Boyd T.J.M., Sanderson J.J. The Physics of Plasmas

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5.5 The solar wind 177

5.5.1 Interaction with the geomagnetic ﬁeld

One of the main aims of satellite observations has been to investigate the interaction

of the solar wind with the magnetic ﬁelds of the planets and with that of the Earth in

particular. We know from (5.33) that the angle ψ between the solar wind magnetic

ﬁeld and ﬂow direction increases with distance from the Sun and decreases with

ﬂow speed. At a mean speed of 430 km s

−1

, particles from the Sun take about four

days to reach the Earth (1 AU = 1.5 × 10

km) during which time the Sun has

executed slightly more than one seventh of its 27 day rotation. A faster stream,

making a smaller angle, will cause turbulence and interplanetary shock formation

as it overtakes a slower stream. In this way events on the Sun, such as solar ﬂares,

lead to major perturbations in the planetary interaction.

The main effect of the solar wind on a planetary magnetic ﬁeld is to create an

asymmetry in the noon–midnight meridian plane. In ideal MHD there can be no

interpenetration of the ﬁelds so the solar wind ﬂows around the planet enclosing its

ﬁeld in a cavity called the magnetosphere. This is compressed on the dayside by

the pressure of the solar wind and stretches out on the nightside in the magnetotail.

The boundary of the magnetosphere is called the magnetopause.

There is, however, another important boundary beyond the magnetopause due

to the fact that the solar wind speed is greater than the fast magnetosonic wave

speed. As we shall see in the next section, in this situation, which is analogous

to supersonic ﬂow around a stationary object, a shock wave is created – the bow

shock. The region between the bow shock and the magnetopause is known as

the magnetosheath. At the bow shock the plasma in the solar wind is slowed,

compressed and heated and it then ﬂows through the magnetosheath and around

the magnetosphere. We shall return to the transition at the bow shock later but for

the moment our interest is in the inner boundary of the magnetosheath, namely, the

magnetopause.

The model described so far, proposed by Chapman and Ferraro in the early

1930s, is illustrated in Fig. 5.22. The magnetopause, being a narrow boundary

layer between oppositely directed magnetic ﬁelds, carries a strong current (the

Chapman–Ferraro current) and, as discussed in Section 5.2, magnetic reconnec-

tion may take place; Dungey (1961) was the ﬁrst to point this out. Reconnection

takes place both at the ‘nose’ of the magnetopause between the northwards magne-

topause ﬁeld and southwards solar wind ﬁeld and in the equatorial plane between

the Earth’s polar ﬁeld lines which are dragged out into the magnetotail by the action

of the solar wind. The effect of reconnection is fundamental because ﬁeld lines

now cross the magnetopause at the nose and in the tail and the magnetosphere is

no longer enclosed. Since particles travel easily along ﬁeld lines this means that

interchange between the solar wind and magnetospheric plasma is possible.

178 Resistive magnetohydrodynamics

bow shock

magnetosheath

magnetopause

magnetosphere

Earth

solar wind

interplanetary

magnetic field

Fig. 5.22. Interaction of solar wind with Earth’s geomagnetic ﬁeld; arrows indicate direc-

tion of magnetic ﬁeld (solid lines) and plasma ﬂow (dashed lines) (after Cowley (1995)).

Detection of the reconnection predicted by Dungey has been discussed by Cow-

ley (1995). One possibility is to detect plasma ﬂow away from the X point (see Fig.

5.4 and the discussion in Section 5.2.1) although the motion of the magnetopause

makes detection difﬁcult. Since the current sheet between the oppositely directed

magnetosheath and magnetosphere ﬁelds is only a few hundred kilometres thick

and the speed of the transverse motion is typically several tens of kilometres

per second, the magnetopause passes across the spacecraft in about ten seconds

thereby requiring resolution of plasma data of just a few seconds. Observations

were made by the Explorer satellites in the late 1970s and, with higher resolution,

by the AMPTE-IRM spacecraft in the mid-1980s. Data show that reconnection

takes place in about half of all magnetopause crossings where the angle between

the ﬁelds is greater than 90

◦

. When reconnection takes place it often does so in a

pulsed manner on a time scale of about ten minutes, creating so-called ﬂux transfer

5.6 MHD shocks 179

events (FTEs) which travel over the magnetopause. Neither the pulsed nature of

FTEs nor the factors that determine when and where reconnection takes place are

well understood. The proximity of oppositely directed ﬁelds is a necessary but not

a sufﬁcient condition.

