Stellar winds and other analogous astrophysical flows can be described, to lowest order, by the familiar one dimensional hydrodynamic equations which, being nonlinear, admit in some instances discontinuous as well as continuous transonic solutions for identical inner boundary conditions. The characteristics of the time dependent differential equations of motion are described to show how a perturbation changes profile in time and, under well defined conditions, develops into a stationary shock discontinuity. The formation of standing shocks in wind type astrophysical flows depends on the fulfillment of appropriate necessary conditions, which are determined by the conservation of mass, momentum and energy across the discontinuity, and certain sufficient conditions, which are determined by the flow's history

Habbal, S. R.

Rosner, R.

Tsinganos, K.

English

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FORMATION OF STANDING SHOCKS IN STELLAR WINDS
AND RELATED ASTROPHYSICAL FLOWS.
K. Tsinganos, S.R. Habbal, R. Rosner
Harvard-Smithsonian Center for Astrophysics
Cambridge, MA, 02138.
ABSTRACT
_tellar winds and other analogous astrophysical flows can be described, to lowest order, by the
familiar role-dimensional hydrodynamic equations which, being nonlinear, admit in some instances
discontinuous as well as continuous transonic solutions for identical inner boundary conditions. We
elaborate on the physics underling the formation of the discontinuous transonic solutions; discuss
cl_ractcriyrics of the time-dependent differential equations of motion to show how a perturbation
changes profile in time and, under well-defined conditions, develops into a stationary shock discon o
t.inuity; and show how the formation of standing shocks in wind-type astrophysical flows depends on
the fulfillment of appropriate necessa_ conditions, which are determined by the conservation of
mass, momentum and energy across the discontinuity, and certain suffwient conditions, which are
determined by the flow's "history'.
L Introduction
The present pa_r continues the study of the solution topologies of the time-dependent isother-
mal wind equations presented by Habbal, Rosner & Tsinganos (198_; Paper I) in these Proceedings;
these two studies are motivated by the desire to understand how the simplest hydrodynamic equations
describing spherically symmetric solar-wind type astrophysical flows (Parker 1965) respond to pertur-
bations which disturb the radial symmetry of the outflow (Kopp & Hoher 1976; Munro & Jackson
1977) or, equivalently (cfo Habbal & Tsinganos 1983), perturb the outflow by means of a localized
non-thermal momentum deposition in the thermal gas (Holzer 1977). In this paper, we consider the
consequences of a localized phenomenological momentum perturbation term of the form (Paper I)
[
D[Do, Rp,C,r] (R, t) = D Otl-e j dyn/gm , (1)
where the free parameters D . R_, o', and 1" characterize the amplitude, location, spatial and tern-
poral width, respectively, of t_'e f_nction D(R,t). The detailed dependence of the steady solutions of
the governing hydrostatic equations on the parameters D o, R_, and e for 'r ,, 0 can be found in
Habbai & Tsinganos (1985). The specific purpose of this noteP_ to further elaborate on the physics
underlying the response of the governing hydrodynamic equations to perturbations of the form (1)
above (discussed in Paper I), and, in particular, to discuss qualitatively the conditions leading to the
formation of shocks in the flow. We shall show that in certain limiting cases much of the relatively
complex behavior can be understoodon the basisof the straightforwardtheoryof characteristics.
2. Nature of the Equations of Motion
In the following we discuss briefly the nature of the equations of motion and how they allow
the development of shocks in the flow. First, it is convenient to write the continuity and momentum
balance equationsin the followingform
op ap w OA 0A
+Av +Ap = p et OR '
0# _, 0v GM
pv-p T2- , (2b)
289
https://ntrs.nasa.gov/search.jsp?R=19840005020 2020-03-22T07:38:49+00:00Z
where c = [dpldp] 112 is the isothermal sound speed and A(R,t) and D(R,t) axe the g_vtn functions of
the flow tube's cross section and non-thermal momentum deposition, respectively.
Eqs. (2) form a system of two quasilinear partial differential equations of the first order for the
functions p and v of the two independent variables R and t, for which solutions in closed form are
not known. However, the qualitative behavior of their solutions and the associated physical effects
can be simply upderstood by exploiting the hyperbolic nature of these equations; the formal theory
