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Rotational discontinuities in an anisotropic plasma
Planet.
Space Sci. 1971, Vol. 19, pp. 1693 to 1699.
ROTATIONAL
Department
Peraamon
Press.
Printed
in Northern
DISCONTINUITIES PLASMA
Ireland
IN AN ANISOTROPIC
P. D. HUDSON of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast, BT7 lNN, Northern Ireland (Received 11 June 1971)
Abstract-A general theory of rotational discontinuities is developed and the changes in the components of the plasma pressure, pll andp,, and in the magnetic induction, B, are found. For a given value of 1 = (pa - pL) 4n,u/B2 upstream only a limited range of downstream anisotropies are possible. If I > 0.6 upstream then isotropy is not possible downstream. Some special solutions are analysed and the identification of rotational discontinuities is the solar wind is discussed. 1. GENERAL
THEORY
Recently Hudson (1970, hereafter called paper I) and Neubauer (1970) have discussed the types of discontinuity which can occur in an anisotropic plasma such as the solar wind. They showed that in general there is a density change across rotational discontinuities and that they propagate at a modified AlfvCn velocity B.n A ’ * = (4.rrPp)1/2
(1)
Here II is the normal to the surface of the discontinuity, B is the magnetic induction, p is the plasma density and p,, and pI are the components of the plasma pressure parallel and perpendicular to the magnetic field. Rotational discontinuities differ from shocks in two essential respects: they propagate relative to both the upstream and the downstream plasma at their respective modified Alfven velocities, and the magnetic field downstream does not lie in a plane containing the upstream field and the normal to the discontinuity. In this they differ from the ‘rotational discontinuity’ discussed by Ivanov (1970) which is a special type of plasma shock which leaves the plasma density unchanged. Since the detailed mechanism of discontinuities in anisotropic plasmas is not fully understood we do not concern ourselves with it but set limit on possible solutions by considering the conservation of mass, momentum and energy. Thus p, p,, and pI contain contributions from the electrons and all the ions. To determine the downstream state we need, in general, an additional relation giving the change in p,, or pI across the discontinuity. However, if we use a classification based on the anisotropy parameter, A= (P,, - ~1)47~W
(2)
we can obtain considerable information about possible solutions. Let the subscripts 1 and 2 denote conditions upstream and downstream of the discontinuity. Then from Equations (48) and (49) of paper I we have PI1 -I- w37v
= P12 + B,2/%
P1(l - 11) =
P2U
1693
-
12).
(3) (4)
P. D. HUDSON
1694
If there is no heat flux and the internal energy density is given by pl + $p,, then we have from Equation (52) of paper I,
B2
; FM + G
(2& + I)
p2.
(5)
Solving Equations (3), (4) and (5) we obtain (3 - 4&)(1 - &)PlZ = p11(3 - 24 + 42, 2 - 513 + B12(tl, - A,)(1 - 211,- 2L2)/4np
(6)
(3 - 4&)(1 - &)B22 = B12(3 - 21, + 41r2 - 51,) + 407r/_~p,,(~,- 1,)
(7)
(3 - 4&)(1 - &)Pilz = P1r(3 - 21, - 6az2 + lo@,-
5&)
- B,2(A,(2L1 - I)(1 - 21,) + 3L2(3L2- 2))/4np.
(8)
Since pL2, plla and B22 must be positive these equations (plus similar ones in terms of pII1, and Blz) imply conditions for the existence of physical solutions. The excluded regions of the LJ, plane are shown in Fig. 1 and the conditions imposed on the ratio kl = 47r,up,,/ B12 in other regions are listed. There are no restrictions on this ratio in region 1 or on the line 1, = 1,. Using Equation (4) we see that the density changes across a rotational discontinuity by less than a factor of 2.5. Some values of (Al, 1,) are possible only when the plasma pressure is large compared to the magnetic pressure and this may be inconsistent with the magnetic orientation of the plasma. Also if 1 < 0, the plasma may suffer from the mirror instability. Both these effects depend upon the pitch angle distribution of the plasma, but as an illustration the dashed line in Fig. 1 shows the boundary of the permitted solutions if we impose the conditions pI < B2/8n,u and pL2 < p,,B2/87r,u on both sides of the discontinuity. The dash-dot line shows the boundary if we also impose the condition p,, < B2/8r,u. The normal to the plane of a rotational discontinuity is not completely determined by our conservation equations, but it lies in the plane containing B, x B, and (B, - B,) x (B, x B,). The inclination to the magnetic field must be determined by the detailed mechanism of the discontinuity, but the criteria for well-posedness (Lynn, 1970) may put limits on this angle. These are most restrictive for large negative il. If the angle is limited it may impose a minimum value on B, since B, . n = B, . n. If Al > 1, the plasma density decreases across the discontinuity and we have an expansive wave with A,. n > A,. n. If il, > I, we have a compressive wave. The change in magnetic field and plasma pressure are also linked to the sign of (A1 - 12) with (3 - 41&l - L2)(B12- Bz2>/87rp = -(Al - &)(5plr - B12(1 - 22, - 21,)/4rp)
(9)
(3 - 412)(1 - ~2)(~112 -Pact) = -(ii, - n2)(5pll(l - 22,) + B,2(2 + 211, - 32, - 4L,i1,)/47rp).
