Physics Problems

Advection Problem

The solver is configured to investigate the advection equation $\partial n/\partial t + c_s \nabla_\parallel n = 0$ where $\nabla_\parallel$ is the gradient operator along the magnetic field lines. The magnetic geometry is a simple sheared cylinder defined by the safety factor $q(r) = 1 + 2r^2$.

Two primary initial conditions are provided to test different physical phenomena.

1. Isotropic Blob (Shearing Test)

• Initial Condition: A 2D Gaussian blob of density that is not aligned with the magnetic field.
• Physics: Because the blob exists across multiple flux surfaces with different q values, it is subject to magnetic shear. The parts of the blob at larger radii will advect poloidally at a different effective speed than the parts at smaller radii.
• Expected Result: The blob will smear across the radii at different rates.

2. Field-Aligned Blob (Parallel Coherence Test)

• Initial Condition: A blob that is elongated along a single magnetic field line.
• Physics: In an ideal case, a structure that is perfectly aligned with the magnetic field should experience no shearing forces. All parts of the structure should move together coherently.
• Expected Result: The elongated blob should rotate rigidly as a coherent structure. It should not shear apart. This is a stringent test of the accuracy of the parallel gradient operator and the interpolation scheme. Any numerical error in the parallel direction will manifest as artificial shearing.

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