of hyperbolic partial differential equations shows then that there exist two characteristic directions at
every point of the (R,t)-plane which define the characteristics of Eqs. (2) (Courant & Friedrichs 1948),
dR
-- V+C (_)
dt
The physical significance of these characteristics can be understood as follows.
velocity in terms of the potential ,(R,t),
• I_(R,t) 0"(R,t) v 2 Ic2d, GM I
- + D(_) d_[OR - v , 8t 2 p R
Expressing the
(4)
we obtain a single differential equation for ,(R,t),
82 . 02 . 02" _nA GM] a'nA 0 y R+2v_ +(v2-c 2)_ = v[c2 _-_ +D-_ +c2 _ + Ot D(_,t)d_[.
R o
(5)
Consider then a small perturbation 6"(R,t) superimposed on some solution ._(R,t) of Eq. (5).
Following Landau & Lifshitz (197'5), we substitute • ffi "o ÷ _I_ in (5) and obtain toViowest order the
following equation for _,
a2/t' 025" 2 a2_
+ o ot + (Vo - = 0 , (6)
where Vo--fl_o/0R is the '_azkgruund" flow field (see below). We write _I_ ffi Be i_, where B is a
slowly varying function of R and t and the "eikonal" • is almost linear in R and t; the "group velo-
city" of the disturbance is then dR/dt = dco/dk, where k = i3_/i3R and co = 0_liX. We obtain from
(6) for this "group velocity" of the perturbation,
dR
dt - Vo(R't) + c , (7)
a relation which coincides with the slope ($) of the characteristic directions on the plane R-t. From
the physical point of view Eq. (7) gives the velocity of propagation of sound waves in the moving
gas relative to some fixed coordinate system; and since a disturbance of finite amplitude and dura-
tion can be regarded as a superposition of a sequence of small amplitude and duration perturbations,
Eq. (7) describes the propagation of finite perturbations in the atmosphere as well (Courant &
Friedrichs 19'_8, Landau & Lifshitz 1975). It is evident then that the characteristics represent the
paths of all possible disturbances in the (R-t)-plane. The inevitable result of having a non-uniform
velocity field, dvJdR ¢ 0, is that the profile of some initial perturbation changes as it propagates in
the atmosphere. "For example, if dvo/dR < 0 somewhere in the flow, an outward-propagating wave
finds its leading segment travelling flower than its trailing edge; the result is that its amplitude pro-
file "steepens". The perturbation then may "break," i.e., it may steepen enough so that the total den-
sity and velocity are no longer single-valued; in that case the characteristics intersect (Landau &
Lifshitz 1975), and a shock has been formed in the flow. Whether this indeed leads to an equili-
brium solution with a discontinuity depends on two facts: first, on the existence of such equilibrium
solutions involving shocks for the given set of spatial parameters in the steady state equations (Hab-
hal & Tsinganos 1985); and second, on the temporal history of the flow, as discussed below.
290
3. Evolution of the Perturbation and Shock Formation
We would now like to see whether the above discussion of characteristics can shed any light on
an understanding of the behavior of a perturbed isothermal wind flow. We begin with a steady
and isothermal (single critical point, or Parker) wind flow at t-0, and apply a perturbation of the
type given in Eq. (1). As shown in Paper I, which of the (maximally three in the present, simplest
case) distinct steady solutions the perturbed flow finally approaches depends upon the time scale
parameter 1" of the perturbation; whereas the character of these possible alternative steady states is
determined by the remaining parameters of the perturbation I_)_ R_, and o'], which fix the spatial
form and strength of the perturbation (Habbal & Tsinganos ._,. Ii_ow is one of these steady states
selected by virtue of the time scale of the perturbation?
The first point to note is that the flow is characterized by three distinct time scales: the advec-
tion time scale _'affiRIv, the downstream signal propagation time scale 1"+ = Ri(v+c) [corresponding to
propagation along the C + characteristics], and the upstream signal propagation time scale I"_ ffiR/(v-
c) [corresponding to propagation along the C- characteristics];, the cruci_ element is that although I"a
and 1"+ are comparable in the region of interest (i.e., of order of $ x 10" sec near the sonic point), _'.
may be very large;, the question is then how 1" compares with these three flow time scales.
Consider then the evolution of the solutions sketched in Figs. (a)-(d) [see also Fig. 2 of Paper I],
which qualitatively reproduce the essential features of the time-dependent solutions. In each case, we
have sketched the velocity profiles of the initial Parker-type flow field, vin(R,t=0), the final equili-
brium flow field v,,,_(R,t "_) consistent with some given set of the spatial parameters [D o, Rp, o']
