Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mso.anu.edu.au/~geoff/AGD/Magnetic_Forces.pdf Дата изменения: Mon Aug 20 12:26:11 2012 Дата индексирования: Sun Apr 10 03:51:45 2016 Кодировка: Поисковые слова: m 8

Characterisation of Magnetic Forces
1 Introduction The momentum equation dv i p ------ = ­ ­ dt xi xi Bi Bj B2 M ij = ---------- ­ ----- 4 8 M ij + xj
ij

(1)

contains pressure gradient terms and gravitational force terms that we are familiar with together with the divergence of the term M ij that we have referred to as "magnetic stresses". The purpose
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of the following is to come to a better physical understanding of what this term represents physically and what effect it can have on magnetised gas. 2 Aside: the forces on a stretched string Before going further it is helpful to consider the forces acting on a stretched string. This analogy is useful for one part of the magnetic force. Take the tension in a stretched string to be T. This is the force exerted over a cross-section of the string by the rest of the string.

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t ­ Tt
i

i

s m = s

T + T t + ti

Forces on element of a stretched string

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Take t i to be the unit tangent to the string, s to be the arc-length along the string, the mass per unit length to be so that the mass of the element is m = s The force on an element of the string as shown in the diagram is Fi = T s + T ti + ti ­ T s ti = T s t i + Tt i + T s t i ­ T s t i = Tt i + T s t i dt i dT = ----- t i + T s ----- s ds ds

(2)

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Now the Frenet-Serret relations for a curve tell us that dt i ----- = n i ds
(3)

where is the curvature and n i is the unit normal. Hence the equation of motion of the mass element is dv i dT ------ = ----- t i + T n i dt ds
(4)

i.e. there is a force along the string equal to the rate of the change of the tension with arc-length and there is a force in the direction of curvature proportional to the curvature times the tension.

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3 Decomposition of the magnetic forces We can write M ij = xj xj = We now write B i = Bb i
(6)

Bi Bj ---------- ­ 4 x i B 2 ---- 8

B 2 ---- 8
(5)

B j B i ----- ­ 4 xj xi

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where b i is a unit vector in the direction of the magnetic field and is therefore tangent to the magnetic field lines. If x i = x i s are the coordinates of a field line with arclength s , then dx i ------ = b i ds
Density of field lines indicates strength

(7)

b

B

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We can therefore write the magnetic force terms as: M ij 1 = ----- Bb j Bb i ­ 4 xj xj xi Bb j b i B B 2 = ------------+ ----4 xj 4 = bi bj x B 2 ---- 8
(8)

b i B 2 b j ­ ----- x j x i 8 b i bj ­ x j x i B 2 ---- 8

B 2 B 2 ----- + ----- 8 4 j ­ P ij x

The first and third terms can be combined in the form: B 2 ---- 8 j
(9)
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where the projection operator P ij = ij ­ b i b j
(10)

projects vectors into the space normal to the magnetic field. That is, suppose we have a vector U i , then P ij U j is normal to the magnetic field, since, b i P ij U j = b i ij ­ b i b j U j = b j ­ b j U j = 0
(11)

The operator P ij therefore projects the gradient operator per xj pendicular to the magnetic field, ie. the operator P ij is the xj component of the gradient perpendicular to B .
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The second term in M ij = bi bj xj xj can be written: B2 ----4 b i B 2 b i dx j B 2 db i b j = ----- ------ = ----- ------ x j 4 x j ds 4 ds
(13)

B 2 B 2 ----- + ----- 8 4

b i bj ­ x j x i

B 2 ---- 8

(12)

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We use the Frenet-Serret relations for the magnetic field lines in the form: db i ------- = B n i ds
(14)

where B is the curvature of the field line and n i is the normal to the field line. Hence, we express the divergence of the stress tensor in the form: M ij = ­ P ij xj xj B 2 B 2 ----- + ----- n 8 4 B i
(15)

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ie, the sum of a pressure term defined by the gradientof the magnetic energy densitybut also perpendicular to the magnetic field plus a term proportional to the curvature of the magnetic field lines.

