- Email: [email protected]
On the properties of Alfvén waves in relativistic magnetohydrodynamics
11 August 1997
PHYSICS LETTERS A
Physics Letters A 232 (1997) 435-442
ELSEVIER
On the properties of Alfv6n waves in relativistic magnetohydrodynamics S.S.Komissarov l Department of Applied Mathematics, University of Leeds, Leeds L..Y29J7: UK Astrospace Centre. Lebedev Physical Institute, Lininsky Prospect 53, Moscow B-333, 177924 Russian Federation
Received 18 February 1997; accepted for publication 24 April 1997 Communicated by M. Porkolab
Abstract Simple AlfvCn waves and Alfven shocks are considered in the framework of relativistic magnetohydrodynamics. It is found that the tangential components of vector fields trace ellipses instead of the circles of Newtonian MHD. Their properties are studied in the general wave frame, Hoffmann-Teller wave frame, and the general laboratory frame. @ 1997 Elsevier Science B.V. Keywords: Relativistic magnetohydrodynamics; AlfvCn waves
1. Introduction It is now well established that active galactic nuclei ( AGN) can produce powerful relativistic jets of plasma with large Lorentz factors (see, e.g., Ref. [ 1 ] ) . These jets are sources of synchrotron emission, which indicates the presence of significant magnetic fields. Various models of the production of these relativistic jets have been proposed (see, e.g., Ref. [2] ). Although it is extremely difficult to test these models observationally the most plausible ones include strong magnetic fields as a key ingredient. All these factors make relativistic magnetohydrodynamics (RMHD) an essential theoretical tool in studying AGN. Even in Newtonian MHD analytical methods are rarely sufficient and one therefore has to resort to numerical calculations. It is even more difficult to find
’ E-mail: [email protected].
analytic solutions to the equations of RMHD. It is, however, important to find exact solutions that can be used to test computer codes. For such hyperbolic systems as RMHD these solutions should include all permitted types of shocks and simple waves. In this paper we consider the properties of relativistic Alfvtn waves. Quite apart from this, AlfvCn waves are also interesting as particular solutions to the equations of magnetohydrodynamics. It is well known that the Newtonian limit is degenerate, which means that the properties of such solutions in the relativistic theory can be dramatically different from the Newtonian ones. The properties of Newtonian AlfvCn waves can be summarized as follows (see, e.g., Ref. [ 31). The thermodynamic variables (e.g., pressure, density, entropy), magnetic pressure, and normal components of velocity and magnetic field are invariant in both continuous (simple) and discontinuous (shock) AlfvCn waves.
0375.9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII SO37S-9601(97)00376-9
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S.S. Komissarov/Physics Letters A 232 (1997) 435-442
The tangential component of the magnetic field rotates by an arbitrary angle, the magnitude of which describes the wave strength. The tangential component of velocity vector also traces a circle, but in general its centre is shifted from the origin. However, it is possible to find a frame (Hoffman-Teller frame) in which the centre of velocity circle coincides with the origin. We shall see how these properties are modified in the relativistic regime. The mathematical theory of RMHD is now reasonably well understood. The most important results can be found in Refs. [ 4,5] , the latter being the most comprehensive monograph on this subject. Since we shall be making extensive use of the results described in Ref. [ 51 we shall adopt its notation wherever possible.
2. Basic equations 2.1. Evolution
equations of RMHD
For a uniform chemical composition, the equations of RMHD can be written as the following covariant quasi-linear conservation laws [ 651, ~3Tap =O,
(1)
a:F*mp = 0,
(2)
J&u”)
(3)
= 0,
run from 1 to 3 and the signature of the space-time is (- +++). From Eq. (6) we can see that in the fluid rest frame b” reduces to the three-vector of the magnetic field b” = (0,B). 2.2. Constraints Because F*O” G 0, the time component of (2) is not in fact an evolution equation but a differential constraint equation, aiF*” = 0
or
V. B=O,
(8)
which describes the absence of magnetic monopoles. As it is well known (8) may be regarded as a constraint on the Cauchy hypersurface. Indeed, in this case the space components of (2) ensure (8) for the whole space-time. The other two constraints are algebraic u a u(I=-1,
(9)
u,ba = 0.
(10)
If we use these constraints and the equation of state, then Eqs. ( 1 )-( 3) for a one-dimensional motion along the x-axis are reduced to a system of seven differential equations for seven unknowns (e.g., B,,B,,uX,u?,uZ,P,andp).
where 3. Covariant Tap = (w + b2)u”uP + (p + +b2)g@ - tf=bp is the energy-momentum fluid enthalpy, pressure the metric tensor.
equations of Alfvin waves
(4)
tensor. w, p and ua are the and four-velocity and gap is
3.1. Wave vector Suppose that the position of the wavefront is given by
F *@ = b”uP _ bpu”
(5)
is the dual tensor of electromagnetic field in the MHDapproximation where bP is the four-vector of magnetic field. The usual three-vectors of magnetic field, Bi, and electric field, Ei, are related to b” by B.I = biuO _ Ui@ , Ei = vijkduk,
b” = u’Bi,
(6) (7)
where is the Levi-Civita alternating symbol. We assume that greek indexes run from 0 to 3, latin ones vijk
4(x”)
=o.
Let A be the wave speed and II be the unit three-vector in the direction of propagation of the wave front in the local Lorentz frame, I:. Note that this frame always has a locally Minkowski metric even in a curved spacetime. Define a covariant vector & such that in Z 4(I = (-A,n). Obviously, & is the wave vector (a&) normalized to 1 - A2. It may therefore be called a Z-related wave
S.S. Komissarov/Physics
vector. Introduce in 2 5a = (-l,O),
the unit vectors ta and ca such that
t
= (O,n).
One can see that ta is the covariant four-velocity of the frame ZZ, whereas ca can be called a C-related wave direction vector since it reduces to n in this frame. It is easy to check that (11)
4% = A5n + 3a, tata
= -1,
&ly
= 1,
I$l”
= 0.
