A derivation of the vorticity equation from conservation of angular momentum:

A barotropic fluid is one whose local density or density gradient depends only on the pressure. Under those conditions, one can write the vorticity equation as:



where ζ = relative vorticity and f = planetary vorticity; later we will see that ζ has a shear component and what is called a curvature component. h is the depth of our fluid and its time rate of change represents a divergence term though this is weak as it goes as the ln.

In more general terms, the above equation is expressed as:



An axisymmetric column of fluid rotating at a fixed point on the Earth's surface has two contributions to its absolute angular momentum. One is due to its motion around the Earth's axis of rotation (orbital angular momen tum), and the other due to its spin around the vertical axis through its center of mass (spin angular momentum)



If we assume the column to be only infinitesimally wide then its rotation approximates solid body rotation such that L_s = IΩ_s



So we can take the time derivative as:

D/dt(L_s) = 0 = d/dt(IΩ_s)

since both of these variables are function of time, then differentiation (using the dot product rule) leads to:





We can transform this equation into these terms



Which we can compare to our starting situation

Are these the same?

The moment of inertia for a column is 1/2MR² and that reduces to



where U = DR/DT = velocity of particles at radius R (assumed to be edge of the column).





And a planetary wave is then the global response of the need to conserve angular momentum in the midst of networks of smaller rotating fluids embedded in an overall rotating atmosphere (or ocean).

Sample scale calculation:

The total vorticity is the sum of planetary vorticity + shear vorticity + curvature vorticity.

An air parcel is located at 35 degrees north. The parcel spins cyclonically (counterclock wise taking to be positive) at 15 degrees per hour due to curvature. The wind shear produces an anti-cyclonic spin. The change is wind speed over distance is 10 m/s over 300 km. What is the value of absolute vorticity at this point? For convenience we take units of vorticity as 10^-5 seconds.

• Planetary vorticity = coriolis = 2Ωsin(35) = +8.37 units
  
  
• Shear vorticity: this is measured at the 500 mb level. In this case the anti-cyclonic so this is negative vorticity. In this case the shear vorticity is 10/300000 = -3.33 units
  
  
• Curvature vorticity: This basically the angular speed of 15 degrees per hour which is +7.27 units.
  
  Absolute vorticity = 8.37 + 7.27 - 3.33 = +12.31 UNITS
  
  Curvature vorticity is basically cyclostrophic balance and the point of this excersize was to demonstate that, at mid latitudes, the rotating of systems around some axis is a significant component of total vorticity.