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The Astrophysical Journal, 698:461–478, 2009 June 10doi:10.1088/0004-637X/698/1/461
C ?2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
HELIOSEISMIC DATA INCLUSION IN SOLAR DYNAMO MODELS
Andr´ es Mu˜ noz-Jaramillo1, Dibyendu Nandy2, and Petrus C. H. Martens3
1Department of Physics, Montana State University, Bozeman, MT 59717, USA; munoz@solar.physics.montana.edu
2Indian Institute for Science Education and Research-Kolkata, Kolkata, WB 741252, India; dnandi@iiserkol.ac.in
3Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA; pmartens@cfa.harvard.edu
Received 2008 November 20; accepted 2009 April 1; published 2009 May 20
ABSTRACT
An essential ingredient in kinematic dynamo models of the solar cycle is the internal velocity field within the
simulation domain—the solar convection zone (SCZ). In the last decade or so, the field of helioseismology has
revolutionized our understanding of this velocity field. In particular, the internal differential rotation of the Sun
is now fairly well constrained by helioseismic observations almost throughout the SCZ. Helioseismology also
gives us some information about the depth dependence of the meridional circulation in the near-surface layers
of the Sun. The typical velocity inputs used in solar dynamo models, however, continue to be an analytic fit
to the observed differential rotation profile and a theoretically constructed meridional circulation profile that is
made to match the flow speed only at the solar surface. Here, we take the first steps toward the use of more
accurate velocity fields in solar dynamo models by presenting methodologies for constructing differential rotation
and meridional circulation profiles that more closely conform to the best observational constraints currently
available. We also present kinematic dynamo simulations driven by direct helioseismic measurements for the
rotation and four plausible profiles for the internal meridional circulation—all of which are made to match the
helioseismically inferred near-surface depth dependence, but whose magnitudes are made to vary. We discuss
how the results from these dynamo simulations compare with those that are driven by purely analytic fits to
the velocity field. Our results and analysis indicate that the latitudinal shear in the rotation in the bulk of the
SCZ plays a more important role, than either the tachocline or surface radial shear, in the induction of the
toroidal field. We also find that it is the speed of the equatorward counterflow in the meridional circulation right
at the base of the SCZ, and not how far into the radiative interior it penetrates, that primarily determines the
dynamo cycle period. Improved helioseismic constraints are expected to be available from future space missions
such as the Solar Dynamics Observatory and through analysis of more long-term continuous data sets from
ground-based instruments such as the Global Oscillation Network Group. Our analysis lays the basis for the
assimilation of these helioseismic data within dynamo models to make future solar cycle simulations more realistic.
Key words: Sun: activity – Sun: helioseismology – Sun: interior – Sun: magnetic fields – Sun: rotation
Online-only material: color figures
1. INTRODUCTION
The dynamic nature of solar activity can often be traced back
to the presence and evolution of magnetic fields in the Sun. The
more intense magnetic fields on the order of 1000 Gauss (G) are
observed to be concentrated within regions known as sunspots,
which often appear in pairs of opposite magnetic polarities
(Hale 1908). Sunspots have been observed regularly now for
about four centuries starting with the telescopic observations
of Galileo Galilei in the early 1600s. These observations point
out that the number of sunspots on the solar surface varies
in a cyclic fashion with an average periodicity of 11 years
(Schwabe 1844), although there are variations both in the
amplitude and period of this cycle. At the beginning of a cycle
sunspotsappearataboutmidlatitudesinbothhemispheres(with
opposite polarity orientation across the hemispheres) and then
progressively appear at lower and lower latitudes as the cycle
progresses (Carrington 1858) until no sunspots are seen (i.e.,
solar minimum). In the next cycle, the same pattern repeats,
but the new cycle spots have their bipolar magnetic orientation
reversed relative to the previous cycle (in both hemispheres).
So considering sign as well as amplitude, the solar cycle has a
period of 22 years.
There is a weaker, more diffuse component of the magnetic
field outside of sunspots which is seen to have a somewhat
different evolution. This field—whose radial component has
been observable at the solar surface since the advent of the
magnetograph—was believed to be on the order of 10 G. It is
found that this field is concentrated in unipolar patches, which
originate at sunspot latitudes at the time of sunspot maxima,
and then moves poleward with the progress of the sunspot cycle
(Babcock 1959; Bumba & Howard 1965; Howard & LaBonte
1981). The sign of this radial field of any given cycle is opposite
to the old cycle polar field, which it cancels and reverses upon
reaching the poles. The amplitude of this radial field achieves a
maximum (at the poles) at the time of sunspot minimum (i.e.,
with a 90◦phase difference relative to the sunspots). However,
theperiodicityofthecycleofthisfieldmatchesthesunspotcycle
period, underscoring that they are related. Recent observations
by Hinode indicate that this radial field gets concentrated within
unipolar flux tubes with field strength on the order of 103G
(Tsuneta et al. 2008).
Explanations of this observations of the solar magnetic cycle
rely on the field of magnetohydrodynamic dynamo theory,
whichseekstoaddressthegenerationandevolutionofmagnetic
fields as a complex nonlinear process involving interactions
between the magnetic field and plasma flows within the solar
interior. In particular, it is now believed that the solar cycle
involves the generation and recycling (feeding on the energy
availableinplasmamotions)oftwocomponentsofthemagnetic
461
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462 MU˜NOZ-JARAMILLO, NANDY, & MARTENSVol. 698
field—the toroidal component and the poloidal component. In
an axisymmetric spherical coordinate system, the magnetic and
velocity fields can be expressed as
B = Bφˆ eφ+ ∇ × (Aˆ eφ),
v = r sin(θ)Ωˆ eφ+ vp,
(1)
(2)
where the first term on the right-hand side of Equations (1)
and(2)isthetoroidalcomponent(inthecaseofthevelocityfield
thiscorresponds tothedifferentialrotation)andthesecondterm
is the poloidal component of the field (in the case of the velocity
fieldthiscorrespondstothemeridionalcirculation).Thetoroidal
component of the magnetic field is thought to be produced by
stretchingofaninitiallypoloidalfieldbythedifferentialrotation
of the Sun (the dynamo Ω-effect); subsequently, strong toroidal
flux loops rise up due to magnetic buoyancy emerging through
the solar surface as sunspots (Parker 1955a). To complete the
dynamo cycle, the poloidal field (whose radial component is
manifested as the observed vertical field on the solar surface)
has to be regenerated from this toroidal field in a process that is
traditionally called the dynamo α-effect. The first explanation
of this α-effect was due to Parker (1955b) who suggested that
helical turbulent convection in the solar convection zone (SCZ)
would twist rising toroidal flux tubes into the poloidal plane,
recreating the poloidal component of the magnetic field. Much
has changed since this pioneering description of the first solar
dynamo model by Parker, although the basic notion of the
recycling of the toroidal and poloidal components remains the
same.
First of all, simulations of the buoyant rise of toroidal flux
tubespointoutthattomatchtheobservedpropertiesofsunspots
at the solar surface, the strength of these flux tubes at the
base of the SCZ has to be much more than the equipartition
field strength of 104G (D’Silva & Choudhuri 1993; Fan et al.
1993). The classical dynamo α-effect due to helical turbulence
is expected to be quenched for superequipartition field strengths
and therefore other physical processes have to be invoked
as a regeneration mechanism for the poloidal field. One of
the alternatives is an idea originally due to Babcock (1961)
and Leighton (1969). The Babcock and Leighton (hereby BL)
model proposes that the decay and dispersal of tilted bipolar
sunspot pairs at the near-surface layers, mediated by diffusion,
differential rotation, and meridional circulation, can regenerate
the poloidal field. This mechanism is actually observed and is
simulated as a surface flux transport process that can reproduce
the solar polar field reversals (Wang et al. 1989). Therefore,
a synthesis of Parker’s original description along with the
BL mechanism for poloidal field generation is now widely
accepted asaleadingcontender forexplaining thesolardynamo
mechanism (Choudhuri et al. 1995; Durney 1997; Dikpati &
Charbonneau 1999; Nandy & Choudhuri 2001; Nandy 2003),
although there are other alternative suggestions as well. A
description of all of those is beyond the scope of this paper and
interested readers are referred to the review by Charbonneau
(2005).
Second, helioseismology has now mapped the solar internal
rotation profile (Schou et al. 1998; Charbonneau et al. 1999),
whichisobservedtobevarymainlyinthelatitudinaldirectionin
the main body of the SCZ. Helioseismology has also discovered
the tachocline—a region of strong radial and latitudinal shear
beneath the base of the SCZ which is expected to play an
important role in the generation and storage of strong toroidal
flux tubes.
