Navier-Stokes equation in predicting solar wind flow patter

Scarf, F. L.

English

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In our most recent dlscussion of solar Wind theory (Scarf and 
Noble, 1965), no deta i led  predict ion of t h e  f l o w  pattern near t he  earth 
u . 2  was glven. For the best fit e d u t i o n  ay- 1.5 x u) /cm ~ e c  at r = 10% 
and the cantfnui ty  relation, nur = c, then ylelde B I ~  uaanbiguow f lux 
value at the earth, JaV. - 3.2b x 10 ione/cp aec, krt the pxwportlans 
of n and u which make up t h i s  flux must k derived by 6 0 l ~ h g  the dpa- 
mica1 equations. Unfortunately, t he  v r l l d l t y  of t h e  Nsvier-Stokes 
equations Kith conventional t ransport  coef f ic ien ts  becorp.6 questionable 
long before the  earth it3 reached. 
2 
8 2 
The basic  problem has t o  do w i t h  the  aesumption t/L << 1, used 
to derive the  Bvier -S tokes  equation by t he  chspman-&skog technique 
and the correct  L must first be i den t i f i ed .  
L[dN/drf N, then t h e  or ig lna l  so lu t lans  of the conductive heating 
eqyations indicate t h a t  4 
i s  not so e a s i l y  solved, and the camplications become Important much 
c lose r  t o  the  sun. It has already been noted that these conductive heat- 
ing equations possess unphysical so lu t ions  with T(r)  = 0, r < - and 
T ( r )  
hope that these annoying so lu t iaae  were only present  becsuse the viscous 
forces (which presumably -bit large velocity gradients) had been 
a r t i f i c i a l l y  oanitted, Scarf and Roble (1964, 1965) introduced the ap- 
propr ia te  viscous terms i n t o  the XmAer-Stokes equations. 
symmetric f l o w  the complete Mavier-Stokes equations have t h e  form 
If L IS s i q l y  defined by 
L near 75 - 100'Ro . However, t he  problem 
T, > 0, r 4 0; in each case u + at  the singularity. With the 
For spher ica l ly  
and 
2 
(2) 
r2k(T) dT 4 p(T)r2 f du u mu - 5  - - T - + z * T -  2 r c b 3 C  r 
2 CM m - - -  \u - -> = E 
1 
I 
2 
where E sn3 c = cur zre  constasts  of m t i o n .  For full;; Sosized kydrogez, 
LL = LL~(T/+,)~",  with LL(T ' given i n  Eq. (9) of Scarf and Noble (19.5). 
However, unl ike the  condx: ive coef f ic ien t ,  k(T), t h e  Viscous coe f f i c i en t  
does change s ign i f i can t ly  as the  gas composition i s  varied.  The add i t ion  
of 10% 5eiiurn! reduces p t o  about 73% of i t s  v a l u e  for f u l i g  ionized kLdro- 
gen. 
eqaal  t o  8.013. This explains why viscous contr ibut ions were previously 
neglected. 
camparable, t h e  viscous term in t h e  energy equation i s  roughly (1 - 2)s 
of the conductive tern and where JrT > mu /2, t h e  ratio is  even smaller. 
0 :  
%xs f o r  the  s o l a r  = ~ i n d  t h e  Frandt l  number, +q.i!rok, i s  approxi-mtely 
When t h e  t h e m 1  and k i n e t i c  energies  and gradien ts  are 
2 
This kind of srgument is Indeed apPropriate -,&en one discusses  
n o m :  subsonic flow, but it shcrild already be c l e a r  froxr Eq. ( 2 )  t h a t  8 
ser ious  e r r o r  is possible when the flow i s  supersonic. Even though w(T) 
i s  nunerical lg  ver;- small, If u i s  l a rge  and constant,  t he  tern b ! T ) u  2 r/3 
can play a very Important r o l e  In  t h e  energy balance equation as r in-  
CreaSPS .  
