Eur. Phys. J. Plus (2012) 127: 6
https://doi.org/10.1140/epjp/i2012-12006-2

Regular Article

On the linearization of the quasi-geostrophic potential vorticity equation at the ocean basin-scale

¹ ISMAR-CNR, viale Romolo Gessi 2, 34123, Trieste, Italy
² ISAC-CNR, via Fosso del Cavaliere, 100, 00133, Roma, Italy

^* e-mail: fulvio_crisciani@alice.it

Received: 23 September 2011
Revised: 25 November 2011
Accepted: 18 December 2011
Published online: 23 January 2012

Abstract

In the framework of the quasi-geostrophic fluid dynamics, we investigate the distinctive condition which leads to the linear or, alternatively, to the nonlinear quasi-geostrophic potential-vorticity equations at the ocean basin-scale. In most of the quasi-geostrophic systems, a shortening of the time scale with respect to the advective one is associated to a change of the ordering parameter (usually the temporal Rossby number in place of the advective Rossby number) of the primitive equations. As a consequence, by means of a standard asymptotic analysis, linear equations are obtained at the geostrophic level of approximation. Unlike this, at the ocean basin-scale the characteristic time of the system and the ordering parameter appearing in the primitive equations are independent of the possible linear or nonlinear nature of the dynamics and are univocally fixed. In this context, we prove that an alternative relationship, involving the ordering parameter, the advective Rossby number and the stratification parameter, determines the linear or the nonlinear nature of the quasi-geostrophic potential-vorticity equation. Hence, a criterion follows which is able to resolve whether a linear model, rather than a nonlinear one, is fit for supporting a given phenomenology at the considered scale.