characterizing D('R_"_ahd, finally, an intermediate velocity field v(R,_t) which results after, say, a time
at of the order 10"z secs of momentum deposition. The question we would like to address then is:
given the initial state v .(R,t=0) and the parameters [Dn, R_, ¢, ':'] of our momentum deposition,
what is the final state tgards which the intermediate s['ate _(R,St) evolves? That is, which of the
three possible states - the continuous transonic solution, or any one of the allowed (if any) discon-
tinuous transonic solutions -- the system chooses to relax to after a sufficiently long evolution?
In the first case (Fig. a), no difficulty arises: in this case, the asymptotic value of D is suffi-
ciently small that the steady perturbed solution (corresponding to t -_) is unique and continuous, and
always passes through the outer critical point; the eventual evolution of the intermediate solution is
hence unambiguous. This is, however, not the case for the three other examples given, for which
the asymptotic value of D is sufficiently large to resuk asymptotically in three distinct steady tran-
sonic solutions. In the first of the latter cases (Fig. b), I" is large (-40,000 secs) when compared to _'+
and Ca everywhere, and when compared to 1"_ almost everywhere. Hence the flow profile evolves
slowly, and the propagation of disturbances along characteristics may be viewed as occurring on a
background flow field which is relatively steady almost everywhere; the exceptional location is near
the sonic point(s), where the C- characteristic becomes vertical (as can be seen from the sketch of the
C- characteristics in Fig. f, derived from the numerical solutions given in Paper I). Thus, the flow
is near equilibrium essentially during the entire course of its history; but which sequence of equili-
bria is followed? The key point is that this sequence of equilibria (corresponding to a gradually
increasing amplitude for D o, cf. Habbai & Tsinganos 1983) is characterized by always having an
outer sonic point downstream from the (eventual) inner critical point, and hence always has a region
of subsonic flow between these two "x-type" critical points. Hence, once the flow becomes supersonic
near the (eventual) inner critical point, the C- characteristics are "trapped" between the two critical
points, and the downstream-facing C- characteristics emanating just downstream from the inner criti-
cal point must intersect the upstream-facing C- characteristics emanating from the subsonic region
just upstream from the outer critical point; a shock results (Fig. f), and the subsequent sequence of
quasi-steady solutions corresponds to the discontinuous solution branch of the steady equations (for
this choice of q', that corresponding to the upstream shock). Note that from the characteristics one
may readily obtain the time scale for, viz., formation of shocks by locating their first intersection.
The opposite extreme is obtained for very small f (_10 secs; Fig. d); in this case the time scale
for the perturbation is far shorter than any signal propagation time scale in the flow. Hence, the
flow is never near any equilibrium solution during the time of significant change in the perturbing
momentum deposition term; for 0 < t < 1", the flow's evolution is essentially entirely determined by
291
vlc
(a)
2 R 4 6 R
P
v/c
(b)
zZ
.....-" " v (R, Bt)
2 Rp 4 6 R
v/c (c)
I --_ -
i _///A///_ I I
2 Rp 4 6 R
v/c (d)
//7 \
f .... _"-.... "- V_q<R,t--_l
"" 1_,_7////X////7"/>_1 I
2 Rp 4 6 R
t (secs)
6 x 105
3 x 105
2 4 6 R
(e)
Figures (a)-(d)
Figures (e)-(f)
t(secs)
6xlO 5
3xlO 5
V:C
V<C
I
V>C
2 4
(f)
V:C
V>C
6 R
are sketches (taken from the solutions obtained in Paper I) of the solution topologies
for the velocity field of the wind equations (2) with a localized momentum deposition
function similar to the gaussian given by expression (1), for different values of the
parameters D,.,, Rn, o" and 1" and at three representative times: t=0, t-St>0 and _®.
The dotted soluticgn Is the initial velocity v_. (R,t-0), the dashed solution is the inter-
mediate velocJty v(Rjt) and the solid veloc_ curves, %, (R,r_), represent the solu-
t/on topologies of the steady state equations (2). All figures (a)-(d) have the same
values of the parameters Rnand ¢r while figures (b)-(d) correspond to the same set of
the spatial parameters [Dn, rRn, o'] with ¢ decreasing from (b) to (d). The two verti-
cal solid lines in figures (l))-(d_ indtcate the location of the steady state shocks.
are sketches of the C" characteristics. In Fig. (e) are plotted the fan-like characteristics
of the initial flow Vtn(R,t-0); and in Fig. (f)the intersecting characteristics of the
