Equations of motion \begin{align} \text{I:} \quad & -fv = -\frac{1}{\rho_0} \frac{\partial p}{\partial x} + \frac{1}{\rho_0} \frac{\partial \tau^x}{\partial z} \\ \text{II:} \quad & fu = -\frac{1}{\rho_0} \frac{\partial p}{\partial y} + \frac{1}{\rho_0} \frac{\partial \tau^y}{\partial z} \\ \text{III:} \quad & 0 = -\frac{1}{\rho_0} \frac{\partial p}{\partial z} + g \\ \text{IV:} \quad & \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \quad \text{(continuity equation)} \end{align}

Combine equations (I) and (II) \begin{align} \frac{\partial}{\partial x} \big(\text{II}\big) \;-\; \frac{\partial}{\partial y} \big(\text{I}\big): \quad f \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) + \beta v = \frac{1}{\rho_0} \left( \frac{\partial^2 p}{\partial x \partial y} - \frac{\partial^2 p}{\partial y \partial x} \right) + \frac{1}{\rho_0} \left( \frac{\partial^2 \tau^y}{\partial x \partial z} - \frac{\partial^2 \tau^x}{\partial y \partial z} \right) \end{align}

Combine with continuity equation (IV) \begin{align} \beta v = f \frac{\partial w}{\partial z} + \frac{1}{\rho_0} \frac{\partial}{\partial z} \left( \frac{\partial \tau^y}{\partial x} - \frac{\partial \tau^x}{\partial y} \right) \end{align}

If there is no friction below the mixed layer, this simplifies to \begin{align} \beta V = f \frac{\partial w}{\partial z} \end{align} This is the Sverdrup Relation

At the ocean surface and bottom, \( w = 0 \). Taking the depth integral \(\boxed{V_s = \int_{-D}^{0} v \, dz}\) yields \begin{align} V_s = \frac{1}{\beta \rho_0} \left( \frac{\partial \tau^y}{\partial x} - \frac{\partial \tau^x}{\partial y} \right) = \frac{1}{\beta \rho_0} \, \text{curl} \, \boldsymbol{\tau} \end{align}

!!! The Sverdrup Balance!