Eigenoscillations of the Differentially Rotating Sun: I. 22-year, 4000-year, and quasi-biennial modes

18 Pages·0.39 MB·English

by  N. S. Dzhalilov

1
Abstract. Retrogradewaveswithfrequenciesmuchlower
than the rotation frequency become trapped in the solar
radiativeinterior.The eigenfunctionsofthe compressible,
nonadiabatic,Rossby-likemodes(ǫ-mechanismandradia-
tive lossestaken into account) are obtainedby an asymp-
toticmethodassumingaverysmalllatitudinalgradientof
rotation,withoutanarbitrarychoiceofotherfreeparame-
ters.Anintegraldispersionrelationforthecomplexeigen-
frequencies is derived as a solution of the boundary value
problem. The discovered resonant cavity modes (called
R-modes) are fundamentally different from the known r-
2
0 modes: their frequencies are functions of the solar inte-
0 riorstructure,andthe reasonfortheir existenceis notre-
2 lated to geometrical effects. The most unstable R-modes

n are those with periods of 1–3yr, 18–30yr, and 1500–
≈
a 20000yr; these three separate period ranges are known
J
from solar and geophysical data. The growing times of

9
those modes which are unstable with respect to the ǫ-

mechanismare 102,103,and105years,respectively.The
1 ≈
v amplitudes of the R-modes are growing towards the cen-
0 ter of the Sun. We discuss some prospects to develop the
2 theoryofR-modesasadriverofthe dynamics inthe con-
1
vectivezonewhichcouldexplain,e.g.,observedshort-term
1
fluctuationsofrotation,acontrolofthesolarmagneticcy-
0
2 cle, and abrupt changes of terrestrial climate in the past.
0
/ Key words: hydrodynamics – Sun: activity – Sun: inte-
h
p rior – Sun: oscillations – Sun: rotation
-
o
r
t
s
a
:
v
i
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r
a
A&A manuscript no. ASTRONOMY
(will be inserted by hand later) AND
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Eigenoscillations of the differentially rotating Sun: I.
22-year, 4000-year, and quasi-biennial modes
N.S. Dzhalilov1,2, J. Staude2, and V.N. Oraevsky1
1
Instituteof Terrestrial Magnetism, Ionosphere and Radio WavePropagation of theRussian Academy of Sciences,
Troitsk City, Moscow Region, 142190 Russia; E–Mail: [email protected], [email protected]
2
Astrophysikalisches Institut Potsdam, Sonnenobservatorium Einsteinturm, 14473 Potsdam, Germany;
E–Mail: [email protected]
Received ; accepted
1. Introduction growth rate and with the spatial scales required for solar
activity.Instead,manyauthorspointedoutthatthecycle
The 22-year magnetic cycle of solar activity is the most
period of 22 years is hard to explain (Stix 1991; Gilman
prominent phenomenon of several large-scale dynamic
1992;Levy 1992;Schmitt 1993;Brandenburg1994;Weiss
eventsthatoccurintheSun.(Really,themagnetichalfcy-
1994; Ru¨diger & Arlt 2000).
clesorsunspotnumbercyclesvaryinlengthbetween9–13
years,and11yrisanaverageofthe 20half-cyclesavail-
≈ From the solution of the inverse problem of helioseis-
able.) An explanation of the basic mechanism underlying
mology (e.g. Tomczyk et al. 1995) it is known that the
this fameous phenomenon is the fundamental challenge
convective envelope of the Sun is rotating with a latitude
of solar physics. The achievements of the theory of the
dependence of the angular velocity similar to that of the
α-ω dynamo turned out to be a great success. However,
surfacebutalmostrigidinradialdirection.Astrongerra-
neither all observations of magnetic and flow fields nor
dial gradientwhichis requiredfor the α-ω dynamomech-
the radiationfluxeswhicharerelatedtothis phenomenon
anism is located in a shallow layer (thickness 0.05R
andwhich aremeasuredat the surfaceofthe Sun or indi-
≈ ⊙
(Kosovichev1996),whereR isthe solarradius)immedi-
rectly,byhelioseismology,inits interior,canbe explained
⊙
atelybelow theconvectivezone—the tachocline(Spiegel
unambigously in this way. Although our present work is
& Zahn 1992). Below the tachocline up to a depth of
not directly relatedto the dynamo theory, we will outline
at least 0.5R the radiative interior is rotating with an
herethosedifficultieswhichhavecommonpointswithour
⊙
angular velocity law similar to that of a solid-body. The
results.
question arises:what compels the Sun to rotate in such a
strange manner, which is different from the generally ac-
1.1. Some problems of dynamo theory
cepted, theoretically predicted stable rotation law? How
to handle a dynamo theory for which the ‘ω’ area is sep-
.
aratedfrom the ‘α’ area over a large partof the extent of
As a consequence of our imperfect knowledge of basic
theconvectivezone?InordertosolvethisproblemParker
characteristics of turbulent convection as well as merid-
(1993) has put forward the idea of an interface dynamo,
ionalcirculationanddetailsoftherotationoftheSun’sin-
the basic features of which existed already in earlier dy-
terior,thesolutionsofthedynamoequationsbecomefunc-
namo models (Steenbeck et al. 1966). To close the cycle
tions of many free, unknown parameters (e.g. Stix 1976).
of such a stretched dynamo it is necessary to have some
For instance, by clever combinations of these parameters
mechanism delivering toroidal magnetic flux, arising by
it is possible to get from kinematic theory an oscillatory
the shear of differential axisymmetric rotation (Cowling
magneticfieldwitha22-yearperiodandagrowingampli-
1953)inthe tachocline,to the ‘α’dynamo area(e.g.Mof-
tude. However, another choice of these parameters leads
fatt1978;Krause&Ra¨dler1980).Togetasolar-likemag-
to waves of growing amplitude for other periods. So one
netic activity it is necessary to suppose the existence of a
could draw a butterfly diagram not only with an 11-year
huge ( 105G) toroidal magnetic field to create enough
periodicity. It remains still an open question which of the ≈
magnetic buouancy for the leakage of magnetic flux and
clevercombinationsresultingin asolar-like22-yearactiv-
to solve the tilt problem of lifting loops (e.g. Caligari et
ity cycle is realized in the Sun. We could not find a work
al. 1998). Moreover, a high magnetic diffusivity contrast
on the dynamo wave problem, showing that just the 22-
between the convective envelope and the underlying ra-
year period is preferred among others with a maximum
diative core should be assumed to solve the quenching
Send offprint requests to: J.Staude problem of the α effect (see, e.g., Fan et al. 1993; Catta-
N.S.Dzhalilov, J. Staude& V.N.Oraevsky:Long-period eigenoscillations of theSun.I. 3
neo & Hughes 1996). However,it is a major challenge for must deposit their angular momentum before returning
any dynamo model to produce such strong fields. totheconvectivezone,butnotbeforepenetratingfarinto
The idea of the interface dynamo was further devel- the radiative interior.
oped, e.g. by Charbonneau & MacGregor (1997). Later, Forthewavemechanismthequestionofananisotropic
a fit to the real solar rotation profile with its latitudinal propagation relative to the azimuthal rotation is a key
and radial dependencies has been included by Markiel & moment. Fritts et al. (1998) have shown that convec-
Thomas (1999), but so far no satisfactory solar-like oscil- tion, penetrating into the stratified and strongly sheared
latorysolutionsfortheinterfacedynamohavebeenfound. tachocline,canproducepreferentiallypropagatinggravity
Growing wave solutions are suppressed by the latitudinal waves.
shear.
