Mountain and Lee Waves

Nonlinear Terms

The nonlinear terms require that the amplitude of mountains are not small compared to the horizontal length. (Otherwise, the nonlinear terms are considered negligible.) A complete mathematical analysis on the nonlinear effects on mountain waves are incomplete. However, nonlinear effects are responsible for strong downslope winds and clear-air turbulence.

Setting U and N to be constants, one then uses Long's Equation to solve for the streamline displacement δ(x,z).

The only difference between linear and nonlinear solutions for this equation is defined by the lower boundary condition.

Even then, there is only a minor difference from the nonlinear terms unless the mountain amplitude is large or there are variations in N and U. This is much harder to calculate, and will not be looked at in this brief overview.

There is a factor to determine whether nonlinearity is important in an equation. This is defined by the Froude Number, which is best used as an order of magnitude approximation. The Froude Number is:
Where H is the depth scale of motion, and NH is the horizontal speed of the wave.

For Fr <<1, the flow is supercritical, there will be a downslope wind, and nonlinear terms will be important.
For Fr > 1, the flow will be subcritical, and will largely be laminar; nonlinear terms are not significant.