- Email: [email protected]
Criterion for existence of shock waves in relativistic magnetohydrodynamics with a general equation of state
20 October
1997
PHYSICS
ELSEYIER
Physics Letters A 235 (1997)
LETTERS
A
71-75
Criterion for existence of shock waves in relativistic magnetohydrodynamics with a general equation of state V.I. Zhdanov ‘, P.V. Tytarenko Astronomical Observatory of Kyiv Schevchenko University, Observatoma Street 3, 53 Kiev 254053, Ukraine Received
31 January
1997; revised manuscript received 4 June 1997; accepted Communicated by AI? Fordy
for publication
30 June 1997
Abstract Relativistic stationary shock waves in an ideal conducting fluid are studied for the general viscosity arguments to obtain a criterion that selects physically admissible shock transitions convexity of the Poisson adiabats. The relations between the magnetosound speeds and the a consequence of this criterion reveal specific differences between relativistic considerations Elsevier Science B.V.
equation of state. We use small without any supposition about speed of the shock obtained as versus classical ones. @ 1997
Keywords: Relativistic
of state; Evolutionary
magnetohydrodynamics;
Shock waves; Anomalous
1. Introduction This paper concerns stationary shock waves in relativistic magnetohydrodynamics (MGD) without using any convexity supposition for the equation of state (i.e. the convexity of the Poisson adiabats) . This supposition plays an important role in the investigation of discontinuous flows. In particular, the treatment of MGD shocks by Lichnerovicz [ l] also essentially uses this property. The requirement of the convexity is one of the Bethe-Weyl conditions [2,3] for a normal medium in the classical theory of shock waves. It is well known that uniqueness of the solutions in fluid dynamics may be lost in case of discontinuous initial data [ 31, and in this case some additional restrictions are needed to select a unique discontinuous solution. In hydrodynamics with a normal equation of state (in
’ E-mail:
[email protected].
0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PI1 SO375-9601(97)00549-5
media; Nonconvex
equations
conditions
the sense of Bethe and Weyl) such a restriction is due to the entropy growth criterion (that, e.g., prohibits rarefaction shocks and permits compression shocks). However, if the convexity condition does not hold, the entropy criterion is not sufficient and the standard theory of shock waves must be modified (see Refs. [ 3,4] and the references therein). Note that convexity of the Poisson adiabats does not follow from general principles and in fact it may be violated [ 3,4] ; in this case rarefaction shocks and compression simple waves are possible. This has been the reason for special considerations in hydrodynamics of super-dense matter (see, e.g., Refs. [6-l l] and references therein). The general criterion for the existence of relativistic shocks in an ideal fluid without having recourse to convexity arguments has been worked out in Refs. [ 9,111. In this paper we generalize the results of Refs. [ 9,111 to the case of the relativistic version of MGD with perfect electrical conductivity [ 11, when addi-
tional degrees of freedom due to the magnetic field are essential. The problem is to find a criterion that selects physically admissible shock transitions. To do this we use a modification of the small viscosity method of the papers of Refs. [ 9,111. The viscosity tensor [5] is introduced into the equations of the relativistic MGD (11 to analyze stationary viscous flow. The condition for the existence of a viscous profile of the shock allows us to formulate the criterion for admissible shock waves in the form of certain in~ualities involving the equation of state. If this criterion does not hold for a certain shock transition, this shock is considered to be unphysical.
The equations of motion of an ideal relativistic fluid permeated by a magnetic field can be written in the form of conservation laws for the MGD energymomentum tensor [ I 1, - p*gfiy - -&h’h”,
(11
where UP is the 4-velocity of the fluid, hp = -- ~e~@~&p, is the magnetic field, e@‘a is the Levi-Civita symbol, F,p is the tensor of the electromagnetic held; {ggy} = diag( 1, - 1, - I, - 1) , p* =p
+ &hJ2,
/hi2 = -h”h,
E* = E + $-lhi2,
> 0,
is the pressure and E is the proper energy density. The pressure is related to E and to the baryon number density n (or density of some other conserved charge) by a sufficiently smooth equation of state p = p( E, V), V = l/n being the specific volume. We choose the magnetic permeability to be 1; the more general case [ 11 may be easily obtained by resealing hfi. The conservation equations for Tfi” must be complemented by conservation equations for the baryonic charge p
~~(~~~) = 0,
(2)
and by the Maxwell equations which yield in the case of MGD
(3)
In a discontinuous flow the hydrody~~ic quantities on both sides of the shock are related through an integral form of the conservation laws. However, the relations on the discontinuities must be complemented by certain restrictions that are necessary to provide uniqueness of the discontinuous solutions [ 3,4]. These restrictions may be obtained by analyzing the structure of the shock transition in the presence of small dissipative effects. In this paper we smear out the discontinuities by taking into account viscosity. In the presence of viscosity the conservation equation for Tp” must be replaced by +(T@” + 7“‘) = 0,
2. Basic relations
Tfi” = (p* + E*)&/
c?,(ul*hv - u”hp) = 0.
