The steady isothermal solar wind equations are shown to admit, under certain circumstances, mutliple transonic solutions when, for example, momentum deposition gives rise to multiplee critical points in the flow. These multiple solutions consist of a continuous solution and solutions which involve shock transitions between critical solutions. The ambiguity arising from the multiplicity of the solutions can be resolved by following the time evolution of a wind profile with one critical point. Results of the numerical integration of the time-dependent equations with momentum addition show that each of these multiple solutions is physically accessible and depends on the rate of change of momentum deposition. These results suggest that standing shocks are likely to be present in the inner solar wind flow

Habbal, S. R.

Rosner, R.

Tsinganos, K.

English

NASA Technical Reports Server

MULTIPLETRANSONICSOLUTIONSANDA NEWCLASSOF SHOCKTRANSITIONS
IN SOLARANDSTELLARWINDS
Shadia R. Habbal, Kanaris Tsinganos and Robert Rosner
Harvard-Smithsonian Center for Astrophysics
60 Garden Street, Cambridge, MA02138
ABSTRACT
The steady isothermal solar wind equations are shownto admit, under certain
circumstances, multiple transonic solutions when, for example, momentum
deposition gives rise to multiple critical points in the flow. These multiple
solutions consist of a continuous solution and solutions which involve shock
transitions between critical solutions. The ambiguity arising from the
multiplicity of the solutions can be resolved by following the time evolution of
a wind profile with one critical point. Results of the numerical integration of
the time-dependent equations with momentumaddition show that each of these
multiple solutions is physically accessible and dependson the rate of change of
momentumdeposition. These results suggest that standing shocks are likely to be
present in the inner solar wind flow.
I. Introduction
The importance of multiple critical points arising in the solar wind flow as
a result of momentumaddition, heat addition, and/or area divergence was first
pointed out by Holzer [1977]. Leer and Holzer [1980] noted that if the wind
velocity becomessupersonic at an inner critical point, shifting from one further
downstream, the requirement for extended energy addition originating from the
coronal base to the supersonic flow could be relaxed (e.g. waves need not
propagate far out in the corona to be dissipated and add energy to the supersonic
flow). In this paper we point out additional interesting properties of the
transonic wind solutions to the isothermal solar wind equation of motion w_en
multiple critical points arise in the flow. We show that, if as a result of
momentumaddition or rapid area divergence the wind velocity becomessupersonic
at an inner critical point, leaving other critical points further downstream, the
steady solar wind equation of motion can admit multiple transonic solutions, one
of which is continuous, while the others involve a shock transition. Wealso
show that the transonic wind solution with initially one critical point can
evolve in time to either one of these multiple solutions depending on the rate of
change of momentumdeposition to the flow.
2. Mathematical Description
To study the time evolution of an isothermal, one-fluid proton-electron
solar wind with momentumaddition we write the time-dependent mass and momentum
conservation equations for a locally radial flow in the form,
8n I d
+----(nay) = 0 (I)
_t A dr
and
I d I dP GM
-_-(nv) +----( nay2) = + nD
t A dr m dr r 2 "
(2)
213
https://ntrs.nasa.gov/search.jsp?R=19840005015 2020-03-22T07:40:19+00:00Z
where A is the cross-sectional area of a flow-tube, n is the electron (or proton)
density, m is the proton mass, and p = 2nkT is the fluid pressure. The
phenomenologieal term D has the dimensions of force per unit mass, and nmD
represents the volume rate of momentumaddition [see Holzer [1977]]. To discuss
the solutions of the time-dependent equations in connection with those of the
steady state equations we combine the above two equations in the form,
I 3v I Sn M2 - I dM2 2 GM d In f D (3)
(c2St-- nv St 2M2 dr r rsC2r2 dr c2
where M = v/c is the flow Mach number, and c the constant sound speed. The
cross-sectional area of a flow-tube is written as A(r) = As(r/r2)f(r)'s with r
the coronal base, and f(r) a function parameterizing the divergence from spheri_
cal symmetry. The properties of the solution topologies of the steady state
limit (S/_ = 0) are discussed in detail in Habbal and Tsinganos [1983]. We
merely note here the equivalent effects of rapid area divergence (d in f/dr) and
