Magneto-gravity waves and heating of solar coron

Mc Lellan, A., IV

Winterberg, F.

English

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https://ntrs.nasa.gov/search.jsp?R=19690002117 2020-03-12T06:04:27+00:00Z
IC/68/10
IMENATIONAL ATMIC ENERGY AM=
MM ATIORAL CWM FOR TS'JORETICAL PHr3ICS
RACWTO-GRAYM WAVES AM THE HEATING OF THE SOLAR CORMA *
Alden McLellan I9
and
F. Yinterierg
TRSESTE
Januaxy 1968
* Nork supported by t}w Watimmal Aermautica and Spam
Adminietratims wader Reaearoh Grant RC.R-29-001--016.
* On leave of abconoo frm the eeert Research Inaaitute
and Depa t—ment a;
	 "Univer-si#F n* Wev , moo,
Nevada USA.
ABSTRACT
It is generally believed that the heating of the solar corona
is caused by waves originating in the photosphere and propagating into
the corona where their energy is dissipated. The wediva through which
these waves propagate is in general permeated by magnetic fields complicating
the behavi+ef this propagation considerably.
	
2
t+be 61ve motions in a plasma permeated by constant magnetic and gravitational
fields. feral three wave n6odsa were found 	 the
e. teach
	
was fesEj	 strongly
coupled to each of the three kinds of motion! acoustitg gravity O and
hydromagnetic. However, the Alfven mode was found to be separable from
the dispersion relation and therefore independent of compressibility
and gravity. The local dispersion relation is derived and expressed
in nondimensional form independent of the cc.ustants that describe a
particular atmosphere. From the dispersion relation oae can show that
rising waves propagate either with a constant or a growing wive amplitude
depending on the magnitudes and directions of the gravitational fieldq
magnetic field and the wave vector. The variation of the density with
height is takau into account by a generalised W.K^B. matoode 9qwtiona
are found which give the height at which wave reflection oecureq 11ving
the upper bound for possible wave propagation.
.. 1 ..
MAGOTO-aRAVITr
 
UVES AND THE HEATING OF M SOLAR CORONA
I,	 INTRODUCTiGr7
The extremely high temperatures of the solar corona are
generally believed to be due to the transfer of energy fres the
convection zone by waves. The waves that have been considerod are
acoustic waves, gravity waves and hydromagnetic wawa. One very
special situation of magneto-gravity waves has been treated by TUlj.
However, in a magnetised plasma atmosphere, it Is in priaoiple not
oorreot to consider either one of these modes independently of the
others. All modem interaot with each other and must be eonaidered
simultaneously. Therefore, we have investigated plasma wave propagation
within a magnetized atmosphere of infinite conductivity under the influence
of gravity * in the magtxeto-hydrody-namio IN X.D .) approxisationq which
is valid, for low-frequenoy waves. We only consider the propagation of
these waves and therefore neglect dissipatiee effeots arising from
vi.soosity, elsotrioal resistivity and heat oonduativity.
In the equilibrium state, we allow for the exponential deMi-lance
in height (z-direction) of the density and presxure, as is certainly
the case for an isothermal atmosphere. In ' ,ue second part of this Papers
this variation of density with height in the magnetic field term in the
equation of motion is taken into a000unt by a generalized W.K.B. method.
The addition of a magnetic field oomplioates the problem, not
only by the introduction of another wave mode l but also due to the f&ot
that the magnstio field is a veotor, thereby creating a third dixwticnf
whioh can be aligned arbitrarily with respect to the gravity vector 4-
and the wave vector I.
2. FMAMENTAL EQUATIONS
Me fundamental equations necessary to describe the wave action`
► Daussian units)
Euler equatiot
P[a+(v.a v_^
	
.Vp+Px
- -L Hx(VYH)
(2.1)
In this equation v is the velocity vector of a particle of the
oscillating medium, p is the pzessurer p th. density $
 I_the gravity
vector, and H the angaetio field vmotor. For convenience era ohoose
*_to be along the negative s-direotion (dcenmard ) t =d ve oh*-*" to
orient the 000rdinate system so that the arbitrarily directed unperturbed
magnetic field in the absence of wave notion has no y-oompanent.
ContinuitZ equation
a 
at	 +	 V - (Pv) = 0
	 (2.2)
Second lax of thermodynamics ( adiabatic approximation)
ds _	 + v . Vs
	 0 ,
dt	 at
where s in the specifio entropy.
Equation of sta+( differential form)
(2.3)
&P -as	 ds	 aP I ,°^^	 (z.4)
Ohm-Maritell e&Mtion
bt	 (2.5)
For the equilibrium conditions for an ideal Can at constant
temperature 9
 the variation of density and pressurs with height is
exponent{,al. Thusg are have at equilibrium
—WhP - PO e
and
	
