Basic Equations (The Math)

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Consider a two-dimensional Boussinesq fluid with a buoyancy frequency N, viscosity ν, and thermal
dissipation rate μ. We can write the vorticity equation as equation (1):



Here Ψ is the streamfunction and σ = (-gΔρ/ρ) is the buoyancy which is governed by equation (2):
The motion is then separated into mean and fluctuating components such that:



Now we can make some assumptions to simplify the math: 1. Use the mean field approximation

which allows you to neglect terms where the wave interacts with itself, 2. neglect viscous dissipation

since it is much smaller than thermal dissipation, 3. Changes in the mean flow take place over long

time scales compared to the wave motion, 4. The Richardson number is large:

,

5. the dissipation rate is small compared with the Doppler-shifted wave frequency, and 6. The buoyancy frequency is unaffected by
the waves.

With these approximations and the assumption that the fluctuation in the streamfunction has the
solution, ie waves, in equation (3):


Equations (1) and (2) become equation (4):



We also get a momentum flux, equation (5):



We then can use WKB approximation to solve equation (4) and from that we can get a new

version of equation (5), equation (6):



This equation tells us something quite interesting about the momentum flux. If we look at the exponent

we have an definite integral that goes from 0 to z. This shows us that the flux is only dependent on

how the wave is damped below the height z while it is independent of what goes on above it.

The term inside exponent is called the attenuation rate, equation (7):



Main Page Basic Model Attenuation Rate