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On a beta plane, the turning rate of an inertial oscillation is larger on the poleward side of its orbit than on the equatorward side. For initial speed $V$ and $\beta V/f_0^2\ll 1$, this asymmetry produces westward mean drift of magnitude $\beta V^2/(2f_0^2)$.

Substitution of $e^{i(kx+ly-\omega t)}$ gives a homogeneous three-by-three system. Its determinant issoThe zero-frequency component is in geostrophic balance and carries all conserved $P$. The oscillatory components are inertia-gravity waves with zero $P$. For $K\ll |f|/c$, $\Omega \sim |f|$ and rotation produces an inertial oscillation; for $K\gg |f|/c$, $\Omega \sim cK$ and surface gravity dominates. The crossover length $c/|f|$ is the barotropic deformation radius.

IntroduceBecause the speed is $V$, writeThe momentum equations giveUse an asymptotic expansionThe zeroth-order inertial oscillation isAt first order,and integration with the initial conditions givesThereforeThe periodic terms describe a slightly distorted clockwise circle. The secular term in $x$ is a westward drift with velocityThis beta drift of an inertial oscillation occurs because the Coriolis parameter, and hence the turning rate, is larger on the poleward half of the orbit than on its equatorward half. A trajectory sketch therefore consists of clockwise loops whose centers move steadily westward; the parcel starts at the westernmost point of its first loop and initially travels northward.