The outflow of the solar magnetic field energy (the radial component of the Poynting vector) per steradian is inversely proportional to the solar wind velocity. It is a decreasing function of the heliocentric distance. When the magnetic field effect is included in the one-fluid model of the solar wind, the transformation of magnetic field energy into kinetic energy during the expansion process increases the solar wind velocity at 1 AU by 17 percent

Whang, Y. C.

English

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CONVERSION OF MAGNETIC FIELD ENERGY INTO 
KINETIC ENERGY IN THE SOLAR WIND Y.  C. Whang 
ABSTRACT The outflow of the solar magnetic field energy (the radial component of the Poynting 
vector) per steradian is inversely proportional to the solar wind velocity. It is a decreasing 
function of the heliocentric distance. When the magnetic field effect is included in the 
one-fluid model of the solar wind, the transformation of magnetic field energy into 
kinetic energy during the expansion process increases the solar wind velocity at 1 AU by 
17 percent. The predicted solar wind conditions at 1 AU are u = 302 km/sec, 
n =8protons/cm3, T =  1.5X105"K, q =  1.4X10-2 erg/cm2/sec,B=7.3y@= 129S",and 
p = 1.58. They agree very well with the observed quiet solar wind. 
INTRODUCTION 
Since the original application of hydrodynamic equa- 
tions to the solar wind by Parker [1958], a number of 
theoretical models have been developed to study the 
solar wind, notably the one-fluid models studied by 
Noble and Scarf [1963] and by Whang and Chang 
[ 19651 , and the two-fluid model by Sturrock and Hartle 
[1966] and by Hartle and Sturrock [ 19681. The 
one-fluid model assumed that all types of particles 
(electrons, proton, helium, etc.) have the same 
temperature, while the two-fluid model took into 
consideration the different temperatures for protons and 
electrons. The inviscid model of Whang and Chang 
predicted a solar wind velocity of 260 km/sec at 1 AU. 
The solution of Hartle and Sturrock predicted a velocity 
of 250 km/sec at 1 AU. These velocites are about 20 
percent below the observed quiet solar wind condition 
[Hundhausen, 19701 . 
The interplanetary magnetic field has a strong coupling 
with the dynamics of the solar wind. The magnetic field 
organizes the collisionless plasma into a continuum in 
the expansion process of the solar wind. It is well known 
that when the solar wind flows across a shock wave, the 
The author is with the Department of Aerospace and 
Atmospheric Sciences, The Catholic University of America, 
Washington, D. C. 
magnetic field energy fiux and the convective thermal 
energy increase during the compression process at the 
expense of the kinetic energy. In a reverse process, 
during the magnetohydrodynamic expansion of the solar 
wind, the magnetic field energy as well as the thermal 
energy is continuously converted into kinetic energy. 
Therefore, we propose that in a magnetohydrodynamic 
model of the solar wind, which includes an additional 
energy source, the predicted solar wind velocity can be 
increased closer to the observed quiet condition. 
Various mechanisms for the additional energy source 
needed to increase the solar wind velocity have been 
proposed by many authors. Barnes [1968, 1969) 
proposed that a significant amount of energy is carried 
by the hydromagnetic waves. The dissipation of 
hydromagnetic waves adds its energy into the solar wind 
flow. Jokipii and Davis 119691 proposed that the 
observed long-lived velocity streams are the source of 
long-wavelength turbulence, which in turn serves as an 
efficient and variable heat source for the solar plasma. In 
this paper, we will show that the magnetic field energy 
flow per steradian is a decreasing function of the 
heliocentric distance r. The total decrease of field energy 
flow from r = 2ro to r = 1 AU is about 25 percent of the 
convective kinetic energy at 1 AU. Therefore, during the 
expansion process of the solar wind, the conversion of 
236 
https://ntrs.nasa.gov/search.jsp?R=19730002053 2020-03-17T07:22:14+00:00Z
/ 
field energy into kinetic energy increases the solar wind 
velocity by 17 percent. 
The solution of a steady-state magnetohydrodynamic 
one-fluid model of the solar wind is carried out in this 
paper. The model assumes that the solar wind velocity is 
in radial directions away from the center of the sun, and 
its temperature is isotropic. The solar wind conditions 
near the equatorial plane predicted by the present 
solution clearly demonstrate the process of energy 
conversion in the solar wind on a quantitative basis. 