(z.6)
—af h
pe* 
o p  e	 9
where h - a21g f is the "eoale height".
3.	 LMEARI ZATIOH
By assuming that all deviations of the perturbed quantities
from their equilibrium values are saa11 9 we mag put
P(^ t) - [Po + P• L' t )] • 
—a3h
p(,Fj t) s [po + p l (rp t) ) •--a
3h
s(r,t)	 req + s'(Yjt)
mnd
H	 Lio + H1 (19 t)	 f
(3.1)
where Po , to , and H  are oonstant. The quantities Pip p'q a t and
H' 1 together with the velooity v_ - v(rpt) are oonsidered to be
;erturbing quantities of fires order.
By ooneidering the equilibrium oonditions and the linearization
of the basic equatione t eqs. (2.1) 9 (2.2) 9
 (2.3): ( 2 ° 4)v and (2.5).
and upon taking the time derivative of the Suler equation (2.1)9
we have updn substitution
a2 v_- a2
 
sa 
2
+Q[
a2
— 
(? .
 es) p h2   (v • 6s ) 1
a2 	 2
+ — (v • v) e - — (o • e aP d	 ^.h	 hpo	 p	 a
-	 + (V • 1)0(h
l	 r
+	 Hb X{ v x [Vxfz Z )	 a 0
4T-P q
(3.2)
where es is the unit veotor in the a-direction and a is the velocity
of sound. This veotor differential equation ( 3.2) is the lineax-ix-id
equation of notion and it represents a set of three linear homogeneous
differential equations for the unknowns V. , V
	
