12.1: Definitions of Vorticity

In simple words, vorticity is the rotation of the fluid. The rate of rotation can be defined various ways. Consider a bowl of water sitting on a table in a laboratory. The water may be spinning in the bowl. In addition to the spinning of the water, the bowl and the laboratory are rotating because they are on a rotating Earth. The two processes are separate and lead to two types of vorticity.

Planetary Vorticity

Everything on Earth, including the ocean, the atmosphere, and bowls of water, rotates with the Earth. This rotation is the planetary vorticity \(f\). It is twice the local rate of rotation of Earth: \[\boxed{f \equiv 2 \Omega \sin \varphi \ (\text{radians/s}) = 2 \sin \varphi \ (\text{cycles/day})} \nonumber \]

Planetary vorticity is the Coriolis parameter I used earlier to discuss flow in the ocean. It is greatest at the poles where it is twice the rotation rate of Earth. Note that the vorticity vanishes at the Equator and that the vorticity in the Southern Hemisphere is negative because \(\varphi\) is negative.

Relative Vorticity

The ocean and atmosphere do not rotate at exactly the same rate as Earth. They have some rotation relative to earth due to currents and winds. Relative vorticity \(\zeta\) is the vorticity due to currents in the ocean. Mathematically it is: \[\boxed{\zeta \equiv \text{curl}_{z} \mathbf{V} = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}} \nonumber \]

where \(\mathbf{V} = (u, v)\) is the horizontal velocity vector, and where we have assumed that the flow is two-dimensional. This is true if the flow extends over distances greater than a few tens of kilometers. \(\zeta\) is the vertical component of the three-dimensional vorticity vector \(\omega\), and it is sometimes written \(\omega_{z}\). \(\zeta\) is positive for counter-clockwise rotation viewed from above. This is the same sense as Earth’s rotation in the Northern Hemisphere.

Note on Symbols

Symbols commonly used in one part of oceanography often have very different meaning in another part. Here I use \(\zeta\) for vorticity, but in Chapter 10, I used \(\zeta\) to mean the height of the sea surface. I could use \(\omega_{z}\) for relative vorticity, but \(\omega\) is also commonly used to mean frequency in radians per second. I have tried to eliminate most confusing uses, but the dual use of \(\zeta\) is one we will have to live with. Fortunately, it shouldn’t cause much confusion.

For a rigid body rotating at rate \(\Omega\), \(\text{curl} \mathbf{V} = 2\Omega\). Of course, the flow does not need to rotate as a rigid body to have relative vorticity. Vorticity can also result from shear. For example, at a north/south western boundary in the ocean, \(u = 0\), \(v = v(x)\), and \(\zeta = \partial v(x)/\partial x\).

\(\zeta\) is usually much smaller than \(f\), and it is greatest at the edge of fast currents such as the Gulf Stream. To obtain some understanding of the size of \(\zeta\), consider the edge of the Gulf Stream off Cape Hatteras where the velocity decreases by 1 m/s in 100 km at the boundary. The curl of the current is approximately \((1 \ \text{ m/s})/(100 \ \text{km}) = 0.14 \ \text{cycles/day} = 1 \ \text{cycle/week}\). Hence, even this large relative vorticity is still almost seven times smaller than \(f\). A more typical value of relative vorticity, such as the vorticity of eddies, is a cycle per month.

Absolute Vorticity

The sum of the planetary and relative vorticity is called absolute vorticity: \[\boxed{\text{Absolute Vorticity} \equiv (\zeta + f)} \nonumber \]

We can obtain an equation for absolute vorticity in the ocean by manipulating the equations of motion for frictionless flow. We begin with: \[\begin{align} \frac{Du}{Dt} - fv &= -\frac{1}{\rho} \frac{\partial p}{\partial x} \\ \frac{Dv}{Dt} + fu &= -\frac{1}{\rho} \frac{\partial p}{\partial y} \end{align} \nonumber \]