The clearest evidence for the validity of Dungey’s model has been obtained by

an analysis of the ion velocity distributions on either side of the magnetopause

conﬁrming the pattern predicted by Cowley in 1982. The acceleration (by the con-

tracting ﬁeld lines), transmission and reﬂection of ions entering the current sheet

produces characteristic D-shaped velocity distributions in the plane of the magne-

topause on open ﬁeld lines. A spherical distribution of incident ions should produce

a D-shaped distribution of transmitted ions on open lines in the magnetosphere and

a double D-shaped distribution of incident and reﬂected ions on open lines in the

magnetosheath. Such D-shaped velocity distributions have been obtained by Smith

and Rodgers (1991) from AMPTE-UKS spacecraft data.

Further support for Dungey’s model comes from observations of magnetospheric

ﬂow correlations. Given the sensitivity to solar activity there is a wide variation of

ﬁeld-ﬂow angle at the nose of the magnetopause and conditions are most favourable

for reconnection when the ﬁeld points south and least favourable when it points

north. Following dayside reconnection there is excitation of magnetospheric ﬂow

via the open ﬁeld lines from dayside to tail and the subsequent return of closed

ﬁeld lines through the magnetosphere. Experiments carried out by Cowley and

collaborators have shown that dayside ﬂows are excited within about ﬁve minutes

of a switch of the interplanetary ﬁeld from north to south and this has been cor-

related with nightside activity, including intense auroral displays associated with

the consequent change in ﬁeld structure which occur after a period of about 30–45

minutes.

5.6 MHD shocks

Supersonic ﬂow gives rise to the generation of shock waves, a well-known illustra-

tion of this principle being the audible ‘sonic bang’ emanating from an aircraft in

supersonic ﬂight. This is no less true for ﬂows in conducting media than for neutral

ﬂuid ﬂows but, as we shall see, it is considerably more complex. In a neutral ﬂuid

any disturbance, such as that produced by a moving aircraft or a piston at one end

of a tube, causes a wave to propagate through the ﬂuid at the speed of sound c

;

such a wave is called a compression or sound wave. So long as the cause of the

compression wave, the aircraft or piston, is itself moving more slowly than the

speed of sound the wave will propagate ahead of the disturbance and adiabatic

changes may take place in response to it. But if the disturbing agent increases its

speed to c

or greater it begins to catch up with and then overtake the compression

180 Resistive magnetohydrodynamics

wave

profile

Fig. 5.23. Illustration of wave-front steepening in propagation of compression wave.

wave-front with the result that the ﬂuid experiences a sudden, non-adiabatic change

of state; this is what we mean by a shock.

The proﬁle of the shock wave, travelling through the ﬂuid and creating the

change of state, is the result of a balance between convective and dissipative effects.

Since the sound speed c

∝ ρ

γ −1

is greatest at the peak of a ﬁnite amplitude

wave the wave-front steepens, as illustrated in Fig. 5.23. However, as the wave-

front steepens, dissipative effects, which are proportional to gradients in the ﬂuid

variables, become stronger and a steady proﬁle is achieved when the convective

steepening is counter-balanced by the dissipative ﬂattening of the wave-front. It is

this steady wave-front, propagating at supersonic speed through the undisturbed

ﬂuid, which constitutes the shock wave. The smaller the coefﬁcients of dissipation

the more nearly the shock wave aproaches a vertical discontinuity.

