Incompressible, inviscid plasma flow from sun in rotational solar magnetic fiel

Hiroshige, Y.

English

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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
CONTRACT N o .  N A S  7-100 
Technical Memorandum No. 33-96 
Flow of Hasma in a Slightly Rotational 
Magnetic Field 
Y. Hiroshige 
J E T  P R O P U L S I O N  L A B O R A T O R Y  
P A S A D E N A ,  C A L I F O R N I A  
C A L I F O R N I A  INSTITUTE O F  TECHNOLOGY 
August 10, 1962 
JPL TECHNICAL MEMORANDUM NO . 33-96 
-t 
CONTENTS 
1 . Introduction . . . . . . . . . . . . . . . . . . . . .  1 
I I  . Small Perturbation Equations . . . . . . . . . . . . . . .  2 
111 . Solution for Rotational Velocity Field . . . . . . . . . . .  4 
IV . Applications . . . . . . . . . . . . . . . . . . . . .  6 
A . IrrotationalFlow . . . . . . . . . . . . . . . . . . .  6 
B . Rotational Flow . . . . . . . . . . . . . . . . . . .  6 
References . . . . . . . . . . . . . . . . . . . . . . .  7 
Figure 1 . Flow field showing the effect of rotation on streamlines 
(a) a = -0.01, (b) a = 0.0, ( c )  D = 0.01, (d) Q = 0.050 . . . .  8 
t 
JPL TECHNICAL MEMORANDUM NO. 33-96 
ACKNOWLEDGMENTS 
The author wishes to express his appreciation to Dr. F. Yagi of JPL 
for many helpful discussions and to S. Ferrades of the Computer 
Sciences Corporation for numerical work. 
IV 
JPL TECHNICAL MEMORANDUM NO.  33-96 
P 
ABSTRACT a-#'> 
A solution for a flow of plasma away from the Sun in a slightly 
rotational magnetic field has been determined, assuming that the 
plasma is incompressible and inviscid and has infinite electrical con- 
ductivity. The magnetic field vector is shown to be a constant multiple 
.of the velocity vector, and the problem is reduced to finding the solu- 
tions to the vorticity equation and inhomogeneous elliptic-type partial 
differential equation. An analytic solution is obtained for the case of 
constant rotation, and streamlines are plotted showing the influence 
of a rotational field. 
1. INTRODUCTION 
The general magnetic field of the Sun approximates an The assumptions made are as follows: _ _  
axisymmetric dipole with its axis in the north-south direc- 
tion. The deviation from the true dipole may, however, 
be such that the magnetic field takes on a rotational 
character. The steady flow of plasma away from the Sun 
under the influence of a rotational magnetic field is the 
subject of the present study. The irrotational magnetic 
field case was studied previously (Ref. 1). 
(l) 
(2) 
(3) 
It is shown that here, as in the irrotational-flow case, 
the magnetic field vector is a constant multiple of the 
velocity vector; the problem reduces to finding the solu- 
tions to the vorticity equation and inhomogeneous elliptic- 
obtained for the case of constant rotation, and streamlines 
are plotted showing the influence of a rotational field. 
type partial differential equation. An analytic solution is (4) 
The plasma is incompressible and inviscid and has 
infinite electrical conductivity. 
The magnetic field is represented as the sum of 
irrotational and rotational components, where the 
latter is very much smaller than the former. 
The influence of the rotational component of the 
magnetic field on each of the other field variables 
is expressed by small perturbations about the 
irrotational-flow solution. 
The planar-flow case is studied since it simplifies 
the mathematical treatment considerably while 
retaining the physical phenomenon of interest here. 
1 
JPL TECHNICAL MEMORANDUM NO. 33-96 
8 
11. SMALL PERTURBATION EQUATIONS 
The equations describing the steady, continuum flow 
of incompressible and inviscid plasma in a magnetic field 
are given below. All variables are expressed in emu. 
