New QS equilibria

PyAQSC can solve - The full governing equation set for a QS equilibrium. - Or only the magnetic equations for a QS field with no force balance.

Each configuration is managed by a object of the traced class Equilibrium. An example is the return from aqsc.circular_axis(). See background on free parameters for a list of free parameters, their physics definition, dependence, and expansion forms.

This part shows how to create a new QS equilibrium from scratch and iterate to higher orders. For creating a QS magnetic field without force balance, see the next part.

All grid values must be provided at $\phi=0, \frac{1}{n_\text{grid}}\frac{2\pi}{n_\text{field period}}, ..., \frac{n_\text{grid}-1}{n_\text{grid}}\frac{2\pi}{n_\text{field period}}$ .

Creating new configrations

aqsc.leading_orders() creates a QS equilibrium and performs leading order ( $n=2$ ) calculations:

new_equilibrium = aqsc.leading_orders(
    nfp,
    Rc, Rs, Zc, Zs,
    p0
    Delta_0_avg,
    B_theta_20_avg,
    B_alpha_1, 
    B0, B11c, B22s, B20, B22c,
    len_phi,
    static_max_freq,
    traced_max_freq,
    riccati_secant_n_iter
)

Note: B11c must be nonzero due to the inner working of the expansion.

Parameters:

nfp : int (static) - The number of field period.
Rc, Rs, Zc, Zs : list(float) (traced) - $sin(i N_{fp}\Phi)$ , $cos(i N_{fp}\Phi)$ coefficients of axis shape $R_0$ and $Z_0$ w.r.t. cylindrical $\Phi$ . The $i$ -th component is the $i$ -th mode coefficient.
p0 : array (traced) - The on-axis pressure as a function of general Boozer coordinate angle $\phi$ in one field period on grids.
Delta_0_avg : float (traced) - The average on-axis anisotropy.
B_theta_20_avg : float (traced) - The average $\chi$ -independent component of $\bar{B}_{\theta 2}$ .
B_alpha_1 : float (traced) - The 1st order component of flux funtion $B_\alpha$
B0, B11c, B22s, B20, B22c : float (Traced) - Leading components of the magnetic field magnitude $B^-$ .
len_phi (static) - The $\phi$ grid number
static_max_freq : int (static) - The cut-off frequency for the low-pass filter on the results. Tied to array sizes during spectral solve, and lower value drastically increases solving speeds. Changing will result in recompiling.
traced_max_freq (traced) - The cut-off frequency for the low-pass filter on the results. Doesn't impact speed and doesn't require recompiling.
riccati_secant_n_iter (static) - The number of Secant iterations for calculating the leading order Riccati equation.

Returns: - A new Equilibrium.

Iterating to higher orders

To calculate the next two orders, use aqsc.Iterate_2():

new_equilibrium = iterate_2(
    equilibrium,
    B_alpha_nb2,
    B_denom_nm1, B_denom_n,
    iota_new, 
    B_theta_n0_avg,
    static_max_freq=(-1,-1),
    traced_max_freq=(-1,-1),
    max_k_diff_pre_inv=(-1,-1),
    n_eval=None,
)

Each call creates a new Equilubrium.

Parameters: - equilibrium : Equilibrium (traced) - The Equilibrium to iterate. - B_alpha_nb2 : float (traced) - $B_{\alpha n/2}$ - B_denom_nm1 : ChiPhiFunc (traced) - $B^-_{n-1}$ - B_denom_n : ChiPhiFunc (traced) - $B^-_n$ - iota_new : float (traced) - $\bar{\iota}$ - B_theta_n0_avg: float(traced) - The toroidal average of $B_{\theta n0}$ , $\int d\phi B_{\theta n0}$ - static_max_freq : (int,int) (static) - The cut-off frequency for the low-pass filter on the results. - traced_max_freq : (scalar,scalar) (traced) - The cut-off frequency for the low-pass filter on the results. - max_k_diff_pre_inv : (scalar,scalar) (traced) - Cut-off frequency for the off-diagonal filter when solving the looped equations. - n_eval : int (static) - Order to evaluate to. By default evaluates the next two orders, but can also re-evaluate 2 orders from lower $n$ . Must be even and no greater than $n+2$ .

Returns - A new Equilibrium

This function is defined in equilibrium.py.