2. Generic form of local diffusion tensor for a resonant wave–particle interaction

Consider a particle, possibly in the presence of a uniform magnetic field. Such a particle has a constant kinetic energy $K$ and canonical momentum components $\boldsymbol{p}$ , which are often combined in the energy–momentum four-vector $p^\mu \equiv (K,\boldsymbol{p})$ .

To this system, add a monochromatic wave (oscillating electromagnetic field) with frequency $\omega$ and wavevector $\boldsymbol{k}$ . These objects can also be combined into a wave four-vector $k^\mu \equiv (\omega , k^x, k^y, k^z)$ . Then, allow this wave to resonantly interact with the plasma, either through a Landau or gyro resonance.

It is well known (Trubnikov Reference Trubnikov and Leontovich1979; Stix Reference Stix1992; Dodin & Fisch Reference Dodin and Fisch2012; Guan et al. Reference Guan, Dodin, Qin, Liu and Fisch2013a ) that, during the resonant wave–particle interaction, the increments of energy and momentum satisfy the relation

(2.1) \begin{align} \frac {\Delta p^\mu }{k^\mu } = \frac {\Delta p^\nu }{k^\nu }, \quad \forall \mu ,\nu \in \{0,1,2,3\}. \end{align}

This relationship follows from the conservation of the wave action $\mathcal{I}$ , the definition of the Minkowski four-momentum $\mathcal{P}^\mu \equiv k^\mu \mathcal{I}$ and the fact that the Minkowski subsystem and resonant particle subsystem form a conserving system (Ochs & Fisch Reference Ochs and Fisch2021b , Reference Ochs and Fisch2022; Ochs Reference Ochs2024).

In a magnetised system, the relation in (2.1) leads to coupled gyrocentre-energy diffusion, as the change in the gyrocentre position is given by

(2.2) \begin{align} \Delta \boldsymbol{X} = \frac {\Delta \boldsymbol{p} \times \boldsymbol{B}}{B^2},\end{align}

where $\boldsymbol{B}$ is the magnetic field. In other words, diffusion takes place along single line in gyrocentre-energy space – which is the basis for the alpha channelling effect (Fisch & Rax Reference Fisch and Rax1992a ,Reference Fisch and Rax b ). Although the early references focused on diffusion due to high-frequency electrostatic waves, in fact the relation holds very generally for any electromagnetic wave. For instance, figure 1 shows diffusion in energy and gyrocentre space for a particle undergoing diffusion due to interaction with a left-polarised electromagnetic wave at the second cyclotron harmonic of a $\hat {z}$ directed magnetic field. It can be seen that over many successive interactions with the wave, the particle stays on the line defined by (2.1)–(2.2). (The simulation is described in more detail in Appendix A.)

Figure 1.

Simulation of ion interaction with a second-harmonic left-hand-polarised wave over many orbits. The particle diffuses in both normalised energy $\bar {K}$ and gyrocentre position $(\bar {X},\bar {Y})$ (points, with time corresponding to colour), but stays on the line determined by (2.1) (solid line). The normalised variables and details of the simulation can be found in Appendix A.

The relation in (2.1) means that the diffusion operator in four-momentum space must take a very specific form. Generally, the tensor describing one-dimensional diffusion along a vector $V^\mu$ satisfies $D^{\mu \nu } \propto V^\mu V^\nu$ . Since (2.1) shows that wave-induced diffusion occurs along a vector in $p^\mu$ space with components proportional to $k^\mu$ , we immediately find

(2.3) \begin{align} D^{\mu \nu } &= D^{K K} \frac {k^\mu k^\nu }{\omega ^2}. \end{align}

Here, $D^{KK}$ is the on-diagonal kinetic-energy component of the tensor. Equation (2.3) is very powerful, because it allows us to construct a four-momentum-conserving wave–particle interaction tensor from a single known tensor component.

3. Restoring conservation to the Kennel–Engelmann diffusion tensor

Quasilinear diffusion in due to a monochromatic wave in a uniform magnetic field is traditionally modelled by the equation (Kennel & Engelmann Reference Kennel and Engelmann1966; Lerche Reference Lerche1968; Harvey & McCoy Reference Harvey and McCoy1992; Stix Reference Stix1992; Petrov & Harvey Reference Petrov and Harvey2016)

(3.1) \begin{align} \frac {\partial f}{\partial t} &= \frac {1}{\sqrt {g}} \frac {\partial }{\partial \boldsymbol{p}} \boldsymbol{\cdot }\left (\sqrt {g} \, \boldsymbol{D} \boldsymbol{\cdot }\frac {\partial f}{\partial \boldsymbol{p}}\right ), \end{align}

where the volume element $\sqrt {g} = 2\pi p_\perp$ , and $\boldsymbol{D}$ is given by the Kennel–Engelmann diffusion tensor, which can be written

