Lorenz energy cycle (stationary-eddy form). Four reservoirs, four conversions. Reservoirs in MJ m⁻², conversions in W m⁻², all mass-weighted vertically over the 12 pressure levels and area-weighted globally (cosφ).

Reservoirs — available potential + kinetic energy, mean + eddy:

P_M = (R/2g) ∫ T̂² / (γ p) dp        (mean APE)
P_E = (R/2g) ∫ ⟨T*²⟩ / (γ p) dp     (eddy APE)
K_M = (1/2g) ∫ ([u]² + [v]²) dp     (mean KE)
K_E = (1/2g) ∫ ⟨u*² + v*²⟩ dp       (eddy KE)

where T̂ = [T] − T_ref (zonal-mean departure from reference), T* = T − [T] (stationary-eddy departure from zonal mean), γ(p) = κT_ref/p − ∂T_ref/∂p (static stability), [·] is zonal mean, ⟨·⟩ is global area-mean. Leading factor R/(2g) is the Holton–Hakim / Vallis form (using c_p/2g instead overshoots by ~3.5×).

Conversions — each has a clean physical reading:

C(P_M→P_E) = -(R/g) ∫ ⟨v*T*⟩/(γp) · ∂[T]/∂y dp
              -(R/g) ∫ ⟨ω*T*⟩/(γp) · ∂[T]/∂p dp
   ► baroclinic release — eddy heat flux
     down the mean T gradient

C(P_E→K_E) = -(R/g) ∫ ⟨ω*T*⟩ / p dp
   ► eddy buoyancy flux — warm air rising,
     cold air sinking in the eddies

C(K_E→K_M) =  (1/g) ∫ ⟨u*v*⟩ cosφ · ∂([u]/cosφ)/∂y dp
              +(1/g) ∫ ⟨u*ω*⟩ · ∂[u]/∂p dp
   ► eddy momentum flux spinning up the
     zonal-mean jets (barotropic)

C(P_M→K_M) = -(R/g) ∫ [ω][T] / p dp
   ► direct thermal conversion by the
     zonal-mean meridional circulation
     (Hadley cell drives this)

Implementation notes:

• Inputs are the monthly-mean u, v, ω, T fields on 12 pressure levels (10 → 1000 hPa) from ERA5.
• Vertical integration uses each level's half-distance layer (dp = (p_{k-1} + p_{k+1})/2 − (p_{k-1} + p_k)/2), one-sided at top and bottom.
• Meridional derivatives ∂/∂y use centered differences on a = 6.371·10⁶ m; vertical ∂/∂p likewise on the 12-level pressure grid.
• Global ⟨·⟩ uses a cosφ area weight over the full (181 × 360) grid, skipping NaN (land-masked) samples.

Reference-state toggle (controls T_ref, which sets γ and the T̂ used for P_M and C(P_M, K_M)):

• Lorenz (sorted) — the true minimum-PE reference from Lorenz (1955). Every parcel (~780k on the 12-level globe) is adiabatically re-stratified by θ, with highest θ at the top; T_ref(p) is the T at the rank where cumulative mass from the top equals (p / p_s)·total. This is the canonical "available" formulation.
• Area-mean — T_ref(p) = global area-weighted mean of T at each pressure level. The textbook diagnostic approximation. Faster, but it overstates how much PE is "available" — a lot of the meridional T gradient it sees is actually unavailable (non-adiabatic to release).

Why sorted ≠ area-mean, especially for C(P_M, K_M): both formulations share u, v, ω, T — only T_ref (and hence T̂, γ) differ. In area-mean, T̂ still contains the full tropics-to-pole gradient, so the Hadley-cell overturning [ω] correlates strongly with T̂ and C(P_M, K_M) is large. In sorted, the reference has already absorbed most of that gradient; the "excess" T̂ is small and C(P_M, K_M) drops sharply. The factor-of-many difference you see between the two panels is Lorenz's (1955) original pedagogical point in visible form — the area-mean calc systematically overstates convertible energy.

Typical magnitudes (for sanity-checking):

Quantity	P&O total	Stat-only band
P_M (MJ/m²)	4.0	~3.8-4.2 (sorted)
K_M (MJ/m²)	0.8	~0.7-0.9
P_E (MJ/m²)	1.5	~0.2-0.4 (stat)
K_E (MJ/m²)	1.4	~0.2-0.3 (stat)
C(P_M, P_E) (W/m²)	2.2	~0.7-1.1 (stat)
C(P_E, K_E) (W/m²)	1.7	~0.5-0.7 (stat)
C(K_E, K_M) (W/m²)	0.2	~0.06-0.12 (stat)
C(P_M, K_M) (W/m²)	0.2	~0.02-0.1 (sorted, stat)

Left column: Peixoto & Oort (1992) Table 13.1 for the full circulation. Right column: the stationary-only contribution you'd expect to see here, derived from Boer (1989)'s stationary/transient decomposition (~15-30% of total for eddy quantities). DJF values are closer to the top of each band; JJA to the bottom. Reanalysis-to-reanalysis variance alone is ±15-20%, so matches within that range count as good.

Caveats:

• Stationary eddies only. * here is the departure of the monthly-mean field from its zonal mean — it captures planetary waves locked to topography / land-sea contrast. High-frequency (sub-monthly) transient eddies, which dominate baroclinic conversion in published totals, need daily data (currently out of scope).
• C(P_M, P_E) here includes both the meridional and vertical eddy-heat-flux terms; many references quote only the meridional term.
• Expect ~3-5× smaller conversions than Peixoto-Oort "total" numbers because transients are excluded — that gap is the transient-eddy contribution, not a bug.

Use the cycle to illustrate structure — which arrows are large, which way energy flows, how seasonal cycles shift it (swipe through months). Toggle the reference state to see how a methodology choice shifts P_M and C(P_M, K_M) by an order of magnitude — that's the pedagogy in its own right.

References: Lorenz (1955, Tellus); Peixoto & Oort (1974; 1992, Physics of Climate, ch. 13-14); Boer (1989, JCli) for the stationary/transient decomposition; Holton & Hakim (textbook ch. 10) for the canonical derivation.