(12)
Letters A 232 (1997) 435-442
431
PDEs. To overcome this difficulty Anile [ 51 used the constraint equations (8)-( 10) to construct the three missing PDEs. The result is exactly ten independent PDEs in covariant form, which are assumed (but not proven) to be equivalent to the RMHD equations together with the constraints. An inevitable side-effect of such an approach is the introduction of three unphysical waves. In the Anile’s model these consist of one advective wave propagating at the plasma speed and two pseudo-AlfvCn waves propagating at the same speed as the AlfvCn waves. These speeds are the solutions of the equation
This X-related wave vector was introduced by Friedrichs [ 7,8] in order to formulate the definition of quasi-linear hyperbolic systems in a covariant framework. Anile [ 51 used it to analyse systematically the equations of RMHD.
where & = w + b*, a = ua&, f3 = b”&. The corresponding right eigenvectors of both Alfven and pseudo-Alfven waves have the form
3.2. Simple Alfikn
I=
waves
&a* - t3* = 0,
( In order to use this covariant approach one has to reduce the system under consideration into the following quasi-linear form, Arc?&
= f’(U),
(13)
where U is the vector of N unknowns and the A? is (N x N x 4) matrix (I, J = 1,. . . , N). The phase speeds ( A) of the permitted waves relative to the frame C with the speed ,!Ja in the direction 4’a are then given by the (real) eigenvalues of the following eigenvalue problem in a covariant form, Ay&lJ
= 0.
(14)
d”&
= 0,
(16)
>
where d” satisfy the following
constraints,
d”b, = 0.
Since the right eigenvectors determine the variation of independent variables along simple waves (see, e.g., Ref. [9]) SULxl, it would be useful to separate the eigenvectors for the Alfven and pseudo-Alfven waves. This can be easily done by using the constraint equations. From (9) it follows that for the physical Alfven waves we must have dau, 0; (Su)“u,
( Afni)ZJ
Therefore,
To make full use of the power of the tensor calculus the system ( 13) has to be in a coordinate-free form. Therefore, U, and hence the eigenvectors I, should be constructed of scalars and four-vectors and the matrix A of four-tensors. The natural choice of U in RMHD involves any two thermodynamic scalars plus Us and 6”. For example, Anile [5] used U = (u”,bB,p,s), where s is the proper entropy per unit mass. This gives N = 10. However, as we have seen in Section 2, the system ( 1) -( 3) has only seven evolution
)
d~,%i~,O,O a
In the frame Z this becomes the familiar = AA$OIJ.
(15)
= 0.
d” = ~a.PY*~puyb~,
(17)
where #@y” is the Levi-Civita alternating tensor. It is easy to check that (17) provides the correct AlfvCn eigenvectors in the non-relativistic limit. It is worth mentioning that Anile’s model is not unique. Van Putten [ lo] derived a different model of RMHD in constraint-free form. Obviously, this model also provides three unphysical waves. It is not difficult to apply the same analysis as in Ref. [ 51 to van Putten’s
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S.S. Komissarov/Physics Leiiers A 232 (1997) 435-442
model and to show that instead of two pseudo-Alfven waves it has two waves propagating with the speed of light in vacuum. However, the right eigenvectors of the AlfvCn waves are still given by (16) and (17). From (16) it follows that gas pressure and entropy remain constant in a simple Alfvtn wave: 6p=O,
&=O,
Simple waves are the solutions which depend only on the wave phase 4 = &Xa = -A(#)xO
p” = P”(4)
+ nix’.
This turns (25) into
(18)
and this must therefore also hold for any thermodynamic variable. In addition, from (17) it follows that magnetic pressure does not vary either. Indeed,
and, since d,i/dd
Sb= = 2b,(Sb)a
These are exactly the shock equations for the system (25) in the covariant form if A is the shock speed. Therefore, AlfvCn shocks must satisfy the same Eqs. (22)-(24) as AlfvCn simple waves. Indeed, Lichnerowicz [ 41 derived (22) -( 24) directly from the shock equations.
From (15)-(
0: 2abUda a
= 0.
(19)
19) one gets
8(&a= - B2) = 6A [2(laua
- Bbn)ta]
= 0.
[~4Jal
= 0, we get
= 0.
Since ca is arbitrary, this means that 6A = 0.
(20)
Now, from ( 17) one can easily show that 6a =0,
st3=0.
Thus, if the states “+” and “-” simple Alfven wave then
(21) are connected
via a
[P] = [p] = [b=] = [A] =O,
(22)
[a] = [B] =O,
(23)
[u”] = ;[b”],
(24)
where [f]
= f+ - f_.
3.3. Alfv&
shocks
4. Components Alfvh waves
of fields b”, u”, &, and Ei in
4. I. Wave frame As we have pointed out in Section 1, Newtonian AlfvCn waves leave the normal components of magnetic field and velocity unchanged, whereas the tips of the tangential components trace circles, the centre of the magnetic circle being always located at the origin. To compare this with the relativistic case we first study Eqs. (23)) (24) in the wave frame C where A = 0 and the x-axis is normal to the wave front (Obviously, such a frame is not unique.) In this frame a = ux, B = b”, and (23), (24) become
[u’] =0,
[bX] =O,
(26)
[u”l = ,y[b*l,
The fact that the Alfven speed does not change in a simple Alfven wave means that AlfvCn waves are linearly degenerate, just as they are in the Newtonian case. AlfvCn shocks cannot therefore be produced by steepening but only by discontinuous initial conditions. Moreover, this means that the jump equations for AlfvCn shocks and simple waves must be the same [ 11,5]. Indeed, the equations of RMHD have the form
where x = uX/bX. (26) gives us a physical interpretation of (23)) the normal components of the fields U* and b” in the wave frame are not changed by AlfvCn waves. Of course, the normal component of field B is invariant in any frame due to (8). From (7), (27) it follows that
&P" = 0.