Third, more is now known about the meridional circulation,
which is observed to be poleward at the surface (Hathaway
1996). To conserve mass, this circulation should turn equa-
torward in the solar interior. This circulation is deemed to be
important for the dynamics of the solar cycle (see Hathaway
et al. 2003, and the review by Nandy 2004) but the profile of
this in the solar interior remains poorly constrained. Nandy &
Choudhuri (2002) proposed a deep equatorward counterflow in
the circulation (penetrating into the radiative interior beneath
the SCZ) to better reproduce in dynamo simulations the latitu-
dinal distribution of sunspots (because equatorward advective
transport and storage of the deep-seated toroidal field is more
efficient at these depths where turbulence is greatly reduced).
However, Gilman & Miesch (2004), based on a laminar anal-
ysis, argued that the penetration of the circulation would be
limited to a shallow Ekman layer close to the base of SCZ. A
recent and more detailed analysis of the problem by Garaud &
Brummell (2008) suggests that the circulation can in fact pene-
trate deeper down into the radiative interior. At this point there
is no consensus on the profile and nature of the meridional cir-
culation in the solar interior. Helioseismic data do provide some
information about the depth dependence of this circulation at
near-surface layers (Braun & Fan 1998; Giles 2000; Chou &
Ladenkov 2005; Gonz´ alez-Hern´ andez et al. 2006), which, in
conjunction with reasonable theoretical arguments, can be used
to construct some plausible interior profiles of this flow.
Numerous kinematic dynamo models have been constructed
in recent years (see Charbonneau 2005 for a review) incorporat-
ing these large-scale flows (differential rotation and meridional
circulation) as drivers of the magnetic evolution. More recently
such dynamo models (based on the BL idea of poloidal field
generation) have also been utilized to make predictions for the
upcoming cycle (Dikpati et al. 2006; Choudhuri et al. 2007). At
present, all these kinematic dynamo models incorporate the in-
formation on large-scale flows as analytic fits to the differential
rotation profile and a theoretically constructed meridional cir-
culation profile that is subject to mass conservation but matches
the flow speed only at the solar surface (i.e., without incorporat-
ingthedepth-dependentinformationthatisavailable).However,
these large-scale flows are crucial to the generation and trans-
port of magnetic fields; the differential rotation is the primary
source of the toroidal field that creates solar active regions, and
the meridional flow is thought to play a crucial role in coupling
thetwosourceregionsforthepoloidalandtoroidalfieldthrough
advective flux transport. Given this, it is obvious that the next
step in constructing more sophisticated dynamo models of the
solarcycleistomovetowardamorerigoroususeofhelioseismic
data to constrain these models in a way such that they conform
more closely to the best available observational constraints; that
is the goal of this study.
In Section 2, we describe the basic features of the kine-
matic dynamo model based on the BL idea that we use for
our study; in this model, we use fairly standard parameter-
ization (commonly used in the community) of various pro-
cesses such as the diffusivity, dynamo α-effect, and magnetic
buoyancy. In Sections 3.1 and 3.2, we present the method-
ologies for using the helioseismically observed solar differen-
tial rotation and constraining the meridional circulation pro-
files within this dynamo model and describe how they im-
prove upon the commonly used analytic profiles. In Section
4, we present results from dynamo simulations using these
improved helioseismic constraints and conclude in Section 5
with a summary of our main results and their contextual rele-
vance.
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No. 1, 2009HELIOSEISMIC DATA INCLUSION IN SOLAR DYNAMO MODELS463
2. OUR MODEL
We substitute Equations (1) and (2) into the magnetic induc-
tion equation
∂B
∂t
= ∇ × (v × B − η∇ × B),
(3)
and add the phenomenological BL poloidal field source α to
obtain the axisymmetric dynamo equations:
∂A
∂t
+1
s[vp· ∇(sA)] = η
?
∇2−1
+ α(r,θ)F(Bav)Bav,
s2
?
A
(4)
∂B
∂t
+ s
?
?
vp· ∇
∇2−1
?B
?
s
??
+ (∇ · vp)B
B + s??∇ × (Aˆ eφ)?·∇Ω?+1
= η
s2
s
∂(sB)
∂r
∂η
∂r,
(5)
where s = r sin(θ). The terms on the left-hand side of
both equations with the poloidal velocity (vp) correspond to
the advection and deformation of the magnetic field by the
meridional flow. The first term on the right-hand side of both
equations corresponds to the diffusion of the magnetic field.
The second term on the right-hand side of both equations is the
source of that type of magnetic field (BL mechanism for A and
rotational shear for B). Finally, the third term on the right-hand
side of Equation (5) corresponds to the advection of toroidal
field due to a turbulent diffusivity gradient.
Asmentioned before,thegeneration ofpoloidalfield near the
surfaceduetothedecayofactiveregionsismodeledthroughthe
inclusion of a source term, which acts as a source for the vector
potential A. It is localized both in radius and latitude matching
observations of active region emergence patterns and depends
on the average toroidal field at the bottom of the convection
zone (Dikpati & Charbonneau 1999). The radial and latitudinal
dependence of the source is the following:
?
×
?
α(r,θ) =α0
16cos(θ)
?
×
1 + erf
?θ − (90◦+ β)
?r − ral
?θ − (90◦− β)
γ
??
1 − erf
γ
???
??
1 + erf
dal
1 − erf
?r − rah
dah
??
,
(6)
where α0sets the strength of the source term and we set it to the
value in which our system starts having oscillating solutions.
The parameters β = 40◦and γ = 10◦characterize the latitudes
in which sunspots appear. On the other hand, ral= 0.94 R?,
dal = 0.04 R?, rah = R?, and dah = 0.01 R?characterize
the radial extent of the region in which the poloidal field is
deposited(seeFigure1(a)and(b)).Besidesradialandlatitudinal
dependence, we also introduce lower and upper operating
thresholds on the poloidal source that is dependent on toroidal
field amplitude—which is a more realistic representation of
the physics involving magnetic buoyancy (as argued in Nandy
& Choudhuri 2001; Nandy 2002; see also Charbonneau et al.
2005). The presence of a lower threshold is in response to the
fact that the plasma density inside weak flux tubes is not low
enough (compared with the density of the surrounding plasma)
to make them unstable to buoyancy (Caligari et al. 1995) and
those that manage to rise have very long rising times (Fan et al.
1993). On the other hand, if flux tubes are too strong they are
not tilted enough when they reach the surface to contribute to
poloidal field generation (D’Silva & Choudhuri 1993; Fan et al.
1993). The dependence of the poloidal source on magnetic field
is
F(Bav) =
1 + (Bav/Bh)2
where Kae= 1/max(F(Bav)) is a normalization constant and
Bh = 1.5 × 105G and Bl = 4 × 104G are the operating
thresholds (see Figure 1(c)).
Another ingredient of this model is a radially dependent
magnetic diffusivity; in this work we use a double-step profile
(see Figure 2) given by
?
+ηsg− ηcz− ηbcd
2
Kae
?
1 −
1
1 + (Bav/Bl)2
?
,
(7)
η(r) = ηbcd+ηcz− ηbcd
2
1 + erf
?
?r − rcz
1 + erf
dcz
?r − rsg
??
dsg
??
,
(8)
where ηbcd = 108cm2s−1corresponds to the diffusivity at
the bottom of the computational domain, ηcz= 1011cm2s−1
corresponds to the diffusivity in the convection zone, ηsg =
1013cm2s−1corresponds to the supergranular diffusivity,
and rcz = 0.73 R?, dcz = 0.03 R?, rsg = 0.95 R?, and
dsg = 0.05 R? characterize the transitions from one value
of diffusivity to the other. Although it is common these days
to use these sort of profile, a point we note in passing is
that the exact depth dependence of turbulent diffusivity within
the SCZ is poorly, if at all, constrained. Besides the value
of the supergranular diffusivity ηsg, which can be estimated
by observations (Hathaway & Choudhary 2008), the others
parameters are not necessarily well constrained.
Once all ingredients are defined (see Section 3 for the flow
field),wesolvethedynamoequationsusingarecentlydeveloped
and novel numerical technique called exponential propagation
(see the Appendix). Our computational domain is defined in a
250 × 250 grid comprising only one hemisphere. Since we run
oursimulationsonlyinonehemisphereourlatitudinalboundary
conditions at the equator (θ = π/2) are ∂A/∂θ = 0 and
B = 0. Furthermore, since the equations we are solving are
axisymmetric, both the potential vector and the toroidal field
need to be zero (A = 0 and B = 0) at the pole (θ = 0), to avoid
singularities in spherical coordinates. For the lower boundary
condition (r = 0.55 R?), we assume a perfectly conducting
core, such that both the poloidal field and the toroidal field
vanish there (i.e., A,B = 0 at the lower boundary). For the
upper boundary condition, we assume that the magnetic field
has only a radial component (B = 0 and ∂(rA)/∂r = 0); this
condition has been found necessary for stress balance between
subsurface and coronal magnetic fields (for more details refer to
van Ballegooijen & Mackay 2007). As initial conditions we set
A = 0 throughput our computational domain and B ∝ sin(2θ)∗
sin(π ∗ ((r − 0.55 R?)/(R?− 0.55 R?))). After a few cycles,
all transients related to the initial conditions typically disappear
and the dynamo settles into regular oscillatory solutions whose
properties are determined by the parameters in the dynamo
equations.