(3.2) \begin{align} \boldsymbol{D} &= D_{0} \boldsymbol{w} \boldsymbol{w}, \end{align}

(3.3) \begin{align} D_{0} &= \frac {\pi q^2}{2} \sum _n | \psi _{n} |^2 \delta (\omega _{r} - k_\parallel v_\parallel - n\varOmega ), \end{align}

(3.4) \begin{align} \boldsymbol{w} &= \left (1 - \frac {k_\parallel v_\parallel }{\omega _{r}} \right ) \hat {p}_\perp + \left (\frac {k_\parallel v_\perp }{\omega _{r}}\right ) \hat {p}_\parallel . \end{align}

Here,

(3.5) \begin{align} \psi _{n} &\equiv E^+ J_{n-1}(z) + E^- J_{n+1}(z) + \frac {p_\parallel }{p_\perp } E^z J_n(z), \end{align}

where

(3.6) \begin{align} z = \frac {k_\perp v_\perp }{\varOmega } \end{align}

and

(3.7) \begin{align} E^\pm \equiv \frac {1}{2} \left (E_x \pm i E_y \right ) e^{\mp i \theta }, \end{align}

where $E_i$ is the ith component of the oscillating electric field. Here the coordinates have been defined such that $\boldsymbol{B} \parallel \hat {z}$ , and $\theta$ is the angle defined by

(3.8) \begin{align} \boldsymbol{k} &= \hat {x} k_\perp \cos \theta + \hat {y} k_\perp \sin \theta + \hat {z} k_\parallel . \end{align}

(Note: the additional phase $i\theta$ is not included in the definition of $E^\pm$ in references, such as those for the CQL3D code (Harvey & McCoy Reference Harvey and McCoy1992; Petrov & Harvey Reference Petrov and Harvey2016), which employ the ‘Stix frame’, where $\boldsymbol{k}$ lies in the $x-z$ plane and thus $\theta = 0$ .)

Equation (3.1) describes a diffusion process in two coordinates: the parallel momentum $p_\parallel \equiv p_z$ , and the perpendicular momentum $p_\perp \equiv m v_\perp$ . It is important to note that the perpendicular momentum so defined is more precisely an energy variable, having nothing to do with the gyrocentre position. Living in this reduced space, the Kennel–Engelmann diffusion tensor fundamentally cannot capture the full set of conservation relations demanded by (2.1).

Nevertheless, the Kennel–Engelmann tensor does capture the energy–momentum relation in the dimensions that it does model. To see this, we can transform the tensor from coordinates $(p_\perp ,p_\parallel )$ to coordinates $(K,p_\parallel )$ . To do this, it is sufficient to transform the vector $\boldsymbol{w}$ (3.4). Using

(3.9) \begin{align} K = \frac {1}{2m} \left (p_\perp ^2 + p_\parallel ^2\right ) \Rightarrow \frac {\partial K}{\partial p_\perp } = \frac {p_\perp }{m}; \; \frac {\partial K}{\partial p_\parallel } = \frac {p_\parallel }{m}, \end{align}

we find

(3.10) \begin{align} w^K = \frac {\partial K}{\partial p_\perp } w^{p_\perp } + \frac {\partial K}{\partial p_\parallel } w^{p_\parallel } = v_\perp . \end{align}

Comparing with (3.4), we see that this satisfies

(3.11) \begin{align} w^K &= \frac {k_\parallel }{\omega } w^{p_\parallel }, \end{align}

implying (via (3.2)) that the diffusion satisfies the requirement in (2.3) for $K$ and $p_\parallel$ . Indeed, the connection of the Kennel–Engelmann diffusion path to the absorption of photon four-momentum has long been appreciated (Kennel & Engelmann Reference Kennel and Engelmann1966; Stix Reference Stix1992).

To make a momentum-conserving tensor is now trivial. Noting that

(3.12) \begin{align} D^{K K} &= D_{0} v_\perp ^2, \end{align}

with $D_0$ given by (3.3), (2.3) then tells us that in four-momentum space $(K,p^x, p^y,p^z)$

(3.13) \begin{align} D^{\mu \nu } &= D_0 \frac {k^\mu k^\nu v_\perp ^2}{\omega ^2}. \end{align}

The projection of this tensor to $(K,p_\parallel )$ is precisely the Kennel–Engelmann tensor, while the new components capture the absorption of perpendicular momentum.

