New Journal of Physics
The open–access journal for physics
Global solar dynamo models: simulations and
predictions of cyclic photospheric fields and
long-term non-reversing interior fields
M Dikpati1 and P A Gilman
High Altitude Observatory, National Center for Atmospheric Research,
3080 Center Green, Boulder, CO 80301, USA
E-mail: [email protected] and [email protected]
New Journal of Physics 9 (2007) 297
Received 15 January 2007
Published 31 August 2007
Online at http://www.njp.org/
doi:10.1088/1367-2630/9/8/297
We review progress made in simulating and predicting principal
properties of solar cycles using flux-transport dynamo models and predictive tools.
We show that such models provide a consistent plausible theory for the following
solar cycle properties: cycle period, phase relation between poloidal and toroidal
magnetic fields, field symmetry about the equator, cycle 23 polar field amplitudes
and reversal timings, timing of the end of cycle 23, and the relative peaks of cycles
16–23, as well as forecasts for the upcoming peak of cycle 24. The same model
is also used to show that so-called ‘interface’ dynamo solutions do not calibrate
well to solar observations, and that the current solar dynamo may be a significant
source of high amplitude very long lived non-reversing toroidal field in the
solar interior.
Abstract.
1
Author to whom any correspondence should be addressed.
New Journal of Physics 9 (2007) 297
1367-2630/07/010297+24$30.00
PII: S1367-2630(07)41618-4
© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
2
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Contents
1. Introduction and philosophical approach
2. Mathematical formulation
3. Major results since 1999
3.1. What determines the period of the solar activity cycle?. . . . . . . . . . . . . .
3.2. What determines the phase relation in time in a magnetic cycle between
the toroidal field generated near the bottom, and the surface poloidal field,
particularly the polar fields? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. What causes the solar dynamo to operate in a mode predominantly antisymmetric
about the equator, as exemplified by Hale’s polarity laws? . . . . . . . . . . . .
3.4. Can flux-transport dynamos explain features of global solar magnetism specific
to a particular solar cycle?. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5. Can the so-called interface dynamos calibrate well to the Sun? . . . . . . . . .
3.6. Can the currently operating solar dynamo be a significant source of toroidal flux
in the solar interior below the convection zone? . . . . . . . . . . . . . . . . .
3.7. Can the timing of the end of a cycle and the onset of the next one be predicted?
3.8. Can the sequence of observed solar cycle peaks be simulated and predicted? . .
3.9. Other dynamo-based forecast methods . . . . . . . . . . . . . . . . . . . . . .
3.10. Criticisms, future calculations and new results . . . . . . . . . . . . . . . . . .
4. Concluding remarks
Acknowledgments
References
2
6
7
7
8
8
9
9
12
13
14
17
18
22
22
23
1. Introduction and philosophical approach
There has been substantial interest in predicting future solar cycles for at least the past forty
years. In the current era of extensive use of the high atmosphere and neighbouring interplanetary
medium by man, such predictions have considerable practical value. Starting with solar cycle 22
(1986–96), NASA and other Federal Agencies concerned with the effects of solar activity on
human activity have convened cycle prediction panels to assess all predictions and methods used,
and give their best judgment on the amplitude and other properties of the next cycle. For cycles 22
and 23, prediction methods were primarily statistical rather than dynamical. That is, no physical
laws were integrated forward in time, as is done for meteorological and climate predictions. But
for solar cycle 24, to begin in the next two years, the first such ‘dynamo based’ cycle prediction,
which involves integrating forward in time a form of Faraday’s law of electromagnetic theory,
has now been made (Dikpati and Gilman 2006; Dikpati et al 2006).
Given the obvious complexity of the physics of the convection zone and adjacent domains
of the Sun, how is such a prediction possible? To understand how it can be done, one must
consider a wide range of approaches to modelling the solar dynamo, and what constraints need
to be applied. Any such model must be able to simulate the important features of a ‘typical’ solar
cycle (Dikpati et al 2004), such as the ‘butterfly diagram’, and the phase relation in time between
toroidal (sunspot) fields and poloidal (polar) fields. It must do this with differential rotation and
meridional circulation that are close to that observed in the Sun. In other words, the dynamo
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
3
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
model must be ‘calibrated’ with solar observations. This process also constrains the choice
of more poorly known physical parameters, such as the magnetic diffusivity. Once calibrated,
the model can be used to simulate and predict departures from the typical cycle, caused by
variations in various inputted data, such as observed magnetic fields from previous times.
