile and dashed phase inclination).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Computational domain aligned along the GW phase at an inclination of ϕ = sin−1(ω/N) from the horizontal. The domain and geophysical coordinates are (x′, y′, z′) and (x, y, z), respectively; the GW velocity and θ′ fields are initially uniform along x′ (blue velocity profile and dashed phase inclination); streamwise FS velocities are uniform along x (red velocity profile and dashed phase inclination).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Computational domain aligned along the GW phase at an inclination of ϕ = sin−1(ω/N) from the horizontal. The domain and geophysical coordinates are (x′, y′, z′) and (x, y, z), respectively; the GW velocity and θ′ fields are initially uniform along x′ (blue velocity profile and dashed phase inclination); streamwise FS velocities are uniform along x (red velocity profile and dashed phase inclination).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
3. MS flow evolution and structure
To demonstrate the relevance of MS DNS to atmospheric flows, we first compare profiles of several quantities measured by the DataHawk at DPG and predicted by the MS DNS. The MS DNS evolution, structure, and instability and turbulence events are described in the remainder of section 3. Implications of these dynamics for LT, LO, and C are examined in section 4.
a. Comparison of DataHawk and MS DNS vertical profiling
Figure 2 shows vertical profiles of θ, N from “Thorpe reordered” θ, d′, logε, LT, LO, and C−1 at 2-m vertical resolution computed from the DPG DataHawk measurements of temperature and horizontal velocity and from the MS DNS. The DataHawk profiles were obtained from a descending spiral flight from ~50 to 400 m having a flight-path diameter of 900 m, a horizontal velocity of 14 m s−1, and a vertical velocity of −0.3 m s−1. Both sensors yielded sensitivity to spatial scales of ~0.3 m along the flight track. These measurements were averaged for 6 s, yielding a vertical resolution of ~2 m (see BB for further details). Note that the spiraling DataHawk flight path sampled vertically and horizontally, hence the “vertical” profiles presented also include fluctuations accompanying horizontal structure at scales greater than ~80 m.
(top) Vertical profiles of θ, N from reordered θ, d′, logε, LT, LO, and C−1 from ~50 to 400 m obtained from a descending DataHawk at DPG on 11 Oct 2012. DataHawk measurements were averaged for 6 s, yielding a vertical resolution of ~2 m (see text for details). (bottom) Vertical (colored) and slant-path (black) profiles from the MS DNS at 12Tb for a domain depth of 500 m and mean N = 0.021 s−1. The slant path had the same slope as the DataHawk profiles and vertical profiles were at ~250-m spacing horizontally. Note that the measurements and model results are shown with different axes in several cases (see text for details).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
(top) Vertical profiles of θ, N from reordered θ, d′, logε, LT, LO, and C−1 from ~50 to 400 m obtained from a descending DataHawk at DPG on 11 Oct 2012. DataHawk measurements were averaged for 6 s, yielding a vertical resolution of ~2 m (see text for details). (bottom) Vertical (colored) and slant-path (black) profiles from the MS DNS at 12Tb for a domain depth of 500 m and mean N = 0.021 s−1. The slant path had the same slope as the DataHawk profiles and vertical profiles were at ~250-m spacing horizontally. Note that the measurements and model results are shown with different axes in several cases (see text for details).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
(top) Vertical profiles of θ, N from reordered θ, d′, logε, LT, LO, and C−1 from ~50 to 400 m obtained from a descending DataHawk at DPG on 11 Oct 2012. DataHawk measurements were averaged for 6 s, yielding a vertical resolution of ~2 m (see text for details). (bottom) Vertical (colored) and slant-path (black) profiles from the MS DNS at 12Tb for a domain depth of 500 m and mean N = 0.021 s−1. The slant path had the same slope as the DataHawk profiles and vertical profiles were at ~250-m spacing horizontally. Note that the measurements and model results are shown with different axes in several cases (see text for details).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Profiles from the MS DNS at 12Tb during significant instabilities and turbulence were scaled to correspond approximately to the observed S&L scales and fluctuation amplitudes measured by the DataHawk. This implies a DNS domain of ~500 m vertically and ~1 km horizontally. DataHawk measurements yielded a mean θ ~ 291 K and Tb ~ 300 s such that the DNS Re = 100 000 is ~500 times smaller than for the DataHawk measurements assuming no background turbulence. However, a background νturb ≫ ν would significantly reduce the disparity in effective Re. This appears likely, given the minima at this time that are ~10 and 100 times above the minimum values observed on successive flights (see BB) and in the MS DNS.
Profiles from the MS DNS include a slant-path profile along the streamwise direction having the same slope as the DataHawk flight, but displayed as a vertical profile, and four true vertical profiles at 250-m spacing to illustrate the horizontal variability and the influences of slant-path rather than vertical sampling. Common features include the S&L structures at vertical scales of ~20–100 m, comparable fluctuations in θ, N, d′, and comparable peak magnitudes of logε between the measured and modeled quantities. Of these, d′ and logε appear reasonably well correlated, but their correlations with θ and N are expected to depend on the form of the instabilities accounting for these fluctuations. Closer examination of θ′ and velocity correlations in slant-path measurements and modeling by BB suggest local KHI and GW breaking that are consistent with S&L vertical scales. Thus, we conclude that the MS DNS is a reasonable approximation to the MS dynamics observed at DPG. MS DNS results also confirm that horizontal gradients along slanted paths can suggest significant artificial vertical gradients.
b. Flow evolution in the domain-averaged 2D and 3D GW and turbulence fields
We decompose the MS flow to assess the evolutions of the initial GW and mean shear, additional GW modes arising from 2D (l′ = 0) wave–wave interactions, 3D dynamics (k′, l′, m′ ≠ 0) arising as a result of flow instabilities and their resulting turbulence, and the decay of total energy with time. The top panels of Fig. 3 show time series to 40Tb (4TGW) of nondimensional
primary GW amplitude and its opposite at wavenumber (k′, l′, m′) = ±(0, 0, −1);
kinetic and potential energies of other 2D GWs arising as a result of wave–wave interactions;
total 2D energy, including contributions by the primary GW, the mean shear, ±(1, 0, −5), and secondary 2D GWs, ±(k′, 0, m′); and
3D potential energy and component kinetic energies, respectively.
