1. Introduction

Observations of Kelvin–Helmholtz instabilities (KHIs) in the atmosphere, oceans, lakes, and the laboratory spanning over 70 years motivated a diversity of theoretical and modeling studies addressing secondary instabilities of KH billows driving turbulence transitions for various Richardson, Reynolds, and Prandtl numbers, Ri, Re, and Pr, spanning ∼40 years. Of necessity, early three-dimensional (3D) theoretical and modeling efforts addressed instabilities of individual KH billows at very modest Re and Ri (Klaassen and Peltier 1985, 1991; Werne and Fritts 1999). Successive higher-resolution studies addressed increasingly smaller Ri and larger Re but continued to employ streamwise domains including one or several KH billows that were very narrow spanwise; see the reviews by Mashayek and Peltier (2012a,b) and Smyth and Moum (2012). Importantly, these domains specifically precluded descriptions of KHI “tube” and “knot” (T&K) dynamics first identified in laboratory shear-flow studies by Thorpe (1985, 1987; hereafter T85 and T87) seen to exhibit much more rapid turbulence transitions and suggested by Thorpe (2002) to be widespread in the atmosphere.

Only recently were a high-resolution, large-eddy simulation (LES) and several direct numerical simulations (DNSs) of KHI performed employing sufficiently wide domains and high resolution to enable the emergence of T&K dynamics and exploration of their evolutions. Those studies to date include the following:

1. Fritts et al. (2021; hereafter F21) performed a high-resolution compressible LES that accounted for multiple KHI T&K features observed at a high altitude in the atmosphere by Hecht et al. (2021; hereafter H21) and seen in the early laboratory studies (T85, T87).

2. Fritts et al. (2022a,b; hereafter F22a, F22b) described a DNS including two forms of KHI T&K dynamics and their implications for elevated energy dissipation rates ε; however, it employed a noise seed that did not enable several important forms of these dynamics.

3. Fritts et al. (2022c; hereafter F22c) used results of the F22a DNS to approximate observations of apparent KHI T&K features observed in polar mesospheric clouds (PMCs).

4. A companion paper by Fritts and Wang (2023; hereafter FW23) revealed multiple, local KHI T&K events in an idealized multiscale gravity wave–fine structure superposition.

5. Mixa et al. (2023; hereafter M23) performed a high-resolution compressible LES designed to approximate observed KHI dynamics in the lower thermosphere and assess the resulting mixing.

The potential dynamical importance of KHI in the atmosphere and oceans derives from their expected contributions to turbulence and mixing. These roles are widely acknowledged in the oceans, where mixing due to small-scale KHI is expected to be an important, and potentially the dominant, contributor to small-scale mixing accompanying the ocean internal gravity wave (GW) spectrum that drives the ocean meridional overturning circulation (Gregg et al. 2018; Mashayek et al. 2015).

KHIs are also an important component of parameterization efforts addressing mixing in the atmospheric stable boundary layer, e.g., Fernando and Weil (2010). But their potential to contribute significant turbulence, mixing, and transport appears to be much less appreciated in the atmosphere at higher altitudes. This is true in the stratosphere, where observations provide few constraints at small vertical scales and model resolutions are often much coarser than expected KHI events that can extend to very small scales at large Re. It is also true in the mesosphere and lower thermosphere (MLT), where KHI T&K dynamics are prolific (Fritts et al. 2023, hereafter F23; H21), but very few observations can resolve fine structure scales enabling KHI. Additionally, current mixing parameterizations based only on assumed GW breaking appear to be insufficient to account for observed mean constituent profiles in the MLT (Garcia et al. 2014; Liu 2019).

Given the above, our objectives in this paper are as follows:

1. Employing a high-resolution DNS to explore and quantify new KHI T&K dynamics not seen to arise in the many idealized KHI studies to date, nor in the initial KHI T&K DNS described by F22a,b

2. Employing a high-resolution DNS for the same initial conditions in a narrow spanwise domain excluding T&K dynamics to enable the evaluation of T&K ε enhancements

3. Employing DNS of the two cases for decreasing Re in order to (i) assess the impacts on T&K dynamics and energetics and (ii) explore to what degree KHI T&K dynamics enable turbulence transitions at Re for which they would not otherwise occur

4. Comparing and contrasting the KHI T&K DNS described by F22a,b and the similar case presented here to reveal the broader range of KHI T&K dynamics and the influences of a later noise seed approximating the influences of persistent small-scale turbulence

The model employed for this study is described in section 2. DNS of KHI with and without T&K dynamics for Re = 5000 and for decreasing Re are described in sections 3 and 4. Section 5 compares the turbulence transitions in narrow and wide domains and their implications for energy dissipation rates with decreasing Re. Energy dissipation rate probability density functions (PDFs) and spectra for the varying Re are described in section 6. A discussion of these results relative to previous studies and their implications for parameterizations of KHI T&K dynamics are provided in section 7. Our conclusions are presented in section 8.

2. Model description, domain, initial conditions, and visualization methods

a. Model description

Following F22a, we employ the spectral atmosphere model (SAM), a Fourier, pseudospectral, triply periodic model that solves the Boussinesq Navier–Stokes equations written as

$\frac{\partial u_j}{\partial x_j}=0,$(1)

$\frac{\partial u_i}{\partial t}+\frac{\partial u_i{u}_j}{\partial x_j}+\frac{1}{{\rho }_0}\frac{\partial p^{\prime }}{\partial x_i}=\frac{{\theta }^{\prime }}{{\theta }_0}g{\delta }_{i3}+\nu \frac{{\partial }^2{u}_i}{\partial x_j\partial x_j},$(2)

$\frac{\partial \theta }{\partial t}+\frac{\partial \theta u_j}{\partial x_j}=\kappa \frac{{\partial }^2\theta }{\partial x_j\partial x_j}.$(3)

Here, u_i = (u₁, u₂, u₃) are the velocity components along x, y, and z; p, ρ, and θ are the pressure, density, and potential temperature (with θ = T for a Boussinesq flow), respectively; and ν and κ are the kinematic viscosity and thermal conductivity, respectively. Additionally, primes and “0” subscripts denote perturbation and mean quantities, respectively, and successive indices imply summation (see Table 1 for symbols and abbreviations).

