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

\[ \begin{aligned} \dot{y_{0}} &= a f - a y_{0} - y_{1}^{2} - y_{2}^{2} \\ \dot{y_{1}} &= - b y_{0} y_{2} + g + y_{0} y_{1} - y_{1} \\ \dot{y_{2}} &= b y_{0} y_{1} + y_{0} y_{2} - y_{2} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.194,\; +0.005269,\; -0.3565$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.559$

Divergence ∇·f

$\nabla\!\cdot f = - a + 2 y_{0} - 2$
state-dependent

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Hadley()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Lorenz (1984), 'Irregularity: a fundamental property of the atmosphere', Tellus 36A, 98-110

BibTeX

@misc{hadley,
  title = {Hadley system},
  note = {Lorenz (1984), 'Irregularity: a fundamental property of the atmosphere', Tellus 36A, 98-110}
}