Ch 2. Governing equations & the Boussinesq approximation

Overview

The full equations of motion for a rotating, stratified fluid have a dozen terms. At any given scale, most of them are negligible — and this chapter’s entire content is the mechanical procedure for finding out which: nondimensionalize the Boussinesq equations by a length $L$, velocity $U$, and latitude, and the dimensionless coefficient in front of each term answers “does this term matter here?” directly. Every reduced equation set used later in the book (geostrophic balance, quasi-geostrophy, internal waves, Boussinesq convection) is what survives this procedure at that chapter’s particular scale.

The model

Notebook: notebooks/ch02_governing-equations.py → exported to /nb/ch02_governing-equations.html. Governing equations use the symbols in notation.

gfdlib.scaling computes $Ro,\ Ek,\ Fr,\ Re,\ Bu$ and the deformation radius $L_R$ directly from physical inputs ($U,L,H,N,\nu$, latitude) — a pure calculator, no discretization or time-stepping anywhere.

Both fluids

• Atmosphere: midlatitude synoptic weather sits at $Ro\sim0.1$ (geostrophically balanced); a single convective thunderstorm cell, at $L\sim10\,$km, sits at $Ro\gtrsim1$ — genuinely different dynamical regimes governed by the identical Boussinesq equations, distinguished only by scale.
• Ocean: mesoscale eddies sit close to $Bu\sim1$ ($L\sim L_R$), which is exactly why they are the scale most efficiently generated by baroclinic instability (Ch. 16) — the regime map in Ch. 4 makes this connection explicit.

Exercises

1. (analytic) Starting from the dimensional horizontal momentum equation, substitute $x=L\hat x,\ u=U\hat u,\ t=(L/U)\hat t,\ \phi=f_0UL\hat\phi$ and confirm you recover $Ro\,D\hat{\mathbf u}/D\hat t+\hat{\mathbf z}\times\hat{\mathbf u}=-\nabla\hat\phi+\cdots$ with $Ro=U/(f_0L)$ appearing exactly as the coefficient of the acceleration term.
2. (computational) Modify ch02_governing-equations.py to add a sixth preset of your choosing (e.g. a hurricane, $U\sim50\,$m/s, $L\sim200\,$km) and read off which terms the calculator says survive.
3. (exploratory) Using only the sliders, find a combination of $(U,L,H,N,\nu,\text{lat})$ where all five numbers are simultaneously $O(1)$ — i.e. no single simplification applies. Is this an easy region of parameter space to find, or a narrow one? What does that suggest about how often “the fully general equations, no simplification” is actually the right tool in GFD?

Further reading

Vallis, AOFD 2nd ed., §1.1-1.6 (the Boussinesq approximation), §2.1-2.3 (scaling); Pedlosky, Geophysical Fluid Dynamics 2nd ed., §1.1-1.4, §2.1-2.2; Cushman-Roisin & Beckers, Introduction to Geophysical Fluid Dynamics, ch. 2-4.