On the stability of sharply stratified shear flows with inflection-free velocity profiles

Abstract: We study the linear stability of shear flows with sharp stratification ($l \ll L$, where $l$ and $L = 1$ are vertical scales of density and velocity variation respectively) and a monotonic velocity profile $V_x = U(z)$ which has no inflection points and increases from $U = 0$ at the bottom ($z = 0$) to $U = 1$ when $z \to +\infty$, $U'(0) = 1$. We show 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, where $k$ is the wave number and $J$ is the bulk Richardson number. Namely, the domain is bounded by abscissa axis ($J = 0$), dispersion curve $J = J(k, c = 1),$ and the segment of ordinate axis ($k = 0$) connecting them. Here $c$ is the phase velocity of the wave. The role of null-curvature points on the velocity profile (where $U'' = 0$, but does not change its sign) in the transformation of such an instability domain into that of a flow with a piecewise linear velocity profile is discussed.

It is shown that in continuously stratified flows with $0 < l \ll 1$, a countable infinity of oscillation modes appears with $J = J_m(k,c)$, $m = 0,1,2,\ldots$. For any $m$, streamwise propagating (i.e., $y$-independent) waves have instability domain extending from the upper boundary,

$J = J_0^{(+)}(k) = J_0(k,c=1) = O(1)$ or $J = J_m^{(+)}(k) = J_m(k,c=1) = O(m^2l^{-1})$, $m \ge 1$ to the lower one,

$J = J_0^{(-)}(k) > 0$, where $J_0^{(-)}(k) = O(l^2)$ when $l < k < 1$, or

$J = J_m^{(-)}(k) > 0$, where $J_m^{(-)}(k) = O(m^2l)$ when $l^{3/2} < k < l^{1/2}$, $m \ge 1$.

By virtue of Squire's theorem, the lower "stability bands" (between $J_m^{(-)}(k)$ and the $J = 0$ axis) are filled with unstable oblique waves. When $J$ is in the range from $O(l^2)$ to $O(l)$, unstable oblique and streamwise propagating waves (mainly belonging to $m = 0$ mode) successfully compete, and a wide spectrum of three- dimensional unstable waves with close streamwise phase velocities and comparable growth rates is excited.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods