Cylindrical Magneto-Coriolis Waves Solver

The interaction of rotation, stratification and magnetic fields gives rise to a rich spectrum of waves in geophysical and astrophysical fluids. In this project, carried out as part of research work, we create a framework that includes both viscosity and magnetic diffusivity. The aim is to derive and solve the equations governing viscous–diffusive Magneto-Coriolis (MC) waves inside a cylindrical geometry. Using spectral methods (Chebyshev collocation) and a toroidal–poloidal decomposition, we construct the eigenvalue problem and compute the corresponding eigenmodes.

The work is organised in two complementary notebooks:

1. Derivation of equations (SageMath): symbolic manipulation of the governing equations in cylindrical coordinates, separation into poloidal/toroidal components, and derivation of the final system to be solved.
2. Numerical solver (Python): implementation of the spectral method, construction of matrices, eigenvalue computation, and visualisation of the resulting modes.

---

Part 1: Derivation of Poloidal–Toroidal Equations

In the SageMath notebook MHD-SageMath, we perform the algebraic steps needed to derive a well-posed eigenvalue problem:

• Decomposition of the velocity and magnetic fields into poloidal and toroidal potentials.
• Application of the governing equations (hydrodynamic and MHD cases).
• Obtain a compact matrix system suitable for spectral discretisation.

This notebook ensures the mathematical consistency of the model before moving to numerical implementation.

---

Part 2: Numerical Solver for MC Waves

In the Python notebook Macnus, we implement the solver step by step:

• Construction of Chebyshev differentiation matrices and boundary conditions.
• Assembly of the eigenvalue problem matrices based on the derived equations.
• Use of linear algebra routines to compute eigenfrequencies and eigenmodes.
• Visualisation of eigenfunctions (radial profiles, meridional slices, convergence analysis).

The notebook is organised so that each cell can be executed sequentially, from initialisation of parameters to post-processing and plotting.

---

Main Results

• Computation of eigenfrequencies of Magneto-Coriolis modes in cylindrical geometry.
• Visualisation of eigenmodes (velocity and magnetic components).
• Exploration of convergence with respect to the number of collocation points.
• Framework that can be extended to other boundary conditions or parameter regimes.

---

Conclusion

This project demonstrates how symbolic derivation and numerical computation can be combined to tackle complex wave problems in rotating magnetised fluids. The methodology provides a flexible basis for exploring parameter dependence and physical regimes relevant to geophysics and astrophysics.

---

Use

Run the notebooks in order:

• Derivation of equations: MHD-SageMath
• Numerical solver: Macnus

Both require standard scientific Python libraries (numpy, scipy, matplotlib, mpmath) and, for the derivation notebook, SageMath.

Citation

If you use this software, please cite it as:

Alexandre Nuyt. (2025). Alnuyt/Magneto-Coriolis-Waves-Solver: Spectral-MC-Solver (v1.0.0). Zenodo. https://doi.org/10.5281/zenodo.17792073