## 1. Introduction

Shear-induced destabilization of laminar currents and the subsequent generation of small-scale turbulence represents one of the major drivers of diapycnal mixing in the ocean interior (e.g., Smyth and Moum 2012). Motivated by numerous geophysical applications, the problem of instability of vertically sheared parallel flows has long evolved into a broad and active research area of fluid dynamics. Comprehensive reviews of the field can be found in, for example, Peltier and Caulfield (2003), Thorpe (2005), and Ivey et al. (2008). The following discussion is therefore focused on a small subset of results that are directly relevant for the present investigation.

_{cr}= 1/4 (Richardson 1920; Miles 1961; Howard 1961). In the subcritical regime (Ri < 1/4), nonlinear evolution of growing modes is typically characterized by a series of secondary instabilities, which ultimately generate fully turbulent billows—a scenario supported by observational (Woods 1968; van Haren et al. 2014), laboratory (Thorpe 1971; Atsavapranee and Gharib 1997), and numerical (Caulfield and Peltier 2000; Smyth et al. 2001) studies.

The inclusion of dissipative effects can substantially modify the conditions for—and the evolutionary patterns of—KHI. For instance, the critical Richardson number required for instability is sensitive to the assumed diffusivity and viscosity values. Somewhat counterintuitively, both viscous and diffusive effects could be either stabilizing or destabilizing, depending on a particular system (e.g., Balmforth and Young 2002; Thorpe et al. 2013). A recent numerical study of the KHI evolution in the field of preexisting turbulence revealed a tendency for suppression of the flow roll-up relative to the patterns realized for corresponding laminar initial states (Kaminski and Smyth 2019). Another interesting effect is realized in two-component fluids with unequal diffusivities of density components, where weak, dynamically stable shears can destabilize the system by triggering so-called thermohaline-shear instability (Radko 2016).

In the oceanographic context, the vertical shear required for KHI is usually attributed to internal waves. The associated shears are, strictly speaking, neither unidirectional nor steady. However, since the spectrum of internal waves is dominated by relatively slow near-inertial waves (e.g., Garrett and Munk 1972), the steady-state approximation appears to offer a reasonable starting point for the analysis of shear-driven instabilities and it has been frequently used in the past (e.g., Dunkerton 1997). Nevertheless, the intrinsic time-dependence of internal waves has some major consequences for their stability. A fundamentally different class of instabilities arises due to the possibility of resonant triad interactions between the primary internal wave and two low-amplitude harmonics. Several analyses of parametric subharmonic instability—usually the most rapidly amplifying type of instabilities induced by triad interactions—have led to a conclusion that propagating internal waves can be unstable regardless of its amplitude (Mied 1976; Drazin 1977; Klostermeyer 1982; Lombard and Riley 1996). This result appears to be robust; it is also supported by our analysis of time-dependent shear instabilities (TDSI) in the Couette model. However, it should be kept in mind that the instability of a flow field does not always guarantee the transition to turbulence. For instance, the laboratory experiments of Troy and Koseff (2005) indicate that the threshold Richardson numbers for the formation of visible billows in oscillatory shears are actually significantly lower than the canonical value of Ri_{cr} = 1/4.

The objective of the present study of time-dependent shear flows is threefold. First, we embark on a rather complete exploration of the parameter space in order to examine stabilizing/destabilizing effects of planetary rotation, temporal variability pattern of the background flow, and dissipation of buoyancy and momentum. Another defining feature of the present analysis is the chosen framework, which is based on the unbounded Couette flow. This configuration makes it possible to delineate the destabilization caused by uniform shear from (i) effects of resonant triad interactions between harmonic waves and (ii) instabilities associated with the presence of inflection points in vertical velocity profiles. In this regard, the present investigation can be related to the stability analyses of inertial currents by Winters (2008) and of oscillatory flows characterized by solid-body rotation in the vertical plane by Majda and Shefter (1998), although our methodology is substantially different. Finally, an attempt is made to identify parameter regimes where the growth rates are substantial and amplifying perturbations are thus likely to strongly influence the evolution of the flow field. An explicit scaling law is proposed, which captures the dependencies of growth rates on the characteristics of the basic flow.

The manuscript is organized as follows. The model configuration and the technique used in linear stability analyses are described in section 2. Section 3 explores properties of TDSI in a simplified two-dimensional framework. The physical interpretation of TDSI is offered in section 4, where the growth of unstable perturbations is linked to the strain-induced modulation of vertical velocity (the Orr mechanism). Section 5 presents the analysis of a more complicated and realistic three-dimensional system under the influence of planetary rotation. The results are summarized and conclusions are drawn in section 6.