SE, = (u?’ - ,yb’)6bZ
(25)
(27)
- (uz - ,ybZ)GbY.
(28)
S.S. Komissarov/Physics Letters A 232 (1997) 435-442
439
where d = b2 - 6:. Finally, substituting (30) into (32) equation of the locus of (b,, 6, ) , all$
+ (a12 + a2l)b.A
+ ad(
we obtain
the
+ (013 + w)b)
+ (a23 + ad b, + a33 = 0,
(33)
where a22= 1 -a:,
all = 1 -a;,
-2.01
. 0.0
2.0
4.0
6.0
I
Fig. 1.The variations of the normal components of Ei and D; in the wave frame. B is the phase of the tangential component of B;. On the “-” side the plasma parameters are p_ = 1,p_ = 1 ( y = 5/3), ua_ =(1.48,1.0,0.3,0.3),bn_ =-(3.39,3.81,2.0,2.0). E, is not invariant (see Fig. l), except for the special case (see Section 4.2) when z&/b’ = x. On the contrary, using (7,27) one obtains that Thus,
[E,]
= [&I
=o.
(29)
Therefore, Alfvtn waves are not really quite transverse. Eqs. (27) allow us to determine the behaviour of the tangential components of the fields. Now we assume that the state “-” is known and the state “+” is to be determined (For brevity’s sake we shall normally omit the index “+” in what follows.) From (27) one has
a33=-(c2+d),
a12=a21=-ayaz,
a13 = ajl = -cay,
a23 = a32 = -ca,.
(33) is analyzed In order signs of
the general equation for a conic and can be by standard methods (see, e.g., Ref. [ 131). to classify the conic one has to determine the its invariants,
1 = att +a22,
D = alla22
with respect to rotations calculation we get
-at2a2t9
A = det( aii>,
and translations.
By direct
z=l+D, D=
(34)
b2 - b2u2 ’ B; ”
(35)
From (15) it follows that in the wave frame bx2 x --. U2E
Substituting
(37) this into (35), one can see that
lla = u” +x[b”l. Substituting
this into ( 10) gives
b” = a,bv + a,b’ + c,
(30)
where ay =
u:
-xb’
=-
uo_ - xbt
c= UO
xb2 - ,.yb:
=
The third equation
E, B,’
a _ u’ -xE ’
u! -,yb!
Fb’. x
=--
Ev
A=b:(bf-b2(l+u:)) B;
Bx’
From Proposition
(31)
in (22) now gives
-b; + b; + b; = d,
and therefore I > 0. Finally,
(36) can be reduced to
’ 2.3 in Ref. [ 51, it follows that
B2 b2 2 a2+G’ where G = &c#“. In the wave frame this becomes
(32)
b; 6 b2(1 +u;),
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S.S. Komissarov/Physics
Letters A 232 (1997) 435-442
which means that A cannot be positive. Since D > 0, I > 0 and A < 0 the conic (33) is an ellipse. Direct calculation shows that ( 1) The eccentricity is e = (1 - D’)‘/‘;
(38)
(2) The minor and major axes are respectively
(3) The angle fl between y-axis is
the major axis and the
a, + = arctan - = R + z 2’ a, where gential (4) major
(40)
R is the angle between the y-axis and the tancomponent of E; The centre of the ellipse is shifted along the axis from the origin to the point
(b:,bE)
= (i)
(a,,a,).
(41)
Eqs. (27) show that the locus of (up, ui) is obtained from the locus of (by, bZ) by a combination of translation and isotropic scaling. Therefore, the locus of ( uY, uz ) is also an ellipse with the same eccentricity and orientation of the principal axes as the b-ellipse (see Fig. 2). Its centre is shifted along its major axis from the origin to the point (u:,u:>
= [ix+
(u!
-,yb!_)]
(+,a,).
(42)
The length of its major axis differs from that of the b-ellipse by the factor of x. It can be seen that, since U$+Ui2 varies across the wave, so does u” and hence u, (see Fig. 1). Somewhat surprisingly, this tells us that in the rest frame of the undisturbed plasma the post-wave state may have non-zero normal component of velocity. Finally, let us determine the locus of the points (B,,B,). From (6), (27), (30) we have Bi=
(
s
>
[b’-ai(a,b?‘+a,bZ+c)],
(43)
where i = y, z . Since (43) is a linear transformation, (B,, B, ) must also trace an ellipse. We can get its equation by substituting for 6’ from (43) into (33) 51 I B; + 2iinByBz + ii22B; + ii33 =
0,
(44)
1
Y
5.0
Fig. 2. The ellipses of the tangential components of b’,u’, and Bi in the wave frame for the same parameters as in Fig. 1. The continuous curve shows the b-ellipse, the dotted curve shows the u-ellipse, and the dashed curve shows the E-ellipse. The ~ITOW shows the tangential component of electric field which is invariant in the wave frame.
where 511 = 1 -al,
2122= 1 -a;,
a12 = -a,a,,
ii33 = - (B,/b’)’
(c2 + dD).
