Chapter 7: Circulation and Vorticity

Circulation

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Integration is performed in a counterclockwise direction

C is positive for counterclockwise flow!!!

Kelvin’s Circulation Theorem

The rate of change of circulation can be expressed as:

What is the magnitude of each term in this equation?

For a barotropic fluid (density is a function only of pressure):

Geopotential term:

In a frictionless (inviscid) flow the friction term is also zero.

In a barotropic, inviscid fluid the circulation is constant!!!

Bjerknes’ Circulation theorem

Changes in circulation can arise due to friction or baroclinicity.

The sea breeze circulation

How does the temperature over the land and over the ocean vary during the course of a day?

What impact does this have on the thickness of an atmospheric column?

How does the atmosphere respond to this horizontal variation in thickness?

Example: Calculating the circulation associated with a sea breeze

Relative circulation

Circulation in the atmosphere arises due to our rotating frame of reference.

where,

If Cabsolute is conservedwhat does this imply about changes in Crelative for meridional flow?

Vorticity

For solid body rotation:

The earth undergoes solid body rotation with an angular velocity of , so:

What is the sign of earth in the Northern and Southern hemispheres?

The circulation around ABCD can be calculated as:

Then the relative vorticity, , is given by:

What is the sign of for clockwise and counterclockwise flow?

What does this imply about the sign of  for flow around low and high pressure centers?

Example: Calculation of relative vorticity from a weather map

Conservation of Potential Vorticity

What conditions were required for constant circulation according to Kelvin’s circulation theorem?

On a constant potential temperature, , surface:

This is analogous to a barotropic fluid.

Therefore, on a constant potential temperature surface the pressure gradient term is zero and Kelvin’s circulation theorem is satisfied.

This implies that:

Consider an air parcel that is confined between two potential temperature surfaces, and , separated by pressure interval .

The motion of this air parcel will be adiabatic.

The mass of the parcel is given by:

and must be conserved following the motion.

This gives:

Combining this result with gives an expression for Rossby-Ertel potential vorticity, P:

What does the term represent?

Potential vorticity depends on the depth of the fluid and the absolute vorticity.

/ Example: Conservation of potential vorticity and flow over the Rocky Mountains
Air column depth / Increase
/ Decrease
/ Increase
/ Decrease
(return to original value)
Change in
/ Decrease / Increase / Decrease / Increase
(return to original value)
Change in
/ Increase / Decrease / Increase / Decrease
Sign of  / Positive / Negative / Positive / Negative
Resulting motion / northward / southward / lee side trough / southward
Change in / Increase / Decrease / Increase / Decrease

For westerly flow across a mountain range a lee wave will form downstream of the mountain.

Stretching a column of the atmosphere results in generation of cyclonic vorticity. Shrinking of a column of the atmosphere results in generation of anticylonic vorticity.

The Vorticity Equation

Using Navier-Stokes equations scaled for mid-latitude weather systems we can derive an equation for the time rate of change of vorticity.

What causes the relative vorticity, at a fixed location, to change in time?

How does vorticity change for a non-divergent flow?

For quasi-geostrophic flow:

What does this equation tell us about changes in relative vorticity in a quasi-geostrophic framework?