## 2. Formulation

*g*is gravity,

*ν*is the molecular viscosity, and

*k*

_{ρ}is the molecular diffusivity of density. Note that here we make no attempt to represent effects of unequal diffusivities of heat and salt—two major components of seawater density—thereby intentionally excluding from our analysis a set of processes collectively referred to as double-diffusive convection (Stern 1960; Radko 2013).

*d*

^{2}/

*k*

_{ρ},

*k*

_{ρ}/

*d*,

*ν*/

*k*

_{ρ}is the Prandtl number. The pressure field can be inferred diagnostically from the divergence of the momentum equations, and therefore (6) effectively represents the closed set of prognostic equations for

*d*= 0.01 m. The key nondimensional numbers are evaluated accordingly:

*z*dependence of its coefficients (

*A*

_{V}

*z*,

*A*

_{U}

*z*), and the problem is treated using techniques utilized in earlier studies of the unbounded Couette model (Knobloch 1984; Shepherd 1985; Radko 2019). The perturbation fields are represented by a superposition of plane-wave components

*m*in the uniform shear model (e.g., Shepherd 1985) is also a function of time:

*k*,

*l*,

*m*

_{0}), time

*t*, instantaneous values of shear (

*A*

_{U},

*A*

_{V}), their temporal integrals (

*B*

_{U},

*B*

_{V}), and the Prandtl number (Pr).

## 3. Two-dimensional model

*x*,

*z*) version of the model, which is obtained by assuming

*l*= 0,

*f*= 0, and

*A*

_{V}= 0. In this regime, the governing equations can be simplified by introducing streamfunction

*ψ*, such that (

*u*,

*w*) = (−∂

*ψ*/∂

*z*, ∂

*ψ*/∂

*x*). In the vorticity–streamfunction form, system (6) reduces to

*a*

_{U}in (14) as follows:

*k*,

*m*

_{0},

*t*,

*A*

_{U},

*B*

_{U}, Pr). Stability properties of the ODE system in (18) are determined by integrating it in time using random complex initial conditions for

*λ*

_{r}is evaluated by introducing the quadratic norm that is based on the net perturbation energy, which reduces, in our nondimensional units, to

*e*) is used to determine

*λ*

_{r}for each (

*k*,

*m*

_{0}).

*π*, which matches the periodicity of the basic shear 2

*π*/

*ω*. The calculations analogous to that in Fig. 1 were repeated for a wide range of wavenumbers and the overall maximal growth rate

*λ*

_{max}was determined by maximizing

*λ*

_{r}over the wavenumber space:

*k*,

*m*

_{0}) → (−

*k*, −

*m*

_{0}), the growth rate in (20) can be maximized over nonnegative values of

*m*

_{0}without loss of generality.

*ω*could be estimated by recalling that the frequency of internal waves lies in the interval

*ω*~

*f*). Thus, even though planetary rotation is not incorporated in our two-dimensional calculations, we assume that, for baseline parameters in (8), the

*ω*= 0.1 case represents the dominant (inertial) oscillations. It should also be kept in mind that the wave fronts of internal waves become progressively steeper with increasing frequency. Our model, on the other hand, is based entirely on the vertical shear of horizontal velocity and thus implicitly assumes that wave fronts are relatively flat. This implies that the proposed configuration offers a consistent representation of wave-induced shear for as long as

*ω*≪

*N*~ 3.16. The following analysis is therefore restricted to the range

*ω*< 1.

*λ*

_{r}is plotted as a function of (

*k*,

*m*

_{0}). The patterns realized in all configurations are qualitatively similar, taking the form of narrow rays emanating from the point (

*k*,

*m*

_{0}) = (0, 0). This pattern indicates that the relation

*λ*

_{r}(

*k*,

*m*

_{0}) can be more efficiently described in polar coordinates (

*κ*,

*θ*), which are defined as follows:

*κ*,

*θ*). The unstable regions take the form of vertical bands that are located in the vicinity of

*θ*≈

*π*/2 and thus represent amplifying harmonics with almost horizontal wave fronts. The range of unstable wavenumbers in Fig. 3 is limited to

*κ*<

*κ*

_{max}~ 0.1. In terms of dimensional wavelengths

*λ*

_{r}with decreasing wavenumber

*κ*for any given

*θ*. While no formal proof of this property is available at this time, the inspection of numerous cases (only a small fraction of which is presented here) indicates that it is a generic feature of the oscillatory Couette flow. This finding implies that the maximal growth rate

*λ*

_{max}can be obtained by considering the long wavelength limit (

*κ*→ 0), in which case (20) reduces to:

*κ*= 10

^{−6}was used in the following calculation) and maximizing

*λ*

_{r}over one-dimensional interval 0 <

*θ*<

*π*rather than over the entire two-dimensional space of (

*k*,

*m*

_{0}).

