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The absolute vorticity is computed from the air velocity relative to an inertial frame, and therefore includes a term due to the Earth's rotation, the Coriolis parameter. The potential vorticity is absolute vorticity divided by the vertical spacing between levels of constant (potential) temperature (or entropy ).
The vorticity equation of fluid dynamics describes the evolution of the vorticity ω of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity). The governing equation is:
where is the relative vorticity, is the layer depth, and is the Coriolis parameter. The conserved quantity, in parenthesis in equation (3), was later named the shallow water potential vorticity . For an atmosphere with multiple layers, with each layer having constant potential temperature, the above equation takes the form
Here is the relative vorticity, the horizontal wind velocity vector, whose components in the and directions are and respectively, the absolute vorticity +, is the Coriolis parameter, = the material derivative of pressure , ^ is the unit vertical vector, is the isobaric Del (grad) operator, () is the vertical advection of vorticity and ...
is absolute vorticity, with ζ being relative vorticity, defined as the vertical component of the curl of the fluid velocity and f is the Coriolis parameter = , where Ω is the angular frequency of the planet's rotation (Ω = 0.7272 × 10 −4 s −1 for the earth) and φ is latitude.
The absolute vorticity composes the planetary vorticity and the relative vorticity , reflecting the Earth’s rotation and the parcel’s rotation with respect to the Earth, respectively. The conservation of absolute vorticity η {\displaystyle \eta } determines a southward gradient of ζ {\displaystyle \zeta } , as denoted by the red shadow in c .
Circulation can be related to curl of a vector field V and, more specifically, to vorticity if the field is a fluid velocity field, =.. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, [4] = = =
Antisymmetric tensors are commonly used to represent rotations (for example, the vorticity tensor). Although a generic rank R tensor in 4 dimensions has 4 R components, constraints on the tensor such as symmetry or antisymmetry serve to reduce the number of distinct components.