(10)
(3 - 4A2)(l - A2)(p12- pJ
=
Thus zero density change across a rotational = pL2 and Bl = B, (or if Al = Q).
g,,
discontinuity
occurs only if plll =plj2,
ROTATIONAL
DlSCONTINUITIES
IN AN ANISOTROPIC
PLASMA
1695
FIG. 1. ROTATIONAL DISCONTINUITIESARE NOT POSSIBLEIF (A,,A,) LIESIN THE SHADED AREA [(A = 47&]j - pJ~“)l.
There are also restrictions on k, = 4~,up~J& %.In regions 2,3 and 6 we must have kr < (3 211 + 411’ - 5&)/10(& - A,); in regions 2 and 6 we need /cl > (1, - A,) (21, - 2& - 1)/(3 - 21, + 41,’ - 5&); in region 5 we require k, > (222 - 41,21, - jil + 2J.J, - 21, + 3&F)/ (3 - 2& - 6&Z - Sjir + lo&&) and in region 6 we must have k, > - &. There are no restrictions on kl in region 1, but in region 7 we need kl < (5il, - 3 + 2& - 4~,z)/10(J., 2,) if 1, > I,, and k, > (d, - &)(2& + 21, - 1)/(5& - 3 + 2& - 4nzz) if ii, > I,. If PI f B’/S?rfl and *^ pb2 < .,, pI, B2/8nfi then the boundary of permitted solutions shrinks to the __ -1^^L_.l
I:-_
_-_I
1P
_
,
‘“9
ID
:r
2
P.._.l__..
?I___-“_.
I
^_
A^
At._
A ^..I.
,a,.&
I:..^
To remove the effect of variations in density we can use pi/p and p,,/p to define temperatures TL and T,, . Since the plasma is a mixture of electrons and ions TL and T,, do not necessarily correspond to the temperature of any of the components but they illustrate the redistribution of energy. Thus we have Pl(1 - &)(3 - 4&)(P,zlP, - PUlPI) = -(I, - il,)(pL12(1 + 21,) - B?(l - 2& - 2&)/4np) PI(1 - &)(3 - 4MP,,zlPz
(11)
-P,,rM = -(A, - L&J1r2(1 - 31,) + B,2(2 - 3/I, - 1,)/477/J) (12)
If 2 > ;1, > &, the average pressure ($p,, + $p,) and the average temperature decrease across the discontinuity. A necessary condition for the existence of rotational discontinuities is that the entropy downstream should be greater than that upstream. The plasma is a mixture of ions and
1696
P. D. HUDSON
electrons each of which may have a complicated velocity and pitch angle distribution and without these and a detailed knowledge of the structure of the discontinuity the entropy change can not be evaluated. However as an illustration we applied the test PL2P,,YlP2 z P~1P;,Il2IP1
(13)
which is appropriate for a single component gas which is bi-Maxwellian both ahead and behind the discontinuity. This condition can be satisfied in most of regions 2, 3 and 6 and in region 7 below 1, = &, but it cannot be satisfied in region 4, most of region 5 or in region 1 if k, 2 0.1762. However it can be satisfied throughout region 1 if k, is small enough. Although the validity of condition 13 is doubtful it implies for 1, > 0 that downstream isotropy can only be achieved if k, = P~~~~T,LL/B,~ is small. We would expect that scattering etc. would decrease the anisotropy of the plasma as it passes through the discontinuity. So instead of condition 13 we will use the condition (14) to limit the range of possible solutions. The allowed values of il, are illustrated in Fig. 2 for several values of k,. For 0.75 > 1, > 0 we see that there are two types of solution, a mildly compressive wave, and an expansive wave with near isotropy downstream. For negative jz we have only a compressive wave. 2. SPECIAL
2.1
SOLUTIONS
1 > 1, > 0.75
The only solution occurs in region 7 with 1 > 31,> 0.75 and (1, - &I < 0.025. To satisfy condition 14 we need a mildly expansive wave with 2, > R, > (10&k + 3 - 22, + 41,2)/(5 + IOk,). Condition 13 is also satisfied for most of these values. We find that pI, T,, the average pressure, and the average temperature all increase across the discontinuity whilep,,, T,, and B decrease. The density change is slight with p1 > p2 > pr(5 + lOk,)(lOk, + 2 + 4/l,) > 0.83~~ but the changes in pI, p,,, Tl and T,, can be sizeable because the factors on the left hand sides of Equations (9)-(12) are small. A small change in density and ;Z must be difficult to achieve, and the only realistic possibility is that B, is very small. This requires a mechanism which extracts energy from the magnetic field and from T,, and transfers it mainly to TL. If the change in B is large then the normal to the plane of the discontinuity must be nearly at right angles to B, since B, . n = B, . n. 2.2 L, = O-75 The only solution permitted is Iz, = 0.75. The density is unchanged across the discontinuity whilepL, p,, and B can have any change subject to Equations (2) and (3). 2.3 The expansive rotational discontinuity 0.6 > A, > 0 The exact upper bound for this type of solution is 1, = (-5k, which lies between 0.5 and 0.6.
