Ch 20. Wind-driven ocean circulation

Overview

The trade winds and westerlies blow steadily over every subtropical ocean basin, curling clockwise around each gyre. Sverdrup balance — the leading- order, frictionless response — can only satisfy the no-normal-flow condition at one wall of a closed basin, conventionally the eastern one. Something has to happen at the western edge to close the circulation. This chapter builds the two classical answers (Stommel’s bottom drag, Munk’s lateral friction) as a genuinely new kind of numerical problem for this book — a bounded rectangular basin, not a periodic or channel domain — and shows, from first principles, why western boundary currents like the Gulf Stream exist and eastern ones don’t.

The model

Notebook: notebooks/ch20_wind-driven-circulation.py → exported to /nb/ch20_wind-driven-circulation.html. Governing equations use the symbols in notation:

\text{Stommel: } \varepsilon\nabla^2\psi+\psi_x=-\sin(\pi y),\qquad \text{Munk: } -\delta^3\nabla^4\psi+\psi_x=-\sin(\pi y).

gfdlib.circulation solves both problems two independent ways: an exact analytic solution (separating $\psi=f(x)\sin(\pi y)$ reduces each PDE to a linear ODE for $f(x)$ , solved via its characteristic polynomial’s roots), and a general 2D finite-difference direct solve (Kronecker-sum operators, plain numpy.linalg.solve, no SciPy) cross-validated against the analytic solution with clean 2nd-order convergence. Munk’s biharmonic operator is handled by keeping vorticity as an independent field (ch16/17/19’s coupled- state pattern) closed with Thom’s (1933) wall-vorticity formula for no-slip on the east/west walls. An earlier version enforced no-slip on all four walls — matching physical intuition but not the separable-solution assumption — and disagreed with the exact solution by a stubborn ~30% that did not shrink with resolution at any boundary-layer width tested, immediately distinguishing a genuine discretization error from a boundary-condition mismatch; the fix (free-slip north/south, the standard textbook simplification) restored clean convergence.

What to look for

The asymmetry is the phenomenon. In either friction model, look at the streamfunction: a broad, slow interior drift and one thin, intense return current — always on the west wall, never the east, no matter what you do to the sliders. The interior matches the Sverdrup solution $(1-x)\sin(\pi y)$ (overlaid in the mid-basin profile), which already “chose” the eastern wall for its boundary condition; friction is only allowed to fix what remains, at the west.
Two frictions, two boundary layers. Toggle Stommel ↔ Munk at comparable boundary-layer widths: Stommel’s current decays monotonically into the interior, while Munk’s overshoots — a weak countercurrent just east of the main jet (the extra structure a 4th-order operator’s oscillatory roots permit). The Gulf Stream’s observed offshore countercurrent looks distinctly more Munk than Stommel.
Squeeze the layer. Shrink $\varepsilon$ (or $\delta$ ): the western current narrows and its peak velocity grows in inverse proportion, because its job is fixed — it must return exactly the transport the Sverdrup interior carries equatorward, however thin friction makes it. This width–speed trade is why the real Gulf Stream, with its tiny effective friction, is so fast and so narrow.
Numerics you can audit. Flip to the finite-difference solution and compare with the analytic curve at increasing resolution: the error drops cleanly at 2nd order. The convergence study here is the same habit every chapter’s test suite applies — and “The model” section’s story of the all-walls-no-slip bug shows what it catches.

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 primary case): every subtropical gyre shows exactly this asymmetry — broad, slow Sverdrup interior, thin intense western boundary current (Gulf Stream, Kuroshio, Agulhas, Brazil Current) — set by the same $\beta$ -plane balance dialed above, closer to the Munk (lateral- friction) limit than the Stommel one for the real ocean.
Atmosphere: the same linear vorticity balance describes the depth-averaged, time-mean response to Ekman pumping under the subtropical highs, but the atmosphere’s turbulent boundary-layer drag and much larger deformation radius put it in a different part of parameter space, without the ocean’s dramatic western intensification — the same equation, a very different appearance, depending on which term actually dominates.

Exercises

(analytic) Starting from the Sverdrup balance $\psi_x=-\sin(\pi y)$ with $\psi(1,y)=0$ , derive $\psi_{Sv}=(1-x)\sin(\pi y)$ and explain why only the eastern boundary condition can be satisfied without friction.
(computational) Modify ch20_wind-driven-circulation.py to plot the western boundary layer’s peak transport (max of $f(x)$ ) as a function of $\varepsilon$ (or $\delta$ ), swept over the slider’s full range. Does it scale the way you’d expect from the boundary-layer width alone?
(exploratory) Compare the Stommel and Munk profiles at boundary-layer widths chosen so their peak transport roughly matches. Which one overshoots the Sverdrup interior value more, and can you connect that to the order of the underlying ODE (2nd vs. 4th)?