Sphere-filling asymptotics of the barotropic potential vorticity staircase

For barotropic flow in spherical geometry, the ideal potential vorticity staircase with flat steps and vertical risers exhibits a relationship between prograde jet spacing and strength such that, for sufficiently regular spacing, the distance between adjacent jets is given by the Rhines scale times a positive constant equal to the square root of six. This result was obtained previously in the equatorial limit of spherical geometry, and in periodic geometry. An improved asymptotic method has been devised to explain the spacing-strength relationship in sphere-filling solutions. This analysis explains the approximate validity of the equatorial asymptotics and yields new insight on minimum energy states and mode transitions simulated with persistent energy input randomly at high horizontal wavenumber.