+ 1 + (25k12 + 14k, + 1)“2)/4
Condition
14 is satisfied for positive 1, less than
ROTATIONAL
FIG. 2.
THE
RANGE
DISCONTINUITIES
OF ALLOWED THE
VALUES
ROTATIONAL
IN AN ANISOTROPIC
OF A2 FOR WHICH
DISCONTINUITY
lp,,-_P&~DECREASES
IS PLO’ITED
1697
PLASMA
ACROSS
VS. 21.
The dashed line denotes the boundary for kl = 5 the solid line for kl = 1, and the dash-dot line for k, = 0.2. In each case the line I, = 1, forms a boundary and there is a forbidden region near A1= 0. For A1 > 0.75 the curves are indistinguishable. There are four types of solution: (a) an expansive discontinuity with small density change if 1 > A, > 0.75 (b) an expansive discontinuity with near isotropy downstream (2, N_ 0) for 0.6 2 1, > 0, (c) a compressive discontinuity (1, > A,) for 0.75 > A, > 0 and (d) a compressive discontinuity if 0 > 1, > - kl. For each value of 1, the downstream pressures and magnetic field can be found using equations (6)-(g).
min [A,, (k,(3 - 54) + A,(1 - 2&))/(5k, + 2&)(2k, + l)] and the range of negative I, is limited by the requirements that i1,2(2& - 5k3(2k, + 1) + 1,(10k,2il, + k,(4i1,2 - 41, + 3) - &) + A,@,(3 - 511) + A,(1 - 21,)) > 0 and 3A,2(1 + 2k3 + i1,(2k, - lOk,il, - 2 + 21, - 411”) - k1(3 - 513 - A,(1 - 213 > 0. It will also be greatly limited by requirements for mirror stability. Condition 13 can not be satisfied except for very small values of k, so the existence of this type of solution is in doubt. T,, and p,, decrease across the discontinuity and so do the average pressure and average temperature. TL and pI usually decrease but they can increase for small k,. The magnetic field usually increases but for k, < 0.2 it can decrease by up to 6 per cent. The basic mechanism is one in which the energy released by the decrease in T,, is used to increase the flow velocity with a secondary adjustment to Tl and the magnetic energy.
1698
P. D. HUDSON
2.4 The compressive rotational discontinuity for 0.75 > AI > 0 This type of solution is always possible for A1 4 1, < (lOJ,k, + 3 - 21, + 4&e)/ (5 + lOk,). It satisfies condition 14 provided we exclude a region near 1, = 0 bounded by 1, = & and i1,(5k, + 21,)(2k, + 1) - k,(3 + 5R,) - &(I - 2il,) = 0. The magnetic field decreases across the discontinuity andp,, TL, the average temperature and the average pressure increase. This is the same type of change that occurs in slow shocks. The exact change in p,, and T,, can be found from Equations (10) and (12) but in general they increase. The density change is given by pr < pZ < pi(5 + lOk,)/(lOk, + 2 + 4kil,) < 5p1/2. The upper boundary for & corresponds to B, = 0 and values close to this may be unattainable because of Lynn’s well-posedness conditions and because the field may be too weak for the plasma to be magnetically orientated. Thus the only solution for il, = 0 is 3L2= 0. Condition 13 is satisfied everywhere except possibly for some values of I, in 0.5 < A1 < 0.75. In 0.75 > 1, 2 0.6 the compressive rotational discontinuity is the only type possible and it is characterized by T,,, > T,,,. 2.5 The compressive rotational discontinuity for 0 > ;I, > -k, This is the only solution in A1 < 0 which satisfies condition 14 and we must exclude a region near 1, = 0 bounded by the curve 3LZ2(2J,- 5k,)(2k, + 1) + 3L2[10k,21, + k,(4L12 - 4il, + 3) - A,] + Al[k,(3 - 51,) + &(I - 2L,)] = 0. For small kl this splits the region in two. The range of 1, is the same as in Section 2.4 and values close to the upper boundary may be unattainable for the reasons given there. Downstream isotropy is possible only for 0 > il, > -0.9. In practice the range of possible 1, and 1, is severely limited by the requirements for mirror
stability.
