6.1 The barotropic streamfunction

With a rigid lid at the ocean surface, the surface height $\eta =0$. Setting $\eta$ to zero eliminates the very fast (order $200\,\mbox{m}\,\mbox{s}^{-1}$ in water with depth 4000-5000m) external mode gravity waves. As a result of making the rigid lid assumption, the vertically integrated horizontal velocity satisfies

$$\nabla_{h} \cdot {\bf U} {\bf u}_{h}(-H) \cdot \nabla_{h} H + \int^{0}_{-H} dz \; \nabla_{h} \cdot {\bf u}_{h}$$ =

$${\bf u}_{h}(-H) \cdot \nabla_{h} H - \int^{0}_{-H} dz \; w_{z}$$ =

$${\bf u}_{h}(-H) \cdot \nabla_{h} H + w(-H) - w(0)$$ =

= -w(0). (6.1)

To reach this result, Leibnitz's Rule (4.65), the continuity equation (4.3), and the bottom kinematic boundary condition (4.24) were used. Taking the additional assumption

w(z=0) = 0 (6.2)

renders

$$\nabla_{h} \cdot {\bf U} = \nabla_{h} \cdot H \, (\overline{u},\overline{v})$$ = 0. (6.3)

As a result, the external mode velocity can be expressed in terms of the external mode streamfunction

  

$$\overline{u} -\left(\frac{1}{Ha}\right) \psi_\phi$$ = (6.4)

$$\overline{v} \left(\frac{1}{Ha\cdot\cos\phi}\right) \psi_\lambda.$$ = (6.5)

As a vector equation, this relation takes the form

 

$${\bf U} = H \, (\overline{u},\overline{v}) = \hat{z} \wedge \nabla_{h} \psi.$$ (6.6)