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It is very important to understand stochastic diffusion of energetic charged particles in non-uniform background magnetic field in plasmas of astrophysics and fusion devices. Using different methods considering along-field adiabatic focusing effect, various authors derived parallel diffusion coefficient $kappa_parallel$ and its correction $T$ to $kappa_{parallel 0}$, where $kappa_{parallel 0}$ is the parallel diffusion coefficient without adiabatic focusing effect. In this paper, using the improved perturbation method developed by He & Schlickeiser and iteration process, we obtain a new correction $T$ to $kappa_{parallel 0}$. Furthermore, by employing the isotropic pitch-angle scattering model $D_{mumu}=D(1-mu^2)$, we find that $T$ has the different sign as that of $T$. In this paper the spatial perpendicular diffusion coefficient $kappa_bot$ with the adiabatic focusing effect is also obtained.
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Over the past two decades scientists have achieved a significant improvement of our understanding of the transport of energetic particles across a mean magnetic field. Due to test-particle simulations as well as powerful non-linear analytical tools our understanding of this type of transport is almost complete. However, previously developed non-linear analytical theories do not always agree perfectly with simulations. Therefore, a correction factor $a^2$ was incorporated into such theories with the aim to balance out inaccuracies. In this paper a new analytical theory for perpendicular transport is presented. This theory contains the previously developed unified non-linear transport theory, the most advanced theory to date, in the limit of small Kubo number turbulence. For two-dimensional turbulence new results are obtained. In this case the new theory describes perpendicular diffusion as a process which is sub-diffusive while particles follow magnetic field lines. Diffusion is restored as soon as the turbulence transverse complexity becomes important. For long parallel mean free paths one finds that the perpendicular diffusion coefficient is a reduced field line random walk limit. For short parallel mean free paths, on the other hand, one gets a hybrid diffusion coefficient which is a mixture of collisionless Rechester & Rosenbluth and fluid limits. Overall the new analytical theory developed in the current paper is in agreement with heuristic arguments. Furthermore, the new theory agrees almost perfectly with previously performed test-particle simulations without the need of the aforementioned correction factor $a^2$ or any other free parameter.
The processes responsible for the effective longitudinal transport of solar energetic particles (SEPs) are still not completely understood. We address this issue by simulating SEP electron propagation using a spatially 2D transport model that includes perpendicular diffusion. By implementing, as far as possible, the most reasonable estimates of the transport (diffusion) coefficients, we compare our results, in a qualitative manner, to recent observations {at energies of 55 -- 105 keV}, focusing on the longitudinal distribution of the peak intensity, the maximum anisotropy and the onset time. By using transport coefficients which are derived from first principles, we limit the number of free parameters in the model to: (i) the probability of SEPs following diffusing magnetic field lines, quantified by $a in [0,1]$, and (ii) the broadness of the Gaussian injection function. It is found that the model solutions are extremely sensitive to the magnitude of the {perpendicular} diffusion coefficient and relatively insensitive to the form of the injection function as long as a reasonable value of $a=0.2$ is used. We illustrate the effects of perpendicular diffusion on the model solutions and discuss the viability of this process as a dominant mechanism by which SEPs are transported in longitude. Lastly, we try to quantity the effectiveness of perpendicular diffusion as an interplay between the magnitude of the relevant diffusion coefficient and the SEP intensity gradient driving the diffusion process. It follows that perpendicular diffusion is extremely effective early in a SEP event when large intensity gradients are present, while the effectiveness quickly decreases with time thereafter.
The influence of adiabatic focusing on particle diffusion is an important topic in astrophysics and plasma physics. In the past several authors have explored the influence of along-field adiabatic focusing on parallel diffusion of charged energetic particles. In this paper by using the Unified NonLinear Transport (UNLT) theory developed by Shalchi (SH2010) and the method of He and Schlickeiser (HS2014) we derive a new nonlinear perpendicular diffusion coefficient for non-uniform background magnetic field. This formula demonstrates that particle perpendicular diffusion coefficient is modified by along-field adiabatic focusing. For isotropic pitch-angle scattering and weak adiabatic focusing limit the derived perpendicular diffusion coefficient is independent of the sign of adiabatic focusing characteristic length. For two-component model we simplify the perpendicular diffusion coefficient up to second order of the power series of adiabatic focusing characteristic quantity. We find that the first order modifying factor is equal to zero and the sign of the second one is determined by the energy of particles.
Based on Magnetospheric Multiscale (MMS) observations from the Earths bow shock, we have identified two plasma heating processes that operate at quasi-perpendicular shocks. Ions are subject to stochastic heating in a process controlled by the heating function $chi_j = m_j q_j^{-1} B^{-2}mathrm{div}(mathbf{E}_perp)$ for particles with mass $m_j$ and charge $q_j$ in the electric and magnetic fields $mathbf{E}$ and $mathbf{B}$. Test particle simulations are employed to identify the parameter ranges for bulk heating and stochastic acceleration of particles in the tail of the distribution function. The simulation results are used to show that ion heating and acceleration in the studied bow shock crossings is accomplished by waves at frequencies (1-10)$f_{cp}$ (proton gyrofrequency) for the bulk heating, and $f>10f_{cp}$ for the tail acceleration. When electrons are not in the stochastic heating regime, $|chi_e|<1$, they undergo a quasi-adiabatic heating process characterized by the isotropic temperature relation $T/B=(T_0/B_0)(B_0/B)^{1/3}$. This is obtained when the energy gain from the conservation of the magnetic moment is redistributed to the parallel energy component through the scattering by waves. The results reported in this paper may also be applicable to particle heating and acceleration at astrophysical shocks.
The Spatial Parallel Diffusion Coefficient (SPDC) is one of the important quantities describing energetic charged particle transport. There are three different definitions for the SPDC, i.e., the Displacement Variance definition $kappa_{zz}^{DV}=lim_{trightarrow t_{infty}}dsigma^2/(2dt)$, the Ficks Law definition $kappa_{zz}^{FL}=J/X$ with $X=partial{F}/partial{z}$, and the TGK formula definition $kappa_{zz}^{TGK}=int_0^{infty}dt langle v_z(t)v_z(0) rangle$. For constant mean magnetic field, the three different definitions of the SPDC give the same result. However, for focusing field it is demonstrated that the results of the different definitions are not the same. In this paper, from the Fokker-Planck equation we find that different methods, e.g., the general Fourier expansion and perturbation theory, can give the different Equations of the Isotropic Distribution Function (EIDFs). But it is shown that one EIDF can be transformed into another by some Derivative Iterative Operations (DIOs). If one definition of the SPDC is invariant for the DIOs, it is clear that the definition is also an invariance for different EIDFs, therewith it is an invariant quantity for the different Derivation Methods of EIDF (DMEs). For the focusing field we suggest that the TGK definition $kappa_{zz}^{TGK}$ is only the approximate formula, and the Ficks Law definition $kappa_{zz}^{FL}$ is not invariant to some DIOs. However, at least for the special condition, in this paper we show that the definition $kappa_{zz}^{DV}$ is the invariant quantity to the kinds of the DIOs. Therefore, for spatially varying field the displacement variance definition $kappa_{zz}^{DV}$, rather than the Ficks law definition $kappa_{zz}^{FL}$ and TGK formula definition $kappa_{zz}^{TGK}$, is the most appropriate definition of the SPDCs.
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