Helioseismic Probing of Solar Variability

23 Pages·2003·0.38 MB·English

Helioseismic Probing of Solar Variability: The Formalism and Simple
Assessments
W. A. Dziembowski
Warsaw University Observatory and Copernicus Astronomical Center,Poland
Philip R. Goode
Big Bear Solar Observatory, New Jersey Institute of Technology
40386 North Shore Lane, Big Bear City, CA 92314-9672, USA
[email protected] – July 2, 2003 version as submitted to ApJ
ABSTRACT
We derive formulae connecting the frequency variations in the spectrum of solar os-
cillations to the dynamical quantities that are expected to change over the solar activity
cycle. This is done for both centroids and the asymmetric part of the fine structure
(so-called even-a coefficients). We consider the near-surface, small-scale magnetic and
turbulent velocity fields, as well as horizontal magnetic fields buried near the base of the
convective zone. For the centroids we also discuss the effect of temperature variation.
We demonstrate that there is a full, one-to-one correspondence between the expan-
sion coefficients of the fine structure and those of both the averaged small-scale velocity
and magnetic fields. Measured changes in the centroid frequencies and the even-a’s over
the rising phase solar cycle may be accounted for by a decrease in the turbulent velocity
of order 1%. We show that the associated temperature decrease may also significantly
contribute to the frequency increase. Alternatively, the increase may be accounted for
by an increase of the small-scale magnetic field of order 100 G, if the growing field is
predominantly radial.
We also show that global seismology can be used to detect a field at the level of a
few times 105 G, if such a field were present and confined to a thin layer near the base
of the convective envelope.
Subject headings: Sun : Helioseismology, solar variability, Submitted to ApJ
1
1. Introduction The present work represents an advance over
earlier ones (Gough & Thompson ,1990; Dziem-
We study global changes, over the solar cy-
bowski & Goode, 1991) because we make a more
cle, in the sun’s eigenmode frequencies – cen-
explicit and useful formulation by eliminating
troidsandasymmetricfinestructure–inasearch
derivativesoftheunknowndynamicalquantities.
of physical changes occurring beneath the pho-
This improved development allows us to obtain
tosphere. There is abundant phenomenological
more physically revealing formulae. This work
information about the helioseismic changes, but
is also aimed at determining a stringent limit on
there is no satisfactory physical model describ-
the size of a buried toroidal field.
ing the changes. We consider three possible dy-
namical sources of the evolution – changes in the
2. The Helioseismic Data
sub-photosphericsmall-scalemagneticandveloc-
ity fields and a large-scale toroidal field buried Solar frequency data are usually given in the
in a thin layer near the base of the convection form
(cid:88)
zone. Here, we develop the formalism needed to νm = ν¯ + a P(cid:96)(m), (1)
(cid:96)n (cid:96)n k,(cid:96)n k
connect these dynamical changes to frequencies k=1
changes. where the P are orthogonal polynomials (see
In our treatment of the small-scale magnetic Ritzwoller&Lavely1991andSchouetal. 1994).
field, we generalize the method of Goldreich et The remaining symbols (n(cid:96)m), in this equation
al.(1991, GMWK) to include the generalized ef- have their usual meanings. This representation
fect of the small-scale magnetic field on nonra- ensures that the ν¯ are a probe of the spheri-
(cid:96),n
dial modes, while further generalizing to a non- cal structure, while the a – the even-a coeffi-
2k
spherical distribution of the averaged field. Al- cients–areaprobeofthesymmetrical(aboutthe
though it is true that radial modes may ad- equator) part of distortion described by the cor-
equately represent lower degree (up to about respondingP (cosθ)Legendrepolynomials. We
2k
(cid:96) = 60) nonradial p-modes, if the magnetic field note that in lowest order, perturbations that are
effectswereconfinedtotheoutermostlayers,this symmetrical about the equator induce an asym-
is not true for higher degree p-modes or most f- metric change in the fine structure of the oscilla-
modes. Still, the more important generalization tion spectrum.
is that we treat a non-isotropic, non-spherical For the angular integrals, we have
fielddistribution,whichallowsustointerpretthe
(cid:90) (cid:90)
2π 1
observed evolution of the anti-symmetric part of Qm ≡ |Ym|2P dµdφ = S P(cid:96) (m)
k,(cid:96) (cid:96) 2k k,(cid:96) 2k
the fine structure in the spectrum of solar oscil- 0 −1
(2)
lations (so-called even-a coefficients).
where µ = cosθ and
Furthermore, we study effects a small-scale,
random velocity field. A role for the changing (2k−1)!! (2(cid:96)+1)!! ((cid:96)−1)!