DECREASE OF THE MAGNETIC FIELD 
ENERGY FLOW 
Spiral Pattern of the Magnetic Field 
The steady-state solar magnetic field is governed by 
Maxwell's equations 
Here N = r2nu = constant, G is the gravitational 
constant, M, the mass of the sun, qr the radial 
component of the heat flux, and pr the radial 
component of the Poynting vector 
In the right side of (S), F is a constant. Thus equation 
(5) expresses the conservation of the total energy in the 
steady-state expansion of the solar wind. During the 
expansion process, energy can be converted from one 
form into another, but the total energy must be 
conserved. 
The radial component of the Poynting vector is 
Making use of (4), we can express the flow of magnetic 
field energy per steradian as 
V X(uXB)=O (2) 
Spherical coordinates (r, $, o) are used in this paper; $ 
denotes the colatitude measured from the north pole of 
the sun. If B = er B,.(r,$) + e,Bwfr,$), then we can 
write equations (1) and (2) as 
a 
ar 
- (r2 Br) = 0 
a -(nCB,) = 0 
ar 
These two equations can be integrated to give 
B,a(nC)-' (4) 
which represent the spiral pattern of the solar magnetic 
field. 
The Poynting Vector 
The equation of energy conservation can be integrated 
once: 
The solar wind velocity u is an increasing function of the 
heliocentric distance r.  Thus the flow of magnetic field 
energy per steradian is a decreasing function of r.  
Conservation of the total energy flow requires that the 
field energy be transformed into other forms of energy 
flow. 
Mechanism of Energy Conversion 
The equation of motion for the magnetohydrodynamic 
expansion of the solar wind is 
D U  1 m i n E  = ming - V(2nkT) -I- ; JXB 
(7) 
where g = -e,.GM,/r2, In the right hand side of (7), the 
magnetic field exerts a force 
1 - JXB 
C 
on the moving solar wind flow. The radial component of 
this force can be written as 
B2 sin2@ & 
4m dr 
This term is always positive, because u is an increasing 
function of r. The magnetic field exerts a force on the 
solar wind in the direction of the solar wind velocity. 
237 
The rate of work done by this force on a unit volume of, 
the solar wind flow is 
mi u*’ C ; = -  
2 k P  
B2 sin2$ & 
4n dr 
The work done on the solar wind flow is used to increase 
Because the denominator of (8) vanishes at the critical 
radius, we obtain 
the kinetic energy of the solar wind at the expense of 
the field energy. 
* B*’/8n - 1 + 2 sin2@* 
P* C; = 1 +sin @ n*kT* - 
In terms of the dimensionless quantities, we can write 
the two governing equations, (8) and (91, in dimension- 
less form, 
YC; dB 
Z dZ 
MAGNETOHYDRODYNAMIC ONE-FLUID MODEL 
OF THE SOLAR WIND 
Governing Equations 2 e - - - z  - 
(10) a==_ 
dz C; v2 - e  - ( ~ ; - i ) ~  1 The magnetohydrodynamical expansion of the one-fluid 
nonlinear differential equations: The equation of motion 
solar wind is governed by a system of two ordinary 
and 
4kT G‘@ 2krdT @-a- 
u 2  - 2 k T  -~ 
mi r mi dr dZ -2’ B ’” cos2 q!~ 
B2 sin2 @ (8) 
( 1  1) 
dr r 
mi 4nmin 
and the equation of energy conservation 
dT r 2  K cos2@ - dr 
where the thermal conductivity K is proportional to 
T5n. 
Let us denote conditions at the critical radius by a 
superscript asterisk. Then we can introduce the follow- 
ing dimensionless variables 
v =  u p  
e = T/T* 
Z = r/r* 
and dimensionless parameters 
2Nk 
r* K * cos2q!J * C Y =  
F 
N mi u*’ H =  
These two ordinary differential equations will suffice to 
solve for the two dependent variables V and 0 .  
At the critical radius, equation ( 1  1 )  reduoes to 
(%)* = + (+hj +;] (12) 
Both the denominator and the numerator of equation 
(10) must vanish simultaneously at 2 = 1. When the 
numerator is zero, we obtain 
(g)* = 2 - y l  
From the above two equations, we can show that a is 
not an independent parameter: 
YE - 2 
= t ( ~ + H - 3 / 2 ) - 3 / 2  
Therefore, the general solutions for Y and 0 can be 
expressed in the form 
v = v(z,H,r,C;,@*) 
e = e(z, H, 7, c;, @*I 
and 
Making use of the 1’Hospital’s rule, we can calculate 
(dV/dZ)*; the result is 
238 
where 
145- 
2 + CY (5 E y - 5) + ( E  y - 2) ($ E-y + 2 sin2 Cp* - 7) c, = -___ 
(3E - 1) 
6.0/ 
I I I I 1 I 
Numerical Solutions 
For any given value of y, H, E, and $J*¶ we can use (13) 
and (1 6) to calculate the solution for VfZ) and 0 f Z )  in 
the E neighborhood of the singular point at Z = 1. Then 
we can integrate equations (IO) and (1 1) for 2 > 1 + e 
and Z < 1 - e .  The two branches of the numerical 
solutions join smoothly at the singular point. 
