A&A 485, 267-273 (2008)
DOI: 10.1051/0004-6361:200809351
G. Guerrero - E. M. de Gouveia Dal Pino
Astronomy Department, Instituto de Astronomia, Geofísica e Ciências Atmosfêricas, Universidade de São Paulo, Rua do Matão 1226, São Paulo, Brazil
Received 3 January 2008 / Accepted 18 March 2000
Abstract
Context. The turbulent pumping effect corresponds to the transport of magnetic flux due to the presence of density and turbulence gradients in convectively unstable layers. In the induction equation it appears as an advective term and for this reason it is expected to be important in the solar and stellar dynamo processes.
Aims. We explore the effects of turbulent pumping in a flux-dominated Babcock-Leighton solar dynamo model with a solar-like rotation law.
Methods. As a first step, only vertical pumping has been considered through the inclusion of a radial diamagnetic term in the induction equation. In the second step, a latitudinal pumping term was included and then, a near-surface shear was included.
Results. The results reveal the importance of the pumping mechanism in solving current limitations in mean field dynamo modeling, such as the storage of the magnetic flux and the latitudinal distribution of the sunspots. If a meridional flow is assumed to be present only in the upper part of the convective zone, it is the full turbulent pumping that regulates both the period of the solar cycle and the latitudinal distribution of the sunspot activity. In models that consider shear near the surface, a second shell of toroidal field is generated above
at all latitudes. If the full pumping is also included, the polar toroidal fields are efficiently advected inwards, and the toroidal magnetic activity survives only at the observed latitudes near the equator. With regard to the parity of the magnetic field, only models that combine turbulent pumping with near-surface shear always converge to the dipolar parity.
Conclusions. This result suggests that, under the Babcock-Leighton approach, the equartorward motion of the observed magnetic activity is governed by the latitudinal pumping of the toroidal magnetic field rather than by a large scale coherent meridional flow. Our results support the idea that the parity problem is related to the quadrupolar imprint of the meridional flow on the poloidal component of the magnetic field and the turbulent pumping positively contributes to wash out this imprint.
Key words: Sun: magnetic fields - Sun: activity
Flux-dominated Babcock-Leighton (FDBL) solar dynamos are mean field models where the poloidal field is generated at the surface by the transport and decay of bipolar magnetic regions (BMRs) which are formed by twisted buoyant magnetic flux ropes. For this process to occur, differential rotation must be able to develop intense toroidal magnetic fields either at the tachocline or at the convection zone. Numerical simulations have shown that magnetic flux tubes with intensities around 104-105 G are able to become buoyantly unstable and to emerge at the surface to form a bipolar magnetic region with the appropriate tilt, in agreement with Joy's law. One important limitation of this scenario is that 105 G results an energy density that is an order of magnitude higher than the equipartition value, so that a stable layer is required to store and amplify the magnetic fields. This raises another problem with regard to the way in which the magnetic flux is dragged down to deeper layers.
Given the lack of accurate observations of the flow at the deeper layers, numerical simulations have shown that the penetration of the plasma is restricted to only a few kilometers below the overshoot layer (Rüdiger et al. 2005; Gilman & Miesch 2004). Nevertheless, the magnetic fields can be transported, not only downwards, but also longitudinally and latitudinally when strong density and turbulence gradients are present in the medium due to the turbulent pumping (Dorch & Nordlund 2001; Ziegler & Rüdiger 2003).
