AXIALLY SYMMETRICAL MAGNETOHYDRODYNAMIC EQUILIBRIUM CONFIGURATIONS TRANSLATED FROM Z. NATURFORSCH. 12A, 850-4(1957)

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• Document Creation Date: December 27, 2016
• Document Release Date: January 31, 2014
• Sequence Number: 1
• Publication Date: August 18, 1958
• Content Type: REPORT

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AXIALLY SYMMETRICAL MAGNETOHYDRODYNAMIC
EQUILIBRIUM CONFIGURATIONS
By R. List and A. SchLfiter
Z. Naturforschg. 12a: 850-854 (1957)
?
Translated for Oak Ridge National Laboratory
By The Technical Library Research Service
Nader Purchase Order No. WM-37383
Letter Release No. X-30
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4 4-
AXIALLY SYMMETRI6L MAGNETOSYDRODYNAMIC
EQUILIBRIUM CONFIGURATIONS
By R. Last and A. Schlater
From the Max-Planck Institute for Physics, Goettingen
The conditions for magnetohydrostatic equilibrium are
studied in the case of axial symmetry. The magnetic field
is divided into its meridional and its toroidal parts
which are described by the scalar functions F and T re-
spectively. It is shown that the gas pressure p and the
functions F and T have to be functions of each other.
Takilhg in particular p(F) and T(F) as known relations,
a differential equation for F is derived. The cases in
which this differential equation is linear are con-
sidered and explicitly solved if furthermore T(F) =
const. In a special case, the magnetic lines of force
are calculated numerically and shown in a figure. Some
remarks on the stability are added.
A magnetic field exerts forces on a conducting body when
electrical currents flow through the latter intersecting the
lines of flux. Conducting bodies of special interest for
astrophysics and for many terrestrial applications are
plasma, i.e. gaseous conductors. If gravitational effects
are unimportant, then a static equilibrium can exist generally
only if the forces exerted by the magnetic field and the gas
pressure of the plasma are compensated everywhere. Since the
forces eaused by pressure are rotation-free, this equilibrium
"At!WW*F
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EtWatithWtri:e7
oly
2
-
gsiiirlatY cammat be fulfilled but corresponds to a requirement ?
.. , :II- cl . f : ? , ?
ef the cemOsiltation of the magnetic field. A special case
, 4- . - ? ? .._ , ,
Of ' equiklbrilat: UV-eta when the currents in the conduct..gr have '
?
a flow wbtlit 1,1,--sioars?. 1 to the magnetic field'everywhere.
' ' ? " ?-?,.i???- ...; .
..,-1-. , ,'' 'i .--'?:,6...',--..,,"is'./4. 4.'e ? ? f
'c fa4Micorafria:signietic field has..been _discussed by us in
?, _,,, - , ? ??"??*';,:,;47,--..1:r Pricleteg4,17.21;.4.e. _ ? .
4.,,:t ;Eitiiiiiit,:pitoto,34....vg74)111 liiii(b! Oil ity . f, an equilibrium be-
...;40-cp;?i.A.-,..; -1,--,'??,74-:kftrie?I'r
Irn:1,4N--,--7,,FIK4':$?(*:Z51..:%!"-4444h11 ,,_:" ? ''. ? ' ? . . .
?-a,.. zis, ,., . "dAlltite1111111.,9.vc,Sitt.4irliVi-f!Ti:e:s.?...)1uta:gi ifi':.e.ifr e 1 surie_J_t_to be utilized for
' -T,' .4 ),5-* --lik,'"nl? 4 ,..f-,...a.-..&liotrA";.-7>,, .. r .._?.
-
,.:?,,c,t. 1,141.,435 iiirifirt thottios:rovs-r.pflisika. by ae magnetic field
,, ...-11.-? .? A-
.....,---.4t.,0-..-;'''.? 4 ,- .