Time series of (top left)–(top right) primary 2D GW u′ and θ′ amplitudes; 2D (l = 0) secondary GW KE and PE; primary 2D GW, secondary 2D GW, 2D FS, and total 2D energy; and 3D (l ≠ 0) PE and component KE. (middle) Amplitudes of FS and the lowest-order secondary GWs arising from wave–wave interactions. (bottom) Domain-averaged 2D (l = 0) and 3D (l ≠ 0) KE and PE spectra in the (left) streamwise and (left middle) vertical domain coordinates for Re = 50 000 and 100 000 (dashed and solid, respectively), and (right middle),(right) evolutions of domain-mean ε, Riχ, and Reb for the two Re with time. Line colors in (bottom right) are black, blue, and green for Re = 100 000 with ε thresholds of 0.003, 0.002, and 0.001εmax, respectively, and red for Re = 50 000 with an ε threshold of 0.001εmax. All quantities are nondimensional for convenience.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Time series of (top left)–(top right) primary 2D GW u′ and θ′ amplitudes; 2D (l = 0) secondary GW KE and PE; primary 2D GW, secondary 2D GW, 2D FS, and total 2D energy; and 3D (l ≠ 0) PE and component KE. (middle) Amplitudes of FS and the lowest-order secondary GWs arising from wave–wave interactions. (bottom) Domain-averaged 2D (l = 0) and 3D (l ≠ 0) KE and PE spectra in the (left) streamwise and (left middle) vertical domain coordinates for Re = 50 000 and 100 000 (dashed and solid, respectively), and (right middle),(right) evolutions of domain-mean ε, Riχ, and Reb for the two Re with time. Line colors in (bottom right) are black, blue, and green for Re = 100 000 with ε thresholds of 0.003, 0.002, and 0.001εmax, respectively, and red for Re = 50 000 with an ε threshold of 0.001εmax. All quantities are nondimensional for convenience.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Time series of (top left)–(top right) primary 2D GW u′ and θ′ amplitudes; 2D (l = 0) secondary GW KE and PE; primary 2D GW, secondary 2D GW, 2D FS, and total 2D energy; and 3D (l ≠ 0) PE and component KE. (middle) Amplitudes of FS and the lowest-order secondary GWs arising from wave–wave interactions. (bottom) Domain-averaged 2D (l = 0) and 3D (l ≠ 0) KE and PE spectra in the (left) streamwise and (left middle) vertical domain coordinates for Re = 50 000 and 100 000 (dashed and solid, respectively), and (right middle),(right) evolutions of domain-mean ε, Riχ, and Reb for the two Re with time. Line colors in (bottom right) are black, blue, and green for Re = 100 000 with ε thresholds of 0.003, 0.002, and 0.001εmax, respectively, and red for Re = 50 000 with an ε threshold of 0.001εmax. All quantities are nondimensional for convenience.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
The energetics of the initial 2D dynamics are summarized in the top-left three panels in Fig. 3. The initial GW undergoes a rapid amplitude decrease of ~30% (~50% in energy) and an emerging oscillation between kinetic and potential energies within ~5Tb. A further reduction to ~35% of initial GW energy occurs by ~7–8Tb before recovering to ~40%–50% of the initial energy (also driven by wave–wave interactions), where it remains thereafter. The mean FS energy drops by >70% within the first ~1Tb. Over this same interval, secondary GWs rise to their maximum 2D energies at ~6–7Tb. The top-right panel in Fig. 3 reveals that there is no increase in 3D energy associated with instabilities or turbulence until ~5Tb. Hence, the initial reduction in the primary GW amplitude is driven entirely by 2D wave–wave interactions involving the initial GW, the mean FS flow, and their interaction products at early stages. Total energy is nearly conserved over this interval, decreasing by ~3% accompanying thermal and viscous dissipation acting on FS shears that rapidly evolve strong gradients due to 2D GW superpositions.
Three-dimensional energy increases rapidly beginning at ~5Tb as the 2D GW field becomes increasingly complex and strongly sheared. As seen below, local instabilities arise frequently thereafter, primarily on the strong spanwise vorticity sheets ζy with instability structures, orientations, and locations that depend on the local shear, stability, and instability type. The larger-scale instabilities lead to turbulence, and these dynamics reverse the growth of 2D GW energy and strongly erode the total 2D GW energy from ~7 to 14Tb. However, their contributions to total energy are small because of the rapid dissipation by turbulence at the smallest scales.
Two- and three-dimensional energy spectra for Re = 100 000 at 11.5Tb (the time of maximum ε and Reb; see below) are shown in the two bottom-left panels in Fig. 3. Black and red spectra (solid lines) in each case are for motions having l′ ≠ 0 and l′ = 0, respectively, and velocities are evaluated in the computational domain. Dashed lines show the corresponding spectra from F13 for Re = 50 000 and reveal that the 2D results are very similar at larger scales. These exhibit ~1–2-decades-higher 2D (l′ = 0) than 3D (l′ ≠ 0) spectral amplitudes at k′ ≤ 5 and m′ ≤ 10, indicating dominance by 2D GW motions at larger streamwise and vertical scales even when turbulence is strongest. The 2D energy spectrum in each case falls below the 3D spectrum at k′ ~ 10 and m′ ~ 15, above which both spectra are dominated by small-scale turbulence. Three-dimensional spectra for the two cases also exhibit clear similarities, including an apparent inertial range extending a decade or more above the crossover wavenumbers. Major differences are the somewhat extended inertial range, higher spectral amplitudes, and higher ε by ~25% for Re = 100 000 during strong instabilities (see below). Note, however, that these spectra are averaged over a highly inhomogeneous flow.