The continuity equation 
v * u = o  (1) 
The momentum equation 
Maxwell’s equations 
T7 X B = 4xJ 
V * B = O  (4) 
V X E = O  
Ohm’s Law 
J = u (E + u X B) 
where 
u = velocity of fluid 
p = static pressure 
p = density of fluid 
B = magnetic field strength 
J = current density 
u = electrical conductivity 
E = electric field intensity 
All the field variables are assumed representable by 
first-order perturbations about the irrotational-flow solu- 
tion; hence, let 
u = u,, + u’ 
B = B,, + B’ 
/’ = p, ,  -k 1’‘ 
J = J, + J’ = J’ 
E = E,, + E’ = E’ 
where primed quantities denote the small perturbations 
and subscript o represents the irrotational-flow solution. 
Upon substituting these quantities into the continuity, 
momentum, and Maxwell’s equations and the Ohm’s Law 
(neglecting the second-order perturbation terms), the 
following equations result: 
The continuity equation 
V*.U‘=O (7) 
The momentum equation 
( ~ X U ’ ) X U , = - ~  u ’*u , ,+ -  +-J’XB,  (8) ( 9 :  
Maxwell‘s equations 
V X B’= 4xJ’ (9) 
v OB’ = 0 (10) 
VXE’=O (11) 
Ohm’s Law 
J’ = u (E’ + u,, X B’ + u’ X B,) (12) 
For a fluid with infinite electrical conductivity, Ohm’s 
Law simplifies to 
E’ = - (u, X B + U’ X B,J) 
Furthermore, for steady planar flow, E’ = 0; hence, 
u, X B‘ + u’ X B, = 0 (13) 
For the irrotational-flow case, it was found that the mag- 
netic field was a constant multiple of the velocity field, or 
B, = b,, (14) 
where A ,  is a constant. Upon substitution in Eq. (13), the 
following results: 
(X0u’ - B’) X U, = 0 (15) 
Using the infinite eIectrica1 conductivity and planar flow 
conditions, Ohm’s Law given in the form of Eq. (6) can 
also be written as 
B,, + B’ = X (u,, + u’) (16) 
where h is an arbitrary scalar function of position. After 
Substituting for B, from Eq. (14) and simplifying, 
B’ = (X - A,) U,) + Xu’ (17) 
Eliminating B’ from Eqs. (17) and (15), 
2 
JPL TECHNICAL MEMORANDUM NO. 33-96 
[(A, - A) u' - (A - A,) u,] x u, = 0 
(A, - A)u' x u, = 0 
or 
Aside from the special case of the perturbed velocity 
being everywhere parallel to the irrotational-flow velocity, 
this equation is satisfied if 
A = A, 
From Eq. (16), there results 
It is thus concluded that the magnetic field vector is 
everywhere a constant multiple of the velocity vector. 
The vorticity equation is obtained by eliminating J' 
and B' from the momentum equation (Eq. 8) by using 
Eqs. (9), (14), and (18) and then taking the curl of both 
sides. After simplification, the following equation results: 
v x [Q x u,] (1 - "") = 0 (19) 
4XP 
where 
is the vorticity vector. Since A, is arbitrary, Eq. (19) 
reduces to 
Expanding and noting that div u, = 0 and Q = ni,, where 
i, is a unit vector normal to the plane containing the 
velocity vector, Eq. (21) becomes 
The solution to this equation, being a linear partial differ- 
ential equation of the first order, can be expressed in 
terms of its characteristics. 
(23) 
where ur0 and Ua,, the irrotational-flow velocity compo- 
nents, are assumed known. Equation (23) is also rec- 
ognized as the equation for the streamlines of the 
irrotational-flow solution. Letting 
be the parametric solution of Eq. (23) with the parameter 
$, defined by f ( T ~ ,  0,) = +k, the solution of Eq. (22) can 
be expressed as 
0 = n u -  +TI)  (24) 
The vorticity, therefore, remains constant along a stream- 
line. If the vorticity is specified along a boundary, T = T,, 
then its value is given everywhere by Eq. (24). 
3 
JPL TECHNICAL MEMORANDUM NO. 33-96 
111. SOLUTION FOR ROTATIONAL VELOCITY FIELD 
Considering a region bounded by r, 4 r I r b  and 
0 4  0 I T / 2 ,  the problem is reduced to finding the 
velocity at any interior point, knowing the boundary 
conditions and the vorticity distribution in the region. 