There havealsobeen speculations that the rotationof
the coremaybe variable,perhapswithatime scaleofthe
1.2. Spinning-down rotation problem solar cycle (e.g. Gough 1985). The present paper is along
these lines.
.
From our short discussion we conclude that the con-
Mechanismsforbrakingthe solarinternalrotationare
vective envelope and the radiative interior are coupled to
alsounderdiscussion.Thecharacterofthecorerotationis
each other through a certain global agent,resulting in al-
not clear because here the accuracy of helioseismic inver-
most co-rotation.To advance the solution of the problem
sionsgetsworse(Chaplinetal.1999)andtheresultsseem
the dynamo theory should take into account the presence
to be in contradiction with the oblateness measurements
of this global agent. We suppose that really this agent is
(Paterno et al. 1996). There are some suggestions that a
provided by waves with the following properties:
decelerationoftheradiativeinteriordependsonthetrans-
Waves should representlarge-scaleglobal eigenoscilla-
port of angular momentum between this region and the
tions ofthe Sun. Theiroriginmust be relatedto rotation,
convectivezone.Forinstance,Mestel&Weiss(1987)sup-
they must be strongly anisotropic with respect to the az-
posed that even a weak large-scale magnetic field would
imuthal angle. Looking at the characteristics of the so-
be sufficient to couple very efficiently the interior and the
lar cycle we immediately see the high coherency of these
convectivezone,leadingessentiallytosolidbodyrotation.
globalmotions (the constant periods, phase shifts, ampli-
Inthis waythe magnetic torquescanalsoextractangular
tudes,thelatitudeappearence,etc.).Activitygrowsinthe
momentum from the radiative interior (e.g. Charbonneau
first phase with a timescale which is considerably shorter
& MacGregor 1993).
than the decay time in the second phase; this fact and
Thewavemechanismforthesolutionofthisproblemis
the quick eruptive release of energy by the reconnection
more popular. Schatzman (1993), Zahn et al. (1997), and
mechanism indicate that the waves must be unstable.
Kumar & Quataert (1997) have concluded that the solid
It is noteworthy that the inner gravity waves do not
rotation of the radiative interior is a direct consequence
fulfill these requirements. The quesion is whether the r-
of the effect of internal gravitywaves.Gravity wavesgen-
modes do?
erated near the interface between the convective and ra-
diative regions transport retrograde angular momentum
intothe interior,therebyspinning itdown.Here the main 1.3. r-modes
idea is that the isotropically generated gravity waves be-
come anisotropic due to Doppler shifts of frequencies in .
thedifferentiallyrotatingSun.Inthatwayforanisotropic In a non-rotating sphere (Ω=0, where Ω is the angu-
retrograde and prograde waves the radiative damping is lar frequency of solar rotation) the wave motion is subdi-
different, and the residual negative angular wave momen- vided into two non-coupling components: spheroidal p ,
−
tum may compel the solar radiative interior to co-rotate f and g modes (for which the main restoring forces
− −
with the convective zone. This idea has been further de- are pressure gradient and buouancy) and toroidal modes
veloped by Kumar et al. (1999) including a toroidalmag- (e.g. Unno et al. 1989). Toroidal modes are degener-
netic field to explain the existence of the unstable shear ated horizontal eddy motions confined to a spherical sur-
layer‘tachocline’.However,Ringot(1998)hasshownthat face with a radius r for which ω = 0, divv = 0, and
a quasi-solidrotationof the radiativeinteriorcannotbe a v = Qm(r) (0, 1 ∂ , ∂ )Ym(θ,φ). Here Ym is the
l × sinθ∂φ −∂θ l l
direct consequence of the action of internal gravity waves spherical harmonic with a degree l and order m, θ is the
producedintheconvectivezone.Gough(1997)questioned colatitude, φ is the azimuthal angle in the spherical polar
thisideaemphasizingthatthemechanismcanworkonlyif coordinates, v is the fluid velocity field, ω is the angular
the waves are generated with strong amplitudes to trans- frequency of the fluid motion, and Qm(r) is an arbitrary
l
port the required angular momentum. This means, reso- amplitude function. Toroidal modes have zero radial ve-
nance waves are required, but such waves may penetrate locity but have non-zero radial vorticity, (rotv) = 0 (for
r
6
onlytodistanceslessthan10 5R beneaththeconvective thespheroidalmodesitisviceversa).Thesemodesdonot
−
⊙
zone due to the strong radiative damping. These waves alter the equilibrium configuration.
4 N.S.Dzhalilov, J. Staude& V.N.Oraevsky:Long-period eigenoscillations of theSun.I.
Whenaslowrotation(Ω2 <Ω2 =GM/R3)isincluded with convective motions (Wolff 1997; 2000). Because for
g
the spheroidal modes are slightly modified but they keep these modes v 0 and div v 0, their chance to take
r
≈ ≈
theirmainproperties.Degeneracyoftoroidalmodesis re- part in the redistribution of angular rotation momentum
moved only partially by the rotation, and quasi-toroidal in the radiative interior is low. Note that the slow solar
waves – known as r-modes – appear with a non-zero fre- differential rotation does not change the behavior of such
quency of ω 2Ωm/l(l+1) in the rotating frame (Pa- r-modes with m=l 1 (Wolff 1998).