Time series of nondimensional mean ε and Riχ, and Reb, for the two Re are shown in the bottom-right panels in Fig. 3. Mean ε and Riχ closely follow the 3D energy to ~14Tb but decay more quickly thereafter, with only episodic peaks due to later instability dynamics not occurring at Re = 50 000. Mean Riχ for both Re are ~60% of mean ε, which is nearer unity than the ratios of 3D energies. The ε thresholds of 0.0003, 0.001, and 0.003εmax (where εmax is the maximum value of ε throughout the domain for each time and Re), meant to exclude nonturbulent values, yield Reb maxima for Re = 100 000 of ~9–13, 13–22, and 27–30, respectively, during the most active interval for instabilities and turbulence from ~7–14Tb. For reference, the 0.001εmax threshold for Re = 100 000 excludes less than 10% and 30% of values during the strongest turbulence from ~8–9 and ~11–13Tb, respectively. This suggests that this DNS meets the threshold of Reb ~ 20 for sustained strong turbulence for the most active intervals. The corresponding Reb for Re = 50 000 and an ε threshold of 0.001εmax is shown in red and is ~2.5 times smaller than for Re = 100 000. This suggests a minimum Re ~ 100 000 is needed to describe the stronger instability and turbulence dynamics that are relevant to flows at higher true Re for the MS dynamics described here.
c. Evolution of S&L structures
Influences of MS interactions on the quasi-2D flow for Re = 100 000 are illustrated initially in Fig. 4. These panels show nearly vertical (z′) profiles of nondimensional θ and u at the domain center from 0–24Tb at 2Tb intervals. These reveal initial localized overturnings that drive streamwise-aligned, counterrotating instabilities due to FS shear advection where significant dθ′/dx > 0 (<0) and du′/dz > 0 (<0) coincide (see the horizontal lines in the second profiles). Thereafter, other instabilities dominate, including GW breaking, KHI, and fluid intrusions comprising rapid, vertically confined motions having strong shears above and below, primarily in the direction of initial GW propagation.
Profiles along z′ of (top) θ and (bottom) u at the center of the domain from 0 to 24Tb at intervals of 2Tb. Horizontal lines in the second profiles in each panel indicate sites where dθ/dz < 0 owing to horizontal advection where (du/dz)/(dθ/dx) > 0; these sites initiate the first 3D instabilities at later times.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Profiles along z′ of (top) θ and (bottom) u at the center of the domain from 0 to 24Tb at intervals of 2Tb. Horizontal lines in the second profiles in each panel indicate sites where dθ/dz < 0 owing to horizontal advection where (du/dz)/(dθ/dx) > 0; these sites initiate the first 3D instabilities at later times.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Profiles along z′ of (top) θ and (bottom) u at the center of the domain from 0 to 24Tb at intervals of 2Tb. Horizontal lines in the second profiles in each panel indicate sites where dθ/dz < 0 owing to horizontal advection where (du/dz)/(dθ/dx) > 0; these sites initiate the first 3D instabilities at later times.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
The consequences of these dynamics are S&L structures, where sheets comprise strong θ and velocity gradients (thus also large N2, ζy, ε, and Riχ) separated by layers having much weaker gradients, driven by successive mixing events. As seen in Fig. 4, these S&L structures are especially sharp prior to strong instability and turbulence mixing events (~4–12Tb). They become weaker following weaker instability events that drive weaker turbulence and less vigorous mixing as the 2D energy subsides. Occurrences of specific instability and turbulence events and their influences on the evolving N2 fields are explored in greater detail below.
d. Dynamical impacts of instabilities and turbulence on S&L evolution
FW13 found a close correspondence between laminar or restratifying ζy sheets and corresponding enhanced ε and Riχ, even in the presence of strong turbulence that often had large ε. They also inferred that strong mixing events accompanying GW breaking and fluid intrusions yielded weakly stratified layers because of their occurrence largely between ζy sheets and strong reductions in Riχ that occurred where ε was strong.
Here we examine the mutual influences among the major instability types and their N2 environments. For this purpose, four time series for subsets of the full streamwise-vertical (hereafter streamwise) cross sections of logε and N2 are displayed in Figs. 5–8. Thorpe reordering is performed to ensure well-behaved (N2 ≥ 0) cross sections by assigning zero or small positive values where local overturning or significant turbulence occurs. Strong sheets in N2 are unaffected by this process and remain closely correlated with those in ζy, ε, and Riχ extending to late times. Their evolution and persistence are driven both by advection and intermittent turbulent mixing throughout the evolution. However, the character, locations, and consequences of turbulence and mixing depend strongly on instability type and intensity. Because of the strong MS environment, there are many cases where different instabilities are initiated by the same dynamics or one event impacts the evolution of another in an adjacent region.
Streamwise cross sections of spanwise-averaged (left) logε and (right) N2 for Re = 100 000 in the subdomain from z′ = 0.4 to 1.0 and from 7.5 to 13.5Tb at intervals of 0.5Tb. This spans the interval of strongest instability and turbulence events, including strong GW breaking and KHI. The color scales are the same for each panel set and that for logε spans ~5 decades of intensities. Black lines highlight GW breaking events; red lines beginning at 10Tb show two KHI evolutions. The ε values are nondimensional so they can be easily scaled to other applications.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Streamwise cross sections of spanwise-averaged (left) logε and (right) N2 for Re = 100 000 in the subdomain from z′ = 0.4 to 1.0 and from 7.5 to 13.5Tb at intervals of 0.5Tb. This spans the interval of strongest instability and turbulence events, including strong GW breaking and KHI. The color scales are the same for each panel set and that for logε spans ~5 decades of intensities. Black lines highlight GW breaking events; red lines beginning at 10Tb show two KHI evolutions. The ε values are nondimensional so they can be easily scaled to other applications.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Streamwise cross sections of spanwise-averaged (left) logε and (right) N2 for Re = 100 000 in the subdomain from z′ = 0.4 to 1.0 and from 7.5 to 13.5Tb at intervals of 0.5Tb. This spans the interval of strongest instability and turbulence events, including strong GW breaking and KHI. The color scales are the same for each panel set and that for logε spans ~5 decades of intensities. Black lines highlight GW breaking events; red lines beginning at 10Tb show two KHI evolutions. The ε values are nondimensional so they can be easily scaled to other applications.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 5, but for the subdomain from z′ = 0.1 to 0.5 and from 14 to 16.5Tb showing the evolution of a weak GW breaking event (red ovals).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 5, but for the subdomain from z′ = 0.1 to 0.5 and from 14 to 16.5Tb showing the evolution of a weak GW breaking event (red ovals).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 5, but for the subdomain from z′ = 0.1 to 0.5 and from 14 to 16.5Tb showing the evolution of a weak GW breaking event (red ovals).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 5, but for the subdomain from z′ = 0.5 to 1 and from 18.5 to 21Tb showing the evolution of a strong intrusion (red outlines).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 5, but for the subdomain from z′ = 0.5 to 1 and from 18.5 to 21Tb showing the evolution of a strong intrusion (red outlines).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 5, but for the subdomain from z′ = 0.5 to 1 and from 18.5 to 21Tb showing the evolution of a strong intrusion (red outlines).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 5, but for the subdomain from z′ = 0.6 to 1 and from 37.5 to 39.5Tb showing the interaction of two oppositely moving intrusions (red ovals).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 5, but for the subdomain from z′ = 0.6 to 1 and from 37.5 to 39.5Tb showing the interaction of two oppositely moving intrusions (red ovals).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 5, but for the subdomain from z′ = 0.6 to 1 and from 37.5 to 39.5Tb showing the interaction of two oppositely moving intrusions (red ovals).