Table 1.

Symbols and abbreviations.

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SAM is highly optimized for massively parallel computing. It employs a pseudospectral algorithm to compute nonlinear products and removal of aliasing errors using the two-third truncation rule (Patterson and Orszag 1971). SAM uses a low-storage, third-order Runge–Kutta method for time advancement, accuracy, and efficiency (see F22a for additional details). SAM also now employs a “pencil” domain decomposition that allows the computational load to be distributed over many more processors.

In this application, we employ SAM for 1) DNS of multiple, interacting KH billows in a wide domain allowing varying KHI phases ϕ and wavelengths λ_h and T&K dynamics along their axes and 2) corresponding DNS in a narrow domain that precludes T&K dynamics. The initial noise seed was chosen to enable the emergence of KHI that are initially misaligned and discontinuous along y at finite amplitudes, hence features not described by F22a. A single DNS for Re = 5000 having a spatial resolution of (N_x, N_y, N_z) = (432, 1296, 432) sufficient to describe the initial, large-scale, responses for all Re was initialized with a white noise seed and extended to 0.38T_b (with T_b being the buoyancy period at the peak stability) at z = 0 and t = 0 at which time weak, but coherent, KH billows arose. The same procedure was employed for DNS in a small domain. These fields were employed to continue the DNS for Re = 5000, 3000, 2000, 1000, and 500 in order to assess the impacts of decreasing Re on the KHI T&K evolutions and turbulence intensities, their implications for enhanced turbulence, and the potential for turbulence and mixing where it might not occur without T&K dynamics. A maximum of 3600 cores were employed for the KHI T&K Re = 5000 DNS. The smaller Re required smaller computational resources, i.e., 2700, 1920, 1440, and 960 cores, respectively.

b. Initial conditions and domain specification

As in F22b, we specify a shear flow approximating a steepened inertia–GW (IGW) having localized shear and stability maxima, with the mean wind and stability profiles given by

$U(z)=U_0\cos \left(\frac{\pi z}{Z}\right)\tanh \left(\frac{z}{h}\right),$(4)

$N^2(z)=N_0^2+(N_m^2-N_0^2){\mathrm{sech}}^2\left(\frac{z}{h}\right),$(5)

where Z is the domain depth, z = 0 at the domain center, U₀ is the half-shear velocity difference, h is the half-shear depth, and we assume $N_0^2=(g/\theta )d\theta /dz=10^{-4}{\mathrm{s}}^{-2}$ external to the shear layer and $N_m^2=8N_0^2$, because large shear and stability enhancements arise in thin layers due to GW interactions in multiscale environments (Fritts et al. 2016, hereafter F16; FW23). The choice for N_m yields a maximum buoyancy period T_b = 2π/N_m = 222 s.

The remaining parameters include a minimum Richardson number, $\text{Ri}=N_m^2/{(dU/dz)}^2=0.1$, a maximum Reynolds number, Re = U₀h/ν = 5000, and a Prandtl number, Pr = ν/κ = 1, to impose the same resolution requirements for the velocity and temperature fields. Re = 5000 is appropriate for a KHI λ_h ∼ 100 m, a shear depth h ∼ 0.07 λ_h, and ν ∼ 0.15 m² s⁻¹ at an altitude of ∼60 km. In reality, KHI λ_h can exceed ∼300 m in the SBL (Blumen et al. 2001), ∼5 km in the troposphere and stratosphere (Luce et al. 2007, 2018), and ∼5–15 km in the MLT (H21; F23), implying a true Re (assuming no background turbulent viscosity) that can be orders of magnitude larger at all altitudes below ∼80–100 km. Importantly, however, Re = 5000 requires very computationally intensive DNS but enables secondary KHI to compete with secondary convective instabilities (CIs) and T&K dynamics where they arise, thus ensuring a reasonable assessment of enhanced turbulence and mixing accompanying KHI T&K dynamics.

To ensure validity of SAM in these applications, we assume an approximate KHI wavelength λ_h ∼ L for fixed length scale L = 2 km, with wavenumber k_h = 2π/λ_h, a shear half-depth h = 0.07, L = 140 m, and U₀ = 12.5 m s⁻¹. We also specify an elevated “turbulence” viscosity, ν = ν_turb = 0.35 m² s⁻¹, to approximate turbulence influences in the middle/upper stratosphere because small-scale turbulence is expected to be pervasive and turbulence ε decays slowly over multiple T_b (F16, F22c). The initial fields are illustrated in Fig. 1 for a periodic domain of depth Z = 3L and periodic U, dθ/dz, and N², enabling a pseudospectral SAM DNS. Horizontal domain dimensions along the shear and the nominal KH billows are X = 3L and Y = 9L. The choice for h assumed above is somewhat smaller than the linear stability KH λ_h estimate to enable primarily 3, but occasionally 4, initial KH billows to arise along the X domain. The large Y = 9L enables significant KH billow phase variations and misalignments along y; the Z = 3L domain depth enables strong T&K dynamics extending well away from the initial shear.