Fluid properties may change considerably across a shock wave and the shock is

thus a steady transition region between the undisturbed (unshocked) ﬂuid and the

ﬂuid through which the shock has passed. In Fig. 5.24, region 1 is the unshocked

ﬂuid and is said to be in front of the shock or upstream; region 2 is the shocked

ﬂuid, said to be behind the shock or downstream. Usually one regards a shock

as a transition region between two uniform states as in Fig. 5.24(a) although in

practice this is difﬁcult to realize and the situation depicted in Fig. 5.24(b) is more

likely. Here, the shocked ﬂuid is not in a uniform state but is subject to a relief

or expansion wave. This means that the state of the ﬂuid behind the shock does

not persist but changes with time after the shock has passed on. Nevertheless, it is

convenient to take both region 1 and region 2 as uniform and this will be assumed

unless otherwise stated. (The validity of this assumption depends on the time of

relaxation from the state represented by the point A in Fig. 5.24(b) being longer

than other times of interest.) Since the establishment of a new equilibrium state in

a non-conducting ﬂuid can only be achieved by collisions, the width or thickness

of the shock is of the order of a few mean free paths.

5.6 MHD shocks 181

(a) (b)

SHOCK SHOCK

shocked shocked

fluid fluid

undisturbed

112 2

Fig. 5.24. Illustration of shock wave transition between (a) uniform states and (b) uniform

initial state and ﬁnal state subject to expansion wave.

The theory of hydrodynamic shocks is reasonably well understood; see

Lighthill (1956). As usual, the hydromagnetic case is considerably more compli-

cated. For a start, conducting ﬂuids in a magnetic ﬁeld can support two further

modes of wave propagation. Returning to the piston analogy, transverse move-

ments of the piston in a non-conducting ﬂuid have no effect (beyond a boundary

layer where viscous forces maintain a velocity gradient). This means that there

is effectively only one (longitudinal) degree of freedom and, therefore, only one

mode of propagation, with the velocity c

. However, the transverse movement of a

conducting piston in an inﬁnitely conducting ﬂuid carries any longitudinal compo-

nent of a magnetic ﬁeld with it, thereby producing a wave. Thus a conducting ﬂuid

in the presence of a magnetic ﬁeld has three degrees of freedom (one longitudinal

and two transverse) and hence three modes of wave propagation. There are three

propagation speeds, therefore, generally known as fast, intermediate, and slow (see

Section 6.4.2). Intermediate waves are purely transverse and do not steepen to

form shock waves. The fast and slow modes in general contain both transverse

and longitudinal components and these modes give rise to shocks.

More fundamental differences between shocks in neutral ﬂuids and in plasmas

arise when we come to consider shock structure. Particularly striking is the exis-

tence of shocks with thicknesses much less than the collisional mean free path.

These collisionless shocks cannot be MHD shocks (though they may in certain

limits be described by ﬂuid equations) and we postpone their discussion until

Chapter 10. However, for collision-dominated shocks two factors greatly facilitate

discussion of the effects of a shock wave on a ﬂuid. First, the shock transition region

182 Resistive magnetohydrodynamics

SHOCK

v = shock speed in laboratory frame

Fig. 5.25. Shock rest frame.

may for most purposes be approximated by a discontinuity in ﬂuid properties.

Second, the macroscopic conservation equations and the Maxwell equations may

be integrated across the shock to give a set of equations which are independent of

shock structure and relate ﬂuid properties on either side of the shock. This most

useful and straightforward aspect of shock theory is developed ﬁrst.

5.6.1 Shock equations

For simplicity we restrict discussion to plane shocks moving in a direction normal

to the plane of the shock. Let this be the x direction; then all variables are functions

of x only inside the shock and are constant outside the shock (in regions 1 and

2). It is convenient to use a frame of reference in which the shock is at rest, In

this frame, depicted in Fig. 5.25, the plasma enters the shock with velocity u

and emerges with velocity u

. Steady state conditions apply (i.e. all variables are

time-independent) and the Maxwell equation for j is

j =



0, −



(5.46)

The MHD equations do not apply in the shock region since dissipative processes

take place there; however they do apply on either side of the shock, i.e. in regions

1 and 2. Thus from Ohm’s law

− u

× B

− u

× B

(5.47)

5.6 MHD shocks 183

Now from ∇ · B = 0 and ∇ × E =−∂B/∂t = 0weﬁnd

= 0 (5.48)

= 0 (5.49)

= 0 (5.50)