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Figure 5 shows streamwise cross sections of logε and N2 (left and right panels, respectively) for z′ = 0.4–1.0 from 7.5 to 13.5Tb at intervals of 0.5Tb. This series spans the interval of maximum instabilities and turbulence energies and exhibits a number of instability processes, interactions, and influences on adjacent dynamics. Seen at top and slightly right of center at 7.5Tb is a patch of strong turbulence resulting from a previous GW breaking event (red ovals at 7.5Tb). This feature lies above and left of a series of sheets in the logε and N2 fields that exhibit maximum elevations aligned from upper left to lower right (black ovals at 7.5Tb). These are indicative of a GW, and the leftward motions of the crests reveal upward and leftward propagation. The vertical displacements of this and the following two phases of the GW packet (not yet formed at 7.5Tb, but arising to the right as the evolution proceeds) achieve steep leading edges from 8 to 10Tb that yield the following responses (see black ovals at each time):
penetration of weakly stratified and nonturbulent air into the initial turbulent layer (but preserving the logε and N2 sheet between them),
initial GW overturning below the strong N2 sheet at three successive phases at ~9–10Tb,
initiation of small-scale KHI on the highest logε and N2 sheet, primarily on the portions at, and trailing, the maximum upward displacements (see the N2 fields at 8–9.5Tb),
successive plunging motions at all three leading edges toward the lower left that are already turbulent owing to the induced KHI on the sheet at earlier times,
merging of the resulting turbulence patches in the horizontal (~10–12Tb), and
formation of an extended N2 layer and new sheet at its upper edge as turbulence and mixing weaken (see the N2 fields at ~10–13.5Tb).
We now examine occurrences of larger-scale KHI seen to accompany 1) the final stages of the previous GW breaking event (see the red ovals at left in each panel at 10–11Tb) and 2) the steepening of a larger-scale GW above the region of plunging air and strong turbulence (see the red ovals at upper right in each panel at 10.5–11.5Tb) in Fig. 5. In each case, the N2 (and zy) sheet is most distorted and intensified at its maximum upward displacement. This results in reduced Ri and KHI initiation that progresses along the sheet as the GW phase advances to the left thereafter. The second case also exhibits a clear increase in the horizontal wavelength and depth of the KH billows as the KHI progresses to the left.
To interpret the observed KHI evolutions, we note that studies of idealized KHI for an initial velocity profile with U(z) = U0 tanh(z/h) and ζy = (U0/h) sech2(z/h) yield a KH wavelength λKHI ~ 4πh for a sufficiently high Re (e.g., Drazin and Reid 2004). KH billow amplitudes depend strongly on Ri and Re, however, and larger Ri and smaller Re yield small amplitudes, largely laminar responses, and laminar broadening of the initial shear layer. Such small-scale KHI extends throughout the GW breaking event discussed above (see the variations on the N2 sheet extending from 7.5 to 12Tb). Evaluation of the initial Ri at 10.4Tb (not shown) for the second KHI event seen beginning at ~10.5Tb reveals an extended region along the ζy and N2 sheets that has initial Ri ~ 0.10–0.15. This yields KHI having larger λKHI, Re = U0h/ν, and amplitudes owing to the thicker sheets, hence strong turbulence generation and mixing at the initial sheet. An additional indication of these stronger dynamics and mixing is the occurrence of a split sheet for a portion of this event at late stages (see the red ovals in both fields at 13 and 13.5Tb). We believe this is the first direct evidence of this behavior in a MS DNS, though multiple DNSs of more idealized KHI have revealed such features at higher effective KHI Re (e.g., Werne and Fritts 1999, 2001).
Figure 6 illustrates a weaker GW breaking event that follows the strong decreases in 2D and 3D energies discussed above. In this case, overturning has already occurred at the leading GW phase by the first time displayed (14Tb) and instability structures can be seen in the logε field (see red ovals throughout). A similar instability form accompanies breaking as the second GW phase achieves a large amplitude (14.5Tb). The resulting turbulence events merge from ~14.5 to 16Tb and contribute to additional GW breaking events at the packet leading edge thereafter. The event duration, turbulence intensities, and mixing are all much smaller in this case because of the much less energetic 2D GW field and initial KHI at these times.
Other dynamics that are distinct from GW breaking and KHI also contribute to layered instabilities and turbulence and the evolution of S&L structures, but primarily at later times. The most prominent of these have the characteristics of intrusions. Intrusions manifest as quasi-horizontal motions that have initial depths comparable to, or smaller than, the initial FS shears. The dominant intrusions exhibit rapid motions in the direction of primary GW propagation, penetrate into more quiescent air above and below, and arise where the various GWs and FS constituting the quasi-2D motion field superpose constructively. Weaker intrusions also arise that exhibit the opposite direction of motion. However, these are much weaker sources of turbulence and mixing, except where intrusions moving in opposite directions collide.