From this we can deduce the following properties of the B-ellipse: ( 1) Eq. (44) has no linear terms and the centre of the B-ellipse therefore coincides with the origin; (2) The invariants of (44) are f=r,
D=D,
A=AD
and the B-ellipse therefore has the same eccentricity as the b-ellipse but its major axis differs by the factor of fiIB,/bXI; (3) Since iill = a22, 222 = all, and 512 = a12 the major axis of the B-ellipse is parallel to the minor axis of the b-ellipse. 4.2. Hojftnann-Teller frame Hoffmann and Teller [ 121 have shown that the treatment of oblique magnetohydrodynamic shocks is
S.S. Komissarov/Physics
Letters A 232 (1997) 43.5-442
441
greatly simplified in the shock frame where magnetic field B is parallel to the fluid velocity. Such a frame exists only if
‘4 < 1’
cos’
where @ is the angle between the wave normal n and B and A is the wave speed in the rest frame of the fluid on any side of the shock. For Alfven waves
BX2 A2 = ___ B2 + w and this condition is always satisfied. In the Hoffmann-Teller frame the electric field vanishes on the both sides of the wave. As the result uY = a, = 0 and the b-, u-, and B-ellipses degenerate into circles centred on the origin. Finally, in this frame [v,] =o. One can see that in the Hoffmann-Teller frame relativistic AlfvCn waves behave exactly like Newtonian ones. 4.3. Laboratory frame Finally, let us consider the behaviour of the field components in the frame 2’ moving relative to Z with velocity B, in x-direction. In this frame not only does v,’ vary, but so does u”. Indeed, the Lorentz transformation for ux gives u XI = iiO( uJ -
ii,iP)
Since in the wave frame uX_= u: but in general u!! # u”, we find that in general [ fP]
# 0.
A similar argument
shows that in general
-5.01 -5.0
I 0.0
y
5.0
Fig. 3. The ellipses of electric and magnetic field in the laboratory frame. The flow parameters are the same as in Fig. 1 but the reference frame is now moving along the x-axis with the velocity of the plasma on the “-” side. The dashed curve shows the B-ellipse and the continuous one shows the E-ellipse.
Using (6)) (27) the Lorentz transformations of the tangential components of B can be reduced to the following, Bi’ = GoBi+ ii’B,ai. Since this is a combination of isotropic scaling and translation, the locus of (By’, B,‘) is still an ellipse with the same orientation and eccentricity as the Bellipse in the wave frame, but its centre is now shifted from the origin along the major axis of the b-ellipse (see Fig. 3) _Thus, in any reference frame the centres of b-, u-, and B-ellipses and the origin lie on a straight line at an angle # to the y-axis. The Lorentz transformations of the tangential components of E are E,’ = ii”E, - i?B,.,
[b”‘] # 0.
Ey’ = ii”Ey - PB,,
This strengthens the point that Alfvtn waves are not really transverse. In the laboratory frame it is only the three-vector Bi which has invariant normal component (because of (8)). Since the Lorentz transformations do not change the tangential components of four-vectors, all the above conclusions about the properties of the tangential components of b’ and ui are valid for any inertial frame.
Thus, in the laboratory frame the tangential component of E-vector is no longer invariant along Alfvtn wave. The locus of ( Ev’, E,‘) is obtained from the locus of (B,, B, ) via rotation by 7r/2, isotropic scaling, and translation in the direction given by (ET, E, ) _ Therefore, this is an ellipse with the same eccentricity and the position angle as b-ellipse. Its centre is also shifted from the origin but this time along the minor axis of
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S.S. Komissarov/Physics Leiters A 232 (1997) 435-442
the b-ellipse. Fig. 3 shows the B’- and E’-ellipses the same wave as in Fig. 2 and CX= u,_.
for
5. Conclusions In RMHD both the continuous and discontinuous AlfvCn waves have the following properties: - Just as in Newtonian MHD, the thermodynamic parameters (e.g., pressure and proper mass-density), the magnetic pressure, and the wave speed are invariant across the waves. -In the general wave frame, the normal components of the fields ui, b’, and Bi are invariant, but the ones of Ui and Ei are not. In the Hoffman-Teller wave frame the normal components of all vector fields are invariant. In frames moving relative to the wave front only the normal component of Bi remains invariant in general. This is in contrast to the Newtonian case where the normal component of Oi is invariant whatever the reference frame. - In the general wave frame the tangential components of the vector fields b’, ui, and Bi trace ellipses which degenerate into circles in the Newtonian limit. These ellipses have the same eccentricity. The major axis of the b-ellipse coincides with the major axis of the u-ellipse and the minor axis of the B-ellipse and goes through the origin. The B-ellipse is centred on the origin but the centres of the b- and u-ellipses are shifted from the origin along the major axis of the bellipse. The tangential component of Ei is invariant. In the Hoffmann-Teller frame the electric field vanishes and all these ellipses degenerate into circles centred on
the origin. In frames moving relative to the wave front the tangential component of Ei also traces an ellipse of the same eccentricity as the other ellipses. The centres of the b-, u-, and B-ellipses are shifted from the origin along the major axis of the b-ellipse, but the centre of the E-ellipse is shifted along the minor axis of the b-ellipse.
11 A.H. Bridle, Observations
of energy transport, in: Energy Transport in Radio Galaxies and Quasars, eds. PE. Hardee, A.H. Bridle and J.A. Census, ASP Conf. Ser. 100 (1996) 383-394. 121 P.J. Wiita, The production of jets and their relation to active galactic nuclei, in: Beams and Jets in Astrophysics, ed. PA. Hughes (Cambridge Univ. Press, Cambridge, 1991) pp. 379427. A. Jeffrey, Magnetohydrodynamics (Oliver and Boyd, Edinburgh. 1966). A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics (Benjamin, New York, 1967). A.M. Anile, Relativistic Fluids and Magneto-Fluids (Cambridge Univ. Press, Cambridge, 1989) 61 G. Dixon, Special Relativity, the Foundation of Macroscopic Physics (Cambridge Univ. Press, Cambridge, 1978). 71 K.O. Friedrichs, Comm. Pure Appl. Math. 7 (1954) 345. 8 1 K.O. Friedrichs, Comm. Pure Appl. Math. 27 (1954) 749. 9 I H. Cabannes, Theoretical Magnetohydrodynamics (Academic Press, New York, 1970). 1101 M.H.P.M. van Putten, Comm. Math. ed. Liguori, 1982) 169. [ 121F. de Hoffmann and E. Teller, Phys. [ 131 E.M. Hartley, Cartesian Geometry of Univ. Press, Cambridge, 1960).