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Figure 1:
Radial and latitudinal profiles for ![]() ![]() ![]() ![]() |
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In axisymmetric mean field models of the solar cycle, the
effects of turbulent pumping rarely have been considered. A
first approach showing the importance of pumping in the solar
cycle was made by Brandenburg et al. (1992); since then few works have
incorporated the diamagnetic pumping component in the dynamo
equation as an extra diffusive term that provides a downward
velocity (Bonanno et al. 2002; Küker et al. 2001; Bonanno et al. 2006). More recently,
Käpylä et al. (2006b) have implemented simulations of mean field dynamos in
the distributed regime, including all the dynamo coefficients
previously evaluated in magneto-convection simulations
(Ossendrijver et al. 2002; Käpylä et al. 2006a). They have produced butterfly diagrams
that resemble observations. However, to our
knowledge no special efforts have been made to study the pumping
effects in the meridional plane (i.e., inside the convection zone)
or in an FDBL description. The latter has been found to be
particularly successful at reproducing most of the large scale
features of the solar cycle (Dikpati et al. 2004; Guerrero & de Gouveia Dal Pino 2007a,b; Dikpati & Charbonneau 1999, hereafter GDPa,b).
In this work, we explore the effects of turbulent pumping on an FDBL model. In a first approximation, we include the radial turbulent diamagnetism velocity term in the induction equation as described by Kichatinov & Ruediger (1992), and then in a second approach we add the pumping terms calculated in local magneto-convection simulations (Ossendrijver et al. 2002; Käpylä et al. 2006a). This latter approximation includes not only the radial but also the latitudinal contribution of the pumping. We will also discuss the implications of pumping when the near-surface radial shear layer reported by Corbard & Thompson (2002) is considered. In the following sections, we briefly present the model, our results and outline our main conclusions.
Our model solves the mean field induction equation:
We solve Eq. (1) for A and
with
r and
coordinates in the spatial ranges
-
and 0-
,
respectively, in a
grid
resolution (see Guerrero & Muñoz 2004, for details of the numerical model).
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Figure 2:
a) Butterfly diagram and latitudinal
snapshots for the
toroidal b) and the poloidal c) fields. The dark (blue) and light
(red) gray (color) scales represent positive and negative toroidal
fields, respectively; the continuous and dashed lines represent
the positive and negative poloidal fields. For this
model T=13.6 yr,
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Figure 3:
The same as in Fig. 2 but considering the
radial diamagnetic term. For this model T=13.6 year,
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In Fig. 2, the turbulent pumping is not considered. As
has been reported in GDPa and GDPb, the most important
contribution to the toroidal field comes from the latitudinal shear
term and therefore the field that is responsible for the observed
activity begins to be formed inside the convection zone. On the one hand,
it can be seen that the penetration into the stable layer is very weak,
on the other hand, the equartorward velocity is faster, so
that the time that the toroidal field has to amplify to the values
required by the magnetic flux tube simulations is probably short.
The upper panel of Fig. 2 shows the butterfly diagram
after a transient time of
years. At this
time, the toroidal field
has almost reached a quadrupolar parity (i.e. the toroidal
field in both hemispheres has the same sign), which is in
contradiction to Hale's law. In the solar dynamo modelling
this problem is known as the parity problem and we will discuss this
in more detail in Sect. 5. In the next sections we
consider the turbulent magnetic pumping as an alternative mechanism of
penetration. The latitude of emergence of the toroidal field depends
on the stability criterion for the buoyancy
(e.g. see Ferriz-Mas et al. 1994, and their Figs. 1 and 2 for details).
A way to describe the diamagnetic behaviour of a non-homogeneous
plasma based on a first order smoothing approximation (FOSA) was
outlined by Kichatinov & Ruediger (1992). If there is an inhomogeneous diffusivity
in a fluid, it causes transport of the magnetic field with an
effective velocity. We introduce this effect in the model by
changing the term
to
in Eq. (1). This new term increases the poloidal flux
that penetrates beneath the overshoot layer (Fig. 3) and, as a
consequence, more toroidal field is produced at all latitudes. The
toroidal field formed inside the convection zone penetrates the
stable layer where it is amplified by the radial shear at the
tachocline. The equartorward velocity of the magnetic flux inside
the stable layer is lower than the velocity right above the
overshoot interface, so that in the absence of latitudinal pumping,
the toroidal field would last longer at this layer. The radial
velocity corresponding to this effect at the overshoot region is
cm s-1 when a
variation of two orders of magnitude is considered for the
diffusivity in a thin region of
.