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Initial vertical profiles of U/U₀, θ/θ₀, N², and Ri employed for our DNS.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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3. KHI T&K dynamics and energy dissipation rates for Re = 5000

a. KHI T&K dynamics in a wide domain

A SAM DNS enabling KHI T&K dynamics for Re = 5000 is shown with 3D λ₂ fields viewed from above from 2.5 to 4.75T_b after the initial noise seed in Fig. 2. The interval extends from the emergence of initial KH billows to T&K dynamics driving secondary instabilities and initial turbulence transitions. The initial noise seed here enables an expanded range of T&K features relative to the initial DNS study by F22a. The absence of a second noise seed at the onset of turbulence transitions in this case precludes other T&K features described by F22a. Note that the times employed here are ∼0.75T_b later than in F22a at the same evolution stages. Features common to F22a and the DNS described here seen in the full domain in Fig. 2 and the expanded views in Fig. 3 include the following:

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The 3D imaging of λ₂ at z = 0 viewed from above at 2.5, 3.25, 4, and 4.75T_b for Re = 5000. The value λ₂ varying from black to red shows increasing rotation. Labels a–k denote feature described in the text. Large-scale vorticity is along y. The λ₂ color scales extend from near zero to maxima that increase by factors of 100, 140, and 4 at successive times. The major responses occur in a depth of 0.4L centered at z = 0.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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Subdomain views of the λ₂ field shown at (top) 4.5T_b in Fig. 2 at smaller and larger y/L viewed from larger x and z and (bottom) upper-right panel at 4.75T_b in Fig. 2 viewed from above. Feature labels are described in the text. The smallest resolved features have scales ∼L/300.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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1. Weak initial KH billows exhibit varying λ_h and phases along y (sites “a”)

2. Large-scale vortex tubes arise on the vortex sheets between KH billows where they exhibit significant phase variations along y, e.g., large |dϕ/dy| (sites “b”)

3. Vortex tubes that intensify, wrap under and over adjacent KHI billows at smaller and larger x, and “attach” to the billows inducing the formation of vortex “knots” that account for rapid transitions to strong turbulence thereafter (sites “c” at smaller y)

4. Large-scale billow core twist waves induced by the vortex knots (sites “g”)

5. Secondary KHI and CI that drive slower and weaker turbulence transitions thereafter (sites “f” and “h”)

6. Secondary T&K dynamics on intensifying vortex sheets between adjacent billows (sites “i” at 4.75T_b)

7. Smaller-scale twist waves arising due to roughly orthogonal interactions among adjacent vortices in close proximity (sites “j”)

8. Interactions of the secondary CI and KHI where they are advected into close proximity wrapping around the KH billows (sites “k”)

   Features arising in the DNS described here not seen by F22a include the following:

9. Intense KH billow interactions arising where 1 billow “end” emerges adjacent to 2 where they are misaligned along y (sites “c” at larger y at 4T_b)

10. A localized KH billow pairing event also inducing a vortex knot (site “e”)

Also seen emerging by 2.5T_b and intensifying strongly thereafter are KH billow displacements along x that are coherent among adjacent billows at larger x extending from y/L ∼1–6. These are a manifestation of an additional axial KHI instability referred to as a “bending” instability (Rogers and Moser 1992; Kit et al. 2010), which we denote as “crankshaft” instabilities, given their varying axial forms that arise for the range of Re explored here. The vortex dynamics accounting for these crankshaft instabilities are the self-advection of the KH billows where they exhibit a curvature that induces vertical motions as seen for vortex rings. Upward (downward) vertical advection leads to advection toward positive (negative) x in the mean shear flow having dU/dz > 0, yielding intensifying curvature of the KH billows in time.

b. KHI T&K energy dissipation rates in a wide domain

Cross sections of log₁₀ε(x, y) at z = 0 and log₁₀ε(x, z) at representative y for the wide domain Re = 5000 simulation are shown with reference to the corresponding λ₂(x, y) fields at 4, 4.75, and 5.5T_b in Figs. 4–6. Cross sections, rather than volumetric imaging, are used for log₁₀ε to reveal the dynamics accounting for these various responses because volumetric views mask the responses at z = 0. The log₁₀ε(x, z) cross sections at 4T_b in Fig. 4 reveal emerging enhancements of ε accompanying initial T&K interactions of adjacent billow cores at different y that drive the initial twist wave generation and cascades to smaller scales (as initially described by F22a,b for a different initial noise seed). Three of these at top right reveal emerging strong interactions where four initial KH billows link to three at larger or smaller y and have become strongly kinked along y by 4T_b (see these regions from y/L ∼ 4–9 in Figs. 2 and 3). The fourth and fifth log₁₀ε(x, z) cross sections at right reveal only three KH billows that are relatively uniform. A sixth log₁₀ε(x, z) cross section at bottom right in Fig. 4 shows two vortex tubes in cross section at right that have evolved on the intensified vortex sheets linking three KH billows where they exhibit significant initial phase variability.

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(left) The 3D imaging of λ₂(x, y) as in Fig. 2 at 4T_b and the corresponding 2D cross sections of log₁₀ε(x, y) at (center) z = 0 and (right) y indicated by the red arrows for the wide-domain Re = 5000 simulation. The color scale for ε is shown at the center top.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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As in Fig. 4, but at 4.75T_b.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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As in Figs. 4 and 5, but at 5.5T_b.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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Cross sections of these fields at 4.75 and 5.5T_b in Figs. 5 and 6 reveal significant further evolutions of the KHI T&K dynamics in the five significantly disturbed regions (and the exception) highlighted at right in Fig. 4. Specific features of relevance include the following:

1. Adjacent billows merge and entwine by 4T_b (see Fig. 4 at left) where they are initially misaligned along y and yield intense vortex knots, local turbulence, and ε by 4.75T_b; see especially these regions at y/L ∼5–6 and 7–8, and x/L ∼1–2 in Fig. 5.