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464 MU˜NOZ-JARAMILLO, NANDY, & MARTENSVol. 698
00 202040 40 60 6080 80
00
0.20.2
0.4 0.4
0.6 0.6
0.8 0.8
11
α(0.96 ∗ Rs,θ) (normalized)
Latitude (o) Latitude (o)
(a)
Poloidal Source at r = 0.96*Rs
α(0 96 ∗ Rs θ) (normalized)
Poloidal Source at r = 0.96*Rs
0.6 0.70.80.91
0
0.2
0.4
0.6
0.8
1
α(r,π/4) (normalized)
r/Rs
(b)
Poloidal Source at θ = π/4
05 10
0
0.2
0.4
0.6
0.8
1
F(B) (normalized)
Toroidal Field (105 Gauss)
(c)
Quenching Function of the Poloidal Source
Figure 1. Latitudinal (a) and radial (b) dependence of the poloidal source as quantified by the dynamo α-term. (c) The magnetic quenching of the poloidal source term.
3. CONSTRAINING THE FLOW FIELDS
Thelasttwoingredientsofthedynamomodelarethevelocity
fields (differential rotation and meridional circulation), which
are the focus of this work. The differential rotation is probably
the best constrained of all dynamo ingredients but the actual
helioseismology data have never before been used directly
in dynamo model, only an analytical fit to it. We discuss
below how the actual rotation data can be used directly within
dynamo models through the use of a weighting function to
filter out the observational data in the region where it cannot be
trusted. On the other hand, the meridional circulation is one
of the most loosely constrained ingredients of the dynamo.
Traditionally, only the peak surface flow speed is used to
constrain the analytical functions that are used to parameterize
it, in conjunction with mass conservation. In this work, we take
advantage of the properties of such functions and make a fit to
the helioseismic data on the meridional flow that constrains the
locationandextentofthepolardownflowandequatorialupflow,
as well as the radial dependence of the meridional flow near the
surface—thereby taking steps toward better constrained flow
profiles.
3.1. Differential Rotation
As opposed to meridional circulation, there are helioseismic
measurements of the differential rotation for most of the con-
0.60.60.7 0.70.8 0.80.9 0.911
10 10
88
10 10
99
1010
10 10
10 10
1111
10 10
1212
10 10
1313
η (cm2/s)
r/Rs
Magnetic Diffusivity ProfileMagnetic Diffusivity Profile
η (cm2
r/Rs
Figure 2. Radial dependence of the turbulent magnetic diffusivity.
vective envelope which can be used directly in our simulations.
Here we use data from the Global Oscillation Network Group
(GONG; courtesy Dr. Rachel Howe) obtained using the Regu-
larized Least Squares (RLS) inversion mapped onto a 51 × 51
grid (see Figure 3(a)). However, these observations cannot be
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No. 1, 2009 HELIOSEISMIC DATA INCLUSION IN SOLAR DYNAMO MODELS465
Splines Interpolation of RLS Inversion
r/Rs
r/Rs
0 0.20.4 0.6 0.81
0
0.2
0.4
0.6
0.8
1
(nHz)
340
360
380
400
420
440
460
Analytical Profile
r/Rs
r/Rs
0 0.20.4 0.60.81
0
0.2
0.4
0.6
0.8
1
(nHz)
340
360
380
400
420
440
460
(a)(b)
Differential Rotation Composite
r/Rs
r/Rs
0 0.2 0.40.6 0.81
0
0.2
0.4
0.6
0.8
1
(nHz)
340
360
380
400
420
440
460
r/Rs
r/Rs
Composite Mask
0 0.20.4 0.60.81
0
0.2
0.4
0.6
0.8
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(c)(d)
Figure 3. (a) Spline interpolation of the RLS inversion. (b) The analytical profile of Charbonneau et al. (1999). (c) The differential rotation composite used in our
simulations. (d) Weighting function used to create a composite between the RLS inversion and the analytical profile of Charbonneau et al. (1999). For all figures the
red denotes the highest and blue the lowest value and the units are nHz with the exception of the weighting function.
(A color version of this figure is available in the online journal.)
trusted fully in the region within 0.3 R?of the rotation axis
(specifically at high latitudes), because the inversion kernels
have very little amplitude there. Below, we outline a method
to deal with this suspect data by creating a composite rotation
profile that replaces these data at high latitudes with plausible
synthetic data, which smoothly matches to the observations in
the region of trust.
3.1.1. Adaptation of the Data to the Model
In the first step, we use a splines interpolation in order to map
the data to the resolution of our simulation (a grid of 250×250,
see Figure 3(a)). The next step is to make a composite with the
data and the analytical form of Charbonneau et al. (1999; see
Figure 3(b)). The analytical form is defined as
?
× (Ωe− Ωc+ (Ωp− Ωe)ΩS(θ))
× ΩS(θ) = a cos2(θ) + (1 − a)cos4(θ),
where Ωc = 432 nHz is the rotation frequency of the core,
Ωe = 470 nHz is the rotation frequency of the equator,
Ωp= 330 nHz is the rotation frequency of the pole, a = 0.483
ΩA(r,θ) = 2π
Ωc+1
2
?
1 − erf
?r − rtc
wtc
??
?
(9)
is the strength of the cos2(θ) with respect to the cos4(θ) term,
rtc = 0.716 the location of the tachocline, and wtc = 0.03.
We use the parameters defining the tachocline’s location and
thickness as reported by Charbonneau et al. (1999) for a latitude
of60◦.Thisisbecausetheymatchthedatabetterathighlatitudes
(which is the place where the data merge with the analytical
profile) than those reported for the equator. This composite
replaces the suspect data at high latitudes within 0.3 R? of
the rotation axis with that of the analytic profile. However, it
is important to note that at low latitudes, within the convection
zone, the actual helioseismic data are utilized.
In order to make the composite, we create a weighting
function m(r, θ) with values between 0 and 1 for each gridpoint
defining how much information will come from the RLS data
and how much from the analytical form (see Figure 3(d)). We
define the weighting function in the following way:
?
where rm= 0.5 R?is a parameter that controls the center of the
transitionanddm= 0.6 R?controlsthethickness.Theresultant
differential rotation profile, which can be seen in Figure 3(c), is
then calculated using the following expression:
m(r,θ) = 1 −1
2
1 + erf
?r2cos(2θ) − r2
m
d2
m
??
,
(10)
Ω(r,θ) = m(r,θ)ΩRLS(r,θ) + [1 − m(r,θ)]ΩA(r,θ).
(11)
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466 MU˜NOZ-JARAMILLO, NANDY, & MARTENSVol. 698
Data Minus Composite
r/Rs
(a)
r/Rs
00.2 0.40.60.81
0
0.2
0.4
0.6
0.8
1
(nHz)
0
10
10
20
20
30
30
40
40
10
20
30
40
Data Minus Analytical Profile
r/Rs
(b)
r/Rs
0 0.2 0.40.60.81
0
0.2
0.4
0.6
0.8
1
(nHz)
0
10
20
30
40
Figure 4. (a) Residual of subtracting the composite used in this work to the RLS inversion. (b) The residual of subtracting the analytical profile commonly used by
the community to the RLS inversion. The red color corresponds to the highest value and blue to the lowest. Graphs are in units of nHz.
(A color version of this figure is available in the online journal.)
3.1.2. Differences Between the Analytical Profile
and the Composite Data
It is instructive to compare the analytical and composite
profiles with the actual helioseismology data. In Figure 4(a),
we present the residual error of subtracting the analytical profile
of Charbonneau et al. (1999), with rtc= 0.7 and wtc= 0.025,
from the RLS data. In Figure 4(b), we present the residual error
of subtracting our composite fromthe RLS data. As isexpected,
thereisnodifferencebetweenthecompositeandrawdataatlow
latitude, but the residual increases as we approach the rotation
axis—where the RLS data cannot be trusted. For the analytical
profile, the residuals errors are more significant, even at low
latitudes. This demonstrates the ability of our methodology to
usefully integrate the helioseismic data for differential rotation.
3.2. Meridional Circulation
Themeridionalflowprofileremainsratherpoorlyconstrained
in the solar interior even though the available helioseismic data
can be used to constrain the analytic flow profiles that are in
use currently. Here, we present ways to betters constrain this
profile with helioseismic data. In order to do that, we use
data from GONG (courtesy Dr. Irene Gonz´ alez-Hern´ andez)
obtained using the ring-diagrams technique. These data, which
we can see in Figure 6(a), correspond to a time average of
the meridional flow between 2001 and 2006; and it comprises
19 values of r from R?down to a depth of 0.97 R?, and 15
different latitudes between −52.◦5 and 52.◦5. It is important to
note that our work relies heavily in the assumption that the
meridional flow is adequately described by a stream function
with separable variables. This is consistent with the assumption
present implicitly in all work on axisymmetric solar dynamo
modelsuptothisdate.Below,weusethispropertyofourstream
function, along with weighted latitudinal and radial averages of
the data, to completely constrain its latitudinal dependence, as
well as the topmost 10% of its radial dependence. As these data
currently do not constrain the depth of penetration of the flow
in the deep solar interior, we explore two different plausible
penetration depths of the circulation. For reasons described
later, we choose to perform simulations with two different
peak meridional flow speeds, therefore exploring four plausible
meridional flow profiles altogether.