(4)
where
is the relativistic viscosity tensor [ 51. Eq. (4) is considered along with Eqs. (2) and (3). Here we confine ourselves to only one of the possible methods to define a (generalized) discontinuous solution. There may be other dissipative effects besides viscosity (see, e.g., the consideration of classical MGD [ 131); one may use also different definitions of the generalized solution of the hydrodynamic equations of motion [ 31. Here, we use a most simple approach based on the viscosity tensor [ 51; moreover, we suppose that a limiting solution exists for Q -+ 0, 5 -+ 0 and to simplify the consideration we further put q = 0. Nevertheless it will be clear from the below that one viscosity coefficient 5 # 0 is sufficient to obtain a continuous viscous profile of the shock and the resulting restrictions appear to be more stringent than the evolutions conditions.
3. Method of consideration In the limit of small viscosity we expect that the solutions of the hydrodyn~i~~ equations yield the discontinuous flows describing shock waves. For l # 0 the stationary shock wave propagating in a space-Iike direction I, is locally represented in a proper frame
VI. Zhdanov,
PV
Tytarenko/Physics
by a stationary viscous how depending upon the only variable x = npl,. We choose the coordinates in such a way that {ZP) = (0, l,O, 0}, x :=x1. Therefore, Eqs. (2)-(4) yield T’” f ?lv = const,
(6)
a’hV - hluV = H” = const,
(7)
rzul = j = const.
(8)
We assume that for x 4 -cc all the quantities tend to some constant values marked by “0” (the state ahead of the shock) and for x --+ 00 the quantities tend to constant values marked by “1” (behind the shock). Because V’ --+ 0 for x -+ &cc, the solution of Eqs. (6)-( 8) satisfies the shock relations. However, is there a continuous solution of Eqs. (6) (8)) satisfying both the above asymptotics for x -+ 03 and for x --+ -co? We shah investigate this boundary value problem, and existence conditions for a solution of this problem are then interpreted as conditions of admissibility of a shock transition u$,, hy,,, VO, PO -+ $I,, h’;; ), 6, ~1, with the corresponding states satisfying the relations on the shock wave. Further, without loss of generality, we may put us z 0 and h3 z 0 due to the choice of the frame. From Eq. (7) one obtains hP = ;(H’”
- u@WU,).
(9)
multiplying Eq. (6) by uy and taking into account that Y’u,, = 0, we have Cl4’ = T;+,,
(10)
whence E and hp are expressed in terms of u’ and u*. EIiminating the oniy derivative du’/dx from Eq. (6) for p = 1,2, one obtains a relation in terms of the thr~-dimensional velocities 01 = u’/u’ and v2 = U2/lP, (H’LQ - ti,(H*
- H0v2) =‘k.‘,(T:gqU2
-T:,:), (11)
Letters A 235 (1997)
Due to Eq. (8) we can express the velocities as functions of the specific volume, 1 ‘?4 = ~~O~~/~ = &) vlvo,
du’
~~I+(u’)*]~=T”-T~~~.
