Momentum Diffusion

Note

Momentum diffusion is only partially supported now.

We can include an momentum diffusion term to the right side of the Parker transport equation.

(1)\[\frac{\partial f}{\partial t} + (\boldsymbol{V}+\boldsymbol{V}_d)\cdot\nabla f - \frac{1}{3}\nabla\cdot\boldsymbol{V}\frac{\partial f}{\partial\ln p} = \nabla\cdot(\boldsymbol{\kappa}\nabla f) + \frac{1}{p^2}\frac{\partial}{\partial p} \left(p^2D_{pp}\frac{\partial f}{\partial p}\right) + Q,\]

Neglecting the source term \(Q\) in equ_parker_2nd and assuming \(F=fp^2\),

\[\begin{split}\begin{aligned} \frac{\partial F}{\partial t} = & -\nabla\cdot\left[(\nabla\cdot\boldsymbol{\kappa}+\boldsymbol{V}+\boldsymbol{V}_d)F\right] + \nabla\cdot(\nabla\cdot(\boldsymbol{\kappa}F)) + \nonumber \\ & \frac{\partial}{\partial p} \left[\left(\frac{p}{3}\nabla\cdot\boldsymbol{V} - \frac{\partial D_{pp}}{\partial p} - \frac{2D_{pp}}{p}\right) F\right] + \frac{\partial(D_{pp}F)}{\partial p^2}. \end{aligned}\end{split}\]

which is equivalent to a system of SDEs of the Ito type,

\[\begin{split}\begin{aligned} dX & = (\nabla\cdot\boldsymbol{\kappa} + \boldsymbol{V} + \boldsymbol{V}_d)ds + \sum_\sigma\boldsymbol{\alpha}_\sigma dW_\sigma(s) \\ dp & = \left(-\frac{p}{3}\nabla\cdot\boldsymbol{V} + \frac{\partial D_{pp}}{\partial p} + \frac{2D_{pp}}{p}\right)ds + \sqrt{2D_{pp}}dW(s) \end{aligned}\end{split}\]

where \(\sum_\sigma\alpha_\sigma^\mu\alpha_\sigma^\nu = 2\kappa^{\mu\nu}\), \(dW\) is the normalized distributed random number with mean zero and variance \(\sqrt{\Delta t}\), and \(\Delta t\) is the time step for stochastic integration.

Wave-particle interaction

For a 2D problem, reference [Skilling75] shows that for forward and backward propagating Alfvén waves,

\[\begin{split}\begin{aligned} \boldsymbol{u} & = \boldsymbol{v}_0 + \left<\frac{3}{2}(1-\mu^2)\frac{\nu^+ - \nu^-}{\nu^+ + \nu^-}\right>, \text{ the velocity of mean wave frame} \\ \kappa_\parallel & = v^2\left<\frac{1-\mu^2}{2(\nu^+ + \nu^-)}\right>, \text{ parallel spatial diffusion coefficient} \\ D_{pp} & = 4\gamma^2m^2v_A^2\left<\frac{1-\mu^2}{2}\frac{\nu^+\nu^-}{\nu^+ + \nu^-}\right>, \text{ momentum diffusion coefficient} \end{aligned}\end{split}\]

where \(\left<\dots\right>\) donates \(\mu\)-average, \(\nu^+\) and \(\nu^-\) are collision frequency against forward waves and backward waves, respectively. If \(\nu^+\) is equal to \(\nu^-\),

\[D_{pp} = \frac{1}{9}\frac{p^2v_A^2}{\kappa_\parallel}\]

where \(p=\gamma mv\) is particle momentum. Depending on the plasma parameter and wave properties, we may have to use more complicated models [Schlickeiser89] [Schlickeiser98] [LeRoux07]. The corresponding SDE is

\[\begin{split}\begin{aligned} dp & = \left(-\frac{p}{3}\nabla\cdot\boldsymbol{V} + \frac{4pv_A^2}{9\kappa_\parallel}\right)ds + \sqrt{\frac{2p^2v_A^2}{9\kappa_\parallel}}dW(s), \text{if $\kappa_\parallel$ is independent of $p$}\\ dp & = \left(-\frac{p}{3}\nabla\cdot\boldsymbol{V} + \frac{8pv_A^2}{27\kappa_\parallel}\right)ds + \sqrt{\frac{2p^2v_A^2}{9\kappa_\parallel}}dW(s), \text{if $\kappa_\parallel\sim p^{4/3}$} \end{aligned}\end{split}\]