S = (−1)k .
turbulent velocities has been suggested by Kuhn k,(cid:96)
k! (2(cid:96)+2k+1)!!((cid:96)−k)!
(1999). However,afirst-principlestreatmentstill
needs to be made. We give an estimate of the Following our earlier works (see e.g. Goode
associated temperature change and its effect on & Dziembowski, 2002), we use here the following
oscillation frequencies. convenient quantities, γ , through the follow-
k,(cid:96)n
Finally, we consider the effect of a buried ing two relations,
toroidal field, which may be expected to be con-
γ
0,(cid:96)n
fined near the base of the convective envelope. ∆ν¯(cid:96)n = I˜ (3)
(cid:96)n
2
and ontheadiabaticapproximation,whichisadopted
γ
k,(cid:96)n
a = S , (4) throughout our study. The first of the two ap-
2k,(cid:96)n k,(cid:96) I˜
(cid:96)n proaches begins with the linearized equations of
where I˜ is the dimensionless mode inertia cal- fluidmotionaboutasteadyconfiguration(seee.g
(cid:96),n
culated for our reference model. A clear advan- Lynden-Bell & Ostriker, 1967, LBO). The other
tage of the γ’s is that their growth replicates the uses Hamilton’s principle (see e.g. GMWK; De-
growth of other measures of solar activity. For war, 1970). Here, we use the form given by LBO
the p-modes, the 1/I˜ factor takes care of the (cid:96)- withsomesimplificationofthevariationalprinci-
and most of the ν-dependence in ∆ν¯ and in the ple, while adding the all-important contribution
even-a coefficients. The fact that the residual ν- of the magnetic field, as calculated explicitly by
dependenceisweakpointstoalocalizationofthe Dewar (1970). The LBO form is valid for strictly
sourceoftheobservedfrequencychangescloseto steady velocity fields. However, we make certain
to the photosphere. simplifications, which will be explained later, to
make it applicable to statistically steady fields.
The numerical values of the γ’s scale with
With this, we write
the square of the eigenfunction normalization at
the photosphere. the normalization we adopted
in our analyses of the SOHO MDI data (e.g. ω2I = −ωC +D (5)
Goode & Dziembowski, 2002) and which is used
where (cid:90)
throughout present paper, is explained in the I = d3xρ|ξ|2 (6)
next section. With this normalization, the value
of γ reaches up to the 0.3µHz range. The abso- (cid:90)
0
lute values of γ and γ are about twice larger. C = 2i d3xρξ∗·(v·∇)ξ, (7)
1 3
Having determined the set of γ , one may con-
k
and where v represents the velocity field. The
struct seismic maps of the varying sun’s activity
eigenvectors,inaspherically-symmetricandtime
(Dziembowski & Goode, 2002), that is the γ(µ)
independent model of the sun, are expressed in
dependence. In such maps, a rising γ reflects the
the following standard form
local rise of in the activity. The highest values
of γ(µ) are about 1µHz and they are reached at ξ = r[y(r)e +z(r)∇ ]Ym(θ,φ)exp(−iωt). (8)
r H (cid:96)
µ ≈ 0.3 and at the peak of the activity. At activ-
ity minimum the highest γ(µ) ≈ 0.2µHz occurs We adopt some approximations regarding the
in the polar region. eigenfunctions. In addition to adiabaticity, we
assumetheCowlingapproximationisvalid,which
Inthesubsequentsections,wewillconnectthe
iswell-justifiedinourapplicationtosolaroscilla-
γ’s, to magnetic and velocity fields that are ex-
tions. Further, we will make use of the fact that
pectedtochangeinthesunoveritsactivitycycle.
the oscillations are either of high degree or high
To achieve this, we start from a variational prin-
order, which means that
ciple for oscillation frequencies. In our expres-
sions, the (cid:96),n subscripts and the m superscript |ξ| (cid:191) Max(r|ξ |,(cid:96)|ξ|).