To be ‘calibratible’, a solar dynamo model must contain sufficient realism to allow inclusion
of known features of the Sun. For example, spherical geometry is essential, but restriction to
axisymmetry is possible, because many solar cycle properties can be captured from axisymmetric
data. Processes observed to occur with longitude dependence on the Sun, such as the emergence
of magnetic flux in active regions, can be averaged to estimate their effects on longitude-averaged
solar cycle features. A deep spherical shell configuration is also essential for including the full
differential rotation known from helioseismic methods, as well as a closed meridional circulation
that conserves mass. On the other hand, local dynamos in cartesian geometry can be and often are
used in complementary ways to study particular dynamo processes, including relevant properties
of magnetohydrodynamic (MHD) turbulence.
At the same time, there are practical limits to the realism that is possible to include
for predictive purposes. Dynamo action in the Sun no doubt occurs on many space and
timescales, from the global down to granulation scales (10−4 of the solar radius), involving many
turbulent processes. Capturing all or even most of these scales and processes in one numerical
model is obviously impossible. Current three-dimensional (3D) global MHD models for solar
differential rotation, convection and magnetic fields are truncated at larger spatial scales than
supergranulation (10−2 of the solar radius), so they must parameterize all smaller-scale turbulent
processes. It is therefore not possible to do direct numerical simulations (DNS) for the solar
convection zone. They show dynamo action, but no evidence of such basic features as global
field reversals yet (Browning et al 2006). So such models are not currently ‘calibratible’. In
any case, these simulations are so expensive in supercomputing power as to be impractical to
use for multiple cycle simulation, let alone prediction. The earliest such simulation attempts
(Gilman 1983), done with much coarser resolution, did show field reversals but the resulting
butterfly diagrams were not solar-like. Toroidal fields produced in them migrated toward the
poles rather than the equator.
As a consequence of the above considerations, we and others have focussed attention on
a class of dynamo model that has enough physics in it to be calabratible with global solar
observations, while at the same time being integratible using modest computing resources. This
class of dynamo is the ‘flux transport’ dynamo, so named (Wang and Sheeley 1991) because of
the inclusion of meridional circulation as well as differential rotation in the induction equation.
Such models generally parameterize key turbulent processes occurring on scales smaller than the
resolution of the model, such as the turbulent transport of magnetic flux, captured by a turbulent
magnetic diffusivity, and the turbulent lifting and twisting of magnetic field lines, represented
by one or more ‘α-effects’. These models also omit others, such as formation and buoyant lifting
of magnetic flux tubes, turbulent diamagnetic effects and Alfven wave effects. The success of
such a model in simulating and predicting solar cycles can only be judged by studying the output
from it, by comparing to the appropriate solar observations.
Figure 1 shows a sequence of schematic diagrams that depict qualitatively the succession
of processes contained in a flux-transport dynamo solution that leads to cyclic magnetic fields of
the correct dominant symmetry about the equator. The extensive legend briefly describes each of
these processes. Figures 1(g) and (h) show the role of meridional circulation which, in reality, is
transporting poloidal and toroidal flux around the domain during all phases of a magnetic cycle.
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
4
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Figure 1. Schematic of solar flux-transport dynamo processes. Red inner sphere
represents the Sun’s radiative core and blue mesh the solar surface. In between
is the solar convection zone where dynamo resides. (a) Shearing of poloidal field
by the Sun’s differential rotation near convection zone bottom. The Sun rotates
faster at the equator than the pole. (b) Toroidal field produced due to this shearing
by differential rotation. (c) When toroidal field is strong enough, buoyant loops
rise to the surface, twisting as they rise due to rotational influence. Sunspots (two
black dots) are formed from these loops. (d–f) Additional flux emerges (d, e)
and spreads (f) in latitude and longitude from decaying spots (as described in
figure 5 of Babcock 1961). (g) Meridional flow (yellow circulation with arrows)
carries surface magnetic flux poleward, causing polar fields to reverse. (h) Some
of this flux is then transported downward to the bottom and towards the equator.
These poloidal fields have sign opposite to those at the beginning of the sequence,
in (a). (i) This reversed poloidal flux is then sheared again near the bottom by
the differential rotation to produce the new toroidal field opposite in sign to that
shown in (b).