≈ ≫
paloizou&Pringle1978;Brayn1889).Usuallythegovern- Lookingforfurtheranalogiesbetweenwavesconnected
ing equations of the r-modes are obtained by expanding with gravity and with rotation, we remember that beside
the initial physical variables of the equations in the ro- thesurfacegravitywavesthereexistinternalgravitywaves
tating system into power series with respect to the small with ω2 N2k2/(k2 +k2), the frequencies of which de-
≈ x x y
parameter(Ω/Ω )2 ( 10 4.7 fortheSun,e.g.Papaloizou pend on the inner structure (N is the Brunt-Va¨is¨al¨a fre-
g −
≈
& Pringle 1978; Provost et al. 1981; Smeyers et al. 1981; quency). Similar to these waves there exist ‘true’ Rossby
Saio1982).Thesepowerseriespracticallydescribethede- (not deformation) waves, the frequency of which depends
viation of the surface of the star from its initial spher- also on the internal structure.
ical state, resulting from rotation through Coriolis and
centrifugal forces. As a result of the deformation of the
1.4. Rossby waves
spherical surface with a radius r the radial vorticity of
the toroidal modes cause a surface pressure perturbation We include here a short review on the main features of
through the Coriolis force. However, the r-modes practi- Rossby waves; they have been investigated in great de-
cally keep the main properties of toroidal flows: vr 0, tail in geophysics (e.g. Pedlosky 1982; Gill 1982). In the
≈
divv 0. The degeneracy of the r-modes is that their simplestcase,thatisinaplane-parallel,homogeneous,ro-
≈
frequencies hardly depend on Qm(r), i.e. they are inde- tating layer, the dispersion relation for the Rossby waves
l
pendentfromtheinnerstructureofthestar.Forthel =1 is ω 2Ωβk /(k2+k2+k2). Here k is the wavenumber
≈ x x y z x
modes the frequency in the inertial system is again close perpendicular to the rotation axis, k is expressed by the
z
to zero, ω 0 (Papaloizou & Pringle 1978). The r-mode internal deformation radius of Rossby which depends on
≈
equations define the amplitudes Qm(r), and taking into the Brunt-Va¨is¨al¨a frequency, β is the transverse gradient
l
account the next terms with small Ω2/Ω2 in the series (in y direction) of the Coriolis parameter: a vertical com-
g
practically does not change the frequencies. ponent of the ‘planetary’ vorticity 2Ω in the given local
Duetothefactthatther-modesarepracticallysurface point. Unlike the r-modes the Rossby wave frequencies
deformation waves, some similarity of these waves to the arefunctionsoftheinternalstructureandhavemaximum
surface gravity waves or to the f-modes is apparent. For dependence on the gradient β: ω 0 if k 0 and if
x
→ →
highlthef-modesareananalogytosurfacegravitywaves k .
x
→∞
inaplane-parallelfluidwithω2 =gk .IntheCowlingap- Any disturbance of the local flow in a rotating frame
⊥
proximation f-modes with l=1 have zero frequency too, maygeneratewavesoftheRossbytype.Thesewavesexist
ω 0 (Unno et al. 1989). This corresponds to a parallel only if there is a gradient of the potential vorticity Π =
≈
displacementofthe wholestar.Forhighl the f-modefre- (ω Ξ)/ρ. Here an absolute vorticity is the sum of the
a
·∇
quencies arealso independent ofthe inner structure,with relative and the planetary vorticities, ω = rotv+2Ω, Ξ
a
ω2 lΩ2 (Gough1980).So,r-modesarealsofundamental is any conservedscalar quantity, dΞ =0 (for instance, for
≈ g dt
rotating modes with an inertial frequency, ω 2Ω. adiabaticmotionthatcouldbe the entropyorthe density
≤
For the Sun the properties of r-modes have been in- in the case of incompressible plasma). The Rossby wave
vestigatedin greatdetail by Wolff et al. (1986)and Wolff motionisasolutionofthenonlinearequationfortransport
(1998; 2000; and refs. therein). of Π. The potential vorticity is conserved if the medium
Some properties of the r-modes are also similar to is barotropic ( ρ p = 0) and if there are no torques.
∇ ×∇
those of the Rossby waves in geophysics (Pedlosky 1982). The rotationof the frame is added to any vorticity in the
Similar to the Rossby waves and unlike the g-modes the velocityfield.Anymotionwithinarotatingfluidservesas
r-modes are strongly anisotropic.They propagateonly in a potential source for vorticity.
azimuthaldirection,oppositetorotation(i.e.theyareret- The relative vorticity may be evoked by the geomet-
rograde waves in the co-rotating frame). Because we are rical surface as well as by internal gradients. It depends
interested in length scales corresponding to those of large onthe choiceofthe functionΞ(r,θ,φ) andonΩ(r,θ). For
sunspots, we have to consider r-modes with l 100. To example, an unevenness of the ocean bottom causes the
≈
get oscillations with periods of years (ω/Ω 10 2) we topographicRossbywaves,oradependenceoftheCoriolis
−
≈
must choose m l 1, just such r-modes are physically parameteronthe earthlatitude (F =2Ωsinϕ,whereϕis
≈ ≫
more interesting (Lockitch & Friedman 1999). However, the geographiclatitude) is the main cause of atmospheric
in the case of high l the amplitudes of the r-modes will Rossby waves.
be concentrated near the surface of the Sun (Provost et In the solar dynamo context the ability of Rossby
al. 1981; Wolff 2000), and so they can actively interact wavestoinducesolar-likemagneticfieldshasbeenconsid-
N.S.Dzhalilov, J. Staude& V.N.Oraevsky:Long-period eigenoscillations of theSun.I. 5
ered by Gilman (1969). Here the mechanism for sustain- 2.1. Basic equations
ing the Rossby waves is a latitudinal temperature gradi-
Let us investigate global motions with large timescales
entinathin,rotating,incompressibleconvectivezone.To
such that the Rossby number is small, ω/Ω < 1. Before
interpret the dynamical features of large-scale magnetic
the appearence of any disturbances the basic stationary
fields the Rossby vorticies excited within a thin layer be-
state of the rotating star is defined mainly by the bal-
neath the convective zone are considered by Tikhomolov
ance of pressure gradient, gravity force, and forces ex-
& Mordvinov (1996) as the result of a deformation of the
erted by the noninertiality of the motion (Coriolis and
lower boundary of the convective zone.
centrifugal forces). In the case of an incompressible fluid
with a homogeneous rotation rate usually this state is
1.5. R-modes called ‘geostrophic balance’. A star disturbed by an ex-
ternal force tends to return to this basic state. Our aim
From the discussion in Subsection 1.4 we conclude that is to study for the Sun the dynamics of small deviations
just Rossby-like waves could be suitable for our require- from the steady geostrophic balance. For this purpose it
ments.Asthemaindrivingmechanismwechoosealatitu- is natural to write the dynamic equations in a frame ro-
dinal(orhorizontal)differentialrotation,Ω=Ω(θ).Baker tating together with the Sun. The magnetic field will be
& Kippenhahn (1959) have pointed out that the uniform ignored. For arbitrary Ω(r,ϑ) the equation of momentum
rotation of a star is not a typical case. Low frequencies in conventional definitions is given by
(periods of years) could easily be obtained searching for dv 1 dΩ
the eigenoscillationsof the Sun’s radiative interior,where +2Ω v = p+g+r Ω (Ω r)+µv 2v,(1)
dt × −ρ∇ × dt − × × ∇
thegradientoftherotationspeedisclosetozero(inaccor-
where µ is the kinematic/turbulent viscosity coefficient.
dance with the helioseismologyresults).Largescalessuch v
In the next steps this equation will be simplified keep-
as those associated with sunspots (k R 100) decrease
x ⊙ ∼ ing the main features of the motion. We consider linear
the frequencies too. Similar to the r-modes the Rossby
waves without taking into account convective and merid-
wavesarestronglyanisotropic(retrogradewaves),butun-
ional flows, v0 = 0, and we suppose that the angular ve-
likethe r-modesthesewavesareconcentratedclosetothe
locity Ω does not depend on time. For the basic state we
solar center. These results have already been obtained by
have from Eq. (1)
Oraevsky & Dzhalilov (1997), who investigated the trap-
1
pingofadiabatic,incompressibleRossby-likewavesinthe p +g =Ω (Ω r).
