Bridging the gap between collisional and collisionless shock waves
aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n Under consideration for publication in J. Plasma Phys. Bridging the gap between collisional andcollisionless shock waves
Antoine Bret , , and Asaf Pe’er † ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain Instituto de Investigaciones Energ´eticas y Aplicaciones Industriales, Campus Universitario deCiudad Real, 13071 Ciudad Real, Spain Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel(Received ?; revised ?; accepted ?. - To be entered by editorial office)
While the front of a fluid shock is a few mean-free-paths thick, the front of a collision-less shock can be orders of magnitude thinner. By bridging between a collisional anda collisionless formalism, we assess the transition between these two regimes. We con-sider non-relativistic, un-magnetized, planar shocks in electron/ion plasmas. In addition,our treatment of the collisionless regime is restricted to high Mach number electrostaticshocks. We find that the transition can be parameterized by the upstream plasma param-eter Λ which measures the coupling of the upstream medium. For Λ . .
12, the upstreamis collisional, i.e. strongly coupled, and the strong shock front is about M λ mfp , thick,where λ mfp , and M are the upstream mean-free-path and Mach number respectively.A transition occurs for Λ ∼ .
12 beyond which the front is ∼ M λ mfp , ln Λ / Λ thickfor Λ & .
12. Considering Λ can reach billions in astrophysical settings, this allows tounderstand how the front of a collisionless shock can be orders of magnitude smaller thanthe mean-free-path, and how physics transitions continuously between these 2 extremes.
1. Introduction
Shock waves are very common in systems that involve fluid flows. Such systems occuron very different scales - from the microphysical scale to astronomical scales. As such,the properties of the shocks can vary considerably, depending on the environment. Fromthe microphysical point of view, it is useful to discriminate between collisional and colli-sionless shocks. Collisional shock waves, first discovered in the 19 th century (Salas 2007),can occur in any fluid as the result of a steepening of a large amplitude sound wave,or collision of two media (Zel’dovich & Raizer 2002). The front of a collisional shock isnecessarily at least a few mean-free-paths thick, as the dissipation from the downstreamto the upstream occurs via binary collisions.Collisionless shock waves were discovered later and can only form in plasma (Petschek1958; Buneman 1964; Sagdeev 1966). The dissipation is provided by collective plasmaphenomena instead of binary collisions. As a result, the front of such shocks can be ordersof magnitude thinner than the mean-free-path. For example, the front of the bow shockof the earth magnetosphere in the solar wind is some 100 km thick (Bale et al. et al. § † Email address for correspondence: [email protected]
A. Bret and A. Pe’er xy Shockfront
UpstreamDownstream U N T U N T n (x)n (x) xN N z Figure 1.
Setup and notations. regime where its front is a few mean-free-paths thick, to another regime where its front ismillion times smaller? Exploring the intermediate case, bridging between collisional andcollisionless shocks, is the aim of this article.On the collisionless side, shock-accelerated particles which can enhance the densityjump, or external magnetization which can reduce it, will be ignored (Berezhko & Ellison1999; Bret & Narayan 2018, 2019; Bret 2020).The method implemented is explained in Section 2. The big picture is as follows: wefirst present an evaluation of the shock front thickness in the collisional regime, thenin the collisionless regime. The first task is achieved in Section 3 using the Mott-Smith ansatz (Mott-Smith 1951), which writes the distribution function at any place along theshock profile, as a linear combination of the upstream and downstream Maxwellians. Wethen follow Tidman (1967) in Section 4 for the collisionless case before we bridge betweenthe 2 expressions of the front thickness in Section 5, to propose an expression of the frontthickness valid from the collisional to the collisionless regime.