For
instance
pL2 G p,, B2/8r,u requires
1 + 2/l
+ 81, > 4k, > 1 -
41 + 81,. The density change is given by pi < p2 < pi(5 + lOk,)/(lOk, + 2 + 41,) < pi(5 + lOk3/(6k, + 2) < 5p,/2 and T,,, p,, and the average pressure increase across the discontinuity. In general, pL and T,_ increase across the discontinuity and B decreases, but we see from Equations (9) and (11) that these changes can be reversed for small k. However, the increase in B is less than 6 per cent. The mechanism is one in which the energy released by the decrease in the flow velocity is used to increase T,, with a secondary adjustment to TL and the magnetic energy. Condition 13 is satisfied at least in those areas where pI increases. 3. ROTATIONAL
DISCONTINUJTIES
IN
THE
SOLAR
WIND
The thermal energy of the ions in the solar wind is usually of the order of the magnetic energy density (Neugebauer and Snyder, 1967) andp,,/p, for the ions is usually in the range l-5 with a typical value of -2 (Hundhausen et al., 1967). For the electrons p,,/pl is less than that for the ions and is usually in the range 1.1-1.2 (Montgomery et al., 1968), but the electron temperature is 1.5-5 times that of the ions. Using this data we find that a typical value of k = 4npp,/B2 is I.0 but that it can differ from this by at least a factor of five. Figure 2 illustrates solutions for k, = 5, 1 and 0.2. A typical value of 1 = (p,, - p1)4zy/B2 is 0.4, and, although it can vary widely, it is rarely negative. Rotational discontinuities cannot propagate if L > 1, but, as the plasma then suffers from the firehose instability these values will be rare in the solar wind. Separate treatment of the ions and electrons (Kennel and Scarf, 1968) shows that an ion cyclotron resonance instability can also occur. In the solar wind this has a significant growth only if liona 2 4 but it will probably reduce the frequency of occurrence of values of il in the range 0.5-l. Since observationally p,,/pI < 5, ;I, is limited for small k, by 1, < 4k,.
ROTATIONAL
DISCONTINUITIES
IN AN
ANISOTROPIC
PLASMA
1699
A procedure for identifying rotational discontinuities is the solar wind is given in paper I. However, most of the tests are inconclusive because of experimental uncertainties and so the additional results derived in this paper are needed to differentiate between rotational discontinuities and shocks, tangential discontinuities, etc. A rotational discontinuity would be identified if it satisfied Equations (3), (4) and (5) or (6), (7) and (8), but if these can not be evaluated accurately then we must fall back on conditions that are necessary but not sufficient for the discontinuity to be rotational. The solar wind speed is much greater than the propagation velocity of a rotational discontinuity or the velocity jump across it and with most satellite observations it will be difficult to decide whether the upstream or downstream side of the rotational discontinuity has been observed first. Let us use subscript a to denote the low density side of the discontinuity and b the high density side. Then we must have 1 > pa/pb > 0.4, and plo + Ba2/8r,u =plb + Bb2/8r~. If ii, < 0.75 the average pressure and temperature increase with density but the magnetic field decreases (except for the rare cases when L, < 0 or k, < 0.2 when it can increase by up to 6 %). This distinguishes rotational discontinuities from most fast shocks (Neubauer, 1970) where the change in p and B have the same sign. The only rotational discontinuities across which we can have BalBb < 0.94 are those with il > 0.75 (Section 2.1) and with these the density change must be very small. Such large values of 1 may be rare in the solar wind and the difficulty in finding a realistic mechanism means that such rotational discontinuities will probably be uncommon. REFERENCES HUDSON,P. D. (1970). Planet. Space Sci., 18, 1611. HUNDHAUSEN, A. J., BAME,S. J. and NESS,N. V. (1967). J. peophys. Res. 72, 5265. IVANOV,K. G. (1970). Preprint. KENNEL,C. F. and SCARF,F. L., (1968). J.geophys. Res. 73,6149. LYNN, Y. M. (1970). Physics Fluids 13, 1762. MONTGOMERY, M. D. BAME,S. J. and HUNDHAUSEN, A. J. (1968). J. geophys. Res. 73,4999. NEUBAUER,F. M. (1970). Z. Physik 237,205. NEUGEBAUER, M. and SNYDER,C. W. (1967). J.geophys. Res. 72, 1823.