and v j! .
Using the assumption of a perfect gasp the expression given
above for the seals height, and well-4mown thermodynamic expressions
relating the density, pressure, entropy and temperature, one finds
a2 arm\ 
ds^y 
-
^5 D ds — p0 Vy—  1) (3.3)
where Y is the ratio of speoifio heats.
By the substitution of eq. (3.3) into (3.2) 9 we have the
'Linearized equation of notion
a2 v
a2 grad div v - grad (v•,	 -( ^/ 4 l_) diw v
ate
1
+ -- 0 x curl ourl(v x Ho )	 0
41% q (3.4)
4•	 LOCAL DISPERSION RELVION
In the local. situation '
 we assume the wavelength to be ,amal.l
compared to the scale height. This assumption allows us to consider
joel, as a constant in the last term of eq. (3.q ). Thus # this equation
is now a seoond-order differential equation with oonstant ooeffiofents,
which yields to a plane-ware solution. For plane wavess . we bsve
i(k•r-eat)
where v is now a constant vector.
Upon dubstitutipg eq. (4.1) into eq. (3.4) and performing the
mathematioal operations, we have
W v - a2 (i v) k + i (v	 ) k +	 (k v)
1
+	 Ho x kxrk z(v x Ham )	 • a
4^peq	 L (4.z)
The vector equation ( 4.2) rep=eeerts a Bet of three linear
homogeneous equations for the unknowns va , v	 and v7 . The
condition for a non—tri ►ial solution, the vanishing of the determinant
of the coefficients, is properly called the love" dispersion relation
^k2	 CH kzGJ 6
 + Q4 - a2 k2 - i g j ks - o	 l —°4.9 P eq	 Or Pe q
k)	 H(x	 k)8	 —
+ 6 2 g2 (y — 1)(k2 	+ i g 
y (.202
----- ks +	 -- k?
Orpeq
	4w P oil
r^r
W2 a	
(Ha a k )2
4nPe
q
( the Alfven mode)
(^ . !S)2
+ 2 a 2 k 2	 +
4TrPeq
H2 (11 - k )2 =	 HZ (H0 . k)3%
- Z q Y
(4 r?e 2 '^q)	 00 (47rp q )2
k2
2	 (-O - 102 2	
2) s2 k2(-
	 X)4
- 9 (Y-1)	 (k —k$ --	 --- •O.
4W-Peq 	(47rpea)
(4-J')
Eq. (4.3) oat: be factorized into two parts, where one part
turns out to be the 4iapersion relation for the Alfven mode and the
other part is the dispersion relation for two distinct nodes # whioh
involve the intimate coupling of magneto-hydaodynaaic and gravity
wavo motion. By factorization, we obtain
(4.4)
and
H2
la4 + G,2 C i g Ykz
 - a2 k2 -. °	 x`^ + g2 (Y
 1 ) (k2 - k2)
4 npe 
.q
2 (4° • k )2 2	 H8(o • k) 2
+s	 k +iq^r	 k ^ Q
4T P	 E	 4^-pqq
(4.5)
From eq. (4.5),whioh is quadratic in w i ve obtain two ocher modes
which we oall the + mode and the - mode.
K2
	-	
3/2 1
i + • ( K2 + —) + 1 ( )	 K 00ef^^
Y¢	 L 2	 J
r
0	 K4	 1	 1
+ ^^ K4 +	 + Y^  - $ ` ^^ 3 K2 00620	 2 y^( Yr l) K2 ein28
v K4 008 2 ^ + i (y }3 2 	 aae d - (^Y i Goo CosC2
+( 4 ) 000
2^
(4,6)
Eq. (4.6) i.a writzen in a non-dimensional form by the use of a
oharaoterietio frequency and a oharaoteri atio wave numbers defined by
110
CJ—	 (,)c	 (4•7)and k . 
a2(4mpe 9.)^	 a
The nor -dimensional frequency rnd ware number are
r90
k
Y	 and	 Y
G^^	 ko
and furthermore
b
I e4
a	 is the ratio of
r 0	 HU /1,f
hydrostatic press-are to magnetic preseuzv. 9 is the angle betxeen
k 
and¢_ 9 q is the angle betveen Ho and 
^r 
and T is the angle
b e t ire en is and H
—	 —o
In generr,l, the dispersion relation is cemFlex,vhich implies
the existence of wave., with either exponentially increasing or decreasing
wave amplitude, corresponding to rising or ielling waver, respectively.
This behaviotrcan be easily understood. A rising wave propagates into
a medium of decreasing density. As a result, a decreasing number of
fluid elements will participate in the wave motion, so that in order to
conserve ware energy, the wave ar.plitude has to grow.	 By a similar
argument the wave amplitude for falling waves must decrease.	 In the
follo-ing analysis we will omit the falling waves with decreasing wave
amplitude, since they are of no relevance to the heating of the solar
corona.	 The rising waves wit:l growing wave amplitudes will eventually
steepen into shock wavP3 accompanied by a large energy dissipation,
However, under certain conditions the waves can propagate with constant
amplitude, depending on the magnitude and direction of the wave number
vector, if the restoring forces of compressibility, gravity, and
magnetic field tend to interfere.
The peculiar dependence of growing wave amplitudes >>pon the
angles 9 , t7? , and T is absent in treatment+ which neglect the
coupling of hydromagnetic and internal gravity wave motion. For certain
angle confiburationp and ranges of the wave number there is no wave growth
and hence no shock graves.	 In these cases the wave propagates with
constant amplitude; therefor e the imaginary part of the frequency must
be zero.
	 This condition is s:v 1 in nondimensional form by
B . :F Ej(02 + D2
xhere the upper sign rspreaento the + mode and and the lover sign
represents the — mode, and vh.Are
B - (yi43/2 ^goas9,
^a	 ^4
C - } t4 + YI z Y^  - of Oar
 7
^	 i
1/8 ^ -t3 g2 oos`a	 I),?- sin28
snd
3/2	
oosA + ( )^ coo 8 — (2 Y coo Y cas 7] r3
^.	 W X .B . APPROXIMATION
We assumed above that f.) in the magnetic term in the equation of
motion (3.4) could be considered a constant in the lowat apprcmimation
if the wavelength was small oospaxed with the scale height. A better
approzimstic^ is obtained by taking into a000unt the slowly varying
change of the equilibrius density in the vertical directian by a
generalized W.K.B. method
Tow" this end, we take a Fourier transform of eQ. (3.4) for
the a — y — t dependence and than introduune the a-dapendeaoo bly means
of an unkn own funot i cn f (s) . we Yut
i(kx z +
	 — Wt)
D -
—Il.
where v is a o metant veotvr and
-a
	
f(s)	 -	 A(S) a	 (5.2)
A(z) is a slowly varying function of z. We collect the terms
	
