The stability of shearing motion in a rotating fluid

Abstract

This paper is concerned with the stability of a parallel shear flow in an inviscid homogeneous unbounded rotating fluid. A sufficient condition for stability is obtained in terms of the dimensionless parameter N = (cosϕ)/S, where ϕ is the angle between the wave-number K of the disturbance and the axis of rotation, and S is the Rossby number based on the thickness of the shear layer and the change in velocity across the layer. The condition is that infinitesimal disturbances are stable if either $N \ge \frac{1}{2}(1-sin\; \theta)\; \; or\; \; N \le -\frac {1}{2}(1+sin\; \theta)$ Where θ is the angle between k and the direction of streaming. For a shear layer profile of the type U = tanh z, the neutral curves are calculated for various Rossby numbers. These are compared to the stability of a shear layer in a stratified non-rotating fluid. The stability criterion for the large wave-numbers in a cylindrical shear layer is inferred from these results.

Inertial instability of intense stratified anticyclones. Part 1. Generalized stability criterion

“…For vorticity profiles with global minimum vorticity at the core this is the area of the maximum of the absolute value of velocity, V max . This is unlike the barotropic parallel shear configuration (Johnson 1963;Stevens & Ciesielski 1986;Yanase et al 1993;Plougonven & Zeitlin 2009), where the unstable inertial perturbations are located in the region of negative absolute vorticity. For a parallel shear with a zonal mean flow, U = (U(y), 0, 0) and zonally symmetric perturbations, one reaches an equation similar to (2.16), only with fQ(y) = f (f − ∂ y U(y)) instead of χ (see the Appendix by S. D. Griffiths in Kloosterziel & Carnevale 2008).…”

“…For circular, barotropic and inviscid vortex columns a sufficient condition for instability to axisymmetric three-dimensional perturbations is that the square of the absolute angular momentum decreases with the radius, r, somewhere in the flow. This is equivalent to the generalized Rayleigh criterion (Kloosterziel & van Heijst where fQ < 0 (Q is the absolute vorticity), which means that it occurs where the absolute vorticity is the opposite sign of the Coriolis parameter (Johnson 1963;Stone 1966), whereas in the circular case, as mentioned above, it is where χ(r) < 0, which occurs immediately outside the negative absolute vorticity region. This means that in the linear growth stage, in the parallel shear case the instability acts to reduce |Q|, whereas it does not affect the core vorticity in the circular case.…”

An experimental investigation of forced steady rotating turbulence

“…Since the rotation of the reference frame occurs along a single direction, the symmetry between the stabilisation characteristics of eddies with cyclonic and anti-cyclonic vorticity starts to break apart [9,19], resulting in the cyclonic dominance mentioned earlier. Mathematically, this condition of small Ro and high Re, permits the reduction of (1) to a wave-like equation where the Coriolis term gives rise to inertial waves -the linear propagation of which is crucial 30 in explaining the formation of columnar structures [3,2,10].…”

How Large-Scale and Cyclogeostrophic Barotropic Instabilities Favor the Formation of Anticyclonic Vortices in the Ocean

“…The hyperbolic tangent velocity profile, U( y) 5 V 0 tanhy/L, was often used as a generic flow to study the stability of a single two-dimensional shear flow (Drazin and Reid 1981;Johnson 1963). However, in the framework of the RSW equations, the stability of such shear flow may strongly depend on the domain size.…”