Space Sci. 1971, Vol. 19, pp. 1693 to 1699.
ROTATIONAL
Department
Peraamon
Press.
Printed
in Northern
DISCONTINUITIES PLASMA
Ireland
IN AN ANISOTROPIC
P. D. HUDSON of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast, BT7 lNN, Northern Ireland (Received 11 June 1971)
Abstract-A general theory of rotational discontinuities is developed and the changes in the components of the plasma pressure, pll andp,, and in the magnetic induction, B, are found. For a given value of 1 = (pa - pL) 4n,u/B2 upstream only a limited range of downstream anisotropies are possible. If I > 0.6 upstream then isotropy is not possible downstream. Some special solutions are analysed and the identification of rotational discontinuities is the solar wind is discussed. 1. GENERAL
THEORY
Recently Hudson (1970, hereafter called paper I) and Neubauer (1970) have discussed the types of discontinuity which can occur in an anisotropic plasma such as the solar wind. They showed that in general there is a density change across rotational discontinuities and that they propagate at a modified AlfvCn velocity B.n A ’ * = (4.rrPp)1/2
(1)
Here II is the normal to the surface of the discontinuity, B is the magnetic induction, p is the plasma density and p,, and pI are the components of the plasma pressure parallel and perpendicular to the magnetic field. Rotational discontinuities differ from shocks in two essential respects: they propagate relative to both the upstream and the downstream plasma at their respective modified Alfven velocities, and the magnetic field downstream does not lie in a plane containing the upstream field and the normal to the discontinuity. In this they differ from the ‘rotational discontinuity’ discussed by Ivanov (1970) which is a special type of plasma shock which leaves the plasma density unchanged. Since the detailed mechanism of discontinuities in anisotropic plasmas is not fully understood we do not concern ourselves with it but set limit on possible solutions by considering the conservation of mass, momentum and energy. Thus p, p,, and pI contain contributions from the electrons and all the ions. To determine the downstream state we need, in general, an additional relation giving the change in p,, or pI across the discontinuity. However, if we use a classification based on the anisotropy parameter, A= (P,, - ~1)47~W
(2)
we can obtain considerable information about possible solutions. Let the subscripts 1 and 2 denote conditions upstream and downstream of the discontinuity. Then from Equations (48) and (49) of paper I we have PI1 -I- w37v
= P12 + B,2/%
P1(l - 11) =
P2U
1693
-
12).
(3) (4)
P. D. HUDSON
1694
If there is no heat flux and the internal energy density is given by pl + $p,, then we have from Equation (52) of paper I,
B2
; FM + G
(2& + I)
p2.
(5)
Solving Equations (3), (4) and (5) we obtain (3 - 4&)(1 - &)PlZ = p11(3 - 24 + 42, 2 - 513 + B12(tl, - A,)(1 - 211,- 2L2)/4np
(6)
(3 - 4&)(1 - &)B22 = B12(3 - 21, + 41r2 - 51,) + 407r/_~p,,(~,- 1,)
(7)
(3 - 4&)(1 - &)Pilz = P1r(3 - 21, - 6az2 + lo@,-
5&)
- B,2(A,(2L1 - I)(1 - 21,) + 3L2(3L2- 2))/4np.
(8)
Since pL2, plla and B22 must be positive these equations (plus similar ones in terms of pII1, and Blz) imply conditions for the existence of physical solutions. The excluded regions of the LJ, plane are shown in Fig. 1 and the conditions imposed on the ratio kl = 47r,up,,/ B12 in other regions are listed. There are no restrictions on this ratio in region 1 or on the line 1, = 1,. Using Equation (4) we see that the density changes across a rotational discontinuity by less than a factor of 2.5. Some values of (Al, 1,) are possible only when the plasma pressure is large compared to the magnetic pressure and this may be inconsistent with the magnetic orientation of the plasma. Also if 1 < 0, the plasma may suffer from the mirror instability. Both these effects depend upon the pitch angle distribution of the plasma, but as an illustration the dashed line in Fig. 1 shows the boundary of the permitted solutions if we impose the conditions pI < B2/8n,u and pL2 < p,,B2/87r,u on both sides of the discontinuity. The dash-dot line shows the boundary if we also impose the condition p,, < B2/8r,u. The normal to the plane of a rotational discontinuity is not completely determined by our conservation equations, but it lies in the plane containing B, x B, and (B, - B,) x (B, x B,). The inclination to the magnetic field must be determined by the detailed mechanism of the discontinuity, but the criteria for well-posedness (Lynn, 1970) may put limits on this angle. These are most restrictive for large negative il. If the angle is limited it may impose a minimum value on B, since B, . n = B, . n. If Al > 1, the plasma density decreases across the discontinuity and we have an expansive wave with A,. n > A,. n. If il, > I, we have a compressive wave. The change in magnetic field and plasma pressure are also linked to the sign of (A1 - 12) with (3 - 41&l - L2)(B12- Bz2>/87rp = -(Al - &)(5plr - B12(1 - 22, - 21,)/4rp)
(9)
(3 - 412)(1 - ~2)(~112 -Pact) = -(ii, - n2)(5pll(l - 22,) + B,2(2 + 211, - 32, - 4L,i1,)/47rp).