An example of a strong intrusion event is displayed from 18.5 to 21Tb in Fig. 7 (see red lines). This appears initially as a quasi-horizontal layering of horizontally extended, interleaved, and strongly and weakly stratified sheets and layers that are largely laminar and wrapped around a clear leading edge. The intrusion initially moves rapidly to the left relative to the adjacent layers above and below (see the red ovals in the images at 18.5 and 19Tb). Thereafter, it slows, thins, and exhibits instabilities and turbulence largely confined to the layers of small or negative N2 (see images at 19.5 and 20Tb) prior to strong mixing and dissipation (~20.5–21Tb). Multiple other intrusions also occur at later times, but their turbulence intensities are typically ~10 times or more smaller than the initial GW breaking and KHI events discussed above.
The collision of two intrusions occurring from 37.5 to 39.5Tb is illustrated in Fig. 8. At this stage of the flow evolution, sheets are far more diffuse and the interaction is largely confined to a weakly stratified layer (above the dominant sheet in N2 at 37.5Tb) and yields local overturning and the only obvious turbulence at these times (see red ovals). Because these instability dynamics are weak, we display ε rather than logε to show the relative contributions of the instability and the laminar sheets more clearly. Multiple such events occur throughout the MS DNS, but like the intrusions, their collisions contribute only sporadic weaker turbulence.
Finally, we note that the various instabilities discussed above do not occur in isolation but often interact directly and that their forms and effects can vary significantly as a result.
4. Implications of MS turbulence events for relations among ε, Riχ, LT, and LO
We now employ several instability and turbulence events in our Re = 100 000 DNS to explore the relationships among various measures of turbulence intensities and representative event scales. Specifically, we compare spatial distributions of θ, ε, Riχ, LO, and LT and explore the dependence of C = LO/LT on event type and various measures of LO. Our intent is to provide initial guidance for interpretations of such events observed with various in situ instruments throughout the atmosphere.
For these purposes, we have identified two KHI events (early and late stages of the KHI seen at right from ~11 to 12Tb in Fig. 5), an intrusion event at late times (~32–33Tb), and three GW breaking events: two that were energetic accompanying the peak in 3D energy from ~9 to 12Tb (see Fig. 5) and a third during reduced 3D energy (~14–15Tb; see Fig. 6). The KHI events and the intrusion event are illustrated in Fig. 9 with cross sections of spanwise-averaged logε and logRiχ. The same fields are shown for the three GW breaking events in Fig. 10.
Subdomains of streamwise cross sections showing logε and logRiχ for (a),(b) two KHI events and (c) an intrusion at (left)–(right) three times. In each panel set, (top) logε and (bottom) logRiχ are shown. Times for the three events are (a) (left)–(right) 11.1, 11.4, and 11.7Tb, (b) (left)–(right) 12.5, 12.8, and 13.1Tb, and (c) (left)–(right) 32.0, 32.3, and 32.6Tb. Color scales are not shown, as only the relative intensities are relevant to this discussion.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Subdomains of streamwise cross sections showing logε and logRiχ for (a),(b) two KHI events and (c) an intrusion at (left)–(right) three times. In each panel set, (top) logε and (bottom) logRiχ are shown. Times for the three events are (a) (left)–(right) 11.1, 11.4, and 11.7Tb, (b) (left)–(right) 12.5, 12.8, and 13.1Tb, and (c) (left)–(right) 32.0, 32.3, and 32.6Tb. Color scales are not shown, as only the relative intensities are relevant to this discussion.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Subdomains of streamwise cross sections showing logε and logRiχ for (a),(b) two KHI events and (c) an intrusion at (left)–(right) three times. In each panel set, (top) logε and (bottom) logRiχ are shown. Times for the three events are (a) (left)–(right) 11.1, 11.4, and 11.7Tb, (b) (left)–(right) 12.5, 12.8, and 13.1Tb, and (c) (left)–(right) 32.0, 32.3, and 32.6Tb. Color scales are not shown, as only the relative intensities are relevant to this discussion.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 9, but for three GW breaking events at (a) (left)–(right) 9.0, 9.6, and 10.2Tb; (b) (left)–(right) 10.7, 11.1, and 11.5Tb; and (c) (left)–(right) 14.2, 14.6, and 15.0Tb.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 9, but for three GW breaking events at (a) (left)–(right) 9.0, 9.6, and 10.2Tb; (b) (left)–(right) 10.7, 11.1, and 11.5Tb; and (c) (left)–(right) 14.2, 14.6, and 15.0Tb.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 9, but for three GW breaking events at (a) (left)–(right) 9.0, 9.6, and 10.2Tb; (b) (left)–(right) 10.7, 11.1, and 11.5Tb; and (c) (left)–(right) 14.2, 14.6, and 15.0Tb.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
Examining first the KHI cross sections in Figs. 9a and 9b (events 1 and 2), we see clear differences in the spatial distributions of logε and logRiχ. LogRiχ exhibits more filamentary maxima in the outer portions of the KH billows in each event. These arise early in the billow evolutions and persist as the billows break down and approach a horizontally uniform turbulent shear layer. Maxima of logε, in contrast, tend to occur in the billow or turbulence layer interiors (see the maxima of logε in the billow cores at the latter two times in event A where logRiχ is weaker). F13 argued that this was a consequence of strong mixing where logε was largest, causing thermal gradients (and logRiχ) to be mixed to the edges of these events.
Similar spatial distributions of logε and logRiχ are seen for the GW breaking events shown in Fig. 10 (events 4–6). As in the KHI events, large logε values typically occur in the centers of the turbulence fields whereas large logRiχ are often more confined and often occur near the edges of large logε. This is because GW breaking involves plunging of elevated fluid within a region of reduced (or negative) N2, hence large logε in these regions.
