BickleyJet

Systems / ODEs / Climate & geophysics

A meandering Bickley (sech²) jet perturbed by three Rossby-wave modes — a benchmark for Lagrangian coherent structures in geophysical flows.

continuous · ODE3 dimensionsquasiperiodic transport

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y} &= \frac{\ell u \left(0.002354415 \sin{\left(0.313922 x - 2.84367499388 \cdot 10^{-6} z \right)} + 0.09417675 \sin{\left(0.627845 x - 8.0648573785 \cdot 10^{-6} z \right)} + 0.2825301 \sin{\left(0.941767 x - 2.72041640921 \cdot 10^{-5} z \right)}\right)}{\cosh^{2}{\left(\frac{y}{\ell} \right)}} \\ \dot{x} &= \frac{u \left(- 2 \left(0.0075 \cos{\left(0.313922 x - 2.84367499388 \cdot 10^{-6} z \right)} + 0.15 \cos{\left(0.627845 x - 8.0648573785 \cdot 10^{-6} z \right)} + 0.3 \cos{\left(0.941767 x - 2.72041640921 \cdot 10^{-5} z \right)}\right) \tanh{\left(\frac{y}{\ell} \right)} - 1\right)}{\cosh^{2}{\left(\frac{y}{\ell} \right)}} \\ \dot{z} &= \omega \end{aligned} \]

Parameters

Symbol	Default	Role
ell	1.77	jet half-width
omega	1	wave-train phase-advance rate
u	6.266e-05	characteristic jet speed

State variables: y, x, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

n/a — too large to display — pathological rational divergence

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.BickleyJet()
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

Hadjighasem, Karrasch, Teramoto & Haller (2016), Phys. Rev. E 93, 063107
doi:10.1103/physreve.93.063107

BibTeX

@misc{bickleyjet,
  title = {BickleyJet system},
  note = {Hadjighasem, Karrasch, Teramoto & Haller (2016), Phys. Rev. E 93, 063107},
  doi = {10.1103/physreve.93.063107}
}