momentumaddition (D/c2) in producing topological changes in the solutions, and
the possibility for steady shock transitions to occur between critical solutions
as determined by the condition
VlV2 = c2 (4)
3. Results
The changes in the steady transonic wind solution as a result of changes,
f(r), in the areal divergence of the flow tube, where [see Kopp and Holzer[1976]]
(r-rl)/q e(rs-rl)/e
f(r) fmax e +1-(fmax-1) (5)
(r-rl)/O
e +I
are shown as thick solid lines in Figure I for different values of fmax" Two
multiple transonic solutions coexist as soon as the velocity becomes supersonic
at an inner critical point, Figure Id. For a further increase in the areal
divergence of the flow tube, three solutions are possible and consist of one con-
tinuous transonic solution and two transonic solutions involving a steady shock
transition between critical solutions, Figure le-lf.
The temporal behavior of the wind profile with initially one critical point
to a final steady profile with multiple critical points as a result of momentum
addition is shown in Figures 2 and 3. In Figure 2, the strength, DO, is such
that the steady state equations allow for multiple solutions to coexist, such as
in Figure le. The resulting steady state velocity and density solutions are
shown in the top panels, Figure 2a-2b, while the time evolution profiles are
shown below. To vary the rate of momentum addition, we introduce the parameter
T , where
214
--t/T -((r-rp)/_) 2
D(r,t) : D0(I _ e ) e (6)
IOe I f_x: 1.6 ' fmax= 1.8 ' 'fmax= 4 '
r
'[ ' ............ 6F
c 21 _\ ::
IF'U_ _,
0 2 4 6 8 I0 0 2 4 6 8 I0 0 2 4 6 8 I0
R/Re
Figure I. Sequence of solution topologies resulting from an increase in the area
divergence of the flow tube, as indicated by fmax' for r I = 2 rs and _ = 0.1 rs
[see (5)]. Additional critical points arise in the flow as the area function,
dlnf/dr, intersects the curve g= GMsC2r2-2/r. The transonic wind solutions are
drawn as thick solid lines, and the shock transitions, when present, are indi-
cated by the wavy lines satisfying the condition (4). The wind is assumed isoth-
ermal at T = 1.2x106 K with a sonic point at 4.8 rs for a spherically symmetric
flow [from Habbal and Tsinganos [1983]].
The results shown correspond to three different values of T . The integration
time step is 10 s, and the time interval between neighboring curves is 104 s.
The initial state is the solution to the steady state solar wind equation of
motion with no momentum addition for an isothermal wind at T = 1.2xi0 b K, with a
sonic point at 4.8 rs. The momentum is then applied in time either slowly (large
T), or rapidly (small T ). For T = 4XI0 q S, Figure 2c-2d, the velocity and
density profiles evolve to form a standing shock at the inner position predicted
by the steady state calculations, while for t = 2 X 104 S, Figure 2e-2f, the
velocity and density profiles evolve with a standing shock further downstream
215
108 , i _ IOJO , ,
'EE io7
_ I°8
_' jo6 jo 6
_o
q i r I i l i
108 + _ , _ i i , + i i IO I0
E 107 108
> l0 6 106 c
- l -o oc .__.
;'_ 10 8 6 I0 I0
I05 I0 4
,o__ _ _,o_ _- -
_o6-_ z ._°,06 _Y-:----
d) T= 4 I0 s -:_-
T=4 I0% CI 105 0 104- - _ ...... _ •
10 8
--_ • 7 " T i , i ....... r-......... 7-- • ]1010
-_" ?
I m
ro 7
E - 108
:'°+o t t'0+ E loiO_" 108 6
10 4l0 5
,o'E + _,o° i _ -
2 -_ 106
E
e) ol f) r=2 Io%105 104 I [ I I I
i0 8 I0
-- l l l -I i I , I I I0
E 107 108 6 I0 I0 I08 E
! "_>io6 ____ .... %, _ ,06=
o _o8E /J 107 E .-
> 10 4
,o_ ,o_-_ 2 -_-,o6
T=I0s g) = c h) r=10s .... __
I i _ J 105 0 104 I l J I I
0 2 4 6 8 I0 TIME 0 2 4 6 8 I0
R/Ro (xlOSs) R/Ro
Figure 2. Sequence of time evolution of the velocity and density profiles to a
final steady state as a result of momentum addition, with D O = 2.Tx103,a = 0.3,
and rp = 3r_, for different values of T [see (6)]. The sequence shows how each
of the stea_y multiple transonle solutions, shown as dark solid lines in (a)-(b),
are accessible. In (c)-(d), the wind profile evolves in time to the steady state
solution with the standing shock at the inner position for T = 4xI04 s. The
second standing shock develops for a slightly smaller value of T = 2XI04 S,
(e)-(f). Fop w = 10 s, the veloeity and density profiles evolve to continuous
ones.
216
than in the previous case. By reducing T further to T = 10 S, the velocitj and
density profiles evolve in such a way that a shock discontinuity develops only
temporarily, as soon as the velocity becomes sonic closer to the base, but
disperses and propagates outwards. Hence, by changing the rate of momentum depo-
sition, this example illustrates how each of the multiple solutions is physically
