DoubleGyre

Systems / ODEs / Climate & geophysics

A kinematic pair of counter-rotating gyres with a periodically oscillating separatrix — the canonical benchmark for Lagrangian coherent structures and chaotic advection.

continuous · ODE3 dimensionschaotic advection

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Definition

\[ \begin{aligned} \dot{y_{0}} &= - \pi \alpha \sin{\left(\pi \left(eps y_{0}^{2} \sin{\left(y_{2} \right)} + y_{0} \left(- 2 eps \sin{\left(y_{2} \right)} + 1\right)\right) \right)} \cos{\left(\pi y_{1} \right)} \\ \dot{y_{1}} &= \pi \alpha \left(2 eps y_{0} \sin{\left(y_{2} \right)} - 2 eps \sin{\left(y_{2} \right)} + 1\right) \sin{\left(\pi y_{1} \right)} \cos{\left(\pi \left(eps y_{0}^{2} \sin{\left(y_{2} \right)} + y_{0} \left(- 2 eps \sin{\left(y_{2} \right)} + 1\right)\right) \right)} \\ \dot{y_{2}} &= \omega \end{aligned} \]

Parameters

Symbol	Default	Role
alpha	0.1	overall advection speed
eps	0.1	amplitude of the gyre-boundary oscillation
omega	0.62832	boundary oscillation frequency

Properties

Lyapunov spectrum

$+0.09586,\; -0.09586,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3$

Divergence ∇·f

$\nabla\!\cdot f = 0$
constant

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Shadden, Lekien & Marsden (2005), Physica D 212, 271-304

BibTeX

@misc{doublegyre,
  title = {DoubleGyre system},
  note = {Shadden, Lekien & Marsden (2005), Physica D 212, 271-304}
}