Why the bolus velocity deserved to die and how I killed it

Peter Rhines introduced the eddy-induced “bolus velocity” (i.e., the correlation of fluctuations in the thickness of an isopycnal surface with eddy velocity) in his 1982 lectures at the Woods Hole Geophysical Fluid Dynamics Program. This construction has bemused oceanographers ever since. In some versions of the epipycnally-averaged equations of motion, the bolus velocity appears as the difference between the average velocity weighted by the layer thickness and the unweighted average velocity. But exactly how the bolus velocity plays in the mean equations seems almost to be a matter of opinion e.g., in some formulations the Coriolis force acts on the bolus velocity and in others not. Moreover, there are complications because the bolus velocity is not incompressible and certainly not horizontal.

I show that a systematic application of density-coordinate thickness-weighted averaging (TWA), applied to the primitive equations, results in exact equations written completely in terms of the thickness-weighted velocity. That is, the unweighted average velocity and the bolus velocity do not appear anywhere in the final averaged equations of motion. Apart from eddy forcing by the divergence of three-dimensional Eliassen-Palm vectors in the horizontal momentum equations, this TWA formulation is identical to the un-averaged primitive equations. The Eliassen-Palm vectors are second-order in eddy amplitude and their divergence can be expressed in terms of the eddy flux of the Rossby-Ertel potential vorticity derived from the TWA equations. Thus there is a three-dimensional and fully nonlinear generalization of Taylor's identity. Ancillary results include the correct density-coordinate expressions for grad, div and curl using the basis vectors most appropriate to density coordinates. This enables easy and direct translation of the TWA equations from buoyancy coordinates back into Cartesian coordinates.