2. Method
As previously stated, a fluid shock is mediated by collisions while a collisionless shockis mediated by collective effects. For a plasma where only electrostatic fields are active(such is the case for an electrostatic shock, the kinetic equation accounting for both kindsof effects would formally read (Kulsrud (2005), p. 9), ∂F∂t + v · ∂F∂ r + q E m · ∂F∂ v = (cid:18) ∂F∂t (cid:19) c + (cid:18) ∂F∂t (cid:19) w , (2.1)where q and m are the charge and the mass of the species considered. The first term ofthe right-hand-side, namely ( ∂F/∂t ) c , stands for the rate of change of the distribution F due to collisions. It is typically given by the Fokker-Planck operator. The secondterm, ( ∂F/∂t ) w , accounts for the effects of the waves and is given, for example, by thequasi-linear operator. In principle, accounting at once for these two collision terms with ridging the gap between collisional and collisionless shock waves ansatz (Mott-Smith 1951). Initially introduced fora neutral fluid, this ansatz consisted in approximating the molecular distribution func-tion F along the shock profile by a linear combination of the upstream and downstreamdrifting Maxwellians, F ( v ) = n ( x ) (cid:16) m πk B T (cid:17) / exp (cid:16) − m k B T ( v − U ) (cid:17) + n ( x ) (cid:16) m πk B T (cid:17) / exp (cid:16) − m k B T ( v − U ) (cid:17) , (2.2)where T , and U , are the upstream (subscript 1) and downstream (subscript 2) tem-peratures and velocities respectively, determined by the Rankine-Hugoniot (RH) jumpconditions (see figure 1) † . The boundary conditions for the functions n , ( x ) are, n (+ ∞ ) = N , n ( −∞ ) = 0 ,n (+ ∞ ) = 0 n ( −∞ ) = N , , (2.3)where again N , and N , fulfill the RH jump conditions.Taking then the appropriate moments of the dispersion equation gives a differentialequation which allows to determine the respective weights of the 2 Maxwellians in termsof x , hence the shock profile together with its front thickness (Mott-Smith 1951).The method implemented here consists in dealing with the collisional and the colli-sionless regimes separately. • We study the collisional regime in Section 3. There we apply the Mott-Smith ansatz using the BGK collision term (Bhatnagar et al. ∂F/∂t ) c in Eq. (2.1), with ( ∂F/∂t ) w = 0. Notably, Bhatnagar et al. (1954) presented 4 differentcollision operators through their Eqs. (3, 4, 5-6, 15-19). Those given by Eqs. (3, 4, 5-6),like ν ( f − f ) ‡ , have been widely used although they do not conserve all 3 quantities:particle number and/or momentum and/or energy. In Bhatnagar et al. (1954), only theoperator of Eqs. (15-19) does conserve all 3, hence this is the one used here.Tidman (1958) used the Fokker-Planck operator to deal with the problem, consideringEq. (2.1) with ( ∂F/∂t ) w = 0 and, (cid:18) ∂F∂t (cid:19) c = 4 πe m i ln Λ (cid:18) ∂F∂t (cid:19) c,F P , (2.4)where ( ∂F/∂t ) c,F P is the Fokker-Planck collision operator, m i the ion mass, and Λ thenumber of particles in the Debye sphere, that is, the co-called “plasma parameter” whichmeasures the coupling of the plasma. As we shall see in Section 6, the present treatmentprovides a more adequate bridging to the collisionless regime than Tidman (1958)’sFokker-Planck result. • For the collisionless regime we follow in Section 4 the collisionless result of Tidman(1967) who also used the Mott-Smith ansatz . In recent years, the correctness of this † All temperatures are not always considered constant in the main articles cited here(Mott-Smith 1951; Tidman 1958, 1967). Yet, they are considered so when it comes to com-puting the shock profile. ‡ Here ν is a collision frequency, f the distribution function and f the equilibrium distributionfunction. A. Bret and A. Pe’er approximation, namely that the distribution function is well approximated by superim-posed drifting Maxwellians, was validated numerically using Particle-In-Cell simulations(Spitkovsky 2008). Tidman (1967) considered Eq. (2.1) with ( ∂F/∂t ) c = 0, describing( ∂F/∂t ) w = 0 by the quasi-linear operator.Having assessed the width of the shock front in the collisional and the collisionlessregimes, we then bridge between the 2 expressions of the shock thickness in Section 5.
