Group Velocity and the Linear Response of Stratified Fluids to Internal Heat or Mass Sources

Abstract

A steadily maintained line heat or mass source turned on in an unbounded, steadily moving, uniformly stratified flow will in general create ever-increasing vertical displacements of the fluid. Lin and Smith viewed a maintained heat source as a train of heat pulses. A pulse occurring a time T before the observation time creates a negative displacement proportional to T?1 at the heat source position when T is large. They pointed out that superposing the pulse responses leads to a displacement that grows logarithmically with time. This paper uses group velocity arguments to recreate the gravity wave field a time T after a heat pulse. The T?1 decay of the displacement is shown to be a geometrical consequence of dispersion in two dimensions. The growing response to a maintained source can be understood as the result of energy being pumped into the gravity wave modes, whose group velocity is near zero, faster than it can spread in physical space due to dispersion. A steady response is shown to be possible only if the heat source distribution has no projection onto the modes of zero group velocity. If the fluid is bounded both above and below, the vertical wavenumbers of gravity wave modes are quantized. Unless the layer depth is resonantly tuned, there are no normal modes of zero group velocity and a steady response develops. The same arguments allow the work of Smith and Lin to be generalized to more complicated situations, e.g.; when there is either ambient rotation or localization of the heat source in all three dimensions, and show that a steady state will develop in response to a maintained heat source in these cases because the response to a pulse heat source decays faster than T?1. Analogous results hold for a mass source or flow over a ramp. Only very large vertical displacements or wave breaking are likely to alter these conclusions.