2. Canonical GC Hamiltonian for an axisymmetric equilibrium

A general axisymmetric toroidal magnetic configuration consisting of nested toroidal flux surfaces can be represented in White–Boozer (White Reference White2014) coordinates as

(2.1)\begin{equation} \boldsymbol{B}=g(\psi) \boldsymbol{\nabla}\zeta+I(\psi) \boldsymbol{\nabla} \theta+\delta(\psi,\theta) \boldsymbol{\nabla} \psi_p , \end{equation}

where $\zeta$ and $\theta$ are the toroidal and poloidal angles. The toroidal flux $\psi$ is related to the poloidal flux $\psi _p$ through the safety factor $q(\psi )= \textrm {d}\psi / \textrm {d} \psi _p$. The functions $g$ and $I$ are related to the poloidal and toroidal currents and $\delta$ is related to the non-orthogonality of the coordinate system.

The GC motion of a charged particle is described by the Lagrangian (Littlejohn Reference Littlejohn1983) $L=( \boldsymbol {A}+\rho _\parallel \boldsymbol {B} ) \cdot \boldsymbol {v} +\mu \dot {\xi }-H$, where $\boldsymbol {A}$ and $\boldsymbol {B}$ are the vector potential and the magnetic field, $\boldsymbol {v}$ is the GC velocity, $\mu$ is the magnetic moment, $\xi$ is the gyro-phase, $\rho _\parallel$ is the velocity component parallel to the magnetic field, normalized to $B$, and

(2.2)\begin{equation} H=\rho_\parallel^{2}B^{2}/2+\mu B \end{equation}

is the Hamiltonian. The GC motion is given in normalized units where time is normalized to $\omega _0^{-1}$, with $\omega _0=eB_0/m$ being the on-axis gyro-frequency, and distance is normalized to the major radius $R$, so that energy is normalized to $m\omega _0^{2}R^{2}$. According to the ordering of the GC approximation, the gyro-radius is $\rho =v/B<<1$ and the magnetic moment $\mu =v_\perp ^{2}/(2B)$ as well as the cross-field drift are of order $\rho ^{2}$ (White Reference White2014).

The three couples of canonically conjugate variables for this GC Hamiltonian are $(\mu , \xi )$, $(P_\theta , \theta )$ and $(P_\zeta ,\zeta )$ with

(2.3)\begin{equation} \left.\begin{aligned} P_\theta & = \psi+\rho_\parallel I(\psi), \\ P_\zeta & = \rho_\parallel g(\psi)-\psi_p(\psi) \end{aligned}\right\} \end{equation}

providing the relation between the canonical momenta $(P_\theta , P_\zeta )$ and $(\psi ,\rho _\parallel )$ (White Reference White2014). In terms of these canonical variables, (2.2) can be written as

(2.4)\begin{align} H(P_\theta,\theta, P_\zeta,\zeta, \mu, \xi)= \frac{\left[P_\zeta+ \psi_p(P_\theta,P_\zeta)\right]^{2}}{2g^{2}(\psi(P_\theta,P_\zeta))} B^{2}(\psi(P_\theta,P_\zeta),\theta)+\mu B(\psi(P_\theta,P_\zeta),\theta). \end{align}

The gyro-angle $\xi$ does not appear in the GC Hamiltonian which is also independent of $\zeta$ due to axisymmetry of the magnetic field; therefore the corresponding canonical momenta, namely $\mu$ and $P_\zeta$, are constants of the motion and since the Hamiltonian does not depend explicitly on time (autonomous system), the system is integrable.

3. Canonical transformation to drift orbit deviation variables

The GC Hamiltonian describes all particle drifts due to the inhomogeneity of the magnetic field, causing the GC deviation from a field line. A canonical transformation with generating function (Goldstein et al. Reference Goldstein, Poole and Safko2002)

(3.1)\begin{equation} \bar{F}(P_\theta, \bar{\theta}, P_\zeta,\bar{\zeta}, \mu, \bar{\xi})=-\bar{\theta} (P_\theta-P_{\theta 0})-\bar{\zeta}\left(P_\zeta+\int^{P_\theta} \frac{ \textrm{d} P'_\theta}{q\left(\psi(P'_\theta, P_\zeta)\right)} \right) - \bar{\xi} \mu \end{equation}