;r
will not be given unless it is necessary for clarity.
Equivalent approximations were also made by
GMWK but, in addition, they ignored the an-
3. Variationalprincipleforoscillationfre-
gular dependence of the displacement.
quencies
Like LBO, we separate the various contribu-
There are two ways of deriving the variational tion to D,
expression for oscillation frequencies. Both rely
D = D +D +D +D . (9)
p g v M
3
The pressure term,
(cid:90)
Dp = d3xp[Ξ+(Γ−1)|divξ|2], (10) Ξ = λ2|Y(cid:96)m|2+(y2+λz+Λz2)
(Λ|Ym|2+|∇ Ym|2)+[...] +[...] , (13)
(cid:96) H (cid:96) ;θ ;φ
is the same as in LBO. The quantity Ξ is a com-
The explicit expressions for the last two terms
pletely contracted double dyadic product,
will not be needed.
∇ξ∗ : ∇ξ. Adopting the standard summation
The gravity term simplifies to
convention, we have
(cid:90)
g
Ξ = ξ∗ ξ = (ξ∗ξ −ξ∗divξ) +|divξ|2, D = −2 d3xρ |ξ |2, (14)
j;k k;j j k;j k ;k g r r
where the subscript “;” denotes covariant deriva- afterusingtheCowlingapproximation, whilethe
tives. However, with our approximation regard- velocity term,
ing ξ, contributions from the terms involving the
(cid:90)
Christoffelsymbolsarenegligible,andthederiva- D = − d3xρ|(v·∇)ξ|2, (15)
v
tives may be regarded as component derivatives.
In terms of the radial eigenfunctions, y and z,
is the same as in LBO, where it was derived for a
with the adopted approximations we have
steady field velocity field. We will use the same
form in our application to a statistically steady
Ξ ≈ [r(yry −yλ) +λ2]|Ym|2+
;r ;r (cid:96) turbulent field. The expression for the magnetic
r(yz) |∇ Ym|2+[0.5(yrz −zλ)|Ym|2
;r H (cid:96) ;r (cid:96) ;θ term, which is taken from Dewar (1970), is
+z2∇ Ym∗·∇ Ym] +[...] ,
H (cid:96) H (cid:96);θ ;θ ;φ
(cid:90) (cid:183)
1
where D = d3x |(B·∇)ξ|2−
gr ω2r2 M 4π
λ = y −z
c2 c2 2divξ∗B·(B·∇)ξ+
(cid:184)
is radial eigenfunction corresponding to divξ, g 1
|B|2(Ξ+|divξ|2) (16)
is the local gravity, and c is the speed of sound. 2
ThelastterminΞisobtainedfromthepreceding
one by the replacement θ ↔ φ. Further, in the We now perturb eq.[5] about the static, non-
adopted approximation, we have magneticequilibriumstate. The∆denoteschanges
in parameters relative to this state. However, for
ry = λ+Λz, (11) centroid frequencies, ∆ is defined with respect to
;r
activity minimum because we do not have mod-
where Λ = (cid:96)((cid:96)+1) and elsofthesunpredictingcentroidfrequencieswith
(cid:181) (cid:182) µHz precision. Since we want to consider terms
N 2 c2
rz = y−λ . (12) that are quadratic in velocity, in principle, we
;r
ω gr
need to consider a second order perturbational
expression, which is
ThetermcontainingtheBrunt-V¨ais¨al¨afrequency,
N, is of the same order as the first one for p-
C C2/4I −ω∆C +∆D +D +D
s v M
modes only, and only in the outermost layers. ∆ω = − +
2I 2Iω
However, the whole contribution from the term
involving rz is small. Hence, we will ignore the where
;r
term, so that
∆D = ∆(D +D )−ω2∆I. (17)
s p g
4
Actually, we do not calculate the C integral or integrals are treated as perturbations. We may
its perturbation, but only comment on the role see that the integrands do not involve differen-
of the terms in C for various velocity fields. The tiation of the unknown characteristics of the ve-
first term, which is linear in velocity, results locity and magnetic fields, rather the differenti-
fromrotation,andgivesrisetoodd-acoefficients, ation is placed upon the eigenfunctions, which
which we are not treating here. It may be eas- are known. This is clearly advantageous and we
ily shown that both meridional and statistically will apply the same strategy in the evaluations
steady turbulence do not contribute. The second of ∆D .