0
−ρ ∇ × ×
solar interior. In the present work we take into account 0
However,wecanexcludepracticallyeverywhereintheSun
compressibility for the nonadiabatic waves. We look for
centrifugal acceleration and consider it as a small correc-
unstablewaves.Itisclearthatthenecessaryconditionfor
the Kelvin-Helmholtz shear instability, 4N2 (rdΩ)2 <0 tion to g. Then the spherically symmetric basic state is
− dr defined by p ρ g. For the linear oscillations we have
(Ando1985),isnotfulfilledintheradiativeinterior.Then ∇ 0 ≈ 0
from Eq. (1):
we decided to include the thermal ε-mechanism of insta-
∂v
bility which is favoured at low frequencies (Unno et al. +2Ω v+rsinϑ(v Ω)e =
ϕ
1989).To balance the ε-mechanismthe radiativelossesin ∂t × ·∇
1 p
tchael edffiffecutssiownereiggnimoreeatrheeinincflluudeendc.eToofethxecluspdheearlilcagleosmurefatrcie- −ρ0∇p′+g′+ ∇ρ200ρ′+µv∇2v. (2)
atthe givenradius.The dispersionrelationinthe limit of Here eϕ is the unit vector in azimuthal direction, and
adiabatic incompressiblity and at very low frequencies is variableswithaprimeareEulerianperturbations.Eq.(2)
the same as that for Rossby waves in geophysics. In or- is written in rotatingsphericalpolar coordinates(r,ϑ,ϕ).
der to distinguish these rotational body waves from the It coincides with the equation of motion by Unno et al.
r-modes we call them R-modes (Rossby rotation). (1989) which is derived for an inertial frame, if the oper-
ator ∂/∂t is replaced by ∂/∂t+Ω∂/∂ϕ. To simplify our
The governing fourth order equation is obtained from
further discussion we shall use the Cowling approxima-
thebasicequationsinSect.2.Somequalitativeanalysisof
tion, g =0, which has a sufficient degree of accuracy for
the wavecavitytrapping is donefor the simpler adiabatic ′
the analysis of short waves.
case in Sect.3. Using the asymptotic solutions obtained
in Sect.4 the complex boundary value problem is solved The next simplification of Eq. (2) is connected with
in Sect.5. The calculationof the eigenfrequencies andthe the quasi-rigidrotationofthe innerpartofthe Sunbelow
instability analysis are done in Sect.6. The obtained un- the convective zone, which is known from the solution of
stable modes are shortly discussed in Sect.7. the inverse problem of helioseismology. In this case we
can omit the third term of the l.h.s of Eq. (2). Such a
restriction of the gradients of Ω(r,ϑ) requires to obey the
conditions:
2. Setting the problem ∂Ω ∂Ω
2Ω r , 2Ω tan(ϑ) . (3)
≫ ∂r ≫ ∂ϑ
6 N.S.Dzhalilov, J. Staude& V.N.Oraevsky:Long-period eigenoscillations of theSun.I.
Toobtaintheseconditionsweusedthecomponentsofthe It means, we can introduce the constant æ = 8.22
0
×
vector Ω in spherical polar coordinates, Ω = Ωcosϑ, 10 27T5/ρ2 8.2 105. Then we have
−
{ ≈ ×
Ωsinϑ,0 . κ æ T3/2 erg/scmK.
− } R ≈ 0
The nextequationsarethe massandenergyconserva- Now for the right-hand side of Eq. (5) we have
tion equations in the standard form: 2
æ 2T5/2+Q. (8)
0
dρ −L≃ 5 ∇
+ρdivv = 0, (4) The main non-perturbed energetic state of the Sun is de-
dt
fined by the condition = 0. In our case this condition
dT 0
L
cvρ +pdivv = , (5) is
dt −L
2 d2T5/2
where: =Q(ρ,T) q (K T5/2 T). Q = æ 0 . (9)
−L − ∇· R−∇· 0 ∇ 0 −5 0 dr2
Another form of Eq. (5) is
dp dρ
c2 = (γ 1) . (6) 2.2. Choice of the frame of reference
dt − sdt − − L
Hereγ =c /c ,c andc arethespecificheatsatconstant Inthenextstepwetrytogettheanalyticalsolutionsofthe
p v p v
pressure and volume, respectively, c is the sound speed, wave equations and to solve the boundary value problem.
s
the source function Q(ρ,T)is the sum of nuclearand vis- For this aim we shall investigate the short waves (WKB)
cous heat generation rates per unit volume: ρ(ε +ε ), approximationforwhichtheeffectsofcurvatureareunim-
N v
q is the radiative energy flux, K T5/2 is the coefficient portant.Itmeanswecanapplytherotatingplane-parallel
R 0
ofelectronheatconductivity.IntheinterioroftheSunwe stratificationapproximation.Suchanapproachhasitsad-
shall neglect viscous heating (ε = 0). For the power of vantages and disadvantages. The main advantage is that
v
nuclearreactionswe haveQ ρ2Tα.In particularforthe at the end points of the integrationpath (center and pole
≈
p-p reaction Q = ρε 9 10 30χ2 ρ2T4, where ε is oftheSun)wehavenosingularity,andthisgivesachance
pp ≈ × − H pp
given in [erg/gs] and χ =0.73. to find the solutions analytically. The disadvantages are
H
In the limit of incompressible fluid, dρ/dt = 0 or connected with the following: a) long waves are excluded
divv = 0, it follows from Eq. (6) that the condition but they areveryimportant,for instance,for the transfer
c2 is not needed to satisfy dp/dt = 0. Hence, in of angular momentum; b) the lost ‘end’ singularities cor-
asd→issi∞pative ( = 0), incompressible flu6 id sound can- respondtoarealphysicalbehaviorofthewaves;c)inthis
notpropagateiLnst6antaneously.Itmeans,wecannotusethe approach we get two distinct directions, Ω and g (taking
condition of c2 to get the incompressible limit for into account all components of Ω), which are absent in a
nonadiabatic wsa→ves∞. self-gravitating, radially stratified sphere. The deviation
of the direction of stratification of the plane-parallelfluid
Now we will try to simplify the energy loss function
layer from the direction of its axis of rotationshould lead
assuming reasonable approximations for the Sun’s in-
L to additional, physically doubtful results. In geophysical
terior. We use the formula for the heat conductivity of a
hydrodynamicsthis problemis solvedby applying the ‘β-
fully ionized gas to show that in the Sun radiative trans-
plane’ within the frame of the traditional approximation
port of energy is more important than that by particle
(Pedlosky 1982; Shore 1992), where the component of Ω
heat conductivity:
parallel to g in the given surface point (local vertical) is
κ =K T5/2 10 6T5/2 erg/scmK.