The numerical solutions for Z >  1 show the same 
property as the numerical solutions for the inviscid 
model of the solar wind {Whang and Chang, 19651. As 
Z -+ *¶ the solution curve is expected to approach the 
condition V = V ,  and 0 = 0 with 
A solution satisfying this condition when H = 3.0, 
= 1.3, Cp* = 176" and y = 1.8978 is obtained as shown 
in figure 1. The numerical values of this solution in 
dimensionless form V = V(Z) and 0 = e(2) are given in 
table 1. 
Physical interpretation of the numerical solution 
depends on the choice of the ratio r*/r@. When the 
critical radius is chosen at between 5 and 6 rg the solar 
wind conditions at 1 AU predicted by the present 
solution are (fig. 2) u = 287-318 km/sec and 
T = (1.45 - 1.57)X IO5 OK. In the remaining part of this 
paper, we will interpret the physical meaning of the 
solution based on r*/ro = 5.5  (fig. 3). The predicted 
solar wind conditions at 1 AU are 
u = 302kmlsec 
T =  1.5OX1O5 "K 
= 5.7X "K/cm dr 
Cp = 129.5" 
5 = 1.58 
where p is defined as the ratio of 2nkT to B2/8?r. 
1.5 
1.0 
P 
0.5 
0 
NUMERICAL SOLUTION 
FOR H13.0 611.3 
I r-oD7 
1 
V 
2 
Figure 1. The numerical solution, given in table 1 ,  
smoothly passes through the singular point at the critical 
radius (2 = I/, and approaches the condition V = V ,  
and 0 = 0 as Z + m 
r*/ro = 5 0 P - 4  8 155 
! I 1  
t a 'i c 
239 
Table 1. Numerical solution for H = 3.0, = 1.3, $*= 176" and y=  1.8978 
z 
0.100 
0.126 
0.158 
0.200 
0.25 1 
0.316 
0.398 
0.501 
0.63 1 
0.794 
1 .ooo 
1.259 
1.585 
1.995 
2.5 12 
3.162 
3.98 1 
5.012 
6.3 10 
7.944 
10.00 
12.59 
15.85 
19.95 
25.12 
3 1.62 
39.8 1 
50.12 
63.10 
79.44 
100.0 
125.9 
158.5 
4 
171.3 
172.7 
174.0 
175.0 
175.7 
176.2 
176.4 
176.5 
176.5 
176.3 
176.0 
175.6 
175 .O 
174.2 
173.3 
172.1 
170.6 
168.8 
166.5 
163.8 
160.5 
156.6 
152.0 
146.7 
141.0 
134.9 
128.8 
122.9 
117.5 
112.7 
108.5 
105.0 
102.1 
V 
0.046 
0.069 
0.105 
0.159 
0.234 
0.33 1 
0.448 
0.579 
0.719 
0.860 
1 .ooo 
1.134 
1.261 
1.380 
1.489 
1.590 
1.682 
1.766 
1.842 
1.911 
1.974 
2.030 
2.08 1 
2.127 
2.169 
2.207 
2.241 
2.27 1 
2.298 
2.3 22 
2.342 
2.359 
2.372 
e 
3.026 
2.704 
2.415 
2.157 
1.928 
1.724 
1.544 
1.384 
1.241 
1.114 
1 .ooo 
0.898 
0.807 
0.725 
0.65 1 
0.585 
0.525 
0.472 
0.424 
0.38 1 
0.343 
0.308 
0.277 
0.249 
0.223 
0.199 
0.175 
0.152 
0.130 
0.108 
0.087 
0.068 
0.052 
dO/dZ 
-14.76 
-10.53 
- 7.483 
- 5.294 
- 3.733 
- 2.630 
- 1.853 
- 1.309 
- 0.926 
- 0.657 
- 0.467 
- 0.333 
- 0.237 
- 0.169 
- 0.121 
- 8.61X 0-: 
- 6.14 
- 4.38 
- 3.11 
- 2.22 
- 1.58 
- 1.13 
- 8.1OXlO-. 