s
term,∝ C2 duetorotationgivesanegligiblecon- The calculated frequency perturbation for in-
tribution ( Dziembowski and Goode, 1992) to p- dividual ((cid:96)nm)-modes are linked to the γ(cid:48)s de-
mode splitting. The term ∝ ∆C arises from the fined in eqs.[3] and [4] by the following relation
firstorderperturbationoftheeigenfunctionsdue
2π (cid:88)
to the velocity fields. Here the contribution from ∆ωm = γ Qm , (19)
rotationandmeridionalcirculationcanbeshown (cid:96)n I˜ k,(cid:96)n k,(cid:96)
(cid:96)n k=0
to be negligible. The only quadratic effect of ro-
with
tation, which we found to be significant for p-
Im = R5ρ¯ I˜ ,
modesisthatofthecentrifugaldistortion. Thus, (cid:96)n (cid:175) (cid:175) (cid:96)n
(cid:90)
it is included in the D term. For the f-mode
s I˜ = dxx4ρ˜E,
even-a’s, which are not accurately determined, (cid:96)n
the terms involving C2 may be important. The r ρ
x = , ρ˜= ,
alternative approach, which has been used by us R ρ¯
(cid:175)
in all our analyses of the even-a coefficients, is
and
to evaluate the centrifugal contribution and sub-
E = y2+Λz2.
tract it from the data. We neglect contribution
from turbulence to the term ∝ ∆C, because we Theadoptednormalization,leadingtomaximum
include only effects of interaction of oscillations γ’s are in the (0.2−1)µHz range is
with the averaged velocity fields. It has to be
y (r ) = 2×104.
kept in mind, however, that not all effects of tur- (cid:96),n phot
bulence are included in our formalism. So that
4. Dynamical perturbations of the struc-
we are left with the expression
ture
1
∆ω = (∆D +D +D ) (18)
s v M
2Iω Here, we include the dynamical effects of the
In the D term, we consider only effects of the magnetic B, and those of the velocity fields, v.
v
turbulent pressure and we will be interested in We write the condition of mechanical equilib-
the part that may vary with the solar activity. rium, in the presence of perturbing force F, in
We do not have yet observational evidence for the following form,
changes in turbulent velocity but such changes
areexpected. Theonlyglobalchangesinvelocity ∇p+ρge = F, (20)
r
which were definitely detected are the torsional
where
oscillations but their effect on frequencies we es-
(cid:181) (cid:182)
timated to be insignificant. ∂V ∂Heθ
F ≡ − e + = (21)
r
The D integral may be calculated consider- ∂r ∂θ r
s (cid:183) (cid:184)
1 1
ing either Eulerian or Lagrangian perturbations. − ∇B2−(B·∇)B −ρ(v·∇)v
The results must be the same. The D and D 4π 2
v M
5
We neglected the perturbation of the gravita- radial component of eq.[20] we have
tional potential, which is justified because we
(cid:90)
R δ r
consideringperturbingforcesconcentratedinthin k
δ p = −V +4 gρdr ≈ −V (25)
k k k
layers containing little mass. eq.[20] implies r r
and from mass conservation
p = −H+h(r) (22)
dδ r δ ρ δ r δ ρ
k k k k
= − −2 ≈ − . (26)
and
dr ρ r ρ
1 ∂
ρ = (H−V −h). (23)
g∂r
The approximate equality in the preceding
The quantities V and H represent non-gas pres- equation corresponds to neglecting of the pertur-
sures, which in general are anisotropic. It should bation in the mass distribution above the point
be noted that when the non-gas pressure is iso- underconsideration. Thisiscertainlyvalidforall
tropic, thenthemassdistributionremainsspher- of our applications. The approximate equality in
icallysymmetric. Thequantityh(r)mayonlybe eq.[26]isjustthelocalplaneparallelapproxima-