Curvature force

Magnetic pressure force

Illustration of components of magnetic force

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It is the last term, in particular that distinguishes magnetic forces from pure hydrostatic forces. Note also that the component of magnetic force along the field lines is zero: M ij Bi =0 xj
(16)

i.e. both the pressure force and the curvature force are perpendicular to the magnetic field. 4 The magnetic pinch The confinement of a plasma by a toroidal magnetic field is an example of the different forces provided by a magnetic field. We can also analyse the stability of this configuration using the physical concepts derived above.
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a

Hot plasma

B

Confinement of a plasma column by a magnetic field

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4.1 Magnetostatic equilibrium From the above diagram, once can see that it is feasible that the "curvature force" associated with the magnetic "tension" can plausibly confine a hot plasma. To see if this is possible, we analyse the magnetostatic configuration using the momentum equations: dv 1 ----- = ­ p + ----- B B = 0 dt 4 We analyse this situation in cylindrical polars, and take B = B r B B z = 0 B 0
(18) (17)

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so that B r B z B r 1 B z B 1 ^B = -­ r+ ­ + - rB ­ z ^ ^ r r z z r r B 1 =­ r + -- rB z ^^ r r z and B B B B = ­ ----- rB r ­ B z ^ ^ r r z
(20)

(19)

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We now take B to be independent of z and B 1 ----- B B = ­ -------- rB r ^ 4r r 4
(21)

and the force on the plasma is in the inward radial direction if rB increases outwards. Radial magnetostatic equilibrium Because of the limitations we have imposed, we only have to consider the radial force balance which is expressed by the equation: p B ­ ­ -------- rB = 0 r 4r r
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(22)
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There is a wide variety of magnetostatic equilibria that we could envisage. For the sake of simplicity, we consider one in which the current density in the plasma is uniform. Ampere's law becomes: 1 4 -- rB = ----- j z = constant r r c The solution of this is 2 rB = ----c 2 B = ----c 2 = ----c r2 jz + C C rj z + --r rj z
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(23)

(24)

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The constant C is put to zero so that the magnetic field is finite at r = 0 . The magnetostatic equilibrium equation becomes an equation for the pressure: B p = -------- rB ­ r 4r r 2 2 = ----- rj z c2 Hence 22 P = A ­ ---- j z r c2
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(25)

(26)

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where A is a constant which is determined by the condition that the plasma be confined to r a , i.e. p=0 at r = a
2 jz p = ------- a 2 ­ r 2 c2 (27)

Note that, with this solution,
2 B p + ------ = constant 4 (28)

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Magnetic field outside r = a The region outside r = a is envisaged as a vacuum, so that Ampere's law in this region becomes: 1 -- rB = 0 r r rB = constant C B = --r

(29)

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This constant of integration is determined by continuity at r = a . Hence, C 2 B = --- = ----- aj z c a 2 2 C = ----- a j z c 2 B = ----c a2 ----- j z r
(30)

We can also easily derive this form of the solution from the integral form of Ampere's law, viz,

C



4 B d l = ----- j n ds c
A



(31)
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where C encloses the area A. Here we just take C to be a circle of radius 2 r outside the plasma column, so that the above integral formulation reads: 4 2 B = 2 B 2 r = ----- j z a ---- c c a2 ----- j z r
(32)

as before. The radial profile of the toroidal field therefore looks like the following diagram:

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B 1 B r B -r r 4.2 Stability of the magnetic pinch The magnetic pinch is subject to two well-known instabilities ­ the "sausage" or "pinch" instability and the "firehose" instability. With our knowledge of the nature of magnetic forces, we can analyse these instabilities as follows.

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The "sausage" instability (pinch instability)

Consider an equilibrium plasma column which is perturbed by being "squeezed" as indicated. Since the field lines follow the motion they are squeezed as well. Hence the curvature force inB2 creases by virtue of the increased value of ----- and because of the 4
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higher curvature of the field lines. The pinching effect of the field is greater so that the toroidal magnetic field pinches the plasma even further. The end result is a sequence of blobs. The "firehose" instability
Pressure force High

B2 ----8

Pressure force

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Now consider a toroidally confined plasma column which is perturbed in an oscillatory fashion. Again because the field lines follow the motion of the plasma, the resulting perturbation to the field is as shown. The bunching up of field lines causing a magnetic pressure gradient as shown and the direction of this is to enhance the perturbation. The perturbation therefore grows in the manner of a hose with water flowing through it ­ hence the name firehose instability.

