Hadley

Systems / ODEs / Climate & geophysics

The Lorenz-84 caricature of the mid-latitude westerly circulation — a zonal current coupled to superposed eddies that yields irregular, intransitive weather.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= a f - a x - y^{2} - z^{2} \\ \dot{y} &= - b x z + g + x y - y \\ \dot{z} &= b x y + x z - z \end{aligned} \]

Parameters

Symbol	Default	Role
a	0.2	damping rate of the zonal flow
b	4	nonlinear eddy-displacement coupling
f	9	symmetric thermal forcing of the zonal flow
g	1	asymmetric thermal forcing of the eddies

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a + 2 x - 2$
state-dependent

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Hadley()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference

Lorenz (1984), 'Irregularity: a fundamental property of the atmosphere', Tellus 36A, 98-110
doi:10.1111/j.1600-0870.1984.tb00230.x

BibTeX

@misc{hadley,
  title = {Hadley system},
  note = {Lorenz (1984), 'Irregularity: a fundamental property of the atmosphere', Tellus 36A, 98-110},
  doi = {10.1111/j.1600-0870.1984.tb00230.x}
}