Vortex merger

Studied problem

Field setup

We work with cylindrical coordinates \((r,\theta,z)\) on the corresponding orthonormal basis vectors \((e_r, e_\theta, e_z)\).

General case

Conservation of the charge density equation and Maxwell-Poisson equation:

\[ \partial_t \rho + div(\rho v) = 0 \]
\[ \Delta \phi + \frac{\rho}{\varepsilon_0} = 0 \]
\[ v = \frac{E\wedge B}{|B|^2} \]
\[ E = -\nabla \phi \]

where

  • \(\rho = \rho(t,r,\theta)\) is the electric charge density of the particles (electrons with mass \(m_e\) and charge \(q\));
  • \(v\) is the velocity of the particles (\(div(v) = 0\));
  • \(\phi\) is the usual electric potential associated with the total electric field;
  • \((E,B)\) the electromagnetic field such that \(B = B_z(r) e_z\) the external magnetic field

Simplified case

We study a simplified case where \(B = e_z\), \(\varepsilon_0 = 1\). On a circular mapping, it gives \(v = -\nabla\phi \wedge e_z\), \(div(v) = 0\)
and with \((e_r, e_\theta)\) the unnormalized local contravariant base,

\[ \partial_t \rho - \frac{\partial_\theta\phi}{r}\partial_r\rho + \partial_r\phi \frac{\partial_\theta\rho}{r} =0, \]
\[ -\Delta \phi = \rho. \]

Test case - vortex merger

The test case implemented tests a vortex merger (see [1] article).

We suppose as perturbed initial condition

\[ \rho(0,r, \theta) = \rho_{eq}(x,y) + \varepsilon \left( \exp\left[ - \frac{(x - x_1^*)^2 + (y - y_1^*)^2}{2 \sigma^2} \right] + \exp\left[ - \frac{(x - x_2^*)^2 + (y - y_2^*)^2}{2 \sigma^2} \right] \right) \]

with

  • \((x_1^*, y_1^*) = (0.08, -0.14)\),
  • \((x_2^*, y_2^*) = (-0.08, 0.14)\),
  • \(\sigma = 0.08\) and
  • \(\varepsilon = 10^{-4}\).

The equilibrium solution is computed during the initialisation phase in the VortexMergerEquilibria class. The details are documented in initialisation.

References

[1] Edoardo Zoni, Yaman Güçlü, "Solving hyperbolic-elliptic problems on singular mapped disk-like domains with the method of characteristics and spline finite elements", https://doi.org/10.1016/j.jcp.2019.108889, Journal of Computational Physics, 2019.

Useful references

[2] Emily Bourne. “Non-Uniform Numerical Schemes for the Modelling of Turbulence in the 5D GYSELA Code”. PhD thesis. Aix-Marseille Université, Dec. 2022.

[3] Eric Sonnendrucker, Jean Roche, Pierre Bertrand and Alain Gizzo. “The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation”. Journal of Computational Physics (1999).

Contents

  • vortex_merger.cpp : runs a vortex-merger simulation.
  • params.yaml : defines the parameters of the simulation.

Python files: (in post_process/PythonScripts/geometryRTheta folder)

  • animation_rho_phi.py : plot and save the density and electrical potential function in time.
  • mass_conservation.py : plot the relative errors of the mass of the particles.
  • Mesh:
  • r_size: 128 : number of cells in \(r\)-dimension. (Tests in Edoardo Zoni's article.)
  • theta_size: 256 : number of cells in \(\theta\)-dimension. (Tests in Edoardo Zoni's article.)
  • r_min: 0.0 : start of \(r\) domain. (Tests in Edoardo Zoni's article.)

  • r_max: 1.0 : end of \(r\) domain. (Tests in Edoardo Zoni's article.)

  • Time:

  • delta_t: 0.1 : time step. (Tests in Edoardo Zoni's article.)

  • final_T: 10.0: final time of the simulation. (Tests in Edoardo Zoni's article.)

  • Perturbation:

  • eps: 0.0001 : amplitude of the perturbation.
  • x_star_1: 0.08 : \(x\) coordinate of the centre of the initial first vortex.
  • y_star_1: -0.14 : \(y\) coordinate of the centre of the initial first vortex.
  • x_star_2: -0.08 : \(x\) coordinate of the centre of the initial second vortex.
  • y_star_2: 0.14 : \(y\) coordinate of the centre of the initial second vortex.

  • sigma: 0.08 : the standard deviation of the initial Gaussian function.

  • Output:

  • time_step_diag: 5 : number of time steps between two recordings of the data.

Executables

  • vortex_merger : the vortex-merger simulation with the given parameters.

The results of the simulation are saved in a output folder that the executable creates.

The path to the params.yaml must be given in the command line of the executable.