Multiple instability of layered stratified plane Couette flow

Tom Eaves, C. P. Caulfield

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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 ±ΔU. Initially the three layers have equal depth 2h/3 and densities ρ0 + Δρ, ρ0 and ρ0 − Δρ, such that ρ0 » Δρ. 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 RiB = gΔρh/(ρ0ΔU2). For a given bulk Richardson number RiB, 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 = ν/κ = 1, where ν is the kinematic viscosity, and κ is the diffusivity of the density distribution, we find
that the most unstable Taylor instability, maximized across wavenumber and RiB, has a (linear) growth rate which is a non-monotonic function of Reynolds number
Re = ΔUh/ν, with a global maximum at Re = 700 over 50 % larger than the growth rate as Re → ∞. 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.
Original languageEnglish
Pages (from-to)250-278
Number of pages29
JournalJournal of Fluid Mechanics
Early online date17 Jan 2017
Publication statusPublished - 25 Feb 2017


  • Instability
  • Nonlinear instability
  • Geophysical and Geological Flows
  • Stratified flows


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