I I1 I G. Boillat, in: Wave Propagation,
Phys. 141 ( 1991) 63. G. Ferrarese (Naples: Rev. 80 ( 1950) 692. the Plane (Cambridge
PHYSICS LETTERS A
Physics Letters A 232 (1997) 435-442
ELSEVIER
On the properties of Alfv6n waves in relativistic magnetohydrodynamics S.S.Komissarov l Department of Applied Mathematics, University of Leeds, Leeds L..Y29J7: UK Astrospace Centre. Lebedev Physical Institute, Lininsky Prospect 53, Moscow B-333, 177924 Russian Federation
Received 18 February 1997; accepted for publication 24 April 1997 Communicated by M. Porkolab
Abstract Simple AlfvCn waves and Alfven shocks are considered in the framework of relativistic magnetohydrodynamics. It is found that the tangential components of vector fields trace ellipses instead of the circles of Newtonian MHD. Their properties are studied in the general wave frame, Hoffmann-Teller wave frame, and the general laboratory frame. @ 1997 Elsevier Science B.V. Keywords: Relativistic magnetohydrodynamics; AlfvCn waves
1. Introduction It is now well established that active galactic nuclei ( AGN) can produce powerful relativistic jets of plasma with large Lorentz factors (see, e.g., Ref. [ 1 ] ) . These jets are sources of synchrotron emission, which indicates the presence of significant magnetic fields. Various models of the production of these relativistic jets have been proposed (see, e.g., Ref. [2] ). Although it is extremely difficult to test these models observationally the most plausible ones include strong magnetic fields as a key ingredient. All these factors make relativistic magnetohydrodynamics (RMHD) an essential theoretical tool in studying AGN. Even in Newtonian MHD analytical methods are rarely sufficient and one therefore has to resort to numerical calculations. It is even more difficult to find
’ E-mail: [email protected].
analytic solutions to the equations of RMHD. It is, however, important to find exact solutions that can be used to test computer codes. For such hyperbolic systems as RMHD these solutions should include all permitted types of shocks and simple waves. In this paper we consider the properties of relativistic Alfvtn waves. Quite apart from this, AlfvCn waves are also interesting as particular solutions to the equations of magnetohydrodynamics. It is well known that the Newtonian limit is degenerate, which means that the properties of such solutions in the relativistic theory can be dramatically different from the Newtonian ones. The properties of Newtonian AlfvCn waves can be summarized as follows (see, e.g., Ref. [ 31). The thermodynamic variables (e.g., pressure, density, entropy), magnetic pressure, and normal components of velocity and magnetic field are invariant in both continuous (simple) and discontinuous (shock) AlfvCn waves.
0375.9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII SO37S-9601(97)00376-9
436
S.S. Komissarov/Physics Letters A 232 (1997) 435-442
The tangential component of the magnetic field rotates by an arbitrary angle, the magnitude of which describes the wave strength. The tangential component of velocity vector also traces a circle, but in general its centre is shifted from the origin. However, it is possible to find a frame (Hoffman-Teller frame) in which the centre of velocity circle coincides with the origin. We shall see how these properties are modified in the relativistic regime. The mathematical theory of RMHD is now reasonably well understood. The most important results can be found in Refs. [ 4,5] , the latter being the most comprehensive monograph on this subject. Since we shall be making extensive use of the results described in Ref. [ 51 we shall adopt its notation wherever possible.
2. Basic equations 2.1. Evolution
equations of RMHD
For a uniform chemical composition, the equations of RMHD can be written as the following covariant quasi-linear conservation laws [ 651, ~3Tap =O,
(1)
a:F*mp = 0,
(2)
J&u”)
(3)
= 0,
run from 1 to 3 and the signature of the space-time is (- +++). From Eq. (6) we can see that in the fluid rest frame b” reduces to the three-vector of the magnetic field b” = (0,B). 2.2. Constraints Because F*O” G 0, the time component of (2) is not in fact an evolution equation but a differential constraint equation, aiF*” = 0
or
V. B=O,
(8)
which describes the absence of magnetic monopoles. As it is well known (8) may be regarded as a constraint on the Cauchy hypersurface. Indeed, in this case the space components of (2) ensure (8) for the whole space-time. The other two constraints are algebraic u a u(I=-1,
(9)
u,ba = 0.
(10)
If we use these constraints and the equation of state, then Eqs. ( 1 )-( 3) for a one-dimensional motion along the x-axis are reduced to a system of seven differential equations for seven unknowns (e.g., B,,B,,uX,u?,uZ,P,andp).
where 3. Covariant Tap = (w + b2)u”uP + (p + +b2)g@ - tf=bp is the energy-momentum fluid enthalpy, pressure the metric tensor.
equations of Alfvin waves
(4)
tensor. w, p and ua are the and four-velocity and gap is
3.1. Wave vector Suppose that the position of the wavefront is given by
F *@ = b”uP _ bpu”
(5)
is the dual tensor of electromagnetic field in the MHDapproximation where bP is the four-vector of magnetic field. The usual three-vectors of magnetic field, Bi, and electric field, Ei, are related to b” by B.I = biuO _ Ui@ , Ei = vijkduk,
b” = u’Bi,
(6) (7)
where is the Levi-Civita alternating symbol. We assume that greek indexes run from 0 to 3, latin ones vijk
4(x”)
=o.
Let A be the wave speed and II be the unit three-vector in the direction of propagation of the wave front in the local Lorentz frame, I:. Note that this frame always has a locally Minkowski metric even in a curved spacetime. Define a covariant vector & such that in Z 4(I = (-A,n). Obviously, & is the wave vector (a&) normalized to 1 - A2. It may therefore be called a Z-related wave
S.S. Komissarov/Physics
vector. Introduce in 2 5a = (-l,O),
the unit vectors ta and ca such that
t
= (O,n).
One can see that ta is the covariant four-velocity of the frame ZZ, whereas ca can be called a C-related wave direction vector since it reduces to n in this frame. It is easy to check that (11)
4% = A5n + 3a, tata
= -1,
&ly
= 1,
I$l”
= 0.
