Non-conservation of ‘geostrophic mass’ in the presence of a long boundary and the related Kelvin wave

Abstract

The evolution of a localized flow in a half-plane bounded by a rigid wall is analysed when the total mass is not conserved within the equivalent-barotropic quasi-geostrophic (QG) approximation. A simple formula expressing the total geostrophic mass in terms of the QG potential vorticity is derived and used to estimate the range of the geostrophic mass variability. The behaviour of the total mass is analysed for a system of two point vortices interacting with a wall. Distributed localized perturbations are examined by means of numerical experiments using the QG model. Two types of time variability of the total geostrophic mass are revealed: oscillating (the mass oscillates near some mean value) and limiting (the mass tends to some constant value with increasing time). In the framework of a rotating shallow-water model, the QG model is known to describe the slow evolution of the geostrophic vorticity, assuming the Rossby number to be small. Consideration of the next-order dynamics shows that conservation of the total mass and circulation is provided by a compensating jet taking away the surplus or shortage of mass from the localized geostrophic disturbance. The along-wall jet expands with the fast speed of Kelvin waves to the right of the initial perturbation. The slow time-dependent amplitude determines the jet sign and intensity at each instant. The dynamics of the compensating jet are discussed for both oscillating and limiting regimes revealed by the QG analysis. The role of Kelvin waves in establishing the usual Phillips condition for conservation of circulation of the along-wall QG velocity is discussed. In the case of periodic motion or motion in a finite domain, the approximation of an infinitely long boundary can be used if (i) the typical basin scale greatly exceeds the typical size of the localized perturbation and the Rossby scale; and (ii) the time does not exceed the typical time required for the Kelvin wave to travel the typical basin scale. Both these conditions are typical of synoptic variability in the ocean.