The other equations to be integrated across the shock are the equations of conser-

vation of mass, momentum and energy. These are discussed in their most general

form in Section 12.5 but since we shall integrate them across the shock and evaluate

them in the upstream and downstream plasmas, where the MHD approximation is

assumed, the result is the same if we use the conservation equations derived in

Section 4.2. Since variables depend on x only, we have from (4.1)–(4.3)

d(ρu

)

= 0 (5.51)

d

= 0

d

= 0

d

= 0 (5.52)

= 0 (5.53)

Integrating (5.48)–(5.50) and using (5.46) and (5.47) gives, on observing that

gradients are zero in regions 1 and 2,

]

= 0 (5.54)



− u



= 0 (5.55)

[

− u

]

= 0 (5.56)

where [φ]

= (φ

− φ

) in the usual notation. The integration of (5.51)–(5.53) is

trivial and one ﬁnds

[ρu

]

= 0 (5.57)



ρu

+ P + (B

+ B

)/2µ



= 0 (5.58)



ρu

− B

/µ



= 0 (5.59)

[

ρu

− B

/µ

]

= 0 (5.60)



ρu

I + Pu

+ ρu

/2 + u

+ B

)/µ

−B

+ B

)/µ

]

= 0 (5.61)

where the internal energy

I = P/(γ − 1)ρ (5.62)

184 Resistive magnetohydrodynamics

Equations (5.54)–(5.61) relate ﬂuid variables on one side of the shock to those

on the other, and these equations are sometimes called the jump conditions across

the shock. Deﬁning the unit vector

n in the direction of shock propagation, the

jump conditions may be written in general vector form as

[ρu ·

= 0 (5.63)



ρu(u ·

n) + (P + B

/2µ

)

n − (B ·

n)B/µ



= 0 (5.64)



u ·

n{(ρ I + ρu

/2 + B

/2µ

) + (P + B

/2µ

)}

− (B ·

n)(B · u)/µ



= 0 (5.65)



B ·



= 0



n × (u × B)



= 0 (5.66)

The ﬁrst three equations represent the conservation of mass, momentum, and

energy, respectively, for the ﬂow of plasma through the shock. The last pair of

equations gives the jump conditions for the magnetic ﬁeld expressing the continuity

of the normal component of B and the tangential component of E =−u×B. When

B = 0, (5.63)–(5.65) reduce to the corresponding hydrodynamic equations known

as the Rankine–Hugoniot equations.

After considerable manipulation (see Exercise 5.6), the velocity variables u

and

may be eliminated from the energy equation (5.65) and the result is

[I ]

+ P

)[1/ρ]

4µ

{[B]

}

[1/ρ]

= 0 (5.67)

The hydrodynamic equivalent of this equation,

[I ]

+ P

)[1/ρ]

= 0 (5.68)

relates the pressure and density on either side of the shock and is known as the

Hugoniot relation. It assumes the role played by the law Pρ

−γ

= const. in adi-

abatic changes of state. Note that the hydromagnetic Hugoniot (5.67) reduces to

(5.68) not only when the magnetic ﬁeld is zero but also when B

and B

are both

parallel to the direction of shock propagation, since (5.66) then gives B

= B

Before discussing particular solutions of the shock equations, the compressive

nature of shocks (i.e. P

> P

) will be proved, assuming the plasma is a perfect

gas. The proof follows as a consequence of the second law of thermodynamics –

the law of increase of entropy. The entropy S of a perfect gas is given by

S = C

log(P/ρ

) + const. (5.69)

where C

is the speciﬁc heat at constant volume. Thus,

dρ

−

γ C

5.6 MHD shocks 185

which, in terms of the ratios r = ρ

/ρ

, R = P

, may be written

dρ

−

(5.70)

In this equation, we regard ρ

and P

as constants (the given values of ρ and P in

region 1). A straightforward rearrangement of (5.67) gives

R =

(γ + 1)r − (γ − 1) + (γ − 1)(r − 1)b

(γ + 1) − (γ − 1)r

(5.71)

where b

= (B

− B

)