(12)
Letters A 232 (1997) 435-442
431
PDEs. To overcome this difficulty Anile [ 51 used the constraint equations (8)-( 10) to construct the three missing PDEs. The result is exactly ten independent PDEs in covariant form, which are assumed (but not proven) to be equivalent to the RMHD equations together with the constraints. An inevitable side-effect of such an approach is the introduction of three unphysical waves. In the Anile’s model these consist of one advective wave propagating at the plasma speed and two pseudo-AlfvCn waves propagating at the same speed as the AlfvCn waves. These speeds are the solutions of the equation
This X-related wave vector was introduced by Friedrichs [ 7,8] in order to formulate the definition of quasi-linear hyperbolic systems in a covariant framework. Anile [ 51 used it to analyse systematically the equations of RMHD.
where & = w + b*, a = ua&, f3 = b”&. The corresponding right eigenvectors of both Alfven and pseudo-Alfven waves have the form
3.2. Simple Alfikn
I=
waves
&a* - t3* = 0,
( In order to use this covariant approach one has to reduce the system under consideration into the following quasi-linear form, Arc?&
= f’(U),
(13)
where U is the vector of N unknowns and the A? is (N x N x 4) matrix (I, J = 1,. . . , N). The phase speeds ( A) of the permitted waves relative to the frame C with the speed ,!Ja in the direction 4’a are then given by the (real) eigenvalues of the following eigenvalue problem in a covariant form, Ay&lJ
= 0.
(14)
d”&
= 0,
(16)
>
where d” satisfy the following
constraints,
d”b, = 0.
Since the right eigenvectors determine the variation of independent variables along simple waves (see, e.g., Ref. [9]) SULxl, it would be useful to separate the eigenvectors for the Alfven and pseudo-Alfven waves. This can be easily done by using the constraint equations. From (9) it follows that for the physical Alfven waves we must have dau, 0; (Su)“u,
( Afni)ZJ
Therefore,
To make full use of the power of the tensor calculus the system ( 13) has to be in a coordinate-free form. Therefore, U, and hence the eigenvectors I, should be constructed of scalars and four-vectors and the matrix A of four-tensors. The natural choice of U in RMHD involves any two thermodynamic scalars plus Us and 6”. For example, Anile [5] used U = (u”,bB,p,s), where s is the proper entropy per unit mass. This gives N = 10. However, as we have seen in Section 2, the system ( 1) -( 3) has only seven evolution
)
d~,%i~,O,O a
In the frame Z this becomes the familiar = AA$OIJ.
(15)
= 0.
d” = ~a.PY*~puyb~,
(17)
where #@y” is the Levi-Civita alternating tensor. It is easy to check that (17) provides the correct AlfvCn eigenvectors in the non-relativistic limit. It is worth mentioning that Anile’s model is not unique. Van Putten [ lo] derived a different model of RMHD in constraint-free form. Obviously, this model also provides three unphysical waves. It is not difficult to apply the same analysis as in Ref. [ 51 to van Putten’s
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S.S. Komissarov/Physics Leiiers A 232 (1997) 435-442
model and to show that instead of two pseudo-Alfven waves it has two waves propagating with the speed of light in vacuum. However, the right eigenvectors of the AlfvCn waves are still given by (16) and (17). From (16) it follows that gas pressure and entropy remain constant in a simple Alfvtn wave: 6p=O,
&=O,
Simple waves are the solutions which depend only on the wave phase 4 = &Xa = -A(#)xO
p” = P”(4)
+ nix’.
This turns (25) into
(18)
and this must therefore also hold for any thermodynamic variable. In addition, from (17) it follows that magnetic pressure does not vary either. Indeed,
and, since d,i/dd
Sb= = 2b,(Sb)a
These are exactly the shock equations for the system (25) in the covariant form if A is the shock speed. Therefore, AlfvCn shocks must satisfy the same Eqs. (22)-(24) as AlfvCn simple waves. Indeed, Lichnerowicz [ 41 derived (22) -( 24) directly from the shock equations.
From (15)-(
0: 2abUda a
= 0.
(19)
19) one gets
8(&a= - B2) = 6A [2(laua
- Bbn)ta]
= 0.
[~4Jal
= 0, we get
= 0.
Since ca is arbitrary, this means that 6A = 0.
(20)
Now, from ( 17) one can easily show that 6a =0,
st3=0.
Thus, if the states “+” and “-” simple Alfven wave then
(21) are connected
via a
[P] = [p] = [b=] = [A] =O,
(22)
[a] = [B] =O,
(23)
[u”] = ;[b”],
(24)
where [f]
= f+ - f_.
3.3. Alfv&
shocks
4. Components Alfvh waves
of fields b”, u”, &, and Ei in
4. I. Wave frame As we have pointed out in Section 1, Newtonian AlfvCn waves leave the normal components of magnetic field and velocity unchanged, whereas the tips of the tangential components trace circles, the centre of the magnetic circle being always located at the origin. To compare this with the relativistic case we first study Eqs. (23)) (24) in the wave frame C where A = 0 and the x-axis is normal to the wave front (Obviously, such a frame is not unique.) In this frame a = ux, B = b”, and (23), (24) become
[u’] =0,
[bX] =O,
(26)
[u”l = ,y[b*l,
The fact that the Alfven speed does not change in a simple Alfven wave means that AlfvCn waves are linearly degenerate, just as they are in the Newtonian case. AlfvCn shocks cannot therefore be produced by steepening but only by discontinuous initial conditions. Moreover, this means that the jump equations for AlfvCn shocks and simple waves must be the same [ 11,5]. Indeed, the equations of RMHD have the form
where x = uX/bX. (26) gives us a physical interpretation of (23)) the normal components of the fields U* and b” in the wave frame are not changed by AlfvCn waves. Of course, the normal component of field B is invariant in any frame due to (8). From (7), (27) it follows that
&P" = 0.
