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

\[ \begin{aligned} \dot{x} &= u \sin{\left(k \left(b \sin{\left(z \right)} + x\right) \right)} \cos{\left(k y \right)} \\ \dot{y} &= - u \sin{\left(k y \right)} \cos{\left(k \left(b \sin{\left(z \right)} + x\right) \right)} \\ \dot{z} &= \omega \end{aligned} \]

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

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.OscillatingFlow()
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

Solomon & Gollub (1988), Phys. Rev. A 38, 6280-6286
doi:10.1103/physreva.38.6280

BibTeX

@misc{oscillatingflow,
  title = {OscillatingFlow system},
  note = {Solomon & Gollub (1988), Phys. Rev. A 38, 6280-6286},
  doi = {10.1103/physreva.38.6280}
}