(10)
(3 - 4A2)(l - A2)(p12- pJ
=
Thus zero density change across a rotational = pL2 and Bl = B, (or if Al = Q).
g,,
discontinuity
occurs only if plll =plj2,
ROTATIONAL
DlSCONTINUITIES
IN AN ANISOTROPIC
PLASMA
1695
FIG. 1. ROTATIONAL DISCONTINUITIESARE NOT POSSIBLEIF (A,,A,) LIESIN THE SHADED AREA [(A = 47&]j - pJ~“)l.
There are also restrictions on k, = 4~,up~J& %.In regions 2,3 and 6 we must have kr < (3 211 + 411’ - 5&)/10(& - A,); in regions 2 and 6 we need /cl > (1, - A,) (21, - 2& - 1)/(3 - 21, + 41,’ - 5&); in region 5 we require k, > (222 - 41,21, - jil + 2J.J, - 21, + 3&F)/ (3 - 2& - 6&Z - Sjir + lo&&) and in region 6 we must have k, > - &. There are no restrictions on kl in region 1, but in region 7 we need kl < (5il, - 3 + 2& - 4~,z)/10(J., 2,) if 1, > I,, and k, > (d, - &)(2& + 21, - 1)/(5& - 3 + 2& - 4nzz) if ii, > I,. If PI f B’/S?rfl and *^ pb2 < .,, pI, B2/8nfi then the boundary of permitted solutions shrinks to the __ -1^^L_.l
I:-_
_-_I
1P
_
,
‘“9
ID
:r
2
P.._.l__..
?I___-“_.
I
^_
A^
At._
A ^..I.
,a,.&
I:..^
To remove the effect of variations in density we can use pi/p and p,,/p to define temperatures TL and T,, . Since the plasma is a mixture of electrons and ions TL and T,, do not necessarily correspond to the temperature of any of the components but they illustrate the redistribution of energy. Thus we have Pl(1 - &)(3 - 4&)(P,zlP, - PUlPI) = -(I, - il,)(pL12(1 + 21,) - B?(l - 2& - 2&)/4np) PI(1 - &)(3 - 4MP,,zlPz
(11)
-P,,rM = -(A, - L&J1r2(1 - 31,) + B,2(2 - 3/I, - 1,)/477/J) (12)
If 2 > ;1, > &, the average pressure ($p,, + $p,) and the average temperature decrease across the discontinuity. A necessary condition for the existence of rotational discontinuities is that the entropy downstream should be greater than that upstream. The plasma is a mixture of ions and
1696
P. D. HUDSON
electrons each of which may have a complicated velocity and pitch angle distribution and without these and a detailed knowledge of the structure of the discontinuity the entropy change can not be evaluated. However as an illustration we applied the test PL2P,,YlP2 z P~1P;,Il2IP1
(13)
which is appropriate for a single component gas which is bi-Maxwellian both ahead and behind the discontinuity. This condition can be satisfied in most of regions 2, 3 and 6 and in region 7 below 1, = &, but it cannot be satisfied in region 4, most of region 5 or in region 1 if k, 2 0.1762. However it can be satisfied throughout region 1 if k, is small enough. Although the validity of condition 13 is doubtful it implies for 1, > 0 that downstream isotropy can only be achieved if k, = P~~~~T,LL/B,~ is small. We would expect that scattering etc. would decrease the anisotropy of the plasma as it passes through the discontinuity. So instead of condition 13 we will use the condition (14) to limit the range of possible solutions. The allowed values of il, are illustrated in Fig. 2 for several values of k,. For 0.75 > 1, > 0 we see that there are two types of solution, a mildly compressive wave, and an expansive wave with near isotropy downstream. For negative jz we have only a compressive wave. 2. SPECIAL
2.1
SOLUTIONS
1 > 1, > 0.75
The only solution occurs in region 7 with 1 > 31,> 0.75 and (1, - &I < 0.025. To satisfy condition 14 we need a mildly expansive wave with 2, > R, > (10&k + 3 - 22, + 41,2)/(5 + IOk,). Condition 13 is also satisfied for most of these values. We find that pI, T,, the average pressure, and the average temperature all increase across the discontinuity whilep,,, T,, and B decrease. The density change is slight with p1 > p2 > pr(5 + lOk,)(lOk, + 2 + 4/l,) > 0.83~~ but the changes in pI, p,,, Tl and T,, can be sizeable because the factors on the left hand sides of Equations (9)-(12) are small. A small change in density and ;Z must be difficult to achieve, and the only realistic possibility is that B, is very small. This requires a mechanism which extracts energy from the magnetic field and from T,, and transfers it mainly to TL. If the change in B is large then the normal to the plane of the discontinuity must be nearly at right angles to B, since B, . n = B, . n. 2.2 L, = O-75 The only solution permitted is Iz, = 0.75. The density is unchanged across the discontinuity whilepL, p,, and B can have any change subject to Equations (2) and (3). 2.3 The expansive rotational discontinuity 0.6 > A, > 0 The exact upper bound for this type of solution is 1, = (-5k, which lies between 0.5 and 0.6.
