Abstract
The potential vorticity equation has been vertically differentiated and used with the thermal wind equation to obtain the following expression for the absolute horizontal Eulerian velocity vector,
where
where ϕz is the rate of turning with height of the epineutral q contours, ∇nq is the gradient of potential vorticity in the neutral surface, k is the unit vector antiparallel to gravity, and ∇2ρ is the horizontal gradient of in situ density in the geopotential surface; εz, is the vertical derivative of the dianeutral velocity plus the effective vortex stretching due to unsteadiness and the lateral mixing of potential vorticity and V⊥ is the horizontal velocity component normal to the epineutral contours of potential vorticity. A new finding is that the velocity component that mixing induces along the epineutral q contours, V⊥z/ϕz, is equal to the rate of change of the V⊥ component up the cast per unit change in ϕ (measured in radians).
The equation for the absolute velocity vector V is used to explore the conditions required for V = 0. In the absence of mixing processes, V = 0 occurs only where the epineutral gradient of potential vorticity is zero; however, with mixing processes, the lateral velocity can be arrested by mixing without requiring this condition on the gradient of potential vorticity. Conversely, in the presence of mixing, the flow is nonzero at points where ∇nq = 0, and it is shown that at thew points the lateral velocity is determined by the epineutral gradient of qεz. A zero of the lateral Eulerian velocity will occur only at isolated points on any particular surface. Similarly, points of zero three-dimensional velocity will be so rare that one may never expect to encounter such a point on any particular surface.
Mesoscale eddy activity is assumed to transport potential vorticity q in a downgradient fashion along neutral surfaces, and this leads directly to a parameterization of this contribution to the lateral Stokes drift. This extra lateral velocity is typically 1 mm s−1 in the direction toward greater |q| along neutral surfaces, that is, broadly speaking, in the direction away from the equator and toward the poles. This Stokes drift will often make a large contribution to the tracer conservation equations.
