Editing Internal wave (section)

==Mathematical modeling of internal waves==

The theory for internal waves differs in the description of interfacial waves and vertically propagating internal waves. These are treated separately below.

===Interfacial waves===

In the simplest case, one considers a two-layer fluid in which a slab of fluid with uniform density <math>\rho_1</math> overlies a slab of fluid with uniform density <math>\rho_2</math>. Arbitrarily the interface between the two layers is taken to be situated at <math>z=0.</math> The fluid in the upper and lower layers are assumed to be [[flow velocity#irrotational flow|irrotational]]. So the velocity in each layer is given by the gradient of a [[velocity potential]], <math>{\vec{u}=\nabla\phi,}</math> and the potential itself satisfies [[Laplace's equation]]:

:<math>\nabla^2\phi=0.</math>

Assuming the domain is unbounded and two-dimensional (in the <math>x-z</math> plane), and assuming the wave is [[periodic function|periodic]] in <math>x</math> with [[wavenumber]] <math>k>0,</math> the equations in each layer reduces to a second-order ordinary differential equation in <math>z</math>. Insisting on bounded solutions the velocity potential in each layer is

:<math>\phi_1(x,z,t) = A e^{-kz} \cos(kx - \omega t)</math>
and
:<math>\phi_2(x,z,t) = A e^{kz} \cos(kx - \omega t),</math>

with <math>A</math> the [[amplitude]] of the wave and <math>\omega</math> its [[angular frequency]]. In deriving this structure, matching conditions have been used at the interface requiring continuity of mass and pressure. These conditions also give the [[dispersion (water waves)|dispersion relation]]:<ref>{{cite book | first=O.M. | last=Phillips | author-link=Owen Martin Phillips |title=The dynamics of the upper ocean | publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=978-0-521-29801-8 | oclc=7319931 | page=37 }}</ref>

:<math>\omega^2 = g^\prime k</math>

in which the reduced gravity <math>g^\prime</math> is based on the density difference between the upper and lower layers:

:<math>g^\prime = \frac{\rho_2-\rho_1}{\rho_2+\rho_1}\, g,</math>

with <math>g</math> the [[Earth's gravity]]. Note that the dispersion relation is the same as that for deep water [[surface ocean wave|surface waves]] by setting <math>g^\prime=g.</math>

===Internal waves in uniformly stratified fluid===

The structure and dispersion relation of internal waves in a uniformly stratified fluid is found through the solution of the linearized conservation of mass, momentum, and internal energy equations assuming the fluid is incompressible and the background density varies by a small amount (the [[Boussinesq approximation (water waves)|Boussinesq approximation]]). Assuming the waves are two dimensional in the x-z plane, the respective equations are

:<math>\partial_x u + \partial_z w = 0</math>
:<math>\rho_{00} \partial_t u = - \partial_x p</math>
:<math>\rho_{00} \partial_t w = - \partial_z p - \rho g</math>
:<math>\partial_t \rho = -w d\rho_0/dz</math>

in which <math>\rho</math> is the perturbation density, <math>p</math> is the pressure, and <math>(u,w)</math> is the velocity. The ambient density changes linearly with height as given by <math>\rho_0(z)</math> and <math>\rho_{00}</math>, a constant, is the characteristic ambient density.

Solving the four equations in four unknowns for a wave of the form <math>\exp[i(kx+mz-\omega t)]</math> gives the dispersion relation

:<math>\omega^2 = N^2 \frac{k^2}{k^2+m^2} = N^2 \cos^2\Theta</math>

in which <math>N</math> is the [[Brunt–Väisälä frequency|buoyancy frequency]] and <math>\Theta=\tan^{-1}(m/k)</math> is the angle of the wavenumber vector to the horizontal, which is also the angle formed by lines of constant phase to the vertical.

The [[phase velocity]] and [[group velocity]] found from the dispersion relation predict the unusual property that they are perpendicular and that the vertical components of the phase and group velocities have opposite sign: if a wavepacket moves upward to the right, the crests move downward to the right.