3.2.1. Constraining the Latitudinal Dependence
of the Meridional Flow
Meridional circulation has been typically implemented in
these type of dynamo models by using a stream function in
combination with mass conservation, i.e.,
− →vp(r,θ) =
1
ρ(r)
?∇ × (Ψ(r,θ)? eφ).
(12)
The two stream functions that are commonly used were
proposed by van Ballegooijen & Choudhuri (1988) and Dikpati
& Choudhuri (1995). They have in common the separability of
variables and thus can be written in the following way:
Ψ(r,θ) = v0F(r)G(θ),
(13)
where v0 is a constant which controls the amplitude of the
meridional flow.
Using such a stream function the components of the merid-
ional flow become
vr(r,θ) = v0F(r)
rρ(r)
1
sin(θ)
∂
∂θ(sin(θ)G(θ)),
(14)
vθ(r,θ) = −v0
1
rρ(r)
∂
∂r(rF(r))G(θ),
(15)
which can themselves be separated into the multiplication
of exclusively radially and latitudinally dependent functions.
This property allows us to constrain the entire latitudinal
dependence of this family of functions by using the available
helioseismology data for vθ at the surface. This can be done
because the latitudinal dependence of vθ is exactly the same
as that of the stream function and only the amplitude of this
functional form changes with depth, see for example Figure 5
for the latitudinal velocity at different depths used by van
Ballegooijen & Choudhuri (1988). In this work, we assume
a latitudinal dependence like the one they used, i.e.,
G(θ) = sin(q+1)(θ)cos(θ).
(16)
In order to estimate the parameter q, we first take a density-
weighted average of the helioseismic data using the values for
Page 7
No. 1, 2009HELIOSEISMIC DATA INCLUSION IN SOLAR DYNAMO MODELS467
0 5050
0
5
5
10
10
15
15
20
20
Latitudinal velocity at different depths
Vθ(m/s)
Latitude (o)
1Rs
0.9Rs
0.8Rs
0.75Rs
Figure 5. Latitudinal velocity as a function of θ used by van Ballegooijen &
Choudhuri (1988) for different depths. Note that the curves differ from each
other only on their amplitude.
solar density from the solar model S (Christensen-Dalsgaard
et al. 1996), such that
?
¯ vθ(θj) =
ivθ(ri,θj)ρ(ri)
?
iρ(ri)
.
(17)
In Figure 6(b), we plot the meridional flow at different depths
weighted by density, which we add for each latitude in order
to find the average. We then use this average to make a least-
squaresfittotheanalyticalexpression(Equation(16)),whichwe
can see in Figure 6(c). We find that a value of q = 1 fits the data
best. This therefore constrains the latitudinal (θ) dependence of
the flow profile.
3.2.2. Constraining the Radial Dependence of the Meridional Flow
As opposed to the latitudinal dependence, the radial depen-
dence of the meridional flow is less constrained since there is no
data below 0.97 R?. However, at least some of the parameters
can be constrained: we start with the solar density, for which
we perform a least-squares fit to the solar model S using the
following expression:
?R?
we find that values of γ = 0.9665 and m = 1.911 fit the model
best (see Figure 7(c)).
In the second step, we constrain the radial dependence of the
stream function. We begin with the function
ρ(r) ∼
r
− γ
?m
,
(18)
F(r) =1
r(r − Rp)(r − R?)sin
?
πr − Rp
R1− Rp
?a
,
(19)
(a)(b)
(c)
Figure 6. (a) Measured meridional flow as a function of latitude at different depths (courtesy Dr. Irene Gonz´ alez-Hern´ andez), each combination of colors and markers
corresponds to a different depth ranging from 0.97 R?to R?. (b) The meridional flow after being weighted using solar density. If we sum all data points at each
latitude we obtain the average velocity. (c) The normalized average velocity and analytical fit.
(A color version of this figure is available in the online journal.)
Page 8
468 MU˜NOZ-JARAMILLO, NANDY, & MARTENSVol. 698
(a)(b)
(c)
Figure 7. (a) Measured meridional flow as a function of radius at different latitudes (courtesy Dr. Irene Gonz´ alez-Hern´ andez), each combination of colors and markers
corresponds to a different latitude varying from −52.5 to 52.5. (b) The meridional flow after removing the latitudinal dependence, i.e., vθ/G(θ). The horizontal line
with zero latitudinal velocity corresponds to the equator. (c) The radial dependence of the latitudinally averaged meridional flow for our helioseismic data is depicted
as large black dots. Other curves correspond to the radial dependence of the meridional flow profiles used in our simulations and solar density: Set 1 (black dotted)
Rp= 0.64 R?, vo= 12 m s−1; set 2 (magenta solid) Rp= 0.64 R?, vo= 22 m s−1; set 3 (green dash-dotted) Rp= 0.71 R?, vo= 12 m s−1, and set 4 (blue
dashed line) Rp= 0.71 R?, vo= 22 m s−1. The solar density taken from the solar model S (Christensen-Dalsgaard et al. 1996) is depicted as a solid red line. The
left-vertical axis is in units of velocity and the right-vertical is in units of density.
(A color version of this figure is available in the online journal.)
where R?corresponds to the solar radius, Rpto the maximum
penetration depth of the meridional flow, and a and R1control
the location in radius of the poleward peak and the value of the
meridional flow at the surface. In order to constrain them we
use the helioseismic data again, but this time we use the radial
dependence (see Figure 7(a)). We first remove the latitudinal
dependence,whichwedobydividingthedataofeachlatitudeby
G(θ) using the value of q = 1 found in the previous section. In
Figure 7(b), we plot the flow data after removing the latitudinal
dependence, note that there is no longer any sign difference
between the two hemispheres. The next step is to generate the
latitudinalaveragewhichwecanseeasblackdotsinFigure7(c).
It is evident from looking at the radial dependence of the
meridional flow that the velocity increases with depth for most
latitudes, and that the point of maximum velocity is not within
the depth up to which the data extends. Since the exact radial
dependence of the data is too complex for our functional form
to grasp, the features we concentrate on reproducing are the
presenceofamaximuminsidetheconvectionzone,aswellasthe
amplitude of the flow at the near-surface layers. The logic here
is to use the fewest possible parameters and a simple, physically
transparent profile that does a reasonable job of matching the
data. In view of the lack of better constraints, we assume here
that the peak of the return flow is at 0.97 R?(which is the depth
at which the current helioseismic data have its peak).
Following the procedures and steps above, we construct
profiles to fit the depth dependence pointed out in the available
helioseismic data; however, this does not constrain how far the
meridional flow can penetrate and therefore we try two different
penetration depths, one shallow 0.71 R?(i.e., barely beneath
the base of the SCZ) and one deep Rp = 0.64 R?(into the
radiative interior)—both of which match the latitudinal and
radialconstraintsasdeducedfromthenear-surfacehelioseismic
data.Notethatobservedlight-elementabundanceratioslimitthe
depth of penetration of the circulation to about Rp= 0.62 R?
(Charbonneau 2007). Now we know that the meridional flow
speed is highly variable, with fluctuations that can be quite
significant and the measured flow speed can change depending
on the phase of the solar cycle (Hathaway 1996; Gizon &
Rempel2008).Themagneticfieldsarealsoexpectedtofeedback
on the flow (Rempel 2006). Taken together these considerations
point out that the effective meridional flow speed to be used
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No. 1, 2009HELIOSEISMIC DATA INCLUSION IN SOLAR DYNAMO MODELS 469
Table 1
Sets of Parameters Characterizing the Different Meridional Flow Profiles used
in Our Dynamo Simulations
vo(m s−1)
12
22
12
22
Rp(R?)
0.64
aR1(R?)
1.027 Set 1
Set 2
Set 3
Set 4
1.795
0.712.031.03
Note. vo corresponds to the meridional flow peak speed, Rp the maximum
penetration of the flow, and a and R1are the parameters that control the location
of the poleward flow as well as the surface speed.
in dynamo simulations could be less than that implied from the
Gonz´ alez-Hern´ andezetal.data,apossibilitythatisborneoutby
the work of Braun & Fan (1998) and Gizon & Rempel (2008),
whofindmuchlowerpeakflowspeedsintherange12–15ms−1.
Keeping this in mind, we use the same latitudinal and radial
constraintasdeducedearlier,butconsiderinoursimulationstwo
additional profiles with peak flow speeds of 12 m s−1with deep
and shallow penetrations. Therefore, in total we explore four
plausible meridional flow profiles in our dynamo simulations
(see Figure 7(c) and Table 1 for an overview), the results of
which are presented in the next section.
