Kelvin-Helmholtz instability above Richardson number 1 / 4 J P Parker, C P Caulfield, R R Kerswell September 4, 2019
Miles-Howard theorem The local/gradient Richardson number is defined as ∂ρ/∂ z Ri g = g ( ∂ u /∂ z ) 2 . ρ Theorem (Miles-Howard) For a steady, one-dimensional, Boussinesq, inviscid, stratified shear flow, linear stability is guaranteed if Ri g > 1 / 4 everywhere. 2 of 15
Mixing layer models ∂ t u + ( U + u ) ∂ x u + w ∂ z ( U + u ) = − ∂ x p + 1 ∂ 2 x u + ∂ 2 � � z u , Re ∂ t w + ( U + u ) ∂ x w + w ∂ z w = − ∂ z p + 1 ∂ 2 x w + ∂ 2 � � z w + Ri b b , Re 1 ∂ 2 x b + ∂ 2 � � ∂ t b + ( U + u ) ∂ x b + w ∂ z ( B + b ) = z b , PrRe ∂ x u + ∂ z w = 0 . U = tanh z , B = tanh z U = tanh z , B = z 3 of 15
Bifurcation diagram? Re = ∞ Ri 1/4 4 of 15
Bifurcation diagram? Re = ∞ : supercritical Amplitude Ri 1/4 5 of 15
Bifurcation diagram? Re = ∞ : subcritical Amplitude Ri 1/4 6 of 15
Bifurcation diagram? Re = ∞ : subcritical Amplitude Ri 1/4 7 of 15
State tracking Formally define F : ( u ( T ) , b ( T )) = F ( u (0) , b (0)) Look for steady states F ( u , b ) = ( u , b ) Solve this using Newton iteration with GMRES, all built on top of a DNS code. 8 of 15
Holmboe model Re = 4000 0.8 0.6 � X � 0.4 0.2 0 0.244 0.246 0.248 0.25 Ri b 9 of 15
Drazin model Re = 4000 0.8 0.6 � X � 0.4 0.2 0 0.23 0.24 0.25 Ri b 10 of 15
Bifurcation point tracking At pitchfork/saddle-node bifurcation, attempt to solve F ( u , b ) = ( u , b ) J ( u , b ) · Y = Y Y · A = 1 for X , Y and Ri b , where A is some fixed direction. Similar for Hopf bifurcation, with 3 time integrations. 11 of 15
Bifurcation tracking 0.25 Ri b 0.248 0.246 0 0.0005 0.001 1 / Re 12 of 15
Pr > 1 0.31 0.3 0.29 0.28 Ri b 0.27 0.26 0.25 0.24 1 1.5 2 2.5 3 3.5 Pr 13 of 15
Pr > 1 0.8 0.6 � X � 0.4 0.2 0 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 Ri b 14 of 15
Conclusions ◮ Supercriticality for Pr > 1 ◮ Pr = 1 is a degenerate case, with very small subcriticality ◮ Finite amplitude instability for Pr = 3 is possible 15 of 15
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