"Where" does dissipated enstrophy go? We are all familiar with the kinetic energy dissipation and how it is converted into heat which can either be radiated away or go into the internal energy of the system. In the enstrophy transport equation:
\begin{align}
\frac{\partial\Omega^2}{\partial t} + u_j \frac{\partial\Omega^2}{\partial x_j} & = \omega_i S_{ij} \omega_j + \nu \frac{\partial^2\Omega^2}{\partial x_j\partial x_j} - \Phi_0 \\
\Omega^2 & = \frac{1}{2} \omega_i \omega_i \\
\Phi_0 & = \nu \frac{\partial\omega_i}{\partial x_j} \frac{\partial\omega_i}{\partial x_j} \\
S_{ij} & = \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)
\end{align}
there is a dissipation term, $\Phi_0$, very similar to that in the kinetic energy equation. Is there some mechanism or "place" where the dissipated enstrophy goes similar to the KE? Does enstrophy have to be conserved in the same sense that the total energy of a system (KE + PE + IE, etc.) has to be conserved?
Some people have explained it to me that since vorticity is a mathematical construct, then there is no "place" that the dissipated energy has to go. But you can describe velocity in the same sense as it is a construct that we created to represent particle motion in space.
Since the vorticity field is directly related to the velocity field (via the curl operator), then does that mean that the dissipated enstrophy is directly related to the dissipated kinetic energy? I'm currently attempting to reform and rewrite the enstrophy equation in terms of KE ($1/2 \times U_i U_i$) and see if there is any direct relation.
EDIT:
It is possible to rewrite both dissipation terms in terms of the strain rate and rotation rate tensor. This gives a slightly better picture of what's going on though it still doesn't answer my question.
\begin{align}
\omega_i = -\epsilon_{ijk} R_{jk}  \\
\frac{\Phi_0}{\nu} = \epsilon_{ijk} \epsilon_{inp} \frac{\partial R_{jk}}{\partial x_l} \frac{\partial R_{np}}{\partial x_l} = (\delta_{jn} \delta_{kp} - \delta_{jp} \delta_{kn})\frac{\partial R_{jk}}{\partial x_l}\frac{\partial R_{np}}{\partial x_l} = 2\frac{\partial R_{jk}}{\partial x_l}\frac{\partial R_{jk}}{\partial x_l} \\
\frac{\Phi_{KE}}{\nu} = \frac{\partial u_i}{\partial x_j}(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}) = (S_{ij} + R_{ij})(2S_{ij}) = 2S_{ij}S_{ij}  \\
\end{align}
 A: I'd say part of the answer must be that whatever dynamic variable you use, like Enstrophy, Vorticity, their potential analogues, etc. those are always 'filtered' fields.
Filtered in the sense, that you start with the velocity field $\vec v = \sum u_i \vec e_i$ that has full information over the dynamics and then apply some operators (integration and differentiation mostly) on top of that to generate your dynamic variable of interest.
Usually information is lost through that process. Sometimes, you can reconstruct $\vec v$ from the vorticity $\vec \omega$ in the incompressible fluid-case, as an example.
However my point here is, that the dissipation of those constructed variables, is always in the end the expression of dissipation of linear momentum, and therefore generation of heat, just filtered through the construction operator.
A: Wondering if the above contributions are making things rather too complicated.


*

*The notion of filtering is relevant to any computation or measurement, but not to the basic equations (unless I mistake your meaning, in which case, please do explain)

*One basic definition of vorticity states it is a measure of local solid-body motion of the fluid.  Thus, destruction of enstrophy should relate to a cessation of the relative motion associated with local solid-body rotation.  Though he does not mention enstrophy by name, BR Morton ("The generation and decay of vorticity" Geophys. Asotrphys. Fluid Dynamics, 1984, vol. 28, 277-308) states plainly that "the only means of decay or loss of vorticity is by cross-diffusion and annihilation of vorticity of opposite signs."   Since enstrophy is a measure of the intensity of that local rotation rate, we might say that enstrophy destruction arises from this mechanism.

*So, where does the "destroyed" enstrophy go (or better(?), transform into)?  The question presumes enstrophy to be a conserved quantity (like energy or mass - but NOT momentum).  The enstrophy equation itself belies this idea: if enstrophy were conserved, we could simply write d(enstrophy)/dt = 0.
Perhaps I'm oversimplifying.  But a return to basic definitions is a good place to start. Would be grateful for feedback on this!