transforms to a new (barred) variable set, related to the original variables as

(3.2)\begin{equation} \left.\begin{aligned} \bar{P}_{\theta} & =P_\theta-P_{\theta 0}, \\ \bar{\theta} & =\theta-\frac{\bar{\zeta}}{q(\psi(P_\theta,P_\zeta))}, \\ \bar{P}_{\zeta} & =P_\zeta+\int^{P_\theta} \frac{ \textrm{d} P'_\theta}{q(\psi(P'_\theta, P_\zeta))}, \\ \zeta & =\bar{\zeta} \left( 1+\int^{P_\theta}\frac{\partial q^{-1}}{\partial \psi} \frac{\partial \psi(P'_\theta, P_\zeta)}{\partial P_\zeta} \textrm{d} P'_\theta \right), \\ \bar{\mu} & =\mu, \\ \bar{\xi} & =\xi . \end{aligned}\right\} \end{equation}

This canonical tranformation is general and can be applied for either axisymmetric or non-axisymmetric equilibrium magnetic fields. The physical meaning of the new canonical variables becomes obvious for a LAR cylindrical equilibrium described by $g=1$, $I=0$, $\delta =0$ and $B=1-r \cos \theta$, where $r=\sqrt {2\psi }$ and the magnetic field is normalized to its on-axis value (White & Chance Reference White and Chance1984). In this case, we have $\partial \psi / \partial P_\zeta = 0$ and the original canonical momenta, as defined by (2.3), become $P_\theta =\psi$ and $P_\zeta =\rho _\parallel -\psi _p(P_\theta )$, so that $\bar{P}_{\zeta }=\rho _\parallel$ and $\bar {\zeta }=\zeta$. The new variables $(\bar{P}_{\theta },\bar {\theta })$ provide the deviation of the GC orbit from a magnetic field line of reference, for which $\theta =\zeta / q(\psi )$, intersecting the poloidal plane $\zeta =0$ at $(\psi _0,\theta _0)=(P_{\theta 0}, \bar {\theta })$. The new canonical angle $\bar {\theta }$ is directly related to the toroidal precession, defined as $\dot {\zeta }-q\dot {\theta }$. In contrast to previous approaches (Brizard & Duthoit Reference Brizard and Duthoit2014) where an a posteriori construction of a canonically conjugate pair of variables $(\rho _\parallel , s)$ for the LAR equilibrium case is necessary, within the context of our formulation, the canonical variables are a priori defined. Moreover, this also holds, not only for the LAR case, but also for an arbitrary equilibrium magnetic field.

For a magnetic field configuration that can be considered as a higher-order perturbation of the LAR equilibrium, according to the standard tokamak ordering, we can write $\psi =P_\theta +\epsilon ^{2}\hat {\psi }(P_\theta ,P_\zeta )$, with $\epsilon$ being the inverse aspect ratio and $\hat {\psi }(P_\theta , P_\zeta )$ corresponding to corrections due to ellipticity and triangularity (White Reference White2014). Therefore, for the new variable $\bar {\zeta }$ we have

(3.3)\begin{equation} \zeta =\bar{\zeta} \left(1+\epsilon^{2}\int^{P_\theta} \frac{\partial q^{-1}}{\partial \hat{\psi}} \frac{\partial \hat{\psi}(P'_\theta, P_\zeta)}{\partial P_\zeta} \textrm{d} P'_\theta\right) \end{equation}

and to lowest order $\bar {\zeta }=\zeta$, whereas the constant of the motion can be written as

(3.4)\begin{equation} P_\zeta(\bar{P}_{\theta},\bar{P}_{\zeta})=\bar{P}_{\zeta}-\psi_p(P_\theta)-\epsilon^{2} \int^{P_\theta}\frac{\partial q^{-1}(P'_\theta)}{\partial \psi} \psi (P'_\theta,P_\zeta)\, \textrm{d} P'_\theta \end{equation}

and to lowest order

(3.5)\begin{equation} \bar{P}_{\zeta}=P_\zeta+\psi_p(P_{\theta 0}+\bar{P}_{\theta}). \end{equation}

For the case of a smooth $q$ profile we can neglect (locally) higher-order derivatives of $\psi _p$ with respect to $\psi =P_\theta$, which is equivalent to neglecting the radial variation of the safety factor $q$ (magnetic shear) within a GC drift orbit, and applying a Taylor expansion in $\bar{P}_{\theta}/P_{\theta 0}$ results in

(3.6)\begin{equation} \bar{P}_{\zeta}=P_\zeta+\psi_p(P_{\theta 0})+\psi'_p(P_{\theta 0}) \bar{P}_{\theta}, \end{equation}

with prime denoting differentiation of $\psi _p$ with respect to its argument. Since $P_\zeta$ is constant, we can write