The intrusion event in Fig. 9c (event 3) exhibits quite different relationships between logε and logRiχ than seen for KHI and GW breaking. Peak initial logε and logRiχ coincide closely because both occur at the strong initial sheets prior to turbulence generation. As turbulence arises thereafter, logε maxima remain quite layered whereas logRiχ evolve toward more uniform distributions having smaller gradients than logRiχ. We attribute this to interleaved regions of low (or negative) and high N2 constituting the initial intrusion that are similar to those seen in the N2 cross sections for the intrusion shown in Fig. 7. As discussed above, mixing is very efficient when it is initiated in weakly stratified layers; hence, we expect greater layering in logRiχ than in logε at later times.
We now employ the events shown in Figs. 9 and 10 to assess the relationships among various fields within the corresponding 3D volumes. Examples of the fields listed above are shown for KHI event 2, the intrusion event 3, and GW breaking event 5 in Figs. 11–13. These fields include θ, N2, ε, Riχ, |d′|, LT, and LO computed from local ε and ε smoothed over LT.
As in Fig. 9, but for KHI event 2 at 12.7Tb. Shown are (top left)–(bottom left) θ, N2 from reordered θ, ε, and Riχ and (top right)–(bottom right) d′, LT, LOs, and LO. Horizontal and vertical scales show the subdomain location in X′ and Z′, respectively. Linear color scales are used to emphasize the highly localized occurrences of large ε and Riχ.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 9, but for KHI event 2 at 12.7Tb. Shown are (top left)–(bottom left) θ, N2 from reordered θ, ε, and Riχ and (top right)–(bottom right) d′, LT, LOs, and LO. Horizontal and vertical scales show the subdomain location in X′ and Z′, respectively. Linear color scales are used to emphasize the highly localized occurrences of large ε and Riχ.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 9, but for KHI event 2 at 12.7Tb. Shown are (top left)–(bottom left) θ, N2 from reordered θ, ε, and Riχ and (top right)–(bottom right) d′, LT, LOs, and LO. Horizontal and vertical scales show the subdomain location in X′ and Z′, respectively. Linear color scales are used to emphasize the highly localized occurrences of large ε and Riχ.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 11, but for the intrusion event 3 at 32Tb.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 11, but for the intrusion event 3 at 32Tb.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 11, but for the intrusion event 3 at 32Tb.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 11, but for the GW breaking event 5 at 11.1Tb.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 11, but for the GW breaking event 5 at 11.1Tb.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
As in Fig. 11, but for the GW breaking event 5 at 11.1Tb.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
All the above quantities are straightforward to assess as volume averages except for LO, and thus C, because of large variations of ε and N throughout each subdomain. Thus, we perform several estimates involving different combinations of smoothed or unsmoothed ε and N2 fields, and thresholds of ε, to obtain estimates for LO and C. The value of LO cannot be computed locally as N2 < 0 occurs frequently in overturning and turbulent regions. Hence, N2 estimates are performed using the reordered θ field and either 1) the full subdomain average, 2) the local average smoothed over a vertical interval 
εs: the local value of ε in the subdomain smoothed over δz;
εmax: the maximum value of ε in the subdomain, and a threshold εT = 0.01εmax;
LTs: the local value of LT computed by smoothing over δz;
LOl and LOTl: the local value of LO using the mean N2 and ε or ε > εT, respectively;
LOs and LOTs: the local value of LO using the mean N2, and εs or ϵs > εT, respectively; and
LONs: and LONa: the local value of LO using εs and
Estimates of ε, Riχ, LT, and C = LO/LT for various estimates of LO for the events and times shown in Figs. 9–13. Column headings are defined in the text. INT and GWB denote the intrusion and GW breaking events, respectively.
The fields shown for KHI event 2 at 12.67Tb in Fig. 11 reveal KH billows varying in wavelength and amplitude across the event, significant vertical displacements, but active overturning confined to the larger billow cores. Maxima of Riχ occur primarily in the regions of large N2 at the billow edges, whereas maxima of ε occur within the billow cores. Estimates of LTs exhibit much greater horizontal localization than either LOs or LOl. Mean values for C computed from the subdomain means for LOs and LOl are C = 0.85 and 0.75, respectively. The same computations for the three times displayed in Fig. 9 reveal significant variations in C throughout this KHI event. These increase from C ~ 0.8 at earlier times to C > 2 as the event dissipates. KHI event 1 yields smaller estimates, with C varying by a factor of ~3 from C ~ 0.3 to 0.9. We have also estimated C for the thresholded values of εT, LOTs, LOTl, and for the estimates accounting for variable N2, LONs, and LONa, as defined above (see Table 1). These estimates yield similar trends to those noted above; however, those employing 
The same fields are shown for the intrusion event 3 in Fig. 12. Corresponding estimates of C are shown in Table 1. Figure 12 reveals large vertical displacements and deeper and more horizontally extended regions of negative N2 compared to KHI event 2. As a result, spatial distributions of LTs and LOs are in somewhat better agreement than for the KHI event and are also more consistent with the estimates of LOl. Estimates of C for this event are dramatically smaller at early stages, but comparable to the initial values in KHI event 2 at the final time (e.g., C ~ 0.1–1.0 from 32 to 32.6Tb).
GW breaking event 5, shown in Fig. 13, exhibits deeper, but less horizontally extended, regions of overturning than seen for the intrusion event. Also seen at the top edge of the initial event are remnants of small-scale KHI that arose at earlier stages for the reasons described above (see the left panels of Fig. 10b). The ε value due to these decaying KHI surely impacts LO inferred at early times that are unrelated to LT accompanying the GW breaking event. Spatial distributions of LTl and LOs exhibit greater differences than seen for the intrusion event in Fig. 12 because widespread turbulence has not yet arisen accompanying GW breaking. Evaluating C as above yields estimates of C ~ 0.7 for the time shown in Fig. 13 and thereafter, but C ~ 0.3 at the initial time shown in Fig. 10 (see GWB event 2 in Table 1).
