Internal Waves on a Thermocline

What is modelled

An ocean of depth H holds a light upper layer of thickness h1 above a denser lower layer. A wave on the interface between them is an internal wave: the density jump is small, so gravity is effectively reduced and the wave is slow and large. With a rigid lid at the top, the layers move in opposite directions and the sea surface stays almost flat.

Equations

• Dispersion relation ω² = (ρ2 − ρ1) g k / (ρ1 coth(k h1) + ρ2 coth(k h2)), with k = 2π/wavelength.
• Long-wave limit c0 = √(g Δρ / (ρ1/h1 + ρ2/h2)), close to √(g′ h₁ h₂ / H) with reduced gravity g′ = g Δρ / ρ.
• Layer currents from potential flow: u1 = −ω a cosh k(h1 − z) / sinh k h1 and the matching expression below.
• Energy per unit area: kinetic equals potential, g Δρ a² / 4 each.

Checks run in the page

• Finite-difference eigenvalue solution on a smooth density step against the formula.
• Long-wave limit against the shallow-water formula.
• Equal and opposite volume flux in the two layers.
• Kinetic energy equals potential energy.

Honest simplifications

The model is linear, with a rigid lid, no rotation, no mean current, no mixing and a sharp interface. Real internal waves steepen, break and shed solitary packets, and the Earth's rotation changes long waves. The drawing compresses the horizontal scale and exaggerates time.

• Press Strong stratification and see a larger density jump speed the wave up.
• Press Long wave limit and watch the phase and group speed meet the dashed c0 line.
• Shorten the wavelength and see the group speed fall below the phase speed.