2. Other knots arise where billow cores exhibit significant dϕ/dy; these sites further intensify the intermediate vortex sheets due to additional stretching along y that induces larger- and smaller-scale vortex tubes by 4T_b (see Figs. 2 and 4 and F22a).

3. Local T&K site turbulence at 4.75T_b spreads dramatically along the vortex tubes and throughout the billow cores by 5.5T_b, but initially less impacted billow cores remain partially laminar.

4. The initial shear layer becomes largely turbulent by 5.5T_b, and $\overline{\epsilon }$ achieves its maximum at ∼6.2T_b (see the solid red line at left in Fig. 7).

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(a) Temporal and (b) cumulative evolutions of $\overline{\epsilon }$ averaged over the full (x, y) domain and a depth of 0.4L centered at z = 0 for the wide and narrow domains (solid and dashed lines) for Re = 5000 and the smaller Re. Color codes are shown in each panel. Red dots show the corresponding $\overline{\epsilon }$ temporal evolution at 0.5T_b intervals for the F22a,b DNS.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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c. KHI T&K dynamics and energy dissipation rates in a narrow domain

To assess the enhancement of local ε in the simulation enabling KHI T&K dynamics, we performed a narrow-domain simulation having Y/L = 1 to exclude T&K dynamics employing a noise seed of equal amplitude. The results are shown with log₁₀ε cross sections from 4.25 to 5T_b in Fig. 8 in the same formats as in Figs. 4–6. In this case, no significant interactions among adjacent KH billows occur and only secondary CI arises and drives turbulence transitions thereafter, despite extending to later times than shown for the KHI T&K dynamics discussed above. The temporal evolution of $\overline{\epsilon }$ in this case is shown with a dashed red line in Fig. 7 at left, and it achieves ∼95% of its initial maximum value at 5T_b. Specific additional features include the following:

1. The only contributions to ε prior to 4.5T_b are in the laminar vortex sheets between KH billows and the emerging CI having spanwise scales ∼0.1L in the outer billow cores.

2. Intensifying CI expands and begins to interact with the vortex sheets by ∼4.5T_b.

3. These dynamics expand into the billow cores thereafter (see Fig. 8) and yield an initial peak $\overline{\epsilon }$ at ∼5.2T_b (see Fig. 7).

4. Successive, variable peak $\overline{\epsilon }$ occur at ∼7.5 and 10T_b thereafter (see Fig. 7 at left).

5. Cumulative $\overline{\epsilon }$ (red dashed line at right in Fig. 7) is surprisingly only ∼3% less than for the wide domain because peak $\overline{\epsilon }$ is smaller, but large $\overline{\epsilon }$ spans a longer interval.

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The 2D log₁₀ε(x, y) cross sections at z = 0 and (x, z) at y/L = 0 for the narrow-domain Re = 5000 simulation. The color scale is as in Figs. 4–6.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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4. KHI T&K dynamics for decreasing Re

We now explore the influences of decreasing Re on KHI evolutions, the T&K dynamics that arise, and their implications for turbulence transitions and intensities. All cases employ the same initial noise seed to ensure that the initial KH billows have common features along y. Evolutions of the vorticity fields for Re = 3000, 2000, 1000, and 500 are compared with that for Re = 5000 over intervals extending to later times for decreasing Re due to increasing viscous influences. Temporal and cumulative $\overline{\epsilon }$ evolutions for the wide and narrow domains are shown in Fig. 7; ε probability density functions and spectra are compared in section 6.

a. KHI T&K dynamics at Re = 3000

The evolution for Re = 3000 shown at top in Fig. 9 not only closely parallels that for Re = 5000 but also exhibits delayed and/or weaker responses at most sites. The λ₂ color scale varies in time to reveal the key features at each. Initial larger-scale billow interactions are nearly identical to those at Re = 5000 because the increase in viscosity has weak influences on the KH billow scales and initial links and hence only weakly impacts turbulence transitions and scales arising in the initial knots. However, they significantly impact the emergence of secondary KHI, CI, larger- and smaller-scale vortex tubes, and their initial turbulence transitions in the absence of T&K enhancements, causing these responses to be much weaker than seen for Re = 5000 in Fig. 2.

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The 3D imaging of λ₂(x, y) as in Fig. 2 at (top) 2.5, 3.25, 4, and 4.75T_b for Re = 3000 and (bottom) 2.5, 3.5, 4.5, and 5.5T_b for Re = 2000. Labels “a”–“l” are as in Figs. 2 and 3 and denote feature evolutions at both Re values described in the text.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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Significant differences in the Re = 3000 evolution relative to Re = 5000 include 1) delayed knot formation due to weaker and fewer adjacent billow attachments, 2) slower initiation of larger-scale twist waves, 3) weaker, delayed, and less extended vortex tubes, 4) significantly delayed emergence and evolutions of secondary KHI, CI, and their interactions, and 5) fewer and delayed turbulence transitions, especially accompanying smaller-scale vortex tubes; compare sites labeled “a–d” and “f–k” for the two Re.

The crankshaft instability emerging at 4T_b and becoming prominent by 4.75T_b at smaller y for Re = 5000 is seen much more clearly for Re = 3000 due to the weaker secondary CI at 4.75T_b (see sites “l”). The instability wavelength is comparable to that estimated for Re = 5000 at 4.75T_b, i.e., λ_y ∼ 2.25L. This instability also plays a clear role in the occurrence of secondary KHI that will be seen below to become more prominent at smaller Re.