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
5
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Figure 2. Schematic of streamlines of meridional circulation used in dynamo
simulations and predictions. Solid dots denote one-year intervals in advection of
a passive tracer. Maximum flow speed set to reasonable solar value. Flow toward
pole in upper branch, toward equator in lower branch.
How quickly this happens of course depends on the amplitude of the meridional flow. Note that,
because the model contains magnetic diffusion everywhere, reconnection occurs everywhere
that advective and/or diffusive processes bring into close proximity magnetic fields of opposite
signs. Particularly active domains of reconnection are high latitudes for poloidal fields and at the
equator for toroidal fields.
Figure 2 shows a plot of an actual meridional flow used in many of our dynamo calculations,
with the poleward flow near the top in agreement with solar observations. The dots on the
streamlines mark one-year intervals of time of this ‘mean-field’ meridional flow (actual particle
trajectory would be much more complex, of course). Figure 2 then shows that magnetic flux
that starts in low latitudes at the top takes about 20 years to be transported to the poles, down
to the bottom of the convection zone (∼r = 0.7R), and back toward the equator. This time
is in fact an estimate of the ‘memory’ of such a model for past magnetic fields, provided the
magnetic diffusivity is not so large as to ‘short circuit’ the meridional flow transport. This means
the magnetic Reynolds number associated with the mean meridional flow should be 1. In
practical terms (see Dikpati and Gilman 2006), this limits turbulent magnetic diffusivities to
3 × 1011 cm2 s−1 in the bulk of the convection zone.
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
6
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
2. Mathematical formulation
In common with most formulations of the dynamo problem, we begin from the induction equation,
∂B
= ∇ × (U × B − η∇ × B),
∂t
(1)
in which B is the vector magnetic field, U is the specified velocity, and η is the magnetic
diffusivity. Then applying mean field theory to equation (1) in the standard way, we obtain the
dynamo equation (2), written as
∂B
= ∇ × (U × B + αB − η∇ × B),
∂t
(2)
in which B is now the mean magnetic field, U the mean differential rotation and meridional
circulation (for simplicity we omit the averaging overbars in (2) to distinguish it from the original
equation (1)), α a parametric form of production of one component of magnetic field from another
by lifting and twisting, such as by kinetic helicity of the turbulent flow, and η is now a turbulent
magnetic diffusivity.
To simplify the problem further, we assume axisymmetry for the mean fields and decompose
them into toroidal (Bφ (r, θ, t) êφ ) and poloidal (∇ × A(r, θ, t) êφ ) and separate the vector
components of equation (2) into equations for A and Bφ :
∂A
1
1
2
+
(u · ∇)(r sin θA) = η ∇ − 2 2
A + S(r, θ, Bφ ) + αBφ , (3)
∂t r sin θ
r sin θ
∂
∂Bφ 1 ∂
+
(rur Bφ ) + (uθ Bφ )
∂t
r ∂r
∂θ
1
= r sin θ(Bp · ∇) − êφ [∇η × ∇ × Bφ êφ ] + η ∇ − 2 2
Bφ .
(4)
r sin θ
In equation (3) for the poloidal field we have added a surface poloidal field source term S(r, θ, Bφ ),
which represents the production of new poloidal field near the outer boundary that originates
from toroidal field near the bottom that rises to the surface by magnetic buoyancy to form active
regions. In equation (4) (r, θ, t) is the specified differential rotation and in both equations
u = ur êr + uθ êθ is the specified meridional circulation. In equation (3) α now is an α-effect that
arises from instabilities in the overshoot tachocline near the bottom of the domain. The details
of the tachocline α-effect we have used, such as its latitudinal profile, are given in Dikpati and
Gilman (2001). Both the surface poloidal source and the α-effect near the bottom of the domain
are ‘quenched’ above a certain field strength, so that the solutions do not grow without bound.
This is the only nonlinearity in the system.
The boundary conditions we apply to the induced toroidal and poloidal fields in our
simulations are shown in figure 3, for the case of simulations in a single hemisphere. The toroidal
field is set to zero on all boundaries, including the poles, while the poloidal field is fitted to a
potential field above the top boundary. A = 0 at the poles is required by axisymmetry. A = 0 at
the bottom boundary (r = 0.6R) keeps the field out of the deep interior, and ∂A/∂θ = 0 at the
equator ensures that the poloidal field is linked between north and south hemispheres. In full
sphere calculations, the equatorial boundary conditions are omitted.
2
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
7
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Figure 3. Boundary conditions on poloidal and toroidal fields used at inner, outer
boundaries, as well as pole and equator.
