10 - Numerical Methods for Solving Barotropic Equations

Summary

Introduction

Barotropic fluid is defined as a fluid whose density ρ is a function of pressure p only, i.e., ρ = ρ(p). Examples of barotropic fluids are those that are homogeneous fluids (having constant and uniform density), an isothermal ideal gas (having constant temperature), or an isentropic ideal gas (having constant specific entropy). A fluid that is not barotropic is called a baroclinic fluid. For a baroclinic fluid, the fluid density depends on pressure and temperature as well as salinity and constituent concentration. For the barotropic fluid, as ρ = ρ(p), lines of constant density (isopycnal) are parallel to lines of constant pressure (isobar) and, hence, there is no mechanism to generate relative vorticity (relative vorticity vector is defined as curl of relative velocity) in a barotropic fluid. Conversely, for a baroclinic fluid, the lines of constant density and constant pressure are no longer parallel. This gives rise to changes in relative vorticity and associated relative circulation as the net pressure force no longer passes through the center of mass of the fluid element in the case of baroclinic fluid.

As far as the atmosphere is concerned, it is relatively compressible. Hence, if the atmosphere is considered to be a barotropic fluid, it follows from the definition of barotropic fluid,that the only changes in density are brought about by changes in pressure. Moreover, a barotropic atmosphere would require that air temperature be horizontally constant, which leads to an absence of thermal wind and consequent vanishing of the vertical shear of the horizontal wind. As the tropics have relatively smaller horizontal air temperature gradients, the assumption of a barotropic atmosphere is a more reasonable assumption in the tropics than in the mid-latitudes.

As far as the oceans are concerned, water can be assumed to be pretty much incompressible. Although the oceans are stably stratified fluids (lighter water lies above heavier water, with water density increasing with depth), the vertical variation of density with depth over the oceans is very small (4 kg/m³) as compared to the density of the water (1025 kg/m³) itself so that the water density can be considered virtually to be a constant for a barotropic ocean.