T 0 ≈ − retainedonlyinthe governingequations.Herewewilluse
Theradiativefluxisgivenbytheradiativediffusionequa- the same approach.
tion
Let us take an arbitrary point at the surface of the
16σ
q = κ T, κ = ST3, (7) rotating sphere. The position of this point is defined by
R R R
− ∇ 3χ
its radius r, its co-latitude ϑ, and its azimuth angle. We
whereκRistheradiativeheatconductivity,σS =aRc/4= assign to this point a local, left-handed Cartesian system
5.67 10−5erg/cm2K4s is the Stefan-Boltzmann con- ofcoordinates x,y,z ,where the z-axisis directedalong
stant×, and χ is the Rosseland mean opacity. For the the radius (loc{al vert}ical), the direction of the y-axis is
opacity we will use Kramers’ formula: χ = 3.68 meridional (towards the pole), and that of the x-axis az-
×
1022ρ2T−7/2cm−1. Then, for the radiative conductivity imuthal. In this frame of reference Ω = 0,Ωy,Ωz =
we have 0,Ωsinϑ,Ωcosϑ .Strictlyspeaking,thez-{axiscoinc}ides
κ 8.22 10 27T13/2 erg/scmK. {withtherotationa}xisonlyatthepole(ϑ=0).Inthecase
R ≈ × − ρ2 of a homogeneous fluid Ω is included into the wave equa-
Now we find that κ /κ 10 20T4/ρ2 > 103. This con- tion only in the term
R T −
dition is fulfilled in the w≈hole Sun. That means, we can ∂2 ∂2 ∂2
(Ω )2 =Ω2 +2Ω Ω +Ω2 .
excludetheheatconductivitytermfromEq.(5).Thenext ∇ y∂y2 y z∂y∂z z∂z2
simplification is connected with the ratio T5/2/ρ which Here we can neglect Ω (Ω = 0), if the condition
y y
is practically not changed over the radius and 1016. Ω ∂ Ω ∂ isfulfilled. Thisconditionis named‘tra-
≈ | y∂y|≪| z∂z|
N.S.Dzhalilov, J. Staude& V.N.Oraevsky:Long-period eigenoscillations of theSun.I. 7
ditional approximation’. For harmonic motions we have (12) must be completed then by the equation of state,
∂ = ik , ∂ = ik , and the traditional approximation p=p(ρ,T). For an ideal gas we have
∂y y ∂z z
corresponds to |ky|≪|kz|cotϑ. If the spatial scale of the p′ = ρ′ + T′. (13)
wave motion in vertical direction is much smaller than in p ρ T
0 0 0
horizontaldirection(atlatitudesnotclosetotheequator), For the ideal gas p = R ρ T /µ , where R is the gas
0 g 0 0 m g
wecanrestrictourselvestoretainonlyΩ inthegoverning constant, µ is the molecular weight, and the squared
z m
equations. This condition for the traditional approxima- adiabatic sound speed c2 = γc2 is defined by the squared
s
tionremainsvalid,ifaradialstratificationisincluded(Lee isothermal sound speed c2 =p /ρ .
0 0
& Saio 1997). We define
Q æ 1
In order to construct the ‘β-plane’ limit we expand Q˜ = 0 , æ˜ = 0 , N2 =g æ æ ,
0 0 p ρ
Ωz around a fixed ϑ0 (ϑ = ϑ0 +δϑ): Ωz = Ω(ϑ)cosϑ cvρ0T0 cvρ0T0 (cid:18)γ − (cid:19)
≈
(1+βy)Ω0cosϑ0. Here y =R⊙δϑ, Ω0 =Ω(ϑ0), and æT = 1 dT0, æρ = 1 dρ0,
1 ∂Ω0 1 T0 dz ρ0 dz
β = tanϑ .
Ross(cid:18)byΩw0a∂vϑes −are pos0si(cid:19)blRe⊙only if β = 0. The parameter æp = p10ddpz0 =æρ+æT =−cg2.
β is a sum of two terms: the second6 one (tanϑ) is due Here N2(z) is the squared Brunt-Va¨is¨al¨a frequency.
to the geometrical change of the Coriolis parameter with æp,æρ, and æT are the reciprocal pressure, density, and
latitude. This term exists always, even if the rotation is temperatureinhomogeneityscales,respectively,andQ˜ is
0
rigid.Ther-modesareconnectedwiththisterm.Thefirst thereciprocalofthecharacteristicKelvin-Helmholtztime-
terminβ appearsifthedifferentialrotationisconsidered, scale—adeviationfromthethermalbalanceofthestaris
and the R-modes are connected with this term. Close to restoredduring this time. As the coefficients of Eqs.(10)-
the pole (ϑ 0) the first term is dominant. (13) are independent of the time t and of the space coor-
0
→
Note that such a ‘β’-limit is applicable also around dinate x we can set
∂ ∂
the equator plane, where the traditional approximation = iω, =ik . (14)
x
∂t − ∂x
does not fit. The advantage of this limit is the possibil-
Now we exclude ρ from the system of Eqs. (10)-(13) and
ity to include ϑ as a parameter in the Cartesian system. ′
have
In this way we use here an inertial Cartesian system of
p T
′ ′
coordinates (x,y,z) in a frame rotating with an angular iω +æρvz +u = 0, (15)
− p − T
frequency Ω(y,z). All non-perturbed model variables are (cid:18) 0 0(cid:19)
p
functions of z only, and for the gravity acceleration we ik c2 ′ +2Ωv +Dˇv = 0, (16)
x y x
have g = 0,0, g(z) . For the observer from the non- − p0
{ − }
rotatingframethe elements offluid aremovingdue to ro- c2 ∂ p′ 2Ωv +Dˇv = 0, (17)
tation with a velocity V0 =Ω r = Ωy,Ωx,0 ,where − ∂yp0 − x y
× {− }
intheframeofourapproximationΩ≈Ωz.V0x <0means c2 ∂ p′ +gT′ +Dˇv = 0, (18)
that the x-axis is directed opposite to the rotation. z
− ∂zp T
0 0
p
(iω+2Q˜ ) ′ +æ v +γu = fˇ, (19)
0 p z
2.3. Oscillation equations − p0
∂v ∂v
For linearization each physical variable f = f0 + f′ is u=divv =ikxvx+ ∂yy + ∂zz, (20)
decomposed into a mean term f0 and a small fluctuating where the operators Dˇ and fˇare defined as
term f . Neglecting terms of higher order than the first
′ T T
one we get our oscillation equations Dˇ =iω+µ 2, fˇ=(α 2)Q˜ ′+æ˜ 2 T5/2 ′ ,(21)
v∇ − 0T 0∇ 0 T
∂ρ dρ 0 (cid:18) 0(cid:19)
∂t′ +vz dz0 +ρ0divv =0, (10) and the relation p0/(cvρ0T0)=γ−1 is used. In the next
steps the viscosity appears in the coefficients of the equa-
∂v +2Ω v = 1 p +gρ′ +µ 2v, (11) tionsinthe form c2+µ O(1).Forafully ionizedplasma
∂t × −ρ ∇ ′ ρ v∇ ω v·
0 0 the kinematic viscosity is µ 10 16T5/2/ρ [cm2/s]. For
v −
∂T dT T ≈
cvρ0 ∂t′ +vz dz0 +p0divv =æ0∇2 T05/2T′ + the solar situation µv ≈ O(1), if we do not take into ac-
(cid:18) (cid:19) (cid:18) 0(cid:19) countthe turbulentviscosity.Aswe areinterestedinvery
+Q0(cid:18)2ρρ0′ +αTT0′(cid:19)=−L′. (12) cloawn fprueqtuDˇenciesiωth(enocnon-vdiistcioounscωc2as≫e).µNvoiws vEalqids..(T1h6e)-n(1w7e)
≃
Foradiabaticoscillations ′ =0.Weapproachthisregime definethehorizontalcomponentsofvelocityandhenceits
L
by setting formally æ = 0 (Eq. 9). But in the nonadi- two-dimensional divergence div v:
0
abatic case, which is considered here, L′ is the sum of v = kxc2 σ2 1+ ⊥1 ∂ p′, (22)
