Reflection and Shock Drift Acceleration

Shock acceleration basically falls into 3 categories: first-order Fermi mechanism [Lee,1982], by which the so called diffuse ion population near the quasi-parallel shock can be modeled very well;  The second is the reflection and shock drift acceleration , which is usually related to high Mach number (the ratio of plasma flow speed and the local sound speed) collisionless, quasi-perpendicular shocks. In shock drift acceleration, particles gain energy by drifting in the induced electric field along the shock surface. The other mechanism mentioned is Stochastic process (second-order Fermi acceleration). These three mechanisms can be distinguished from each other but usually they work together with different relative contributions depending on the properties of the shock. 

In general, tenuous astrophysical plasmas don’t satisfy the Maxwellian-Boltzman distribution of thermal particles. Instead, they are observed with a power law distribution, where individual particle energies can be extremely large. 

                             N = N p                  
Where N is the particle density in space and p is the scalar momentum , and σ =(r+2)/(r-1) is the spectral index. 

In most environment where energetic populations are observed, the bulk flow speed is much greater than the typical sound speed, which makes shock easier to develop. As a matter of fact,  shocks are associated with most energetic populations seen in the space. Shocks and energetic particles are intimately related. Shocks provide an important mechanism to accelerate energetic particles of planetary, solar, and even galactic origin.
On the other hand, particle acceleration is a way for dissipating the incoming flow energy and producing the irreversible transition from upstream region to downstream region, which characterizes a shock. That is to say, particle acceleration is essential to form a shock.  

Different approaches for studying shock acceleration are: test particle method, individual particles method, and the diffusion-convection method, the “two fluid “ approach (thermal particles and energetic particles, two components). 
 
Assumptions: Shock is thin compared with Larmor radius (gyro-radius). Shock front is planar. The jump in the field and plasma may be arbitrary large but are homogenous in both upstream and downstream region.

After we have a general picture given above, now we will concentrate on reflection and shock drift acceleration of ions. 

Reflection  and shock drift acceleration

Before we talk about the shock drift acceleration, first we have to see where the induced electric field which is crucial in this acceleration mechanism comes from.
For all oblique (we define oblique to be all shocks with an angle between the upstream magnetic field and the shock normal greater than 0, i.e., θ>  0) shocks, the magnetic field takes an active role and influences both the shock jump conditions and particle acceleration. If  B, u, and n are coplanar upstream from the shock then B, u, and n remain in the same coplanar plane in the downstream side of the shock. An important property of oblique shock is that, when viewed in the shock rest frame, an electric field is produced by the motion of the plasma across the magnetic field. This field is given by 
                       E = -                   (1)

And will play an important role in the shock drift acceleration of the particles in oblique shocks. 
This electric field is also referred to interplanetary electric field. 
Definition for quasi-parallel and quasi-perpendicular shocks is:
 45 > θ> 0, quasi-parallel; 45 < θ < 90, quasi-perpendicular.

Ion reflection is a general characteristic of high Mach number collisionless shocks and provides an important source of free energy in the region ahead of the shock. Ahead of an oblique shock, there are the so-called reflected ions observed well upstream of the shock. they stream away from the shock along the magnetic field (field aligned beams , FABs) (Asbridge et al., 1968; Gosling et al, 1978; Paschmann et al., 1980, 1981). While in the foot region of nearly perpendicular shock, the ions gyrating transverse to the magnetic field were observed [Paschmann et al. 1982].
    
Important aspects of the physics of quasi-perpendicular shocks can be deduced from observations of FABs observed upstream of the quasi-perpendicular Earth bow shock [Paschmann et al, 1981]. The foreshock region contains electrons and ions streaming away from the shock (FABs) as well as wave activity and the cold solar wind (T-10 - 10K) streaming toward the shock.  It is clear that in most cases the FABs result from some type of reflection process of the solar wind off the shock: they are either specularly reflected when they hit the shock front because of the potential barrier across the shock (the potential barrier across shock is resulted from different gyroradii of electrons and ions) ; or they first transmit into the downstream region and get turned back to the shock through the combined action of the induction field and the Lorentz force.  These beams carry information on how thermal particles interact with quasi-perpendicular shocks.  Most FABs come from shocks with 30 < θ<60and they typically have a number density approximately 1% of that of the solar wind. 

