Boozer transform

Lets apply the Boozer transform to the equilibrium computed in the elliptic stellarator tutorial.

In GVEC, we use a poloidal angle \(\vartheta\) and toroidal angle \(\zeta\), from which we can easily sample quantities on a regular grid. The transform to Boozer angles is

\[\begin{align*} \vartheta_B(\rho,\vartheta,\zeta) &= \vartheta + \lambda(\rho,\vartheta,\zeta) + \iota(\rho)\nu(\rho,\vartheta,\zeta)\\ \zeta_B(\rho,\vartheta,\zeta) &= \zeta + \nu(\rho,\vartheta,\zeta) \end{align*}\]

The Boozer transform is described in the theory section. In short:

• \(\lambda\) is recomputed on a higher Fourier resolution (by default MNfactor=5 is used).

• \(\nu\) can be deduced from \(\lambda\), and is computed with the same resolution as \(\lambda\).

• Evaluations on a regular grid in Boozer angles \((\vartheta_B)_i,(\zeta_B)_j\) means that we have to first find the corresponding \((\vartheta)_{ij},(\zeta)_{ij}\) positions, using a 2D Newton search, and then evaluate on these positions.

import matplotlib.pyplot as plt
import numpy as np

import gvec

Load previously computed state

We load the state from the previously computed equilibrium by specifying a runpath which we provide to gvec.find_state.

# path to the previous run
runpath = "run_ellipstell"
state = gvec.find_state(runpath)

Boozer transform

Now we can construct the grid we want to evaluate on, and call gvec.evaluate_sfl, which computes the Boozer transform and evaluates on a regular grid in Boozer angles.

nfp = state.nfp
# select 4 radial positions
rho = [0.2, 0.5, 0.8, 1.0]
theta_B = np.linspace(0, 2 * np.pi, 51)
zeta_B = np.linspace(0, 2 * np.pi / nfp, 51)
varlist = ["mod_B", "B_contra_t", "B_contra_z", "B_contra_t_B", "B_contra_z_B"]

evb = state.evaluate_sfl(*varlist, rho=rho, theta=theta_B, zeta=zeta_B, sfl="boozer")

Visualization in Boozer angles

Lets visualize first the transform as a grid in \(\vartheta_B\) and \(\zeta_B\).

fig, axs = plt.subplots(2, 2, figsize=(8, 8), sharey=True, sharex=True, tight_layout=True)
for i, ax in enumerate(axs.flatten()):
    evi = evb.isel(rad=i)
    ax.contour(
        evi.theta_B / (2 * np.pi),
        evi.zeta_B / (2 * np.pi),
        evi.theta.T,
        np.linspace(-2 * np.pi, 2 * np.pi, 20),
        colors="black",
    )
    ax.contour(
        evi.theta_B / (2 * np.pi),
        evi.zeta_B / (2 * np.pi),
        evi.zeta.T,
        np.linspace(-2 * np.pi / 3, 2 * np.pi / 3, 20),
        colors="red",
    )
    ax.set(
        xlabel=r"$\vartheta_B/(2\pi)$",
        ylabel=r"$\zeta_B/(2\pi)$",
        title=f"$\\rho = {evi.rho.data:.2f}$",
    )
fig.legend(
    handles=[
        plt.Line2D([0], [0], color="black", label=r"$\vartheta=$const."),
        plt.Line2D([0], [0], color="red", label=r"$\zeta=$const."),
    ]
)
fig.suptitle(
    "Contours of logical GVEC coordinates ($\\vartheta$,$\\zeta$)\n in Boozer coordinates ($\\vartheta_B$,$\\zeta_B$)"
);

Finally, we can visualise the \(|B|\) contours in Boozer coordinates.

fig, axs = plt.subplots(2, 2, figsize=(8, 8), sharey=True, sharex=True, tight_layout=True)
for i, ax in enumerate(axs.flatten()):
    evi = evb.isel(rad=i)
    c = ax.contour(evi.theta_B / (2 * np.pi), evi.zeta_B / (2 * np.pi), evi.mod_B.T, levels=20)
    fig.colorbar(c, ax=ax)
    ax.set(
        xlabel=r"$\vartheta_B/(2\pi)$",
        ylabel=r"$\zeta_B/(2\pi)$",
        title=f"$\\rho = {evi.rho.data:.2f}$",
    )

fig.suptitle("Contours of |B| in Boozer coordinates ($\\vartheta_B$,$\\zeta_B$)");