Are the Beta Gyres Really Normal Modes?

When an axisymmetric vortex is represented in a cylindrical coordinate system whose origin is displaced from the axis of rotation, some of the flow projects onto wavenumber 1. The apparent asymmetry is a normal mode of the linearized governing equations. It is sometimes called the "pseudo-mode" since it represents invariance of the vortex under relocation of the reference frame. Because a gradient-balance vortex is a steady solution of the governing equations, the mode's natural frequency is zero. Growth of the "pseudo-mode" as a result of asymmetric forcing is also the way that the linearized equations represent translation of the vortex. Thus, "translational mode (TM)" is a more apt term than "pseudo-mode".

The TM vorticity is the radial gradient of axisymmetric vorticity dotted into the displacement of the vortex axis from the origin. In a hurricane-like vortex, its magnitude is greatest just inward from the Radius of Maximum Wind (RMW). Beta effect forcing of the wavenumber 1 vorticity equation is strongest at the RMW because it is proportional to the swirling wind. It is constant in time and therefore projects strongly onto the TM spatial structure and frequency.

When an otherwise unforced vortex is constrained to move through a quiescent environment, vorticity conservation prevents the surrounding fluid from entering the vortex. In a barotropic nondivergent fluid, the environmental streamlines diverting around the vortex define curves of zero absolute vorticity. A volume of fluid contained within the translating vortex also has no asymmetric vorticity.

The asymmetric vorticity is concentrated in interlocking crescents at the edge of the vortex. These features are standing vortex-Rossby waves that maintain constant azimuth by propagating against the swirling flow. They are immune to vorticity filamentation because the radial gradient of the swirling flow is small. Both the relative current around the vortex and the uniform flow inside it are free-vortex flows. The streamfunction field ---which arises solely from the motion of the vortex--- is identical with the "beta gyres". Once the motion and asymmetry are established in the linear system defined by wavenumber 1 asymmetries on a moving but otherwise unchanging vortex, they can persist for long times with zero forcing.

On a beta plane, this linear system predicts steady poleward and westward acceleration. Initially, the beta effect causes the TM to grow. The "beta gyres" then appear in response to the vortex motion. If at any instant the beta effect were turned off, the motion would continue. The singular "engine" of the acceleration is growth of the TM because of strong projection of the forcing onto it. This interpretation contrasts with explanation of the beta gyres through conservation of absolute vorticity on quasi-circular trajectories around the vortex. In a nonlinear model, loss of energy through wave-wave interaction causes the motion to reach an asymptotic value. A similar, but spurious, limit occurs under the Asymmetric Balance (AB) approximation, which fails to maintain gradient balance under relocation of the reference frame.