radiativedamping andtheε-mechanismterms.Eqs.(10)- x − ω 1 σ2 σk ∂y p
− (cid:18) x (cid:19) 0
8 N.S.Dzhalilov, J. Staude& V.N.Oraevsky:Long-period eigenoscillations of theSun.I.
kxc2 σ σ ∂ p′ assume the rotation rate Ω(y,z) Ω˜(z)(1 βy ), where
vy = i ω 1 σ2 1+ k ∂y p , (23) Ω˜(z) O(Ω ). Then we have in E≈q. (28) − ∗
div⊥v = c2ikxσω′−Mˇ pp0′,(cid:18) x (cid:19) 0 (24) Mˇ ≈≈M =1⊙+ kxω2Ωk⊥2′(y), sˇ=s≈1−Mc22kωx∂∂yΩ1, (32)
where where k2 =k2+k2 .
Mˇ =1+ 2σ ∂ + σ2 ∂2 k2 , σ = ω , σ = ∂σ. Our⊥next sxtep isyto derive one differential equation for
kx ∂y kxσ′ (cid:18)∂y2 − x(cid:19) 2Ω ′ ∂y the temperature perturbations. The variable vz is easily
Excluding u=div v+∂vz/∂z by using Eq.(15)we have excludedfromEqs.(25)-(27).Thenwegetforthepressure
⊥
the system of equations perturbations
iωvz =c2∂∂zpp′ −gTT′, (25) ap′ = T′ + ω 2q 1 N2 T′ + fˇ + (33)
0 0 p T cd − ω2 T γ
T p 1 ∂ 0 0 (cid:16) (cid:17) (cid:20)(cid:18) (cid:19) 0 ∗(cid:21)
′ sˇ(y,z) ′ + æ + v =0, (26) ∂ T fˇ 1
T0 − p0 iω (cid:18) ρ ∂z(cid:19) z + æρ+ ∂z T′ + γ d(z),
p 1 N2 T 1 (cid:18) (cid:19)(cid:18) 0 ∗(cid:19)
q ′ + v = ′ + fˇ, (27)
z where
p iω g T γ
where the dim0ensionless quantit0es are∗ ωq 2 ∂ q N2 æp
a=s+ + æ + , d= = æ .(34)
ρ ρ
σ γ 1 2Q˜ cd ∂z d g γ −
sˇ=1−c2kxω2′Mˇ, γ∗ =γiω, q = −γ − γ 0. (28) The par(cid:16)amet(cid:17)erd(cid:18)ispositive(cid:19)inthe solarradiativeinterior,
Now we use the condition σ 1 and receive∗ d > 0, while we have d 0 in the convective zone. The
c2 ∂ p ≪ pressure perturbations (E≤q. 33) have a singularity at a=
′
v = , (29)
x −2Ω∂yp 0.However,it will be shownbelow that this singularityis
0
removable.
c2 p
′
vy = ikx . (30) Introducing a new dependent variable
2Ω p
0 5/2
In the case of rigid rotation, ∂Ω = 0, it follows immedi- T′ T0
∂y Θ=κ , κ= , T00 =const, (35)
ately from Eqs. (29)-(30) that div v = 0. It means, that T0 (cid:18)T00(cid:19)
the incompressible case (divv = 0⊥) for which everywhere we receive the final equation of fourth order
Θ +A Θ +A Θ +A Θ +A Θ=0, (36)
v = 0 (from the boundary condition at the center) is ′′′′ 3 ′′′ 2 ′′ 1 ′ 0
z
notsointerestingforastrophysicalsituations.Inthis case where Θ =dΘ/dz,ε =æ˜ T5/2/γ and
′ 0 0 0
ρ = 0 and the equation of heat conductivity, Eq. ((12), aω2 N2 ∗
′
A = a a +a ε a , (36)
is separated from the equation of motion. In geophysical 0 1 2 ′2− ε c2 ω2 − 0 3 ′
0 (cid:18) (cid:19)
situations just this case is interesting, when geostrophic
eddies are investigated. For our task waves with v = 0 A1 = a1a3+a2+a′3,
z 6 d
are more important. A = a a b b , A = ln(p T ad2),
2 3− 1 2− ′2 3 −dz 0 0
Our next step is to separate the z and y dependence
a d qω2
of the variables in the governing equations to have finally a = ′ + ′ +æ + ,
oneordinarydifferentialequation.Inthegeneralcasesuch − 1 a d p dc2
1 5
asimsepplaeractaisoenwishneonttphoessfiubnlec,taionndw∂∂ye(cΩ1o)nsiisdeinrdheepreentdheenvteroyf a2 = ε0 (cid:18)b1− 2æT(cid:19)+æ′1+b2k˜⊥2,
y. Only in this case it is possible to separate the equa- a = 1 k˜2, k˜2 =k2+k2 æ =k2 æ ,
tions with respect to the variables y and z and we can 3 ε0 − ⊥ ⊥ x y− 1 ⊥− 1
write ∂∂y22 =−ky2. Here ky should be determined from the b = æp d′ qω2 N2 1 , b =æ + d′ qω2,
boundaryconditions,andacomplexky isnotexcluded.In 1 γ − d − dc2 ω2 − 2 T d − dc2
thiswaythe systemofpartialdifferentialequationsinthe (cid:18) (cid:19)
2 κ γ 1 4 κ
′′ ′′
plane-parallel approximation is reduced to ordinary dif- æ = (2 α) , q = − + ε .
