VallisElNino

Systems / ODEs / Climate & geophysics

A Lorenz-like conceptual model of the tropical Pacific ocean–atmosphere that makes El Niño's aperiodicity intrinsic, with no stochastic forcing needed.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= b y - c p - c x \\ \dot{y} &= x z - y \\ \dot{z} &= - x y - z + 1 \end{aligned} \]

Parameters

Symbol	Default	Role
b	102	wind-driven ocean coupling gain
c	3	wind-anomaly relaxation rate
p	0	steady external wind forcing

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - c - 2$
constant

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.VallisElNino()
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

Vallis (1988), J. Geophys. Res. 93, 13979-13991
doi:10.1029/jc093ic11p13979

BibTeX

@misc{valliselnino,
  title = {VallisElNino system},
  note = {Vallis (1988), J. Geophys. Res. 93, 13979-13991},
  doi = {10.1029/jc093ic11p13979}
}