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For an arbitrary material volume $V(t)$, Newton's second law equates the rate of change of linear momentum to the resultant contact force, since there is no body force:Use the Reynolds transport theorem on the left. Expanding the product and using the continuity equation leavesOn the right, the divergence theorem applied to each row of the Cauchy stress tensor givesConsequently the volume integral of $\rho D{u}_i/Dt-{\partial }_j{\sigma }_{ij}$ vanishes on every material volume. Localizing as in part (i) gives the Cauchy momentum equation:The left side is mass density times material acceleration; the right side is the divergence of the Cauchy stress tensor.

For a homogeneous incompressible flow, strict geostrophic balance means that the Coriolis acceleration is exactly balanced by the pressure gradient, after conservative body-force potentials have been absorbed into the pressure per unit density $\pi$. It neglects material acceleration rather than merely assuming that its size is small. With constant vertical rotation, ${\pi }_z=0$ and taking the curl gives ${\mathbf{u}}_z=0$, the Taylor–Proudman theorem. Nonzero cross-contour motion over sloping rigid boundaries requires small departures from this strict limit.

A vertically mixed gravity current with horizontally varying reduced gravity $b$ obeys $h_t+(hu{)}_x=0$ and $(hu{)}_t+(h{u}^2+b{h}^2/2{)}_x=0$. Consequently the material acceleration is $Du/Dt=-b{h}_x-h{b}_x/2$. Advected density variables add repeated contact characteristic curves of speed $u$ to the gravity-wave characteristic curves of speeds $u\pm \sqrt{bh}$.