.S.f tatt,;-al-%7t-he7. piaspa with material -
tvrIv.441,;,.,..,? ????;?,.. ? , 4
erclic, en,- thisipiitimma-Ars:',.i.WOWIred, however, .lad, the
. ,??? :-., A.
tf-J- ? -,v...;;.\,,-? ,!?."1,-- 4."..1---
4;4; .-...?;iTf.-1,. :,...:---.i or4p-,,,V47?VrA.' -4,'"1!. 4.:,.." ' :. ,- -. ?
iiiiial ' *gain! britiat p..rarblem:?Mustk,be investigated. We
.o...,... ! "41' -''''' ?Z-.'1'-'i'. '-' ' : ? -" ' ''',- v-
. . .. .
piiikairraltila;'1,n this- Calle t4.- -axial.ly=sysaa. ittiNiCil arrange-
-ezz,,,',1'.4, '.4....4 t?.:-..t.-,. ? . - ? *. ? ? ..
?4t, 4- ' ''.... .. ? - ..
initivansionaAler batissially selatttOni-whieh Can be solved
f.41-',, ?.*..';'.; ...1''',?: "7..4 .... ,y,? ?
02stailctlf-iiii:lyticill.', ?-\.,' t sillier axial lye.symmetrical
- NI ?-'.., 4 ? - '' t ?
ratit...4!" `'...?,,:i.';'.,,...,..1s+i?': ',,. - -: ? : . ,'''z-''.' ' . : ' , ...- ' .
e,se:r --,,,-- "Z --11.;`,hai bnimenwsumed that allcurrents flaw on
? ?
il?z-,:-..4_,... A-?,4.--?-,f.? . ..- , ? ; ,. ,../. ? ? , ,
itintrf -,.-?,k os?-jilaliaii, a intut iv* -by a' series expansion
0, then S(s;;k) increases exponentially for large
values of s, while S(s)1) for A< 0 is proportional to15
when s moves towards infinity. In this case the magnetic
field moves towards infinity for large values of IZI.
For A= 0 we obtain specially:
S(5; 0) = Ds2 + E
it
and
Z(s; 0) =Gz+ K
(30a)
(30b)
where D, E, G are integration constants.
Solutions which are period-ical in z, for example, will
be discussed in greater detail in the following.
In this
case )k 0 and the integration constant G 0. Then the
value of the integration constant E is negligible. Then we
have, for the flux function F(s, z0.) according to Equations
(19), (27), (29), (30a) and (30b):
b 4
F(s,z;)0 = Alisii(ila,$) cos (1a,z) + 2 s (31)
O
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In the above the factor i has been chosen in such a manner
that AI is a real integration constant. B1 is also an inte-
gration constant. An additional free integration constant ,
has been set equal to zero which denotes only a determination
of the phase position with respect to z:
A field resuLyt4. from this-function F(s,z00 has been.
-------- -
shown in Fig. 1. The parameters have been chosen in such.a
manner that the gas pressure
p = -(044) F + coast'
is maximum on the axis and always decreases for all
vicinity of s = 0 for increasing s. For the special para-
meters of Ftg. 1 (Al = 1, B1 = 1, b = 1) it is the clue for
the vicinity of the axis up to the line of force on which
,
F,1:19.6. By a suitable choice of constants available in the .
pressure (= gas pressure onit is then possible
to obtain a positive pressure everywhere in the tube thus
formed and assume a given value, for example, p = 0 en an -"
arbitrary line of flux. This line of flux can then be idew-
tined by the wall of the vessel in the interior of ultick,Illa
magnetic field holds the plasma entirely (for p = 0 on thi
wall) and partially together and there our equations are no
longer valid on its exterior; in contrast to the above, the
magnetic field is formed by a corresponding arrangenent'of
coils.
Fig. 2 shows the graph of the magnetic field strength
. _
and of the gas pressure p on the lines z = 0, t21, ... end
Li 5y
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eP's
? ?.k.t?Cf.
6-4
? 1 ?
11 ?
z w
tit, t311... as a function of the distance from the
(B2Alf-)+ o
symmetry axis. In addition, the function
(= "total pressure" = "magnetic pressure" 4 gas pressure)
has been plotted also. In the case of an extended magnetic
field, this function would be constant while in this case it
shows the influence of curvature.
