Renormalized Green's function for nonlinear circulation on the beta plane.

Abstract

The nonlinear response of a barotropic quasigeostrophic fluid to a 5 forcing by the rotation of the wind stress is discussed in terms of the symmetry properties related to the multipole structure of the response's relative vorticity field and the gradient of planetary vorticity. It is shown that by a global and local renormalization of the P plane, introducing an effective absolute vorticity gradient by means of the global derivatives of the relative vorticity distribution, the three basic symmetry properties, as they are known from numerical simulations, can be explained analytically. This is achieved by means of a renormalized perturbation series which gives the factors that govern the symmetry properties as a function of the parameter measuring the strength of the nonlinearity. These properties are (1) the turning of the symmetry axis for increasing nonlinearity in the direction of the rotation of the wind stress, (2) the concurrent weakening of the symmetry breaking for increasing nonlinearity, and (3) the maximum strength of symmetry breaking around the vorticity dipole axis for intermediate nonlinearity. The first two properties are related to the vorticity distribution s dipole character and the third one to its quadrupole character. The shape of the induced circulation is shown to vary from an oval-shaped pattern with an east-west symmetry axis for weak nonlinearity over a completely asymmetrical swirl for intermediate nonlinearity to a butterfly pattern for the almost-free, strongly nonlinear, inertial mode that is again asymptotically symmetric around the north-south axis.