(3.7)\begin{equation} \bar{P}_{\zeta}=\rho_{\parallel 0}+q^{-1}(P_{\theta 0})\bar{P}_{\theta}, \end{equation}

with $\rho _{\parallel 0}=P_{\zeta }+\psi _p(P_{\theta 0})$ depending on both the invariant $P_{\zeta }$ and the flux surface of reference related to $P_{\theta 0}$. This relation imposes a constraint on the two canonical momenta $\bar{P}_{\zeta}$ and $\bar{P}_{\theta}$ due to the axisymmetry of the magnetic field and the corresponding invariance of $P_\zeta$. It is worth mentioning that for the case of drift orbits spanning a larger radial distance, we can also include the second-order derivative $\psi _p''(P_{\theta 0})$, related to the magnetic shear $(q')$, that would result in a quadratic term with respect to $\bar {P}_\theta$ in (3.6) and (3.7). As a result, the magnetic shear can modify the GC dynamics and orbital spectrum (Shaing Reference Shaing2015; Albert et al. Reference Albert, Heyn, Kapper, Kasilov, Kernbichler and Martitsch2016).

In the new canonical variables the Hamiltonian is given as

(3.8)\begin{align} H(\bar{P}_{\zeta},\bar{\zeta}, \bar{P}_{\theta},\bar{\theta} ; \bar{\mu})&= \frac{\bar{P}_{\zeta}^{2}}{2g^{2}(\psi(P_{\theta 0}+\bar{P}_{\theta}))} B^{2}\left(\psi(P_{\theta 0}+\bar{P}_{\theta}), \frac{\bar{\zeta}}{q(\psi(P_{\theta 0}+ \bar{P}_{\theta}))}+\bar{\theta}\right) \nonumber\\ &\quad +\bar{\mu} B\left(\psi(P_{\theta 0}+\bar{P}_{\theta}), \frac{\bar{\zeta}}{q(\psi(P_{\theta 0}+\bar{P}_{\theta}))}+\bar{\theta}\right) + O(\epsilon^{2}). \end{align}

It is clear that in the new variables there is no cyclic angle, so that neither $\bar{P}_{\theta}$ nor $\bar{P}_{\zeta}$ is a constant of the motion. However, the quantity $P_\zeta$ is still a constant of the motion, due to axisymmetry, restricting $\bar{P}_{\theta}$ and $\bar{P}_{\zeta}$ according to (3.7), so that integrability is preserved.

By substituting either of the canonical momenta as a function of the other from (3.7), the Hamiltonian system can be readily described in one degree of freedom either as $H(\bar{P}_{\zeta},\bar {\zeta };\rho _{\parallel 0},P_{\theta 0};\bar {\mu })$ or as $H(\bar{P}_{\theta},\bar {\theta };\rho _{\parallel 0}, P_{\theta 0}; \bar {\mu })$. In the former case, since the Hamiltonian does not depend on $\bar{P}_{\theta}$, its canonically conjugate angle $\bar {\theta }$ appears as an additive phase constant that can be omitted, whereas the same holds for $\bar {\zeta }$ in the latter case. This duality directly relates the radial and poloidal drift with the toroidal momentum and precession.

The Hamiltonian (3.8) provides a canonical description of the GC motion for a generic axisymmetric magnetic field equilibrium to the lowest order with respect to the standard tokamak ordering and drift orbit deviation from a magnetic field line. Higher-order magnetic field equilibria, with respect to the inverse aspect ratio $\epsilon$, can be considered by keeping higher-order terms in (3.4). The one-degree-of-freedom Hamiltonian $H(\bar{P}_{\zeta},\bar {\zeta };\rho _{\parallel 0},P_{\theta 0};\mu )$ can be readily used to calculate the action

(3.9)\begin{equation} J^{(b,t)}_\zeta=\frac{1}{2 {\rm \pi}} \oint \bar{P}_{\zeta}(\bar{\zeta}; E,\bar{\mu},\rho_{\parallel 0},P_{\theta 0})\, \textrm{d}\bar{\zeta}, \end{equation}

as well as the respective frequencies

(3.10)\begin{equation} \hat{\omega}^{(b,t)}_\zeta=\frac{\partial H}{\partial J^{(b,t)}_\zeta}, \end{equation}

with $E=H$ being the constant energy value of each orbit and the index $(b,t)$ corresponding to the cases of trapped (bounce) or passing (transit) orbits. The mixed-variable generating function (Goldstein et al. Reference Goldstein, Poole and Safko2002)