In a convectivelly unstable rotating plasma, the magnetic field is
not advected in the vertical direction only. The diamagnetic effect
may have components in all directions. Also, another pumping
effect due to density gradients can develop and in some conditions
can produce an upward transport that can balance the diamagnetic
effect (Ziegler & Rüdiger 2003). To investigate a more general
pumping advection, we will consider in this section the integration
of Eq. (1) with
given by Eq. (2) and
given by both contributions. The radial and latitudinal components
of this total
were computed numerically from
three-dimensional magneto-convection simulations by
Ossendrijver et al. (2002); Käpylä et al. (2006a). Similarly to Käpylä et al. (2006b), we use
the following profiles approximately fitted from the numerical
simulations:
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(3) |
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= | ![]() |
(4) |
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Figure 4:
The same as in Fig. 2 for a model with the full turbulent
pumping terms obtained from magneto-convection simulations. For this
model T=8.2 yr,
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Figure 4 shows that besides advecting the magnetic fields towards the stable regions, the pumping terms lead to a distinct latitudinal distribution of the toroidal fields when compared with the results of Figs. 2 and 3. The turbulent and density gradient levels present in a convectively unstable layer cause pumping of the magnetic field both down and equartorward, allowing its amplification within the stable layer and its later emergence at latitudes very near the equator. This result is important for dynamo modeling because it suggests that the pumping can not only solve the problem of the storage of the toroidal fields in the stable layer, but it can also help to provide a latitudinal distribution that is in agreement with the observations.
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Figure 5:
Meridional flow streamlines and the butterfly diagram for a
model with the full pumping term, but with a shallow meridional flow
penetration with a depth of only
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As the pumping and the meridional flow are both advective
terms and in some regions inside the convection zone their radial
and latitudinal components have the same sign, when the total
pumping is considered the period of the cycle is strongly affected.
It goes from 13.6 yr in the models of Figs. 2 and 3, to 8.2 yr
in the model of Fig. 4. One way to reproduce the solar
period is to decrease the value of the diffusivity at the convection
zone. Another possibility is to decrease the depth of penetration of
the meridional flow. This is supported by recent helioseismic
results (Mitra-Kraev & Thompson 2007) that suggest that the return point of the
meridional circulation can be at
.
At lower regions,
beneath
,
a second weaker convection cell or even a
null large scale meridional flow can exist. In Fig. 5, we have
decreased the depth of penetration of the flow and found that the
period increases to the observed value at the same time that the
toroidal fields become more concentrated at lower latitudes. If we
further decrease the depth of penetration, the equatorward
concentration of the toroidal fields becomes larger and the period
longer. The reason is that no net magnetic flux is moving
poleward at the lower regions of the convective layer since it is
all moving to the equator with the pumping velocity. This result is
in agreement with Ossendrijver et al. (2002) who suggest than the equartorward
motion of the magnetic activity could not be the result of
meridional bulk motion, but due to the latitudinal pumping of the
toroidal mean magnetic field. A parametric analysis of the
simulations performed under the conditions above gives:
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(5) |
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Figure 6:
Butterfly diagram for a model with the same
parameters as
Fig. 2, but with near-surface shear layer. For this model T=15.6 yr
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Helioseismology inversions have identified a second
radial shear layer located below the solar photosphere in the upper
35 Mm of the sun (Corbard & Thompson 2002). It is possible that the solar
dynamo is operating in this region, as has been discussed
by Brandenburg (2005). The more attractive features
of an
dynamo operating in this region are, among
others: (i) the intensity of the magnetic flux tubes in
this region does not need to be as large as 105 G in order to
form sunspots with the observed magnitudes, but 103 G is
sufficient; and (ii) with a near-surface
effect it is
possible to explain the
coincidence of the angular velocity of the sunspots in the
photosphere with the rotation velocity at
(see Fig. 2 of Brandenburg 2005), as well as the apparent disconnection between
the sunspot and its roots (Kosovichev 2002). The contribution
of a near-surface radial shear has been investigated in
interface-like dynamos (Mason et al. 2002), in distributed dynamos with a
turbulent
effect (Käpylä et al. 2006b), and also in advection
dominated dynamos (Dikpati et al. 2002). The latter authors have
discarded the radial shear layer
since it generates butterfly diagrams in which a positive toroidal
field gives rise to a negative radial field, which is exactly the
opposite to that observed. In this section, we include the radial
shear term in our FDBL model in order to explore the
contribution of pumping to this new configuration. We use the
analytical expression given in Eqs. (1)-(3) of Dikpati et al. (2002). The
near-surface shear described by these equations gives a
negative shear below
and a positive shear above
this latitude (see Fig. 1 of Dikpati et al. 2002)
.