(12)
(13)
and, therefore, we reduce the problem to the investigation of V(x) by means of the only first order differential equation ( 12). Eqs. (9)-( 13) completely define the structure of the shock transition. Suppose that we are on the connected component of this curve. The differential equation ( 12) may be written as
ig$ =p(I!E(V))
(14)
-P(V),
where the function P(V) = [ 1 + (z&2] -1 x
T& + -$H”u,12
- T;&d >
- --&~)-2[(~~u,~2
- HW,]
is determined through Eqs. (7), (8), ( 11) and
(cf. Eq ( 10)) where the magnetic field is defined by Eq. (9). The right hand side of Eq. (14) equals zero for V = Vo and V = V,. Let, e.g., V, > V,, and the r.h.s. of Eq. ( 14) be smooth and positive for V E ( VO,V, ) , Then it is easy to see that the solution V(x) of Eq. ( 14) tends to Vo for x -+ --oo and to VI for x --+ 0;). If the right hand side of Eq. ( 14) changes sign at some point between VO and VI, then there is no continuous solution of Eq. (14) in (--00, co) connecting Vo and VI. Care is required in order to check that du’ /dvz does not change sign along the solution. This can be proved under the requirement that the qualitative behavior of the solution must be stable under small changes of the equations of state. As a result we obtain the following criterion ofadnukibility
which is linear in ur and quadratic in 02. Eq. (6) for ,u = 1 yields
73
71-7.5
of a stationary shock transition,
(v, - v,)[P(Y~(V)>
-P(V)1
3 0,
(15)
for all V between VOand VI, where we suppose that Eq. ( 11) has a regular solution between vi) and Vi.
74
VI. Zhdanov, I% TytarenkolPhysics
We have included here also the case when the right hand side of Rq. ( 15) equals zero between & and V, , but it does not change sign. This case is allowed by the small viscosity limit and must be treated separately. In the case of zero magnetic field the criterion ( 15) agrees with the results of Refs. [9,1 I 1. 4. Consequences
of the criterion (15)
Here we outline the most important results of our consideration on account of the criterion ( 15). If the shock satisfies the criterion ( 15)) then it also satisfies the entropy criterion. It is known that the reverse statement in the general case is not true; the corresponding examples of this behavior of relativistic shocks (without magnetic field) may be found in Ref. [ 121. It may be shown that the inequality ( 15) does yield the entropy growth during the shock transition. Clearly, this statement may be obtained directly from the entropy consideration for a viscous flow. We have obtained an equivalent form of the criterion in terms of the shock adiabat. For a given initial state u/;b,, h$,, k& po the shock adiabat PH ( V) [ 1] defines the states that are related to the initial one by means of the conservation equations. Under the assumption that PH( V) is a single-valued function, the inequality (15) is equivalent to the condition (VI -V,)[PH(V)
-P(V)1
20.
(16)
In the neighborhood of the initial state this can be proved by a straightforward calculation making use of the definition of the shock adiabat by Lichnerovich [ 11. Then we observe that the points where Eq. (15) turns into an equality lie on the shock adiabat. This yields an extension of Eq. ( 16) to all the states between “0” and “1”. This result also agrees with Refs. [ 9,111 in the case of a zero magnetic field. In case of a non-single-valued shock adiabat the criterion may be fo~ulated in terms of the absence of intersections of ~,v( V) and p(V) between VOand Vi, It is interesting to compare the speed of the shock !&, to the relativistic fast and slow magnetosound speeds uf, us and the Alfven speed VA] 1 ] . The aim is to obtain inequalities which are analogous to the well-known evolution conditions [ 51. In order to do this we need the relation that follows identically for the solutions of Eqs. (9)-( 11) in the case of the tensor (I ),
Letters A 235 (1997) 71-75
where R = (p* + E*)(u’)~ - (?~‘)~/47r, and k2 is the transverse component of the magnetic field in the frame where u2 = 0, h2 = (h2 - u#)/~~. For continuous solutions it follows from Eq. ( 17) that in the nontrivial case ii* and R do not change signs on (-co, co). This agrees with the results of Lichnerowicz [ I] who has shown the nonexistence of switch-on and switch-off relativistic shocks. This is an example of the relativistic specificity of MGD, because these shocks may exist in the nonrelativistic case. The relation (~a) 2 = ( h’)2/47r(p* + E*) defines the 4-velocity component of an Alfven wave moving along the x-axis, so the sign of (u’)~ - (MA)* is conserved under a shock transition. The case R > 0 corresponds to a fast shock and R < 0 - to a slow shock. Another condition for the velocities follows from the fact that the sign of du’/dq is conserved along an admissible trajectory. Ahead of the shock (X -+ -oc) where ufo) = 0, we have (du’/duz)j(ot
