Ertel's potential vorticity theorem in physical oceanography

Abstract

Under certain circumstances the equations of motion of a fluid yield theorems that provide powerful aids for comprehension of the character and physics of a wide variety of motions. Perhaps the most powerful of such theorems, especially in geophysical fluid dynamics, are the vorticity theorems that specify how the angular velocity of fluid particles changes with time and position. For large‐scale water motions the relevant form of vorticity is what is called potential vorticity, which incorporates inhomogeneities of the constituent elements of seawater. Ertel's theorem specifies the dynamical evolution of potential vorticity. Most other vorticity theorems can be derived from it. For ideal one‐component fluids, potential vorticity is materially conserved, that is, it is conserved when a particle of fluid is followed as its location changes, which is a powerful constraint for analyzing fluid motion. Seawater is a two‐component fluid with its components being water and salt. Potential vorticity becomes materially conserved for ideal oceanic motions when the thermobaric coefficient (dependent upon the variations with pressure of the thermal expansion and haline contraction coefficients) is assumed to be zero as, for example, for incompressible seawater (incompressibility being a quite valid assumption for many oceanic motions). Approximate forms of potential vorticity are illustrated for standard oceanographic approximations, including a flat Earth with zero, constant, or varying rotation; a spherical Earth; shallow water; stratification; quasi‐geostrophy; and others. The concept of potential vorticity naturally defines forms of motion (“vortical” modes) which exhibit nonzero amounts of potential vorticity. The best known forms of “vortical” modes are planetary geostrophic motion (where Coriolis and pressure forces are in approximate balance over the globe), quasi‐geostrophic motion (where the forces deviate somewhat from the geostrophic balance), and two‐dimensional stratified turbulence. The absence of potential vorticity defines the inertia‐gravity mode of motion, that is, gravity waves dependent upon the ocean's stratification. An arbitrary flow can be separated into a part that carries linear motionally induced potential vorticity and a part that does not.