4. DYNAMO SIMULATION: RESULTS AND
DISCUSSIONS
4.1. Analytic versus Helioseismic Composite Differential
Rotation
Wefirstcomparedynamosolutionsfoundusingthehelioseis-
mologycompositeofthedifferentialrotation(Section3.1.1)and
the analytical profile of Charbonneau et al. (1999). In doing so,
we perform simulations with a model as described in Section 2
(see also numerical methods in the Appendix) and generate
field evolution maps for the toroidal and poloidal fields. From
thesimulatedbutterflydiagramsfortheevolutionofthetoroidal
field at the base of the SCZ and the surface radial field evolution
(Figure 8), we find that large-scale features of the simulated so-
lar cycle are generally similar across the two different rotation
profiles (even with different meridional flow penetration depths
and speeds), specially for the shallowest penetration (sets 3 and
4 with Rp= 0.71 R?).
In order to understand this similarity, it is useful to look
at Figures 10 and 11 corresponding to simulations using the
composite differential rotation, and the meridional flow sets 1
(Rp = 0.64 R?, vo = 12 m s−1) and 4 (Rp = 0.71 R?,
vo= 22 m s−1). The first two columns, from left to right of both
figures show the evolution of the shear sources (Br· ∇rΩ) and
(Bθ· ∇θΩ)—which contribute to toroidal field generation by
stretching of the poloidal field. It is evident that the location and
strength of these sources is different—the radial shear source is
mainly present near the surface whereas the latitudinal shear is
spread throughout the convection zone. It can also be seen that
the radial shear source is roughly five times stronger than the
latitudinal. However, if attention is paid to the evolution of the
toroidal field (third column from the right in both figures), it is
clear that this radial shear term has no significant impact on the
structure and magnitude of the toroidal field (a similar result
was found by Dikpati et al. 2002). The reason is that the upper
boundary condition (B = 0 at r = R?), in combination with
thehighturbulentdiffusivity(andthusshortdiffusivetimescale)
there, imposes itself very quickly on the toroidal field generated
by the radial shear—washing it out. This greatly reduces the
relativeroleofthesurfaceshearasasourceoftoroidalmagnetic
field, effectively making the surface dynamics very similar
across simulations using the analytical rotation profile (without
any surface shear) or the composite helioseismic profile.
Once the surface layers are ruled out as important sources
of toroidal field generation we are faced with the fact that the
strongest source of toroidal field is the latitudinal shear inside
the convection zone (and not the tachocline radial shear), as is
evident in Figures 10 and 11. This goes against the commonly
held perception that the tachocline is where most of the toroidal
field is produced. However, the importance of the latitudinal
shear term in the SCZ is clearly demonstrated here where we
have plotted the shear source terms, which is not normally done
by dynamo modelers.
The establishment of the SCZ as an importance source region
of toroidal field is of relevance when regarding the similarity
of the solutions obtained using the composite data and the
analytical profile. This is because the region where the shear
of the analytical profile and the helioseismic composite data
differ most (both for radial and latitudinal shear) is in the
tachocline.ThisisevidentinFigure9whereweplottheresidual
of subtracting the radial and latitudinal shear of the analytical
profile from the shear of the composite data. The similarity
between solutions is specially important for shallow meridional
flow profiles with low penetration—which does not transport
any poloidal field into the deeper tachocline, thereby further
diminishing the role of the tachocline shear.
4.2. Shallow versus Deep Penetration of the Meridional Flow
Inthesecondpartofourwork,wecomparedynamosolutions
obtained for each of the four different meridional flow profiles
withtwodifferentpenetrationdepthandwithtwodifferentpeak
flowspeeds.First,fromFigure8itisevidentthattheshapeofthe
solutions changes with varying penetration depth; this is caused
by the increasing role of the tachocline shear in generation
and storage of the toroidal field as the penetration depth of the
meridional flow increases. This is apparent when one compares
Figure 10 (deep penetration) to Figure 11 (shallow penetration).
If we look at the inductive shear sources it is evident that for
the shallowest penetration no field is being generated inside the
tachocline. In the poloidal field plots (right column) we see that
no poloidal field is advected into the tachocline region for the
shallowest penetration, but some is advected for the deepest.
Second, we compare the periods of our solutions in Table 2:
we find that most solutions have a sunspot cycle (i.e., half-
dynamo cycle) period that is comparable to that of the Sun,
with the exception of the fast flow with deep penetration and the
slow flow with shallow penetration, which have, respectively,
a comparatively smaller and larger period. As the meridional
flow is buried deeper, one expects the length of the advective
circuit to increase, thereby resulting in larger dynamo periods.
However, it is evident that as we increase the penetration,
the period decreases—even if the length of the flow loop
that supposedly transports magnetic flux increases; this is
counterintuitive but has a simple explanation. Our simulations
andexhaustiveanalysispointsoutthatitisnothowdeeptheflow
penetrates that governs the cycle period, but it is the magnitude
of the meridional counterflow right at the base of the SCZ
(Rp= 0.713 R?) that is most relevant. This is because most of
the poloidal field creation at near-surface layers is coupled to
buoyanteruptionsoftoroidalfieldfromthislayerofequatorward
migrating toroidal field belt at the base of the SCZ (see third
Page 10
470MU˜NOZ-JARAMILLO, NANDY, & MARTENS Vol. 698
Figure 8. Butterfly diagram of the toroidal field at the bottom of the convection zone (color) with radial field at the surface (contours) superimposed on it. Each row
corresponds to one of the different meridional circulation sets. The left column corresponds to runs using the helioseismic composite and the right one to runs using
the analytical profile.
(A color version of this figure is available in the online journal.)
column in Figures 10 and 11) and it is the flow speed at this
regionthatgovernsthedynamoperiod.InFigure7,itisclearthat
the speed of the counterflow in this convection zone–radiative
interiorinterfaceincreasesastheflowbecomesmorepenetrating
(a consequence of the constraints set by mass conservation and
the fits to the near-surface helioseismic data), thereby reducing
the dynamo period.
Overall, an evaluation of the butterfly diagram (Figure 8),
points out that the toroidal field belt extends to lower latitudes
(wheresunspotsareobserved)fordeeperpenetratingmeridional
flow, although there is a polar branch as well. For the shallow
flow, we find that the toroidal field belt is concentrated around
midlatitudes with almost symmetrical polar and equatorial
branches—a signature of the convection zone latitudinal shear
Page 11
No. 1, 2009HELIOSEISMIC DATA INCLUSION IN SOLAR DYNAMO MODELS 471
(a) (b)
Figure 9. (a) Residual after subtracting the radial shear of the analytical profile commonly used by the community from the radial shear of our composite data.
(b) The residual of subtracting the latitudinal shear of the analytical profile commonly used by the community from the latitudinal shear of our composite data.
(A color version of this figure is available in the online journal.)
Figure10.Snapshotsoftheshearsourcetermsandthemagneticfieldoverhalfadynamocycle(asunspotcycle).Eachrowisadvancedbyaneightofthedynamocycle
(a quarter of the sunspot cycle), i.e., from top to bottom t = 0, τ/8, τ/4, and 3τ/8. The solution corresponds to the composite differential rotation and meridional
flow of set 1 (deepest penetration with a peak flow of 12 m s−1).
(A color version of this figure is available in the online journal.)
Page 12
472MU˜NOZ-JARAMILLO, NANDY, & MARTENSVol. 698
Figure11.Snapshotsoftheshearsourcetermsandthemagneticfieldoverhalfadynamocycle(asunspotcycle).Eachrowisadvancedbyaneightofthedynamocycle
(a quarter of the sunspot cycle), i.e., from top to bottom t = 0, τ/8, τ/4, and 3τ/8. The solution corresponds to the composite differential rotation and meridional
flow of set 4 (shallowest penetration with a peak flow of 22 m s−1).
(A color version of this figure is available in the online journal.)
Table 2
Simulated Sunspot Cycle Period for the Different Sets of Meridional Flow
Parameters
Rp(R?)
0.64
vo(m s−1)
12
22
12
22
τ—HS Data (yr)
9.67
5.63
14.67
11.85
τ—Analytical (yr)
10.00
5.67
14.02
12.85
Set 1
Set 2
Set 3
Set 4
0.71
Notes.Rpcorrespondstothemaximumpenetrationdepthofthemeridionalflow,
voto the peak speed in the poleward flow, and τ is the period of the solutions in
units of years. For the rest of the parameters in each set please refer to Table 1.
producing most of the toroidal field (as in the interface dynamo
models, see, e.g., Parker 1993; Charbonneau & MacGregor
1997).
4.3. Dependence of the Solutions on Changes in the Turbulent
Diffusivity Profile
Although a detailed exploration of the turbulent diffusivity
parameter space is outside the scope of this work, we study two
special cases in which we vary a single parameter while leaving
the rest fixed.