d A ^ d	 &Z ^
	d z 1 d	 and A d 	
into one equation and the terms
	
A ; A d	 and Af d	 into a second equation. In the spirit
of the W.K.B. approximation (short wavelength approximation)p we
neglect terms multiplied by dA and AZ —A . Both of these equations
d t	 dva-
are three-dimensional vector equation s for -'VC and the condition for
a aon-trivial solution is determined by setting the determinant of
the coefficients cf each system of equations equal to zero. From the
first equation ve obtain a first-order differential equation with the
solution
1
CL	 —^
A - const (	 (5.3 )\ d,
wkioh is the expeoted relation between the amplitude and phase from
employing the W.K.B. method. From the second mquation we obtain a first-
ordor differential equation of the sixth degree for the phase li(s).
This differential
 equation oan be faotorised into two differential equatiom
of the second and fourth degree in a aimilar spanner as ibe local dispersion
relation was factorised into two algebraic equations of the second and
fourth order. These two differential equations are
-12.
d j 2	 (^ • kl) d	 (^ k 1)2
+Z1
d	 (Ho ems )	 d s(10 	5)2
4?t w 2
9
(Ro • P-S
(5.4)
d 4	 d	 3	 (^ ° kl)	 6 Y
2
d	 ` CR [d2	 w2 Ho 	 2
+	 1°ea(s) ^	 v kl
d s
	
(Roo eS)2	 2(40 a )2
k o2	 $ Y ( ' kl) 	d	 g I kl
(^ s a )2	 e2	 (10 o s )	 d s	 e2
S Ya 2 4I'	 2 (^ kl^ t
e (.,,o Is )	 ^-•o -•s
and
-13-
4 N 4 4	 247 W kl
+	 (^,^ (s)
	 -	 2	
f eq( s )
W	 Hoki	 4x g2(Y-1)kl
a2
(
	
0 )2	 a2(4 .,S)2
(^	 k )2	 k2	 (Ho k 1)
=o —1+ k2	 + t
(20	 Is
Paq(5)
a 0
(5.5)
where kl - k h e x + k %kt- 9 and $X and g 'j, ex°e unit vectors in the
x- and y-direction, respectively. The solution of the differential
eq. (5.4) gi•res the phiwe for the X.K.B. aolution for the Alfvan mode,
The solution of the differential eq. (5.5) gives the pLase for the
WiC.B. solution of the two remaining modes. Both of these differential
equations San be brought into a form that can be solved by the separation
of variableev after first solving Both equations as algebraic 9WAtions
of the second ani fourth de gree in the unlmown d1 . This can shays
be done in closed form for es, algebraic equation up to the fourth deVes.
By putting ^ - i ks in eq. (5.5), we obtain the seas dispersion relation
as eq. (4.5), which therefore justifies calling eq. (4.5) the local
dispersion relation.
The ®ondition	 0 implies that t1ne 3-component of the
^►e ^►ti ^ie^ ''ve^otor is eras ^'ht gi a the odhaitfeil -f6 tho ` tni .naition
of the wate^A" pi-opa 4on in the	 ireofi ^' h t{ ®e th®` f^ ctioa"1` r :
ri tt%i'r ' te^pth. .At thia gein ' •^ k
	
Se tiaa'	 °^" `	 '+^q• X5:4
and using the definition of:
	
- •, •
g.pQV the ` i^ir^t equation off' ( ^ 06 ^ ^'' ^."a
upon ®oI.Tiag . for tilis J'Ahatration deptnt' s :
Po
kA)
x re h is the soale height * This is the penetration depth for the
A3.fv6 n mode. Setting	 0 in eq.,
	
`gives upon aolVing for
amplitudes. These growth rates may easily be determined from our
dispersion relation. One mode l identical with the Alfv®n node, has
always constant amplitude and is independent of o.))mpressibilitg and gravity.
From the dispersion rolations fo3z the + and — modes, it can be
seen that there is a large anisotropy with regard to constant amplitude
wave propagation. This behaviour does not occur in a treatment
neglecting the wave mode coupling. Therefore, our results suggest that
a search be made for this anistropy in gravity rave propagation in
the`buter solar atmosphere.
A(MOeIMCa'.ENTS
This work has been carried out with the support of the National
Aeronautics ana Space AdministratioL,USA, while the authors were at
the Internationa-1 Centre for Theoretioal Physics of the international
Atomic Energy Agency in Trieste, Italy. Special thanis are given to
Professor Abdur :slam and the IARA for the kind hospitality during our
stay.
REFS RMCE
1) C.P. YU, Phys. Fluids 8, 650 (1960.
--!-I
—1


Magneto-gravity waves and the heating of the solar corona

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