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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:
Vorticity is useful for understanding how ideal potential flow solutions can be perturbed to model real flows. In general, the presence of viscosity causes a diffusion of vorticity away from the vortex cores into the general flow field; this flow is accounted for by a diffusion term in the vorticity transport equation. [9]
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
The quasi-geostrophic vorticity equation can be obtained from the and components of the quasi-geostrophic momentum equation which can then be derived from the horizontal momentum equation D V D t + f k ^ × V = − ∇ Φ {\displaystyle {D\mathbf {V} \over Dt}+f{\hat {\mathbf {k} }}\times \mathbf {V} =-\nabla \Phi } (3)
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] = = =
Formula for vorticity can give another explanation of the Stokes' Paradox. The functions (), > belong to the kernel of and generate the stationary solutions of the vorticity equation with Robin-type boundary condition. From the arguments above any Stokes' vorticity flow with no-slip boundary condition must be orthogonal to the obtained ...
Mathematically, the vorticity is defined as the curl (or rotational) of the velocity field of the fluid, usually denoted by and expressed by the vector analysis formula , where is the nabla operator and is the local flow velocity.
The governing equation for an inviscid fluid with a conservative body force is ... The vorticity of a velocity field in fluid dynamics is defined by: