The zonal drift associated with time-dependent particle motion on the earth

Abstract

The zonal drift associated with the time-dependent particle velocity on the rotating earth, such as the inertial motion–westward in midlatitudes and eastward in the tropics–is shown to result from the conservation of angular momentum. This finding follows from a transformation of Newton's second law of motion on the earth into a dynamical system where the angular momentum is a system variable, a procedure similar to that used in the description of a spinning top's dynamics. In near-geostrophic zonal motion (i.e. time-dependent motion in the presence of a fixed meridional pressure gradient), where the angular momentum is conserved, and for angular momentum that is below a threshold value, the dynamical system possesses two elliptic points located off the equator. The fixed points of the dynamical system correspond to the steady, geostrophic, zonal motion so the time-dependent velocity of the small-amplitude motion (when the latitudinal particle excursion is not too large) deviates only slightly from the geostrophic velocity. The explicit expressions derived for the drift near the elliptic points imply that in near-geostrophic, eastward (westward) motion in midlatitudes the geostrophic, steady, speed provides an overestimate (underestimate) of the temporally averaged drift. This difference between the steady motion and the average of the periodic motion characterizes nonlinear systems. At large values of the angular momentum a single elliptic point exists on the equator and the drift is directed eastwards. This drift, too, can be quantified near the fixed point associated with a fixed zonal velocity on the equator, where the Coriolis force vanishes. At large oscillation amplitude in midlatitudes, when the oscillation encompasses a large latitude band and the instantaneous velocity deviates significantly from the geostrophic velocity, the zonal drift is strongly affected by the presence of a hyperbolic point at the equator. Explicit expressions for the zonal drift are not available in this case, but numerical results show that the effect of this nonlinearity is to greatly enhance the westward drift in midlatitudes compared with the small-amplitude case.