Lorenz-96 Model

The Lorenz-96 model is a chaotic, continuous-in-time, discrete-in-space, dynamical system that was proposed by Lorenz in 1996 as a toy model for weather dynamics¹.
$$ \frac{dx_j}{dt} = ( x_{j+1} - x_{j-2} ) x_{j-1} - x_j + F(t). $$ Even though the equations have been used and studied for a long time, their chaotic behavior has been mathematically proven only recently².

The system is implemented in helpers.py in class Lorenz.

• N: Number of grid points, defaults to 40.
• dt: Time step, defaults to 0.05. This is hard-coded for forecasts, so be careful when you change it. The error growth time scale is assumed to be such that 0.05 corresponds to 6 hours in an operational weather forecast system.
• forcing: External forcing function, defaults to the traditional choice of 8, where the system behaves chaotically.

• rk4 is a 4th order Runge-Kutta integrator for solving the Lorenz system of ODEs.
• rhs implements the right hand side using vectorized numpy expressions.

dotx[2:-1] = (x[3:]-x[0:-3])*x[1:-2] - x[2:-1] + forcing[2:-1]

Periodic boundary conditions are imposed on the outer grid points.

• solve solves the system of ODEs for a number of days with given initial data. The method populates the sol attribute by integrating the system using a 4th order Runge-Kutta method. The core portion of the method is

self.sol[0] = init_data
for i in range(1, self.nt+1):
    self.sol[i] = self.rk4(self.sol[i-1])
    self.t += self.dt

The total number of time steps nt is determined from the days and the time step via self.nt = int(4*0.05/self.dt*days) as dt=0.05 corresponds to 6 hours.

• animate is provided to view the dynamic solution in time.

The default system can be initiated with random data, solved for 90 days, and visualized by

N=40; F=8;
initial_data=np.random.normal(0.25*F,0.5*F,N)
Default=Lorenz()
Default.solve(days=90,init_data=initial_data)
anim=Default.animate()