The enstrophy equation describes the production and destruction of enstrophy Ω = 1 2 | ω → | 2 {\displaystyle \Omega ={\frac {1}{2}}|{\vec {\omega }}|^{2}}
D Ω D t = ω i ω j e i j ⏟ vortex tilting/stretching − g ρ 0 ω → ⋅ [ ∇ × ( ρ k ^ ) ] ⏟ baroclinic − ν ( ∇ × ω → ) 2 ⏟ viscous destruction + ν ∇ ⋅ [ ω → × ( ∇ × ω → ) ] ⏟ boundary production {\displaystyle {\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.
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69f103d567947ff17dffc0d9688e241af43887d4)