Results of the assessments described above for all events shown in Figs. 9–13 are listed in Table 1. The greatest similarities across events are the rough magnitudes of C estimated in various ways, suggesting that estimates of LO are relatively insensitive to the specific method. In other respects, however, there is little or no consistency in the estimates of C within or across event types. Both KHI events exhibit increasing C from early to late times, as noted by Smyth and Moum (2000), but yield magnitudes of C that vary by factors of ~2–3 within and between them. The GW breaking events yield values of C that vary by ~2–5 times (and oppositely with time) and from being comparable with to much smaller than estimates of C for the KHI events. Finally, the intrusion event exhibits greater variability than the GW breaking events, C ~ 0.08–0.80 or larger, with a tendency for increasing C with time as seen for the KHI events.
These results yield an overall mean (standard deviation) for C of 0.70 (0.56), and KHI, GW breaking, and intrusion means (standard deviations) of 0.96 (0.72), 0.49 (0.42), and 0.48 (0.23) employing LOs and mean N2. Hence, our DNS suggest that C is not as uniform and universal as has been suggested, at least for local instability and turbulence events in a MS environment. Estimates using 
(top) Estimates of C = LO/LT from 4 to 24Tb for the full model domain using mean N2, ε, LO, and LT for all ε (solid line) and ε > 0.001εmax (dashed line). (bottom) Semilog distribution of C vs ε for the individual events at each time (small text) and for the event means (large text). Numbers refer to events and numerals refer to sequential event times. Note in the top panel that the domain-mean C remains small as 3D energy is increasing (prior to 11.5Tb) and fluctuates between C ~ 0.4 and 2 or greater as instability and turbulence events subside.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
(top) Estimates of C = LO/LT from 4 to 24Tb for the full model domain using mean N2, ε, LO, and LT for all ε (solid line) and ε > 0.001εmax (dashed line). (bottom) Semilog distribution of C vs ε for the individual events at each time (small text) and for the event means (large text). Numbers refer to events and numerals refer to sequential event times. Note in the top panel that the domain-mean C remains small as 3D energy is increasing (prior to 11.5Tb) and fluctuates between C ~ 0.4 and 2 or greater as instability and turbulence events subside.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
(top) Estimates of C = LO/LT from 4 to 24Tb for the full model domain using mean N2, ε, LO, and LT for all ε (solid line) and ε > 0.001εmax (dashed line). (bottom) Semilog distribution of C vs ε for the individual events at each time (small text) and for the event means (large text). Numbers refer to events and numerals refer to sequential event times. Note in the top panel that the domain-mean C remains small as 3D energy is increasing (prior to 11.5Tb) and fluctuates between C ~ 0.4 and 2 or greater as instability and turbulence events subside.
Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-14-0343.1
We also show in Fig. 14 (top) C based on LOs and mean N2 for the entire model domain versus time for all ε and for ε >εT = 0.001εmax. These show that C remains near zero throughout the strong initial instability and turbulence growth prior to ~7Tb, remains below ~0.4 until the peaks in turbulence kinetic and potential energy at ~11.5Tb, and attains maxima of C ~ 1–2 or above accompanying and following the strong turbulence decay beginning ~12Tb.
5. Discussion and conclusions
Our goals defined above include the following: 1) characterize the evolutions of instability and turbulence events for a higher-Re MS flow than previously performed, 2) explore the roles of these events and larger-scale wave–wave interactions in the formation and evolution of S&L structures, and 3) perform an assessment of C = LO/LT for various stages of several of the instabilities yielding significant turbulence. These goals were addressed through analysis of an MS flow performed for Re = 100 000 that is twice that described previously by F13. This yielded Reb ~ 20–30 and sustained instability dynamics and turbulence during the most active phase of the MS flow evolution.
Relevance of our study was demonstrated by comparing profiles of θ and N from Thorpe-reordered θ and magnitudes of d′ and logε from the MS DNS and those measured by a DataHawk UAV at DPG under stable conditions. These reveal close agreement in the character of S&L structures, perturbation amplitudes, instability scales, and turbulence magnitudes. Further evidence of this MS interpretation is provided by BB, who present a more extensive analysis of the DPG data, including correlations of θ and velocity fluctuations and more detailed comparisons of measurements with MS DNS results that suggest embedded KHI and GW breaking events. We believe that these results, together with multiple previous measurements from the SBL into the stratosphere and above (e.g., Dalaudier et al. 1994; Luce et al. 2002; Balsley et al. 2013; Schneider et al. 2015), are strong evidence for the ubiquity of such flows and of the small-scale instability dynamics that result throughout the atmosphere.
Our MS DNS results provide some new insights into MS dynamics relative to previous results for a lower Re (e.g., F13, FW13). Increasing Re by 2 times results in essentially the same instability dynamics and spatial and temporal scales and nearly the same turbulence character and evolution for strong events. Anticipated differences include increased 〈ε〉 and Reb (by ~25% and ~2.5 times, respectively), both of which are expected for a doubling of Re. Also seen are additional, weaker instability events at later times that did not occur for the smaller Re. Extrapolation of these results to even higher Re suggests that the strong earlier instability dynamics will not change significantly but that further increases in Re will allow the new events that occurred for Re = 100 000 to intensify further, and may allow others to also arise.
The major instabilities that occur throughout the MS DNS include wave–wave interactions, GW breaking, KHI, and intrusions. Wave–wave interactions are active throughout the evolution but play the major role in defining the initial S&L structures at early stages before local instabilities arise. They do not account directly for energy dissipation at small scales, apart from their contributions to strong intensification of S&L structures and conditioning of the flow for subsequent local instabilities.
GW breaking plays major roles in the flow evolution in the energetic S&L environment created by the initial wave–wave interactions (e.g., after ~7Tb). The spatial scales and intensities of these events depend on the background flow, and both typically decrease with time as the total energy in the initial flow subsides. Strong GW breaking events can span several S&L structures, whereas weaker events (especially at later times) are more localized. In each case, the initial instabilities typically occur on the strongly stratified sheets and can take several forms, including direct overturning and/or initial small-scale KHI. Throughout, large ε occurs on the initial sheets and in the adjacent layers, whereas large χ is much more confined to the initial sheets. The resulting mixing expands (and sometimes splits) the initial sheet and contributes to the regeneration of the adjacent layers.