(3.11)\begin{equation} \hat{F}(\bar{\zeta},\bar{\theta},\bar{\xi};J_\zeta,J_\theta,J_\xi)= \bar{\xi} J_\xi+\bar{\theta}J_\theta+\int^{\bar{\zeta}} \bar{P}_\zeta(\bar{\zeta}';J_\zeta,J_\theta,J_\xi)\, \textrm{d}\bar{\zeta}' \end{equation}

provides the canonical transformation to action-angle variables also for the remaining canonical variable pairs:

(3.12)\begin{equation} \left.\begin{aligned} (\bar{P}_\zeta,\bar{\zeta}) & \rightarrow (J_\zeta,\hat{\zeta}), \\ (\bar{P}_\theta,\bar{\theta}) & \rightarrow (J_\theta,\hat{\theta}),\\ (\bar{\mu},\bar{\xi}) & \rightarrow (J_\xi,\hat{\xi}). \end{aligned}\right\} \end{equation}

It is worth emphasizing that in the other two pairs of canonical variables, the new momenta (actions) are identical to the old momenta, i.e. $J_\theta =\bar {P}_\theta$ and $J_\xi =\bar {\mu }=\mu$, but the new positions (angles) differ from the old ones, due to the last term of the generating function (3.11) (Kaufman Reference Kaufman1972). The transformation to action-angle variables allows for the calculation of the orbital frequencies in the remaining two degrees of freedom. Therefore,

(3.13)\begin{equation} \hat{\omega}_\theta=\frac{\partial H}{\partial J_\theta}=-\frac{\partial H}{\partial J_\zeta} \frac{\partial J_\zeta}{\partial J_\theta} \end{equation}

and

(3.14)\begin{equation} \hat{\omega}_\xi=\frac{\partial H}{\partial J_\xi}=-\frac{\partial H}{\partial J_\zeta} \frac{\partial J_\zeta}{\partial J_\xi} \end{equation}

or

(3.15)\begin{equation} \frac{\hat{\omega}_\theta}{\hat{\omega}_\zeta^{(b,t)}}=-\frac{\partial J_\zeta^{(b,t)}}{\partial J_\theta} \end{equation}

and

(3.16)\begin{equation} \frac{\hat{\omega}_\xi}{\hat{\omega}_\zeta^{(b,t)}}= -\frac{\partial J_\zeta^{(b,t)}}{\partial J_\xi}. \end{equation}

The above equations show that the bounce/transit actions $J_\zeta ^{(b,t)}$ (with a sign change) can be considered as the Hamiltonian functions providing the canonical equations for the other degrees of freedom when time is normalized with respect to the inverse bounce/transit frequency $\hat {\omega }_\zeta ^{(b,t)}$ (White Reference White2014). Along with the analytical expressions obtained in the next section, these equations provide compact formulas for bounce/transit-averaged dynamics under low-frequency electromagnetic fluctuations in the context of a bounce/gyro-kinetic theory (Brizard Reference Brizard2000; Duthoit, Brizard & Hahm Reference Duthoit, Brizard and Hahm2014).

The frequency $\hat {\omega }_\theta$ is directly related to the bounce/transit-averaged toroidal drift as shown in the following expression:

(3.17)\begin{align} \frac{\hat{\omega}_\theta}{\hat{\omega}_\zeta}&= -\frac{1}{2 {\rm \pi}}\oint \frac{\partial}{\partial J_\theta} \bar{P}_{\zeta}(\bar{\zeta};E,\bar{\mu},\rho_{\parallel 0},P_{\theta 0})\, \textrm{d}\bar{\zeta}= \frac{1}{2 {\rm \pi}}\oint \frac{\partial \bar{P}_{\zeta}}{\partial E} \frac{\partial E}{\partial J_\theta} \textrm{d}\bar{\zeta} \nonumber\\ &= \frac{1}{2 {\rm \pi}}\oint \left(\frac{ \textrm{d}\bar{\zeta}}{ \textrm{d} t}\right)^{-1} \frac{\partial E}{\partial \bar{P}_\theta} \textrm{d}\bar{\zeta}=\frac{1}{2 {\rm \pi}}\oint \left(\frac{ \textrm{d}\bar{\theta}}{ \textrm{d} t}\right) \textrm{d} t=\frac{1}{2 {\rm \pi}} \left({\rm \Delta} \bar{\theta}\right)_{T_\zeta=2{\rm \pi}/\hat{\omega}_\zeta}, \end{align}

where $({\rm \Delta} \bar {\theta } )_{T_\zeta =2 {\rm \pi} / \hat {\omega }_\zeta }$ is the variation of $\bar {\theta }=\theta -\zeta /q$ in the time interval of a period $T_\zeta =2{\rm \pi} /\hat {\omega }_\zeta$. Similarly, the frequency $\hat {\omega }_\xi$ corresponds to the bounce/transit-averaged gyro-frequency (Kaufman Reference Kaufman1972):