SE, = (u?’ - ,yb’)6bZ
(25)
(27)
- (uz - ,ybZ)GbY.
(28)
S.S. Komissarov/Physics Letters A 232 (1997) 435-442
439
where d = b2 - 6:. Finally, substituting (30) into (32) equation of the locus of (b,, 6, ) , all$
+ (a12 + a2l)b.A
+ ad(
we obtain
the
+ (013 + w)b)
+ (a23 + ad b, + a33 = 0,
(33)
where a22= 1 -a:,
all = 1 -a;,
-2.01
. 0.0
2.0
4.0
6.0
I
Fig. 1.The variations of the normal components of Ei and D; in the wave frame. B is the phase of the tangential component of B;. On the “-” side the plasma parameters are p_ = 1,p_ = 1 ( y = 5/3), ua_ =(1.48,1.0,0.3,0.3),bn_ =-(3.39,3.81,2.0,2.0). E, is not invariant (see Fig. l), except for the special case (see Section 4.2) when z&/b’ = x. On the contrary, using (7,27) one obtains that Thus,
[E,]
= [&I
=o.
(29)
Therefore, Alfvtn waves are not really quite transverse. Eqs. (27) allow us to determine the behaviour of the tangential components of the fields. Now we assume that the state “-” is known and the state “+” is to be determined (For brevity’s sake we shall normally omit the index “+” in what follows.) From (27) one has
a33=-(c2+d),
a12=a21=-ayaz,
a13 = ajl = -cay,
a23 = a32 = -ca,.
(33) is analyzed In order signs of
the general equation for a conic and can be by standard methods (see, e.g., Ref. [ 131). to classify the conic one has to determine the its invariants,
1 = att +a22,
D = alla22
with respect to rotations calculation we get
-at2a2t9
A = det( aii>,
and translations.
By direct
z=l+D, D=
(34)
b2 - b2u2 ’ B; ”
(35)
From (15) it follows that in the wave frame bx2 x --. U2E
Substituting
(37) this into (35), one can see that
lla = u” +x[b”l. Substituting
this into ( 10) gives
b” = a,bv + a,b’ + c,
(30)
where ay =
u:
-xb’
=-
uo_ - xbt
c= UO
xb2 - ,.yb:
=
The third equation
E, B,’
a _ u’ -xE ’
u! -,yb!
Fb’. x
=--
Ev
A=b:(bf-b2(l+u:)) B;
Bx’
From Proposition
(31)
in (22) now gives
-b; + b; + b; = d,
and therefore I > 0. Finally,
(36) can be reduced to
’ 2.3 in Ref. [ 51, it follows that
B2 b2 2 a2+G’ where G = &c#“. In the wave frame this becomes
(32)
b; 6 b2(1 +u;),
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S.S. Komissarov/Physics
Letters A 232 (1997) 435-442
which means that A cannot be positive. Since D > 0, I > 0 and A < 0 the conic (33) is an ellipse. Direct calculation shows that ( 1) The eccentricity is e = (1 - D’)‘/‘;
(38)
(2) The minor and major axes are respectively
(3) The angle fl between y-axis is
the major axis and the
a, + = arctan - = R + z 2’ a, where gential (4) major
(40)
R is the angle between the y-axis and the tancomponent of E; The centre of the ellipse is shifted along the axis from the origin to the point
(b:,bE)
= (i)
(a,,a,).
(41)
Eqs. (27) show that the locus of (up, ui) is obtained from the locus of (by, bZ) by a combination of translation and isotropic scaling. Therefore, the locus of ( uY, uz ) is also an ellipse with the same eccentricity and orientation of the principal axes as the b-ellipse (see Fig. 2). Its centre is shifted along its major axis from the origin to the point (u:,u:>
= [ix+
(u!
-,yb!_)]
(+,a,).
(42)
The length of its major axis differs from that of the b-ellipse by the factor of x. It can be seen that, since U$+Ui2 varies across the wave, so does u” and hence u, (see Fig. 1). Somewhat surprisingly, this tells us that in the rest frame of the undisturbed plasma the post-wave state may have non-zero normal component of velocity. Finally, let us determine the locus of the points (B,,B,). From (6), (27), (30) we have Bi=
(
s
>
[b’-ai(a,b?‘+a,bZ+c)],
(43)
where i = y, z . Since (43) is a linear transformation, (B,, B, ) must also trace an ellipse. We can get its equation by substituting for 6’ from (43) into (33) 51 I B; + 2iinByBz + ii22B; + ii33 =
0,
(44)
1
Y
5.0
Fig. 2. The ellipses of the tangential components of b’,u’, and Bi in the wave frame for the same parameters as in Fig. 1. The continuous curve shows the b-ellipse, the dotted curve shows the u-ellipse, and the dashed curve shows the E-ellipse. The ~ITOW shows the tangential component of electric field which is invariant in the wave frame.
where 511 = 1 -al,
2122= 1 -a;,
a12 = -a,a,,
ii33 = - (B,/b’)’
(c2 + dD).
From this we can deduce the following properties of the B-ellipse: ( 1) Eq. (44) has no linear terms and the centre of the B-ellipse therefore coincides with the origin; (2) The invariants of (44) are f=r,
D=D,
A=AD
and the B-ellipse therefore has the same eccentricity as the b-ellipse but its major axis differs by the factor of fiIB,/bXI; (3) Since iill = a22, 222 = all, and 512 = a12 the major axis of the B-ellipse is parallel to the minor axis of the b-ellipse. 4.2. Hojftnann-Teller frame Hoffmann and Teller [ 121 have shown that the treatment of oblique magnetohydrodynamic shocks is
S.S. Komissarov/Physics
Letters A 232 (1997) 43.5-442
441
greatly simplified in the shock frame where magnetic field B is parallel to the fluid velocity. Such a frame exists only if
‘4 < 1’
cos’
where @ is the angle between the wave normal n and B and A is the wave speed in the rest frame of the fluid on any side of the shock. For Alfven waves
BX2 A2 = ___ B2 + w and this condition is always satisfied. In the Hoffmann-Teller frame the electric field vanishes on the both sides of the wave. As the result uY = a, = 0 and the b-, u-, and B-ellipses degenerate into circles centred on the origin. Finally, in this frame [v,] =o. One can see that in the Hoffmann-Teller frame relativistic AlfvCn waves behave exactly like Newtonian ones. 4.3. Laboratory frame Finally, let us consider the behaviour of the field components in the frame 2’ moving relative to Z with velocity B, in x-direction. In this frame not only does v,’ vary, but so does u”. Indeed, the Lorentz transformation for ux gives u XI = iiO( uJ -
ii,iP)
Since in the wave frame uX_= u: but in general u!! # u”, we find that in general [ fP]