3. Collisional regime: applying Bhatnagar et al. (1954) collision termto the Mott-Smith ansatz
We switch to the reference frame of the shock and assume steady state in this frame. Asin Tidman (1958), we consider the distributions are functions of ( v x , v y , v z ) and assumequantities only vary with the x coordinate. We therefore set ∂ y,z,t = 0 and E y,z = 0 sothat equation (2.1), with ( ∂F/∂t ) w = 0, reads for the ion distribution F , v x ∂F∂x + eE x m i ∂F∂v x = (cid:18) ∂F∂t (cid:19) c,BGK , (3.1)where m i is the ion mass. The BGK collision term now reads (Bhatnagar et al. (cid:18) ∂F∂t (cid:19) c,BGK = 1 σ (cid:0) N Φ − N F (cid:1) , (3.2)which vanishes for a Maxwellian distribution. According to Bhatnagar et al. (1954), “ σ − × a density” is a collision frequency ν . In the present setting we define, N , σ = ν ∼ v thi , λ mfp , ⇒ σ = N , λ mfp , v thi , , (3.3)where v thi , and λ mfp , are the upstream thermal velocity and mean-free-path respec-tively. Then N and Φ are given by Eqs. (15-19) of Bhatnagar et al. (1954), N = Z F d v = n ( x ) + n ( x ) , (3.4)Φ = (cid:18) m i πk B T (cid:19) / exp (cid:18) − m i k B T ( v − q ) (cid:19) , (3.5) q = 1 N Z v F d v, (3.6)3 k B Tm i = 1 N Z ( v − q ) F d v, (3.7)where F has been considered of the form (2.2). Multiplying equation (3.1) by v y andintegrating over d v (see detailed calculation reported in Appendix A) gives an exact,simple result, U k B T m i ∂n ∂x + U k B T m i ∂n ∂x = 13 σ ( U − U ) n n . (3.8)This differential equation is structurally identical to the ones found in Mott-Smith(1951); Tidman (1958). We show in Appendix B how it yields density profiles like theones pictured in Figure 1, of the form, n ( x ) = N ,
11 + e − x/ℓ , ridging the gap between collisional and collisionless shock waves Figure 2.
Shock front thickness in units of the mean-free-path, ℓ/λ mfp , , in terms of theupstream Mach number M . n ( x ) = N , e − x/ℓ e − x/ℓ , (3.9)implicitly defining the shock width ℓ . From Eq. (B 7) we find the thickness of the shockaccording to the present formalism, ℓ = 3 σ k B ( T − T ) U N , m i ( U − U ) = 3 N , λ mfp , v thi , k B ( T − T ) U N , m i ( U − U ) . (3.10)It is now convenient to use the RH jump conditions to express ℓ in terms of the up-stream quantities, like the upstream Mach number and mean-free-path. The calculationsreported in Appendix C give, ℓ = λ mfp , U v thi , ( M + 3)(5 M ( M + 2) − M ( M − , = λ mfp , ( M + 3)(5 M ( M + 2) − M ( M − , (3.11)where M is the upstream Mach number † . We eventually obtain the following limits forthe shock width ℓ , ℓ = λ mfp , × (cid:26) ( M − − for M ∼ , M for M → ∞ . (3.12)The function ℓ/λ mfp , is plotted in Figure 2 in terms of the Mach number. It reachesa minimum for M = 3 .
53 with ℓ/λ mfp , = 6. Such a “U” shape has also been found inTidman (1958). We shall further comment on Tidman (1958) in Section 6.
4. Collisionless regime
Our expression (3.12) of the shock width cannot be used to bridge all the way tocollisionless shocks since it has been derived from the kinetic equation (2.1) without † Here we set v thi , ∼ c s in order to write U /v thi , ∼ M , where c s is the upstream soundspeed. An exact calculation only changes the end result by a factor of order unity. Moreover,the same factor also modifies the collisionless shock width (4.2). Therefore, the critical plasmaparameter Λ c defined by Eq. (5.1) for the collisional/collisionless transition, remains unchangedwhen considering v thi , ∼ c s . A. Bret and A. Pe’er Figure 3.