(3.18)\begin{equation} \frac{\hat{\omega}_\xi}{\hat{\omega}_\zeta}= \frac{1}{2 {\rm \pi}} \left({\rm \Delta} \bar{\xi} \right)_{T_\zeta=2{\rm \pi}/\hat{\omega}_\zeta}. \end{equation}

The ZDW approximation, under which the GC is considered as being fixed on a given flux surface $P_{\theta 0}=\psi _0$, can be readily obtained by setting $\bar{P}_{\theta}=0$. Moreover, for a relatively small drift width $\bar{P}_{\theta} << P_{\theta 0}$, $\psi$ can be Taylor-expanded around $P_{\theta 0}$ and with $\bar{P}_{\theta}$ substituted from (3.7) we have a single-degree-of-freedom Hamiltonian for the canonical variables $(\bar{P}_{\zeta},\bar {\zeta })$.

For a LAR equilibrium, the FDW Hamiltonian is written as

(3.19)\begin{align} H_{\textrm{FDW}}(\rho_\parallel,\zeta; \rho_{\parallel 0}, \mu; \psi_0) = \frac{\rho_\parallel^{2}}{2}\left[1-\sqrt{2\psi} \cos\left(\frac{\zeta}{q\left( \psi \right)} \right) \right]^{2} +\mu \left[1-\sqrt{2\psi} \cos\left(\frac{\zeta}{q\left( \psi \right)} \right) \right], \end{align}

where we have substituted $\bar{P}_{\zeta}=\rho _\parallel$ and dropped bar in $\zeta$ for simplicity. It is worth emphasizing that obtaining the Hamiltonian (3.19) is enabled by the utilization of the normalized parallel GC velocity $\rho _\parallel$ (normalized to the magnetic field), instead of the regular parallel GC velocity used in previous works (Brizard Reference Brizard2011). Moreover, according to (3.7), $\psi$ is taken as a function of $\rho _\parallel$ from

(3.20)\begin{equation} \psi= \psi_0 +q(\psi_0)(\rho_\parallel-\rho_{\parallel 0}), \end{equation}

with $\psi _0=P_{\theta 0}$ and $\rho _{\parallel 0}=P_{\zeta 0}+\psi _p(\psi _0)$.

4. Zero drift width approximation

We consider the ZDW approximation under which the GC motion is considered fixed on a given flux surface $\psi =\psi _0$ and the frequencies of $\theta$ and $\zeta$ are simply related as $\omega _\zeta =q(\psi _0)\omega _\theta$. The Hamiltonian is written as

(4.1)\begin{align} H_{\textrm{ZDW}}(\rho_\parallel,\zeta; \psi_0, \mu)= \frac{\rho_\parallel^{2}}{2}\left[1-\sqrt{2\psi_0} \cos\left(\frac{\zeta}{q\left( \psi_0 \right)} \right) \right]^{2} + \mu \left[1-\sqrt{2\psi_0} \cos\left(\frac{\zeta}{q\left( \psi_0 \right)} \right) \right], \end{align}

or equivalently as

(4.2)\begin{align} & H_{\textrm{ZDW}}(\rho_\parallel,\zeta; \psi_0, \mu) = \frac{\rho_\parallel^{2}}{2} \nonumber\\ &\quad +\mu\left\{1-\left[ 1+ \frac{\rho_\parallel^{2}}{\mu} -\frac{\rho_\parallel^{2}}{2\mu} \sqrt{2\psi_0} \cos\left(\frac{\zeta}{q\left( \psi_0 \right)} \right) \right] \sqrt{2\psi_0} \cos\left(\frac{\zeta}{q\left( \psi_0 \right)} \right) \right\}. \end{align}

Further approximations can be based on the relative magnitude of the three terms in the square brackets of the above equation. For $\rho _\parallel ^{2} /\mu << 1$, corresponding to particles with large pitch angles $\alpha =\tan ^{-1}(v_\perp /|v_\parallel |)$, the Hamiltonian is reduced to

