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SUMMARY:S.M. Churilov (Institute of Solar-Terrestrial Physics\, Irkutsk)
DTSTART:20221124T110000Z
DTEND:20221124T120000Z
DTSTAMP:20260423T005819Z
UID:mmandim/47
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/mmandim/47/"
 >On the stability of sharply stratified shear flows with inflection-free v
 elocity profiles</a>\nby S.M. Churilov (Institute of Solar-Terrestrial Phy
 sics\, Irkutsk) as part of Mathematical models and integration methods\n\n
 \nAbstract\nWe study the linear stability of shear flows with sharp strati
 fication ($l \\ll L$\, where $l$ and $L = 1$ are vertical scales of densit
 y and velocity variation respectively) and a monotonic velocity profile $V
 _x = U(z)$ which has no inflection points and increases from $U = 0$ at th
 e bottom ($z = 0$) to $U = 1$ when $z \\to +\\infty$\, $U'(0) = 1$. We sho
 w that such a flow with step density variation ($l = 0$) and $U'' < 0$ has
  the instability domain of an universal form on the ($k$\, $J$) plane\, wh
 ere $k$ is the wave number and $J$ is the bulk Richardson number. Namely\,
  the\ndomain is bounded by abscissa axis ($J = 0$)\, dispersion curve $J =
  J(k\, c = 1)\,$ and\nthe segment of ordinate axis ($k = 0$) connecting th
 em. Here $c$ is the phase velocity\nof the wave. The role of null-curvatur
 e points on the velocity profile (where\n$U'' = 0$\, but does not change i
 ts sign) in the transformation of such an instability\ndomain into that of
  a flow with a piecewise linear velocity profile is discussed.\n\nIt is sh
 own that in continuously stratified flows with $0 < l \\ll 1$\, a countabl
 e\ninfinity of oscillation modes appears with $J = J_m(k\,c)$\, $m = 0\,1\
 ,2\,\\ldots$. For any $m$\,\nstreamwise propagating (i.e.\, $y$-independen
 t) waves have instability domain\nextending from the upper boundary\,\n\n$
 J = J_0^{(+)}(k) = J_0(k\,c=1) = O(1)$ or $J = J_m^{(+)}(k) = J_m(k\,c=1) 
 = O(m^2l^{-1})$\, $m \\ge 1$\nto the lower one\,\n\n$J = J_0^{(-)}(k) > 0$
 \, where $J_0^{(-)}(k) = O(l^2)$ when $l < k < 1$\, or\n\n$J = J_m^{(-)}(k
 ) > 0$\, where $J_m^{(-)}(k) = O(m^2l)$ when $l^{3/2} < k < l^{1/2}$\, $m 
 \\ge 1$.\n\nBy virtue of Squire's theorem\, the lower "stability bands" (b
 etween $J_m^{(-)}(k)$ and the\n$J = 0$ axis) are filled with unstable obli
 que waves. When $J$ is in the range from\n$O(l^2)$ to $O(l)$\, unstable ob
 lique and streamwise propagating waves (mainly\nbelonging to $m = 0$ mode)
  successfully compete\, and a wide spectrum of three-\ndimensional unstabl
 e waves with close streamwise phase velocities and\ncomparable growth rate
 s is excited.\n
LOCATION:https://researchseminars.org/talk/mmandim/47/
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