The equations to be solved simultaneously for the velocity 
components are the continuity equation and the defining 
equation for vorticity which are, in component form, 
l a  1 all,, 
- - ( r u , )  + - - = 0 r ar r a/? 
i a  1 atc 
r ar r a0 
- - ( T I C o )  - - 2= R 
By solving for uo between the two equations, the follow- 
ing inhomogeneous partial differential equation of the 
elliptic type results: 
(27) 
where 
(28)  
air 
f ( r ,  0)  = r2 - + 2rfl 
ar 
The boundary conditions are 
where (0)  is assumed known for the time being and 
rj, is taken large enough so that the velocity is essentially 
radial beyond this radius. 
To show explicitly the dependence of the solution on 
the boundary conditions and the vorticity distribution, 
Green's method is used in solving Eq. (27). Green's func- 
tion, G (r ,  0 I r,,, O , , ) ,  satisfies the following differential equa- 
tion: 
(30) 
where the subscript o denotes the source coordinates and 
6, the Dirac delta function. The boundary conditions are 
G ( r ,  0 1 r,, 0,) = 0 
G ( r ,  8 I ror 0) = 0 
G ( r ,  0 I rb, e,,) = 0 
G ( r ,  0 I ror x / 2 )  = 0 (31) 
After multiplying Eq. (27) by G ( r ,  0 I I,, 0,) and Eq. (30) 
by u0 (I, 0)  and subtracting, then the resulting equation 
is integrated over the source coordinates, keeping in mind 
Green's theorem and the symmetry property of Green's 
function. After insertion of the boundary conditions and 
simplification, the 0 component of the velocity becomes 
(32)  
It is assumed that us,{ (0)  is known; however, in most 
physical situations (0)  is given instead. The two vari- 
ables are connected by the continuity equation. Inserting 
Eq. (32) into the continuity equation and integrating, the 
following relation is obtained: 
(33) 
where g ( e )  is an arbitrary function. Letting r = r,, in 
Eq. (33) and interchanging the order of integration, thc 
following linear integral equation results: 
TO solve for Green's function, which is a solution of 
Eq. (30), it is assumed that the 0 dependence is cxprc,ssi- 
ble as a Fourier series, or 
4 
~ 
J P L  TECHNICAL MEMORANDUM NO. 33-96 
" 1 .  rarb 
G (r, 0 I ro, 6,) = n=1 - ni: sin 2n0, sin 2nO 
where the coefficients g n  are functions of T only. Substi- 
tuting in Eq. (30) and simplifying, 
6 (r - ro) 6 (0 - e,) - 
r 
After multiplying both sides by sin2mO and integrating 
over the range 0 to r/2,  a linear nonhomogeneous differ- 
ential equation of the Sturm-Liouville type results. 
The boundary conditions follow from Eq. (31) and are 
Following Morse and Feshbach (Ref. 2) the solution 
to Eq. (36) is given by 
where yl ( r )  and y2 ( r )  are two independent solutions of 
the following homogeneous equation: 
and A ( yl y.) is the Wronskian. These values are given by 
Y1 (x) = r a ~  - zn  r-l + 2n - r-l + 2n 7-1 - r n  
a 
A(YlY2) = +[(;>'" r r,rb - (97 
Substitute in Eq. (38) and then in Eq. (35). The final 
form of Green's function becomes 
(40) 
and the kernal occurring in Eq. (34) becomes 
I ae ar, I r, = r. dr ra a'G (ru, 0 I To, e,) 
[($ + ( ; ) 2 n ]  
r: [(rb)'" - ($7 
= m 4 .  -sin2nB,cos2n6' 
n=1  X 
In conclusion, it is seen that the specification of vor- 
ticity distribution and tangential or radial component of 
velocity on the inner boundary determines the problem 
completely. The computational procedure is summarized 
as follows: 
(1) If given for the inner boundary, the vorticity dis- 
tribution is computed everywhere in the region 
using Eq. (24). 
(2) The source function, f (r, e), is next computed from 
Eq. (28). 
(3) The velocity distribution is then computed from 
Eqs. (32) and (33). If the radial velocity is specified 
on the inner boundary, Eq. (34) is used to generate 
the tangential component of velocity. 
