Shear Alfvén Wave Field Classes

Given an equilibrium field \(\textbf{B}_0\), a shear Alfvén wave is modeled through the perturbed electrostatic potential, \(\Phi\), and parameter \(\alpha\) defining the perturbed vector potential \(\delta \textbf{A} = \alpha \textbf{B}_0\). The perturbed electric and magnetic fields then satisfy:

\[ \begin{align}\begin{aligned}\delta \textbf{E} = -\nabla \Phi - \frac{\partial \alpha}{\partial t}\\\delta \textbf{B} = \nabla \times \left(\alpha \textbf{B}_0 \right)\end{aligned}\end{align} \]

The parameter \(\alpha\) is related to \(\Phi\) given ideal Ohm’s law:

\[\nabla_{||} \Phi = -B_0 \frac{\partial \alpha}{\partial t}\]

In the ShearAlfvenWave classes, the equilibrium field is prescribed as a BoozerMagneticField class in addition to input parameters determining \(\Phi\).

ShearAlfvenHarmonic

This class initializes a Shear Alfvén Wave with a scalar potential of the form:

\[\Phi(s, \theta, \zeta, t) = \hat{\Phi}(s) \sin(m \theta - n \zeta + \omega t + \phi_0)\]

where \(\hat{\Phi}(s)\) is a radial profile, \(m\) is the poloidal mode number, \(n\) is the toroidal mode number, \(\omega\) is the frequency, and \(\phi_0\) is the phase shift. The perturbed parallel vector potential parameter, \(\alpha\), is then determined by ideal Ohm’s law. This representation is used to describe SAWs propagating in an equilibrium magnetic field \(\textbf{B}_0\).

Usage Example

from simsopt.field import BoozerAnalytic, ShearAlfvenHarmonic
import numpy as np

# Create equilibrium field
eq_field = BoozerAnalytic(B0=1.0, iota0=0.5)

# Define radial profile (constant amplitude)
Phihat = 0.01

# Create shear Alfvén wave
saw = ShearAlfvenHarmonic(
    Phihat,  # radial profile amplitude
    m=2,     # poloidal mode number
    n=1,     # toroidal mode number
    omega=1.0,  # frequency
    phase=0.0,  # phase shift
    B0=eq_field  # equilibrium field
)

ShearAlfvenWavesSuperposition

Class representing a superposition of multiple Shear Alfvén Waves (SAWs).

This class models the superposition of multiple Shear Alfvén waves, combining their scalar potentials \(\Phi\), vector potential parameters \(\alpha\), and their respective derivatives to represent a more complex wave structure in the equilibrium field \(\textbf{B}_0\).

The superposition of waves is initialized with a base wave, which defines the reference equilibrium field \(\textbf{B}_0\) for all subsequent waves added to the superposition. All added waves must have the same \(\textbf{B}_0\) field.

See Paul et al., JPP (2023; 89(5):905890515. doi:10.1017/S0022377823001095) for more details.

Usage Example

from simsopt.field import ShearAlfvenWavesSuperposition
import numpy as np

# Create base wave
Phihat1 = 0.01

wave1 = ShearAlfvenHarmonic(
    Phihat1, m=2, n=1, omega=1.0, phase=0.0, B0=eq_field
)

# Create superposition
saw_super = ShearAlfvenWavesSuperposition(wave1)

# Add additional waves
Phihat2 = 0.005

wave2 = ShearAlfvenHarmonic(
    Phihat2, m=3, n=1, omega=1.5, phase=0.0, B0=eq_field
)

saw_super.add_wave(wave2)

Wave Evaluation

Both wave classes provide methods to evaluate the perturbed fields. First, set the evaluation points, then evaluate the wave quantities:

# Set evaluation points - must be in shape (npoints, 4)
points = np.array([
    [0.5, 0.0, 0.0, 0.0],  # [s, theta, zeta, t]
    [0.6, 0.1, 0.1, 0.1],
    [0.7, 0.2, 0.2, 0.2]
])
saw.set_points(points)

# Now evaluate wave quantities
phi = saw.Phi()  # scalar potential
alpha = saw.alpha()  # vector potential parameter

# Get derivatives
dphi_dpsi = saw.dPhidpsi()  # derivative with respect to psi
dphi_dtheta = saw.dPhidtheta()  # derivative with respect to theta
dphi_dzeta = saw.dPhidzeta()  # derivative with respect to zeta
dphi_dt = saw.Phidot()  # time derivative of phi
dalpha_dt = saw.alphadot()  # time derivative of alpha

# For single point evaluation
single_point = np.array([[0.5, 0.0, 0.0, 0.0]])  # shape (1, 4)
saw.set_points(single_point)
phi_single = saw.phi()[0]  # get first (and only) value

Radial Profiles

The radial profile \(\hat{\Phi}(s)\) can be specified in two ways:

1. Constant Profile (Uniform) A single float value representing a uniform amplitude across all radial positions:

# Uniform amplitude across all s
Phihat = 0.01
saw = ShearAlfvenHarmonic(Phihat, m=2, n=1, omega=1.0, phase=0.0, B0=field)

2. Varying Profile (Tabulated) A tuple of two lists defining the radial dependence: (s_values, Phihat_values):

# Define varying radial profile
s_values = [0.0, 0.3, 0.5, 0.7, 1.0]
Phihat_values = [0.0, 0.005, 0.01, 0.005, 0.0]

# Create wave with varying profile
saw = ShearAlfvenHarmonic(
    (s_values, Phihat_values),
    m=2, n=1, omega=1.0, phase=0.0, B0=field
)

Note

The s_values must be in the range [0, 1] and will be automatically sorted. For non-zero poloidal mode numbers (m ≠ 0), the profile is automatically set to zero at s = 0.