determined by utilizing the condition of thermal tion. This is valid for most of applications con-
equilibrium. For the non-spherically symmetric sidered here. The only possible exception will be
partsoftheforce,thepressureanddensityfollow discussed briefly in subsection 5.1. Both approx-
from the condition of mechanical equilibrium. imations were adopted in GMWK. We stress,
From now on, we treat F as a small perturb- however,theyarenotneededforderivingexpres-
ing force. Primed letters denote Eulerian pertur- sions for γk, except for k = 0. For k > 0, it is
bations of the respective structure parameters, onlyimportantand,infactwelljustified,forseis-
letters preceded by δ denote Lagrangian pertur- mic determination of the aspherical part of the
bations, and letters without such symbols imply subphotospheric temperature changes. To this
unperturbed variables. We use the standard re- aim, we first derive an expression for δkr from
lation the relation between p(cid:48)k and δkp,
df
δf = f(cid:48)+ δr
dr Vk −Hk
δ r = . (27)
k
and we adopt δM = 0, for both spherical and gρ
r
aspherical perturbations.
Then, usingthelinearizedp(ρ,T)relationweob-
Note that if we make a Legendre expansion
tain
in even orders, P (cosθ), of H and V, all the
2k (cid:183) (cid:179)
expansion coefficients p(cid:48) and ρ(cid:48), starting from δkT 1 d
k k = − Hk + χρ +
k = 1 are completely specified. Here we are T χTp dlnp
(cid:180) (cid:184)
considering only even order polynomials because dlnT
χ (V −H ) . (28)
T k k
these are the ones that contribute to the even- dlnp
a coefficients. The anti-symmetric (odd-order)
Here, we used a standard notation in astro-
polynomials average out. From eqs.[22] and [23],
physics, e.g. χ’s denote derivatives of logp with
we get for k > 0,
respect of the logρ and logT. We do not put,
1 d however, a subscript 1 on Γ, so as to avoid con-
p(cid:48) = −H and ρ(cid:48) = (H −V ). (24)
k k k gdr k k fusion with the expansion coefficients.
Weseethatforthenon-sphericalpart, allper-
The expansion coefficients for the Lagrangian
turbationsofthermodynamicalquantitiesarede-
perturbationsarecalculatedasfollows. Fromthe
termined by H and V. This is not true for k = 0,
6
where one of the thermodynamical parameters is Insertingthisexpressionintoeq.[31]andusing
left free. Choosing the temperature, we have eq.[22], we get
(cid:181) (cid:182)
δρ0ρ = −χ1 Vp0 +χTδT0T , (29) Vk = ρTkV, and Hk = 12ρTkH. (33)
ρ
4.2. Small-scale random magnetic field
or, if we choose the entropy per mass variation,
δ0S, in place of δ0T, Our treatment of the small-scale magnetic
field is analogous to that of the turbulent veloci-
δ ρ 1 V χ δ S
0 0 T 0
= − − . (30) ties. That is, the correlation matrix for the field
ρ Γ p χ c
ρ p
components is represented in the form of the fol-
4.1. Turbulent pressure lowing Legendre polynomial series,
(cid:88)
Thelarge-scaleaverageoftheReynold’sstress, BiBj = δij [δjrMVk(r)+
F = −ρ(v·∇)v, due to the turbulent velocity, k=0
1
is evaluated in the local Cartesian system with MH(r)(δ +δ )]P (cosθ).(34)
2 k jθ jφ 2k
axes parallel to e = (e ,e ,e ) (effects of cur-
r θ φ
vature are negligible for this small-scale velocity Components of the Lorentz force, treated locally
field). Then, we have as Cartesian, are given by
(cid:181) (cid:182)
1 (cid:88) 1
F = −(ρv v ) +(ρv ) v . F = B B − (B2) .