Eq. (16). 
(4) The magnetic field is finally computed from 
5 
9 
JPL TECHNICAL MEMORANDUM NO.  33-96 
IV. APPLICATIONS 
A. Irrotational Flow 
On the inner boundary, T = r,, it is assumed that tlie 
rotation vector 9 is zero, and the radial velocity compo- 
nent is given by 
\diere is the strength o f  a two-dimensional dipole. The 
outer boundary is placed far enough from the inner 
boundary so that the flow field is essentially radial beyond 
this ratliiis. Here, it is placed at ~b = 41.,,. 
The rotation vector is seen to be zero everywhere from 
Eq. (24); hence, the source function, f ( T ,  e ) ,  is also zero 
from Eq. (28). 
The tangential component of the velocity on tlie inner 
boundary, u0,, ( e ) ,  is computed from Eq. (34). Inserting 
th t  value of the kernal (Eq. 41) into Eq. (34), the follow- 
ing linear integral equation is obtained: 
(42) 
Assume that tioiI (B , , )  can be expanded in a Fourier serics. 
(43) 
Substituting for tin,, (0 , ) )  in Eq. (42) and performing the 
integration, thc following cquation results: 
Multiplying both sides by cos 2nO and integrating from 
0 = 0 to ~ ~ 2 ,  the coefficient A,, becomes 
6 
1 - ($'* 
1 + ($" 
4 P  1 A,, = -~ T r ;  1 - 4n2  
and 
21" 
g = -  Tr,' 
Hence, the tangential component of velocity on the inner 
boundary becomes 
The velocity distribution is finally obtained by insert- 
ing Eq. (44) into Eqs. (32) and (33) and performing the 
indicated operations. The velocity componmts become 
8. Rotational Flow 
Thc inner boundary is assumed to have a tangential 
velocity distribution ( a s  given by Eq. 44) e q i d  to that 
for the irrotational-flo\i~ cascl. Also, on the, ii1nc.r l ~ ~ i n t l a r y  
the rotation vector is assulncd constant, which iiwans 
that it is constant cwxy\vhcw by Eq. (24). Tlw oiitc'r 
boundary is again taken at TI ,  41.". 
Substituting Eqs. (98),  (39), (40), and (44) into Eqs. (32) 
and (33) and performing thcl indicatcd operations, thc 
velocity components become 
. 
J P L  TECHNICAL MEMORANDUM 
$ 
($ - (:)'" 
1 + ($4n 
sin 2nO 1 
--2 
ue(rr0) = +- 
=, rr; 4n-' - 1 
2RA, 
+ 2  sin 2nO (47) 
noaar[l - (;)'"I 
2p * 4p 1 
iir(r,@) = - + 2 :- rr,,r ,, = r,r (1 - 4n') 
, 40 B ,  - 2  cos 2nO 
where 
NO. 33-96 
A, = [l - ($'"3&+ [l- (q+7(&)- 
[ (;)2n-']( -- n )r"+' 
+ 1 -  - n2 - 1 r r n + l  ?I> I 
+ [ ( z ) ~  Inr, - Inrb - - -ln-- r', rb 1 2 273 r,, 
The flow field showing the effect of rotation on the 
streamlines obtained from Eqs. (47) and (48) is shown 
in Fig. 1 (a-d). Here, p and r, were taken to be unity. 
REFERENCES 
1 .  Hiroshige, Y., and F. Yagi, Fluid Issuing From the Center of a Magnetic DiRole. - 
Technical Memorandum No. 33-63, Jet Propulsion Laboratory, Pasadena, October 
30, 1961. 
2. Morse and Feshbach, "Methods of Theoretical Physics," McGraw-Hill Book Co., 
p. 826, 1953. 
J P L  TECHNICAL MEMORANDUM NO.  33-96 
I 
-0.01 
= 0.01 D = 0.050 80 r- 7 ---.rJ 
Fig. 1.  Flow field showing the effect of rotation on 
streamlines (a) R = -0.01, (b) R = 0.0, 
(c) R = 0.01, (d) = 0.050 
8 


Flow of plasma in a slightly rotational magnetic field

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