evolving flow field (Fig. b). The effect of this momentum addition with a large
value of ,r is to "turn" downstream the characteristics just downstream of the inner
critical point such that they intersect the upstream facing characteristics just upstream
of the outer critical point.
292
the local acceleration due to the imposed perturbation. Thus, by the time the perturbation has
reached its final amplitude and form, the flow has been locally accelerated (by the external force) to
virtually its final (steady) spatial form throughout the region of application of the perturbation.
Note that the discontinuity which temporarily forms is advected downstream from the position of the
second shock because the C- characteristics immediately downstream from this position face down-
stream; the continuous transonic solution is obtained.
Finally, in the intermediate case sketched in Fig. (c), the intermediate solution v(R,_t) finds itself
close to the equilibrium solution which involves the second (downstream) shock. Subsequent propa-
gation along the C- characteristics leads to the steepening of the velocity profile and the intersection
of the characteristics at a location between the two equilibrium shocks; the intersection is convected
downstream until it finds an equilibrium at the position of the second shock, and the system relaxes
to the equilibrium solution which involves this second shock.
In all these cases the discontinuity forms because of the dispersive properties of the non-
uniform background flow which lead to the gradual intersection of the C- characteristics. The value
of the time constant of the perturbation _" determines how soon and where in the flow, in relation to
the position of the equilibrium shocks, this intersection occurs. In other words, I" determines how
rapidly the fan-like C characteristics of the unperturbed flow (Fig. e) will be "turned" so as to "face"
one another, Le., converge, and inevitably intersect before the changes in the background flow are
convected downstream past the positions of the equilibrium shocks (Fig. f). Notice that propagation
along the C + characteristics does not lead to any interesting effects since the changes are advecr_
away relatively fast due to the smaller value of ,r..
4. Summary
We have shown that the formation of standing shocks in wind-type astrophysical flows with
some effective, localized, non-thermal momentum deposition depending on the spatial parameters D_
R_ and o" - which determine its amplitude, location and width, respectively - and the temporal
p_meter _" - which determines iB temporal width - requires both necessa_ and suffic_en_ condi-
tions. The _cessm,,y conditions are simply the Rankine-Hugoniot relations for the conservation of
mass, momentum and energy - but not entropy - across the shock discontinuity, which in :urn depend
on the set of the spatial parameters [Do, Ro, _]. The _ff_tt conditions are determined by the
detailed temporal evolution of the non-thend-al momentum addition and, in particular, on the inter-
section of the C- characteristics of the flow upstream of the position of the equilibrium shocks,
which in turn depends on the fourth parameter I". It is expected that such standing shocks might be
formed in astrophysical flows such as the solar and stellar winds, or flows in astrophysical jets, when
these necessary and sufficient conditions are satisfied.
Acknowledgments: This work was supported by NASA grants NAGW-2_t9 (S.RJ-I) and NAGW-79
(R.R & K.T) at the Harvard-Smithsonian Center for Astrophysics.
5. References
Courant, R. and K.O. Friedrichs, Sup_ersortic Flow and Shock Waves. Interscience, New York, 1948.
Habbal, S.R., R. Rosner and K. Tsinganos, Multiple Transonic Solutions and a New Class of Shock
Transitions in Solar and Stellar Winds, these P_r._qfy,_lRl_ 1988, ('Paper I).
Habbal, S.R. and K. Tsinganos, Multiple Transonic Solutions and a New Class of Shock Transitions
in Steady Isothermal Solar and Stellar Winds, 1. Geophys. Res.. in press, 1985.
Holzer, T.E., Effects of Rapidly Diverging Flow, Heat Addition and Momentum Addition in the
Solar Wind and Stellar Winds, J. Gfophys. Re#_ 82. 2_, 1977.
Kopp, R.A. and T. Holm, Dynamics of Coronal Hole Regions. I. Steady Polytropic Flows with Mul-
tiple Critical Points, Solar Phys.. 49. 45, 1976.
Landau, L.D. and E.M. Lifshitz, Fluid Mechanics. Pergamon Press, 1975.
Munro, R.H. and B.V. Jackson, Geometry and Density of a Polar Coronal Hole, _ 874, 1977.
Parker, E.N., Interp_lanet_ry Dynamical Processes. Interscience, New York, 1965.
293


Formation of standing shocks in stellar winds and related astrophysical flows

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