Student(s):
Aaron Smith
Dates of Involvement:
Spring 2008 – Present
Faculty Advisor(s):
Keith Julien
Graduate Mentor:
Instabilities of Internal Gravity Waves
Background
When a fluid parcel in a stably stratified fluid is displaced from equilibrium gravity acts as a restoring force on the parcel returning it to equilibrium. If the parcel overshoots the equilibrium level the buoyancy force pushes the parcel back up to equilibrium. The resulting motion is oscillations called internal gravity waves. Internal gravity waves are frequently observed in both the ocean and atmosphere. Perturbations in the wave field lead to instabilities in the waves and can eventually grow and cause the wave to break. Momentum flux from breaking waves can alter flows in the upper atmosphere ultimately effecting weather systems. Gravity wave breaking is also a cause of clear air turbulence Predicting where and when clear air turbulence happens could be useful for air traffic. These are two simple applications that motivate the study of gravity wave instabilities.
The motion of the fluid particles can be described mathematically by the continuity equation, momentum equation and energy equation in Boussinesq fluid.
Where 
is the velocity vector, 
is the deviation from hydrostatic pressure, 
is the Boussinesq density, g is the acceleration of gravity, 
is a diffusion coefficient and 
is the kinematic viscosity.
By following the analysis used by Lombard and Riley [1], the equations of motion can be rotated into a coordinate system where the wave is periodic in only one direction, 
, which is aligned with the wave phase velocity. To assess stability characteristics a base wave solution can be perturbed according to
where 
are eigenfunctions of , u, v, w and ; 
and 
are perturbation wave numbers and 
is a complex eigenvalue associated with the growth rate of the perturbation. Using a numerical eigenvalue finding scheme the growth rate for the instabilities can be determined. Analysis of the growth rates can determine the conditions for which the wave will break.
In the atmosphere, as the wave propagates it would encounter different background flows in different layers. The results presented in Lombard and Riley assumes that the wave is not propagating in a background flow. The ultimate goal of this project is to superimpose a background fine structure shear flow given by
and analyze the growth rate of the perturbations. High-resolution simulations of wave breaking in the same background shear flow were performed at Colorado Research Associates (CoRA). The linear stability analysis results will eventually be compared with, and used to better understand, the results of these simulations.
About Aaron Smith:
Aaron Smith is currently in his 3rd year at the University of Colorado, where he is majoring in Applied Mathematics with an emphasis in Aerospace Engineering. Aaron has a strong interest in computational fluid dynamics and geophysical fluids and hopes to peruse a graduate degree in Applied Mathematics with an emphasis in fluid mechanics.
References:
[1] Instability and breakdown of internal gravity waves. I. Linear stability analysis Peter N. Lombard and James J. Riley, Phys. Fluids 8, 3271 (1996), DOI:10.1063/1.869117