(4.3)\begin{equation} H'_{\textrm{ZDW}}(\rho_\parallel,\zeta; \psi_0, \mu)=\frac{\rho_\parallel^{2}}{2}+\mu \left[1-\sqrt{2\psi_0} \cos\left(\frac{\zeta}{q\left( \psi_0 \right)} \right)\right], \end{equation}

which resembles the Hamiltonian of a pendulum, with the trapped and passing orbits corresponding to a libration and rotation type of motion (Shaing, Chu & Sabbagh Reference Shaing, Chu and Sabbagh2009; Brizard Reference Brizard2011; Brizard & Duthoit Reference Brizard and Duthoit2014). It is worth mentioning that, under the ZDW approximation, both $H_{\textrm {ZDW}}$ and $H'_{\textrm {ZDW}}$ scale with $\mu$ when $\rho _\parallel ^{2}$ is also divided by $\mu$, which clearly is not the case for the FDW Hamiltonian (3.19).

The above approximations have significant quantitative and qualitative differences. They provide different values for the total energy of a specific orbit and also result in different separatrices between trapped and passing orbits in the phase space, as shown in figure 1. Therefore, a particle that is described as being trapped according to $H'_{\textrm {ZDW}}$ can be actually passing according to $H_{\textrm {ZDW}}$ and vice versa. Both ZDW Hamiltonians describe GC orbits that are symmetric with respect to $\rho _\parallel =0$, whereas this is not the case with the FDW Hamiltonian, according to which positive (co-passing) and negative (counter-passing) orbits are not symmetric. These differences are not uniform across the phase space of the system and depend strongly on the pitch angle and the flux surface of reference $\psi _0$.

Figure 1. Phase space $(\rho _\parallel ,\zeta )$ of the Hamiltonian $H_{\textrm {FDW}}$ (3.19) for $q=1$, $\mu =10^{-4}$ and (a) $\psi _0=0.01$ and (b) $\psi _0=0.09$. The separatrices between bounce and transit motion according to $H_{\textrm {ZDW}}$ (4.1) and $H'_{\textrm {ZDW}}$ (4.3) are depicted by blue and red lines, respectively.

The ZDW Hamiltonians are very useful for obtaining analytical forms for the frequencies of the GC motion and their dependence on the constants of the motion, parametrizing each orbit. These frequencies determine the orbital spectrum of different particle species, including bulk and energetic particles, as well as their resonance conditions with any type of non-axisymmetric perturbations. The latter are crucial for the energy, momentum and particle transport in a toroidal magnetic field configuration.

For both $H_{\textrm {ZDW}}$ and $H'_{\textrm {ZDW}}$ bounce and transit motion is characterized by $0 \leq k < 1$ and $1 < k$, respectively, with $k$ being the trapping parameter, defined as

(4.4)\begin{equation} k=\frac{E-\mu(1-r)}{2\mu r}, \end{equation}

where

(4.5)\begin{equation} r=\sqrt{2\psi_0}. \end{equation}

The bounce/transit frequencies and actions corresponding to $H'_{\textrm {ZDW}}$ are given as follows (appendix A):

(4.6)\begin{align} \omega'_b&=\frac{{\rm \pi} \sqrt{\mu r}}{2 q(\psi_0) K(k)}=\frac{{\rm \pi}}{2K(k)}\omega'_{b0}, \end{align}

(4.7)\begin{align} \omega'_t&=\frac{{\rm \pi} \sqrt{\mu r}\sqrt{k}}{q(\psi_0)K(k^{-1})}=\frac{{\rm \pi}\sqrt{k}}{K(k^{-1})}\omega'_{b0} \end{align}

and

(4.8)\begin{align} J'_b&=\frac{8q(\psi_0)\sqrt{\mu r}}{{\rm \pi}}[E(k)+(k-1)K(k)] \end{align}

(4.9)\begin{align} J'_t&=\frac{4q(\psi_0)\sqrt{\mu r}}{{\rm \pi}}\sqrt{k}E(k^{-1}), \end{align}

where $K$ and $E$ are the complete elliptic integrals of the first and second kind (Gradshteyn & Ryzhik Reference Gradshteyn and Ryzhik2007), and

(4.10)\begin{equation} \omega'_{b 0}=\frac{{\rm \pi} \sqrt{\mu r}}{2 q(\psi_0) K(0)}= \frac{ \sqrt{\mu r}}{q(\psi_0)} \end{equation}

is the frequency of the deeply trapped bounce motion, corresponding to $k=0$. These are the standard analytical expressions for the frequencies and actions obtained from the pendulum-like Hamiltonian $H'_{\textrm {ZDW}}$ (Shaing et al. Reference Shaing, Chu and Sabbagh2009; Brizard Reference Brizard2011; White Reference White2014).