KHI occurs only on the stratified sheets where divergent horizontal GW and FS flows along the sheets cause local Ri reductions. This can occur both on intensifying sheets without small-scale GW perturbations and locally accompanying initial GW steepening that yield smaller KHI scales. Mixing due to KHI acts most strongly within the KH billows at early times. As KHI events restratify, ε remains large throughout but χ maxima are confined to the new sheets. Strong KHI events can cause a splitting of the initial sheets and a clear separation of ε and χ maxima.
Intrusions play more significant roles following the strong initial instabilities and extending to late times. Major intrusions comprise superpositions of larger horizontal-scale motions, often including the primary GW, given their tendency to be much stronger along the direction of primary GW propagation. They appear distinct from GW breaking owing to the absence of apparent vertical phase structure. However, they often exhibit smaller-scale features resembling GW breaking and/or KHI at their edges during the transitions to turbulence.
Importantly, these various instabilities often do not occur in isolation within the MS flow. As just noted, KHI can accompany GW breaking, and both GW breaking and KHI can accompany intrusions. Even in cases where the underlying dynamics are distinct, different instability types (or events) readily occur in close proximity and thus cannot avoid some mutual interaction effects, especially when instability tendencies are strong.
Despite the significant evidence for, and importance of, small-scale instability dynamics that drive turbulence and mixing throughout the atmosphere and oceans, very few DNS studies have been performed that address relevant Re, achieve sufficient Reb, and properly resolve these instability and turbulence dynamics. Even fewer DNSs have been performed that specifically address such dynamics embedded within larger-scale MS flows.
As noted by Moin and Mahesh (1998), Pope (2000), Zikanov (2011), Chung and Matheou (2012), and others, such DNSs require a minimum resolution given by Δx/LK ~ 1.5–2.1. Various studies have also identified a threshold Reb ~ 20 that is required for sustained strong turbulence (e.g., Smyth and Moum 2000). Until recently, the required resolution has been costly or impossible to achieve. However, this resolution requirement has been met or exceeded in recent DNS studies of KHI at various initial Re and Ri (e.g., Werne and Fritts 1999, 2001; Werne et al. 2005; Fritts et al. 2014), GW breaking for several GW amplitudes (e.g., Fritts et al. 2009a,b), MS flows of various character (e.g., F13; FW13), and idealized turbulence flows without larger-scale geophysical sources (e.g., Moin and Mahesh 1998; Chung and Matheou 2012; and references therein). Importantly, this resolution requirement has not been met in multiple previous studies, many of which made no attempt to describe instability and turbulence dynamics directly, but also others that did. As an example, the study of the mixing transition by Caulfield and Peltier (2000) employed only 100 grid points per KHI wavelength, thus describing the initial secondary instabilities, but not the transition to turbulence or mixing. Similarly, Smyth et al. (2001) employed several DNSs, the highest resolution of which (their simulation 3) addressed a KHI event for Re = 4978 with 512 grid points per KHI wavelength. This simulation employed a maximum N ~ 0.01s−1, ν = 10−6 m2 s−1, and yielded a maximum Reb ~ 400, implying ε ~ 4 × 10−8 m2 s−3, LK ~ 2.4 × 10−3 m, and Δx/LK ~ 4. By comparison, our MS DNS achieved Reb ~ 20 (the threshold for sustained turbulence) using a resolution of Δx/LK ~ 1.6 or better. Given the importance of these dynamics throughout the atmosphere and oceans, future studies of instabilities and turbulence in stratified flows will need to employ these constraints on model resolution. This is especially onerous for MS DNSs, but these goals are now within reach using the largest available computers.
The final goal of this study was to assess turbulence parameters, especially LO, LT, and C = LO/LT for various instability types in the MS DNS. Despite significant variability in the sampling of individual events (Clayson and Kantha 2008; Schneider et al. 2015), quantification of the dependence of C on specific event types and/or character, and the averaging required to allow application of an approximately universal value for assessments of turbulence influences, would be very beneficial. Our ability to now perform DNSs of MS flows appears to provide this capability.
For these purposes, our DNS results revealed highly variable LT, LO, and C for different events and types. The value of LO varied most when a local rather than a mean N was used. Because these estimates have a potential for significant biases, they were not used for our assessments of C.
The value of C tends to be small at early stages (~0.1–0.4), because LT is large accompanying incipient instability before ε and LO have become significant. Typically, C increases throughout the event (to ~0.8–3) as a result of increasing and persistent ε and LO and strong decreases in LT as turbulence arises. Thus, we should expect an evolution of C from near zero to large values throughout individual turbulence events and perhaps also for overall flow evolutions exhibiting increasing instability dynamics followed by strong turbulence decay. A similar evolution was noted by Smyth et al. (2001) for a DNS of an individual KHI event performed at a much higher Re than describes the KHI in our MS DNS.
Evolutions of C observed in our DNS for various events and types suggest a potential to define a mean C based on event type and character, with suitable averaging. If this is found to be the case in further DNSs and observational studies, it may justify applications to high-resolution radiosonde profiling (and similar profiling at higher altitudes) for semiquantitative turbulence statistics assessments, as proposed by various authors. We note, for example, a mean C ~ 0.8–1 over the interval of significant instability dynamics in Fig. 14 (top) from ~12 to 20Tb, with variability about the mean of a factor of ~2. We also note that estimates employing thresholded values of ϵ did not differ significantly from those that did not, suggesting that existing instrumentation is sufficient for these assessments.
A final issue that impacts assessments of ε, LO, and C is the disparity between true and modeled Re. As noted by FW13, current MS DNSs are now able to achieve true Re for observed GW and instability scales in the MLT because of large ν ~ 1 m2 s−1 near ~80-km altitude. As noted above, however, the current MS DNS 
Acknowledgments
Support for this research was provided by ARO under Contracts W911NF-12-C-0097 and W911NF-12-2-0075; NSF under Grants AGS-1242943, AGS-1242949, AGS-1261623, AGS-1041963, and AGS-1101258; and NASA under Contract NNH09CF40C. We gratefully acknowledge access to large computational resources provided by the DoD HPCMO for our various studies.
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