Ch 21. Buoyancy-driven / overturning circulation

Overview

The Hadley cell and the ocean’s meridional overturning circulation (MOC) look nothing alike on a map, but they share a single mechanism: heat a fluid unevenly along one boundary and it organizes into one overturning cell — rising where it’s heated, sinking where it’s cooled — not many small convective rolls. This chapter closes Part VI (and the book’s physically-motivated main path) by building that mechanism directly, and by deriving Munk’s (1966) classic “abyssal recipe” — the simple balance that first let oceanographers infer the deep ocean’s overturning rate from its observed stratification.

The model

Notebook: notebooks/ch21_overturning.py → exported to /nb/ch21_overturning.html. Governing equations use the symbols in notation:

w\frac{\partial T}{\partial z}=\kappa\frac{\partial^2T}{\partial z^2}, \qquad \frac{\partial\theta}{\partial t}+J(\psi,\theta)=\nabla^2\theta+Q(x,z).

gfdlib.overturning.abyssal_profile solves Munk’s balance exactly (a linear 2nd-order ODE, general solution $A+Be^{wz/\kappa}$ , boundary constants from a $2\times2$ numpy.linalg.solve) — large $w/\kappa$ compresses the transition from bottom to surface values into a thin layer near the top, the real ocean’s near-uniform abyssal temperature capped by a thin thermocline. gfdlib.overturning.rhs_overturning reuses ch. 14’s Boussinesq vorticity-streamfunction machinery (convection.ChannelGrid) unchanged, replacing uniform bottom heating with a differential, surface-concentrated heating field — Rossby’s (1965) “horizontal convection” mechanism — and was verified to produce exactly the expected circulation sense: rising motion under the heated column, sinking under the cooled one, in every configuration tested.

What to look for

One ratio controls the profile. In Part A, only $w/\kappa$ matters (verify: doubling both leaves the curve unchanged). Crank the ratio up and the exponential’s e-folding depth $\kappa/w$ shrinks: a nearly uniform abyss capped by a thin thermocline — the observed shape of essentially every mid-ocean temperature profile. Munk’s famous move was to run this logic backwards: from the observed thermocline depth and an estimate of $w$ , infer the ocean’s interior mixing rate $\kappa\sim10^{-4}\,$ m $^2\,$ s $^{-1}$ .
One cell, not many. Run Part B: differential heating along one boundary produces a single domain-filling overturning cell — rising over the heated end, sinking over the cooled end — rather than Ch. 14’s array of counter-rotating rolls. The forcing’s own asymmetry organizes the flow at the largest available scale.
No threshold this time. Sweep $Ra$ downward: unlike Ch. 14, the circulation never switches off — it just weakens smoothly. Horizontal convection has no conduction-only rest state to bifurcate from, because a horizontal temperature gradient along a boundary cannot be balanced by pure conduction (exercise 3 makes you say why).
Hunt the asymmetry. Horizontal-convection theory predicts the two branches of the cell are not mirror images: the surface-cooled end is convectively destabilized and should concentrate into a narrower, faster sinking branch as you raise $Ra$ , while the heated end stays broad and gentle. Test it across the $Ra$ slider’s range. The real counterparts are extreme versions of this fingerprint — the MOC’s few localized deep-water-formation sites versus basin-wide diffuse upwelling.

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

Atmosphere: the Hadley cell is the thermally-direct overturning circulation between the heated tropics and the cooler subtropics — rising near the equator, sinking near 30° latitude, with the trade winds and subtropical highs as its surface signature. Held & Hou’s (1980) angular-momentum-conserving theory extends this same “single overturning cell” logic with rotation added.
Ocean: the meridional overturning circulation sinks dense water at high latitudes (North Atlantic Deep Water, Antarctic Bottom Water), spreads it through the deep ocean, and slowly upwells it back toward the surface elsewhere — set by exactly the diffusion/upwelling balance in Part A, and driven by exactly the differential (here: buoyancy-, not just heat-) forcing mechanism in Part B.

Exercises

(analytic) Solve $w\,dT/dz=\kappa\,d^2T/dz^2$ from scratch (general solution $T=A+Be^{wz/\kappa}$ , then the two boundary conditions) and confirm it matches gfdlib.overturning.abyssal_profile.
(computational) Modify ch21_overturning.py’s Part B to use a heating pattern with TWO full cycles across the domain ( $\cos(4\pi x/L_x)$ instead of $\cos(2\pi x/L_x)$ ). Does the flow organize into four cells, or does it prefer to merge into fewer, larger ones — and if so, why might that connect to the inverse cascade of Ch. 18?
(exploratory) In Part B, hold $Q_0$ fixed and sweep $Ra$ from low to high. Is there a sharp threshold before any circulation appears (like Ch. 14’s $Ra_c$ ), or does the overturning strength grow smoothly from zero? What does that tell you about how “horizontal convection” differs from Rayleigh-Bénard convection at a basic level?