If we expand the substantial derivative, and if we subtract \(\partial/\partial y\) of \((\PageIndex{4})\) from \(\partial/\partial x\) of \((\PageIndex{5})\) to eliminate the pressure terms, we obtain after some algebraic manipulations: \[\boxed{\frac{D}{Dt} (\zeta + f) + (\zeta + f) \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right) = 0} \nonumber \]

In deriving \((\PageIndex{6})\) we used: \[\frac{Df}{Dt} = \frac{\partial f}{\partial t} + u \frac{\partial f}{\partial x} + v \frac{\partial f}{\partial y} = \beta v \nonumber \]because \(f\) is independent of time \(t\) and eastward distance \(x\).

Potential Vorticity

The rotation rate of a column of fluid changes as the column is expanded or contracted. This changes the vorticity through changes in \(\zeta\). To see how this happens, consider barotropic, geostrophic flow in an ocean with depth \(H(x, y, t)\), where \(H\) is the distance from the sea surface to the bottom. That is, we allow the surface to have topography (Figure \(\PageIndex{1}\)).

Figure \(\PageIndex{1}\): Sketch of fluid flow used for deriving conservation of potential vorticity. After Cushman-Roisin (1994: 55).

Integrating the continuity equation \((7.7.3)\) from the bottom to the top of the ocean gives (Cushman-Roisin, 1994): \[\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right) \int_{b}^{b+H} dz + w|_{b}^{b+H} = 0 \nonumber \]

where \(b\) is the topography of the bottom, and \(H\) is the depth of the water. Notice that \(\partial u/\partial x\) and \(\partial v/\partial y\) are independent of \(z\) because they are barotropic, and the terms can be taken outside the integral.

The boundary conditions require that flow at the surface and the bottom be along the surface and the bottom. Thus the vertical velocities at the top and the bottom are: \[\begin{align} w(b+H) &= \frac{\partial (b+H)}{\partial t} + u \frac{\partial (b+H)}{\partial x} + v \frac{\partial (b+H)}{\partial y} \\ w(b) &= u \frac{\partial (b)}{\partial x} + v \frac{\partial (b)}{\partial y} \end{align} \nonumber \]

where we used \(\partial b/\partial t = 0\) because the bottom does not move, and \(\partial H/\partial z = 0\). Substituting \((\PageIndex{8})\) and \((\PageIndex{9})\) into \((\PageIndex{7})\) we obtain \[\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right) + \frac{1}{H} \frac{DH}{Dt} = 0 \nonumber \]

Substituting this into \((\PageIndex{6})\) gives: \[\frac{D}{Dt} (\zeta + f) - \frac{(\zeta + f)}{H} \frac{DH}{Dt} = 0 \nonumber \]

which can be written: \[\frac{D}{Dt} \left(\frac{(\zeta + f)}{H}\right) = 0 \nonumber \]

The quantity within the parentheses must be constant. It is called potential vorticity \(\Pi\). Potential vorticity is conserved along a fluid trajectory: \[\boxed{\text{Potential Vorticity} = \Pi = \frac{\zeta + f}{H}} \nonumber \]

For baroclinic flow in a continuously stratified fluid, the potential vorticity can be written (Pedlosky, 1987: §2.5): \[\Pi = \frac{\zeta + f}{\rho} \cdot \nabla \lambda \nonumber \]

where \(\lambda\) is any conserved quantity for each fluid element. In, particular, if \(\lambda = \rho\) then: \[\Pi = \frac{\zeta + f}{\rho} \frac{\partial \rho}{\partial z} \nonumber \]

assuming the horizontal gradients of density are small compared with the vertical gradients, a good assumption in the thermocline. In most of the interior of the ocean, \(f >> \zeta\) and \((\PageIndex{12})\) is written (Pedlosky, 1996, eq 3.11.2): \[\Pi = \frac{f}{\rho} \frac{\partial \rho}{\partial z} \nonumber \]

This allows the potential vorticity of various layers of the ocean to be determined directly from hydrographic data without knowledge of the velocity field.