Ch 12. Internal gravity waves

Overview

Stratified fluids support a wave family with a genuinely strange dispersion relation: frequency depends only on the angle of the wavevector, not its magnitude. Shake a stratified tank at one point and the response isn’t a circular ripple — it’s four narrow beams radiating out at a fixed angle, carrying energy in a direction perpendicular to their own phase propagation. This chapter builds that dispersion relation from the linear equations of motion, then uses it to explain two things every internal wave does: it picks its own propagation angle from $\omega$ , $N$ , and $f$ alone, and it reflects wherever the local stratification $N(z)$ drops below its own frequency — a mechanism identical in spirit to a light ray hitting total internal reflection, except here the “refractive index” is the local buoyancy frequency.

The model

Notebook: notebooks/ch12_internal-gravity-waves.py → exported to /nb/ch12_internal-gravity-waves.html. Governing equations use the symbols in notation:

\frac{\partial^2 q}{\partial t^2} = -\Big(N^2(z)\,\psi_{xx}+f^2\,\psi_{zz}\Big)+S, \qquad q=\nabla^2\psi,\qquad u=\psi_z,\ w=-\psi_x.

The $(x,z)$ plane reuses gfdlib.spectral.Grid exactly as the horizontal chapters do, with the grid’s $y$ -axis standing in for depth. A leapfrog scheme (gfdlib.internalwaves.step_leapfrog) evaluates $\psi_{xx},\psi_{zz}$ spectrally but multiplies by $N^2(z)$ in physical space, so — unlike the textbook two-layer or constant- $N$ treatments — the stratification is free to vary with depth without any special-case machinery. Two presets: uniform $N$ (the classic St. Andrew’s Cross) and a cosine-shaped thermocline that creates a wave duct, bounded above and below by genuine turning-level reflections the solver produces on its own.

What to look for

Four beams, one angle. Run the St. Andrew’s Cross preset: the point forcing radiates four beams whose inclination is set by $\omega/N$ alone — steepen them toward vertical by raising $\hat\omega$ toward $\hat N$ , flatten them by lowering it. No property of the forcing (its size, its strength) moves that angle; only the frequency does.
Phase across, energy along. Zoom your attention onto one beam and scrub consecutive frames: the crests march across the beam while the beam itself extends along its own axis. That is $\mathbf c_g\perp\mathbf k$ seen directly — energy propagating perpendicular to phase, the signature strangeness of internal waves.
The waveguide. Switch to the thermocline duct preset: where the local $N(z)$ falls below $\omega$ , the beam bends back — a turning level. The solver was never told about reflection; it emerges from $N^2(z)\psi_{xx}$ alone. Watch the reflected beam interfere with the incident one inside the duct.
Rotation closes the window from below. Turn $\hat f$ up from zero: nothing dramatic happens until $\hat\omega$ approaches $\hat f$ , and then the beams flatten toward horizontal and the response stalls — the other edge of the internal-wave frequency window $f<\omega<N$ . Waves exist only inside it; both presets live or die by where $\hat\omega$ sits in that window.

Prefer to run it yourself? Download as a Jupyter notebook — it installs its own dependencies in the first cell. The sliders are static in Jupyter; edit value= and re-run to change a parameter.

Both fluids

Atmosphere: mountain waves are internal gravity waves forced by flow over topography; they propagate obliquely upward at the angle this chapter’s dispersion relation predicts, and can reflect at a critical/turning level where a wind reversal or a stratification change intervenes — the same geometry as this chapter’s duct, with a different cause.
Ocean: the barotropic tide flowing over ocean-ridge topography generates internal tides — low-mode internal waves radiating away at a fixed angle set by the tidal frequency and the local $N(z)$ profile; their breaking supplies roughly 90% of the energy for deep-ocean diapycnal mixing.

Exercises

(analytic) From $\omega^2=N^2\cos^2\theta+f^2\sin^2\theta$ , show that $\omega$ is bounded between $f$ and $N$ for any real $\theta$ , and that group velocity is perpendicular to phase velocity (differentiate $\omega(k,m)$ and dot the result into $\mathbf k=(k,m)$ ).
(computational) Modify ch12_internal-gravity-waves.py to force with two point sources at different depths instead of one. Do the resulting beam patterns superpose linearly, as the governing equation’s linearity predicts?
(exploratory) In the thermocline-duct preset, sweep $\hat\omega$ upward at fixed $\hat N_{mean},\hat N_{amp}$ : find the value at which the two turning levels merge with the domain boundary and the wave stops being trapped.