=~~~~o~~~o~(~~o~)-~,
wheretheexpressionR* = (p+~)(~‘)~(l+(u*)~)( 114~) ( h’ ) * does not change its form under motion of an inertial frame parallel to the plane x = const. The sign of du’/dq is also conserved under the above motions, so R* defines this sign in any such frame and the signs of RTo, ahead of and Rf,, behind the shock are equal. In the nonrelativistic limit R’ tends to R. In order to find relations between the shock and magnetosound speeds we expand Eq. (15) in the vicinity of the initial and fmal states. On account of Eqs. (9)-( 1I) in the neighborhood of “0” we obtain
where
D(d) = Q(d)/R*(u’),
+
w)2C: 4a(p + E) ’
rl. Zhdanov, f?E TyfarenkolPhysics l.&ters A 235 (1997) 71-75
and cf = (3p/&), is the speed of sound. Analogous results can be obtained in the neighborhood of “1”. A s~~ghtforw~d c~culation yields Q(t) z ( 1 cz) (t* - 6;) (t2 - [t,), where & and tSl are the 4velocities of the fast and slow magnetosound waves. Now the condition (15) in the neigh~rh~ of the points “0” and “1” gives the inequalities “(5)1&&,
> 0,
WE) I&f,, < 0,
yielding obvious restrictions on the shock velocities that involve a new p~~eter - the 4-velocity component that turns R” into zero, UN=
1 z
1[(
1+
112 _ 1
(h’)2 dP
+ E) >
magnetic field an analogous consideration [ 111 leads to relations between the speed of the shock and the sound speed that are equivalent to analogous consequences of the evolution conditions. The presence of a magnetic field changes the result so that it is different from the latter. In the nonrelativistic limit %?$j + VA,so that the inequalities transform into the standard evolution conditions [ 141. Therefore, this is one more example of the relativistic specificity of MGD shocks in addition to the results of Lichnerowicz [ 1f. out a
References
“2
II*
These restrictions combined with the above results concerning the signs of R* and R lead to relations for the three-dimensional speeds &,hof the admissible shock and fast and slow magnetosound speeds uf, 0~1. For fast shock positions in the initial state “0” f&h(O) > UffO) > UN(O) > uA(O) > %(O)r
and in the final state “1” uf(1) > Ush(l) > *N(l) > uA(1) > us(l).
The relations for the final state appear to be more restrictive than the standard evolution criterion that does not involve UN2 UA.For a parallel shock UA= UN;for a pe~endicul~ one both velocities vanish. For the slow shocks we have the inequalities uA(0) > %h(O) > &l(O),
75
us(l) > &.h(l)t
that are the same as the evolution conditions. With-
Ill A.
Lichnerowicz, Relativistic hydrodynamics and magnetohydrodynamics (Benjamin, New York, 1967). 121H. Weyl, Comm. Pure Appl. Math. 2 (1949) 103. 131 B.L. Rozhdestvensky and N.N. Yanenko, Systems of quasilinear equations (Moscow, Nauka, 1978, in Russian). [41 R. Menikoff and B.J. Plobr, Rev. Mod. Phys. 61 ( 1989) 75. IS1 L.D. Landau and EM. Lifshits. Hydrodynamics (Moscow, Nauka. 1986) Iin Russian]. [61 H.W. Barz, B. Kampfer and B. Lukacs, Phys. Rev. R 32 (1985) 1234. 171 P. Danielewicz and P.V. Ruuskanen, Phys. Rev. D 35 ( 1987) 344. [81 J.l? Blaizot and J.Y. Ollitrault, Phys. Rev. D 36 (1987) 1234. 191 K.A. Bugaev, M.1. Gorenstein and V.I. Zhdanov, Z. Phys. C 39 (1988) 365. [ 101 K.A. Bugaev, M.I. Gorenstein, B. Kampfer and V.I. Zhdanov, Phys. Rev. D 40 (1989) 2903. 1111 K.A. Bugaev, M.I. Gorenstein and V.I. Zhdanov, Tear. Mat. Fiz. 80 (1989) 138. I121 M.I. Gorenstein and V.I. Zhdanov, Z. Pbys. C 34 ( 1987) 79. 1131 N.N. Kuznetsov, Zh. Eksp. ‘Ieor. Fiz. 88 (1985) 470. f141 A.I. Achiezer, G.Ya. Lubarsky and R.V. Polovin, Zh. Eksp. Tear. Fiz. 35 (1958) 731; L.D. Landau and E.M. Lifshits, Electrodynamics of continuous media (Moscow, Nauka, 1992) l in Russian]
1997
PHYSICS
ELSEYIER
Physics Letters A 235 (1997)
LETTERS
A
71-75
Criterion for existence of shock waves in relativistic magnetohydrodynamics with a general equation of state V.I. Zhdanov ‘, P.V. Tytarenko Astronomical Observatory of Kyiv Schevchenko University, Observatoma Street 3, 53 Kiev 254053, Ukraine Received
31 January
1997; revised manuscript received 4 June 1997; accepted Communicated by AI? Fordy
for publication
30 June 1997
Abstract Relativistic stationary shock waves in an ideal conducting fluid are studied for the general viscosity arguments to obtain a criterion that selects physically admissible shock transitions convexity of the Poisson adiabats. The relations between the magnetosound speeds and the a consequence of this criterion reveal specific differences between relativistic considerations Elsevier Science B.V.
