Step 1: Differentiate the zonal momentum equation with respect to \(y\) \[\frac{\partial}{\partial y} \left( \textcolor{#002eff}{\frac{\partial u}{\partial t}} \right) + \frac{\partial}{\partial y} \left( \textcolor{#002eff}{u \frac{\partial u}{\partial x}} \right) + \frac{\partial}{\partial y} \left( \textcolor{#002eff}{v \frac{\partial u}{\partial y}} \right) - \frac{\partial}{\partial y} \left( \textcolor{#002eff}{f v} \right) = -g \frac{\partial^2 \eta}{\partial x \partial y}\] Step 2: Differentiate the meridional momentum equation with respect to \(x\) \[\frac{\partial}{\partial x} \left( \textcolor{#004d24}{\frac{\partial v}{\partial t}} \right) + \frac{\partial}{\partial x} \left( \textcolor{#004d24}{u \frac{\partial v}{\partial x}} \right) + \frac{\partial}{\partial x} \left( \textcolor{#004d24}{v \frac{\partial v}{\partial y}} \right) + \frac{\partial}{\partial x} \left( \textcolor{#004d24}{f u} \right) = -g \frac{\partial^2 \eta}{\partial x \partial y}\] Step 3: Subtract the meridional \(x\)-derivative from the zonal \(y\)-derivative, subtracting these eliminates the pressure gradient terms \[ \left( \frac{\partial}{\partial y} \textcolor{#002eff}{\frac{\partial u}{\partial t}} - \frac{\partial}{\partial x} \textcolor{#004d24}{\frac{\partial v}{\partial t}} \right) + \left( \frac{\partial}{\partial y} \textcolor{#002eff}{u \frac{\partial u}{\partial x}} - \frac{\partial}{\partial x} \textcolor{#004d24}{u \frac{\partial v}{\partial x}} \right) + \left( \frac{\partial}{\partial y} \textcolor{#002eff}{v \frac{\partial u}{\partial y}} - \frac{\partial}{\partial x} \textcolor{#004d24}{v \frac{\partial v}{\partial y}} \right) - \left( \frac{\partial}{\partial y} \textcolor{#002eff}{f v} + \frac{\partial}{\partial x} \textcolor{#004d24}{f u} \right) = 0 \] Step 4: Group into vorticity form by the defination of relative vorticity \( \boxed{\zeta = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}}\)
Group the term as time derivative of vorticity \(\boxed{\frac{\partial}{\partial t} \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) }\), nonlinear advection of vorticity \(\boxed{\frac{\partial}{\partial x} \left( u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) - \frac{\partial}{\partial y} \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right)}\) and coriolis terms (using \(f = f_0 + \beta y\)) \(\boxed{f_0 \left( \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \right) + \beta v}\)
Final Result: Vorticity Equation \[\frac{\partial}{\partial t} \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) + \frac{\partial}{\partial x} \left( u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) - \frac{\partial}{\partial y} \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) + f_0 \left( \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \right) + \beta v = 0\]