- 5.88 
- 4.35 
- 3.27 
- 2.51 
- 1.96 
- 1.53 
- 1.18 
- 8.78X10- 
- 6.16 
- 3.87 
P 
0.02 1 
0.020 
0.019 
0.017 
0.017 
0.017 
0.018 
0.019 
0.022 
0.027 
0.033 
0.041 
0.052 
0.067 
0.089 
0.1 18 
0.157 
0.21 1 
0.283 
0.378 
0.503 
0.66 1 
0.850 
1.063 
1.277 
1.464 
1.588 
1.626 
1.567 
1.424 
1.222 
0.997 
0.781 
The values of u, T, dTfdr, @, and p calculated above are the present solution depend on the value of the con- 
independent of the constant K used in the expression for stant K (table 2). When the value of K is about half of 
the thermal conductivity K = KT5'2. However, the that used in the ordinary expression for the thermal 
proton number density, the magnitude of the mag- conductivity of fully ionized hydrogen, the predicted 
netie field and the heat flux at 1 AU predicted by n, B ,  and q agree very well with the observed solar 
240 
100 
r? 
I 
Y 
180 
- 9 1 
9 -  
- 100- 
- 
0 
i 160 lobo 
r / r o  
Figure 3. The solar wind temperature, velocity, direc- 
tion angle, and pvalue as functions of the heliocentric 
distance, at I AU u = 302 kmfsec, T =  I . 5 O X I O 5  O K ,  
4 = 129.5” and P = 1.58. 
1 10 100 1000 
r /ra  
Figure4 The proton density and the magnitude of 
the magnetic field as functions of the heliocentric 
distance. 
Table 2. The proton density, the magnitude of the 
magnetic field, and the heat flux at 1 AU predicted by 
the present solution 
,04 
I 0’ 
ergs cm-’ protons gammas ergs cm-2 sec-’ G 1 ~ 2  
w; 10 
K/,”, n, B, q, 
W sec- ’ deg-7’2 cm-3 ln 
6 . 0 0 ~  10-7 13.1 9.3 2.23X10-’ v) 
4 . 5 8 ~  10-7 10.0 8.1 1.70X10-’ ;; 
3 10.’ 
4 . 0 0 ~  10-7 8.7 7.6 1.49X10-’ LI t g 10-2 
3 . 6 6 ~  10-7 8 .O 7.3 1.36X10-’ 
2 . 7 5 ~  10-7 6 .O 6.3 1.02x10-2 
wind conditions [Hundhausen, 1970; Ness, 19671. 
For K = 3.66X10-7 TS” erg cm-’ sec-’ deg-’, the 
predicted results at 1 AU are n = 8 protons/cm3, r/ro 
B = 7.3 y and q = 1.36X lo-’ erg/cm2/sec. The pre- 
dicted n, B, and q as functions of the heliocentric 
distance r are plotted in figures 4 and 5. The density 
W 
= I  W 
10-5 
I IO  IO‘ I 0 3  
The heat fsux as a function of the heliocen- Figure 5. 
tric distance, q = 1.4X 1F2 ergfcm’fsec at I AU. 
24 1 
curve agrees reasonably well with the observed data in 
the region 2 r e  < r < 20 re [van de Hulst, 1950, 1953; 
Michard, 1954; Blackwell, 19561. In the region r <  2 ro, 
the present solution is not expected to be accurate, 
because the structure of the solar magnetic field is more 
complicated than the simple model described by 
equations (3) and (4). 
CONCLUDING REMARKS 
The solar wind conditions predicted by the present 
magnetohydrodynamic model show that when the effect 
of the solar magnetic field is included in the one-fluid 
model, we obtain an increase of 17 percent in the solar 
wind velocity or 37 percent in the convective kinetic 
energy at 1 AU. The variations of the magnetic field 
energy flow and the convective kinetic energy flow per 
steradian as functions of the heliocentric distance r are 
plotted in figure 6. The magnetic field energy flow per 
steradian is a monotonically decreasing function of r .  
Using the convective kinetic energy per steradian at 
1 AU as a unit, we can see that the total decrease of the 
magnetic field energy flow from r = 2 r ,  to r = 1 AU is 
about 0.35. Conversion of the magnetic field energy into 
the kinetic energy certainly plays an important role in 
the expansion process of the solar wind. 
Urch [ 19691 has studied the magnetohydrodynamic 
one-fluid model of the solar wind. The equations of 
motion and energy conservation were integrated in a 
quite different way. He has obtained the numerical 
solutions for the solar wind after several iterations. The 
temperatures at 1 AU predicted by Urch are 3 to 
4 X  lo5 O K ,  which are too high compared with the 
observed data. 