*ω*. The growth rate pattern is visibly different in the narrow region of relatively low Richardson numbers

*λ*

_{max}(

*ω*) relation for

*λ*

_{max}∝

*ω*, as apparent from Fig. 5a. This property can be rationalized by the following argument. The calculation in Fig. 4 is based on the limit

*κ*→ 0, which represents relatively large spatial scales. On such scales, the direct influence of molecular dissipation is expected to be negligible and the linear system (13) reduces to

*C*≪ 1, which represents an increase in the frequency of oscillations relative to some nominal configuration. As a result, (25) further reduces to

*λ*

_{max}is expected to be well described by the relation

*λ*

_{max}/

*ω*as a function of

*ω*, spanning the interval from 0.15 to 1. Remarkably, all 23 curves in Fig. 5b are visually indistinguishable from each other.

*λ*

_{norm}on

*c*

_{1},

*c*

_{2}) are obtained from the best fit of (29) to the pattern in Fig. 5b:

*λ*

_{S}< 0.1. This finding implies that, in terms of its ability to rapidly extract the energy from the basic shear, TDSI is somewhat less effective than KHI (

## 4. Physical interpretation

*ς*′ = ∇

^{2}

*ψ*′ is the vorticity perturbation, all dissipative processes are neglected, and

*D*/

*Dt*represents the linearized material derivative:

_{1}) and upward in others (e.g., point A

_{3}). The response of the system to this perturbation is apparent from the vorticity equation in (34). The tilt of the density interface [(∂

*ρ*′/∂

*x*) > 0] at point A

_{2}results in the clockwise torque [(

*Dς*′/

*Dt*) > 0], which reflects the tendency of buoyancy forces to flatten inclined density interfaces. This torque acts in the manner opposing the initial displacement, thereby providing the restoring force that eventually reverses the tilt of the density interface. This process then repeats over and over, which maintains the oscillatory motion of invariable magnitude.

*w*′ = ∂

*ψ*′/∂

*x*to

*ς*′ = ∇

^{2}

*ψ*′ as follows:

*w*′ and

*ς*′:

Thus, for any given amplitude of vorticity *m*^{2}, which happens when wave fronts become more upright. Now let us consider for a moment a configuration where this strain-induced increase in vertical velocity occurs in upwelling locations—such as the point A_{3} in the upper panel of Fig. 7. In this case, the perturbation will gain energy and its amplitude will increase. In order for this mechanism to produce persistent long-term growth, the periodicity of the buoyancy-driven oscillating mode should match the periodicity of strain-induced forcing. The maximum of *m*^{2} which can be achieved either once or twice per shear period, depending on relative values of *m*_{0} and *A*_{u}. For instance, in the case presented in Fig. 1, the dominant period of the perturbation is twice that of the background shear. Thus, this realization of TDSI can be placed into a broader class of parametric subharmonic instabilities of time-dependent flows (e.g., Drazin and Reid 1981).

Of course, the mechanism illustrated in Fig. 7 demands that the strain-induced upwelling occurs in regions where the motion of buoyancy-driven oscillatory mode is also upward for much of the oscillation period and vice versa. One can easily imagine configurations in which it is not the case. For instance, if these two components of vertical velocity are anticorrelated, then the buoyancy-driven oscillatory mode would decay rather than grow. The phase difference between the strain-induced velocity forcing and the perturbation pattern is ultimately set by the initial value of the vertical wavenumber (*m*_{0}) or—to be more precise—by the initial inclination of wave fronts, as measured by *m*_{0}/*k*. For some values of *m*_{0}, the strain-induced forcing is stabilizing but for others it is destabilizing. This sensitivity of the stability/instability of individual harmonics to *m*_{0} is reflected very clearly in the growth rate patterns *λ*_{r}(*k*, *m*_{0}), such as shown in Fig. 2. Nevertheless, it should be kept in mind that the overall instability of any system is ultimately controlled by its most unstable component. Therefore, the mechanism of TDSI is based fundamentally on the emergence of unstable harmonics in an oscillating shear for certain values of *m*_{0}.

