Ch 4. Scaling & the dimensionless numbers

Overview

Ch. 2 computed five dimensionless numbers from raw physical inputs. Two of them, $Ro$ and $Bu$, do almost all the work of predicting which of this book’s reduced models applies at a given scale. This chapter turns that observation into a map: place any flow on the $(Ro,Bu)$ plane and read off directly whether it’s unbalanced, baroclinically active, or columnar/barotropic — and see exactly where every earlier chapter’s example flow actually lands.

The model

Notebook: notebooks/ch04_scaling.py → exported to /nb/ch04_scaling.html. Governing equations use the symbols in notation.

gfdlib.scaling.classify_regime labels every $(Ro,Bu)$ point unbalanced / QG-baroclinic / QG-barotropic; the notebook renders it as a filled region map with five reference points from earlier chapters’ example flows placed directly on it.

Both fluids

• Atmosphere: synoptic weather sits near $Bu\sim1$ — not a coincidence, since that is exactly the scale at which baroclinic instability (Ch. 16) most efficiently converts available potential energy into storms.
• Ocean: the wind-driven gyre interior (Ch. 20) sits deep in $Ro\ll1$, but its western boundary current is controlled by friction, a physics this $(Ro,Bu)$-only map cannot see — a useful reminder that even a good map has a domain of validity.

Exercises

1. (analytic) Show algebraically that $Bu=(Ro/Fr)^2$ (both are ratios built from the same $U,L,H,N,f_0$; note $Fr/Ro=f_0L/(NH)=L/L_R$), so the $(Ro,Bu)$ map is equivalent to an $(Ro,Fr)$ map — confirm this against the definitions in gfdlib.scaling.
2. (computational) Modify ch04_scaling.py to add a sixth reference point for Ch. 12’s internal-wave beams, using that chapter’s typical $\hat N,\hat f,\hat\omega$ to estimate an effective $Ro,Bu$ (this requires some judgment — internal waves aren’t really a “balanced flow” scale in the same sense as the other five points, which is itself worth noting in your answer).
3. (exploratory) Sweep the marker along the $Bu=1$ line from very small to very large $Ro$. At what $Ro$ does the marker cross into “unbalanced”? Does that crossing point depend on where along the $Bu=1$ line you started? (It shouldn’t — re-read classify_regime’s logic to see why the $Ro$ boundary is checked independently of $Bu$.)

Further reading

Vallis, AOFD 2nd ed., §2.7-2.8 (regime diagrams); Pedlosky, Geophysical Fluid Dynamics 2nd ed., §6.1-6.3; Cushman-Roisin & Beckers, Introduction to Geophysical Fluid Dynamics, ch. 4-5.