Fluid model

ShearAlfven

Shear-Alfvén part of LinearMHD with zero-flow equilibrium.

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Overview

ShearAlfven is the incompressible magnetic-tension subsystem extracted

from linear MHD. It is useful when one wants to isolate Alfvénic propagation from the full linear MHD model.

Use cases

This model is appropriate for:

isolated shear-Alfvén wave propagation
verification of the ShearAlfven propagator
reduced linear-MHD studies without compressibility

Governing equations

PDEs solved by model:

Momentum:

ρ_{0} \frac{\partial U ~}{\partial t} = (\nabla \times B ~) \times B_{0}

Induction:

\frac{\partial B ~}{\partial t} - \nabla \times (U ~ \times B_{0}) = 0

Normalization

All velocities are normalized by the Alfvén speed,

U^= v^_{A} .

Discretization

Time integration is performed by the following propagators (in sequence):

ShearAlfvenPropagator

Diagnostics

The following scalars are tracked during simulation:

Perturbed kinetic energy: en_U
Perturbed magnetic energy: en_B
Background magnetic energy: en_B_eq
Total magnetic energy including background: en_B_tot
Perturbation total energy: en_tot
Total energy including background contribution: en_tot2

Examples

Standing shear-Alfvén wave

A single transverse velocity mode splits into two counter-propagating Alfvén waves, i.e. a standing wave. Its energy oscillates between the kinetic and the magnetic channel at twice the wave frequency, while the structure-preserving discretization keeps the sum constant. The launch condition is $u_x(z,0)=0.01\sin(\pi z/5),\qquad \delta\mathbf{B}(z,0)=0,$ in a periodic interval of length $L_z=20$ , with $\mathbf{B}_0=\mathbf{e}_z$ and $n_0=1$ .

Shear-Alfvén wave dispersion

Excite a broadband transverse velocity perturbation and recover the shear-Alfvén dispersion relation ω = v_A k with Struphy’s linearized MHD solver. Both transverse velocity components are seeded with coefficient-noise amplitude $A=0.05$ . The background is $\mathbf{B}_0=\mathbf{e}_z,\qquad n_0=1,\qquad L_z=20,$ so the reference Alfvén speed is $v_A=B_0/\sqrt{n_0}=1$ .