3. Major results since 1999
We and colleagues have used solutions of equations (3) and (4) to answer an important sequence
of questions about basic properties of the solar cycle. To make the model as solar-like as possible,
we specify the differential rotation and meridional circulation with guidance from helioseismic
and solar surface doppler measurements. We also constrain the poloidal sources and diffusivity
values by both observations and theory, as specified in the various papers we cite below.
3.1. What determines the period of the solar activity cycle?
A principal result of Dikpati and Charbonneau (1999) was to show that in flux transport dynamos,
the dynamo period is approximately inversely proportional to the speed of the meridional flow.
By contrast, it is only very weakly dependent on the turbulent diffusivity. In particular it is the
speed of the equatorward return flow near r = 0.7R that determines this period, by advecting
both poloidal and toroidal field from high latitudes to low there. Agreement between the model
cycle period and that of the Sun requires an equatorward return flow of a few m s−1 near the
base of the convection zone. While this flow cannot be observed directly, it is consistent with a
poleward flow near the solar surface of 15–20 ms−1 , which is observed (Hathaway et al 1996).
The flow is larger near the top because of the substantially lower fluid density there. By contrast,
the cycle period in flux-transport dynamos is rather insensitive to the amplitude of the differential
rotation, α effects, or the diffusivity. This is quite the opposite of the so-called interface dynamos
(Parker 1993) and classical α-ω dynamo (Stix, 1973).
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
8
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
3.2. What determines the phase relation in time in a magnetic cycle between the toroidal field
generated near the bottom, and the surface poloidal field, particularly the polar fields?
It is well known that the polar fields of the Sun reverse near solar maximum, while they are near
maximum at the minimum of sunspots. Naval Research Laboratory (NRL) scientists first showed,
using a ID flux-transport model applied to the Sun’s surface (Wang et al 1989), that advective–
diffusive transport of the Sun’s weak magnetic flux arising from the decay of active regions
can explain quantitatively the poleward drifts of these flux and the polar reversal that follows.
Simulations from a complementary 2D flux-transport model (Dikpati 1996) for a meridional cut in
the Sun’s convective envelope confirmed the aforementioned calculations. In a detailed Babcock–
Leighton flux-transport dynamo calculation with solar-like differential rotation, Dikpati and
Charbonneau (1999) showed that this phase relation is a natural consequence of the meridional
flow of observed amplitude on the Sun. Here, it is the surface poleward-flowing branch that is
responsible. Too fast a flow, and the polar fields reverse before sunspot maximum is reached; too
slow a flow, and the polar reversals are delayed until well past sunspot maximum. The observed
meridional circulation leads to reversal near sunspot maximum. The key to getting the correct
phase relation between the peak in the toroidal fields and sunspots, and the surface poloidal fields,
particularly the polar field reversals, is having the primary creation of toroidal fields occurring
near r = 0.7R, while the poloidal flux is created at the surface and transported toward the pole by
surface processes.
3.3. What causes the solar dynamo to operate in a mode predominantly antisymmetric
about the equator, as exemplified by Hale’s polarity laws?
This question arose as soon as Hale’s polarity rules were discovered in the early 20th century,
but until much more recently has not received very much attention. In building dynamo models
applied to the Sun, it has been common practice to solve the dynamo equations for a single
hemisphere, with symmetry conditions imposed at the equator. Dikpati and Gilman (2001)
addressed the symmetry selection problem in detail. By doing full spherical shell simulations
without imposing equatorial symmetry conditions, they showed that the combination of a
meridional flow toward the equator at the bottom of the convection zone, together with an
α-effect there, favours the antisymmetric mode. Simulations with meridional flow and a surface
Babcock–Leighton type poloidal source choose the opposite symmetry, even when started from a
previously obtained antisymmetric solution. Furthermore, if the model includes only an α-effect
near r = 0.7R, the base of the convection zone, and no meridional flow, so the dynamo is a
classical α–ω interface type dynamo, then the symmetric mode is still favoured. Therefore both
a meridional flow and an α-effect near r = 0.7R are needed to ensure the correct symmetry.