i j i ;j j ;j i i 4π j i;j 2 j ;i
j(cid:54)=i
We use the following relations
Averaging over wide zonal areas and making use
∂ρ of Bi;i = 0, we get
ρv v = δ ρv2 and (ρv ) v = − v = 0,
j i ji i j ;j i ∂t i
1∂B2
B B +B B = −B (B +B ) = r.
where double-subscripted δ is, as usual, the Kro- θ r;θ φ r;φ r θ;θ φ;φ 2 ∂r
necker symbol. That is, we assume uncorrelated
Thus,
velocity components and the rate of density fluc- 1 ∂ (cid:179) (cid:180)
F = B2−B2 ,
tuations. Hence, we have r 8π∂r r H
which is the same expression that was obtained
F = −(ρv2) no summation over i ! (31)
i i ;i by GMWK. It yields the net effect of the vertical
Further, we allow the vertical component (r) to component of the random field magnetic on the
be statistically different from the two horizontal verticalstructure,anditisoppositetothatofthe
(θ and φ) ones. In this, values of ρv2 are treated horizontal components. The radial component
j
as functions of depth, and slowly varying func- acts as a negative pressure, when it rises the gas
tions of the co-latitude. The latter dependence pressure must rise too.
is represented in the form of a Legendre polyno- To evaluate the horizontal force, we use
mial series,
1
B B = −B B = (B2) +B B ,
ρv v = ρδ (cid:88)[δ TV(r)+ φ θ;φ φ;φ θ 2 θ ;θ r;r θ
i j ij jr k
to get
k=0
1 (cid:183) (cid:179) (cid:180) (cid:184)
2TkH(r)(δjθ +δjφ)]P2k(cosθ), (32) Fθ = 41π (BrBθ);r + 21 Bθ2−Bφ2 −Br2 ;θ =
whereweincludedonlytermsthataresymmetric (B2)
r ;θ
− .
about equator.
8π
7
Finally, for the coefficients in the expansion of 5. The term arising from the structural
the magnetic pressure, we obtain perturbation, ∆D
s
MH −MV MV The frequency perturbation arising through
V = k k and H = k . (35)
k 8π k 8π the perturbation of the structure for all forces
considered by us is, typically, of the same order
Just as in the case of the turbulent velocity field
as that arising directly from the forces. We will
(see eq.[33]), isotropy implies V = H , hence no
k k
express now perturbation the structural term D
density perturbation, but effects of a departure s
in terms of V and H calculated in the previous
from isotropy are clearly different.
section. The variational principle ensures that
we may keep ξ (not y and z !) unperturbed.
4.3. Large-Scale Toroidal magnetic field
The large-scale field, B = Bt(r,θ)eφ, gives 5.1. Calculations of ∆Ds for centroid fre-
rise to the Lorentz force quency shift
F = −∇(Bt2r2sin2θ) ≈ −∇(Bt2sin2θ). Here, using the Lagrangian formulation of the
8πr2sin2θ 8πsin2θ perturbations is more convenient. Since we have
δ(drr2ρ) = 0, there is no contribution from
The last approximation is valid for a field con-
∆I. Furthermore, with our approximation for
fined to a narrow layer, which we will assume
the eigenfunctions, the contribution from D is
here, so that we have g
negligible. For the present application, it is con-
(cid:90)
B2 1 dθ ∂(B2sin2θ) venient to write eq.[10] in the form,
V = t and H = t .