1 0
5 − κ γ 5 κ
ferential equations. Now we assume the following formula
In order to solve Eq. (36) we normalize it to get a
for the rotation profile
dimensionlessequation.Forthispurposewenormalizethe
Ω˜(z)
Ω(y,z)= , (31) radial distance to the solar radius: z z/R and get
1+βy/R ⇒ ⊙
finally the following equation
where β > 0. This⊙profile reveals that the rotation rate
εΘ +εϕ Θ +(1+εϕ )Θ +
decreasesatthepole.Ify thenΩ 0.Helioseismol- ′′′′ 3 ′′′ 2 ′′
→∞ → +(ψ +εϕ )Θ +(ψ +εϕ )Θ=0, (37)
ogyinversionspredictΩ(ϑ)intheconvectivezonewhichis 1 1 ′ 0 0
almostthesameasthatatthesurface(β O(1)),butbe- wherewekeptthenotationk˜2 R2k˜2,theinversescale
low the bottom of the convective zone (at≈the tachocline) heights æp,T are defined as ⊥ab⇒ove ⊙(be⊥low Eq. (13)) but
rotation is close to a solid-body rotation (β 1). As with a normalized z, and
y = y/R 1, in the deepest layers of the Su≪n we can ψ = aω˜2(N2/ω2 1+εk˜2)+b (a +æ +b /b ),
∗ ⊙ ≤ 0 − − ⊥ 1 1 T ′1 1
N.S.Dzhalilov, J. Staude& V.N.Oraevsky:Long-period eigenoscillations of theSun.I. 9
d d
ψ = ln(T3/2ad2), ϕ = ln p T ad2 , æp 0, d æρ, Λ and instead of Eq. (41) we
1 −dz 0 3 −dz 0 0 obta→in → − | | → ∞
ϕ0 = a1(æ′1+b2k˜2)+(æ′1+b2k˜2)′, (cid:0) (cid:1) (37′) I βk¯ ω N2 1 5æ æ 1 5æ æ 2,(42)
ϕ = k˜2ϕ + 4æ⊥ κ′′ +2æ , ⊥ ≈ x2Ω˜ (cid:18)ω2 − (cid:19)−2 T Ω−4(cid:18)2 T − Ω(cid:19)
1 − ⊥ 3 5 p κ ′1 wherek¯x =kxR andæΩ =dlnΩ˜/dz.Heretherearethree
ϕ = k˜2 +a b +b ω˜2a, possibilities: ⊙
− 2 1 2 ′2−
⊥ 1) absolute rigid body rotation, β =æ =0. In this case
a d q γ 1 4 κ Ω
a1 = ′ + ′ +æp+ω˜2 , q = − + ε ′′, I <0 and we have no cavity solutions;
− a d d γ 5 κ
2) rigid rotation with respect to latitude (β = 0), but
æ 5 d q N2
b = p æ ′ ω˜2 1 , ‘vertical’ differential rotation with respect to the radius
1 γ − 2 T − d − d ω2 − (æ = 0). It follows from Eq. (42) that in this case for
(cid:18) (cid:19) Ω
b = æ + d′ ω˜2q, M =1 ω k¯2 , oscill6ating solutions we must have æΩ > 0 as æT < 0
2 T d − d − 2Ω˜ βk⊥¯x is obeyed in the inner part of the Sun. It means that
solar rotation must have a decreasing speed toward the
q 2 ∂ q
a = 1 Λ+ ω˜ + æ + , center (dΩ/dz > 0). Then for I > 0 the condition
ρ
− d ∂z d
ε æ(cid:16)T5/(cid:17)2 1(cid:18) 1 (cid:19) κ 31)0T|æhTe|æmΩos>t r(e5aæliTst/i2c−caæseΩ)fo2rmtuhsetSbuenfuislfialleddiff;erential ro-
ε = 0 = 0 0 = R
R2 c ρ T γ R2 γ R2 c ρ tation in both directions (β = 0 and dΩ/dz = 0). In this
v 0 0 v 0 6 6
⊙ ∗ ⊙ ∗ ⊙ casewehavevariouschancestogettrappedwaves.Weare
k c2M
k¯ = k R , ω˜ =ωR /c, Λ=β x . interestedin oscillations with ω 10 8 s 1 (years)and
x,⊥ x,⊥ ⊙ ⊙ 2Ω˜ωR k¯ 102 (λ 3 104km, e.g. su≈nspo−ts)×. F−or solar condi-
Remind that γ = γiω. Here the main param⊙eter is Λ, it x ≈ ≈ ×
tions(Ω 2.86 10 6s 1 andN2 6 10 6s 2)we
includes the ro∗tation rate gradient (β=const). have k¯ ω⊙/≈2Ω 1×and−N2− /ω2 m1a0x10≈. So×the−dom−inant
Because we are interested in very low frequency oscil- x ≈ max ≈
terminEq.(42)isthe firstone.Tohaveanoscillatingso-
lations with periods of 1–20 years, we take ω 10 8s 1,
Tρ050/2 ≈ const= 1016, cv ≈ 2×108, æ0 ≈ 8×1≈05, a−nd−we lduitffieornen(Itia>l r0o)tiattiisonsu,ffiβc≥ien1t0−to8h≈avOe(aωv[se−ry1]s)l.owlatitudinal
have a small parameter for our task
For waves running in opposite direction to rotation in
T
ε(z) 10−8 c . (38) the azimuth (βk¯x > 0), the cavities (trapped wave area)
≈ T (z)
0 can form between the bottom of the convective zone and
Here T is the temperature of the solar center. For the
c almost the center of the Sun as well as in the outer part
whole Sun this parameter ε is changed in the interval
of the Sun where N2 >0.
10 8 ε 10 5. ε characterizes the degree of nonadia-
− −
≤ ≤ Wavespropagatingparalleltorotationmaybetrapped
baticity ofthe waves,it is definedas the ratioofthe wave
only in the convective zone (N2 < 0). To the outer and
period to the reciprocal of the Kelvin-Helmholtz time.
innersidesfromtheconvectivezonetheamplitudeofthese
waves decrease exponentially.
3. Adiabatic case
For the incompressible case it is easy to solve the
For idealized adiabatic waves (ε = 0) we have a second eigenvalue problem of the cavity oscillations, because
order equation, Eq. (42) has no singularity. Such a task has been solved
Θ′′+ψ1Θ′+ψ0Θ=0. (39) by Oraevsky& Dzhalilov (1997).However,in the nonadi-
Introductinganewvariableitmaybewritteninstandard abatic case we cannot apply the limit c2 . Therefore
→∞
form we haveto investigatethe morecomplicatedcompressible
Θ=Y T3/2ad2, Y (z)+I(z)Y =0, (40) case.