-%?erica1 in t eg ra t ion  of Zqs. (l), (2) confirms this expectat ion,  
bxt %e w a : ~  i n  which t l e  viscous s t r e s s e s  modlf : ;  t h e  flow appears a t  f i r s t  
glance, 50 be .mpk:i-sical. 
t yp ica l  r e s u l t s  derived b;: in tegra t ing  i n  koth d i r ec t ions  from tr-e cross-  
over. 73e parameter E i s  e s sen t i a l ly  the Prandt l  ncxriher times 
Figure 4 of Scarf and Noble ( l?k5)  shows some 
r 
d A = 2k(T0)GK m/& T c where Tn i s  the temperature a t  t>e  coronal base. 
t h e  case shown there,  A = 200, T 
€3 0 is  t h e  one derived earlier, @-th no viscous terms a t  a l l .  The 
ve loc i ty  p r o f i l e  labeled B = 2.46 is t h e  numerical so lu t ion  of t h e  Navler- 
Stokes equation w i t h  t h e  appropriate  Coeff ic ient  of v i scos i ty  f o r  t h e  
coronal gas, and t o  t h e  accuracy shown i n  t h i s  f igure ,  t h e  temperature 
d i s t r fbu t ion  i s  unchanged by t h e  addi t ion  of viscos i ty .  
For 
0 0 " 
= 1.5 x loD-& and t h e  curve labe led  0 
It +:as note5 by Scarf and Noble t h a t  In  t h e  subsonic region t h e  
viscous terms produce a negl ig ib le  change i n  u(r) . 
because& << mk and (1/2)m S3rT  here.  it can a l s o  be seen t h a t  t h e  
viscous terms do inzuce a s lgn i f i can t  change In  u(rf i n  t h e  supersonic 
region.  
This w a s  expected 
2 
Again, this i s  not  ilnexpected; although w/mk remiins small, 
2 
(1/2)m 
energy equation i s  conparable t o  t h e  conductlve t e r n .  Hovever. it was 
found t h a t  u(r,B) i s  greeter than u ( r , B  E 0 ) .  Physical ly ,  one would expect 
viscous d i s s ipa t ion  t o  y i e l d  a lower f l o w  speed r e l a t i v e  t o  the  sun, and 
not a bigher one. 
becomes l a r g e  compared t o i r ~ ,  mid t he  v iscom cont r ibu t ion  t o  tke  
The as-ver ts t h i ~  paradox Was found by tryiug t o  vary t h e  Francit1 
number t o  approach B = 0. 
cuhted using artificial and unphysical d u e s  for w(T0)? keeping k(To) 
fired. 
at any given -us does Indeed decrease in a physically sensible m e r .  
However, as E-+ 0, we  do not approach the  curve B 
of t he  non-viscous conductive heat ing equations.  
The curves for B = 0.1, 1.0, and 3.56 w e r e  cal- 
Xt was determined that as p(To) (or B) is increased, t h e  flow speed 
0 which i s  t h e  s o l u t i o n  -
In f a c t ,  t h e  "paradox" has an exact a n a l o a  when thermal conduction 
alone i s  considered. If I.L and k a r e  both neglected,  then t h e  flow i s  
ad iaba t i c  and T ( r )  +I- -4/3 . 
then T(r)  -. (k(To)r)  -2/5 or  (r -*I7 + Ck(To) -1 r -'/7 + . . . I .  me so lu t ions  
wi th  f i n i t e  thermal conductivity do not y i e l d  ad iaba t i c  flow i n  the  l i m l t  
k(To) -. 0, and s imi la r l - - ,  t he  so lu t ions  w i t h  small but f i n i t e  u(To) do 
not  go over t o  t h e  u(T ) 0 solut ions as *(To) 0.  From a mithematical 
po in t  of view, t h i s  i s  a l so  clear .  The d i f f e r e n t i a l  equations are of t h e  
form dT/dr = G/k( T) , dx/dr = F,/LL.(T) and k 3 0; u E 0 represent  siagu- 
lar i t ies .  
Rarefied Gases," Schaaf and Ckambre' (1961) renark on page 33, "Solutions 
are thus sing?dar i n  t h e  v i scos i ty  (o r  equivalent ly ,  t h e  mean f r e e  p a t h ) ,  
and t i e  boundary l a y e r  solut ion ( f o r  ci - 0)  cannot be obtained by pertur- 
bat ion schemes s t a r t i n g  with t he  Invisc id  so lu t ion ."  
However, i f  only the viscous terms are omitted,  
-
0 -
m e  r e s u l t  i s  well known t o  aerodynamicists. I n  "Flow of 
I f  t h e  B = 2.46 curve i n  Figure 4 of Scarf and Noble (1965) is 
regarded as  t h e  "correct"  one, then t h i s  so lu t ion  can be used t o  i n v e s t i -  
ga t e  tl?e limits of t h e  continurn or  f l u i d  m o d e l s  of t h e  corona. ?lie first 
question has t o  do with $he precise  meaning of t he  r e s t r i c t i o n  , t / ~  << 1. 
I n  p a r t i c u l a r ,  it must be nscertained which s c a l e  length i s  involved. 
can be determined b:; exsminlng t h e  higker order te,rms i n  t h e  Chapman- 
Ehskog treatment.  