## 5. Three-dimensional model

*x*and

*y*directions:

*a*

_{U},

*a*

_{V}) are determined by the mean Richardson number and the wave frequency as follows:

*e*), where the net perturbation energy

*e*in 3D takes the form

Taking planetary rotation into account makes the shear instability problem dynamically richer and more interesting. Figure 8 presents typical patterns the growth rate *λ*_{r} as a function of (*k*, *l*) for various values of *m*_{0} = 0.01. The three-dimensional visualization of a typical *λ*_{r}(*k*, *l*, *m*_{0}) relation is shown in Fig. 9, revealing a rather intricate growth rate pattern realized in the presence of planetary rotation.

*λ*

_{max}now requires maximization of

*λ*

_{r}over the three-dimensional wavenumber space (

*k*,

*l*,

*m*

_{0}). This difficulty, however, is alleviated by adopting the spherical coordinate system (

*κ*,

*θ*,

*γ*) defined as follows:

*λ*

_{r}monotonically increases with decreasing

*κ*, which implies that the maximal growth rate can be obtained using

*κ*= 10

^{−6}) and maximize

*λ*

_{r}over the two-dimensional interval 0 <

*θ*<

*π*, 0 <

*γ*< 2

*π*. The results are shown in Fig. 10, where we plot the growth rates

*λ*

_{max}as a function of frequency

*ω*for

*λ*

_{max}monotonically increases with increasing

*ω*. The growth rates realized at the larger Richardson number (

*ω*≥ 0.2), significant differences are apparent at near-inertial frequencies (

*ω*≈ 0.1). The latter observation is readily confirmed by inspecting the

*ω*= 0.1 and

*ω*= 0.2 in Fig. 11. For

*ω*= 0.2, the 2D and 3D patterns are generally similar, with 3D growth rates exceeding the 2D ones by 20%–30%. The inertial shear case, on the other hand, is marked by cardinal dissimilarities between the 2D and 3D solutions—the growth rates in 3D are now less, by an order of magnitude, than the corresponding 2D values.

The differences between 2D and 3D stability characteristics at low frequencies can be attributed to (i) the influence of planetary rotation on the temporal pattern of the background shear in (40) and/or (ii) the direct effects of the Coriolis parameter in the perturbative system (6). To glean some insight into the relative significance of these effects, a series of calculations in Fig. 11 were reproduced using a hybrid model, which retains the Coriolis parameter in the calculation of basic shear in (40) while neglecting planetary rotation in the perturbative system (6). These calculations, which are also included in Fig. 11, indicate that much of the destabilizing influence of planetary rotation can be ascribed to its direct effect on the perturbative system. This influence can be substantial and, somewhat unexpectedly, it can be either destabilizing (*ω* = 0.1 case) or stabilizing (*ω* = 0.2 calculation).

The significance of taking into account the Coriolis effect in stability analyses of (6) is revealed perhaps most clearly by the dissimilar growth patterns *λ*_{r}(*k*, *l*) realized in rotating and hybrid models, such as those shown in Fig. 12. The original model, which fully accounts for planetary rotation, produces the “tropical butterfly” pattern (Fig. 12a). The “alien spider” structure in Fig. 12b was obtained using the hybrid model that ignores the Coriolis effect in the perturbative system (6). Both tropical butterfly and alien spider patterns are remarkably intricate, visually striking, and rich in fine details. This complexity is surprising, given the minimal character of systems being considered in the present investigation. While the maximal growth rates in Figs. 12a and 12b differ by a mere ~30%, the Coriolis effect completely changes the symmetry characteristics of the growth rate patterns. The alien spider (Fig. 12b) is perfectly symmetric relative to the *k* axis, which is oriented in the propagation direction of the basic internal wave (the appendix). This symmetry is visibly violated in the tropical butterfly pattern, implying that the growth rate relation in Fig. 12a is no longer invariant with respect to *l* → −*l* transformation.

## 6. Discussion

This study presents the stability analysis of spatially uniform but time-dependent vertical shears in two and three dimensions. The overarching conclusion that permeates all aspects of our investigation is that of a profound influence of temporal variability of shear flows on their stability. The extensive exploration of the parameter space, which spans a wide range of frequencies and Richardson numbers, reveals that temporally varying shears are unconditionally unstable. In this sense, the steady basic state model, commonly used in stability analyses of parallel flows, can be viewed as a singular limit of corresponding time-dependent systems.

The growth rates realized in our model for moderately supercritical Richardson numbers

The proposed physical interpretation of TDSI is based on the strain-induced modulation of vertical velocity, commonly referred to as the Orr effect. The unstable TDSI modes can be described as plane internal waves that are resonantly forced by time-dependent basic shear flow. The Orr effect modifies their dynamics by providing the energy input into the perturbation field during each oscillation cycle. This amplification necessarily requires appropriate phase alignment between the strain-induced forcing and the perturbation pattern, which, in turn, is realized for proper initial orientations of the perturbation wave fronts.

