The full vorticity equation rewritten with streamfunction $\psi$ is \begin{align} &\overbrace{ \underbrace{\tfrac{\partial}{\partial t} (\nabla^2 \psi)}_{\text{Local change of relative vorticity } \zeta = \tfrac{\partial v}{\partial x} - \tfrac{\partial u}{\partial y}} + \underbrace{J(\psi, \nabla^2 \psi)}_{\text{Jacobian, Nonlinear advection of vorticity}} }^{\text{Material derivative } \tfrac{D\zeta}{Dt}} + \underbrace{\beta \psi_x}_{\text{Meridional advection of planetary vorticity}} \notag \\ &= \underbrace{\tfrac{f_0}{D} w_E}_{\text{Column Stretching/compression by Ekman pumping}} - \underbrace{r \nabla^2 \psi}_{\text{Linear bottom friction}} + \underbrace{k_H \nabla^4 \psi}_{\text{Horizontal friction}} \end{align} with \( J(A, B) = A_x B_y - A_y B_x \)

Stommel's assumption

• Include only linear bottom drag: keep $-r \nabla^2 \psi$
• Ignore $\nabla^4 \psi$ and nonlinear terms

\begin{align} &\overbrace{ \cancel{ \underbrace{\tfrac{\partial}{\partial t} (\nabla^2 \psi)}_{\text{Local change of relative vorticity } \zeta}} + \cancel{ \underbrace{J(\psi, \nabla^2 \psi)}_{\text{Nonlinear advection of vorticity}}} }^{\text{Material derivative } \tfrac{D\zeta}{Dt}} + \underbrace{\beta \tfrac{\partial \psi}{\partial x}}_{\text{Meridional advection of planetary vorticity}} \notag \\ &= \underbrace{\tfrac{f_0}{D} w_E}_{\text{Ekman pumping (stretching/compression)}} - \underbrace{r \nabla^2 \psi}_{\text{Linear bottom friction}} + \cancel{ \underbrace{k_H \nabla^4 \psi}_{\text{Horizontal friction}}} \end{align} $$ \frac{\partial^2 \psi}{\partial x^2} \gg \frac{\partial^2 \psi}{\partial y^2} \quad \Rightarrow \quad \boxed{ \beta \frac{\partial \psi}{\partial x} = \frac{f_0}{D} w_E - r \frac{\partial^2 \psi}{\partial x^2} } $$

Munk's extension

• Adds horizontal friction using biharmonic term $+k_H \nabla^4 \psi$
• Important for resolving realistic boundary layers

\begin{align} &\overbrace{ \cancel{ \underbrace{\tfrac{\partial}{\partial t} (\nabla^2 \psi)}_{\text{Local change of relative vorticity } \zeta}} + \cancel{ \underbrace{J(\psi, \nabla^2 \psi)}_{\text{Nonlinear advection of vorticity}}} }^{\text{Material derivative } \tfrac{D\zeta}{Dt}} + \underbrace{\beta \tfrac{\partial \psi}{\partial x}}_{\text{Meridional advection of planetary vorticity}} \notag \\ &= \underbrace{\tfrac{f_0}{D} w_E}_{\text{Column Stretching/compression by Ekman pumping}} - \underbrace{\cancel{r \nabla^2 \psi}}_{\text{Linear bottom friction}} + \underbrace{k_H \nabla^4 \psi}_{\text{Horizontal friction}} \end{align} Applying the western boundary layer assumption $ \nabla^4 \psi \approx \frac{\partial^4 \psi}{\partial x^4} \quad \Rightarrow \quad \boxed{ \beta \frac{\partial \psi}{\partial x} = \frac{f_0}{D} w_E + k_H \frac{\partial^4 \psi}{\partial x^4} } $