For the first case, we lower the diffusivity in the convection
zone, ηcz, from 1011cm2s−1to 1010cm2s−1. As can be seen in
Figure 12, this introduces two important changes in the dynamo
solutions: the first one is an overall increase in magnetic field
magnitudeduetothereductionindiffusivedecaywhilekeeping
the strength of the field sources constant. The second is a drastic
increase in the dynamo period (which can be seen tabulated
in Table 3) for the solutions that use a meridional flow with
a penetration of Rp = 0.71 R?. The reason behind such a
change resides in the nature of the transport processes at the
bottom of the convection zone, which are a combination of
Page 13
No. 1, 2009HELIOSEISMIC DATA INCLUSION IN SOLAR DYNAMO MODELS 473
Figure 12. Butterfly diagram of the toroidal field at the bottom of the convection zone (color) with radial field at the surface (contours) superimposed on it when using
a low diffusivity in the convection zone (ηcz= 1010cm2s−1). Each row corresponds to one of the different meridional circulation sets. The left column corresponds
to runs using the helioseismology composite and the right one to runs using the analytical profile.
(A color version of this figure is available in the online journal.)
both advection and diffusion. In the case of flow profiles with
deep penetration the velocity at the bottom is high enough for
downward advection to transport flux into the tachocline. On
the other hand, in the case of low penetration, the last bit of
downward transport into the tachocline is done by diffusive
transport and thus dominated by diffusive timescales. Because
of this, by decreasing turbulent diffusivity by an order of
magnitude, we drastically increase the period of the solutions.
In the second parameter space experiment, we lower the
supergranular diffusivity ηsgfrom 1013cm2s−1to 1011cm2s−1.
Page 14
474 MU˜NOZ-JARAMILLO, NANDY, & MARTENSVol. 698
Figure 13. Snapshots of the magnetic field over half a dynamo cycle (a sunspot cycle) when using a low supergranular diffusivity (ηcz= 1011cm2s−1). Each row
is advanced by an eighth of the dynamo cycle (a quarter of the sunspot cycle), i.e., from top to bottom t = 0, τ/8, τ/4, and 3τ/8. The solutions correspond to the
meridional flow of set 2 (deepest penetration with a peak flow of 12 m s−1) and analytic differential rotation (left) and composite data (right).
(A color version of this figure is available in the online journal.)
This was done to study the impact of the surface radial shear
under low diffusivity conditions. However, as can be seen
in Figure 13, there is very little difference between the two
solutions. This means that even after reducing the supergranular
diffusivity by 2 orders of magnitude, the radial shear has very
little impact on the solutions and the upper boundary conditions
still play an important role in limiting the relative contribution
from the near-surface shear layer.
5. CONCLUSIONS
In summary, we have presented here methods which can
be used to better integrate helioseismic data into kinematic
dynamo models. In particular, we have demonstrated that using
a composite between helioseismic data and an analytical profile
Table 3
Simulated Sunspot Cycle Period for the Different Sets of Meridional Flow
Parameters when using a Low Diffusivity in the Convection Zone
(ηcz= 1010cm2s−1)
Rp(R?)
Set 1 0.64 12
Set 2 22
Set 30.7112
Set 4 22
vo(m s−1)
τ—HS Data (yr)
12.46
6.47
87.78
70.48
τ—Analytical (yr)
12.86
6.55
90.92
74.41
Note. Rpcorresponds to the maximum penetration depth of the meridional flow,
voto the peak speed in the poleward flow, and τ is the period of the solutions in
units of years.
for the differential rotation, we can directly use the helioseismic
rotationdataintheregionoftrustandsubstitutethesuspectdata
Page 15
No. 1, 2009HELIOSEISMIC DATA INCLUSION IN SOLAR DYNAMO MODELS 475
by smoothly matching it to the analytical profile where the data
are noisy. This paves the way for including the helioseismically
inferredrotationprofiledirectlyindynamosimulations.Wehave
also shown how mathematical properties of the commonly used
analytic stream functions describing the meridional flow can be
fitted to the available near-surface helioseismic data to entirely
constrain the latitudinal dependence of the meridional flow, as
well as weakly constrain the radial (depth) dependence.
In our simulations, comparing the helioseismic data for the
differential rotation with the analytical profile of Charbonneau
et al. (1999), with four plausible meridional flow profiles, we
find that there is little difference between the solutions using the
helioseismic composite and the analytical differential rotation
profile—specially for shallow penetrations of the meridional
flow and even at reduced supergranular diffusivity. This is
becausetheimpactofthesurfaceradialshear,whichispresentin
thehelioseismiccompositebutnottheanalyticprofile,isgreatly
reduced by the proximity of the upper boundary conditions.
Also, for the shallow circulation, the toroidal field generation
occurs in a region located above the tachocline with mainly
latitudinal shear, where the difference between the composite
data and the analytical profile is not significant.
The main result from this comparative analysis is that the
latitudinal shear in the rotation is the most dominant source
of toroidal field generation in these type of models that are
characterized by high diffusivity at near-surface layers, but
lower diffusivity within the bulk of the SCZ—specially near
the base where most of the toroidal field is being created. Since
this latitudinal shear exists throughout the convection zone, an
interestingquestioniswhethertoroidalfieldscanbestoredthere
long enough to be amplified to high values by the shear in
the rotation, without being removed by magnetic buoyancy. If
this were to be the case, i.e., the latitudinal shear is indeed
confirmed to be the dominant source of toroidal field induction,
we anticipate then that downward flux pumping (Tobias et al.
2001; see also Guerrero & de Gouveia Dal Pino 2008)—which
tends to act against buoyant removal of flux, may have an
important role to play in this context. This could also call into
question the widely held view that the solar tachocline is where
most of the toroidal field is created and stored (see Brandenburg
2005 for arguments favoring a more distributed dynamo action
throughout the SCZ).
Our attempts to integrate helioseismic meridional flow data
into dynamo models and related simulations have uncovered
points that are both encouraging and discouraging.
On the discouraging side, we find that the currently available
observational data are inadequate to constrain the nature and
exact profile of the deep meridional flow, especially the return
flow. Neither do the simulation results and their comparison
with observed features of the solar cycle clearly support or rule
out any possibility. A recent analysis on light-element depletion
due to transport by meridional circulation indicates that solar
light-element abundance observations restrict the penetration to
0.62 R?(Charbonneau 2007); however, this analysis does not
necessarily suggest that the flow does penetrate that deep. Also
vexing is the fact that different inversions, involving different
helioseismic techniques such as ring-diagram or time–distance
analysis recovers different profiles and widely varying peak
meridional flow speeds (Giles et al. 1997; Braun & Fan 1998;
Gonz´ alez-Hern´ andezetal.2006;Gizon&Rempel2008).Inour
analysis, we chose to use the Gonz´ alez-Hern´ andez et al. data
because at present, this provides the (relatively) deepest full
inversion of the flow within the SCZ. Chou & Ladenkov (2005)
reported time–distance diagrams reaching a depth of 0.79 R?
but have not yet reported a full inversion that could be used on
our simulations.
We point out that there is an important consequence of the
presence of the flow speed maximum inside the convection
zone—which is related to mass conservation: if the maximum
poleward flow speed is found to be deeper inside the convection
zone this would result in a stronger mass flux poleward, which
needs to be balanced by a deeper counterflow subject to mass
conservation; the density of the plasma increases rapidly as one
goes deeper, e.g., the density at 0.97 R?is 10,000 times larger
than at the surface. Although that is not achieved currently, our
extensiveeffortstofitthedatapointoutthatstrongerconstraints
on the return flow may be achieved even with data those do
not necessarily go down to where this return flow is located, a
fact that may be usefully utilized when better depth-dependent
helioseismic data on meridional circulation become available.
Although the depth of penetration of the circulation is an
important constraint on the flow itself, our results indicate that
the period of the dynamo cycle does not in fact depend on
this depth. Rather, our simulations point out that the period
of the dynamo cycle is more sensitive to changes on the
speed of the counterflow than changes anywhere else in the
transport circuit, as this is where the dynamo loop originates.
An accurate determination of the average meridional flow speed
over this loop closing at the SCZ base is very important in
the context of the field transport timescales. As shown by
the analysis of Yeates et al. (2008), the relative timescales
of circulation and turbulent diffusion determine whether the
dynamo operates in the advection or diffusion dominated
regime—two regimes which have profoundly different flux
transport dynamics and cycle memory (the latter may lead
to predictability of future cycle amplitudes). Getting a firm
handle on the average meridional flow speed is therefore very
important and that is not currently achieved from the diverging
helioseismic inversion results on the meridional flow.
Thissuggeststhataconcerted effortusingdifferenthelioseis-
mic techniques on data for the meridional flow over at least a
complete solar cycle (over the same period of time) may be nec-
essary to generate a more coherent picture of the observational
constraint on this flow profile. It is important to note that even
though we used time-averaged data, nothing prevents one from
using the same methods to assimilate time-dependent helioseis-
mic data at different phases of the solar circle, allowing us to
study the impact of time varying velocity flows on solar cycle
properties and their predictability.
