Multiple instability of layered stratified plane Couette flow

Abstract

We present the linear stability properties and nonlinear evolution of two-dimensional plane Couette flow for a statically stable Boussinesq three-layer fluid of total depth$2h$between two horizontal plates driven at constant velocity$\pm \unicode[STIX]{x0394}U$. Initially the three layers have equal depth$2h/3$and densities$\unicode[STIX]{x1D70C}_{0}+\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$,$\unicode[STIX]{x1D70C}_{0}$and$\unicode[STIX]{x1D70C}_{0}-\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$, such that$\unicode[STIX]{x1D70C}_{0}\gg \unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$. At finite Reynolds and Prandtl number, we demonstrate that this flow is susceptible to two distinct primary linear instabilities for sufficiently large bulk Richardson number$Ri_{B}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}h/(\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x0394}U^{2})$. For a given bulk Richardson number$Ri_{B}$, the zero phase speed ‘Taylor’ instability is always predicted to have the largest growth rate and to be an inherently two-dimensional instability. An inherently viscous instability, reminiscent of the ‘Holmboe’ instability, is also predicted to have a non-zero growth rate. For flows with Prandtl number$Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}=1$, where$\unicode[STIX]{x1D708}$is the kinematic viscosity, and$\unicode[STIX]{x1D705}$is the diffusivity of the density distribution, we find that the most unstable Taylor instability, maximized across wavenumber and$Ri_{B}$, has a (linear) growth rate which is a non-monotonic function of Reynolds number$Re=\unicode[STIX]{x0394}Uh/\unicode[STIX]{x1D708}$, with a global maximum at$Re=700$over 50 % larger than the growth rate as$Re\rightarrow \infty$. In a fully nonlinear evolution of the flows with$Re=700$and$Pr=1$, the two interfaces between the three density layers diffuse more rapidly than the underlying instabilities can grow from small amplitude. Therefore, we investigate numerically the nonlinear evolution of the flow at$Re=600$and$Pr=300$, and at$Re=5000$and$Pr=70$in two-dimensional domains with streamwise extent equal to two wavelengths of the Taylor instability with the largest growth rate. At both sets of parameter values, the primary Taylor instability undergoes a period of identifiable exponential ‘linear’ growth. However, we demonstrate that, unlike the so-called ‘Kelvin–Helmholtz’ instability that it superficially resembles, the Taylor instability’s finite-amplitude state of an elliptical vortex in the middle layer appears not to saturate into a quasiequilibrium state, but is rapidly destroyed by the background shear. The decay process reveals$Re$-dependent secondary processes. For the$Re=600$simulation, this decay allows the development to finite amplitude of the co-existing primary ‘viscous Holmboe wave instability’, which has a substantially smaller linear growth rate. For the$Re=5000$simulation, the Taylor instability decay induces a non-trivial modification of the mean velocity and density distributions, which nonlinearly develops into more classical finite-amplitude Holmboe waves. In both cases, the saturated nonlinear Holmboe waves are robust and long-lived in two-dimensional flow.

Diapycnal mixing in layered stratified plane Couette flow quantified in a tracer-based coordinate

“…In order to examine the linear stability of these mean profiles, the viscous, diffusive and stratified eigenvalue problem, e.g. as described in equations (3.6)-(3.7) of Eaves & Caulfield (2017), is solved numerically using the procedure described in Smyth et al (2011). shown in figure 5(c), the mean profiles analysed are all unstable to instabilities which can be identified as being of Holmboe-type.…”

Instability of sheared density interfaces

“…Eventually, the secondary billows drift sideways to become consumed by the larger primary billow. In the Pr = 10 simulations, however, a large amplitude, long wavelength, parasitic nonlinear Holmboe wave develops on top of the array of billows, much as in the earlier computations of Balmforth et al (2012) and Eaves & Caulfield (2017). This is illustrated in figure 6(d); the parasitic Holmboe wave is evidenced by the filaments of baroclinic vorticity drawn out from the main billow towards the right-hand side of the domain.…”

“…In addition, the higher Reynolds number of these simulations permitted the observation of secondary instabilities at late times that took the form of parasitic nonlinear Holmboe waves. Such secondary features were confirmed at finite-wavelength by Eaves & Caulfield (2017). However, the saturated states of the primary TCI modes were quite different in the higher bulk Richardson number simulations of that study, resembling propagating, cuspy nonlinear waves rather than billows.…”

“…We further fix the shear stress rather than the horizontal velocity at y = ±1 to minimise the effect of the confining walls and avoid viscous boundary layer instabilities (cf. Eaves & Caulfield 2017). In this situation, we take the initial horizontal velocity and density deviation to be u = ±1 and ρ = ∓1 at y = ±1 (so the channel walls are initially moving at a dimensional speed of 2U with respect to one another and 2∆ρ is the initial density difference).…”

“…This renders large three-dimensional simulations extremely expensive in terms of grid resolution and time-stepping. Moreover, Eaves & Caulfield (2017) observed a strong dependence of the secondary instabilities on the presence of confining walls, demanding that simulation domains be relatively deep. For these reasons, we conduct two-dimensional simulations to flesh out the nonlinear dynamics of TCI over a wide range of parameter settings, rather than targeting a relatively small number of three-dimensional cases.…”