Sometimes a sheared flow is stably stratified (d theta / dz > 0). If this flow b

Question

Sometimes a sheared flow is stably stratified (d theta / dz > 0). If this flow becomes unstable and the fluid column is mixed, the potential energy of the fluid column will increase. Complete mixing is only possible when enough kinetic energy is released from the background environment to support the increase in potential energy. The following questions lead you to the solution for the flow conditions under which this complete mixing is possible and explore Kelvin-Helmholtz instability. The fluid column is stably stratified (rho_1

Explanation / Answer

Instability of a Stratified Shear Flow

Consider a sharp discontinuity in density and velocity. In actual environmental systems, there exists a zone of gradual changes from one layer to the other, and the analysis becomes complicated because of the existence of nonconstant coefficients.

For a continuous stratification ¯(z), corresponds the stratification frequency N(z) defined from

N2 = (g/0)(d/dz) > 0

and accompanied by a sheared horizontal and unidirectional flow ¯u(z).

A small perturbation of this system are

u/x + w/z = 0

When a wave is capable of extracting energy from the system. It does so by drawing either kinetic energy from pre-existing motion or potential energy from background stratification. In either case the wave amplitude grows over time, and the wave is said to be unstable. This chapter reviews the most common types of instability in environmental fluid systems, progressing from smaller billows to weather systems. Instabilities are a prelude to mixing and turbulence.

From the perspective of energy conservation, it is clear that the billowing motion involves lifting of denser fluid and sinking of lighter fluid, both of which cause a rise of the center of gravity which in turn demands a supply of potential energy. This potential energy can only be provided by conversion of kinetic energy present in the shear flow. Indeed, not all of the kinetic energy that is released goes into potential energy because some is consumed by turbulent dissipation.

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