Lorenz 96

class otp.lorenz96.Lorenz96Problem

Bases: otp.Problem

A chaotic system modeling nonlinear transfer of a dimensionless quantity along a cyclic one dimensional domain.

The \(N\) variable dynamics [Lor96] are represented by the equation,

\[ y_i' = -y_{i-1} (y_{i-2} - y_{i+1}) - y_i + f(t), \qquad i = 1, …, N, \]

where \(y_0 = y_N\), \(y_{-1} = y_{N - 1}\), and \(y_{N + 1} = y_2\), exhibits chaotic behavior for certain pairs of values of the dimension \(N\) and forcing function \(f\).

Notes

Type	ODE
Number of Variables	\(N\) for any positive integer four or greater
Stiff	no

Example

>>> problem = otp.lorenz96.presets.Canonical('Forcing', @(t) 8 + 4*sin(t));
>>> sol = problem.solve();
>>> problem.movie(sol);

Parameters

class otp.lorenz96.Lorenz96Parameters

Bases: otp.Parameters

Parameters for the Lorenz ‘96 problem.

Constructor Summary

Lorenz96Parameters(varargin)

Create a Lorenz ‘96 parameters object.

Parameters:: varargin – A variable number of name-value pairs. A name can be any property of this class, and the subsequent value initializes that property.

Property Summary

F: A forcing function, scalar, or vector.

Presets

class otp.lorenz96.presets.PopovSandu

Bases: otp.lorenz96.Lorenz96Problem

A preset that has a cyclic forcing function that is different for every variable. This preset was created for [PS19]. This preset uses time span \(t ∈ [0, 720]\), \(N = 40\), and initial conditions of \(y_i = 8\) for all \(i\) except for \(y_{⌊N/2⌋}=8.008\). The forcing as a function of time is given by

\[ f(t) = 8 + 4\cos(2 π ω (t+\text{mod}(i - 1, q)/q)) \]

where by default \(q=4\) is the number of partitions, and \(ω = 1\) is a forcing period of one time unit.

Constructor Summary

PopovSandu(varargin)

Create the Popov–Sandu Lorenz ‘96 problem object.

Parameters:

varargin –

A variable number of name-value pairs. The accepted names are

N – The size of the problem as a positive integer.
Partitions – The number of partitions into which to divide the variables.
ForcingPeriod – The period of the forcing function in radians per unit time.

class otp.lorenz96.presets.Canonical

Bases: otp.lorenz96.Lorenz96Problem

Original Lorenz ‘96 preset presented in [Lor96] which uses time span \(t ∈ [0, 720]\), \(N = 40\), \(F=8\), and initial conditions of \(y_i = 8\) for all \(i\) except for \(y_{⌊N/2⌋}=8.008\).

Constructor Summary

Canonical(varargin)

Create the canonical Lorenz ‘96 problem object.

Parameters:

varargin –

A variable number of name-value pairs. The accepted names are

N – The size of the problem as a positive integer.
Forcing – The forcing as a scalar, vector of N constants, or as a function.