Normalization

In order to normalize the equations of the various models implemented in this package, we start at the level of the Lagrangian. In the following, this is explained exemplary for the charged particle Lagrangian, however, it generalizes straightforwardly to other systems like the Pauli particle or the guiding center system.

Charged Particle Lagrangian

Consider the phasespace Lagrangian

\[L (q, \dot{q}, v) = ( mv + A (x) ) \cdot \dot{x} - \frac{m}{2} \vert v \vert^2 - e \phi (x) ,\]

and, in full generality, introduce the following normalizations:

\[t = \hat{t} t' , \quad x = \hat{x} x' , \quad v = \hat{v} v' , \quad A = \hat{A} A' , \quad \phi = \hat{\phi} \phi' , \quad L = \hat{L} L ' .\]

This leads us to

\[L' = \frac{L}{\hat{L}} = \frac{1}{\hat{L}} \, \big( m \hat{v} v' + e \hat{A} A' \big) \cdot \frac{\hat{x}}{\hat{t}} \dot{x}' - \frac{m \hat{v}^2}{\hat{L}} \frac{\vert v' \vert^2}{2} - \frac{e \hat{\phi}}{\hat{L}} \phi' .\]

Let us choose the following normalizations (and note that others are possible and may be more appropriate, depending on the problem at hand):

\[\begin{aligned} \hat{A} &= \hat{l} \hat{B} , & \hat{\phi} &= \hat{l} \hat{E} , & \hat{L} &= m \hat{v}^2 = e \hat{\phi} , \end{aligned}\]

with the characteristic length $l$. The normalized Lagrangian becomes

\[L' = \bigg( \frac{\hat{x}}{\hat{t} \hat{v}} \, v' + \underbrace{\frac{e \hat{B}}{m}}_{\hat{\omega}_c} \frac{\hat{x} \hat{l}}{ \hat{t} \hat{v}^2} \, A' \bigg) \cdot \dot{x}' - \frac{\vert v' \vert^2}{2} - \phi' ,\]

where $\hat{\omega}_c = e \hat{B} / m$ is the characteristic gyration frequency. This suggests to set

\[\begin{aligned} \hat{t} &= \hat{\omega}_c^{-1} , & \hat{x} &= \hat{l} , & \hat{v} &= \frac{\hat{x}}{\hat{t}} = \hat{l} \hat{\omega}_c , \end{aligned}\]

so that

\[L' = ( v' + A' ) \cdot \dot{x}' - \frac{\vert v' \vert^2}{2} - \phi' .\]

We thus have obtained the normalized Lagrangian.

Strongly Magnetized Plasmas

Often, especially when simulating an ensemble of particles, it is more appropriate to normalize the velocity to the thermal velocity and to choose different length scales for $\hat{x}$ and $\hat{l}$, specifically

\[\begin{aligned} \hat{v} &= v_{\mathrm{th}} , & \hat{t} &= \hat{\omega}_c^{-1} , & \hat{A} &= \hat{l} \hat{B} , & \hat{\phi} &= \hat{l} \hat{E} , & \hat{L} &= m v_{\mathrm{th}}^2 , \end{aligned}\]

where $v_{\mathrm{th}} = \sqrt{ W / m}$ denotes the thermal velocity and $W = k_B T$ the thermal energy. Consequently,

\[L' = \bigg( \frac{\hat{x}}{\hat{t}} \frac{m \hat{v}}{m v_{\mathrm{th}}^2} \, v' + \frac{\hat{\omega}_c^2}{v_{\mathrm{th}}^2} \, \hat{x} \hat{l} \, A' \bigg) \cdot \dot{x}' - \frac{m \hat{v}}{m v_{\mathrm{th}}^2} \frac{\vert v' \vert^2}{2} - \frac{e \hat{\phi}}{m v_{\mathrm{th}}^2} \phi' .\]

Further, set

\[\begin{aligned} \hat{x} &= \hat{v} \hat{t} = \frac{v_{\mathrm{th}}}{\hat{\omega}_c} = \hat{\rho}_{\mathrm{th}} , \end{aligned}\]

where $\hat{\rho}_{\mathrm{th}}$ is the characteristic gyro radius, and the normalized Lagrangian becomes

\[L' = \bigg( v' + \frac{\hat{l}}{\hat{\rho}_{\mathrm{th}}} \, A' \bigg) \cdot \dot{x}' - \frac{\vert v' \vert^2}{2} - \frac{e \hat{\phi}}{m v_{\mathrm{th}}^2} \phi' .\]

This leaves room for different orderings and consequently different normalizations. For example, in drift kinetics, we have $\hat{\rho}_{\mathrm{th}} / \hat{l} \sim \epsilon$ and $e \hat{\phi} / m v_{\mathrm{th}}^2 \sim 1$, while in gyro kinetics, we have $\hat{\rho}_{\mathrm{th}} / \hat{l} \sim 1$ and $e \hat{\phi} / m v_{\mathrm{th}}^2 \sim \epsilon$.