Plot of ℓ ( M λ mfp , ) − in terms of Λ. For a collisional plasma with small Λ, the frontthickness ℓ is given by Eq. (3.12). The collisionless thickness is given by Eq. (4.2). The widthof the front for any plasma parameter Λ is given by the red curve. Only valid for strong shock(see end of Section 4). the ( ∂F/∂t ) w collision term. Yet, collisionless shocks are sustained by the mechanismdescribed by this very term.Tidman (1967) treated the problem of a collisionless shock by setting ( ∂F/∂t ) c = 0in Eq. (2.1) and considering the quasi-linear operator for ( ∂F/∂t ) w . The Mott-Smith ansatz was also implemented in this study. Tidman (1967) could not derive an equationof the form (3.8) allowing to extract an analytical shock profile. Further analysis inBiskamp & Pfirsch (1969) and Tidman & Krall (1969) concluded that the quasi-linearformalism is not non-linear enough to fully render a shock.Yet, Tidman (1967) could derive the following estimate of the width of the front, ℓ = A U ω pi, = A U v thi , v thi , ω pi, = A M λ Di , , (4.1)where λ Di , is the upstream ionic Debye length and A is a parameter expected to be oforder O (10). We can eventually cast this result under the form, ℓ = A M ln ΛΛ λ mfp , , (4.2)where Λ is the plasma parameter already introduced in Eq. (2.4) and we have used(Fitzpatrick (2014), p. 10), λ mfp , = Λln Λ λ Di , . (4.3)Notably, Tidman (1967) only addressed high Mach numbers turbulent shocks triggeredby electrostatic instabilities. The forthcoming bridging between the 2 regimes is thereforeonly valid for such shocks. Weibel shocks sustained by electromagnetic instabilities aretherefore excluded (Stockem et al. et al.
5. Bridging between the 2 regimes
Figure 3 shows the collisional and collisionless expressions of ℓ ( M λ mfp , ) − from Eqs.(3.12, 4.2). For upstream Mach number M > a few (4-5), these 2 expressions intersect ridging the gap between collisional and collisionless shock waves (cid:1) c Figure 4.
Value of Λ c for which Eqs. (3.12 & 4.2) intersect, in terms of A defined through Eq.(4.1). for a critical plasma parameter Λ c defined by, λ mfp , M = A M ln Λ c Λ c λ mfp , ⇒ A ln Λ c Λ c , (5.1)fulfilled for Λ c ∼ .
12 and then for Λ c ∼
35 (for A = 10).For Λ < .
12, the upstream is strongly coupled, that is, collisional, and the width of thefront will be given by the collisional result (3.12). For Λ > .
12, the upstream is weaklycoupled, that is, collisionless, and the relevant front width is therefore the collisionlessresult (4.2). Hence, the larger value of Λ c ∼
35 where the 2 expressions intersect againis not physically meaningful. For such values of Λ, the upstream is collisionless so thatthe collisionless result applies.The transition between the 2 regimes occurs therefore for a critical plasma param-eter Λ c = 1 .
12, coinciding with the transition of the upstream from the strongly cou-pled/collisional regime, to the weakly coupled/collisionless regime. Although this valueof Λ c has been computed for A = 10, Figure 4 shows it is poorly sensitive to A as longas A = O (10).Note that this value of Λ c = 1 .
12 is only indicative. For example, Lee & More (1984)developed an electron conductivity model for dense plasmas requiring ln Λ >
2, i.e,Λ > e = 7 .
39. Therefore, while Λ = 35 probably pertains to weakly collisional plasmas,the value Λ c = 1 .
12 only gives a general idea of where the transition occurs.The width of the front for any plasma parameter Λ is eventually given by the redcurve in Figure 3. Simply put, the nature of the shock is the same as the nature of theupstream. Both are collisional or collisionless together.The non-monotonic behavior in the collisionless regime is just the consequence of thenon-monotonic variation of the mean-free-path in terms of the plasma parameter. Thefunction g ( x ) = A ln x/x reaches a max for x = e with g ( e ) = 3 .
67, still for A = 10.