/2µ

. Now differentiating (5.71) with respect to r and

substituting in (5.70), we get

dρ

γ(γ

− 1)(r − 1)

+ (γ − 1)[γ(γ − 1)r

− 2(γ

− 1)r + γ(γ + 1)]b

r[(γ + 1) − (γ − 1)r][(γ + 1)r − (γ − 1) + (γ − 1)(r − 1)b

]

(5.72)

The next step is to show that dS

/dρ

is positive and we do this by proving that

both numerator and denominator of the expression in (5.72) are positive. Since

γ>1, it is easy to verify that this statement is true for r = 1. Writing (5.71) as

R = (A + Cb

)/D

it follows for r < 1 that C < 0 and D > 0. Since the pressure ratio R must be

positive, A must also be positive, and

r >(γ − 1)/(γ + 1) (5.73)

Likewise, if r > 1 then C > 0, A > 0 and, therefore, R > 0 implies D > 0, i.e.

r <(γ + 1)/(γ − 1) (5.74)

Combining (5.73) and (5.74),

γ − 1

γ + 1

< r <

γ + 1

γ − 1

(5.75)

Thus, in the expression for R, the denominator D is positive. Therefore, R > 0

implies that the numerator (A +Cb

) is also positive. Since the denominator of the

right-hand side of (5.72) is simply rD(A + Cb

), it is therefore positive.

Turning now to the numerator in (5.72), it is clear, since γ>1, that the ﬁrst

term is positive. The remaining term is quadratic in r and positive for r = 1. If this

term is to be negative for some r it must pass through zero. However, equating it

to zero one ﬁnds imaginary roots for r , which proves that the term is positive for

186 Resistive magnetohydrodynamics

all real r . The proof that dS

/dρ

> 0 is thus complete, showing that S

and ρ

increase or decrease together.

If ρ

= ρ

, it follows from (5.67) that I

= I

and hence P

= P

. Then, from

(5.69), S

= S

. This is the limiting case in which no shock is present. Now since

the second law of thermodynamics requires S

≥ S

, and S

and ρ

change in the

same sense, it follows that ρ

≥ ρ

and (5.75) must be replaced by

1 ≤ r <(γ + 1)/(γ − 1) (5.76)

Then, from (5.71), discounting r = 1 (no shock)

= R > 1 +

(r − 1)(γ − 1)b

(γ + 1) − (γ − 1)r

i.e. shocks are compressive, conﬁrming the qualitative arguments used earlier. Al-

though this proof applies only to a perfect gas it seems that for all gases shocks

are compressive. This may be proved quite generally for weak shocks (Landau and

Lifshitz (1960)) and with some further assumptions (Ericson and Bazar (1960)) for

shocks of arbitrary strength.

We shall now discuss some particular shocks but before we do this some general

observations are helpful. From the jump conditions, provided there is non-zero

mass ﬂux through the shock (ρu ·

n = 0) it is easy to show that

n, B

and B

are

coplanar (see Exercise 5.7). It then follows from (5.66) that

B ·

n[u]

= [(u ·

n)B]

Thus, if B ·

n = 0andu has a component perpendicular to the plane of

n and B it

must be the same on both sides of the shock. If B ·

n = 0 the same result is obtained

from (5.63) and (5.64). This means that we may choose a frame of reference in

which

n, B and u are coplanar. If this is the (x, y) plane all z components in the

jump conditions (5.54)–(5.61) are zero and (5.56) and (5.60) are satisﬁed trivially.

The angle θ between B

and

n is used to classify shocks as parallel (θ = 0),

perpendicular (θ = π/2) and oblique (0 <θ<π/2). We shall begin with

the simple cases of parallel and perpendicular shocks for which, without loss of

generality, we may set u

= (u

, 0, 0), i.e. the unshocked ﬂuid ﬂow is normal to

the stationary shock front. This means that we have chosen a frame of reference

moving with the shock speed in the x direction and the tangential ﬂow speed u

of the unshocked ﬂuid in the y direction.

5.6.2 Parallel shocks

Here both u

and B

are parallel to

n and we have noted already that one possibility

is that B

= B

. In this case it follows from (5.66) that u

is also parallel to

n and