Ch 11. Stratification & the buoyancy frequency

Overview

Ch. 10 split a two-LAYER fluid exactly into barotropic and baroclinic parts. A continuously stratified fluid admits the same trick, generalized: any $N^2(z)$ profile has its own complete, orthogonal set of vertical normal modes, each behaving as an independent single-layer QG system with its own deformation radius. This chapter solves for those modes directly, and shows the two-layer model used everywhere earlier in the book is exactly the crudest possible truncation of this richer theory — mode 0 and mode 1 only.

The model

Notebook: notebooks/ch11_stratification.py → exported to /nb/ch11_stratification.html. Governing equations use the symbols in notation.

\frac{d}{dz}\!\left[\frac{1}{N^2(z)}\frac{d\Phi_n}{dz}\right] +\frac{1}{c_n^2}\Phi_n=0,\qquad \Phi_n'(0)=\Phi_n'(H)=0,\qquad L_n=\frac{c_n}{f_0}.

gfdlib.stratification.vertical_modes discretizes this Sturm-Liouville problem by the finite-volume method and solves it as a genuinely symmetric eigenproblem (numpy.linalg.eigh), validated against the exact constant- $N$ solution $\Phi_n=\cos(n\pi z/H)$ , $c_n=N_0H/(n\pi)$ .

What to look for

Trust, then verify. Start with the constant- $N$ profile: the computed modes should lie exactly on the analytic $\cos(n\pi z/H)$ curves with $c_n=N_0H/(n\pi)$ — the notebook overlays both. (With the default $N_0=10^{-2}\,$ s $^{-1}$ , $H=1000\,$ m, $f_0=10^{-4}\,$ s $^{-1}$ , that puts $L_1=c_1/f_0\approx32\,$ km — already the right order for midlatitude ocean eddies.)
Realistic stratification bends the modes. Switch to the pycnocline profile: the zero crossings and extrema of every mode get pulled toward the high- $N^2$ layer, and mode 1’s velocity structure (which follows $\Phi_1$ ) concentrates its shear across the pycnocline — the vertical structure Ch. 10’s two layers were caricaturing.
Sharpen the pycnocline toward the two-layer limit. Narrow the thickness slider at fixed $N_{min},N_{max}$ : $L_1$ converges toward the two-layer $\sqrt{g'H_1H_2/(H_1+H_2)}/f_0$ value, while the higher modes — which the two-layer model simply does not have — keep changing. What survives truncation and what doesn’t is exercise 3’s subject.
Latitude matters, stratification radius doesn’t care. Slide $f_0$ across its range (equatorward to poleward): every $c_n$ is untouched (they are properties of $N^2(z)$ alone) but every $L_n=c_n/f_0$ shrinks toward the pole — the reason observed eddy scales shrink from $\sim$ 200 km in the subtropics to $\sim$ 10 km in the subpolar oceans (Chelton et al. 1998, in the further reading).

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

Ocean: the first baroclinic deformation radius, computed from a realistic pycnocline profile, sets the observed $\sim30$ – $50\,$ km scale of mesoscale eddies at midlatitudes — smaller than the constant- $N$ formula would suggest, because real stratification concentrates in a thin near-surface layer rather than spreading uniformly through the water column.
Atmosphere: the tropopause plays the ocean pycnocline’s role — a region of sharply enhanced $N^2$ near which the first baroclinic mode’s structure concentrates, setting the $\sim1000\,$ km scale of synoptic weather systems.

Exercises

(analytic) Verify by direct substitution that $\Phi_n=\cos(n\pi z/H)$ solves $d/dz[(1/N_0^2)d\Phi/dz]+(1/c_n^2)\Phi=0$ with $c_n=N_0H/(n\pi)$ , and that $\Phi_n'(0)=\Phi_n'(H)=0$ for every integer $n$ (matching gfdlib.stratification’s docstring and the notebook’s built-in validation panel).
(computational) Modify ch11_stratification.py to add a two-pycnocline profile (a shallow seasonal thermocline stacked above a deeper permanent one) and see whether a genuinely new SHAPE of mode 2 emerges — does it develop an extra local extremum near the shallow feature?
(exploratory) Using the pycnocline profile, sweep the thickness slider from thick to thin at fixed $N_{min},N_{max}$ , and record $L_1$ at each setting. Does $L_1$ approach a two-layer-like limit as the pycnocline sharpens? Compare to a matching Ch. 16 $F=f_0^2/(g'H)$ calculation with $g'$ estimated from your $N_{max},$ thickness.