Guiding-Center dynamics in plasma physics

• gc2ds_classes.py: contains the GC classes and main functions defining the GC dynamics

• run_gc2ds.py: example of a run file to reproduce the computations done in [REF]

Once run_gc2ds.py has been edited with the relevant parameters, run the file as

python3 run_gc2ds.py

or

nohup python3 -u run_gc2ds.py &>gc2d.out < /dev/null &

The list of Python packages and their version are specified in requirements.txt

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Main parameters of the class GC2Ds

• A: Amplitude of the electrostatic potential
• M: Number of modes in the electrostatic potential
• seed: Seed for the random phases $\phi_{nm}$ of the electrostatic potential (optional; default=27)

$$ V(x,y,t) =\sum_{n, m=1}^M \frac{A}{(n^2+m^2)^{3/2}} {\rm e}^{i(n x+my +\phi_{nm} -t) } $$

Example

params = {"A": 1.0, "M": 16, "seed": 42}
gc = GC2Ds(params)
z0 = gc.initial_conditions(100, type="random")

Key methods

Since GC2Ds is s subclass of HamSys (from the python package pyhamsys), it inherits all its methods, including:

• integrate: Integrate numerically the trajectories of the system defined by the element of the class GC2Ds from the initial conditions defined by the function initial_conditions.

• compute_lyapunov: Compute the Lyapunov spectrum.

• save_data: Save simulation results to a .mat file with metadata.

In addition, GC2Ds has the following key methods:

• initial_conditions: Generate starting (x, y) positions—random or on a regular grid.

• y_dot: Time derivative of positions for integration.

• k_dot: Scalar diagnostic of the potential field.

• potential: Potential value at time t and position z=(x, y), and its first and second derivatives, obtained by specifying (dx, dy).

• hamiltonian: Total Hamiltonian (sum of the potentials for each trajectory).

• y_dot_lyap: Extended system (equations of motion and tangent flow) for Lyapunov-exponent calculations.

• plot_sol: 2-D plot of a solution obtained by the function integrate.

initial_conditions

Generate initial 2-D coordinates for trajectories on a periodic domain.

Usage

initial_conditions(
    n_traj: int = 1,
    x: tuple[float, float] | None = None,
    y: tuple[float, float] | None = None,
    type: str = "fixed",
    seed: int | None = None
) -> xp.ndarray

Parameters

• n_traj: Number of points. For "fixed", rounded to a perfect square for a square grid.
• x, y: (min, max) ranges; default (0, 2π).
• type: "random" for uniform random samples, "fixed" for a regular grid.
• seed: Random seed when type="random".

Returns

1-D xp.ndarray of length 2*n_traj: all x's followed by all y's.

Example

z0 = gc.initial_conditions(50, type="random", seed=123)
z0 = gc.initial_conditions(100, x=(0, np.pi), y=(0, np.pi), type="fixed")

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Reference:

For more information: cristel.chandre@cnrs.fr