Page 1

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

Page 9

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