Ch 19. Eddy transport & mixing

Overview

Ch. 18’s turbulent eddies don’t just cascade energy and enstrophy — they stir anything riding along with the flow: heat, salt, chemical tracers, potential vorticity itself. This chapter builds a passive tracer directly into ch. 18’s turbulence machinery and asks the practical question every coarse ocean or atmosphere model has to answer: if you can’t afford to resolve the eddies, what single number stands in for everything they do to a tracer? The answer is an effective diffusivity — and this notebook computes one directly from a turbulent simulation, rather than taking it as a free parameter.

The model

Notebook: notebooks/ch19_eddy-transport.py → exported to /nb/ch19_eddy-transport.html. Governing equations use the symbols in notation:

\frac{\partial c'}{\partial t}+J(\psi,c')=-\Gamma v+\kappa\nabla^2c', \qquad K_{eff}=-\frac{\overline{v'c'}}{\Gamma}.

gfdlib.mixing.rhs_coupled stacks vorticity and tracer perturbation into one state (the same two-field pattern as ch. 16’s two-layer model), reusing ch. 18’s exact turbulence machinery for $\zeta$ unchanged, with the tracer’s mean gradient split off as an ordinary source term — the identical “mean + periodic perturbation” trick used for $\beta y$ (ch. 18) and mean shear (ch. 16). Because the perturbation equation is linear in $c'$ , $K_{eff}$ must be exactly independent of the imposed gradient $\Gamma$ for a fixed flow — checked numerically (two runs differing only in $\Gamma$ agree to machine precision) rather than assumed, alongside a control run with the velocity field set identically to zero, which gives $K_{eff}=0$ exactly (no possible eddy flux without a flow).

What to look for

Stirring before mixing. Watch the tracer field in the first hundreds of steps: the turbulence doesn’t blur the tracer, it folds it — drawing $c'$ into ever-thinner filaments that wrap around the vortices. Only at filament scale does the (tiny) molecular $\kappa$ finally act. Stirring cascades tracer variance downscale; mixing destroys it there — two distinct processes, visible as two stages.
The flux finds the down-gradient direction on its own. Nothing in the equations tells $\overline{v'c'}$ which way to point, yet the $K_{eff}$ time series rises from zero as the filaments develop and then fluctuates about a clearly positive value: eddies move tracer from high mean concentration toward low. Note that it never settles to a perfectly flat plateau — it tracks the turbulence’s own slow evolution — a warning about diagnosing eddy diffusivities from short records.
Linearity, exploited. Double $\Gamma$ and rerun with the same seed: the tracer field doubles but $K_{eff}$ lands on the same number — the closure’s one honest free parameter is a property of the flow, not the gradient (the test suite checks this to machine precision). This independence is precisely what licenses coarse models to use one $K$ for heat, salt, and carbon alike.
A mixing-length reality check. Compare the measured $K_{eff}$ against the eyeball estimate $u_{rms}\times\ell$ with $\ell$ the dominant eddy size in the vorticity panel. Prandtl’s argument usually lands within a factor of two — exercise 2 turns this comparison into a habit.

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: eddy mixing of potential vorticity and chemical tracers by synoptic-scale storms sets the effective diffusivity used in every coarse-resolution climate and chemistry-transport model — the same flux-gradient closure computed directly here.
Ocean: mesoscale eddies (ch. 16’s baroclinic instability, at ocean scales) are the dominant mechanism transporting heat, carbon, and nutrients across ocean fronts and gyre boundaries; every non-eddy- resolving ocean climate model parameterizes this transport with an eddy diffusivity of exactly the kind measured in this notebook (the Gent-McWilliams scheme, in practice).

Exercises

(analytic) Substitute $C_{total}=\Gamma y+c'$ into the advection- diffusion equation and show the mean-gradient term becomes the source $-\Gamma v$ in the perturbation equation for $c'$ , with no other change.
(computational) Modify ch19_eddy-transport.py to report the time-mean $K_{eff}$ over the last quarter of the run (instead of just plotting the full time series) and print it alongside $u_{rms}\sqrt{2E}$ for a direct mixing-length comparison.
(exploratory) Run at two different resolutions ( $n=64$ and $n=128$ , same $k_0$ , seed, and total simulated time) and compare the late-time $K_{eff}$ . Does resolving finer-scale filamentation change the measured diffusivity much, or does it converge quickly?