Sonnerup (1968) offered a simple explanation for the observed ion acceleration in terms of a displacement along the interplanetary electric field during the reflection process. The theory used only geometric relations to calculate the energy of the reflected particles. 
In that theory, the acceleration occurs because the particles are displaced along the interplanetary electric field during reflection, as a result of, for example, a gradient B drift. The strength of the theory lies in the fact that it makes no assumptions on the nature of the reflection process and does not require any orbit calculations. Instead it describes the resulting beam energy in very simple geometric relations. The basic assumption of the theory is that the particle energy is conserved in the special frame of reference where the interplanetary electric field vanishes. (deHoffman –Teller frame). His calculation can be considered as a macroscopic picture. 

Closely following the original derivation of Sonnerup but using a more general geometry, Paschmann et al. (1981) give a theoretical prediction about the observed ion acceleration. 

Figure 1 shows the basic situation. 

Figure 1. Vector diagram illustrating the relation between the guiding center velocity vof the incident solar wind, the interplanetary magnetic field B, and the shock normal n. The guiding center velocity vcan be decomposed into a component v parallel to B and a component valong the shock front. The component vrepresents the transformation velocity between a reference frame S fixed with respect to the shock and a frame S, where the solar wind flows along B with velocity v. In S there is an electric field E=- (v× B)/c; while in Sthe electric field vanishes.

A solar wind particle with guiding center velocity v is incident on the shock at an angle φ with respect to the shock normal n. The interplanetary field B forms an angle θ with n.     
Since n, B, and v are in general not in the same plane, the angle ψ between B and vmay be different from θ+φ. For ψ ≠ 0, there will be an electric field E = -(v× B)/c  in the shock frame of reference S. Except for θ = 90, one can always find a coordinate system S(it Is also called deHoffman-Teller frame) where the solar wind flows along the magnetic field  In this frame, the electric field vanishes everywhere except in the shock structure itself, and the particle conserves energy upon reflection if no other acceleration processes are present. 
The coordinate system Sis found by decomposing v into a component v along B and another component v along the shock front. v is just the required transformation velocity between S and S.

If a particle is reflected at the shock, its guiding center velocity will be reversed in S no matter what kind of reflection mechanism is. If the magnetic moment of the particle is conserved in the reflection ( as is assumed in Figure1.), the velocity after the reflection v should be equal to - v, if not, v = -δ · v, where δ is a positive constant. Transformation back to the fixed coordinate system S, we can get the following relations.

                       v+ v = v    (2)
                       v + v = v  (3)
                       vcosθ = vcosф  (4)
In the reference frame Sthe energy is the same after reflection as before. Therefore 
                       v+ v= v+ v≈ v  (5)
where v are the velocity components perpendicular to B. Assuming the incident solar wind ion thermal speed is very small, we have v≈ 0. The magnitudes of the incident and reflected particle velocities are related by
                       v= δ · v             (6)
If δ =1, it implies that v= v, i.e., the magnetic moment is conserved. 
The incident and reflected energies in the S frame are given by 
                     E= ( v+ v) ≈v         
                     E= ( v+ v)                     (7)
Using (2), (3), and (6), we have
                     v = v+ (1+ δ) v-2(1+ δ) v vcosψ    (8)
Equation (5) and (6) give
        
                      v = v+ (1- δ) v≈ (1- δ) v          (9) 
 Using (4) to eliminate v in (8) and (9), it finally gives 
   = 1+2 (1+ δ)                             (10)
Equation (10) tells us about the wide range of energy ratios that the theory allows. The acceleration of the particle is essentially from the particle drifting along the induction electric field . but in the macroscopic picture here, it is hidden in the reflection and the transformation of two different frames.
Sonnerup’s calculation doesn’t apply to the shocks when θ = 90because the transformation to H-T frame is impossible. 

Now we can use simple ion orbit calculation to understand microscopically (looking at it more closely to the shock) what the physics is for the ion acceleration by shocks. 

Figure 2 shows three different situations that can happen in the shock drift acceleration.  

Figure.2. Shock drift acceleration: different sample of trajectories in the shock rest frame of a quasi-perpendicular shock with θ= 80. 

