The potential vorticity equation has been vertically differentiated an
d used with the thermal wind equation to obtain the following expressi
on for the absolute horizontal Eulerian velocity vector, [GRAPHICS] wh
ere [GRAPHICS] where phi(z) is the rate of turning with height of the
epineutral q contours, del(n)q is the gradient of potential vorticity
in the neutral surface, k is the unit vector antiparallel to gravity,
and del(2)p is the horizontal gradient of in situ density in the geopo
tential surface; epsilon(z) is the vertical derivative of the dianeutr
al velocity plus the effective vortex stretching due to unsteadiness a
nd the lateral mixing of potential vorticity and V perpendicular to is
the horizontal velocity component normal to the epineutral contours o
f potential vorticity. A new finding is that the velocity component th
at mixing induces along the epineutral a contours, V1/2/phi(z), is equ
al to the rate of change of the V perpendicular to component up the ca
st per unit change in phi(measured in radians). The equation for the a
bsolute velocity vector V is used to explore the conditions required-f
or 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 witho
ut requiring this condition on the gradient of potential vorticity. Co
nversely, in the presence of mixing, the flow is nonzero at points whe
re del(n)q = 0, and it is shown that at these points the lateral veloc
ity is determined by the epineutral gradient of q epsilon(z). A zero o
f the lateral Eulerian velocity will occur only at isolated points on
any particular surface. Similarly, points of zero three-dimensional ve
locity will be so rare that one may never expect to encounter such a p
oint on any particular surface. Mesoscale eddy activity is assumed to
transport potential vorticity q in a downgradient fashion along neutra
l; surfaces, and this leads directly to a parameterization of this con
tribution 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 e
quator and toward the poles. This Stokes drift will often make a large
contribution to the tracer conservation equations.