# 0.
A similar argument
shows that in general
-5.01 -5.0
I 0.0
y
5.0
Fig. 3. The ellipses of electric and magnetic field in the laboratory frame. The flow parameters are the same as in Fig. 1 but the reference frame is now moving along the x-axis with the velocity of the plasma on the “-” side. The dashed curve shows the B-ellipse and the continuous one shows the E-ellipse.
Using (6)) (27) the Lorentz transformations of the tangential components of B can be reduced to the following, Bi’ = GoBi+ ii’B,ai. Since this is a combination of isotropic scaling and translation, the locus of (By’, B,‘) is still an ellipse with the same orientation and eccentricity as the Bellipse in the wave frame, but its centre is now shifted from the origin along the major axis of the b-ellipse (see Fig. 3) _Thus, in any reference frame the centres of b-, u-, and B-ellipses and the origin lie on a straight line at an angle # to the y-axis. The Lorentz transformations of the tangential components of E are E,’ = ii”E, - i?B,.,
[b”‘] # 0.
Ey’ = ii”Ey - PB,,
This strengthens the point that Alfvtn waves are not really transverse. In the laboratory frame it is only the three-vector Bi which has invariant normal component (because of (8)). Since the Lorentz transformations do not change the tangential components of four-vectors, all the above conclusions about the properties of the tangential components of b’ and ui are valid for any inertial frame.
Thus, in the laboratory frame the tangential component of E-vector is no longer invariant along Alfvtn wave. The locus of ( Ev’, E,‘) is obtained from the locus of (B,, B, ) via rotation by 7r/2, isotropic scaling, and translation in the direction given by (ET, E, ) _ Therefore, this is an ellipse with the same eccentricity and the position angle as b-ellipse. Its centre is also shifted from the origin but this time along the minor axis of
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S.S. Komissarov/Physics Leiters A 232 (1997) 435-442
the b-ellipse. Fig. 3 shows the B’- and E’-ellipses the same wave as in Fig. 2 and CX= u,_.
for
5. Conclusions In RMHD both the continuous and discontinuous AlfvCn waves have the following properties: - Just as in Newtonian MHD, the thermodynamic parameters (e.g., pressure and proper mass-density), the magnetic pressure, and the wave speed are invariant across the waves. -In the general wave frame, the normal components of the fields ui, b’, and Bi are invariant, but the ones of Ui and Ei are not. In the Hoffman-Teller wave frame the normal components of all vector fields are invariant. In frames moving relative to the wave front only the normal component of Bi remains invariant in general. This is in contrast to the Newtonian case where the normal component of Oi is invariant whatever the reference frame. - In the general wave frame the tangential components of the vector fields b’, ui, and Bi trace ellipses which degenerate into circles in the Newtonian limit. These ellipses have the same eccentricity. The major axis of the b-ellipse coincides with the major axis of the u-ellipse and the minor axis of the B-ellipse and goes through the origin. The B-ellipse is centred on the origin but the centres of the b- and u-ellipses are shifted from the origin along the major axis of the bellipse. The tangential component of Ei is invariant. In the Hoffmann-Teller frame the electric field vanishes and all these ellipses degenerate into circles centred on
the origin. In frames moving relative to the wave front the tangential component of Ei also traces an ellipse of the same eccentricity as the other ellipses. The centres of the b-, u-, and B-ellipses are shifted from the origin along the major axis of the b-ellipse, but the centre of the E-ellipse is shifted along the minor axis of the b-ellipse.
11 A.H. Bridle, Observations
of energy transport, in: Energy Transport in Radio Galaxies and Quasars, eds. PE. Hardee, A.H. Bridle and J.A. Census, ASP Conf. Ser. 100 (1996) 383-394. 121 P.J. Wiita, The production of jets and their relation to active galactic nuclei, in: Beams and Jets in Astrophysics, ed. PA. Hughes (Cambridge Univ. Press, Cambridge, 1991) pp. 379427. A. Jeffrey, Magnetohydrodynamics (Oliver and Boyd, Edinburgh. 1966). A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics (Benjamin, New York, 1967). A.M. Anile, Relativistic Fluids and Magneto-Fluids (Cambridge Univ. Press, Cambridge, 1989) 61 G. Dixon, Special Relativity, the Foundation of Macroscopic Physics (Cambridge Univ. Press, Cambridge, 1978). 71 K.O. Friedrichs, Comm. Pure Appl. Math. 7 (1954) 345. 8 1 K.O. Friedrichs, Comm. Pure Appl. Math. 27 (1954) 749. 9 I H. Cabannes, Theoretical Magnetohydrodynamics (Academic Press, New York, 1970). 1101 M.H.P.M. van Putten, Comm. Math. ed. Liguori, 1982) 169. [ 121F. de Hoffmann and E. Teller, Phys. [ 131 E.M. Hartley, Cartesian Geometry of Univ. Press, Cambridge, 1960).
I I1 I G. Boillat, in: Wave Propagation,
Phys. 141 ( 1991) 63. G. Ferrarese (Naples: Rev. 80 ( 1950) 692. the Plane (Cambridge