Differing KHI interaction dynamics and secondary instabilities occurring for Re = 3000 are seen in Fig. 7 to delay but enable, a higher, more transient, peak $\overline{\epsilon }$, and a cumulative $\overline{\epsilon }$ only ∼1% smaller than occurs for Re = 5000.

c. KHI T&K dynamics at Re = 1000

The Re = 1000 evolution shown at the top in Fig. 10 exhibits significant departures from those at higher Re despite sharing the initial noise seed. As discussed above, all higher Re fields reveal links between adjacent billows at small amplitudes leading to sites where 1) one billow links to two where their initial “ends” are misaligned and 2) adjacent billows link via larger- and/or smaller-scale vortex tubes emerging in regions of significant phase variability along y.

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As in Fig. 9, but for (top) Re = 1000 and (bottom) Re = 500. Note the later times from 3 to 6.5T_b and from 7 to 12T_b, respectively, due to the more slowly evolving KHI and T&K features in the increasingly viscous environments. See the text for details.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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In contrast, the Re = 1000 evolution reveals links among the misaligned initial responses yielding continuous adjacent billows exhibiting similar phase variability along y that link via small-scale vortex tubes thereafter (cf. sites “a–c” for Re = 1000 in Fig. 10 with those for the larger Re). These lead to limited bands at y/L ∼ 1 and 6 in which small-scale vortex tubes between adjacent billows lead to relatively weak knots at 6.5T_b (sites “c”) in which the weak initial turbulence is largely confined.

Also seen are crankshaft instabilities emerging by 4.5T_b that exhibit strong, roughly square-wave responses by 6.5T_b (sites “l”). The strong billow distortions induce additional, smaller-scale vortex tubes where adjacent KH billows are closely spaced (sites “d”), secondary KHI where billow curvature is strong (sites “f”), and large-scale twist waves in the distorted billow cores (sites “g”). These vortex dynamics and associated turbulence are delayed relative to those arising for Re = 2000 and larger. Importantly, however, the turbulent knots seen at 6.5T_b have already yielded a broad turbulence spectrum and are almost certainly unable to account for the peak $\overline{\epsilon }$ for this case, which is ∼6 times larger and occurs ∼2T_b later. The only remaining potential sources of stronger turbulence at later times are the intensified and highly distorted KH billows (see the discussion in section 5 and Fig. 12).

5. Comparisons of turbulence transitions in wide and narrow domains

Figure 7 at left reveals that turbulence transitions driving $\overline{\epsilon }$ increases are slower for Re decreasing from 5000 to 500 in both wide and narrow domains. T&K dynamics are less constrained by decreasing Re because they arise at the KH billow scales. Secondary KHIs are not expected for the specified Ri and Re but are enabled where billow curvature intensifies the vortex sheets between adjacent billows for all Re examined here. Secondary CIs, in contrast, are expected but arise at spatial scales that increase significantly with decreasing Re.

Figure 11 shows narrow domain responses for Re decreasing from top to bottom at the last times shown for these cases in Figs. 2, 9, and 10. These include 3D imaging of λ₂(x, y) in a volume of depth Δz = 0.4L centered at z = 0 (left) and log₁₀ε(x, y) and (x, z) cross sections at z = 0 and y = 0 at center and right, respectively. Whereas Fig. 2 for Re = 5000 reveals widespread secondary CI where the billows do not exhibit a strong curvature (see the relatively uniform billow at x/L ∼ 2 and y/L ∼ 8), those seen for decreasing Re in Figs. 9 and 10 reveal strongly suppressed or absent secondary CI compared to the narrow domain fields in Fig. 11. Thus, the expectations that secondary CIs are the major driver of turbulence for uniform KH billows at smaller Re, and that secondary KHIs only contribute for larger Re, are the opposite of what occurs in KHI T&K environments if the wide-domain results described here are representative. Importantly, both play much less significant roles than the larger-scale T&K dynamics.

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As in Figs. 4–6, but for the narrow domain for Re = 5000 to 500 (from top to bottom) at the times shown at right. (left) The λ₂ viewed from above and log₁₀ε (middle) horizontal and (right) vertical cross sections at z = 0 and y = 0.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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Corresponding log₁₀ε(x, y) cross sections at z = 0 for the wide-domain KHI T&K dynamics for decreasing Re at 4.75, 5, 5.5, 7.5, and 12T_b are shown from left to right in Fig. 12. These times are slightly later for Re = 3000 and 1000 in order to show all fields when the large domain $\overline{\epsilon }$ is increasing rapidly due to intense, large-scale interactions among adjacent KH billow cores at ∼40%–50% of their peak magnitudes thereafter. Times at peak $\overline{\epsilon }$ are not shown because they are after the breakdown of all coherent, initial KH billows in the wide domains.

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Cross sections of log₁₀ε(x, y) at z = 0 for the wide-domain KHI T&K dynamics at Re = 5000, 3000, 2000, 1000, and 500 (from left to right) at 4.75, 5, 5.5, 7.5, and 12T_b, respectively. Note the common ε color scale for all panels.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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Comparing the Re = 5000 field in Fig. 12 with the top center image in Fig. 11 reveals that secondary CIs in the wide domain are significantly less advanced at all sites where billows are even weakly influenced by T&K dynamics. They also have little or no time to arise in regions exhibiting strong initial T&K dynamics, hence contributing only very weakly to turbulence transitions and $\overline{\epsilon }$ in this case. The same conclusions arise by examination of the corresponding fields for Re = 3000 and 2000.