Recently Wolff et al. E19711 have studied the 
magnetohydrodynamic two-fluid model of the solar 
wind; their predicted solar wind conditions at 1 AU are 
very close to the results obtained in this paper. Barnes et 
al. [ 19711 have also included the magnetic field in their 
two-fluid model of the solar wind. Because they have 
dropped out the JXB/c force term in their equation of 
motion, the magnetic field force is not doing work on 
the expanding solar wind. The conversion of magnetic 
field energy into lunetic energy is not included in their 
work. They proposed that an energy source due to 
dissipation of hydromagnetic waves supplies the energy 
needed to increase the solar wind velocity. Their source 
of energy is completely different from the magnetic field 
energy flow (the Poynting vector) discussed here. 
ACKNOWLEDGMENTS 
This work was supported by National Aeronautics 
and Space Administration under Grant No. 
NGR-09-005-063. 
REFERENCES 
Barnes, A.: Collisionless Heating of the Solar Wind 
Plasma, I ,  Theory of Heatingof a Collisionless Plasma 
by Hydromagnetic Waves. Astrophys. J., Vol. 154, 
1968, p. 75 1. 
Barnes, A.: Collisionless Heating of the Solar Wind 
Plasma, 11, Application of the Theory of Plasma 
Heating by Hydromagnetic Waves. Astrophys. J., Vol. 
155,1969,p.311. 
Barnes, A.; Hartle, R. E.; and Bredekamp, J. H.: On the 
Energy Transport in Stellar Winds. Astrophys. J. 
Ltrs., Vol. 166, 1971, p. L-53. 
Blackwell, D. E.: A Study of the Outer Solar Corona 
from a High Altitude Aircraft at Eclipse of 1954, 
June 30. Mon. Not. Roy. Astron. SOC., Vol. 116, 
1956, p. 57. 
"0' 1 1 0  100 
r /re 
Figure 6. The radial flow of magnetic field energy per 
steradian is a decreasing function of the heliocentric 
distance. The field energy is converted into the kinetic 
energy during the expansion of the solar wind. The 
convective kinetic energy flow per steradian at I AU is 
chosen as unity in this fiwre. 
242 
Hartle, R. E.; and Sturrock, P. A.: Two-Fluid Model of 
the Solar Wind. Astrophys. J., Vol. 151, 1968, 
Hundhausen, A. J.: Composition and Dynamics of the 
Solar Wind Plasma. Rev. Geophys. Space Phys., Vol. 
8,1970, p. 729. 
Jokipii, J..R.; and Davis, L. Jr.: Long Wavelength 
Turbulence and the Heating of the Solar Wind. 
Astrophys. J., Vol. 156, 1 9 6 9 , ~ .  1101. 
Michard, R: Densitie’s E’lectroniques Dans la Couronne 
Externe du 25 FLvrier 1952. Ann. Astrophys., Vol. 
17,1954, p. 429. 
Ness, N. F.: Observed Properties of the Interplanetary 
Plasma. Ann. Rev. Astron. Astrophys., Vol. 6, 1967, 
Noble, L. M.; and Scarf, F. L.: Conductive Heating of 
the Solar Wind, I. Astrophys. J., Vol. 138, 1963, 
p. 1169. 
Parker, E. N.: Dynamics of the Interplanetary Gas and 
Magnetic Fields. Astrophys. J. ,  Vol. 128, 1958, 
p. 664. 
p. 1155. 
p. 79. 
Sturrock, P.A.; and Hartle, R. E.: Two-Fluid Model of 
the Solar Wind. Phys. Rev. Letters, Vol. 16, 1966, 
p. 628. 
Urch, I.H.: A Model of the Magnetized Solar Wind. 
SolarPhys., Vol. 10, 1969, p. 219. 
Whang, Y.C.; and Chang, C.C.: An Inviscid Model of 
the Solar Wind. J. Geophys. Res., Vol. 70, 1965, 
p. 4175. 
Wolff, C. L.; Brandt, J. C.; and Southwick, R. G.: A Two 
Component Model of the Quiet Solar Wind with 
Viscosity, Magnetic Field, and Reduced Heat 
Conduction. Astrophys. J., Vol. 164, Pt. I, 1971, 
p. 181. 
van de Hulst, H. C.: The Electron Density of the Solar 
Corona. Bull. Astron. Inst. Neth., Vol. 11, 1950, 
p. 136. 
van de Hulst, H. C.: The Chromosphere and the Corona, 
in The Sun, edited by 6. P. Kuiper, Univ. of Chicago 
Press, Chicago, 1953, p. 259. 
243 


Conversion of magnetic field energy into kinetic energy in the solar wind

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