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Figure 7:
The same as in Fig. 5 for a model with
near-surface shear
action. For this model T=16.3 yr
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With the assumption that the sunspots are formed in
the upper layers, the Babcock-Leighton poloidal source term, which
is concentrated in the same region (above
), does
not have to be non-local any longer. For the same reason, the values of
both the radial and the toroidal fields in the butterfly diagram
can be taken at the same radial point (
). Using the
same parameters as in the model of Fig. 2, but considering a
near-surface shear, the results of Fig. 6 show two main branches
in the butterfly diagram. One is migrating poleward (at high
latitudes) and one is migrating equatorward (below
). This
result is expected if the Parker-Yoshimura sign rule
(Yoshimura 1975; Parker 1955)
is considered. We note that the resulting parity is quadrupolar but
with the correct phase lag between the fields, which is opposite to
the result obtained in Dikpati et al. (2002). This difference probably
arises from the fact that we are
using a lower meridional circulation amplitude. Anyway, the
polar branches are strong enough to generate undesirable sunspots
close to the poles. The period increases to 15.6 y, which is due
to the fact that the dominant dynamo action at the surface goes in
the opposite direction to the meridional flow.
In Fig. 7 the same parameters as in Fig. 5 have been
used, but this time considering the radial shear near the
surface. As the radial pumping has its maximum amplitude close to the
poles (see the dashed line in Fig. 1), the toroidal fields created
there are efficiently pushed down
before reaching a significant amplitude, so that only the
equatorial branches below
survive. This
scenario requires that the pumping be dominant over the buoyancy at
such latitudes. Also, the phase relation of
obtained in the model of Fig. 7 seems to be the one
observed, at least at the latitude of
activity. However, there is some overlap between one cycle and the
next. Results in better agreement with the observations
may be achieved if the parameters are finely tuned.
We note that the introduction of the radial
shear
close to the surface when a meridional flow cell penetrating down to
is considered, as in the model of Fig. 4, requires
an increase of the amplitude of
.
This result is in
agreement with that found by Käpylä et al. (2006b).
Even though it already has been explored by several
authors, the anti-symmetry (dipolar parity) or symmetry
(quadrupolar parity) of the toroidal magnetic fields across the
solar equator still constitutes one of the most challenging
questions in solar dynamo theory. This
is mainly because the resulting parity in a model is very sensitive
to a large parameter space. The solar-like (antisymmetric) solution
could result from the effective diffusive coupling of the poloidal
field in both hemispheres (Chatterjee et al. 2004), but it may also depend on
the position and amplitude of the
effect
(Bonanno et al. 2002; Dikpati & Gilman 2001), or be the result of the imprint of the
quadrupolar form of the meridional flow on the poloidal magnetic
field, as argued by Charbonneau (2007). Small variations in the parameter
space can switch one solution from a dipolar to a quadrupolar
one. Although the main goal of this work was not to
study the parity problem itself, but the contribution of the turbulent
magnetic pumping, it is interesting to take advantage of the full
sphere integration in order to see how the pumping affects the
parity.
