Vorticity and Quasi-Geostrophy

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Rossby Waves

Parcels of higher potential vorticity tend to be found as you move northward, as \(f\) always increases in this direction. In fact, PV tends to be negative in the southern hemisphere, while it tends to be positive in the northern hemisphere – just like \(f\).

From NCAR.

Let’s say we are in the jet stream in the midlatitudes of the northern hemisphere. You will tend to see crests that meander northward and troughs that meander southward from the primary direction of the stream. For the crests, the increase in \(f\) leads to a decrease in \(\zeta\), resulting in an additional clockwise component on its relative vorticity. For the troughs, the opposite happens: there is an increase in relative vorticity, inducing an additional counter-clockwise component. This leads to a westward phase velocity, even though the jet stream itself moves eastward. You can see this by noting how slowly the crests propagate compared to the bulk of the flow:

Note the importance of the \(\beta y\) term in generating this wave. Rossby waves cannot be generated on an f-plane, as variations in latitude are necessary. We will demonstrate this mathematically in a future blog post on waves.

Topographic Waves

Instead of varying the latitude (e.g. \(f\)), let us instead vary the height \(h\). The deeper the water, the larger the value of \(h\). Consider a stream of water moving along a coastline. If the water meanders closer to shore, a negative vortical component will be generated. If it meanders away from shore, a positive vortical component will be generated. This results in a wave that propagates alongshore, with the coast on the righthand side. This is analogous to the Rossby wave, where north is to south as the shallow direction is to the deep direction.


Western Boundary Current Intensification

We can use our understanding of potential vorticity conservation to explain why currents along western boundaries of ocean basins tend to be faster than those along eastern boundaries.

Consider the trade winds, which blow westward near the equator, and the westerlies, which blow eastward in the midlatitudes.

From Wikipedia.

The curl of the wind stress between these two prevailing winds points downward, thus creating a downwelling in the ocean. This results in a decrease in the height of the water column. By potential vorticity conservation, this means that we need a corresponding decrease in vorticity. The geostrophic flow thus moves equatorward to compensate.

We can see this mathematically by looking at the vorticity equation we derived earlier, but assuming geostrophic balance so the advective term is small:

$$(f_0 + \beta y)\nabla \cdot \vec{u} + \beta v = 0$$

If we replace \(\nabla \cdot \vec{u}\) with \(-\frac{\partial w}{\partial z}\), we get:

$$ \beta v = f \frac{\partial w}{\partial z}$$

When there is downwelling, \(\frac{\partial w}{\partial z}\) becomes negative, resulting in a negative meridional velocity (i.e. equatorward flow).

However, by conservation of mass, this must be balanced by an ageostrophic poleward flow somewhere else. This implies that the flow should be narrow and fast (i.e. outside the typical scales that define geostrophic flow). We expect that this will take the form of a boundary current. The question is: should it be a westward or eastward boundary current?

If a parcel moves north, then \(f\) increases. To conserve PV, it must be true that the relative vorticity is negative. If the current is on the eastern boundary, then \(\frac{\partial v}{\partial x} > 0\) and the relative vorticity is positive. But if the current is on the western boundary, we achieve the desired result of negative PV.

On the next page we’ll begin talking about Quasi-Geostrophy, the last part of Paul Spence’s lecture.


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