The physical reason that the antisymmetric mode is favoured in this case is that the
α-effect near the base of the convection zone at ∼r = 0.7R produces new poloidal field there
that is promptly swept toward the equator by the meridional flow. Turbulent diffusion near the
equator annihilates any oppositely directed radial field coming from higher latitudes, resulting
in a latitudinal field that connects north and south hemispheres. Then latitudinal differential
rotation produces antisymmetric toroidal field there, ensuring that the α-effect subsequently
produces antisymmetric poloidal field. By contrast, if the α-effect in the flux transport dynamo
is at the top, such as produced by the Babcock–Leighton process of decay of active regions, then
the meridional flow has to carry this flux to the poles, down to ∼r = 0.7R and back toward the
equator, by which time it has lost much of its strength. In this case therefore the antisymmetric
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
9
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
mode is not favoured. However, a Babcock–Leighton poloidal source is observed to exist, so it
cannot be ignored. Therefore, two α-effects are needed to get the correct symmetry.
3.4. Can flux-transport dynamos explain features of global solar magnetism specific to a
particular solar cycle?
Dikpati et al (2004) was the first to apply a flux-transport dynamo to explain specific features
of a particular solar cycle, that of cycle 23. That paper focussed on the behaviour of the polar
fields in both north and south hemispheres, using time-dependent information about both the
surface radial magnetic flux and the meridional circulation. As in section 3.3, we compute the
polar field patterns in a full-spherical shell simulation of the flux-transport dynamo model. The
equator is not a boundary in this case and therefore no symmetry conditions about the equator
were imposed. The model was allowed to pick its symmetry about the equator. There are no other
possible symmetries in the problem, and all other boundary conditions are the same as shown in
figure 3.
The model was first calibrated against an average solar cycle, and then initialized with
magnetic data from cycle 22. Dikpati et al (2004) found that the 1.5 year delay in polar field
reversal in cycle 23 compared to an average cycle was primarily due the weaker surface poloidal
field sources in cycle 23 compared to 22. When transported to the poles by the meridional
circulation, these weaker sources took longer to cancel out the polar field created during cycle
22. These weaker sources also caused the post-reversal polar field of cycle 23 to be weaker than
in cycle 22, and take longer to build up.
But the meridional circulation seen during cycle 23 also played a role in the behaviour
of the polar fields. In particular, the systematic decrease in meridional field amplitudes during
1996–2002 (Haber et al 2002; Basu and Antia 2003) caused the polar transport of surface radial
flux also to slow down, contributing to the ∼1.5 year delay in polar field reversal cited above.
In addition, the reversal of the north pole field ∼1 year ahead of that of the south pole is a
consequence of the appearance of a submerged high latitude reversed meridional flow cell in the
north during 1998–2001 (Haber et al 2002). In the model simulation, this reversed cell caused
a faster cancellation of the remnant polar fields in the north from cycle 22 by the fields of cycle
23, by carrying those remnant fields to a lower latitude where they were cancelled sooner by the
new fields.
3.5. Can the so-called interface dynamos calibrate well to the Sun?
Parker (1993) defined and applied the first interface dynamo to the Sun. In its simplest form this
dynamo contains a radial differential rotation characteristic of the solar tachocline, together with
an α -effect just above the tachocline. The magnetic diffusivity is taken to be much smaller in the
tachocline than above it. Parker showed that it is capable of producing large amplitude toroidal
fields in the tachocline due to the low diffusivity, where they can be stored long enough to reach
the strength needed to be the source of sunspots seen in middle and low latitudes.
But does such a solution calibrate well with solar observations? For example, does it produce
a butterfly diagram from toroidal field near r = 0.7R and surface radial field that agrees well
with that of the real Sun? Markiel and Thomas (1999) performed a detailed calculation of an
interface dynamo with a realistic solar rotation profile, and obtained a solar-like low-latitude
butterfly diagram. But they did not show what the time–latitude diagram for the surface radial
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
10
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
fields would look like and whether it is possible to calibrate simultaneously the low-latitude
sunspot butterfly and the high-latitude surface radial field time–latitude diagrams in this class of
models.
Also, Garaud (1999) showed that the skin-depth of cyclically reversing poloidal fields near
the base of the convection zone can be severely limited by the rapidly declining magnetic
diffusivity there. What if conditions are such that these poloidal fields cannot penetrate
the tachocline enough to be sheared by the radial rotation gradient there? In that case, a
pure interface dynamo might fail completely. Dikpati et al (2005) examined these questions
in detail.