The respective frequencies and actions for $H_{\textrm {ZDW}}$ are (appendix A)

(4.11)\begin{align} \omega_b&=\frac{{\rm \pi} (1-r)\sqrt{\mu r}}{2 q(\psi_0)\varPi\left(\eta k, k\right)}= \frac{{\rm \pi} (1-r)}{2 \varPi\left(\eta k, k\right)}\omega'_{b 0}, \end{align}

(4.12)\begin{align} \omega_t&=\frac{{\rm \pi} \sqrt{k} (1-r)\sqrt{\mu r}}{q(\psi_0)\varPi\left(\eta, k^{-1}\right)}= \frac{{\rm \pi} \sqrt{k} (1-r)}{\varPi\left(\eta, k^{-1}\right)}\omega'_{b 0} \end{align}

and

(4.13)\begin{align} J_b&=\frac{8 q(\psi_0)\sqrt{\mu r}}{{\rm \pi} \eta (1-r)} \left[ (\eta k-1)\varPi\left(\eta k,k\right)+ K(k) \right], \end{align}

(4.14)\begin{align} J_t&=\frac{4 q(\psi_0)\sqrt{\mu r}}{{\rm \pi} \eta (1-r)}\left[ \frac{\eta k-1}{\sqrt{k}}\varPi \left(\eta,k^{-1}\right)+ \frac{K(k^{-1})}{\sqrt{k}} \right], \end{align}

with $\varPi$ being the complete elliptic integral of the third kind (Gradshteyn & Ryzhik Reference Gradshteyn and Ryzhik2007) and

(4.15)\begin{equation} \eta=-\frac{2r}{1-r}. \end{equation}

For a given set of values of the magnetic moment $(\mu )$ and the flux surface $(\psi _0)$, the value of the trapping parameter $(k)$ is determined by the total energy $(E)$ of the particle. The dependence of the bounce and transit frequencies on the energy $E(k)$ according to $H'_{\textrm {ZDW}}$ and $H_{\textrm {ZDW}}$, as given by (4.6), (4.7) and (4.11), (4.12), respectively, is depicted in figures 2 and 3. The bounce and transit frequencies $\dot {\theta }$ according to the FDW Hamiltonian in the original canonical variable set, given in (2.4), are also shown. Note that under the ZDW approximation $\dot {\theta }=\dot {\zeta }/q=\dot {\bar {\zeta }}/q=\omega _{b,t}/q=\hat {\omega }_\zeta /q$, and we have taken $q=1$ for simplicity. The analytical formula for $\omega _b$ (4.11) shows a remarkable agreement with the numerically calculated frequencies based on the FDW Hamiltonian and significantly deviates from the analytical formula for $\omega '_b$ (4.6) with the deviation being more pronounced for larger values of $\psi _0$, as shown in figure 2. For the transit frequencies, shown in figure 3, the FDW Hamiltonian describes asymmetric orbits and consequently different frequencies for co-passing $(\rho _\parallel >0)$ and counter-passing $(\rho _\parallel <0)$ orbits. The analytical formula for $\omega _t$ (4.12) corresponds to an intermediate frequency with respect to the two branches of the numerically calculated FDW frequencies, whereas the analytical formula for $\omega '_t$ (4.7) tends to follow one of the two branches.

Figure 2. Analytically (solid lines) and numerically (dashed lines) calculated bounce frequencies according to $H'_{\textrm {ZDW}}$ (blue line), $H_{\textrm {ZDW}}$ (red line) and $H$ (2.4) (dashed line) for $q=1$, $\mu =10^{-4}$ and (a) $\psi _0=0.01$ $(\eta =-0.33)$ and (b) $\psi _0=0.09$ $(\eta =-1.47)$.

Figure 3. Analytically (solid lines) and numerically (dashed lines) calculated transit frequencies according to $H'_{\textrm {ZDW}}$ (blue line), $H_{\textrm {ZDW}}$ (red line) and $H$ (2.4) (dashed line) for $q=1$, $\mu =10^{-4}$ and (a) $\psi _0=0.01$ $(\eta =-0.33)$ and (b) $\psi _0=0.09$ $(\eta =-1.47)$.

