Baroclinic vector

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Beginning with the equation of motion for a fluid (say, the Euler equations or the Navier-Stokes equations) and taking the curl, one arrives at the equation of motion for the curl of the fluid velocity, that is to say, the vorticity.

In a fluid that is not all of the same density, a source term appears in the vorticity equation whenever surfaces of constant density (isopycnic surfaces) and surfaces of constant pressure (isobaric surfaces) are not aligned. This term (denoted by the subscript $\mathbf{bc}$ below) is known as the baroclinic vector. (Note that $\vec \omega$ is the vorticity vector, $P$ is pressure, and $ρ$ is density):

$\left( \partial_t \vec \omega \right)_{\mathbf{bc}} = \frac{1}{\rho^2} \nabla \rho \times \nabla P$

This vector is of interest both in compressible fluids and in incompressible (but inhomogenous) fluids. Internal gravity waves as well as unstable Rayleigh-Taylor modes can be analyzed from the perspective of the baroclinic vector. It is also of interest in the creation of vorticity by the passage of shocks through inhomogenous media, such as in the Richtmeyer-Meshkov instability.

Divers may be familiar with the very slow waves that can be excited at a thermocline or a halocline; these are internal waves. Similar waves can be generated between a layer of water and a layer of oil. When the interface between these two surfaces is not horizontal and the system is close to hydrostatic equilibrium, the gradient of the pressure is vertical but the gradient of the density is not. Therefore the baroclinic vector is nonzero, and the sense of the baroclinic vector is to create vorticity to make the interface level out. In the process, the interface overshoots, and the result is an oscillation which is an internal gravity wave. Unlike surface gravity waves, internal gravity waves do not require a sharp interface. For example, in bodies of water, a gradual gradient in temperature or salinity is sufficient to support internal gravity waves driven by the baroclinic vector.