This 
!%e successive approximations a r e  
3 
-u 
(Schaaf and Chambre', 1961, p. 30) and thus t h e  N8~5er-Stokes equations 
should be va l id  i f  p(du/dr)/&T << 1. Since @ fil mn1i2(kT)1i26, t h i s  can 
be wr i t t en  ES 
and tke  zppropriate sca le  ler&% % x n s  out t o  be t h e  one associated with 
t h e  streaming ve loc i t -  . 
3 e  c o m i e t e  expressioc for p'2' 5:  contaim temperature gradients  a s  vel;. 
L-Ae 3avier-Stokes eq:atlons become questionable wFenever one of these  
t e rns  i s  s igrAf ican5y l a rge .  In our case,  trie ve loc i ty  gradients  are 
s teep  beyond t h e  crossover k;File t h e  thermal grad ien ts  remain moderate.) 
By inspect ion of the  numerical solut ion,  Scarf and Noble (1964) pointed 
out  t h a t  tl.e above izequal i t l -  i s  not s a t i s f i e d  be:ond r = 20 R 
(mis i s  n o t  a statement of general  vai idi t : . .  
7% 
. 
IC t h e  lar,gJage of aerodwamics, w e  pass from the cont imun 
region t o  t h e  "s l ip  flo-6'' region when tr.e appropriate  lbudsen m a h e r  ap- 
proackes -zit?. 
t h e  region of contimxm f l o w  (& << L) and free molecular flow ( A  >> L). 
Actually, because t k e  solar wind I s  6, magnetized p l a s m ,  there i s  no 
region i n  ;-fiich free flow occurs i n  t h e  classicsl 4 4  gasdpamlcs sense.  
Nevertheless, we can define t h e e  regions.  
Navler-Stokes equatlons tire val id .  For p ( ? * ~ ) / & T  - 1, but CidT/dr' << T, 
t h e  Navier-Stokes equations are  not  va l id ,  but  there are enough c o l l i s i o n s  
t o  maintain a s t a t i s t l c s l  d i s t r i b u t i o n  funct ion Kt,th a well-defined 
t e p e r a t u r e .  For &x&T!c?r 2 T, ordinary two-bdy c o l l i s i o n s  a r e  unimpor- 
tas: and t h e  c h a r e c t e r i s t i c s  of the  p a r t i c l e  d i s t r ibGt lon  funct ions are 
For I nes t ra l  gas, t h e  s l i p  f lov  regime i s  bounded b) 
For (pV-u/nLrT) << 1, t h e  f u l l  
4 
In ordinar: gss?;knam5cs no r igorous  f o r m l a t i o n  of ??.e eqilations 
of motion i s  ava i lab le  f o r  s:ip flow. Tv'arious complex systems of 
4 
equations ( t h e  b m e t t  equations, t h e  B i r t e e n  Moment equations) have k e n  
proposed t o  replace t h e  Xavier-Stokes equations f o r  moderate ?Gmdsec 
nlunbers, but these  replacerneats introfiuce severe difficT;lties, and Con- 
s iderable  doubt e x i s t s  about t h e i r  physical  va1idit;T. 
Ckambre’, 1951, pp. 31-34.) 
w i l l  r e a l l y  be of use kere, sc that  t h e  flow patterns ’have t o  be obtained 
by solving t h e  Boltzmann equation i tself .  However. it appears t h a t  t h e  
TTavier-Stokes equations, with some modifications, provt-de a better des- 
cription of‘ the gas i n  t h e  s l i p  f l o w  regime than  t h e  higher  order  systems. 
(Schaaf and 
It seems l i k e l y  t h a t  no moment equations 
This discusslor, indicates  t%t at  presen? no rigorous treatment 
of t 5 e  so l a r  wind flow i s  possible  beyont! (15 - 2 0 )  Re . 
