Take the curl of \(f \times \mathbf{u} = -\nabla \phi + \frac{1}{\rho_0} \frac{\partial \boldsymbol{\tau}}{\partial z}\) (cross differentiate its \( x \) and \( y \) components) and integrate over the depth of the ocean to give
\[ \int f \nabla \cdot \mathbf{u} \, dz + \frac{\partial f}{\partial y} \int v \, dz = \text{curl}_z (\boldsymbol{\tau}_T - \boldsymbol{\tau}_B) \]where the operator \( \text{curl}_z \) is defined by \( \text{curl}_z \mathbf{A} \equiv \frac{\partial A^y}{\partial x} - \frac{\partial A^x}{\partial y} = \mathbf{k} \cdot \nabla \times \mathbf{A}, \) and the subscripts \( T \) and \( B \) are for top and bottom
\(
\because f \nabla \cdot \mathbf{u} = \nabla \cdot (f \mathbf{u}) - (\nabla f) \cdot \mathbf{u}
\)
\(f = f(y) \Rightarrow \nabla f = \frac{\partial f}{\partial y} \hat{\mathbf{y}}, \quad \mathbf{u} = u \hat{\mathbf{x}} + v \hat{\mathbf{y}} \)
\(
\therefore (\nabla f) \cdot \mathbf{u} = \frac{\partial f}{\partial y} v
\)
\(
\therefore \int (\nabla f) \cdot \mathbf{u} \, dz = \frac{\partial f}{\partial y} \int v \, dz
\)
\(
\int f \nabla \cdot \mathbf{u} \, dz = \int \left[ \nabla \cdot (f \mathbf{u}) - \frac{\partial f}{\partial y} v \right] dz
\)
\(\Rightarrow \int f \nabla \cdot \mathbf{u} \, dz = \int \nabla \cdot (f \mathbf{u}) \, dz - \frac{\partial f}{\partial y} \int v \, dz\)
\(
\because \int f \nabla \cdot \mathbf{u} \, dz + \frac{\partial f}{\partial y} \int v \, dz = \text{curl}_z(\boldsymbol{\tau}_T - \boldsymbol{\tau}_B)
\)
\(\therefore \left( \int \nabla \cdot (f \mathbf{u}) \, dz - \cancel{\frac{\partial f}{\partial y} \int v \, dz} \right) + \cancel{\frac{\partial f}{\partial y} \int v \, dz} = \text{curl}_z(\boldsymbol{\tau}_T - \boldsymbol{\tau}_B)\)
\(
\because \int_B^T \nabla \cdot (f \mathbf{u}) \, dz = \nabla \cdot \left( \int_B^T f \mathbf{u} \, dz \right) - \left[ f \, \mathbf{u} \cdot \hat{\mathbf{z}} \right]_B^T, \quad \mathbf{u} \cdot \hat{\mathbf{z}} = 0 \Rightarrow \text{no vertical flux of horizontal velocity}
\)
\(
\therefore \int \nabla \cdot (f \mathbf{u}) \, dz = \nabla \cdot \left( \int f \mathbf{u} \, dz \right)
\)
\(
\because \text{Continuity equation \( \nabla \cdot \mathbf{v} = 0 \), where \(\mathbf{v} = (u, v, w)\)}
\)
\(
\Rightarrow \int_B^T \nabla \cdot \mathbf{u} \, dz = -\int_B^T \frac{\partial w}{\partial z} \, dz = -w|_T + w|_B
\)
\(
\therefore \text{if \( w = 0 \) at both the surface and bottom (assume rigid lid and flat bottom):}
\)
\(
\int \nabla \cdot \mathbf{u} \, dz = 0 \Rightarrow \int f \nabla \cdot \mathbf{u} \, dz = 0
\)
\(
\Rightarrow \frac{\partial f}{\partial y} \int v \, dz = \text{curl}_z(\boldsymbol{\tau}_T - \boldsymbol{\tau}_B)
\)
\(
\because \frac{\partial f}{\partial y} = \beta \Rightarrow \beta \int v \, dz = \text{curl}_z(\boldsymbol{\tau}_T - \boldsymbol{\tau}_B)
\)
\(
\therefore \beta \int v \, dz = \text{curl}_z(\boldsymbol{\tau}_T - \boldsymbol{\tau}_B) \Rightarrow \boxed{\beta \overline{v} = \text{curl}_z(\boldsymbol{\tau}_T - \boldsymbol{\tau}_B)}
\)