= Internal wave
{wiki}
= Internal gravity wave
{synonym}
= IGW
{c}
{synonym}
An <internal wave> is an oscillation within a stably stratified <fluid>, where a displaced parcel experiences restoring <buoyancy>. Such waves can propagate through the interior of an <ocean> or <atmosphere>, unlike waves confined to a free surface. The local <buoyancy frequency> $N$ measures the stratification's restoring strength.
For an inviscid nonrotating <fluid> in the <Boussinesq approximation> with constant $N>0$, write horizontal and vertical <velocities> as $u,w$, pressure divided by reference density as $p$, and the <buoyancy> perturbation as $b$. The linear two-dimensional equations are
$$
u_t=-p_x,\qquad w_t=-p_z+b,\qquad
b_t=-N^2w,\qquad u_x+w_z=0.
$$
A <plane wave> proportional to $e^{i(kx+mz-\omega t)}$ therefore obeys
$$
\omega^2=\frac{N^2k^2}{k^2+m^2}.
$$
This is the nonrotating internal-gravity-wave <dispersion relation>. It gives $|\omega|\leq N$ and depends on the direction of the <wavevector>, rather than its magnitude. Consequently <group velocity> and <phase velocity> need not point together; in this uniform idealization the <group velocity> is perpendicular to the <wavevector>. Rotation, compressibility and varying stratification modify the relation. <Internal waves> can transport <energy> away from their source without transporting each oscillating parcel along the entire propagation path.
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