OscillatingFlow

Systems / ODEs / Climate & geophysics

A kinematic array of convection rolls with laterally oscillating boundaries whose time-dependent separatrices chaotically transport tracers between adjacent cells.

continuous · ODE3 dimensionschaotic advection

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= u \sin{\left(k \left(b \sin{\left(y_{2} \right)} + y_{0}\right) \right)} \cos{\left(k y_{1} \right)} \\ \dot{y_{1}} &= - u \sin{\left(k y_{1} \right)} \cos{\left(k \left(b \sin{\left(y_{2} \right)} + y_{0}\right) \right)} \\ \dot{y_{2}} &= \omega \end{aligned} \]

Parameters

Symbol	Default	Role
b	0.48	amplitude of the roll-boundary oscillation
k	1	roll wavenumber (roll size)
omega	0.49	roll oscillation frequency
u	0.72	overall flow speed

Properties

Lyapunov spectrum

$+0.1235,\; -0.1235,\; 0$
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.OscillatingFlow()
traj = sys.integrate(final_time=100.0, dt=0.01)

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

Reference

Solomon & Gollub (1988), Phys. Rev. A 38, 6280-6286

BibTeX

@misc{oscillatingflow,
  title = {OscillatingFlow system},
  note = {Solomon & Gollub (1988), Phys. Rev. A 38, 6280-6286}
}