AtmosphericRegime

Systems / ODEs / Climate & geophysics

Two coupled oscillators with widely spaced frequencies whose timescale separation drives intermittent bursts between quasi-stationary weather regimes.

continuous · ODE3 dimensionsintermittent

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= \mu_{1} y_{0} + \sigma y_{0} y_{1} \\ \dot{y_{1}} &= \alpha y_{1} y_{2} + \beta y_{2}^{2} + \mu_{2} y_{1} + \omega y_{2} - \sigma y_{0}^{2} \\ \dot{y_{2}} &= - \alpha y_{1}^{2} - \beta y_{1} y_{2} + \mu_{2} y_{2} - \omega y_{1} \end{aligned} \]

Parameters

Symbol	Default	Role
alpha	-2	quadratic slow–fast coupling coefficient
beta	-5	quadratic slow–fast coupling coefficient
mu1	0.05	linear growth/damping of the slow mode
mu2	-0.01	linear growth/damping of the fast oscillator
omega	3	natural frequency of the fast oscillator
sigma	1.1	bilinear slow–fast feedback strength

Properties

Lyapunov spectrum

$+0.03871,\; +0.01594,\; -0.9632$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.057$

Divergence ∇·f

$\nabla\!\cdot f = \alpha y_{2} - \beta y_{1} + \mu_{1} + 2 \mu_{2} + \sigma y_{1}$
state-dependent

Equilibria

2 equilibria
0 stable · 2 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Tuwankotta (2006), Int. J. Non-Linear Mech. 41, 180-191

BibTeX

@misc{atmosphericregime,
  title = {AtmosphericRegime system},
  note = {Tuwankotta (2006), Int. J. Non-Linear Mech. 41, 180-191}
}