which are normalized to

\[\begin{split}\begin{aligned} d\tilde{p}_n & = \left(-\frac{\tilde{p}_n}{3}\tilde{\nabla}\cdot\tilde{\boldsymbol{V}} + \frac{4\tilde{p}_n\tilde{v}_A^2}{9\tilde{\kappa}_\parallel}\right)d\tilde{s} + \tilde{p}_n\tilde{v}_A\sqrt{\frac{2}{9\tilde{\kappa}_\parallel}}dW(\tilde{s}), \text{if $\kappa_\parallel$ is independent of $p$}\\ d\tilde{p}_n & = \left(-\frac{\tilde{p}_n}{3}\tilde{\nabla}\cdot\tilde{\boldsymbol{V}} + \frac{8\tilde{p}_n\tilde{v}_A^2}{27\tilde{\kappa}_\parallel}\right)d\tilde{s} + \tilde{p}_n\tilde{v}_A\sqrt{\frac{2}{9\tilde{\kappa}_\parallel}}dW(\tilde{s}), \text{if $\kappa_\parallel\sim p^{4/3}$} \end{aligned}\end{split}\]

where \(\tilde{p}_n=\tilde{p}\tilde{p}_{n0}=p\tilde{p}_{n0}/p_0\), where is \(\tilde{p}_{n0}\) is the numerical value for particles with \(p_0\) in the code (e.g., 0.1 as often used), \(\tilde{\nabla}=L_0\nabla\), \(\tilde{\boldsymbol{V}}=\boldsymbol{V}/v_{A0}\), \(\tilde{v}_A=\tilde{v}_{A0}\), \(\tilde{\kappa}_\parallel=\kappa_\parallel/\kappa_0\), \(\kappa_0=L_0v_{A0}\), \(\tilde{s}=s/t_0\), and \(t_0=L_0/v_{A0}\). These are all given in the code.

Flow shear

For isotropic particle distributions, the flow shear introduces another momentum diffusion term. If there is no average magnetic field [Earl88].

\[\begin{split}\begin{aligned} D_{pp} & = \Gamma\tau p^2, \\ \Gamma & = \frac{1}{30}\left(\frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i}\right)^2 - \frac{2}{45}\frac{\partial U_i}{\partial x_i}\frac{\partial U_j}{\partial x_j} = \frac{2}{15}\sum_{ij}\sigma_{ij}^2 \end{aligned}\end{split}\]

where \(\Gamma\) is the coefficient of viscous momentum transfer, \(\sigma_{ij}=(\partial_iU_j + \partial_jU_i - 2\nabla\cdot\boldsymbol{U}\delta_{ij}/3)/2\) is the shear tensor, \(\tau\) is the relaxation time for particle scattering. According to [Webb18], \(\tau\) is related particle diffusion coefficient \(\kappa_\parallel=v^2\tau/3\). The corresponding SDE is

\[\begin{aligned} dp = \left(-\frac{p}{3}\nabla\cdot\boldsymbol{v} + \frac{\Gamma}{p^2}\frac{\partial(p^4\tau)}{\partial p}\right)ds + \sqrt{2\Gamma\tau p^2}dW(s) \end{aligned}\]

For \(\tau\sim\tau_0(p_0/p)^\alpha\),

\[\begin{aligned} dp = \left(-\frac{p}{3}\nabla\cdot\boldsymbol{v} + \frac{\Gamma\tau_0p_0^\alpha}{p^2}(4-\alpha)p^{3-\alpha}\right)ds + \sqrt{2\Gamma\tau_0 p_0^\alpha p^{2-\alpha}}dW(s) \end{aligned}\]

which is normalized to

\[\begin{aligned} d\tilde{p}_n = \left(-\frac{\tilde{p}_n}{3}\tilde{\nabla}\cdot\tilde{\boldsymbol{v}} + (4-\alpha)\tilde{\Gamma}\tilde{\tau}_0\tilde{p}_n^{1-\alpha}\tilde{p}_{n0}^\alpha\right)d\tilde{s} + \sqrt{2\tilde{\Gamma}\tilde{\tau}_0\tilde{p}_n^{2-\alpha}\tilde{p}_{n0}^\alpha}dW(\tilde{s}) \end{aligned}\]

