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Warm salty water is lighter overall but loses heat 10x faster than salt. A sinking parcel cools, keeps its salt and sinks on: the finger instability.
Scenario and view
Model checks
The science
What is modelled
A dimensionless 2D Boussinesq box with an impermeable lid, floor and walls. Temperature and salinity are advected and diffused, buoyancy depends on both, and a projection step keeps the flow divergence-free.
Equations
- Density
rho = beta S - alpha T(relative). - Diffusivity ratio
tau = kS / kT; fingering needs1 < R_rho < 1/tauwithR_rho = alpha dT / (beta dS). - Advection-diffusion of T and S plus buoyancy force on the vertical velocity.
Checks run in the page
- Mean salinity conserved in the closed box.
- Mean temperature conserved with no source.
- Warm-salty-over-cool-fresh column is statically stable but in the finger window.
- The stable control has top density below bottom density.
Honest simplifications
The solver is a coarse 72 x 96 semi-Lagrangian scheme in two dimensions with a global conservation correction and tau = 0.1 rather than the ocean value near 0.01, so finger widths are qualitative. Not an oceanographic tool.
What to try
- Press Equal diffusivity: with tau = 1 the fingers stop forming.
- Raise viscosity: fingers become fat and slow.
- Choose Stable: nothing overturns, only slow diffusion.