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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
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.
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 ...
Thus, as a fluid parcel moves equatorward (βy approaches zero), the relative vorticity must increase and become more cyclonic in nature. Conversely, if the same fluid parcel moves poleward, (βy becomes larger), the relative vorticity must decrease and become more anticyclonic in nature.
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 .
The three terms of equation (17) are, from left to right, the geostrophic relative vorticity, the planetary vorticity and the stretching vorticity. Implications