To answer the questions posed above, Dikpati et al (2005) performed a series of numerical
experiments, the results of which are shown in condensed form in figure 4. They started with a
pure interface dynamo qualitatively similar to that used by Parker (1993). Figure 4(a) shows the
resulting butterfly diagram for this case, in which the relative locations of the tachocline radial
shear and the diffusivity drop with depth are such that the induced poloidal field can penetrate
the tachocline from above. This pattern shows at least two major differences from the Sun: the
cycles are very short and highly overlapping, implying that several Sunspot bands of opposite
polarity would be present at the same time; and surface poloidal fields migrate toward the equator
at all latitudes, while on the Sun in mid and high latitudes they in fact migrate toward the poles.
Both these non-solar like features are very hard to eliminate by tuning the interface model. They
are in fact very typical of this type of dynamo. These results cast doubt on the ‘calibratability’
of the interface dynamo to solar observations.
Figure 4(b) shows what happens when the relative locations of the tachocline shear and
diffusivity drop are such that the induced cyclic poloidal field cannot penetrate the tachocline.
Here the solutions remain cyclic, with a much longer and more solar-like period, but essentially
no latitudinal propagation at all. The lack of latitudinal migration is because only latitudinal
rotation gradients are inducing toroidal fields, not radial gradients. So this case also does not
apply well to the Sun. Figure 4(c) shows what happens when a meridional flow whose amplitude
at the top is solar-like (peak flow ∼15 ms−1 poleward) is added. The period becomes too long
for the Sun and there is still very little migration of the toroidal fields toward the equator with
time. So this case is also deficient for the Sun.
In Figure 4(d), we have doubled the amplitude of the meridional flow to show what is
required, with these dynamo ingredients, to get a solar-like dynamo period. In this case, we
also produce a modest phase-shift with latitude in the timing of the peak toroidal field. But this
effect is too small for the Sun, even when the meridional flow in the model is twice a reasonable
solar value.
In Figure 4(e), a surface Babcock–Leighton source of radial fields is added, in accordance
with solar observations, and the meridional flow is reset to a solar value, and a much more solarlike butterfly diagram results, with a reasonable period. Here while the toroidal field migrates
toward the equator at all latitudes, the surface radial field shows a high-latitude poleward branch,
and an equatorward branch in low latitudes. The shorter period originates due to the presence
of Babcock–Leighton effect that ‘short circuits’ the equatorward transport of toroidal field and
low latitude upward transport of poloidal field. Through the combined actions of rising buoyant
fluxtubes, creating active regions together with their subsequent decay, puts new poloidal field
at the outer boundary instantly. The poleward migration of surface radial fields is due to the
meridional flow advecting poleward the surface radial fields newly deposited in sunspot latitudes
by the Babcock–Leighton process.
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
11
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Figure 4. Time–latitude plots of sequence of dynamo solutions, adapted from
figure 3 of Dikpati et al (2005). Contours denote toroidal field amplitude at bottom,
shading radial field at top. (a) Interface dynamo including tachocline radial shear;
(b) interface dynamo in which poloidal field does not reach tachocline because
of low magnetic diffusivity there; (c) interface dynamo of (b) with addition of
solar like meridional circulation; (d) same as (c) but with meridional circulation
doubled; (e) full flux-transport dynamo with Babcock–Leighton surface poloidal
source added, solar-like meridional circulation, but no tachocline radial shear.
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
12
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Dikpati et al (2005) concluded from these simulations that an interface dynamo will not work
for the Sun, whether or not the induced poloidal fields reach tachocline depths. By contrast, fluxtransport dynamos, particularly those with both a surface Babcock–Leighton source for poloidal
fields and an α-effect near r = 0.7R, are very calibratible (Dikpati et al 2004) and therefore quite
robust for the Sun, even if poloidal fields do not reach tachocline depths. Thus the skin-effect
illustrated by Garaud (1999) is not limiting in the flux transport case.
3.6. Can the currently operating solar dynamo be a significant source of toroidal flux in the
solar interior below the convection zone?
One of the remaining mysteries concerning radiative stellar interiors is whether they contain
magnetic fields, and if so, what kind and of what magnitude. Two concepts of longstanding are
that such fields are primordial (Cowling 1945; Wrubel 1952) or that they are remnant fields
from extinct dynamos (Schussler 1975; Parker 1981). More recently, Mestel and Weiss (1987)
made the suggestion that convection zone dynamos with cycles fluctuating in amplitude could
create net interior fields. But Dikpati et al (2006) have now shown that flux-transport dynamos
producing very regular cycles in the convection zone can create substantial non-reversing toroidal
fields in the outer part of the stellar interior, provided there is an α-effect near the bottom of the
convection zone. They solved the problem using the Sun as a well observed example, but their
results should apply to stellar interiors much more broadly.