As noted in the discussion of Fig. 10, T&K dynamics for Re = 1000 and 500 exhibit 1) emerging crankshaft instabilities and 2) knot formation driving larger-scale turbulence at later times. These dynamics for Re = 1000 progress rapidly from 6.5 to 7.5T_b (see Fig. 6 at the top and Fig. 12) due to smaller-scale vortex tubes and secondary KHI driving additional turbulence transitions. Surprisingly, they also yield the largest $\overline{\epsilon }$ occurring ∼1T_b later (see Fig. 7), accompanying T&K dynamics for all Re explored here.

Evidence of the T&K dynamics accounting for the largest and narrowest $\overline{\epsilon }$ peak for Re = 1000 at ∼8.5T_b is seen in log₁₀ε (x, y) at ∼7.5T_b in Fig. 12. The log₁₀ε(x, y) maxima at y/L ∼ 0–1.8 and 5–6.5 correspond closely to the strongest responses in the 3D λ₂(x, y) field at 6.5T_b at the top in Fig. 10. Also seen in Fig. 12 are additional bands of enhanced log₁₀ε (x, y) centered at y/L ∼ 2.2, 3.3, 4.4, and 8.5, all of which correspond to |dϕ/dy| maxima at the transitions between the advanced and retarded phases of the crankshaft instability along x. The implication is that T&K dynamics of these distorted billow cores initiate additional turbulence transitions due to the crankshaft instability in these regions that progress very rapidly.

Corresponding wide-domain results for Re = 500 in Fig. 12 are surprisingly very different. Here, the strong crankshaft instability, and more limited turbulence transitions in the evolving knots, results in a smaller peak $\overline{\epsilon }$ than accompanies secondary CI in a narrow domain. This is in sharp contrast to the trend toward increasing peak and cumulative peak $\overline{\epsilon }$ enhancements with decreasing Re for all larger Re. The cause appears to be suppression of secondary CI by the crankshaft instability in this case (compare the Re = 500 panels in Figs. 11 and 12).

We also explore the causes of peak $\overline{\epsilon }$ in the narrow domains occurring from ∼8.5 to 10T_b. Figure 13 shows log₁₀ε (x, z) fields at y = 0 for Re decreasing from 5000 to 1000 at 8 and 10T_b. They are significantly advanced relative to those shown at the right in Fig. 11, where KH billows are well defined and exhibit initial or emerging turbulence transitions. The fields in Fig. 13 reveal initial billow pairing prior to 8 T_b for Re = 2000–5000 but with decreasing efficiency for smaller Re. Pairing appears to begin for Re = 1000 by 8T_b but apparently does not occur thereafter. All these cases exhibit similar responses and peak $\overline{\epsilon }$ at 10T_b but with that for Re = 2000 smaller by ∼30% and that for Re = 2000 exhibiting much weaker $\overline{\epsilon }$ at earlier times.

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As in Fig. 11, but for the narrow domain for Re = 5000 to 1000 (from top to bottom) at the times shown at right. Corresponding $\overline{\epsilon }$ values are shown at left in Fig. 7.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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6. Energy dissipation rate PDFs and spectra

We now examine the evolutions of the KHI T&K responses for the five Re revealed in ε PDFs and streamwise spectra. The assessments include all data in a layer of depth 0.4L centered at z = 0 because this region contains the larger majority of KHI T&K dynamics and turbulence throughout the five evolutions.

PDFs and ε(k) and kε(k) spectra are shown from left to right for decreasing Re from top to bottom in Fig. 14. Results for the five Re are shown at 1) the onset of turbulence at times of ∼4, 4, 4.5, 5.5 T_b, and 11T_b, 2) the final times shown for each case in Figs. 2, 9, and 10, and 3) extending to the peak $\overline{\epsilon }$ seen at ∼6.2, 6.5, 6.9, 8.5, and 13.4T_b in Fig. 7, respectively.

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(left) PDFs of log₁₀ε and streamwise spectra of (center) ε(k) and (right) kε(k) assessed in wide domains for decreasing Re from top to bottom. Blue, green, and red lines show the first, second, and third times for each Re. Red dashed lines at the left show $\overline{\epsilon }$ at the peak times. Units at center and right are MKS.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0085.1

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7. Discussion

KHI T&K dynamics were first identified in laboratory shear-flow studies by T85 and T87. Thorpe (2002) also first noted frequent evidence of these dynamics in tropospheric clouds. Subsequent observations in the atmosphere and oceans reviewed by F23 reveal these dynamics to be widespread, perhaps even ubiquitous, in multiscale GW environments. Of these, OH airglow and PMC imaging at high altitudes were the first to confirm T&K dynamics accompanying KH billows (H21; Kjellstrand et al. 2022).

Initial modeling in wide domains confirmed KHI dynamics to enable new and accelerated T&K pathways to turbulence and enhanced wide-domain enhanced $\overline{\epsilon }$ maxima. F21 specified misaligned initial billows that linked to form multiple knots. F22a,b,c employed a weak initial noise seed yielding vortex tubes linking adjacent KH billows where they exhibit varying phases along their axes. A second noise seed was added in the latter studies prior to turbulence transitions to approximate the influences of pre-existing small-scale turbulence.

KHI dynamics described here were specified to enable intercomparisons of the various T&K dynamics and evolutions and their contributions to enhanced $\overline{\epsilon }$ relative to KHI evolutions in their absence for Re varying from 5000 to 500 employing the same initial conditions. The T&K dynamics include both 1) vortex knots arising due to misaligned billow cores addressed by F21 and 2) larger and smaller vortex tubes linking adjacent KH billows addressed by F22a,b,c. An initial Ri = 0.1 enabled a maximum Re = 5000 sufficient to ensure strong secondary CI in the billow cores but no secondary KHI in the vortex sheets between billows for the specified noise seed in the absence of T&K dynamics. The same initial conditions were employed for Re of 3000, 2000, 1000, and 500 to evaluate their impacts on T&K dynamics, ε(k) spectra, and peak and temporally varying $\overline{\epsilon }$.

