Editing Vorticity equation (section)

==Physical interpretation==

* The term {{math|{{sfrac|''D'''''ω'''|''Dt''}}}} on the left-hand side is the [[substantive derivative|material derivative]] of the vorticity vector {{math|'''ω'''}}. It describes the rate of change of vorticity of the moving fluid particle. This change can be attributed to [[steady state flow|unsteadiness]] in the flow ({{math|{{sfrac|∂'''ω'''|∂''t''}}}}, the ''unsteady term'') or due to the motion of the fluid particle as it moves from one point to another ({{math|('''u''' ∙ ∇)'''ω'''}}, the ''[[convection]] term'').
* The term {{math|('''ω''' ∙ ∇) '''u'''}} on the right-hand side describes the stretching or tilting of vorticity due to the flow velocity gradients. Note that {{math|('''ω''' ∙ ∇) '''u'''}} is a vector quantity, as {{math|'''ω''' ∙ ∇}} is a scalar differential operator, while {{math|∇'''u'''}} is a nine-element tensor quantity.
* The term {{math|'''ω'''(∇ ∙ '''u''')}} describes [[vortex stretching|stretching of vorticity]] due to flow compressibility. It follows from the Navier-Stokes equation for [[continuity equation|continuity]], namely <math display="block">\begin{align}
  \frac{\partial\rho}{\partial t} + \nabla \cdot\left(\rho \mathbf u\right) &= 0 \\
  \Longleftrightarrow \nabla \cdot \mathbf{u} &= -\frac{1}{\rho}\frac{d\rho}{dt} = \frac{1}{v}\frac{dv}{dt}
\end{align}</math> where {{math|1=''v'' = {{sfrac|1|''ρ''}}}} is the [[specific volume]] of the fluid element. One can think of {{math|∇ ∙ '''u'''}} as a measure of flow compressibility. Sometimes the negative sign is included in the term.
* The term {{math|{{sfrac|1|''ρ''<sup>2</sup>}}∇''ρ'' × ∇''p''}} is the [[baroclinity|baroclinic term]]. It accounts for the changes in the vorticity due to the intersection of density and pressure surfaces.
* The term {{math|∇ × ({{sfrac|∇ ∙ ''τ''|''ρ''}})}}, accounts for the diffusion of vorticity due to the viscous effects.
* The term {{math|∇ × '''B'''}} provides for changes due to external body forces. These are forces that are spread over a three-dimensional region of the fluid, such as [[gravity]] or [[electromagnetic force]]s. (As opposed to forces that act only over a surface (like [[drag coefficient|drag]] on a wall) or a line (like [[surface tension]] around a [[Meniscus (liquid)|meniscus]]).

=== Simplifications ===
* In case of [[conservative force|conservative body forces]], {{math|1=∇ × '''B''' = 0}}.
* For a [[barotropic|barotropic fluid]], {{math|1=∇''ρ'' × ∇''p'' = 0}}. This is also true for a constant density fluid (including incompressible fluid) where {{math|1=∇''ρ'' = 0}}. Note that this is not the same as an [[incompressible flow]], for which the barotropic term cannot be neglected. 
** This note seems to be talking about the fact that conservation of momentum says {{math|1=<math>\frac{D \rho}{D t} + \rho (\nabla \cdot \mathbf u) = \frac{\partial \rho}{\partial t} + \mathbf u \cdot \nabla \rho + \rho (\nabla \cdot \mathbf u) = 0</math>}} and there's a difference between assuming that ρ=constant (the 'incompressible fluid' option, above) and that <math>\nabla \cdot \mathbf u = 0
</math> (the 'incompressible flow' option, above). With the first assumption, conservation of momentum implies (for non-zero density) that <math>\nabla \cdot \mathbf u = 0
</math>; whereas the second assumption doesn't necessary imply that ρ is constant. This second assumption only strictly requires that the time rate of change of the density is compensated by the gradient of the density, as in:{{math|1=<math>\frac{\partial \rho}{\partial t} = - \mathbf u \cdot \nabla \rho</math>}}. You can make sense of this by considering the ideal gas law  {{math|1=''p'' = ''ρRT''}} (which is valid if the Reynolds number is large enough that viscous friction becomes unimportant.) Then, even for an adiabatic, chemically-homogenous fluid, the density can vary when the pressure changes, e.g. with Bernoulli.
* For [[inviscid]] fluids, the viscosity tensor {{mvar|τ}} is zero.

Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to
: <math>\frac{D}{Dt} \left( \frac{\boldsymbol \omega}{\rho} \right) = \left( \frac{\boldsymbol\omega}{\rho} \right) \cdot \nabla \mathbf u </math>

Alternately, in case of incompressible, inviscid fluid with conservative body forces, 
: <math>\frac{D \boldsymbol \omega}{Dt} = \frac{\partial \boldsymbol \omega}{\partial t} + (\mathbf u \cdot \nabla)\boldsymbol \omega = (\boldsymbol \omega \cdot \nabla) \mathbf u </math><ref>{{cite book |last1=Fetter |first1=Alexander L. |last2=Walecka |first2=John D. |title=Theoretical Mechanics of Particles and Continua |date=2003 |publisher=Dover Publications |isbn=978-0-486-43261-8 |page=351 |edition=1st}}</ref>

For a brief review of additional cases and simplifications, see also.<ref>{{cite web| first=K.&nbsp;P.| last=Burr |title=Marine Hydrodynamics, Lecture 9| website=MIT Lectures| url=https://ocw.mit.edu/courses/mechanical-engineering/2-20-marine-hydrodynamics-13-021-spring-2005/lecture-notes/lecture9.pdf}}</ref> For the vorticity equation in turbulence theory, in context of the flows in oceans and atmosphere, refer to.<ref>{{cite web | first=Richard L. | last=Salmon | title=Lectures on Geophysical Fluid Dynamics, Chapter 4 | website=Oxford University Press; 1 edition (February 26, 1998) | url=http://pordlabs.ucsd.edu/rsalmon/chap4.pdf}}</ref>