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Finite amplitude Kelvin-Helmholtz billows at high Richardson number J P Parker, C P Caulfield, R R Kerswell November 19, 2018 Stratified shear flow 4 u t + u u = p + Ri b b e z + 1 Re 2 u 2 b t + u b =

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Finite amplitude Kelvin-Helmholtz billows at high Richardson number

J P Parker, C P Caulfield, R R Kerswell

November 19, 2018

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Stratified shear flow

∂u ∂t + u · ∇u = −∇p + Ribbez + 1 Re ∇2u ∂b ∂t + u · ∇b = 1 Re ∇2b ∇ · u = 0 (Pr = 1)

0.5 1.0

2 4

u b

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Richardson Number

The Richardson number Ri is the non-dimensional ratio of buoyancy to shear. It is important to distinguish the gradient (local) Richardson number Rig = ∂b/∂z (∂u/∂z)2 Rib from the bulk Richardson number Rib, a parameter.

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Miles-Howard Theorem

For a steady, one-dimensional, Boussinesq, inviscid, stratified shear flow, linear stability is guaranteed if Rig > 1/4 everywhere. “Sufficiently strong stratification enforces stability.” Ri = 1/4 is often seen as a magic number in oceanography, and is used in parameterisations, despite the limited scope of the theorem.

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Forced equations

∂u ∂t + u · ∇u = −∇p + Ribbez + 1 Re ∇2u + 2 Re tanh (z) sech2 (z)ex ∂b ∂t + u · ∇b = 1 Re ∇2b + 2 Re tanh (z) sech2 (z) ∇ · u = 0 Steady solution u = ex tanh z b = tanh z

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State tracking

Formally define F: (u(T), b(T)) = F(u(0), b(0), T) Look for steady states F(u, b, T) = (u, b), ∀T. For simplicity, let X = (u, b). Pick arbitrary T, find zeros of G(X) ≡ F(X) − X

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Newton-GMRES

To solve G(X) = 0 use Newton iteration GX(Xn) · (Xn+1 − Xn) = −G(Xn). To solve linear system at each step using GMRES, need only know GX(X) · Y ≈ G(X + ǫY ) − G(X) ǫ

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Results (Re = 1000)

X Rib

stable

ne unstable direction

two unstable directions

Stability analysis performed with Arnoldi iteration.

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Results (Re = 1000)

?

X Rib

stable

ne unstable direction

two unstable directions

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Bifurcation point tracking

At pitchfork/saddle-node bifurcation, attempt to solve F(X, Rib) = X FX(X, Rib) · Y = Y Y · A = 1 for X, Y and Rib, where A is some fixed direction. Similar for Hopf bifurcation, with 3 time integrations.

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Results

Rib 1/Re

1/500 1/1000 1/2000 1/5000

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Conclusions

◮ Non-trivial steady states and complex behaviour are possible at

Ri > 1/4.

◮ States only just go past 1/4.

X Rib

stable

ne unstable direction

two unstable directions

jpp39@cam.ac.uk

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Open questions

◮ How relevant is forced system? ◮ What are the effects of Pr?

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