On the conjugate behaviour of weak along-shore flows

Abstract

Using a simple analytical model containing a step shelf in a rotating ocean, confirmation is found for the idea that unstratified positively directed currents (poleward on western boundaries, equatorward on eastern boundaries) may be arranged in one of two structural forms each of which transports the same flux of each water type. In other words, the flux of water of each potential vorticity (or Bernoulli head) occurring within the current is the same for both structural forms. These structural forms are referred to as conjugate structural forms of the current. In agreement with an earlier study, it is shown that the wider of these two conjugate forms is subcritical to shelf waves and the narrower form is supercritical to the same waves. The present study seeks to understand the behaviour of these structures in the limit of weak flow. It is shown that as the velocity is decreased in a subcritical structure of fixed width, the conjugate supercritical structure continues to exist but with decreased width as well as velocity. At rest, conditions may be obtained by letting the flow tend to zero in either the subcritical structure of fixed width or its conjugate supercritical structure. However, the branch of the dispersion curve that comes to correspond to shelf waves in the at rest state depends on whether this state is approached as the weak flow limit of a subcritical or supercritical flow.