6. Comparison with Tidman (1958)
A calculation parallel to the one performed in Section 3 for the collisional regime wasachieved in Tidman (1958). However, as we show here, the bridging it provides to thecollisionless regime is inadequate.For the ion distribution function F , Tidman (1958) used the Fokker-Planck operator A. Bret and A. Pe’er for ( ∂F/∂t ) c in Eq. (2.1), set ( ∂F/∂t ) w = 0, and found for strong shocks † , ℓ T = α c s N , Γ M , (6.1)where α = 29 . / π and Γ = πe m i ln Λ. We can recast this result under the form, ℓ T = 4 πα λ mfp , M ∼ . λ mfp , M , (6.2)where we have used Eq. (4.3).As a consequence, bridging the collisional result of Tidman (1958) with the collisionlessresult of Tidman (1967), that is, bridging Eq. (6.2) with Eq. (4.2), implicitly defines acritical plasma parameter Λ c through,4 παA M = ln Λ c Λ c , (6.3)yielding a Mach number-dependent value of Λ c and having no solution if the left-hand-sideis larger than the maximum of the right hand-side, that is, for M > .
51 (considering A = 10).As opposed to that, the ∝ M scaling of the collisional ℓ given by BGK-derived Eq.(3.12) is essential to give a value of Λ c independent of the upstream Mach number M ,with a switch from the collisional to the collisionless regime when the upstream becomescollisionless.We therefore find that BGK provides a better bridging to the collisionless regimethan Fokker-Planck. Hazeltine (1998) already noted the capacity of the BGK opera-tor to behave adequately in the collisionless limit. Computing the moments of the ki-netic equation with the BGK operator, he could derive a non-local expression of theheat flux in the collisionless regime, as expected when the mean-free-path becomes large(Hammett & Perkins 1990; Hazeltine 1998). Indeed, the BGK operator was specificallydesigned to provide an operator capable of giving an adequate description of low-densityplasmas (Bhatnagar et al. F = N φ , where φ is a Maxwellian (see Eq. 3.5).In contrast, the Fokker-Planck operator does not assume any a priori form of the equi-librium distribution function. It can even be used to prove that such a function is aMaxwellian. Yet, the collision rate is implicitly assumed large compared to the dynamicterms v/L in the Fokker-Planck equation (Kulsrud (2005), p. 213) since the derivation ofthe Fokker-Planck operator involves a Taylor expansion in time, implicitly assuming col-lisions are frequent enough (Kulsrud (2005), Eq. 29-30, p. 204 or Chandrasekhar (1943), § II.4).Therefore, when collisions become scarce, the BGK formalism keeps forcing, by design,a Maxwellian equilibrium, while Fokker-Planck progressively loses validity.
7. Conclusion
We propose a bridging between collisional and collisionless shocks. The collisional“leg” is worked out using the Moot-Smith ansatz (Mott-Smith 1951) with the “full” BGK † See Eq. (6.6) of Tidman (1958) where V is the sound speed and K is the Mach number. ridging the gap between collisional and collisionless shock waves et al. . .
12, the strong shock is collisional with a front thickness ∼ M λ mfp , given by Eq. (3.12). From Λ & .
12, the shock switches to the collisionlessregime, with a front thickness ℓ ∼ M λ mfp , ln Λ / Λ, given by Eq. (4.2).We show that the BGK treatment of the collisional regime provides a better bridgeto the collisionless regime than the Fokker-Planck model. Nevertheless, a confusing fea-ture remains: in the collisional limit, one would expect the BGK and the Fokker-Plancktreatments to merge. Yet, they don’t, as evidenced by their different M scaling for thestrong shock width ( ∝ M for BGK vs. ∝ M for Fokker-Planck). The reason for thiscould be that the collision frequency used in BGK (Eq. 3.3) does not depend on theparticle velocity. However, this is still unclear to us.A smoother transition between the 2 regimes could be assessed from Eq. (2.1) consid-ering both ( ∂F/∂t ) c and ( ∂F/∂t ) w at once, whereas we here switched them on and offaccording to the regime considered. The Mott-Smith ansatz could still be applied, whileusing BGK for ( ∂F/∂t ) c and the operator proposed by Dupree (1966) (as suggested inTidman (1967)) or Baalrud et al. (2008), for ( ∂F/∂t ) w .Although the present theory is formally restricted to high Mach number, un-magnetized,electrostatic shocks, it may help understand how the value of Λ ∼ observed in thesolar wind (see for example Fitzpatrick (2014), p. 8) yields an earth bow shock thicknessorders of magnitude shorter than the mean-free-path.