Since the effective magnetic diffusivity in the solar tachocline is unknown, constrained
only by convection zone values above (1010 –1012 cm2 s−1 ) and radiative interior atomic values
below (102 –103 cm2 s−1 ), Dikpati et al varied the diffusivity in the tachocline with multiple
profiles between these extremes, assuming core diffusivity values from 109 down to 103 cm2 s−1 .
Particularly for the lower values, they ran the dynamo model for more than 105 years to obtain
clear convergence. Typical field patterns from their results are shown in figure 5, which is adapted
from figure 4 of their paper. The three meridional cross-sections shown depict three phases of
one sunspot cycle after ∼105 years of simulation. In this particular calculation, the core magnetic
diffusivity is 106 cm2 s−1 , an intermediate value. Shading denotes the toroidal field, solid and
dashed lines the poloidal field. Comparing the left and right cross-sections, we see clearly the
reversal in all the fields in the domain above the tachocline (bounded by the pair of dotted circular
contours), while the fields below are the same in the three frames.
To get the build-up of non-reversing field below the tachocline requires a small α-effect
in the tachocline, which can be produced by one or more MHD instabilities likely to occur
there. Depending on the details of the diffusivity profile with depth in and near the tachocline,
and particularly how small a diffusivity is attained within it, both reversing and non-reversing
toroidal fields as large as 102 or even 103 kG can be produced. Since the dynamo model used is
kinematic, including dynamical jxB type feedbacks might limit these fields to somewhat lower
values, but for all diffusivity values taken they will still be so large in the tachocline that the
global dynamics of this shell must be that of full MHD with timescale of one solar cycle or even
much less. Thus the much longer timescale dynamics invoked in Gough and McIntyre (1998) to
understand the existence of the tachocline do not apply. Therefore, the so-called ‘slow’tachocline
(Gilman 2000) does not exist.
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
13
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Figure 5. Three phases of toroidal (shading), poloidal (contours) fields from flux
transport dynamo simulation run ∼105 years, adapted from figure 4 of Dikpati
et al (2006). Note that the fields below the dashed circle at r = 0.7R are steady,
while those above are reversing after evolving through half a magnetic cycle.
3.7. Can the timing of the end of a cycle and the onset of the next one be predicted?
Since for flux-transport dynamo models the dynamo period is set primarily by the amplitude
of the meridional flow, if that flow changes from one cycle to the next or within a given cycle,
the period of that cycle will be affected. Time variations in meridional flow on the Sun have
been observed only since about 1996 (Basu and Antia 2003) so only the timing of the end of
the current cycle 23 can be studied. It was observed that between the years 1996 and 2003 the
surface poleward meridional flow slowed down by a few tens of percent. Dikpati (2004, 2005)
performed a series of numerical experiments starting from cycle 22 and allowing for different
meridional flow amplitudes during cycle 23. The resulting time trace of the induced toroidal field
in the model tachocline for the different meridional flows is shown in figure 6. Here we see that,
compared to a meridional flow set by the average cycle period for cycles 12–22 (10.75 years), if
the meridional flow decelerates during 1996–2003 and then remains low, the onset of cycle 24
will be delayed by about 12 months leading to the event of new cycle spots exceeding the old
cycle spots in late 2007 or early 2008. If the meridional flow instead accelerates after 2003, some
but not all of the delay can be made up, resulting in a net delay of about six months. Previously
the end of cycle 23 was predicted for late 2006 (Hathaway and Wilson 2004) by other methods.
This prediction of a late minimum between cycles 23 and 24 is now supported by a variety
of observational evidence. For example, the observational butterfly diagram from NASA shown
in figure 6(c) shows clearly that we have not yet (at the start of 2007) reached minimum. There
is still significant cycle 23 activity in the southern hemisphere. In addition, the corona is not yet
in minimum phase. This we can see in figure 6(b), which shows that the corona at total eclipse
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
14
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Figure 6. a) Time trace of integrated toroidal flux in bottom shear layer for
simulations beginning in 1995 with three different meridional flow variations
with time; (b) coronal images at eclipse, showing 2006 corona not at
minimum phase; (c) observed butterfly diagram from David Hathaway’s website
(http://solarscience.msfc.nasa.gov/images/bfly.gif).
of 29 March 2006 looked very much like the corona of November 1994, not the dipolar corona
of early 1996, during the last minimum. As the onset of a new cycle occurs ∼1 year after the
minimum, both of these observations are consistent with the next solar cycle onset being delayed
until late 2007 or early 2008.