All the simulations presented in the previous sections
evolved 107 time steps up to 104 years. All
started with antisymetric (A) or with symmetric (S) toroidal
magnetic fields, but we have also performed tests with random (R)
fields. The parity of the solution is calculated, following
Chatterjee et al. (2004), with the equation below:
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(6) |
where BN and BS are the values of the
toroidal magnetic field at
,
and
and
,
respectively,
and
are their
respective temporal averages over one period. The value of P should
be between +1 (symmetric) and -1 (anti-symmetric) depending on the
parity of the fields. The results of our simulations with regard to the
parity can be summarized as follows:
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Figure 8: Parity curves for the three classes of models considered, i.e., a) for models without pumping (as e.g., in Fig. 2); b) for models with full pumping (as e.g., in Fig. 5); and c) for models with near-surface shear (as e.g., in Figs. 6 and 7). In the panels a) and b), the continuous, dashed and dot-dashed lines correspond to symmetric, anti-symmetric and random initial conditions, respectively. In panel c), the continous line is used for the model with turbulent pumping while the dashed line is for the model without pumping. |
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We have performed 2D numerical simulations of BLFD solar dynamo models including the turbulent pumping. Our first set of simulations include a solar rotation profile but without the a near-surface radial shear layer. The results show that the pumping transport effect is relevant in solar dynamo modelling, since it can solve two important problems widely discussed in the literature: the storage of the toroidal field at the stable layer and its latitudinal distribution. A new class of dynamo is proposed in which the meridional flow is important only near the surface layer in order to make the Babcock-Leighton mechanism operate over the toroidal fields, while in the inner layers, the advection is dominated by the pumping velocity. Our results support the idea that the equatorward migration of the sunspot activity is related to the latitudinal pumping velocity at the overshoot layer and the convection zone. Another attractive feature of this model is that a large coherent meridional flow is no longer required.
In a second set of simulations, we have
included the shear layer
found by Corbard & Thompson (2002) in the upper 35 Mm of the sun. The results
show the formation of a second shell of a strong toroidal field just
below the photosphere when the full pumping is absent. The branches of
this field obey the
Parker-Yoshimura sign rule for a positive
effect, i.e., they move poleward at high latitudes and equatorward
below
.
The role of the pumping in this kind of models is
also interesting since it reduces the amplitude of the polar toroidal
fields, pushing them inwards (Fig. 7). These
models work better if a shallow meridional circulation profile
is used. When a deeper meridional flow going down to the tachocline is
considered, a strong
effect is required in order to excite
the dynamo.
With regard to the parity problem, our
results show that a
simple
dynamo with the
effect concentrated
near thesurface leads to a quadrupolar parity, although the switch
from dipolar to quadrupolar
parity takes longer than in previous studies.
The models with full pumping conserve the initial parity, and
when the initial condition is random, the system tends to switch to a
dipolar parity. All the models that combine full pumping with
near-surface shear prefer dipolar parity solutions too.
In summary, our results have demonstrated the importance of the pumping in the solar dynamo, and suggest that this effect must be included in subsequent studies, even in those that employ multiple convection cells (Jouve & Brun 2007; Bonanno et al. 2006). Also it decreases the influence of the meridional flow in two important aspects: the period of the cycle and the latitudinal distribution of the toroidal fields. On the other hand, our results indicate that in the presence of full pumping there are two possible solutions to the question of where the dynamo operates: it could be either at the convection zone with the magnetic flux tubes emerging from the overshoot layer, or it could be at the layers near the surface. Both possibilities have their pros and cons, however a kinematic dynamo model only is not sufficient to reach a definitive answer and we will explore this in more detail in forthcoming work.
Acknowledgements
This work was supported by CNPq and FAPESP grants. G. Guerrero thanks the MPI in Garching and the ALFA project for their kind hospitality and support during the production of part of the present paper. We would like to thank the anonymous referee for his/her suggestions that have enriched this work.