realizable. For a further increase in D O (Figure 3), the condition for the
existence of steady shocks is no longer satisfied. The velocity and density pro-
files evolve to continuous steady solutions for either value of T , despite the
temporary formation of a shock discontinuity.
4. Conclusion
In the present study we have illustrated how the multiplicity of the tran-
sonic solutions to the steady isothermal wind equations arises as a result of
changes in the areal divergence of the flow tube and/or as a result of momentum
addition. By changing the rate of momentum deposition, and by following the
corresponding temporal evolution pattern of the wind profile with initially one
critical point, we have shown how each one of the multiple solutions is physi-
cally realizable and corresponds to different intermediate temporal states of the
solar wind. In particular, we have shown how a transonic solution with a stand-
ing shock could occur in the wind flow as predicted by Holzer [1977] (albeit
without the restriction of an isothermal flow).
The multiplicity of the transonic solutions to the steady isothermal solar
wind equations, however, is not unique to this study. Multiple solutions have
originally been found by Leer et al. [1982] in their study of Alfv4n wave driven
stellar winds. They find two continuous transonic solutions characterized by
different velocities at the base, for a certain range of wave damping lengths.
In our case, the steady multiple transonic solutions exhibit some form of degen-
eracy since they all have the same velocity at the base, although the continuous
solution and those involving a shock transition have different asymptotic flow
speeds.
Finally, as an example based on our results, we note that if the rate of
change of momentum addition or the temporal change in the area divergence of a
flow tube, such as in a coronal hole, occurs over time scales of the order of
104 s, and the final value reached is maintained for time scales of the order of
105 s, i.e. days, a standing shock can develop within 5 rs from the coronal base
for a solar wind at say 1.2xi06 K. In a realistic solar wind model the forma-
tion of a standing shock in the solar wind could lead to a reduction in the
energy flux per particle, since the temperature rise in the shock will result in
energy loss by thermal conduction to the base if the shock occurs within I or 2
R s of the inner critical point [E. Leer, private communicaton [1983], see also
Holzer and Leer [1980] for a detailed discussion of conductive solar wind
models]. For an isothermal wind, however, and an isothermal shock, although the
infinite thermal conductivity is implicit, the loss of energy flux to the wind as
a result of shock formation is apparent in the lower asymptotic flow speed, yet
there is no corresponding increase in particle flux at the base when compared
with the particle flux associated with the continuous transonic solution.
Acknowledgements. We thank Dr. Egil Leer for stimulating discussions. This
work was supported by NASA grants NG-249 (S. R. H.) and NG-79 (K. T. and R. R).
217
10 8
I
_n 7
E Io
O
_10 6
C]
E-
10 5
b)
I I
=I-
105 0 10 4
T: 2 104s C)
1 J 1
d) T: 2 104!
I I
I I 1 I " q I I
'° ,o,C
107 E u 108
6 > c:: 6
,o i
5
I0 0 I0 41
TIME 0
(xl0Ss)
T: lOs e)
I .L I
0 2 4 6 8 10 2 4
R/R e
f) T=10s
I 1
i
10 4
I __.J__ J
6 8 10
R/R o
I ...... • _ _ I
I0
I V 1 0
10 8 'E
C
10 6
IZl
C
10 4
Figure 3. Same as Figure 2 for a strength of momentum addition such that the
steady state equations do not admit multiple solutions. The temporal behavior of
the wind profile is such that shock discontinuities form temporarily in the flow
but disperse and propagate outwards.
References
Habbal, S. R., and K. Tslnganos, Multiple Transonic Solutions and a New Class of
Shock Transitions in Steady Isothermal Solar and Stellar Winds, to appear in
J. Geophys. Res., 1983.
HoTzer, T. E._--Effects of Rapidly Diverging Flow, Heat Addition and Momentum
218
Addition in the Solar Wind and Stellar Winds, J. Geophys. Res., 82, 23, 1977.
Holzer, T. E., and E. Leer, Conductive Solar Wind Models in Rapidly Diverging
Flow Geometries, J. Geophys. Res., 8_55,4665, 1980.
Kopp, R. A., and T. E. Holzer, Dynamics of Coronal Hole Regions. I. Steady
Polytropic Flows with Multiple Critical Points, Solar Ph_slos, 49, 43, 1976.
Leer, E., Holzer, T. E. and T. FI_, Acceleration of the Solar Wind, Space Sci.
Rev., 33, 161, 1982.
Lee_,E. and T. E. HolzeP, Energy Addition in the SolaP Wind, J. Geophys. Res.,
8_.55, "681, 1980.
219


Multiple transonic solutions and a new class of shock transitions in solar and stellar winds

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