Vorticity equation

The vorticity equation is an important "prognostic equation" in the atmospheric sciences. Vorticity is a vector, therefore, there are three components. The equation of vorticity (three components in the canonical form) describes the total derivative (that is, the local change due to local change with time and advection) of vorticity, and thus can be stated in either "relative" or "absolute" form.

The more compact version is that for "absolute vorticity", component eta, using the pressure system:

:frac{d eta}{d t} = -eta abla_h cdotmathbf{v}_h - left( frac{partial omega}{partial x} frac{partial v}{partial z} - frac{partial omega}{partial y} frac{partial u}{partial z} ight) - frac{1}{ ho^2} mathbf{k} cdot ( abla_h p imes abla_h ho )

Here, ho is density, "u", "v", and omega are the components of wind velocity, and abla_h is the 2-dimensional (i.e. horizontal-component-only) del.

The terms on the RHS denote the positive or negative generation of "absolute vorticity" by divergence of air, twisting of the axis of rotation, and baroclinity, respectively.

Fluid dynamics

The vorticity equation describes the evolution of the vorticity (vec omega) of a fluid element as it moves around. The vorticity equation can be derived from the conservation of momentum equation. [ Derivation of the vorticity equationIn the absence of any concentrated torques and line forces, the "momentum conservation equation" gives,

:frac{D vec V}{D t} = frac{partial vec V}{partial t} + vec V cdot vec abla vec V = - frac{1}{ ho} vec abla p + ho vec B + frac{vec abla cdot underline{underline{ au}{ ho}

Now, vorticity is defined as the curl of the velocity vector ( vec omega = vec abla imes vec V). Taking curl of momentum equation yields the desired equation. The following identities are useful in derivation of the equation, : vec V cdot vec abla vec V = vec abla ( frac{1}{2} vec V cdot vec V) - vec V imes vec omega :vec abla imes (vec V imes vec omega ) = -vec omega (vec abla cdot vec V) + (vec omega cdot vec abla ) vec V - vec V cdot (vec abla vec omega ):vec abla imes phi = 0 , where phi is a scalar.:vec abla cdot vec omega = 0

] In its general vector form it may be expressed as follows,

:egin{align}frac{Dvecomega}{Dt} &= frac{partial vec omega}{partial t} + vec V cdot (vec abla vec omega) \&= (vec omega cdot vec abla) vec V - vec omega (vec abla cdot vec V) + frac{1}{ ho^2}vec abla ho imes vec abla p + vec abla imes left( frac{vec abla cdot underline{underline{ au}{ ho} ight) + vec abla imes vec Bend{align}

where, vec V is the velocity vector, ho is the density, p is th pressure, underline{underline{ au is the viscous stress tensor and vec B is the body force term.

Equivalently in tensor notation,

:egin{align}frac{Domega_i}{Dt} &= frac{partial omega_i}{partial t} + V_j frac{partial omega_i}{partial x_j} \&= omega_j frac{partial V_i}{partial x_j} - omega_i frac{partial V_j}{partial x_j} + e_{ijk}frac{1}{ ho^2}frac{partial ho}{partial x_j}frac{partial p}{partial x_k}+ e_{ijk}frac{partial}{partial x_j}left(frac{1}{ ho}frac{partial au_{km{partial x_m} ight)+ e_{ijk}frac{partial B_k }{partial x_j}end{align}

where, we have used the Einstein summation convention, and e_{ijk} is the Levi-Civita symbol.

Physical Interpretation

* The term frac{Dvecomega}{Dt} = frac{partial vec omega}{partial t} + vec V cdot (vec abla vec omega) is the material derivative of the vorticity vector vec omega . It describes the rate of change of vorticity of a fluid particle (or in other words the angular acceleration of the fluid particle). This can change due to the unsteadiness in the flow captured by frac{partial vec omega}{partial t} (the unsteady term) or due to the motion of the fluid particle as it moves from one point to another, vec V cdot (vec abla vec omega) (the convection term).

* The first term on the RHS of the vorticity equation, (vec omega cdot vec abla) vec V , describes the stretching or tilting of vorticity due to the velocity gradients. Note that this is a tensor with nine terms.

* The next term, vec omega (vec abla cdot vec V), describes stretching of vorticity due to flow compressibility.The flow "continuity equation" states that,:frac{partial ho}{partial t} + vec abla cdot( ho vec V) = 0 This can be rewritten as, : vec abla cdot vec V = -frac{1}{ ho} frac{D ho}{Dt} = frac{1}{v} frac{Dv}{Dt}

where v = frac{1}{ ho} is the specific volume of the fluid element. Thus one can think of vec abla cdot vec V as a measure of flow compressibility.] Sometimes the negative sign is included in the term.

* The third term, frac{1}{ ho^2}vec abla ho imes vec abla p is the baroclinic term. It accounts for the changes in the vorticity due to the intersection of density and temperature surfaces.

* vec abla imes left( frac{vec abla cdot underline{underline{ au}{ ho} ight), accounts for the diffusion of vorticity due to the viscous effects.

* vec abla imes vec B provides for changes due to body forces. [ A body force is one which is proportional to mass/volume/charge on a body. Such forces act over the whole volume of the body as opposed to a surface forces which act only on the surface. Examples of body forces are gravitational force, electromagnetic force, etc. Examples of surface forces are friction, pressure force, etc. Also there are line forces, like surface tension.]

Simplifications

# In case of conservative body forces, vec abla imes vec B = 0 .
# For a barotropic fluid, vec abla ho imes vec abla p = 0. This is also true for a constant density fluid where vec abla ho = 0. [Note that incompressible fluid (constant density fluid) is not same as incompressible flow and the barotropic term can not be neglected in case of incompressible flow. ]
# For inviscid fluids, underline{underline{ au = 0.

Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to, [ We use the continuity equation to get to this form.] : frac{D}{Dt} left( frac{vec omega}{ ho} ight) = left( frac{vecomega}{ ho} ight) cdot (vec abla vec V)

Alternately, in case of incompressible, inviscid fluid with conservative body forces, : frac{D vec omega}{Dt} = vec omega cdot (vec abla vec V)

Notes

ee also

* Vorticity
* Barotropic vorticity equation


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