8. Acknowledgments
A.B. acknowledges support by grants ENE2016-75703-R from the Spanish Ministe-rio de Econom´ıa y Competitividad and SBPLY/17/180501/000264 from the Junta deComunidades de Castilla-La Mancha.A. P. acknowledges support from the European Research Council via ERC consolidat-ing grant
Appendix A. Proof of Eq. (3.8)
Equation (3.8) is the v y moment of Eq. (3.1). The left-hand-side is calculated in Tidman(1958). Note that the term proportional to E x vanishes in this moment. We only detailhere the calculation proper to the present work, that is, that of the right-hand-side. Forthis we need Φ, hence q and T defined by Eqs. (3.4-3.7).According to Eq. (3.6), q is given by, q = 1 N Z v F d v = 1 n + n Z v F d v. (A 1)Since F is the sum of 2 drifting Maxwellians given by Eq. (2.2), we find for q , q = n U + n U n + n ≡ q e x , (A 2)0 A. Bret and A. Pe’er where e x is the unit vector of the x axis. For T we then get from (3.7) † ,3 k B Tm i = 1 n + n Z ( v − q ) F d v = 1 n + n Z [( v x − q ) + v y + v z ] F d v = 1 n + n Z ( v x − q ) F d v + 2 n + n Z v y F d v (A 3)= 1 n + n Z ( v x − q ) F d v + 2( k B T n + k B T n ) m i ( n + n ) ⇒ k B T = n k B T + n k B T n + n + n n n + n ) m i ( U − U ) . (A 4)Let us now write explicitly the v y moment of the right-hand-side ( rhs ) of Eq. (3.1), rhs = 1 σ Z v y ( − N F + N Φ) d v = N σ Z v y Φ d v | {z } − Nσ Z v y F d v | {z } . (A 5)From (3.4) we see N does not depend on v . It can therefore be taken out of the integrals.Computing we find, Nσ Z v y F d v = n + n σ (cid:18) n k B T m i + n k B T m i (cid:19) , = k B m i σ ( n + n )( n T + n T ) . (A 6)Then we compute . N σ Z v y Φ d v = ( n + n ) σ Z v y (cid:18) m i πk B T (cid:19) / exp (cid:18) − m i k B T ( v − q ) (cid:19) d v, = ( n + n ) σ (cid:18) m i πk B T (cid:19) / Z v y exp (cid:18) − m i k B T ( v − q ) (cid:19) d v | {z } . For we get, = (2 π ) / (cid:18) k B Tm i (cid:19) / , (A 7)so that gives, = ( n + n ) σ (cid:18) m i πk B T (cid:19) / (2 π ) / (cid:18) k B Tm i (cid:19) / = ( n + n ) σ k B Tm i . (A 8)Finally, Eq. (A 5) simplifies nicely and reads, rhs = ( n + n ) σ k B Tm i − k B m i σ ( n + n )( n T + n T ) , = 13 σ n n ( U − U ) , (A 9) † The factor 2 in the second term of Eq. (A 3) comes from R v z F = R v y F . ridging the gap between collisional and collisionless shock waves Appendix B. Derivation of the density profiles (3.9) from Eq. (3.8)
Let us define α, β, γ from Eq. (3.8) by, U k B T m i | {z } α ∂n ∂x + U k B T m i | {z } β ∂n ∂x = 13 σ ( U − U ) | {z } γ n n . (B 1)Consider now the matter conservation equation obtained equating the v x moments of(2.2) between any x and x = + ∞ , n ( x ) U + n ( x ) U = N , U . (B 2)Differentiate with respect to x gives, ∂n ( x ) ∂x U + ∂n ( x ) ∂x U = 0 ⇒ ∂n ( x ) ∂x = − ∂n ( x ) ∂x U U , (B 3)and use the result to eliminate ∂n /∂x in (B 1), ∂n ∂x (cid:18) α − β U U (cid:19) = γn n ⇒ ∂n ∂x n n = γα − β U U . (B 4)Making now use again of the conservation equation (B 2) to write, n = ( N , − n ) U U , (B 5)one gets, U U ∂n ∂x n ( N , − n ) = U U ∂n ∂x N , (cid:18) n + 1 N , − n (cid:19) = γα − β U U . (B 6)We eventually obtain, ∂n ∂x (cid:18) n + 1 N , − n (cid:19) = N , γα − β U U U U ≡ − ℓ − , (B 7)where ℓ is the shock thickness since the solution accounting for the boundary conditions(2.3) is, n ( x ) = N ,
