ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence September 23, 2016, Axel Brandenburg

Handout 9: Stratified shear Flows

• Stabilizing buoyancy force due to shear. Relevant in meteorology.

• Taylor, G. I. (1931) and Goldstein (1931) published first work on this in the same issue of Proc. Roy. Soc.

1 Goldstein-Taylor equation

We now swap directions and consider a shear flow of the form U_{x}(z). In Rayleigh’s instability equation, this only leads to a change of ∂/∂x → ∂/∂z.

Rayleigh’s instability equation,

(U − c) (∂_{z}^{2} − k^{2}) ψ̂ − U′′ψ̂ = 0 (no gravity yet),  (1)

where primes denote now z derivatives and k denotes the x component of the wavevector; see KCD for details. Including gravity in the curled momentum equation for the y component of the vorticity ω_{y} = −∇^{2}ψ with stream function ψ and u = ∇ × (ψŷ) the term

ŷ · ∇ × (−g ρẑ) = g ∂ρ/∂x.  (2)

Changes in ρ are caused by changes in the specific entropy as fluid moves up and down. Those changes are governed by

\frac{DS}{Dt} = 0,  (3)

Linearizing this eqation (S = S_{0}(z) + S_{1}(x, z, t)) about a stably stratified entropy gradient

\frac{dS_{0}}{dz} = N^{2}c_{p}/g > 0  (4)

yields

\frac{∂S_{1}}{∂t} + U_{x} \frac{∂S_{1}}{∂x} = −N^{2}c_{p}/gu_{z},  (5)

where u_{z} is expressed in terms of ψ as u_{z} = ∂ψ/∂x. Assuming all linearized fields to vary like S_{1} = Ŝ_{1}(z) e^{k(x−ct)}, this yields

ik(U − c) Ŝ_{1} = N^{2}(c_{p}/g)ikψ̂,  (6)

or

(U − c) Ŝ_{1} = N^{2}(c_{p}/g)ψ̂.  (7)

Assuming rapid pressure equilibration, we can replace entropy changes by negative density changes in Equation (2). Inserting gravity into yields

(U − c) (∂_{z}^{2} − k^{2}) ψ̂ − U′′ψ̂ + \frac{N^{2}}{U − c} ψ̂ = 0.  (8)

Upon similar manipulations as before (details next time), one can show that

c_{i} ∫ \frac{N^{2} − \frac{1}{4}(U′)^{2}}{|U − c|^{2}} |ψ|^{2} = −c_{i} ∫ (|ψ′| + k^{2}|ψ|^{2}) dx  (9)

with c = c_{r} + ic_{i}. This shows that the flow cannot be unstable if N^{2} − \frac{1}{4}(U′)^{2} > 0 everywhere in the domain. In terms of the gradient Richardson number Ri(z) = N^{2}/(U′)^{2}, this means Ri > 1/4 for stability.

In Figure 1, we show numerical solutions to the approximate eigenvalue problem

[U (∂_{z}^{2} − k^{2} + N^{2}/U^{2}) ψ̂ − U′′] ψ̂ = c (∂_{z}^{2} − k^{2} − N^{2}/U^{2}) ψ̂.  (10)

Figure 1: Eigenvalues to the approximate Taylor–Goldstein equations for |c/U| ≪ 1 for U = tanh x

in −5 < x < 5; see http://lcd-www.colorado.edu/~axbr9098/teach/ASTR_5410/lectures/9_
Richardson_crit/idl/.

References

Goldstein, S., “On the stability of superposed streams of fluids of different densities,” Proc. Roy. Soc. Lond. 132, 524-548 (1931).

Taylor, G. I., “Effect of variation in density on the stability of superposed streams of fluid,” Proc. Roy. Soc. Lond. 132, 499-523 (1931).

$Header: /var/cvs/brandenb/tex/teach/ASTR_5410/9_Richardson_crit/notes.tex,v 1.4 2016/09/23 16:50:51 brandenb Exp $