+ 1 + (25k12 + 14k, + 1)“2)/4
Condition
14 is satisfied for positive 1, less than
ROTATIONAL
FIG. 2.
THE
RANGE
DISCONTINUITIES
OF ALLOWED THE
VALUES
ROTATIONAL
IN AN ANISOTROPIC
OF A2 FOR WHICH
DISCONTINUITY
lp,,-_P&~DECREASES
IS PLO’ITED
1697
PLASMA
ACROSS
VS. 21.
The dashed line denotes the boundary for kl = 5 the solid line for kl = 1, and the dash-dot line for k, = 0.2. In each case the line I, = 1, forms a boundary and there is a forbidden region near A1= 0. For A1 > 0.75 the curves are indistinguishable. There are four types of solution: (a) an expansive discontinuity with small density change if 1 > A, > 0.75 (b) an expansive discontinuity with near isotropy downstream (2, N_ 0) for 0.6 2 1, > 0, (c) a compressive discontinuity (1, > A,) for 0.75 > A, > 0 and (d) a compressive discontinuity if 0 > 1, > - kl. For each value of 1, the downstream pressures and magnetic field can be found using equations (6)-(g).
min [A,, (k,(3 - 54) + A,(1 - 2&))/(5k, + 2&)(2k, + l)] and the range of negative I, is limited by the requirements that i1,2(2& - 5k3(2k, + 1) + 1,(10k,2il, + k,(4i1,2 - 41, + 3) - &) + A,@,(3 - 511) + A,(1 - 21,)) > 0 and 3A,2(1 + 2k3 + i1,(2k, - lOk,il, - 2 + 21, - 411”) - k1(3 - 513 - A,(1 - 213 > 0. It will also be greatly limited by requirements for mirror stability. Condition 13 can not be satisfied except for very small values of k, so the existence of this type of solution is in doubt. T,, and p,, decrease across the discontinuity and so do the average pressure and average temperature. TL and pI usually decrease but they can increase for small k,. The magnetic field usually increases but for k, < 0.2 it can decrease by up to 6 per cent. The basic mechanism is one in which the energy released by the decrease in T,, is used to increase the flow velocity with a secondary adjustment to Tl and the magnetic energy.
1698
P. D. HUDSON
2.4 The compressive rotational discontinuity for 0.75 > AI > 0 This type of solution is always possible for A1 4 1, < (lOJ,k, + 3 - 21, + 4&e)/ (5 + lOk,). It satisfies condition 14 provided we exclude a region near 1, = 0 bounded by 1, = & and i1,(5k, + 21,)(2k, + 1) - k,(3 + 5R,) - &(I - 2il,) = 0. The magnetic field decreases across the discontinuity andp,, TL, the average temperature and the average pressure increase. This is the same type of change that occurs in slow shocks. The exact change in p,, and T,, can be found from Equations (10) and (12) but in general they increase. The density change is given by pr < pZ < pi(5 + lOk,)/(lOk, + 2 + 4kil,) < 5p1/2. The upper boundary for & corresponds to B, = 0 and values close to this may be unattainable because of Lynn’s well-posedness conditions and because the field may be too weak for the plasma to be magnetically orientated. Thus the only solution for il, = 0 is 3L2= 0. Condition 13 is satisfied everywhere except possibly for some values of I, in 0.5 < A1 < 0.75. In 0.75 > 1, 2 0.6 the compressive rotational discontinuity is the only type possible and it is characterized by T,,, > T,,,. 2.5 The compressive rotational discontinuity for 0 > ;I, > -k, This is the only solution in A1 < 0 which satisfies condition 14 and we must exclude a region near 1, = 0 bounded by the curve 3LZ2(2J,- 5k,)(2k, + 1) + 3L2[10k,21, + k,(4L12 - 4il, + 3) - A,] + Al[k,(3 - 51,) + &(I - 2L,)] = 0. For small kl this splits the region in two. The range of 1, is the same as in Section 2.4 and values close to the upper boundary may be unattainable for the reasons given there. Downstream isotropy is possible only for 0 > il, > -0.9. In practice the range of possible 1, and 1, is severely limited by the requirements for mirror
stability.
