The governing equations with hydrostatic, rigid-lid, and Boussinesq approximations are

 

where  is the vorticity of the vertically averaged flow,  is the (volume transport) streamfunction, f is the Coriolis parameter on the β plane, g’ is the reduced gravity, h₁ and h₂ are the layer thickness of the upper and lower layers, respectively, τ is the wind stress, and  are the difference of the upper-layer velocity () and lower-layer velocity () in x and y directions, respectively, and J(A,B) is the Jacobian (≡A_xB_y-A_yB_x); otherwise notation is standard. The vorticity and streamfunction are defined as