Magnetic moment nonconservation in magnetohydrodynamic turbulence models

The fundamental assumptions of the adiabatic theory do not apply in the presence of sharp field gradients or in the presence of well-developed magnetohydrodynamic turbulence. For this reason, in such conditions the magnetic moment $\mu$ is no longer expected to be constant. This can influence particle acceleration and have considerable implications in many astrophysical problems. Starting with the resonant interaction between ions and a single parallel propagating electromagnetic wave, we derive expressions for the magnetic moment trapping width $\Delta \mu$ (defined as the half peak-to-peak difference in the particle magnetic moments) and the bounce frequency ${\omega }_b$. We perform test-particle simulations to investigate magnetic moment behavior when resonance overlapping occurs and during the interaction of a ring-beam particle distribution with a broadband slab spectrum. We find that the changes of magnetic moment and changes of pitch angle are related when the level of magnetic fluctuations is low, $\delta B/B_0=({10}^{-3},{10}^{-2})$, where $B_0$ is the constant and uniform background magnetic field. Stochasticity arises for intermediate fluctuation values and its effect on pitch angle is the isotropization of the distribution function $f\left(\alpha \right)$. This is a transient regime during which magnetic moment distribution $f\left(\mu \right)$ exhibits a characteristic one-sided long tail and starts to be influenced by the onset of spatial parallel diffusion, i.e., the variance $\langle {\left(\Delta z\right)}^2\rangle$ grows linearly in time as in normal diffusion. With strong fluctuations $f\left(\alpha \right)$ becomes completely isotropic, spatial diffusion sets in, and the $f\left(\mu \right)$ behavior is closely related to the sampling of the varying magnetic field associated with that spatial diffusion.

Figure 1

Gyroresonant interaction between a circularly polarized wave and a particle with $v=100v_A$ and $\alpha =1/8$: cosine of pitch angle $\alpha$ (top row), particle magnetic moment $\mu$ (middle row), and parallel component of the induced electric field (bottom row). Different columns correspond to different wave amplitudes: $\delta b=0.001$ (first column), $\delta b=0.01$ (second column), $\delta b=0.1$ (third column), and $\delta b=1.0$ (fourth column).

Figure 2

Time evolution of the cosine of the pitch angle $\alpha$ (first row) and its distribution function $f\left(\alpha \right)$ (second row), and time evolution of the magnetic moment $\mu$ (third row) and its distribution function $f\left(\mu \right)$ (fourth row) of a resonant ($\alpha =0.125$, left column) and a nonresonant particle ($\alpha =0.5$, right column). $v=100v_A$.

Figure 3

$f\left(\alpha \right)$ (first row) and $f\left(\mu \right)$ (second row) at the end of the simulation for an initial distribution of 1000 resonant (left column) and nonresonant (right column) particles randomly distributed in the simulation box. $\delta b=0.01$.

Figure 4

Transition from nonoverlapping to overlapping resonances: $\alpha$ (left column) and $\mu$ (right column) profiles as the wave amplitude is varied: $\delta b=0.001$ (first row), $\delta b=0.01$ (second row), $\delta b=0.1$ (third row), and $\delta b=1.0$ (fourth row).

Figure 5

Transition from nonoverlapping to overlapping resonances with varying wave amplitude: $\delta b=0.001$ (first row), $\delta b=0.01$ (second row), $\delta b=0.1$ (third row), and $\delta b=1.0$ (fourth row). The distribution functions $f\left(\alpha \right)$ (left column), $f\left(\mu \right)$ (central column), and $f\left(\delta z\right)$ (right column) are averaged over time.

Figure 6

Power spectrum of the turbulent magnetic field. $k$ is normalized to the coherence length $l_z$.

Figure 7

Distribution functions of the cosine of the pitch angle $f\left(\alpha \right)$ (left column), the magnetic moment $f\left(\mu \right)$ (central column), and the particle displacements relative to the initial position $f\left(\delta z\right)$ (right column) at $20{\tau }_c$ for different values of $\delta b$: $\delta b=0.001$ (first row), $\delta b=0.01$ (second row), $\delta b=0.1$ (third row), and $\delta b=1.0$ (fourth row). The blue and the green (light gray) lines indicate the initial value and the mean value of each distribution. Particle parameters at injection: $v=10v_A$ and ${\alpha }_0=0.125$.

Figure 8

Statistics for $v=10v_A$. Variances for the cosine of the pitch angle $\alpha$ (a), magnetic moment $\mu$ (b), and particle displacements relative to the initial position $\delta z$ (c) at different values of $\delta b$: $\delta b=0.001$ [black (1) line], $\delta b=0.01$ [purple (2) line], $\delta b=0.1$ [red (3) line], and $\delta b=1.0$ [blue (4) line]. The $s$ values indicate the different slopes for the scalings $\langle {\left(\Delta \alpha \right)}^2\rangle ,\langle {\left(\Delta \mu \right)}^2\rangle$, and $\langle {\left(\Delta z\right)}^2\rangle \propto t^s$. The variances are fitted with the gray lines: the dotted line is $s=2$, dot-dashed line $s=1$, dashed line $s=0.8$, and three-dot-dashed line $s=0.7$.

Figure 9

Statistics for $v=10v_A$ and $\delta b=0.5$. Magnetic moment variance $\langle {\left(\Delta \mu \right)}^2\rangle$ (first plot) and mean square displacement $\langle {\left(\Delta z\right)}^2\rangle$ (second plot) versus time, and magnetic moment distribution function $f\left(\mu \right)$ after $0.01{\tau }_c$ (third plot) and at the end of the simulation (fourth plot). The $s$ values indicate the different slopes for the scaling of the variances $\langle {\left(\Delta \mu \right)}^2\rangle$ and $\langle {\left(\Delta z\right)}^2\rangle \propto t^s$. The variances are fitted with the red dot-dashed lines and the blue dotted lines.

Figure 10

Standard deviation ${\sigma }_{\mu }/{\mu }_{\min }$ (blue triangle, left plot) and variation in magnetic moment $\Delta \mu //{\mu }_{\min }$ (red circle, right plot) versus $\delta b$ at $20{\tau }_c$ for $v=100v_A$ and $\alpha =0.125$.