~ k - a i ~  soTe bound to t-.e range of flow pa t te rns ,  Scarf and N o ~ S e  (19t2t) 
In  order t o  
r n r . 4  + _  +p-i - A  -ly a s s u e d  t h a t  t h e  3avier-Stokes equations are s t r i c t l y  va l id  
~p t o  1” RG, , w i t h  ad iaba t ic  continuurr flow beyond 17 Ea . 
f o r  a q ~ i e t  s o l a r  hind are shown in Table I. Although these  numbers 
obviously are subject  t o  considerable rev is ion  as more r e a l i s t i c  treat-  
~ n t s  of t h e  region beyond (15  - 20) R a re  considered, t he  e n t r i e s  can 
be  xse5 t o  eva lsa te  roughly t k  range of importance of two-body col-  
l’sions. i;e find P . ( T , ~ )  &T/* f o r  r - 75 3o , and two-body col-  
lisiocs cannot serve t o  maintain t h e  & ~ l t z m , m  d i s t r i b u t i o n  f i n c t i o n  
~ : S > ; Z ~ : Y ~ ~ Z I - Z ~ ~ J S .  The r a d i s l  flow itself tends t o  b t r o d u c e  a severe 
z.,niso:ro~:, i n  the ebsefice of a randomizing mechanism (5  .e., T, i s  gen- 
The r e s u l t s  
0 
e r a l l y  m t  eqLa1 t o  T f o r  co l l i s ion le s s  spher ica l  flow becacse t h e  
p a r t i c l e s  x i t h  v parallel t o  the mean flow d i r e c t i o n  tend t o  migrate 
f a r t h e r  than those w i t h  f i n i t e  vI; in t h e  presence of a cen t r a l  fo rce  
f i e l d  tLis snisotropy i s  even more marked, because t h e  p a r t i c l e s  t r a v e l  
on b a l l i s t i c  o r b i t s ) .  Thus, t h e  c lose  agreement between t p e  d i s t r ibu -  
t ? m s  f requent ly  observed a t  the  ea r tk  and a Boltzmann d i s t r i b u t i o n  
in5icstes t n a t  some mechanisn; other  than two-part ic le  s c a t t e r i n g  binds 
t h e  solar w h d  i n t o  a f h i d  and maintains the  s t a t i s t i c a l  spectrum 
>e;, and (‘-3 - 100) R . 
1 
4 
0 
One apprsach t o  t h i s  problem has been attempted b:*. Sturrock and 
Ysrtle (1s-5). 
d e c l h e ,  the e lec t rons  and protons become thermally decoupled. and tney 
The)- point out  t h a t  as t k e  dens i ty  and t e q e r a t u r e s  
5 
scggest use of a two-fluid model i n  the  outer  corona. 
protori temperature near t he  e a r t h  i s  then  a tnu t  3 x 1 0 3 ~  K and T~ - 13 
Altkougk T 
!.Lr,r)m from measurements on Mariner 2, LMP 1, 2 ,  and VELA 2, 3 t h a t  T i s  
general ly  much higher than t M s  value (? - 2 x I O 5  OK). 
particle-wave scattering iii-~t heet the protons.  
The predic ted  
6 0  K. 
has not been w e l l  measured i n  in te rp lane tary  space, it i s  
P 
e 
??IUS, t h e  
P 
Four kinds of i n s t a b i l i t y  vfiich generate waves i n  e c o l l i s i o n l e s s  
plasm have been discussed i n  the l i t e r a t u r e .  For nK(Ti -TL) > B2/4n, t h e  
"firehose" i n s t s b i l i t y  (Parker, 1958) allows fifven waves t o  grow. 
nK(Tl)L/T; > B /B, t h e  s:;stem w i l l  be i n s t a b l e  with respect t o  t h e  growth 
of magnetosonic waves. 30th of these  i n s t a b i l i t i e s  T o d d  e a s i l y  arise 
i n  z5e absence of c o i l i s i o n s  because only the  wave s c a t t e r i s g  m i n t a i n s  
T zz T, . Hoxever, ve ioc i ty  space an iso t ropies  a lso allow l o w  frequency 
e l e c t r o s t a t i c  p l a s m  o s c i l l a t i o n s  t o  grow t o  l a rge  amplitudes (Harris, 
i3tl; 3osenbluth an3 Post ,  lq6:). F ina l ly ,  i f  f l uc tua t ions  i n  tke  s o l a r  
wind speed produce e l e c t r i c  currents  o r  longi tudina l  e l e c t r i c  f i e l d s ,  ion  
wave d r i f t  i n s t a b i l i t i e s  xi11 ke t r iggered .  The in te rp lane tary  e l e c t r i c  
f i e l d ,  E = -U x E/C, i s  normally about 10 ' grea te r  tkan t h e  rmawa: f i e l d .  
8 1  
If 
2 .. 
I 
1 
- 
4 * 4  
% = Ee//e-L, where 4, i s  tr,e mean free p a t h ,  
s ince  it i s  normal t o  E. However, :t voul:! 
+ 
4 4  
E/1 E? I yie l5 ing  e fl>ict.:ating e i e c t r i c  f i e12  
leed  T o  a runaway sitcat!.on. 
