Stability of gravity currents generated by finite-volume releases

Abstract

We generalize the linear stability analysis of the axisymmetric self-similar solution of gravity currents from finite-volume releases to include perturbations that depend on both radial and azimuthal coordinates. We show that the similarity solution is stable to sufficiently small perturbations by proving that all perturbation eigenfunctions decay in time. Moreover, asymmetric perturbations are shown to decay more rapidly than axisymmetric perturbations in general. An asymptotic formula for the eigenvalues is derived, which indicates that asymptotic rates of decay of perturbations are given by $t^{-\sigma}$ where $0\,{<}\,\sigma\,{<}\,\frac{1}{4}$ as the Froude number decreases from $\sqrt2$ to $0$. We demonstrate that this formula agrees closely with numerically calculated eigenvalues and, in the absence of azimuthal dependence, it reduces to an expression that improves on the asymptotic formula obtained by Grundy & Rottman (1985). For two-dimensional (planar) currents, we further prove analytically that all perturbation eigenfunctions decay like $t^{-{1/2}}.$