# 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.