11 + e − x/ℓ . (B 8)From (B 2) one then obtains for n ( x ), n ( x ) = N , e − x/ℓ e − x/ℓ . (B 9) Appendix C. Derivation of Eq. (3.11) from Eq. (3.10)
We first cast Eq. (3.10) under the form, ℓ = 3 σ k B T (1 − T /T )( U /U ) U N , m i U (1 − U /U ) . (C 1)2 A. Bret and A. Pe’er
We then use the RH jump conditions (see for example Fitzpatrick (2014) p. 216, orThorne & Blandford (2017) p. 905), (cid:18) U U (cid:19) − = N , N , = γ + 1 γ − M − , (C 2)and, T T = P P N , N , (C 3)with, P P = 2 γ M − γ + 1 γ + 1 . (C 4)Substituting these ratios and setting, M = U γP /N , (C 5)we get to Eq. (3.11) with γ = 5 / REFERENCESBaalrud, S. D., Callen, J. D. & Hegna, C. C.
Physics of Plasmas (9), 092111. Bale, S. D., Mozer, F. S. & Horbury, T. S.
Phys. Rev. Lett. , 265004. Balogh, A. & Treumann, R.A.
Physics of Collisionless Shocks: Space Plasma ShockWaves . Springer New York.
Berezhko, E. G. & Ellison, Donald C.
Astrophysical Journal (1), 385–399.
Bhatnagar, P. L., Gross, E. P. & Krook, M.
Phys. Rev. , 511. Biskamp, D. & Pfirsch, D.
Physicsof Fluids (3), 732–733. Bret, Antoine
Astrophysical Journal (2), 111.
Bret, Antoine & Narayan, Ramesh
Journal of Plasma Physics (6),905840604. Bret, A. & Narayan, R.
Physics of Plasmas (6), 062108. Buneman, O.
Physics of Fluids (11), S3–S8. Chandrasekhar, S.
Rev. Mod. Phys. ,1–89. Dupree, T. H.
Physics of Fluids (9), 1773–1782. Fitzpatrick, R.
Plasma Physics: An Introduction . Taylor & Francis.
Gross, E. P. & Krook, M.
Phys. Rev. , 511.
Hammett, Gregory W. & Perkins, Francis W.
Phys. Rev. Lett. ,3019–3022. Hazeltine, R. D.
Physics of Plasmas (9),3282–3286. ridging the gap between collisional and collisionless shock waves Kulsrud, Russell M
Plasma physics for astrophysics . Princeton, NJ: Princeton Univ.Press.
Lee, Y. T. & More, R. M.
Physicsof Fluids (5), 1273–1286. Mott-Smith, H. M.
PhysicalReview (6), 885–892. Petschek, H. E.
Reviews of Modern Physics , 966–974. Ruyer, C., Gremillet, L., Bonnaud, G. & Riconda, C.
Physics of Plasmas (4), 041409. Sagdeev, R. Z.
Re-views of Plasma Physics , 23. Salas, Manuel D.
Shock Waves (6), 477–487. Schwartz, Steven J., Henley, Edmund, Mitchell, Jeremy & Krasnoselskikh,Vladimir
Phys. Rev.Lett. , 215002.
Spitkovsky, Anatoly
Astrophys. J. Lett. , L5–L8.
Stockem, A., Fiuza, F., Bret, A., Fonseca, R. A. & Silva, L. O.
Scientific Reports , 3934. Thorne, K.S. & Blandford, R.D.
Modern Classical Physics: Optics, Fluids, Plasmas,Elasticity, Relativity, and Statistical Physics . Princeton University Press.
Tidman, D. A.
Physical Review (6), 1439–1446.
Tidman, D. A.
Physics of Fluids (3), 547–564. Tidman, Derek A. & Krall, Nicholas A.
Physics of Fluids (3), 733–735. Zel’dovich, Ya B & Raizer, Yu P