8π 8π sin2θ ∂θ (cid:90) (cid:181) (cid:182)
dZ
D = drr2p +λ2 ,
We now put Bt(r,θ) in the form of the follow- p dr
ing series,
where Z = 2r(Λyz −y2) and, which after inte-
(cid:115)
(cid:88) 2j +1 dP (cosθ) gration by parts becomes
j
B (r,θ) = B (r) .
t t,j
2j(j +1) dθ (cid:90) (cid:183)(cid:181) (cid:182) (cid:184)
j 2p
(36) Dp = drr2 gρ− Z +pΓλ2 .
r
Note that with this representation, B is the
t,j
surface averaged intensity of the field component In the whole solar envelope the second term in
at a distance r from the center. Considering the coefficient at Z is much less than the first
only first two terms in the expansion, we get the one and it will be ignored. Now we calculate
following non-zero components of the Legendre ∆D ≈ ∆D using
s,0 p
polynomials expansion for V
δg δZ δr
= 2 = −2
B2 +B2 g Z r
t,1 t,2
V = , (37)
0
16π
and
(cid:181) (cid:182)
1 5 3B2
V = −B2 + B2 , V = − t,2 (38) δ(c2) δp δρ
1 16π t,1 7 t,2 2 28π = (1+Γ )− (1−Γ ),
c2 p p ρ ρ
and for H
1 (cid:181) 5 (cid:182) 9B2 where we denoted by Γp and Γρ logarithmic
H = − B2 + B2 , H = − t,2. derivativesofΓ. Further, weuseeq.[25]toelimi-
1 8π t,1 7 t,2 2 56π
nateδpandeq.[29]toeliminateδρ. Finally, with
(39)
8
ourapproximationsregardingtheeigenfunctions, 5.2. Calculations of∆D for the splittings
s
we get
In the present application, it is more conve-
(cid:90) (cid:183) (cid:181) (cid:182)(cid:184)
δT δr nient to treat the perturbations of the structural
∆D = drr2 D V +p D +D
s,0 isoth 0 T T r r parameters as being Eulerian. We consider dis-
(40) tortions proportional to P . We will see that
2k
where within our approximation, all the angular inte-
(cid:34)(cid:195) (cid:33) (cid:35)
grals appearing in ∆I, ∆D and ∆D reduce
1+Γ 2Λzλ p g
D = −Γ 1+Γ + ρ λ2+ , to Q . These factors take care of the k and m
isoth p χ χ k
ρ ρ dependence. The property is self-evident in the
(41)
case of D . From eq.[14] with the use of the
g
definitions given in eqs.[8] and [2], we get
χ
D = − TΓ[(1+Γ )λ2+2Λzλ], (42) (cid:90)
T χρ ρ ∆Dg = −2Qk drr3gρ(cid:48)ky2
and
and with eq.[24] after one integration by parts,
grρ
D = 2pΓΛzλ+6 (y2−Λyz). and use of eq.[11], we get
r
p
(cid:90)
The relative roles of the temperature and ra- ∆Dg = 4Qk drr2(Vk −Hk)y(λ+Λz). (46)
dius depends on the character of perturbation
and mode. As pointed out by Dziembowski, The cases of Ip and Dp are more involved. We
Goode and Schou (2001, heretoforward DGS), first note that
the latter may become dominant for f-modes, if
the magnetic perturbation is predominantly be- (cid:90) (cid:90)
1 2π
low the region sampled by these modes. For f- d(cosθ) dφ|∇Ym|2 ≈ ΛQ .
(cid:96) k
modes, to a very good accuracy, we may use −1 0
grρ The approximation assumes (cid:96) (cid:192) k, which is
D = −6 Λyz (43)
r p not valid for low degree modes. However the
termsinvolvingthisfactoraresignificantforsuch
In Section 9 of the present paper, we will dis-
modes only in the core, which we assume is un-
cussingreaterdetailstheroleoftemperatureand
perturbed. Thus, for ∆I we have approximately
radius variation in the f- and p-mode frequency
(cid:90)
changes.