-e 
but E does n o t  produce ruzawa) 
seem t 5 a t  f luct ,Jat ions In 
$E = -U x 6E/C could e?cjlS 
-4 4 -. 
The ava i lab le  evidence sxggests rather s t rongly that. mgnetosonic 
waves do not have l a rge  anpl i tudes i n  t h e  solar wind. Holzer. McLed, and 
Smith (1965) found a veg. rapid decrease I n  s p e c t r a l  i n t e n s i t y  above 
(0.2 - 0.5)  cps when OGO-1 was i n  in te rp lane tary  space. An i nd i r ec t  sup- 
For t  fo r  t r i s  contect lon comes from tne  IMP-1 magnetometer r e s u l t s  (Ness, 
e t  ~l., 1?&4), s ince  detect ion of l a r g e  variances i n p l i e s  t F L a t  t! e t i l t e d  
f l zxga te  i s  contaminated by s ign i f i can t  high frequent) noise  (Creenstadt ,  
l Q t 5 ;  Fre2ricks and Sonnet, L 3 r 5 ) .  The general  absence of l a rge  5.40 
n i x t e  vwiznces  i n  i c t e rp l ane ta ry  space %?erefore suggests that  l i t t l e  
h i g h  frequency E g n e t i c  ioise was present .  Thus, the  thermalizing waves 
a re  probakly long period magnetic o s c i l l a t i o n s  of t h e  Alfven v a r i e t y ,  o r  
e l e c t r o s t a t i c  plasma o s c i l l a t i o n s .  
, 
5 
I 
9-2 p l a s m  o r c i l l a t i s ? ~  a re  the  m e s  which can accs.int f o r  proton 
F-ezting zost e a s i l y  - - t ~  e s 5 x k a s t i c  cyclotron acce lera t ion  process (S t ix ,  
- / -  lQ;r,. J SCE”’ -, e t  a l . ,  1965; 3-e5r icks,  e t  a i . ,  1965; Sturrock, 1965), and 
e l ec t rons  cen also %e Eccelerated >J these  waves. 
l ? t e A ~ l m & s r y  o k s - n a t i o n s  ( V m  Allen m e  X?k5gis, 1965) of about kl kev 
exergy e lec t rons  s t rongly suggest t s a t  31; i n t e - p l a n e t t r y  acce lera t ion  pro- 
cess is ~ p r ~ t i v e ,  2s GriginaLiy suggested by Parker (1 6 5 ) .  
fu r the r  experimental study of t h e  electromappetic and e ! e c t r o s t a t i c  no ise  
spectrum i n  t h e  solar wind i s  c l e a r l y  needed. 
Indeed, the Mariner 4 
However, 
. . . . . . . . . .  
F. L. Scarf mid L. M. Noble, A I A A  2c,rnal, - 2 ,  1156 - 11~0, 1305. 
F. L. Scarf and L. M. Noble, Astrophls ical  Journal,  - 1L1, 147.1 - 1431, 19~5. 
S. A .  Schaaf and P. L. Ckambre‘, B i n c e t o n  U n i v f * s i t y  Press,  (Princeton, 
F. A .  Sturrock and R .  E.  B r t i e ,  Sublnitted t o  Phys. Rev. L e t t .  
5. N .  Parker, Phys. Rev. - 109, 1874, 1955. 
E. G.  Brrls, J. Nucl. Ehergy, P t .  C, - 2, 138, 1961. 
M. N.  Rosefibluth and R .  F. Post, Phys. Fluids,  - 8, 547, 2965. 
R. E. Holzer, M. G. McLeod, and E.  J. Smitt., subbniftted t o  J. Geop’r?:cs. R e s .  
Ti’. Ness, e t  al.,  J. Geophys. Res. - 69, 3531, 1 9 3 .  
z. 3.  <>reenstadt,  Astroph;:s. J. ,  4n Tress. 
3 .  W. Fredrlcks and C .  F. So-met, to be Fz51ished. 
N . J . ) ,  1961. 
- 
r I. 2. Stix,  Phys. Fla ids ,  1, 2.960, I?&. 
F. L. Scarf, e t  al.,  J. Geophys. R e s .  - 70, 9, 1965. 
R. W. -e&icks, e t  el., J. Geophys. Res., - 70, 21, 19b5. 
P. A .  Sturrock, Phys. Rev., i n  press 
E. N.  Parker, Phys. Rev. Lett. 


Conductive heating of the solar wind. III - Breakdown of the Navier-Stokes equations, 4 June - 4 December 1965

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