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5 Sunspots
Corona

z

B x
Low magnetic field

Chromosphere

Sunspot region
High magnetic field

Sunspots are a classic example of the simple application of magnetostatics and provide us with an example of the importance of magnetic pressure. They are regions of the solar photosphere which are much cooler than average. (T sunspot 3800 K as opAstrophysical Gas Dynamics: Magnetic Forces 28/59


posed to T 5780 K for the rest of the sun's photosphere.) Consider a model of a sunspot as indicated above. Equilibrium in the horizontal (x) direction implies for B = B x i + B z k p 0=­ + x x
2 Bx ------ + 4 z

Bx Bz ----------- ­ 4 x

2 2 Bx + Bz ------------------- 8

(33)

In this model we neglect B x in the region below the sunspot so that
2 Bz p + ------ = constant . 8 (34)

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Therefore if we envisage the sunspot as having a high magnetic field inside and comparatively negligible field outside, then
2 Bz = p photosphere p + ------ 8 sunspot (35)

Since, in our model, we assume that the magnetic field is independent of height, then p = p z sunspot z photopshere
(36)

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Now consider the vertical equilibrium. This is expressed by the equation: p 0 = ­ + gz + x z Bx Bz + ---------- z 4
2 Bz ------ ­ 4 z

B 2 ---- 8

(37)

where g z is the local acceleration due to gravity. Since B x « B z below the photosphere boundary and there is no dependence of B z on height (why?), then p = gz z
(38)

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in both sunspot and the surrounding photosphere. Since we have shown that the pressure gradient is the same in both, then the density must be the same in both regions. Hence the equation for horizontal equilibrium becomes
2 B sunspot nkT sunspot + ------------------- = nkT photosphere 8 2 B sunspot ------------------- = nk T photosphere ­ T sunspot 8

(39)

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Typical parameters n 10 17 cm ­ 3 T photosphere 5780 K T sunspot 3800 B 830 Gauss

(40)

This is typical of the magnetic field that is observed from Zeeman measurements of the sun's photosphere.

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6 The parameter Magnetic forces are important when the magnetic field is "large". What does large mean? We parameterize the relative importance of magnetic and thermal forces via the parameter, defined by: p gas nkT = ---------------------- = --------------------p m agnetic B2 8 Thus "low " means a stong magnetic field.
(41)

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7 Evolution of the Magnetic Field 7.1 Development of evolution equation So far we have only considered the effect of the magnetic field on the fluid. We also need to know how the magnetic field evolves with time as a result of the motion of the fluid. In principle this is given by Maxwell's equations. However, there are some simplifications in the MHD approximation which have some interesting consequences. Let us first examine the consequences of the high conductivity of magnetised gases, without assuming that the conductivity is infinite. We still assume an Ohm's type law for the conduction current:
Ji = Ei
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(42)
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where is the conductivity and the prime refers to the rest frame of the gas. Remembering the behaviour of electric fields under a Galilean transformation, we have vj E i = E i + ijk --- B k c
(43)

where V j is the velocity of the gas in the lab frame. Hence, vj J i = E i + ijk --- B k (44) c

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We now use the following two of Maxwell's equations, neglecting the displacement current in the first, B k 4 4 1 E i ijk = ----- J i + -= ----- J i c c t c xj E k 1 B i ijk + -=0 xj c t

(45)

Using the expression for the current to solve for the magnetic field, gives vj E i = ­ 1 J i ­ ijk --- B k c
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(46)

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We also neglect the displacement current so that we can substic tute ----- B for the current to obtain 4 vj c E i = --------- ijk B k j ­ ijk --- B k c 4 c v E = --------- B ­ -- B 4 c and we substitute this into Faraday's law to obtain vl c 1 B i ijk --------- jlm B m l ­ jlm --- B m = ­ -4 c c t c v 1 B --------- B ­ -- B = ­ -4 c c t
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(47)

(48)

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B Solving for , t B i + ijk klm B l v m = ­ ijk t xj xj c2 --------- - klm B m l 4

(49)

B c2 + B v = ­ --------- B 4 t We denote the electrical resistivity by c2 = --------4
(50)

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The "curl curl" term on the right can be simplified as follows: B m l B m l Bj Bi ijk klm = il jm ­ im jl = ­ xj xj x j x i x j x j Bi =­ x j x j that is, B = ­ 2B since B = 0 .
(52) 2 2 2

(51)

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Hence for = constant , B i Bi + ijk klm V l B m = t x j x j B + V B = 2B t 7.2 Diffusion time scale Obviously, if v = 0 , then we have a diffusion equation for B , viz, B = 2B t
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(53)

(54)

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The diffusion time scale, t D , associated with this is determined by order of magnitude estimate of each side of this equation. Let the length scale of the magnetic field be L , then B B L2 ----- ------ t D ---- tD L 2
(55)

Normally, for astrophysical plasmas, the length scale is so long and the conductivity is so high that this time scale is very long. For many phenomena we are interested in, the timescales are much less than the characteristic time, t D . (Estimates of times in various regions are given in the exercises.)

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8 Alfven's flux-freezing theorem When the conductivity is infinite, the equation for the evolution of the magnetic field becomes: B i + ijk klm B l v m = 0 t xj B + B v = 0 t The implications of this are extremely interesting: The flux through a comoving loop is conserved, as we now show.