equation of state. We use small without any supposition about speed of the shock obtained as versus classical ones. @ 1997
Keywords: Relativistic
of state; Evolutionary
magnetohydrodynamics;
Shock waves; Anomalous
1. Introduction This paper concerns stationary shock waves in relativistic magnetohydrodynamics (MGD) without using any convexity supposition for the equation of state (i.e. the convexity of the Poisson adiabats) . This supposition plays an important role in the investigation of discontinuous flows. In particular, the treatment of MGD shocks by Lichnerovicz [ l] also essentially uses this property. The requirement of the convexity is one of the Bethe-Weyl conditions [2,3] for a normal medium in the classical theory of shock waves. It is well known that uniqueness of the solutions in fluid dynamics may be lost in case of discontinuous initial data [ 31, and in this case some additional restrictions are needed to select a unique discontinuous solution. In hydrodynamics with a normal equation of state (in
’ E-mail:
[email protected].
0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PI1 SO375-9601(97)00549-5
media; Nonconvex
equations
conditions
the sense of Bethe and Weyl) such a restriction is due to the entropy growth criterion (that, e.g., prohibits rarefaction shocks and permits compression shocks). However, if the convexity condition does not hold, the entropy criterion is not sufficient and the standard theory of shock waves must be modified (see Refs. [ 3,4] and the references therein). Note that convexity of the Poisson adiabats does not follow from general principles and in fact it may be violated [ 3,4] ; in this case rarefaction shocks and compression simple waves are possible. This has been the reason for special considerations in hydrodynamics of super-dense matter (see, e.g., Refs. [6-l l] and references therein). The general criterion for the existence of relativistic shocks in an ideal fluid without having recourse to convexity arguments has been worked out in Refs. [ 9,111. In this paper we generalize the results of Refs. [ 9,111 to the case of the relativistic version of MGD with perfect electrical conductivity [ 11, when addi-
tional degrees of freedom due to the magnetic field are essential. The problem is to find a criterion that selects physically admissible shock transitions. To do this we use a modification of the small viscosity method of the papers of Refs. [ 9,111. The viscosity tensor [5] is introduced into the equations of the relativistic MGD (11 to analyze stationary viscous flow. The condition for the existence of a viscous profile of the shock allows us to formulate the criterion for admissible shock waves in the form of certain in~ualities involving the equation of state. If this criterion does not hold for a certain shock transition, this shock is considered to be unphysical.
The equations of motion of an ideal relativistic fluid permeated by a magnetic field can be written in the form of conservation laws for the MGD energymomentum tensor [ I 1, - p*gfiy - -&h’h”,
(11
where UP is the 4-velocity of the fluid, hp = -- ~e~@~&p, is the magnetic field, e@‘a is the Levi-Civita symbol, F,p is the tensor of the electromagnetic held; {ggy} = diag( 1, - 1, - I, - 1) , p* =p
+ &hJ2,
/hi2 = -h”h,
E* = E + $-lhi2,
> 0,
is the pressure and E is the proper energy density. The pressure is related to E and to the baryon number density n (or density of some other conserved charge) by a sufficiently smooth equation of state p = p( E, V), V = l/n being the specific volume. We choose the magnetic permeability to be 1; the more general case [ 11 may be easily obtained by resealing hfi. The conservation equations for Tfi” must be complemented by conservation equations for the baryonic charge p
~~(~~~) = 0,
(2)
and by the Maxwell equations which yield in the case of MGD
(3)
In a discontinuous flow the hydrody~~ic quantities on both sides of the shock are related through an integral form of the conservation laws. However, the relations on the discontinuities must be complemented by certain restrictions that are necessary to provide uniqueness of the discontinuous solutions [ 3,4]. These restrictions may be obtained by analyzing the structure of the shock transition in the presence of small dissipative effects. In this paper we smear out the discontinuities by taking into account viscosity. In the presence of viscosity the conservation equation for Tp” must be replaced by +(T@” + 7“‘) = 0,
2. Basic relations
Tfi” = (p* + E*)&/
c?,(ul*hv - u”hp) = 0.
