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What is enstrophy and how can we generalize that notion to higher dimensions? Comment: a formulation using differential forms (and/or tensors/multivectors) is preferred!

riemannium
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    Is there some specific aspect of the Wikipedia article that confuses you? Granted, it's light on detail, but the strict answer to "what is X" is just a definition of X, and the fewer footholds you give in terms of what kind of answer you're looking for, the less useful this thread will be both for you and for future visitors. – Emilio Pisanty Jan 04 '18 at 17:06
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    More on enstrophy. – Qmechanic Jan 04 '18 at 17:06
  • Dear Emilio Pisanty, the wikipedia article is an old known of mine, but the idea of enstropy and other potential densities is quite unknown for many physicists. And I think is or it could be USEFUL to get a broad and general formulation of that concept. Indeed, I supposed correctly that someone could confuse the name with entropy!!!! – riemannium Jan 04 '18 at 17:39

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If $$u ~=~ u_i~\mathrm{d}x^i \tag{1}$$ is the flow velocity 1-form in $n$ spatial dimensions, let $$ \omega ~:=~ \mathrm{d}u \tag{2}$$ be the vorticity 2-form. The enstrophy in the region $R$ is then defined as $$ {\cal E}~:=~\int_R \! \omega \wedge {\star} \omega.\tag{3}$$ Definitions (1)-(3) are analogous to the gauge potential $A$, the field strength $F$, and the Maxwell action $S$ in E&M, respectively.

Qmechanic
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  • OK, but can we generalize the enstropy idea to other potential forms in a non-abelian sense as well? 2. Therefore, enstrophy is not a squared thing in higher dimensions?
    • – riemannium Jan 04 '18 at 18:49
  • Yeah, why not? 2. It is still a non-negative quantity (up to possible sign conventions).
    • – Qmechanic Jan 04 '18 at 18:51
  • My remark was to point out what I realized reading the article on enstrophy...Enstrophy is essentially the Yang-Mills action! (Of course, giving up the issue of units and matching magnitudes). A dictionary could be helpful! It is amazing the fluidic analogy of fields... – riemannium Jan 04 '18 at 21:41
  • Dictionary fluid-field theory (I): – riemannium Jan 04 '18 at 21:49
  • Velocity field <-> Gauge field – riemannium Jan 04 '18 at 21:49
  • Vorticity 2-form <-> Field strength – riemannium Jan 04 '18 at 21:50
  • Enstrophy <-> Action (Yang-Mills) – riemannium Jan 04 '18 at 21:50
  • How to generalize this stuff to non-abelian p-branes and other types of action or theories, not necessarily Yang-Mills type? – riemannium Jan 04 '18 at 21:51