A more accurate form of the guiding center drift

Equation. (61) can also be written as

$$\left( 1 + \frac{v_{\parallel}}{\Omega} \ensuremath{\boldsymbol{b... ... t} + \frac{v_{\parallel}^2}{\Omega} \nabla \times \ensuremath{\boldsymbol{b}},$$ (67)

which can be arranged in the following form

$$\left( 1 + \frac{v_{\parallel}}{\Omega} \ensuremath{\boldsymbol{b... ...{\boldsymbol{b}} \times \frac{\partial \ensuremath{\boldsymbol{b}}}{\partial t}$$ (68)

As discussed above, the last term $v_{\parallel} / \Omega \ensuremath{\boldsymbol{b}} \times \partial \ensuremath{\boldsymbol{b}} / \partial t$ is usually ignored. Thus, the above equation is written as

$$\left( 1 + \frac{v_{\parallel}}{\Omega} \ensuremath{\boldsymbol{b... ...mbol{b}} \times \left( - Z e \ensuremath{\boldsymbol{E}} + \mu \nabla B \right)$$ (69)

Define

$$B^{\star} = B \left( 1 + \frac{v_{\parallel}}{\Omega} \ensuremath{\boldsymbol{b}} \cdot \nabla \times \ensuremath{\boldsymbol{b}} \right),$$ (70)

which is related to $\ensuremath{\boldsymbol{B}}^{\star}$ defined in Eq. (17) through $B^{\star} = \ensuremath{\boldsymbol{b}} \cdot \ensuremath{\boldsymbol{B}}^{\star}$, then Eq. (69) is written as

$$\dot{\ensuremath{\boldsymbol{X}}} = \frac{v_{\parallel}}{B^{\star... ...+ \frac{1}{m \Omega B^{\star}} \ensuremath{\boldsymbol{B}} \times \mu \nabla B,$$ (71)

Define

$$\ensuremath{\boldsymbol{v}}_{\parallel}^{\star} \equiv \frac{v_{\... ... \frac{v_{\parallel}}{\Omega} \nabla \times \ensuremath{\boldsymbol{b}} \right)$$ (72)

Eq. (71) is written as

$$\dot{\ensuremath{\boldsymbol{X}}} = \ensuremath{\boldsymbol{v}}_{... ... + \frac{1}{m \Omega B^{\star}} \ensuremath{\boldsymbol{B}} \times \mu \nabla B$$ (73)

which agrees with Eqs. (8)-(14) in Todo's paper[4]. Note that in this form of the guiding center drift, the curvature drift is included in $\ensuremath{\boldsymbol{v}}_{\parallel}^{\star}$ (the second term in Eq. (72)). Compared with Eq. (66), Eq. (73) is more accurate because it does not use the approximation that $\ensuremath{\boldsymbol{b}} \cdot \nabla \times \ensuremath{\boldsymbol{b}} \approx 0$. The numerical results form my numerical code indicate that Eq. (73) can conserve the toroidal angular momentum more accurately than Eq. (66).