The planetary-geostrophic equations for a Boussinesq fluid are:
\( \underbrace{\frac{Db}{Dt} = \dot{b}}_{\text{Thermodynamic equation}}, \quad \underbrace{\nabla_3 \cdot \mathbf{v} = 0}_{\text{Mass continuity equation}}, \quad \underbrace{f \times \mathbf{u} = -\nabla \phi + \frac{1}{\rho_0} \frac{\partial \tau}{\partial z}}_{\text{Horizontal momentum equation}}, \quad \underbrace{\frac{\partial \phi}{\partial z} = b}_{\text{Vertical momentum equation}} \)
\(
\because f \times \mathbf{u} = -\nabla \phi + \frac{1}{\rho_0} \frac{\partial \boldsymbol{\tau}}{\partial z}
\)
\(
\because \text{Take horizontal curl \( \hat{\mathbf{z}} \cdot \nabla \times (\cdot) \) to both sides of the equation}
\)
\(
\therefore \nabla \times (-\nabla \phi) = 0 , \hat{\mathbf{z}} \cdot \left[ \nabla \times (f \times \mathbf{u}) \right] = \frac{1}{\rho_0} \hat{\mathbf{z}} \cdot \left[ \nabla \times \frac{\partial \boldsymbol{\tau}}{\partial z} \right]
\)
\(
\because \nabla \times (f \mathbf{u}) = f \nabla \cdot \mathbf{u} + (\nabla f) \times \mathbf{u}
\)
\(
\because \hat{\mathbf{z}} \cdot [(\nabla f) \times \mathbf{u}] = \frac{\partial f}{\partial y} v
\)
\(
\therefore f \nabla \cdot \mathbf{u} + \frac{\partial f}{\partial y} v = \frac{1}{\rho_0} \hat{\mathbf{z}} \cdot \left[ \frac{\partial}{\partial z} (\nabla \times \boldsymbol{\tau}) \right]
\)
\(
\Rightarrow f \nabla \cdot \mathbf{u} + \frac{\partial f}{\partial y} v = \frac{1}{\rho_0} \frac{\partial}{\partial z} \left( \nabla \times \boldsymbol{\tau} \right)_z
\)
\(
\because \text{Integrate vertically from } z = B \text{ to } z = T:
\)
\(\int_B^T \left[ f \nabla \cdot \mathbf{u} + \frac{\partial f}{\partial y} v \right] dz = \frac{1}{\rho_0} \int_B^T \frac{\partial}{\partial z} \left( \nabla \times \boldsymbol{\tau} \right)_z dz\)
\(
\int_B^T f \nabla \cdot \mathbf{u} \, dz + \frac{\partial f}{\partial y} \int_B^T v \, dz = \frac{1}{\rho_0} \int_B^T \frac{\partial}{\partial z} \left( \nabla \times \boldsymbol{\tau} \right)_z \, dz
\)
\(
\therefore \int_B^T \frac{\partial f}{\partial y} v \, dz = \frac{\partial f}{\partial y} \int_B^T v \, dz
\)
\(
\Rightarrow \int_B^T \frac{\partial}{\partial z} \left( \nabla \times \boldsymbol{\tau} \right)_z dz = \left( \nabla \times \boldsymbol{\tau} \right)_z \Big|_T - \left( \nabla \times \boldsymbol{\tau} \right)_z \Big|_B
\)
\(
\because \left( \nabla \times \boldsymbol{\tau} \right)_z = \frac{\partial \tau_y}{\partial x} - \frac{\partial \tau_x}{\partial y} \equiv \text{curl}_z(\boldsymbol{\tau})
\)
\(
\frac{1}{\rho_0} \left[ \left( \nabla \times \boldsymbol{\tau} \right)_z \right]_T^B = \frac{1}{\rho_0} \left[ \left( \frac{\partial \tau_y}{\partial x} - \frac{\partial \tau_x}{\partial y} \right)\Big|_T - \left( \frac{\partial \tau_y}{\partial x} - \frac{\partial \tau_x}{\partial y} \right)\Big|_B \right] = \text{curl}_z\left( \frac{1}{\rho_0} \boldsymbol{\tau}_T - \frac{1}{\rho_0} \boldsymbol{\tau}_B \right)
\)
\(
\therefore \int f \nabla \cdot \mathbf{u} \, dz + \frac{\partial f}{\partial y} \int v \, dz = \frac{1}{\rho_0} \left[ \left( \nabla \times \boldsymbol{\tau} \right)_z \right]_T^B
\)
\(
\Rightarrow \boxed{\int f \nabla \cdot \mathbf{u} \, dz + \frac{\partial f}{\partial y} \int v \, dz = \text{curl}_z (\boldsymbol{\tau}_T - \boldsymbol{\tau}_B)}
\)