∆I = Q drr4ρ(cid:48)E.
k k
If instead of eq.[29], we use eq.[30], then we
get an alternative expression for ∆D which is
s,0 Again, we make use of eq.[24] and integrate by
particularlyusefuliftheperturbingforceislocal-
parts, and with eqs.[11] and [12], to obtain ap-
ized in the deeper layers, which may be regarded
proximately
adiabatic on the eleven-year scale,
(cid:90)
(cid:90) (cid:183) (cid:195) (cid:33)(cid:184) r3
∆D = drr2 D V +p D δ0s +D δ0r , ∆I = 2Qk dr g (Vk −Hk)(yλ+2Λzy). (47)
s,0 ad 0 T r
c r
p
(44) In calculating ∆D , we first note that the [...]
p φ
where termineq.[13]doesnotcontribute,whichfollows
from the assumed axial symmetry of the pertur-
D = −[Γ(1+Γ )+1+Γ ]λ2−2Λzλ. (45)
ad p ρ bation. The contribution from the [...] term is
θ
9
nonzero, but it is small, as may be justified as eq.[32]. We note that
follows. Integrating by parts over θ, one gets the (cid:181) (cid:182)
(cid:88) 1
k(k+1)Q factor from the angular integral and ρ|(v·∇)ξ|2 = ρ TVAV + THAH P ,
k k 2 k 2k
the whole contribution from this term is of the k
(51)
same order as the one neglected above. Thus, we
where
have
(cid:90) (cid:104) (cid:179) (cid:181) dy(cid:182)2 (cid:181) dz(cid:182)2
∆D = Q drr2 Γ p(cid:48)(1+Γ )+ AV ≈ r |Ym|2+ r |∇Ym|2
p k k p dr (cid:96) dr (cid:96)
(cid:180) (cid:105)
pΓ
ρ(cid:48) ρ λ2+p(cid:48)2Λ(E +zλ) . and
k ρ k
AH ≈ y2|∇Ym|2+z2[|(Ym) |2
(cid:96) (cid:96) ;θ;θ
Thequantityρ(cid:48) isagaineliminatedbyintegra-
k +2|(Ym) |2+|(Ym) |2].
(cid:96) ;θ;φ (cid:96) ;φ;φ
tion by parts. The use is also made of eqs.[11]
and [12]. The result is The radial derivatives in AV are eliminated with
(cid:90) the help of eqs.[11] and [12]. The angular inte-
∆D = −Q drr2{H [Γ(1+Γ )λ2+ grals are evaluated by parts keeping only deriva-
p k k p
tives of the spherical harmonics. This approxi-
2Λ(E +zλ)]+(H −V )ψ},
k k mation justifies, in particular, the replacement
where 2|(Ym) |2 → 2(cid:60)[(Ym) (Ym) ].
(cid:96) ;θ;φ (cid:96) ;θ;θ (cid:96) ;φ;φ
(cid:183)(cid:181) (cid:182) (cid:181) (cid:182)(cid:184)
dlnc2 ω2r
ψ ≡ Γρ 2−Γ λ2+2λ Λz− y In this way, we get the contribution to Dv from
dlnp g
the P component of the turbulent pressure,
2k
dΓ
−Γ ρ λ2. (cid:90) (cid:104)
dlnp D = −Q drr2ρ TV(λ2+2Λzλ+ΛE)
v,k k k
(cid:184)
Combining this with eq.[47] in eq.[17] (∆D is Λ
g +TH E (52)
negligible), we obtain k 2
(cid:90)
ThecontributiontoD fromtheinducedchange
∆D = Q drr2(DVV +DHH ), (48) s,0
s,k k s k s k in the gas pressure at constant temperature and
radius is given by
where we denoted
(cid:90)
∆D = drr2ρD TV,
DV = −2ζ +ψ, (49) s isoth 0
s
ω2r which follows from eqs.[33] and [40]. Using these
ζ = (yλ+2Λzy), two expressions in eq. [18], we get
g
(cid:90)
and (∆ω) = 1 drr2ρ(RV TV+RH TH),
v,isoth 2ωI v,isoth 0 v,0 0
DH = 2ζ−ψ−Γ(1+Γ )λ2−2Λ(E +zλ). (50) (53)
s p
where
6. Frequency change to due varying tur-
RV = D −(λ2+2Λzλ+ΛE)
bulent pressure v,0 isoth
and
ForevaluatingDv accordingtoeq.[15], weuse Λ
RH = − E.
the random velocity field representation given in v,0 2
10