0 ′′
where q To investigatethe function I(z)givenby Eq.(41)ina
I =ψ 1 d ln(T3/2ad2) 2+ 1 d2 ln(T3/2ad2). (41) compressibleplasmaweneedtheordersofthequantitities
0− 4 dz 0 2dz2 0 enteringthe functionI(z).Toestimate these values letus
(cid:20) (cid:21)
ψ anda aredefined by Eqs.(37’) andd=R N2/g.The consideralinearprofileoftemperature,T T (1 β z),
0 0 c T
⊙ ≈ −
behavior of the function I(z) gives a possibility to ana- where the gradient β = 1 T /T 1, T is the
T eff c eff
− ≈
lyze qualitatively the waves in the solar interior and the effective temperature. Then we have a limit for the pa-
boundariesfromwhichthewavesarereflectedandbecome rameter æ from the center (z = 0) up to the sur-
T
trapped.IfI >0wehaveoscillatingsolutionsandifI 0 face (z = 1): 1 æ 103. The other parameters
T
≤ ≤ − ≤
the waves are exponentially decreasing (evanescent) with have the same order, æ O(æ ). Then we get also:
p,ρ T
≈
z. Now we shall breafly discuss the incompressible and æ æ2, æ æ æ , æ æ (æ +æ ). Now we
′T ≈ − T ′p ≈ p T ′ρ ≈ T p T
compressible cases. can estimate the sign behavior of I(z). We shall consider
As we consider adiabatic waves, the transition to the morecharacteristicplaces ofthe Sun.Inthe followingthe
incompressible case can be done by c2 . In this case condition βk¯ >0 will be supposed.
x
→ ∞
10 N.S.Dzhalilov, J. Staude& V.N.Oraevsky:Long-period eigenoscillations of theSun.I.
At the center, where z 0, g 0, N2 0, æ
p
→ → → →
0, c 350km/s, if Λ>1 we have
≈
2
ω 3 1 1 1
I βk¯ æ æ æ æ2 <0.
≈− x2Ω˜ − 2 Ω− 4 2 T − Ω − 2 T I(z)
(cid:18) (cid:19)
In the middle part of the Sun (between the edge of
the core and the bottom of the convective zone), where
N2 N2 andæ const 10,the dominanttermin cavity convective surface
≈ max ρ ≈ ≈− zone
the function I(z) is the first term of function ψ . Hence
0
we have I >0 such as in the incompressible case .
The area around the convective zone is more compli-
cated. The function I(z) has singularities at the points
z/R
where d = 0 and a = 0; d is connected with the Brunt-
Va¨is¨al¨afrequency anda is defined mainly by the rotation
gradient, Eqs. (37’). The two points of d = 0 correspond
to the bottom and the upper boundary of the convective
zone. The function a(z) is defined mainly by two terms
γ 1 d
′
a Λ − . (43) Fig.1. Scheme of the dependence of the wave potential
≈− − γ d2
I(z) defined by Eq. (41) on the distance normalized to
It is clear that as d < 0 at the lower boundary of the
′
the solar radius. Zeros of this function I(z) = 0 (turn-
convective zone, the two zeros of a(z) are located around
ing points) divide the wavezone into transparent(cavity)
this boundary where d=0. We can easily conclude that
3 a 2 and opaque (tunnel) parts. In the upper part of the Sun
′
limI(z) , therearesingularpointsofthefunctionI(z)whicharelo-
a 0 ≈−4 a →−∞
→ (cid:18) (cid:19) catedbetweentheturningpoints:thecirclescorrespondto
2
d′ a(z)=0 and the boxes correspond to N2(z)=0 (bound-
limI(z) 2 + . (44)
d 0 ≈ d → ∞ ariesof the convectivezone).The narrowareaatthe bot-
→ (cid:18) (cid:19)
In the convective zone (N2 < 0) we have I < 0. At tomoftheconvectivezonebetweencirclesandboxesisthe
the surface of the Sun we have again N2 >ω2 and hence tachocline. The main internal cavity comprises the whole
I >0. radiative interior as the convective zone becomes opaque
Thus if we approach the convective zone from below for the waves.
there exists a sequence of special points: four times the
function I(z) crosses zero (turning points), and between
if z z . It means, that the boundaries of the convec-
d
these zerosthe singularpoints areplaced(see Fig.1).The →
tive zone are not singular levels for the initial physical
turningpointsaredeterminedpracticallybythecondition
variables.
of N2(z) ω2 =0 and the singularities by the conditions
Anothersituationexistsatthepointz =z wherea=
− a
a=0 (circles in Fig.1) and N2 =0 (boxes in Fig.1). Due
0. If we denote now x as x=z z , our Eq. (40) around
a
totheverylowfrequencyandthesharpdecreaseofN2(z) thepointx 0isx2Y Y3/4=−0,thesolutionsofwhich
′′
the turning point of the main inner cavityzt is very close are Y = x3≈/2, Y = x −1/2. Hence, for these solutions we
1 2 −
to the first singular point where a=0. have Θ x2, Θ const. Then we get from Eq. (33)
1 2
In this way for waves with kx > 0 a main large in- that the s≈econd solu≈tion in p′/p0 diverges at a=0.
ternal cavity is placed practically between the center and Thereexistmethodstoconstructasymptoticsolutions
the bottom of the tachocline. The solar atmosphere is a of differential equations of second order with a singu-
wave-propagatingzone. Between the inner cavity and the lar turning point. However, we are now returning to our
solarsurfaceadarkconvectivetunnelisplaced;averynar- fourth order Eq. (37) for two reasons: our intent is to
row wave-trapping zone around the bottom of convective consider the instability problem of the eigenmodes, and
zone is also possible. It is clear that a tunneling of waves consequently in the complex ω plane the singularity at
acrossthemagnetizedandturbulentconvectiveareatothe a(z,ω)=0 is removed from the real z axis.
surface is probably possible. For waves with k < 0 the −
x
convective zone becomes a cavity. In this case the waves
4. Asymptotic solutions
cannot be propagating at the solar surface.
Not allsingularitiesofthe wavepotentialI(z)aresin- The existence of the small parameter ε in Eq. (37) allows
gularlevels of the physicalvariables.Around the pointzd us to apply asymptotic methods to solve this equation.
whered(zd)=0wecanwrited d′(z zd).Thenwehave Here we shall construct the inner cavity solutions only.
≈ −
the equation x2Y′′+2Y =0, where x=z zd. The solu- As it has beendiscussedabovethe coefficientsof Eq.(37)
−
tions of this equation are Y = √xx i√7/2. As a d 2 vary over a wide range. Very crudely, we have the fol-
1,2 ± −
∼
we get from Eqs. (33 + 40) that p/p Θ Y 0 lowing estimates from the center of the Sun to the bot-
′ 0 1,2
∼ ∼ →