From the left panel of Fig.2, we can see that when the particle hits the shock, it starts to drift along the induction electric field which lies in the shock front. Therefore, it gains energy. Depending on pitch angle and phase of the gyration, some upstream particles can drift along the shock front for long distances and receive large energy gains. Also depending on the pitch angle and θ, some particles will gyrate in such a way that they can turn back to the upstream region. That is to say, they can be reflected. When the particles penetrate briefly to the downstream region and come back to the shock front, you can explain it as the reflection occurs via magnetic mirroring since the magnetic moment is conserved. Since the shock thickness is small compared to gyroradius and the particle velocity is bigger than the flow velocity, the particle can make many gyrations while drifting through the shock. Shock drift acceleration occurs because the induction electric field accelerates the particle when it gyrates in a large orbit in the upstream region and decelerates it when it gyrates in a small orbit in the downstream region.  The positive work on the large circle exceeds the negative work on the small circle and so a net acceleration results [Jones and Ellison, 1991].  Particles gyrate in a big orbit in the upstream region and small orbit in the downstream region because the magnetic field is bigger in the downstream region than that of the upstream region for fast shock. (Travelling interplanetary shocks in general and planetary bow shocks in particular always are fast MHD shocks.)
With the increasing energy, its velocity component which is along the shock normal increases too. Finally when that velocity component becomes larger than the shock speed, the particle can escape to the upstream region. The details of what the particle trajectory is and whether it is transmitted or reflected strongly depend on its initial energy and pitch angle. After the acceleration, an initially isotropic particle distribution in the upstream and downstream region usually ends up with a very strong field-aligned beam in the upstream region and to a smaller beam perpendicular to the field in the downstream region. 
If the particle can drift along the induction field for longer time, the energy gain from that will be bigger. How long the particle can drift along the electric field depends on the particle’s speed perpendicular to the shock. If it is small, the particle sticks to the shock; if it is big, the particle escapes before it can gain a large amount of energy. The particle energy relative to the shock is determined by particle speed, shock speed, pitch angle and 
θ. With increasing particle speed, the range of the other three parameters which allow for efficient acceleration will become narrow, thus the efficiency of the acceleration will become small. 
The average energy gain is a factor of  1.5-5. It is obvious that such an energy gain is too small to accelerate particles out of the solar wind to energies of some tens of keV or even some of MeV. Acceleration up to higher energy would require repeated interactions between particles and shock. If a particle is in the downstream region, further acceleration is not difficult because the turbulence created by the shock makes stochastic acceleration or scattering particles back to the shock front possible. This statement is true for scattering free ( i.e. particles move helical in the uniform magnetic field with no waves or turbulence) conditions usually assumed with the shock drift acceleration. In fact, space plasmas are usually turbulent and particles get scattered. Thus the high energy gain is possible. [Kallenrode, 1998] 

Different acceleration mechanisms usually work together in space plasmas. Which one dominates in the shock acceleration usually depends on the properties of the shock. SDA was observed in the quasi-perpendicular Earth bow shock.[Paschmann et al., 1982]

After the particles get accelerated, these energetic particles can have some effect on the shock itself. Some nonlinear effects, such as the smoothing of the shock due to the diffusion of the accelerated particles , the increasing of the overall compression of the shock because of the escape of particles and the particles becoming relativistic, and the production of magnetic turbulence by the accelerated particles, have to be taken into account. 
In a word, physics of the shock and particle acceleration by the shock can not be separated.  
 
References:

1.	Jones, F.C., D.C. Ellison, The Plasma Physics of Shock Acceleration, Space Science  Reviews, 58, 259,346, 1991 
2.	Kellenrode, M. B., Space Physics, An Introduction to Plasmas and Particles in the 
Heliosphere and Magnetospheres, Springer-Verlag, Berlin, New-York, 1998
3.   Paschmann, G ,  N. Sckopke et al., Energization of Solar Wind Ions by Reflection  
      From  the Earth Bow Shock, Journal of Geophysical Research, Vol. 85, NO. A9,  
      p4689-4693, September 1, 1980
4.   Paschmann, G, N. Sckopke  et al., Observations of Gyrating Ions in the Foot of the  
      Nearly Perpendicular Bow Shock, Geophysical Research Letters, Vol., 9, NO. 8,  
      p881-884, 1982
5.   Sonnerup, B. U. O, Acceleration of Particles Reflected at a Shock Front, J. Geophys. 
      Res., 74, 1301-1304, 1969.