Magnetic shift

Remove magnetic shift from flutter terms

$A_\parallel$ is not a pure fluctuation in our full-f model: it contains a part that overlaps with the magnetic equilibrium, which should not be double-counted by the flutter operators Scott 2006. This overlapped part is called magnetic shift, represented by $A_s$. Therefore in flutter terms only the fluctuating part $A_1$ shall be used, i.e. $\tilde{\mathbf{b}}\cdot\nabla f = \frac{\delta\beta_0}{B_\varphi} \left[A_1, f\right]_{R,Z}$, where $A_1$ is obtained by removing the magnetic shift $$\begin{align*} &A_1=A_\parallel-\langle A_s \rangle_\varphi ,\qquad \text{for axisymmetric geometry}\\ &A_1=A_\parallel-A_s,\qquad \text{for non-axisymmetric geometry}, \end{align*}$$ The toroidal average $\langle \rangle_\varphi$ is applied in axisymmetric geometries e.g. tokamaks.

Evolution equation of magnetic shift

$$\begin{equation} \frac{\partial A_s}{\partial t} = \frac{A_\parallel-A_s}{\lambda} . \label{eq:con-As} \end{equation}$$ $\lambda$ is an input parameter with the unit of time. MHD events with a time scale longer than $\lambda$ will be filtered out as $A_s$ is removed. The initial value of $A_s$ must be prescribed.

Eqn. (\ref{eq:con-As}) is valid only if $\lambda>0$. If $\lambda =0$, the evolution equation will be chosen as
$$\begin{equation} A_s = A_\parallel. \end{equation}$$

Eqn. (\ref{eq:con-As}) will degenerate into the exponentially weighted moving average Lucas 1990 with the finite uniform timestep $\Delta \tau$ $$\begin{equation} A_s[n] = (1- \frac{\Delta \tau}{\lambda})A_s[n-1] + \frac{\Delta \tau}{\lambda} A_\parallel[n] , \label{eq:dis-As} \end{equation}$$