On the encouraging side, our dynamo simulations show that
it is relatively straightforward to use the available helioseismic
data on the differential rotation (on which there is more con-
sensus and agreement across various groups) within dynamo
models. Also encouraging is the fact that the type of solar dy-
namo model presented here is able to handle the real helioseis-
mic differential rotation profile and generate solarlike solutions.
Moreover, as evident from our simulations, this dynamo model
also generates plausible solarlike solutions over a wide range of
meridional flow profiles, both deep and shallow, and with fast
and slow peak flow speeds. This certainly bodes well for assim-
ilating helioseismic data to construct better constrained solar
dynamo models—building upon the techniques outlined here.
We are grateful to Irene Gonz´ alez-Hern´ andez and Rachel
Howe at the National Solar Observatory for providing us with
helioseismic data and useful counsel regarding its use. We also
Page 16
476 MU˜NOZ-JARAMILLO, NANDY, & MARTENSVol. 698
thank Jørgen Christensen-Dalsgaard of the Danish AsteroSeis-
mology Center for data and discussions related to the solar
model S. Finally, we want to thank Paul Charbonneau and an
anonymous referee for useful comments and recommendations.
The simulations presented here were performed at the com-
puting facilities of the Harvard-Smithsonian Center for Astro-
physics and this research was funded by NASA Living With
a Star Grant NNX08AW53G to the Smithsonian Astrophysical
Observatory. D.N. acknowledges support from the Department
of Science and Technology of the Government of India through
the Ramanujan Fellowship.
APPENDIX
NUMERICAL METHODS
In order to use exponential propagation, we transform our
system of partial differential equations (PDEs) into a system of
coupled ordinary differential equations (ODEs) by discretizing
the spatial operators using the following finite difference oper-
ators. For advective terms?∂A
⎧
+6Ai+1− Ai+2)
v
6?x(2Ai+1+ 3Ai
−6Ai−1+ Ai−2)
For diffusive terms
diffusion coefficient, we use a second-order space-centered
scheme:
∂t= −v∂A
∂x+ χ(x)?, where v is the
velocity, we use a third-order upwind scheme:
v
6?x(−2Ai−1− 3Ai
v∂A
∂x=
⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
if v < 0
if v ? 0
+ O(?x3).
(A1)
?
∂A
∂t= η∂2A
∂x2+ χ(x)
?
, where η is the
η∂2A
∂x2=
η
(?x)2(Ai−1− 2Ai+ Ai+1) + O(?x2).
For other first derivative terms
second-order space-centered scheme:
(A2)
?∂A
∂t=∂B
∂x+ χ(x)?
we use a
∂B
∂x=
1
2?x(Ai−1− Ai+1) + O(?x2).
(A3)
Here, χ(x) corresponds to all the additional terms a PDE
might have on the right-hand side besides the term under
discussion and Ai = A(x0+ i?x),i = 1, 2, ..., Nx is our
variable evaluated in a uniform grid of Nxelements separated
by a distance ?x.
A.1. Exponential Propagation
After discretization and inclusion of the boundary conditions
we are left with an initial value problem of ODEs:
∂U(t)
∂t
= F(U(t)),
(A4)
U(t0) = U0,
(A5)
where U is the solution vector in RN. Provided that the Jacobian
∂F(U(t)) exists and is continuous in the interval [t0,t0+Δt], we
can linearize F(U(t0+ Δt)) around the initial state to obtain
∂U(t)
∂t
= F(U0) + ∂F(U0)(U(t0+ Δt) − U0) + R(U(t0+ Δt)),
(A6)
where R(U(t0+ Δt) are the residual high-order terms. The
solution to this equation can be written as
U(t0+ Δt) = U0+eAΔt− I
where A = ∂F(U0). Neglecting higher order terms leaves us
with a scheme that is second-order accurate in time and is
an exact solution of the linear case. However, there is a way
of increasing the time accuracy of this method by following
a generalization of Runge–Kutta methods for nonlinear time-
advancement operators proposed by Rosenbrock (1963). The
combination of exponential propagation with Runge–Kutta
methods was first proposed by Hochbruck & Lubich (1997)
and then generalized by Hochbruck et al. (1998). In this work,
we use a fourth-order algorithm which goes to the following
intermediary steps to advance the solution vector between time
steps (Un) → Un+1)):
?1
k2= Φ(ΔtA)F(Un),
w3=3
u3= Un+ Δtw3,
d3= F(u3) − F(Un) − ΔtAw3,
k3= Φ
?
A = F(Un),
A.2. Krylov Approximation to the Exponential Operator
A
+ O(Δt2),
(A7)
k1= Φ
2ΔtA
?
F(Un),
8(k1+ k2),
?1
2ΔtA
?
k2+16
d3,
Un+1= Un+ Δt
27k3
Φ(ΔtA) =eAΔt− I
?
,
A
.
Without any further approximation, this method is very
expensive computationally due to the need of continuously
evaluating the matrix exponential. However, it is possible to
make a good approximation by projecting the operator into a
finite dimensional Krylov subspace
SKr= span{U0,AU0,A2U0,...,Am−1U0}.
In order to do this, we first compute an orthonormal basis for
this subspace using the Arnoldi algorithm (Arnoldi 1951).
1. v1= U0/|U0|2.
2. For j = 1,...,m do
a) for i = 1,...,i compute hi,j= vT
b) calculate w = Avj−
i=0
c) evaluate h(j + 1,j) = |w|2,
d) if hj+1,j < ? stop, else compute the next basis vector
vj+1= w/hj+1,j,
where ? is the parameter that sets the error tolerance in this
approximation.Oncewehavefinishedcomputingthealgorithm,
we have the relationship
(A8)
i∗ Avj,
j?
hi,jvi,
AVm≈ VmH,
(A9)
where Vm = [v1,...,vm] and H is a matrix whose elements
are hi,j = vT
i∗ Avj. The validity of this approximation
Page 17
No. 1, 2009 HELIOSEISMIC DATA INCLUSION IN SOLAR DYNAMO MODELS 477
0 50 100150 200250 300
1.9
2
2.1
2.2
2.3
2.4
Case C’ Jouve et al. 2008
Cs critical
Amount of gridpoints in both r and θ directions
0 50 100150200 250300
520
530
540
550
560
570
ω = 2π/T
Amount of gridpoints in both r and θ directions
Figure 14. Simulation results (dots) of running case C?of the dynamo
benchmark by Jouve et al. (2008). The top figure shows how the minimum
value of Cα= α0R?/ηczfor which the dynamo has stable oscillations depends
on resolution. The bottom figure shows how ω = 2πR2
resolution. In order to make the plot comparable to the benchmark, we include
the mean value found by the different codes and a shaded area corresponding
to 1 standard deviation around this mean. This run was made using the same
meridional flow and poloidal source profiles as in the benchmark study (for
details see Jouve et al. 2008).
?/(Tηcz) depends on
depends on thedimension of theKrylov subspace butnumerical
experiments have found that 15–30 Krylov vectors are usually
enough (see Hochbruck & Lubich 1997; Tokman 2001). Since
{v1,...,vm} is an orthonormal basis of the Krylov subspace
SKr then VT
a projector from RNonto SKr. Using this projector, we can
find an approximation to any matrix vector multiplication
by projecting them onto the Krylov subspace by calculating
Ab ≈ VmVT
Ab ≈ VmHVT
In fact we can approximate the action of any operator Φ(A),
that can be expanded on a Taylor series, on the vector U0using
the Krylov subspace projection:
mVM is a m × m identity matrix and VmVT
mis
mAVmVT
mb. After using Equation (A9) this becomes
mb.
(A10)
Φ(A)U0≈ VmΦ(H)VT
where e1 = [1 0... 0] and we used v1 = U0/|U0|2and
VT
the Krylov approximation effectively reduces the size of the
matrix operator; this makes the use of the exponential operator
relatively inexpensive computationally.
These two algorithms, in combination with a robust error
control and an adaptative time-step mechanism strategy, form
the core of the SD-Exp4 integrator (for more details see
Hochbruck & Lubich 1997; Hochbruck et al. 1998; Tokman
2001).
In order to verify the performance standards of the SD-Exp4
code, we ran case C?of the dynamo benchmark by Jouve et al.
(2008). This case is similar in nature to the simulations done in
this work with the following differences.
1. The meridional flow profile has different radial and latitu-
dinal dependence.
2. The poloidal source term has no quenching term and a
different radial and latitudinal dependence.
mU0= |U0|2VmΦ(H)e1,
(A11)
mv1 = e1. As we can see in Equation (A11), the use of
3. The turbulent diffusivity profile consists of only one step
and reaches a peak value of 1011cm2s−1.
4. Theanalyticdifferentialrotationhasnocos4(θ)dependence
and uses a thinner tachocline.
