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Abstract
We present the linear stability properties and nonlinear evolution of twodimensional plane Couette flow for a statically stable Boussinesq threelayer 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 twodimensional instability. An inherently viscous instability, reminiscent of the ‘Holmboe’ instability, is also predicted to have a
nonzero 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 nonmonotonic 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 twodimensional 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 socalled ‘Kelvin–Helmholtz’ instability that it superficially resembles, the Taylor instability’s finiteamplitude 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 Redependent secondary processes. For the Re = 600 simulation, this decay allows the development to finite amplitude of the coexisting primary
‘viscous Holmboe wave instability’, which has a substantially smaller linear growth
rate. For the Re = 5000 simulation, the Taylor instability decay induces a nontrivial modification of the mean velocity and density distributions, which nonlinearly
develops into more classical finiteamplitude Holmboe waves. In both cases, the saturated nonlinear Holmboe waves are robust and longlived in twodimensional flow.
Original language  English 

Pages (fromto)  250278 
Number of pages  29 
Journal  Journal of Fluid Mechanics 
Volume  813 
Early online date  17 Jan 2017 
DOIs  
Publication status  Published  25 Feb 2017 
Keywords
 Instability
 Nonlinear instability
 Geophysical and Geological Flows
 Stratified flows
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Eaves, T. (Member)
3 Mar 2021Activity: Other activity types › Public engagement and outreach  public lecture/debate/seminar