Interpretation of Physical Terms of the Transport EquationShock AccelerationPhys 954, Solar Wind and Cosmic Rays Section VE. MöbiusV.3 Acceleration Processes(Consequences of Transport Effects)BasicsIn the following the fundamental transport equation will be the basis of acceleration processes. Transport and acceleration are intimately linked. V.3_1The results from the preceding discussion (i.e. that particle distributions are assimilated into a moving rest frame) is the reason for one of the fundamental acceleration pro-cesses, driven by the interplay of scattering and a substantial bulk flow between 2 me-dia:1) diffusive or 1st order Fermi accelerationIn the equation it is the Convection term along with the pitch-angle scattering that do the trick.The second process also involves bulk flow, magnetic fields (and thus the induced elec-tric fields) to accelerate particles (in a deterministic manner along the trajectories in E and B fields).- shock drift accelerationIn the equation it is the Drift term that is the driver.Finally, we will see that electromagnetic fluctuations, which move at random in the plasma rest frame, just by means of scattering, will lead to a stochastic energy gain, i.e. acceleration.- 2nd order Fermi accelerationThis is based on the Diffusion term on the right hand side. However, it is the term of the p component, if we represent p = (p, µ).Ultimately we will see that in all these cases electric fields play a fundamental role, at least on the micro-level, although sometimes hidden by the description of the process. Because electric fields in nature are generally produced by induction due to relative mo-tion w.r.t. to a magnetic field, these will be mostly electric fields perpendicular to B. I will consider acceleration in E||B as a special case (very simple case) of the second ex-ample. In such cases particles gain the total energy by falling through an electric poten-tial, which is limited by the gyroradii in each step of drift acceleration. Generally, we 1/14/19 92 ∂f∂t + r v ∇f + Q(r v xr B o)∂f∂r p = 〈∂∂r p (D∂f∂r p )〉 Convection Drift Momentum DiffusionPhys 954, Solar Wind and Cosmic Rays Section VE. Möbiusneed to add one more process, magnetic pumping (based on dB/dt), but this can also beinterpreted as an induced E-field.Because in many cases in astrophysical plasmas also vbulk >> vA, acceleration based on vbulk (1st order Fermi) is often more important than that based on turbulent waves with vA(2nd order Fermi), which in addition scales with the square of the speed. This is gener-ally true for all energetic particle populations in the solar wind. Here also the energy density is smaller than that of the solar wind.Interpretation of Physical Terms of the Transport Equation ∂f∂t + r v ∇f + Q(r v xr B o)∂f∂r p = ∂∂μ[(1− μ2)Dμ∂f∂μ] + ∂∂p(Dp∂f∂p)(V3_2)The distribution function f can be inhomogeneous, i.e. have a gradient, and we allow moving frames. Scattering is treated as diffusion in p and µ. Let us make the assumptionthat momentum diffusion, i.e. diffusion that changes the total energy of the particles is inefficient. So we drop this term, but we keep pitch angle diffusion.Let us further use, where convenient and if not addressed otherwise fo, i.e. the isotropic main part of f. If needed, we will add a small anisotropic part, i.e. f will always be nearly isotropic (diffusion approximation). We can define the one dimensional phase space F(p) density of fo as:F(p) = 4πp2fo with n = F(p)dp∫We recall that we derived (V3_2) originally from conservation of particles in phase space: ∂F∂t + ∇r S x + ∂∂pr S p = 0(j = S∫xdp) (V3_2a)where Sx is the streaming (phase space flux) in configuration space and Sp the streaming in momentum (p) space. The change of the flux in momentum space thus must be re-lated to work done on particles between p and p+dp. We know that the solar wind does work on a particle population according to: work = r V sw⋅∇P where P = 13pvF(p)dp∫Note: 1/3 comes from isotropic F, 3 degrees of freedomHence:Sp = V⋅∇(13pF(p)) where V any bulk flow(V.3_3)Now Sx comes from (1) convection, (2) drift, and (3) diffusion.(1) Assume f isotropic in frame flowing with |V| << v. Then we can set in the observer frame:1/14/19 93Phys 954, Solar Wind and Cosmic Rays Section VE. Möbiusf = fo(p) – V(p/v)∂fo/∂p (linear expansion)Then:j = nV = 4π3p2vf(p)dp ≈ ∫−V4π3p3∂fo∂pdp ∫or: Sx(Convection) = −V4π3p3∂fo∂p(V.3_4)This is in essence the Compton-Getting effect that we already discussed.(2) Drifts are produced by inhomogeneous magnetic fields and inhomogeneous distri-butions perpendicular to magnetic fields. I.e. we have to look at motion solely perpen-dicular to B. This term is independent of any scattering. In principle, we only need to keep the 2nd and 3rd term on the left-hand side of (V.3_2). Still this is a relatively complex series of vector and matrix operation. Let a plausibility argument suffice. We have en-countered gradient drift in our discussion after the Compton-Getting effect about re-maining anisotropies. We arrived heuristically at a relation: jD = rciv∇n = mv2QB∇nwith similar arguments about an isotropic distribution:S⊥ = pv3Qr B oBo2×∇F(V.3_5)(3) Finally, diffusion in configuration space is connected to pitch angle diffusion. We takethe resulting net motion as ||B. This is true as long as scattering from field line to field line or a tangling of field lines (spaghetti), as lately assumed for interplanetary field, are unimportant. I.e. the dominant term is:vz∂f∂z = ∂∂μ[(1−μ2)Dμ∂f∂μ]Now we use a deviation from the isotropic distribution fof = fo + g where still: ∫gdμ =0(V.3_6)We compute the flux j by integrating over p space, noting that this is only an integral over µ, where vz = vµ. Integrating over the first equation (V.3_6) we get:vμ22∂fo∂z = (1−μ2)Dμ∂g∂μ + A(V.3_7)1/14/19 94Phys 954, Solar Wind and Cosmic Rays Section VE. Möbiusg drops out of the integration on the left hand side with integration by parts and gdµ =∫0. On the right hand side the term with fo is 0, because fo is isotropic and not subject to diffusion. Evaluating (V.3_7) at µ±1 yields: A = v/2∂fo/∂z. Now we get:∂g∂μ = −v2Dμ∂fo∂z after integration :g = −v2∂fo∂zdμ'Dμ0μ∫ + CNow we get S|| = 4πp2v µgdµ. After integration by parts