where \(\tilde{p}_n=\tilde{p}\tilde{p}_{n0}=p\tilde{p}_{n0}/p_0\), where is \(\tilde{p}_{n0}\) is the numerical value for particles with \(p_0\) in the code (e.g., 0.1 as often used), \(\tilde{\nabla}=L_0\nabla\), \(\tilde{\boldsymbol{v}}=\boldsymbol{v}/v_{A0}\), \(\tilde{\Gamma}=\Gamma t_0^2\), \(\tilde{\tau}_0=\tau_0/t_0\), \(\tilde{s}=s/t_0\), and \(t_0=L_0/v_{A0}\). For \(\tau\sim\tau_0(p_0/p)^2\) [Earl88],

\[\begin{aligned} d\tilde{p}_n = \left(-\frac{\tilde{p}_n}{3}\tilde{\nabla}\cdot\tilde{\boldsymbol{v}} + \frac{2\tilde{\Gamma}\tilde{\tau}_0\tilde{p}_{n0}^2}{\tilde{p}_n}\right)d\tilde{s} + \sqrt{2\tilde{\Gamma}\tilde{\tau}_0\tilde{p}_{n0}^2}dW(\tilde{s}) \end{aligned}\]

For \(\tau\sim\tau_0(p_0/p)^{2/3}\) [Giacalone99],

\[\begin{split}\begin{aligned} d\tilde{p}_n & = \left(-\frac{\tilde{p}_n}{3}\tilde{\nabla}\cdot\tilde{\boldsymbol{v}} + \frac{10}{3}\tilde{\Gamma}\tilde{\tau}_0\tilde{p}_{n}^{1/3}\tilde{p}_{n0}^{2/3}\right)d\tilde{s} + \sqrt{2\tilde{\Gamma}\tilde{\tau}_0\tilde{p}_n^{4/3}\tilde{p}_{n0}^{2/3}}dW(\tilde{s}) \\ \tau_0 & = 3\kappa_{\parallel 0} / v_0^2 \end{aligned}\end{split}\]

If there is an average magnetic field, the equation is more complicated (see [Williams91] [Williams93]).

[Earl88] (1,2)

Earl, J.A., Jokipii, J.R. and Morfill, G., 1988. Cosmic-ray viscosity. The Astrophysical Journal, 331, pp.L91-L94.

[LeRoux07]

Le Roux, J.A. and Webb, G.M., 2007. Nonlinear cosmic-ray diffusive transport in combined two-dimensional and slab magnetohydrodynamic turbulence: a BGK-Boltzmann approach. The Astrophysical Journal, 667(2), p.930.

[Schlickeiser89]

Schlickeiser, R., 1989. Cosmic-ray transport and acceleration. I-Derivation of the kinetic equation and application to cosmic rays in static cold media. II-Cosmic rays in moving cold media with application to diffusive shock wave acceleration. The Astrophysical Journal, 336, pp.243-293.

[Schlickeiser98]

Schlickeiser, R. and Miller, J.A., 1998. Quasi-linear theory of cosmic ray transport and acceleration: the role of oblique magnetohydrodynamic waves and transit-time damping. The Astrophysical Journal, 492(1), p.352.

[Skilling75]

Skilling, J., 1975. Cosmic Ray Streaming—II effect of particles on alfvén waves. Monthly Notices of the Royal Astronomical Society, 173(2), pp.245-254.

[Webb18]

Webb, G. M., Barghouty, A. F., Hu, Q., & le Roux, J. A. 2018, The Astrophysical Journal, 855, 31

[Williams91]

Williams, L.L. and Jokipii, J.R., 1991. Viscosity and inertia in cosmic-ray transport-Effects of an average magnetic field. The Astrophysical Journal, 371, pp.639-647.

[Williams93]

Williams, L.L., Schwadron, N., Jokipii, J.R. and Gombosi, T.I., 1993. A unified transport equation for both cosmic rays and thermal particles. The Astrophysical Journal, 405, pp.L79-L81.