(4)
where
is the relativistic viscosity tensor [ 51. Eq. (4) is considered along with Eqs. (2) and (3). Here we confine ourselves to only one of the possible methods to define a (generalized) discontinuous solution. There may be other dissipative effects besides viscosity (see, e.g., the consideration of classical MGD [ 131); one may use also different definitions of the generalized solution of the hydrodynamic equations of motion [ 31. Here, we use a most simple approach based on the viscosity tensor [ 51; moreover, we suppose that a limiting solution exists for Q -+ 0, 5 -+ 0 and to simplify the consideration we further put q = 0. Nevertheless it will be clear from the below that one viscosity coefficient 5 # 0 is sufficient to obtain a continuous viscous profile of the shock and the resulting restrictions appear to be more stringent than the evolutions conditions.
3. Method of consideration In the limit of small viscosity we expect that the solutions of the hydrodyn~i~~ equations yield the discontinuous flows describing shock waves. For l # 0 the stationary shock wave propagating in a space-Iike direction I, is locally represented in a proper frame
VI. Zhdanov,
PV
Tytarenko/Physics
by a stationary viscous how depending upon the only variable x = npl,. We choose the coordinates in such a way that {ZP) = (0, l,O, 0}, x :=x1. Therefore, Eqs. (2)-(4) yield T’” f ?lv = const,
(6)
a’hV - hluV = H” = const,
(7)
rzul = j = const.
(8)
We assume that for x 4 -cc all the quantities tend to some constant values marked by “0” (the state ahead of the shock) and for x --+ 00 the quantities tend to constant values marked by “1” (behind the shock). Because V’ --+ 0 for x -+ &cc, the solution of Eqs. (6)-( 8) satisfies the shock relations. However, is there a continuous solution of Eqs. (6) (8)) satisfying both the above asymptotics for x -+ 03 and for x --+ -co? We shah investigate this boundary value problem, and existence conditions for a solution of this problem are then interpreted as conditions of admissibility of a shock transition u$,, hy,,, VO, PO -+ $I,, h’;; ), 6, ~1, with the corresponding states satisfying the relations on the shock wave. Further, without loss of generality, we may put us z 0 and h3 z 0 due to the choice of the frame. From Eq. (7) one obtains hP = ;(H’”
- u@WU,).
(9)
multiplying Eq. (6) by uy and taking into account that Y’u,, = 0, we have Cl4’ = T;+,,
(10)
whence E and hp are expressed in terms of u’ and u*. EIiminating the oniy derivative du’/dx from Eq. (6) for p = 1,2, one obtains a relation in terms of the thr~-dimensional velocities 01 = u’/u’ and v2 = U2/lP, (H’LQ - ti,(H*
- H0v2) =‘k.‘,(T:gqU2
-T:,:), (11)
Letters A 235 (1997)
Due to Eq. (8) we can express the velocities as functions of the specific volume, 1 ‘?4 = ~~O~~/~ = &) vlvo,
du’
~~I+(u’)*]~=T”-T~~~.