.rStiiility of Axial'-Symmetrical Fields
In. conclusion we will mention briefly the stability of
the meridional fields under consideration here. In an
earlier report,7 the stability of general equilibrium con-
figurations has been investigatCd. In the case of meridion-
al magnetic fields, we obtain the following from the cited
Equation (23):
-02S f 1102dt -S p( div 11)2 4. ,-Tr 4(rot
1 '
(11 in2jdr
+01 grad r)2 d211F) 4t (32)
5
dr
+S(1 (11. r11)(reirl-grad (1 grad F))
dp
- (*
t.
,, ..)
grad F)(rozrlagrad 1 ?Ltzr,1 i)))&t.
where 1 is the velocity of the plasma, riir the ratio_ of the
specific beat and tr. is the element of volume. (For the
derivation Of Equation (32) it has been assumed that the
soma compossats of and 111 at the surface of the considered
volume will disappear.) The stability of an equilibrium con-
figuration is determined by the sign of ol2, ',hereby 02 < 0
demotes instabklity. It can be seen from Equation (32) that
?4/57
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the first inte-iral always results in a stable part. The
se.2-orid integral will also always result In a stab!! part .oro-
vided that d?p(lr)./d!-' J everywhere. The sign of the. last
term can 1).e positive as well as negative. Put it can be shown
that the integral 'disappears if the disturbance 13 Is inde-
poriden,t of the azimuth (I) . Meridional fields will be stable
to those disturbances in case d2p(r)/di. J. For he field
described by Equation ('1) (see Fil. 1) the Equation (11a) is
also d2p/dF2 = 0. This field is thus stable to distgrbances
Which do not depend upon op.
We wish to express our gratitude to Mr. A. Kurau for
the mathematical calculations which have been perfor4med
with the electronic calculator 32.
Footnotes
1) R. Last and A. Scnlater, Z. Astropnys.
2) L. Rlerman, K. Hain, K. Jardens and R.
Naturforschg. 12a, 826 (1957)..
3) R. Last and A. Schlater, 2. Astrophys. Ili, 190 (1955).
4) S. Chandrasekhar, Proc. Nat. Acad. Sci. 12_, 1 (1956).
5) S. Chandrasekhar and K. H. Prendergast, Proc. Nat.
Acad. Sci. IL, 5 (106).
6) E. Kamke, Gewahnliche DiOterentialgleichungen (Ordin-
ary Differential Equations), Akad. Verlagsgesellschaft,
Leipzig 1943, P. 427.
7) K. Hain, R. Last and A. Schlater, Z. Naturforschg.
12a, 833 (1957).
3L4, 26 (195u).
Lest, Z.
/.3
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Fig. 1.--Course of the magnetic field which is defined
by the flux function F according to Equation (31) wIth the
parameter values A1 = B1 = b = 1. The numbers at the flux
lines are a measure for tsnetic flux which passes through
the circular cross-section between the corresponding flux
line and the z-axis.
Fig. 2.--The magnetic field strength B (solid cla\t.we),
the gas pressure p (dotted line) and the "total pressure"
p B2/81'(dot-dash curve) in relation to the distance s
from the symmetry axis for V-0?kz = 0, ?21... and for
V-5:z = ? ---a; ---P, B2/81 P.
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i
Fig. 1.--Course of the magnetic field whi
by the flux function F according to Equation (31) with the,
parameter values A1 = Bl = b = 1. The numbersi;at the flux
lLaes are a measure for likagnatic flux which passes through
6
the circular cross-section between the corres nclIng flux
line and the z-axis.
Fig. 2.--The,mAgnetic field stren th B solid curve),
the gas pressure p (dotted line) and tile "to a
p + B2/81'(dot-dash curve) in relatfonito th d1stan.53.0
1
from the symmetry axis for VTz = 0, /121r. . and for
V-Tz = ?-w, ?3ir ---B; ---P, 2/811+ P.
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