Enstrophy equation

The enstrophy equation describes the production and destruction of enstrophy $\Omega ={\frac {1}{2}}|{\vec {\omega }}|^{2}$

${\frac {D\Omega }{Dt}}=\underbrace {\omega _{i}\omega _{j}e_{ij}} _{\text{vortex tilting/stretching}}-\underbrace {{\frac {g}{\rho _{0}}}{\vec {\omega }}\cdot \left[\nabla \times (\rho {\hat {k}})\right]} _{\text{baroclinic}}-\underbrace {\nu (\nabla \times {\vec {\omega }})^{2}} _{\text{viscous destruction}}+\underbrace {\nu \nabla \cdot \left[{\vec {\omega }}\times (\nabla \times {\vec {\omega }})\right]} _{\text{boundary production}}$

From left to right, the terms are production/destruction due to vortex tilting/stretching, baroclinic production, viscous destruction, and boundary production.