(12)
(13)
and, therefore, we reduce the problem to the investigation of V(x) by means of the only first order differential equation ( 12). Eqs. (9)-( 13) completely define the structure of the shock transition. Suppose that we are on the connected component of this curve. The differential equation ( 12) may be written as
ig$ =p(I!E(V))
(14)
-P(V),
where the function P(V) = [ 1 + (z&2] -1 x
T& + -$H”u,12
- T;&d >
- --&~)-2[(~~u,~2
- HW,]
is determined through Eqs. (7), (8), ( 11) and
(cf. Eq ( 10)) where the magnetic field is defined by Eq. (9). The right hand side of Eq. (14) equals zero for V = Vo and V = V,. Let, e.g., V, > V,, and the r.h.s. of Eq. ( 14) be smooth and positive for V E ( VO,V, ) , Then it is easy to see that the solution V(x) of Eq. ( 14) tends to Vo for x -+ --oo and to VI for x --+ 0;). If the right hand side of Eq. ( 14) changes sign at some point between VO and VI, then there is no continuous solution of Eq. (14) in (--00, co) connecting Vo and VI. Care is required in order to check that du’ /dvz does not change sign along the solution. This can be proved under the requirement that the qualitative behavior of the solution must be stable under small changes of the equations of state. As a result we obtain the following criterion ofadnukibility
which is linear in ur and quadratic in 02. Eq. (6) for ,u = 1 yields
73
71-7.5
of a stationary shock transition,
(v, - v,)[P(Y~(V)>
-P(V)1
3 0,
(15)
for all V between VOand VI, where we suppose that Eq. ( 11) has a regular solution between vi) and Vi.
74
VI. Zhdanov, I% TytarenkolPhysics
We have included here also the case when the right hand side of Rq. ( 15) equals zero between & and V, , but it does not change sign. This case is allowed by the small viscosity limit and must be treated separately. In the case of zero magnetic field the criterion ( 15) agrees with the results of Refs. [9,1 I 1. 4. Consequences
of the criterion (15)
Here we outline the most important results of our consideration on account of the criterion ( 15). If the shock satisfies the criterion ( 15)) then it also satisfies the entropy criterion. It is known that the reverse statement in the general case is not true; the corresponding examples of this behavior of relativistic shocks (without magnetic field) may be found in Ref. [ 121. It may be shown that the inequality ( 15) does yield the entropy growth during the shock transition. Clearly, this statement may be obtained directly from the entropy consideration for a viscous flow. We have obtained an equivalent form of the criterion in terms of the shock adiabat. For a given initial state u/;b,, h$,, k& po the shock adiabat PH ( V) [ 1] defines the states that are related to the initial one by means of the conservation equations. Under the assumption that PH( V) is a single-valued function, the inequality (15) is equivalent to the condition (VI -V,)[PH(V)
-P(V)1
20.
(16)
In the neighborhood of the initial state this can be proved by a straightforward calculation making use of the definition of the shock adiabat by Lichnerovich [ 11. Then we observe that the points where Eq. (15) turns into an equality lie on the shock adiabat. This yields an extension of Eq. ( 16) to all the states between “0” and “1”. This result also agrees with Refs. [ 9,111 in the case of a zero magnetic field. In case of a non-single-valued shock adiabat the criterion may be fo~ulated in terms of the absence of intersections of ~,v( V) and p(V) between VOand Vi, It is interesting to compare the speed of the shock !&, to the relativistic fast and slow magnetosound speeds uf, us and the Alfven speed VA] 1 ] . The aim is to obtain inequalities which are analogous to the well-known evolution conditions [ 51. In order to do this we need the relation that follows identically for the solutions of Eqs. (9)-( 11) in the case of the tensor (I ),
Letters A 235 (1997) 71-75