In order to compare the performance of the different codes
in the benchmark study, two quantities are used: Ccrit
αcrit
relative to dissipation that yields stable oscillations and ω =
2πR2
cycle relative to the diffusion timescale. The dependence of this
quantities is then plotted versus resolution for all the different
codes (see Figure 11 of Jouve et al. 2008). In order to compare
our code, we perform simulations with the same parameters and
modelingredientsasincaseC?ofJouveetal.(2008)andplotthis
two quantities versus resolution. We also plot the location of the
mean found by the other codes and a shaded area encompassing
one standard deviation around the mean. As can be seen in
Figure 14, we find lower values of Ccrit
in Jouve et al. (2008; 1.86 as opposed to an average 2.46) for
oscillatory solutions; this may be due to a lower amount of
numerical diffusion in our code. Moreover, we find values of
ω in very good agreement with those found in Jouve et al.
(2008; 540 as opposed to an average of 538.2). We also find that
for our code this quantities are less sensitive to resolution when
comparedwithothercodesusedinthebenchmark.Thisislikely
due to the fact that we try to avoid numerical differentiation
whenever we can do it analytically.
As a final remark, Table 4 of Jouve et al. (2008) specifies the
different time steps that are used by the different codes, which
range from values of 10−8to 10−5code time units. Thanks
to the use of Krylov approximations, we are able to use time
steps as large as 10−2in code time units while solving case C?.
However, this does not translate to a performance improvement
of several orders of magnitude (due to the added computations
in the Krylov approximation). In order to quantify the relative
performance improvement, we also made comparisons with the
Suryacode,whichhasbeenstudiedextensivelyindifferentcon-
texts (e.g., Nandy & Choudhuri 2002; Chatterjee et al. 2004;
Choudhuri et al. 2007), and is made available to the public
on request. In comparison to the Surya code, we find that the
SD-Exp4 code achieves a performance improvement that re-
duces runtime from a half to a tenth of the total runtime depend-
ing on the particularities of the simulation.
s
=
0R?/ηczwhichquantifiestheminimumstrengthofthesource
?/(Tηcz) which quantifies the frequency of the magnetic
s
than the values obtained
REFERENCES
Arnoldi, W. E. 1951, Q. Appl. Math., 9, 17
Babcock, H. D. 1959, ApJ, 130, 364
Babcock, H. W. 1961, ApJ, 133, 572
Brandenburg, A. 2005, ApJ, 625, 539
Braun, D. C., & Fan, Y. 1998, ApJ, 508, L105
Bumba, V., & Howard, R. 1965, ApJ, 141, 1502
Caligari, P., Moreno Insertis, F., & Schussler, M. 1995, ApJ, 441, 886
Carrington, R. C. 1858, MNRAS, 19, 1
Charbonneau, P. 2005, Living Rev. Sol. Phys., 2, 2
Charbonneau, P. 2007, Adv. Space Res., 39, 1661
Charbonneau, P., Christensen-Dalsgaard, J., Henning, R., Larsen, R. M., Schou,
J., Thompson, M. J., & Tomczyk, S. 1999, ApJ, 527, 445
Charbonneau, P., & MacGregor, K. B. 1997, ApJ, 486, 502
Charbonneau, P., St-Jean, C., & Zacharias, P. 2005, ApJ, 619, 613
Chatterjee, P., Nandy, D., & Choudhuri, A. R. 2004, A&A, 427, 1019
Chou, D.-Y., & Ladenkov, O. 2005, ApJ, 630, 1206
Choudhuri, A. R., Chatterjee, P., & Jiang, J. 2007, Phys. Rev. Lett., 98, 131103
Choudhuri, A. R., Schussler, M., & Dikpati, M. 1995, A&A, 303, L29
Christensen-Dalsgaard, J., et al. 1996, Science, 272, 1286
Page 18
478MU˜NOZ-JARAMILLO, NANDY, & MARTENSVol. 698
Dikpati, M., & Charbonneau, P. 1999, ApJ, 518, 508
Dikpati, M., & Choudhuri, A. R. 1995, Sol. Phys., 161, 9
Dikpati, M., Corbard, T., Thompson, M. J., & Gilman, P. A. 2002, ApJ, 575,
L41
Dikpati, M., de Toma, G., & Gilman, P. A. 2006, Geophys. Res. Lett., 33, 5102
D’Silva, S., & Choudhuri, A. R. 1993, A&A, 272, 621
Durney, B. R. 1997, ApJ, 486, 1065
Fan, Y., Fisher, G. H., & Deluca, E. E. 1993, ApJ, 405, 390
Garaud, P., & Brummell, N. H. 2008, ApJ, 674, 498
Giles, P. M. 2000, PhD thesis, Stanford Univ.
Giles, P. M., Duvall, Jr. T. L., Scherrer, P. H., & Bogart, R. S. 1997, Nature, 390,
52
Gilman, P. A., & Miesch, M. S. 2004, ApJ, 611, 568
Gizon, L., & Rempel, M. 2008, Sol. Phys., 251, 241
Gonz´ alez-Hern´ andez, I., Komm, R., Hill, F., Howe, R., Corbard, T., & Haber,
D. A. 2006, ApJ, 638, 576
Guerrero, G., & de Gouveia Dal Pino, E. M. 2008, A&A, 485, 267
Hale, G. E. 1908, ApJ, 28, 315
Hathaway, D. H. 1996, ApJ, 460, 1027
Hathaway, D. H., & Choudhary, D. P. 2008, Sol. Phys., 250, 269
Hathaway, D. H., Nandy, D., Wilson, R. M., & Reichmann, E. J. 2003, ApJ,
589, 665
Hochbruck, M., & Lubich, C. 1997, SIAM J. Sci. Comput., 34, 1911
Hochbruck, M., Lubich, C., & Selhofer, H. 1998, SIAM J. Sci. Comput., 19,
1552
Howard, R., & Labonte, B. J. 1981, Sol. Phys., 74, 131
Jouve, L., et al. 2008, A&A, 483, 949
Leighton, R. B. 1969, ApJ, 156, 1
Nandy, D. 2002, Astrophys. Space Sci., 282, 209
Nandy, D. 2003, in ESA Special Publication, Vol. 517, GONG+ 2002. Local
and Global Helioseismology: the Present and Future, ed. H. Sawaya-Lacoste
(Noordwijk: ESA), 123
Nandy, D. 2004, in ESA Special Publication, 559, GONG 2004. Helio- and
Asteroseismology: Towards a Golden Future, ed. D. Danesy (Noordwijk:
ESA), 241
Nandy, D., & Choudhuri, A. R. 2001, ApJ, 551, 576
Nandy, D., & Choudhuri, A. R. 2002, Science, 296, 1671
Parker, E. N. 1955a, ApJ, 122, 293
Parker, E. N. 1955b, ApJ, 121, 491
Parker, E. N. 1993, ApJ, 408, 707
Rempel, M. 2006, ApJ, 647, 662
Rosenbrock, H. 1963, Comput. J., 5, 329
Schou, J., et al. 1998, ApJ, 505, 390
Schwabe, M. 1844, Astron. Nachr., 21, 233
Tobias, S. M., Brummell, N. H., Clune, T. L., & Toomre, J. 2001, ApJ, 549,
1183
Tokman, M. 2001, PhD thesis, California Inst. of Tech.
Tsuneta, S., et al. 2008, ApJ, 688, 1374
van Ballegooijen, A. A., & Choudhuri, A. R. 1988, ApJ, 333, 965
van Ballegooijen, A. A., & Mackay, D. H. 2007, ApJ, 659, 1713
Wang, Y.-M., Nash, A. G., & Sheeley, Jr., N. R. 1989, ApJ, 347, 529
Yeates, A. R., Nandy, D., & Mackay, D. H. 2008, ApJ, 673, 544
Page 19
The Astrophysical Journal, 707:1852, 2009 December 20 doi:10.1088/0004-637X/707/2/1852
C ?2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
ERRATUM: “HELIOSEISMIC DATA INCLUSION IN SOLAR DYNAMO MODELS” (2009, ApJ, 698, 461)
Andr´ es Mu˜ noz-Jaramillo1, Dibyendu Nandy2, and Petrus C. H. Martens3
1Department of Physics, Montana State University, Bozeman, MT 59717, USA; munoz@solar.physics.montana.edu
2Indian Institute for Science Education and Research-Kolkata, WB 741252, India; dnandi@iiserkol.ac.in
3Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA; pmartens@cfa.harvard.edu
In our paper, the following statement is made in Section 3: “The differential rotation is probably the best constrained of all dynamo
ingredients but the actual helioseismology data have never before been used directly in dynamo model, only an analytical fit to it.”
Following the publication of our paper, it has been pointed out to us by Dr. David Moss that their group has been using an
interpolation of the helioseismically measured solar differential rotation profile in their dynamo model since 2000 (please see Covas
et al. 2004 and references therein).
REFERENCES
Covas, E., Moss, D., & Tavakol, R. 2004, A&A, 416, 775
1852
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