(56)

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S3

v dt n1 v dt S1 dl

S2

n2 B B B

n 3 dA = dl v dt

Conservation of magnetic flux through a comvoving surface
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In the above diagram, the surface S 1 evolves to the surface S 2 in the element of time dt due to the motion of the fluid. The magnetic field is transported by the fluid according to the transport equation above. In the figure the magnetic field is shown at the time t + d t . The flux through the moving surface is given by t =

S1



B r t n dA

t + dt =

B r t + d t n dA

S2

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The directed area formed by the sides of the tube generated by the motion of the fluid is n 3 dA = d l v dt .The flux through the sides of the volume generated by the moving surface is given to first order in dt by sides =

S3



B t dl v dt = ­ d t

S3



B t v dl

(57)

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Now, since div B = 0 , the total flux through the correctly oriented surfaces S 1 , S 2 and S 3 at a fixed time, is zero, since these surfaces enclose a fixed volume. Hence,

S2



B r t + d t n dA ­

S1



B r t + d t n dA ­ dt =0

S3



B t v dl
(58)

The integral through S 1 can be expanded to first order in dt to ­

S1



B r t + d t n dA = ­

S1



B r t n dA ­ dt

S1



B n dA (59) t
(60)

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and, using Green's theorem

S3



B t v dl =

S1



B v n dA

so that we end up with t + d t ­ t ­ dt

S1



B + B v n dA = 0 t

However, the integral over S 1 is zero because of the induction equation, so that t + dt ­ t = 0 i.e.
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d ------- = 0 dt where the time derivative refers to the time derivative following the motion of the loop. This elegant result is known as Alfven's flux-freezing theorem. 8.1 Motion of the field lines There is another way to characterize the motion of the field lines when diffusion is negligible. We expand B i + ijk klm B l v m = 0 t xj
(61)

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to

B i + il jm ­ im jl B l v m = 0 t xj B i + Bi vj ­ B j v i = 0 p t xj xj B i B i v j B j v i + vj + Bi ­ vi ­ Bj =0 t xj xj xj xj Using B j =0 xj and v j 1 d = ­ -- ----- xj dt
(63)
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(62)

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gives dB i B i d v i ------- ­ ---- ----- = B j -dt dt xj d ---dt Bi B j v i ---- = ---- x j

(64)

To consider the implications of this equation, we need to make a slight diversion into the theory of two-dimensional congruences of curves.

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t = constant

u C

ne mli ea S tr

U i = tangent t x i = x i t u

line am Stre

Ui u

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We consider streamlines, originating from a curve, C, so that the congruence of streamlines defines a two dimensional space, x i = x i t u with the velocity along each streamline being defined by v i = x i t u t We define the separation vector Ui = x i t u u
(67) (66) (65)

which is a tangent vector to the curves formed by t = constant .

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Sometimes it is intuitively easier to think in terms of the infinitesimal separation between two neighbouring streamlines. This is U i u and motivates the use of the term separation vector for Ui . Now consider the rate of change of the separation vector with respect to time, as we move along a trajectory. This is U i v i = x i t u = x i t u = t t u u t u t
(68)

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Now the rate at which the components of the velocity, v i , change at a fixed time, is given by their spatial derivatives, i.e. v i v i v i x j t u = = Uj u xj u xj Hence, U i v i = Uj t xj
(70) (69)

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The operator at fixed u represents differentiation following t the motion, so that we have dU i -------- = v i j U j dt In terms of the infinitesimal separation vector x i = U i u x i = v i j x j t
(72) (71)

since u is constant between neighbouring streamlines. These equations show us that the way in which points on neighbouring streamlines separate is determined by the gradient of the velocity.
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Note that the separation vector satisfies the same equation as the magnetic field divided by the density and this leads to the folBi lowing interpretation. Consider the vector U i ­ ---- . This satisfies the equation, Bi Bj d ---- U i ­ ---- = v i j U j ­ ---- dt
(73)

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Bt B0 u C
line am e Str

ne m li a Stre

Ui u

t

x i = x i t u

If we now take our initial curve C such that it is tangent to a magnetic field line and choose the parametrization of that curve so Bi that U i = ---- , then from the above equation we can see that

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Bi U i ­ ---- = 0 always. Therefore the line t = constant formed by the evolution of fluid elements along the magnetic field will remain parallel to B . In other words, the magnetic field remains parallel to the curve defined by the new positions of the fluid elements. Thus, the magnetic field is carried along by the fluid.

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