Streamfunction

streamfunction

A parameter of two-dimensional, nondivergent flow, with a value that is constant along each streamline.

For flow in the (x,y) plane, the streamfunction ψ is related to the respective coordinate velocities u and v by the equations
In meteorology the most common application of the streamfunction is in the assumption of geostrophic equilibrium. If variations in the Coriolis parameter f are ignored, the streamfunction in a constant-pressure surface is proportional to the geopotential gz, that is, ψ = gz/f; in an isentropic surface, it is ψ = (gz + c_pT)/f, called the Montgomery streamfunction, where c_p is the specific heat at constant pressure and T the Kelvin temperature.
Stokes's streamfunction (
also called current function): If the flow is three-dimensional but is axisymmetric (i.e., the same in every plane containing the axis of symmetry), a Stokes's streamfunction ψ will exist such that
where v_s is the speed in an arbitrary direction s, r is the distance from the axis of symmetry, and n is normal to the direction s, increasing to the left.

Note that Stokes's streamfunction has dimensions of volume per time. Streamfunctions can also be defined for more complex three-dimensional flows.