In terms of underlying physics of TDSI, the present model should be contrasted with the stability analysis of free oscillatory flows characterized by solid-body rotation in the vertical plane (Majda and Shefter 1998, 2000). While the latter “rocking vorticity” basic states also exhibit instability for arbitrary large Richardson numbers, their dynamics are substantially different. In particular, the strain—the essential destabilizing ingredient of the time-dependent Couette model—plays no role in the instability of solutions analyzed by Majda and Shefter (1998). On the contrary, adding finite strain to the rocking vorticity patterns always makes them linearly stable. Of course, both time-dependent Couette and rocking vorticity models are interesting and bring complementary insights into shear flow instability. However, it should be kept in mind that oceanic shears are, by and large, associated with an active internal wave field. The spectrum of large-scale internal waves is dominated by slow modes with nearly flat wave fronts, predominantly horizontal particle displacements, and finite strain. Therefore, it is our belief that typical oceanic conditions may be better represented by the time-dependent Couette model.

The spatial scales of unstable TDSI modes are relatively large (

One of the distinguishing features of the present investigation is its framework, which assumes the basic states with spatially uniform shear. Aside from a series of analytical simplifications afforded by this configuration, it also helps us to clarify several aspects of the internal wave dynamics. In particular, the uniform gradient model excludes the possibility of destabilization through the resonant triad interaction, which is often invoked in stability analyses of internal waves (e.g., Mied 1976; Lombard and Riley 1996). The present work, therefore, draws attention to the existence of alternative routes to instability and dissipation. Likewise, the uniform shear model has led us to the conclusion that the presence of inflection points in vertical velocity profiles may not be essential for triggering instabilities of large-scale internal waves. This finding is particularly suggestive in view of qualitative dissimilarities in the stability properties of inflected and noninflected shears, which have been reported for various models of parallel flows (Drazin and Reid 1981; Rayleigh 1880). For instance, steady uniform unbounded shears are stable regardless of how low the Richardson numbers are (Knobloch 1984), whereas the inflected shears are unstable for Ri < 1/4 (e.g., Hazel 1972). Considering the general tendency of inflected shears to be less stable than their noninflected counterparts, it is plausible that the estimates in our study can serve as a lower bound for the growth rates realized for more irregular shear patterns, such as expected to occur in the ocean.

This project can be further developed in a number of promising directions. For instance, it could prove beneficial to extend our linear stability analyses beyond the monochromatic shear model. The temporal pattern of the basic shear can be represented by a superposition of Fourier components conforming to the Garrett–Munk spectrum with random initial phase distribution (e.g., Radko et al. 2015). Such an investigation would help to answer intriguing questions with regard to the role of various temporal harmonics in triggering TDSI. Our results consistently indicate that high-frequency modes are more effective in destabilizing shear flows than low-frequency harmonics. The spectrum of internal waves, on the other hand, is dominated by slow, near-inertial oscillations (Garrett and Munk 1972). Therefore, the relative significance of slow and fast spectral components in determining the stability properties of wave-induced shears is not clear a priori. The analysis of time-dependent inflected shears also represents a natural extension of the present investigation. Another potentially profitable avenue of exploration is modeling of the nonlinear evolution of TDSI and transition to turbulence. The assessment of the associated vertical fluxes of seawater properties is expected to improve our understanding of wave-induced mixing in the ocean.

The author thanks the editor Paola Cessi and the anonymous reviewers for helpful comments. Support of the National Science Foundation (Grant OCE 1756491) is gratefully acknowledged.

# APPENDIX

## Internal Wave Model

*w*is used to emphasize that the field variables in (A1) represent free large-scale internal waves rather than their small-scale instabilities. We also assume that the molecular dissipation is negligible on the scale of these waves and seek plane-wave solutions satisfying (A1) as follows:

*f*-plane model, we can assume that the wave is propagating in the positive

*x*direction without loss of generality, which accounts for the absence of

*y*dependence in the assumed form (A2). When (A2) is substituted into the linear system (A1) and the amplitudes of

*S*in terms of

*S*, in turn, is directly related to the mean Richardson number:

*x*,

*z*) are equivalent in terms of shear characteristics (except for different phases). Thus, a model of temporal variability of the wave-induced shear can be constructed, without loss of generality, by assuming (

*x*,

*z*) = (0, 0), which reduces (A7) to (39) and (40). The latter system is used in section 4 to represent the temporal variation in basic shear associated with large-scale internal waves.

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