Results for Re = 5000 reveal that the T&K dynamics driving the earliest transitions to the most intense vortex knots are those where 2 KH billows link to 1, or 3 KH billows link to 2, either directly or via large-scale vortex tubes. Regions exhibiting multiple, smaller-scale vortex tubes emerging on vortex sheets between KH billows having large |dϕ/dy| initiate secondary T&K dynamics that approach comparable local ε. Additional turbulence transitions accompany a local billow pairing event and interactions between secondary KHI and CI where they are advected into close proximity over and under the KH billows. However, these latter transitions are much less significant than the earlier, larger-scale, T&K dynamics.

These same sites drive the more rapid and intense turbulence transitions for Re = 3000 and 2000 but at successively later times due to the slower evolutions of the links among adjacent billows. Turbulence transitions for Re = 1000 and 500 are weaker and delayed due to an absence of misaligned billows and initial links via smaller-scale vortex tubes where they arise. Those for Re = 1000 occur in similar, but more confined, regions along y, than for larger Re. Those for Re = 500 exhibit no clear correlations with the transitions at larger Re, weak and widespread transitions, and slow progressions strongly constrained by viscosity. Also seen is a systematic decrease in the contributions of secondary CI to the turbulence transitions in the presence of T&K dynamics with decreasing Re relative to their occurrence in narrow domains without T&K dynamics. This suggests that secondary CI is suppressed by impacts on KH billow intensities by the increasing billow distortions due to T&K dynamics at later times.

As expected, the larger Re = 5000, 3000, and 2000 yield earlier, wide-domain $\overline{\epsilon }$ peaks, and these arise at successively later times because the initial vortex knots (sites “c” in Figs. 2 and 10) change character somewhat and evolve more slowly for decreasing Re. Surprisingly, the largest peak $\overline{\epsilon }$ occurs for Re = 1000. This can be traced to delayed turbulence transitions initiated by small-scale vortex tubes as the KH billows for Re = 1000 approach their largest amplitudes. The delayed transition enables peak KH billow velocity variances in wavenumbers 2 and 3 at ∼8.5T_b to be significantly larger than occur for larger Re. The result is a higher, but narrower, $\overline{\epsilon }$ peak and a cumulative $\overline{\epsilon }$ only ∼4%–8% smaller than for the larger Re.

Peak $\overline{\epsilon }$ for Re = 500 is ∼3 times smaller and ∼5–7T_b later because the larger viscosity causes significant dissipation prior to instabilities driving turbulence. Despite these differing dynamics and time scales, all the wide-domain turbulence transitions can be attributed to links between initially misaligned KH billows or via larger and smaller vortex tubes at decreasing Re. T&K dynamics cumulative $\overline{\epsilon }$ decrease for decreasing Re. We expect enhanced turbulence intensities to yield enhanced mixing, as inferred by M23 for a KHI T&K simulation addressing mixing in the thermosphere, but that assessment will be reported elsewhere.

Temporally varying $\overline{\epsilon }$ for the narrow-domain cases exhibits less variability for Re = 1000–5000 because they all initiate turbulence via secondary CI having roughly similar spanwise scales. Their cumulative $\overline{\epsilon }$ decreases relative to the large y domains for decreasing Re. If these results are broadly representative of KHI events in geophysical flows, they suggest a potential for augmented mixing and transport for a significant range of geophysical parameters accompanying KHI T&K dynamics. The exception to this trend is Re = 500, which exhibits delayed temporal $\overline{\epsilon }$ evolutions and cumulative $\overline{\epsilon }$ peaks for both the wide and narrow domains. In this case, small-scale vortex dynamics arise at comparable times at which laminar dynamics already contributed significantly to $\overline{\epsilon }$ in both domains.

We noted at the beginning of this section that the temporal evolution of $\overline{\epsilon }$ described here differs from that discussed by F22b in several ways. For comparison, that in F22b is shown with red dots at 0.5 T_b intervals at the left in Fig. 7. The second noise seed in F22b is seen to significantly accelerate turbulence transitions where initial interactions between vortex tubes and billow cores arise. These drive an initial maximum $\overline{\epsilon }\sim 0.0285{\mathrm{m}}^2{\mathrm{s}}^{-3}$ at ∼4.5T_b in the layer of depth 0.4L centered at z = 0 as these dynamics expand throughout the billow cores spanning these sites. A higher maximum $\overline{\epsilon }\sim 0.038{\mathrm{m}}^2{\mathrm{s}}^{-3}$ occurs at ∼7T_b in the F22b case as the turbulence transitions expand throughout the billows at all locations and $\overline{\epsilon }$ remains higher than seen in the Re = 5000 case described here until late times. This results in a cumulative $\overline{\epsilon }$ that is ∼2.5% larger in the F22b case at 20T_b. By comparison, the present case exhibiting knots driven by misaligned initial billows achieves a peak $\overline{\epsilon }\sim 0.051{\mathrm{m}}^2{\mathrm{s}}^{-3}$ that is ∼35% larger, but having a full width half maximum ∼35% smaller. Implications for mixing in these very different evolutions have not yet been performed.