For
instance
pL2 G p,, B2/8r,u requires
1 + 2/l
+ 81, > 4k, > 1 -
41 + 81,. The density change is given by pi < p2 < pi(5 + lOk,)/(lOk, + 2 + 41,) < pi(5 + lOk3/(6k, + 2) < 5p,/2 and T,,, p,, and the average pressure increase across the discontinuity. In general, pL and T,_ increase across the discontinuity and B decreases, but we see from Equations (9) and (11) that these changes can be reversed for small k. However, the increase in B is less than 6 per cent. The mechanism is one in which the energy released by the decrease in the flow velocity is used to increase T,, with a secondary adjustment to TL and the magnetic energy. Condition 13 is satisfied at least in those areas where pI increases. 3. ROTATIONAL
DISCONTINUJTIES
IN
THE
SOLAR
WIND
The thermal energy of the ions in the solar wind is usually of the order of the magnetic energy density (Neugebauer and Snyder, 1967) andp,,/p, for the ions is usually in the range l-5 with a typical value of -2 (Hundhausen et al., 1967). For the electrons p,,/pl is less than that for the ions and is usually in the range 1.1-1.2 (Montgomery et al., 1968), but the electron temperature is 1.5-5 times that of the ions. Using this data we find that a typical value of k = 4npp,/B2 is I.0 but that it can differ from this by at least a factor of five. Figure 2 illustrates solutions for k, = 5, 1 and 0.2. A typical value of 1 = (p,, - p1)4zy/B2 is 0.4, and, although it can vary widely, it is rarely negative. Rotational discontinuities cannot propagate if L > 1, but, as the plasma then suffers from the firehose instability these values will be rare in the solar wind. Separate treatment of the ions and electrons (Kennel and Scarf, 1968) shows that an ion cyclotron resonance instability can also occur. In the solar wind this has a significant growth only if liona 2 4 but it will probably reduce the frequency of occurrence of values of il in the range 0.5-l. Since observationally p,,/pI < 5, ;I, is limited for small k, by 1, < 4k,.
ROTATIONAL
DISCONTINUITIES
IN AN
ANISOTROPIC
PLASMA
1699
A procedure for identifying rotational discontinuities is the solar wind is given in paper I. However, most of the tests are inconclusive because of experimental uncertainties and so the additional results derived in this paper are needed to differentiate between rotational discontinuities and shocks, tangential discontinuities, etc. A rotational discontinuity would be identified if it satisfied Equations (3), (4) and (5) or (6), (7) and (8), but if these can not be evaluated accurately then we must fall back on conditions that are necessary but not sufficient for the discontinuity to be rotational. The solar wind speed is much greater than the propagation velocity of a rotational discontinuity or the velocity jump across it and with most satellite observations it will be difficult to decide whether the upstream or downstream side of the rotational discontinuity has been observed first. Let us use subscript a to denote the low density side of the discontinuity and b the high density side. Then we must have 1 > pa/pb > 0.4, and plo + Ba2/8r,u =plb + Bb2/8r~. If ii, < 0.75 the average pressure and temperature increase with density but the magnetic field decreases (except for the rare cases when L, < 0 or k, < 0.2 when it can increase by up to 6 %). This distinguishes rotational discontinuities from most fast shocks (Neubauer, 1970) where the change in p and B have the same sign. The only rotational discontinuities across which we can have BalBb < 0.94 are those with il > 0.75 (Section 2.1) and with these the density change must be very small. Such large values of 1 may be rare in the solar wind and the difficulty in finding a realistic mechanism means that such rotational discontinuities will probably be uncommon. REFERENCES HUDSON,P. D. (1970). Planet. Space Sci., 18, 1611. HUNDHAUSEN, A. J., BAME,S. J. and NESS,N. V. (1967). J. peophys. Res. 72, 5265. IVANOV,K. G. (1970). Preprint. KENNEL,C. F. and SCARF,F. L., (1968). J.geophys. Res. 73,6149. LYNN, Y. M. (1970). Physics Fluids 13, 1762. MONTGOMERY, M. D. BAME,S. J. and HUNDHAUSEN, A. J. (1968). J. geophys. Res. 73,4999. NEUBAUER,F. M. (1970). Z. Physik 237,205. NEUGEBAUER, M. and SNYDER,C. W. (1967). J.geophys. Res. 72, 1823.