9: Vorticity
Before the discussion of ocean currents, there is one more key concept that needs to be introduced: vorticity. Essentially, vorticity is the same as rotation in a horizontal plane. Mathematically, the vorticity \(\zeta\) is defined as:
\[\zeta=\dfrac{dv}{dx}-\dfrac{du}{dy} \label{8.1}\]
Which is also just the curl of velocity.
\[\zeta=\vec{∇⃗} \times \vec{v}\]
Positive vorticity means that the fluid rotates counterclockwise, whereas negative vorticity implies clockwise rotation. Using the horizontal momentum balance equations \((1.2a)\) and \((1.2b)\) from Section 1 , a vorticity equation can be constructed, describing the time development of the vorticity of a fluid parcel. Under the assumption that the density of the fluid is constant, this equation becomes:
\[\dfrac{d\zeta}{dt}=\dfrac{d\left(\dfrac{dv}{dt}\right)}{dx}-\dfrac{d\left(\dfrac{du}{dt}\right)}{dy}= -f\left(\dfrac{du}{dx}+\dfrac{dv}{dy}\right)-\beta v+K_h \left(\dfrac{d^2\zeta}{dx^2}+\dfrac{d^2\zeta}{dy^2}\right)+K_v\dfrac{d^2\zeta}{dz^2} \label{8.2} \]
with \(\beta=\dfrac{df}{dy}\). The terms on the right-hand side of the equation can be interpreted as follows: \(-f\left(\dfrac{du}{dx}+\dfrac{dv}{dy}\right)\) says that horizontal divergence or convergence of the flow leads to rotation due to the Coriolis force ; \(\beta v\) is the so-called \(\beta\)-effect: as a parcel moves in the meridional (North-South) direction, it tends to spin up, because the Coriolis force is stronger on one side of the parcel than on the other side; the further terms simply indicate turbulent diffusion of vorticity.