where R = (p* + E*)(u’)~ - (?~‘)~/47r, and k2 is the transverse component of the magnetic field in the frame where u2 = 0, h2 = (h2 - u#)/~~. For continuous solutions it follows from Eq. ( 17) that in the nontrivial case ii* and R do not change signs on (-co, co). This agrees with the results of Lichnerowicz [ I] who has shown the nonexistence of switch-on and switch-off relativistic shocks. This is an example of the relativistic specificity of MGD, because these shocks may exist in the nonrelativistic case. The relation (~a) 2 = ( h’)2/47r(p* + E*) defines the 4-velocity component of an Alfven wave moving along the x-axis, so the sign of (u’)~ - (MA)* is conserved under a shock transition. The case R > 0 corresponds to a fast shock and R < 0 - to a slow shock. Another condition for the velocities follows from the fact that the sign of du’/dq is conserved along an admissible trajectory. Ahead of the shock (X -+ -oc) where ufo) = 0, we have (du’/duz)j(ot
=~~~~o~~~o~(~~o~)-~,
wheretheexpressionR* = (p+~)(~‘)~(l+(u*)~)( 114~) ( h’ ) * does not change its form under motion of an inertial frame parallel to the plane x = const. The sign of du’/dq is also conserved under the above motions, so R* defines this sign in any such frame and the signs of RTo, ahead of and Rf,, behind the shock are equal. In the nonrelativistic limit R’ tends to R. In order to find relations between the shock and magnetosound speeds we expand Eq. (15) in the vicinity of the initial and fmal states. On account of Eqs. (9)-( 1I) in the neighborhood of “0” we obtain
where
D(d) = Q(d)/R*(u’),
+
w)2C: 4a(p + E) ’
rl. Zhdanov, f?E TyfarenkolPhysics l.&ters A 235 (1997) 71-75
and cf = (3p/&), is the speed of sound. Analogous results can be obtained in the neighborhood of “1”. A s~~ghtforw~d c~culation yields Q(t) z ( 1 cz) (t* - 6;) (t2 - [t,), where & and tSl are the 4velocities of the fast and slow magnetosound waves. Now the condition (15) in the neigh~rh~ of the points “0” and “1” gives the inequalities “(5)1&&,
> 0,
WE) I&f,, < 0,
yielding obvious restrictions on the shock velocities that involve a new p~~eter - the 4-velocity component that turns R” into zero, UN=
1 z
1[(
1+
112 _ 1
(h’)2 dP
+ E) >
magnetic field an analogous consideration [ 111 leads to relations between the speed of the shock and the sound speed that are equivalent to analogous consequences of the evolution conditions. The presence of a magnetic field changes the result so that it is different from the latter. In the nonrelativistic limit %?$j + VA,so that the inequalities transform into the standard evolution conditions [ 141. Therefore, this is one more example of the relativistic specificity of MGD shocks in addition to the results of Lichnerowicz [ 1f. out a
References
“2
II*
These restrictions combined with the above results concerning the signs of R* and R lead to relations for the three-dimensional speeds &,hof the admissible shock and fast and slow magnetosound speeds uf, 0~1. For fast shock positions in the initial state “0” f&h(O) > UffO) > UN(O) > uA(O) > %(O)r
and in the final state “1” uf(1) > Ush(l) > *N(l) > uA(1) > us(l).
The relations for the final state appear to be more restrictive than the standard evolution criterion that does not involve UN2 UA.For a parallel shock UA= UN;for a pe~endicul~ one both velocities vanish. For the slow shocks we have the inequalities uA(0) > %h(O) > &l(O),
75
us(l) > &.h(l)t
that are the same as the evolution conditions. With-
Ill A.
Lichnerowicz, Relativistic hydrodynamics and magnetohydrodynamics (Benjamin, New York, 1967). 121H. Weyl, Comm. Pure Appl. Math. 2 (1949) 103. 131 B.L. Rozhdestvensky and N.N. Yanenko, Systems of quasilinear equations (Moscow, Nauka, 1978, in Russian). [41 R. Menikoff and B.J. Plobr, Rev. Mod. Phys. 61 ( 1989) 75. IS1 L.D. Landau and EM. Lifshits. Hydrodynamics (Moscow, Nauka. 1986) Iin Russian]. [61 H.W. Barz, B. Kampfer and B. Lukacs, Phys. Rev. R 32 (1985) 1234. 171 P. Danielewicz and P.V. Ruuskanen, Phys. Rev. D 35 ( 1987) 344. [81 J.l? Blaizot and J.Y. Ollitrault, Phys. Rev. D 36 (1987) 1234. 191 K.A. Bugaev, M.1. Gorenstein and V.I. Zhdanov, Z. Phys. C 39 (1988) 365. [ 101 K.A. Bugaev, M.I. Gorenstein, B. Kampfer and V.I. Zhdanov, Phys. Rev. D 40 (1989) 2903. 1111 K.A. Bugaev, M.I. Gorenstein and V.I. Zhdanov, Tear. Mat. Fiz. 80 (1989) 138. I121 M.I. Gorenstein and V.I. Zhdanov, Z. Pbys. C 34 ( 1987) 79. 1131 N.N. Kuznetsov, Zh. Eksp. ‘Ieor. Fiz. 88 (1985) 470. f141 A.I. Achiezer, G.Ya. Lubarsky and R.V. Polovin, Zh. Eksp. Tear. Fiz. 35 (1958) 731; L.D. Landau and E.M. Lifshits, Electrodynamics of continuous media (Moscow, Nauka, 1992) l in Russian]