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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:
Hence from the knowledge of vorticity field, the operator can be inverted and the stream function can be calculated. In this particular case (equation 21), vorticity gives all the information needed to deduce motions, or streamfunction, thus one can think in terms of vorticity to understand the dynamics of the fluid.
In the context of meteorology, a solenoid is a tube-shaped region in the atmosphere where isobaric (constant pressure) and isopycnal (constant density) surfaces intersect, causing vertical circulation. [1] [2] They are so-named because they are driven by the solenoid term of the vorticity equation. [3]
In the 1950s, the first successful programs for numerical weather forecasting utilized that equation. In modern numerical weather forecasting models and general circulation models (GCMs), vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a prognostic equation.
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 barotropic vorticity equation assumes the atmosphere is nearly barotropic, which means that the direction and speed of the geostrophic wind are independent of height. In other words, there is no vertical wind shear of the geostrophic wind. It also implies that thickness contours (a proxy for temperature) are parallel to upper level height ...
In meteorology, [2] helicity corresponds to the transfer of vorticity from the environment to an air parcel in convective motion. Here the definition of helicity is simplified to only use the horizontal component of wind and vorticity, and to only integrate in the vertical direction, replacing the volume integral with a one-dimensional definite integral or line integral:
In general, the evolution of vorticity can be broken into contributions from advection (as vortex tubes move with the flow), stretching and twisting (as vortex tubes are pulled or twisted by the flow) and baroclinic vorticity generation, which occurs whenever there is a density gradient along surfaces of constant pressure.