The two analytical formulas for the bounce frequencies have also a significant qualitative difference. In fact, in contrast to $\omega '_b$, $\omega _b$ is non-monotonic with respect to $k$ (energy) and, in accordance with FDW numerical calculations, predicts a local maximum of the frequency of the trapped orbits with respect to the energy, indicating the existence of two trapped orbits with different energy but equal frequency, as shown in figure 2(b). The location of this local maximum in the space $(E(k), \psi _0)$ is depicted in figure 4. The non-monotonic dependence of the frequency on the particle energy has important implications for particle and momentum transport in the presence of non-axisymmetric perturbations that break the integrability of the system. In such cases, the locations of the phase space regions where orbits are significantly modified due to perturbations are determined by resonance conditions. The non-monotonicity suggests that the same resonance can modify two regions of the phase space leading to extended transport under resonance overlap conditions. Moreover, the existence of energy values where the derivative of the frequency with respect to energy is zero can be related to an intrinsic degeneracy condition which is crucial for the effect of perturbations (Lichtenberg & Lieberman Reference Lichtenberg and Lieberman1992).

Figure 4. (a) Bounce frequency $\omega _b$ dependence on energy $(k)$ and $\psi _0$. (b) Maximum bounce frequency as a function of $\psi _0$. (c,d) Energy and $k$ corresponding to the maximum bounce frequency; for small $\psi _0$, the maximum frequency corresponds to the deeply trapped $k\simeq 0$ particles.

The bounce-averaged toroidal precession and gyration frequencies can be analytically caclulated according to (3.15) and (3.16) with the utilization of either $J_b$ (4.13), $\omega _b$ (4.11) or $J'_b$ (4.8), $\omega '_b$ (4.6). The resulting expressions are too lengthy to be given here; their dependence on energy $(k)$ is depicted in figures 5 and 6. A sign reversal in the bounce-averaged toroidal precession frequency is shown in figure 5. The analytical results based on $H_{\textrm {ZDW}}$ show a remarkable agreement with the numerical results based on the FDW Hamiltonian, where the expression (3.17) has been utilized. The differences between analytical results based on $H_{\textrm {ZDW}}$ and $H'_{\textrm {ZDW}}$ significantly differ for larger values of $\psi _0$. These differences imply different phase space regions where resonant interactions with low-frequency electromagnetic perturbations take place (Duthoit et al. Reference Duthoit, Brizard and Hahm2014). Similar differences are also shown for the bounce-averaged gyration frequencies in figure 6, determining the conditions for resonant interactions with high-frequency waves.

Figure 5. Analytically (solid lines) and numerically (dashed lines) calculated bounce-averaged toroidal precession frequencies according to $H'_{\textrm {ZDW}}$ (blue line), $H_{\textrm {ZDW}}$ (red line) and $H$ (2.4) (dashed line) for $q=1$, $\mu =10^{-4}$ and (a) $\psi _0=0.01$ $(\eta =-0.33)$ and (b) $\psi _0=0.09$ $(\eta =-1.47)$.

Figure 6. Analytically (solid lines) and numerically (dashed lines) calculated bounce-averaged gyro-frequencies according to $H'_{\textrm {ZDW}}$ (blue line), $H_{\textrm {ZDW}}$ (red line) and $H$ (2.4) (dashed line) for $q=1$, $\mu =10^{-4}$ and (a) $\psi _0=0.01$ $(\eta =-0.33)$ and (b) $\psi _0=0.09$ $(\eta =-1.47)$.

5. Summary and conclusions

The GC motion in an axisymmetric magnetic field is analysed under a Hamiltonian formulation in canonical variables. The ZDW approximation has been described in the context of the canonical formulation through a canonical transformation to variables measuring the deviation of the GC from a magnetic field line of reference. The latter allows for the expression of the system in action-angle variables and the systematic dynamical reduction to lower-dimensional phase spaces, depending on the specific physical problem under consideration, such as low- or high-frequency perturbations.

A novel ZDW Hamiltonian has been obtained along with compact analytical formulas for all the GC orbital frequencies, namely bounce/transit frequency and bounce-averaged toroidal precession and gyration frequency. The analytical results significantly differ from those corresponding to the widely used pendulum-like Hamiltonian and show a remarkable agreement with numerically calculated frequencies for the FDW Hamiltonian.

The knowledge of the orbital frequencies is crucial for determining the resonance conditions under particle interaction with non-axisymmetric perturbations that affect energy, momentum and particle transport in toroidal plasma configurations. Moreover, the transformation to action-angle variables allows for the application of standard canonical perturbation methods as well as the systematic dynamical reduction and the formulation of a bounce gyro-kinetic description.