Though not a focus of this study, our results appear to address predictions of secondary KHI and stagnation point instability (SPI) by Mashayek and Peltier (2012a,b, 2013), whose chosen Pr, Re, and Ri are the same or similar to several of our cases here and those of F21 and F22a. The most relevant cases are the narrow-domain results without T&K dynamics discussed above. However, these exhibit neither secondary KHI on the vortex sheets nor SPI for the specified profiles, parameters, and noise seed. Our wide-domain results do exhibit secondary KHI on the vortex sheets, but these are attributable to sheet deformation and intensification adjacent to regions of KH billow curvature. Secondary KHIs also arise near the stagnation points and elsewhere on the vortex sheets in both the F21 and F22a simulations, for which the initial noise seeds were significantly larger than employed in the wide-domain cases described here. In these cases, all secondary KHIs that arose on the vortex sheets between KH billows that were mis-aligned or had varying phases along y can also be attributed to T&K dynamics. Secondary CIs also arose but were significantly suppressed by billow curvature and interactions with other instability features, especially secondary KHI, but they did not compete with the stronger T&K dynamics. These results do not demonstrate that the potential secondary KHI and SPI identified by Mashayek and Peltier (2012a,b, 2013) cannot occur. However, they did not arise for the environments and initial conditions posed here.

8. Conclusions

We cannot draw broad conclusions regarding KHI T&K implications for mixing and transport based on sensitivity to Re for one Ri and a single specification of initial conditions, nor can we assert such influences without assessing them directly, as done by M23. We can, however, draw confident conclusions with respect to the parameters explored here and infer likely extensions to smaller Ri and larger Re for which there have been many previous analytic and numerical KHI studies in idealized, small domains (see, e.g., Mashayek and Peltier (2012a,b) and many previous studies).

Our results reveal that KHI T&K dynamics and enhanced $\overline{\epsilon }$ can span a decade of Re for Ri that would enable only secondary CI in their absence without strong initial, and/or subsequent, noise seeds (i.e., F22a). If such results are found to occur for wide ranges of Re, Ri, and Pr, multiscale GW environments, and KHI T&K forms, as implied by the observations reported by F23, and the multiscale modeling described by FW23, they will likely have important implications for parameterizations of mixing and transport in the atmosphere and the oceans, given the evidence for their occurrence described by F23. Several of the results described here, and implications for likely T&K dynamics for smaller Ri and larger Re, are worth noting:

1. Our finding that KHI T&K dynamics enhance $\overline{\epsilon }$ at constrained Re and Ri will likely also occur for smaller Ri and larger Re, suggesting widespread $\overline{\epsilon }$ enhancements, given the range of KHI scales, environments, and Re described by F23 and FW23.

2. High-resolution atmospheric profiling reveals background turbulence to be nearly always present extending from the stable boundary layer into the MLT (Lübken et al. 2002; Strelnikov et al. 2003; Rapp et al. 2004; Balsley et al. 2006; Schneider et al. 2015); a turbulent viscosity elevated by ∼1–3 decades implies a lower effective Re; however, stronger noise seeds likely favor T&K dynamics and induce competitive, potentially faster, transitions to turbulence at smaller scales (F22a).

3. Results shown here, and evidence of significant GW influences on KHI scales and their T&K dynamics demonstrated by F23 and FW23, suggest that KHI T&K dynamics arise easily as Ri decreases in intensifying multiscale GW environments; such environments are virtually always present, often emerge with decreasing Ri over short intervals, but are neither resolved nor described adequately in regional or global models.

4. Additional instabilities of thin sheared and stratified layers, such as Holmboe and Taylor–Caulfield instabilities that arise for differing shear and stratification profiles, will likely also exhibit T&K dynamics due to environmental modulations of such layers by superpositions of various waves on multiple spatial and temporal scales.

5. Expanded explorations of the dependence of T&K dynamics on varying multiscale, and time-dependent, environments, Re, Ri, Pr, and noise seeds are required to assess their broader implications for potentially enhanced turbulence and mixing, and the parameterization of these dynamics, in geophysical fluids.

Finally, our results are specifically relevant to the “zoo” of KHI secondary instabilities described by Mashayek and Peltier (2012a,b) that arise for idealized KHI in narrow domains. Importantly, those evaluations did not enable secondary instability dynamics arising due to misaligned or varying KH billows along their axes. The apparent widespread occurrence of KHI T&K dynamics in the atmosphere and oceans described by F23 and seen to arise serendipitously in the multiscale GW DNS described by FW23 suggests that multiple new species need to be added to the zoo. Most significant are three large pachyderms: specifically vortex knots driven by 1) misaligned initial KH billows, 2) large-scale vortex tubes linking KH billows exhibiting large, local |dϕ/dy|, and 3) coherent “crankshaft” instabilities arising along adjacent billow cores that have major influences on smaller-scale dynamics and turbulence transitions that arise in their absence, at least at smaller Re.

Smaller, but significant, additional species include 1) multiple, smaller-scale tubes linking adjacent billows in regions of smaller |dϕ/dy| that also drive intense vortex knots at slightly later times, 2) local KH billow pairing that leads to somewhat weaker knots, and 3) secondary KHI T&K dynamics that arise on the vortex sheets between adjacent KH billows and intensify strongly prior to their entrainment around the respective billows. Last, KHI T&K dynamics appear to suppress the importance of secondary CI and elevate the importance of secondary KHI in such environments extending to significantly smaller Re. Enhanced secondary KHI may then contribute to the suppression of primary KH billow pairing, as suggested by Mashayek and Peltier (2013) and seen to play a minor role in the KHI T&K results described above.

Data availability statement.

Our DNS modeling data reside on the DoD High Performance Computing Modernization Program systems, where all analyses and imaging were performed. These data comprise huge volumes; a single field at the peak resolution at one time is >500 GB in single precision, binary form, and these DoD systems are not open access. Hence, these data are not publicly available. The authors can provide limited such data for specific research interests.