Ch 10. Two-layer QG & available potential energy

Overview

A two-layer fluid decomposes uniquely into a barotropic mode (the depth mean, obeying exactly Ch. 7/18’s ordinary barotropic vorticity equation) and a baroclinic mode (the inter-layer difference, obeying exactly Ch. 8’s single-layer QG equation, and carrying all the available potential energy). This chapter derives that exact decoupling, then breaks it on purpose: seeding a disturbance purely in the baroclinic mode and watching nonlinear self-advection leak a barotropic component into existence where none was placed.

The model

Notebook: notebooks/ch10_two-layer-qg.py → exported to /nb/ch10_two-layer-qg.html. Governing equations use the symbols in notation.

q_{bt}=\frac{q_1+q_2}{2}=\nabla^2\psi_{bt},\qquad q_{bc}=\frac{q_1-q_2}{2}=\nabla^2\psi_{bc}-2F\psi_{bc}.

gfdlib.baroclinic.to_modes/from_modes convert between layer space and mode space; pv_2layer is the forward PV operator (exact inverse of invert_2layer); the time evolution reuses rhs_2layer from Ch. 16 unchanged, with zero mean shear — an unforced initial-value problem, not an instability calculation.

What to look for

Start pure, watch it leak. The initial condition lives entirely in the baroclinic mode ( $\psi_{bt}=0$ exactly, to machine precision — the mode-energy panel confirms it at $t=0$ ). Run, and watch the barotropic energy curve lift off from zero: the linear decoupling is exact, so every bit of barotropic flow you see was manufactured by the nonlinear term $J(\psi_{bc},q_{bc})$ , which projects back onto the barotropic equation.
Two vortices, two personalities. Look at the two mode fields side by side as the run evolves: the baroclinic part stays compact (screened at the scale $1/\sqrt{2F}$ , Ch. 8’s mechanism with $L_R$ replaced by the baroclinic deformation radius), while the growing barotropic part spreads — it has no deformation radius to stop it.
$F$ is the knob that matters. Raise $F$ (thinner layers / weaker stratification): the baroclinic mode becomes more tightly screened and the leak into the barotropic mode changes character. Compare the final barotropic-energy fraction at $F=0.5$ and $F=3$ before reading exercise 3.
Read the two panels as two earlier chapters. The snapshot browser shows $q_{bt}$ and $q_{bc}$ side by side: the left panel is a flow obeying exactly Ch. 7’s barotropic vorticity equation, the right panel one obeying exactly Ch. 8’s single-layer QG equation — and at frame 0 the left panel is identically zero. What couples them afterward is nothing in the linear physics, only advection. That change of glasses — from two similar-looking layers to two familiar, decoupled equations — is the whole point of the chapter.

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: newly-formed mesoscale eddies are overwhelmingly baroclinic at birth (generated by Ch. 16’s instability, which draws only on APE); the slow barotropization this notebook’s leak illustrates in miniature is a leading theory for how eddy energy ultimately escapes into the basin-scale, nearly-barotropic circulation.
Atmosphere: the storm track’s baroclinic eddies feed the barotropic-dominated jet stream through essentially this same nonlinear route — Ch. 18’s inverse cascade needs a source, and baroclinic instability plus this mode-coupling leak is where its energy actually comes from.

Exercises

(analytic) Starting from $q_1=\nabla^2\psi_1+F(\psi_2-\psi_1)$ , $q_2=\nabla^2\psi_2+F(\psi_1-\psi_2)$ , substitute $\psi_1=\psi_{bt}+\psi_{bc}$ , $\psi_2=\psi_{bt}-\psi_{bc}$ and confirm by hand that the $F$ terms cancel in $q_1+q_2$ and double in $q_1-q_2$ , reproducing the boxed result above.
(computational) Modify ch10_two-layer-qg.py to seed the initial condition purely in the barotropic mode instead ( $q_{bc}(t{=}0)=0$ ). Does a baroclinic component leak into existence the same way, or does a purely barotropic vortex stay purely barotropic? (Hint: think about whether a purely barotropic flow’s Jacobian can ever produce a nonzero $q_{bc}$ tendency, given $q_{bc}$ ’s own equation has no barotropic-mode source term.)
(exploratory) Track the ratio (barotropic KE)/(baroclinic KE + APE) at the final time as you vary $\beta$ . Does stronger $\beta$ enhance or suppress the leak — and can you connect your answer to how $\beta$ changes the baroclinic mode’s own Rossby-wave dispersion (Ch. 8)?