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

\[ \begin{aligned} \dot{y_{0}} &= b y_{1} - c p - c y_{0} \\ \dot{y_{1}} &= y_{0} y_{2} - y_{1} \\ \dot{y_{2}} &= - y_{0} y_{1} - y_{2} + 1 \end{aligned} \]

Parameters

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

Properties

Lyapunov spectrum

$+0.5238,\; +0.02375,\; -5.548$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.099$

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Vallis (1988), J. Geophys. Res. 93, 13979-13991

BibTeX

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