4. Constants-of-motion space diffusion

A typical application of quasilinear diffusion theory is to bounce-averaged Fokker–Planck theory. Such theories describe diffusion in the space of particle orbits, which can be specified by the constants of motion (COM) of the orbit. It is thus instructive to examine the effects of the new terms in the diffusion tensor on the particle diffusion in COM space.

A typical choice of the COM in an axisymmetric system is $(\epsilon ,\mu ,p_\phi )$ , where

(4.1) \begin{align} \epsilon &\equiv \frac {1}{2m} \left(p_\perp ^2 + p_\parallel ^2\right) + \psi (\boldsymbol{r}), \end{align}

(4.2) \begin{align} \mu &\equiv \frac {p_\perp ^2}{2m |B|}, \end{align}

(4.3) \begin{align} p_\phi &\equiv (\boldsymbol{r} \times \boldsymbol{p}) \boldsymbol{\cdot }\hat {z} + q r A_\phi (r). \end{align}

Here, $\boldsymbol{r} \equiv (r,\phi ,z)$ is the cylindrical coordinate vector, $\psi (\boldsymbol{r})$ is a position-dependent potential energy and $A_\phi (\boldsymbol{r})$ is the $\phi$ -component of the vector potential. The $p_\phi$ component is closely related to the particle flux surface, and is often taken to define the flux surface. In these coordinates, the diffusion tensor again has the form from (3.2), and the task is to calculate the diffusion path vector components $w^\epsilon$ , $w^\mu$ and $w^{p_\phi }$ .

It is clear that $w^\epsilon$ and $w^\mu$ are unchanged by the new momentum-conserving terms in the diffusion tensor. However, $w^{p_\phi }$ is certainly changed, as

(4.4) \begin{align} w^{p_\phi } &= \frac {\partial p_\phi }{\partial p_i} w^{p_i}, \end{align}

which involves a sum over the $\phi$ -projection of all three momentum components, rather than just the parallel momentum. In a linear-confinement axisymmetric system such as a magnetic mirror, where $\partial p_\phi /\partial p_\parallel = 0$ , the difference to the traditional Kennel–Engelmann theory is particularly striking, as the new components make the difference between non-existent cross-flux-surface transport, versus cross-flux-surface transport due to perpendicular momentum absorption.

Furthermore, we can see easily that the new terms make good sense. Plugging in $w^{p_i} = k^i v_\perp / \omega$ (see e.g. (2.3), (3.2) and (3.11)) to (4.4), we find

(4.5) \begin{align} w^{p_\phi } = \left (\boldsymbol{r} \times \frac {\boldsymbol{k} v_\perp }{\omega } \right ) \boldsymbol{\cdot }\hat {z} = \frac {\boldsymbol{r} \times \boldsymbol{k}\boldsymbol{\cdot }\hat {z}}{\omega } w^{K} , \end{align}

the natural relation between the absorbed photon energy and angular momentum. Thus, we see that the change in the diffusion tensor modifies the momentum absorption so as to enforce angular momentum conservation.

For completeness, we can now fully derive the form of the diffusion tensor in COM space. It is clear that $w^\epsilon = w^K$ . Equation (4.5) can be expressed in more familiar form by making use of the azimuthal mode number of the wave $n_\phi$ , from whence

(4.6) \begin{align} w^{p_\phi } = \frac {n_\phi }{\omega } w^\epsilon . \end{align}

Then the final task is to find $w^\mu$ . Noting that

(4.7) \begin{align} \mu = \frac {1}{|B|} \left (K - \frac {1}{2m} p_\parallel ^2 \right ) \end{align}

and making use of the $\delta$ -function in (3.3), we find

(4.8) \begin{align} w^{\mu } &= \frac {1}{|B|} \left ( 1 - \frac {k_\parallel v_\parallel }{\omega } \right ) w^\epsilon = \frac {n \varOmega }{|B| \omega } w^{\epsilon }. \end{align}

Thus, in $(\epsilon ,\mu ,p_\phi )$ space, the diffusion tensor is given by (3.2), but with

(4.9) \begin{align} \boldsymbol{w} &= v_\perp \begin{pmatrix} 1 \\[5pt] \dfrac{n \varOmega }{|B| \omega } \\[12pt] \dfrac{n_\phi }{\omega } \end{pmatrix}. \end{align}

This has the same form of the diffusion path found in the action-angle space quasilinear theory (see e.g. Herrmann (Reference Herrmann1998, (4.67)), but now calculated with a diffusion coefficient from the conventional bounce-averaged theory.