3.8. Can the sequence of observed solar cycle peaks be simulated and predicted?
The first truly dynamo-based simulation of the sequence of peaks of past solar cycles and a
forecast of solar cycle 24 was published by Dikpati and co-workers (Dikpati and Gilman 2006;
Dikpati et al 2006). These simulations and forecast were done by starting from the calibrated flux
transport dynamo of Dikpati et al (2004) and making certain modifications. First, in equation
(3) for the poloidal field in terms of vector potential A, the link between the toroidal field in the
dynamo at the bottom and the poloidal field at the surface is replaced by a surface forcing term
that is derived from the waxing and waning of the observed poloidal fields created by the decay of
tilted bipolar active regions in previous solar cycles. This represents a form of 2D (time–latitude)
data assimilation in the model, a very common practice for meteorological forecast models, but
perhaps the first time it has been used in a solar forecasting problem.
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
15
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Figure 7. (a) Sunspot area data stretched and compressed (dashed curve) in time
to equalize all solar cycle periods to 10.75 years, compared to original data (solid
curve); (b) Gaussian profiles used to represent latitude width of sunspot zone at
various phases of a sunspot cycle.
As a result of the above modification, no quenching of the Babcock–Leighton poloidal
source, and no buoyancy mechanism are included in the model. However the nonlinear quenching
in the α-effect near the base of the convection zone is retained. Since this α-effect is much smaller
compared the Babcock–Leighton surface α-effect, in effect the model is mathematically changed
from being a self-contained nonlinear system to a primarily linear system that is forced at the
upper boundary by a measure of the Sun’s past surface fields. The resulting equation for A is
given by
1
Bφ
1
∂A
2
,
+
(u · ∇)(r sin θA) = η ∇ − 2 2
A + F (r, θ, t) + α
∂t r sin θ
[1 + (Bφ /B0 )2 ]
r sin θ
(5)
in which, B0 is the quenching field strength, set to 30 kG.
The inclusion of the surface forcing function, F , representing the time variation of past
magnetic fields of the Sun imposes on the model the (variable) period of past solar cycles. But
as described above and in Dikpati and Charbonneau (1999), flux-transport dynamos already
have their own intrinsic period, determined primarily by the strength of the meridional flow.
We do not know directly from velocity observations prior to 1996 what the amplitude of this
meridional flow was. In our initial approach, we estimate an average amplitude of meridional
flow from the average period of as many past cycles as we have high quality sunspot data for,
which takes us back to cycle 12. From this data we get an average period of 10.75 years. A
more sophisticated approach might involve an ensemble of 12 cycle simulations with different
or time-varying meridional circulation, but that remains for the future.
If we use a meridional flow amplitude determined in this way, but force the model with
data that has the observed period of each sunspot cycle, a phase mismatch in the link between
the toroidal and poloidal fields grows to an unacceptably large value over several cycles’ time
integration. This makes simulation and forecast of cycle peaks impossible. To avoid this problem,
Dikpati et al stretched and compressed in time the observed data so that each cycle in the surface
forcing had the same period, 10.75 years, as the intrinsic period of the dynamo, so that no
such destructive phase mismatches would occur. As a result, this method cannot be used to
simulate or predict the period of an upcoming cycle, but it can be used to predict the amplitude.
Figure 7(a) compares the stretched and compressed sunspot data with the original data. Then
we represent the latitude distribution of the poloidal source within each cycle by a Gaussian
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
16
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Figure 8. (a) Observed sunspot area data for cycles 12–23; (b) integral from 0 to
45◦ latitude of simulated toroidal magnetic flux in bottom shear layer for cycles
12–24, plus two forecasts for cycle 24; (c) scatterplot of sunspot area peaks versus
peak of toroidal magnetic flux integral.
with half-width 6◦ latitude (figure 7(b)) whose peak matches with spot area curve for that cycle,
which migrates toward the equator at a fixed rate. Clearly this representation is not unique, and
others should also be tried, particularly ones that capture more of the information present in the
observational data.
Strictly speaking, the surface poloidal forcing should come from synoptic observations of
the photospheric magnetic flux that originates from the decay of active regions, averaged over
longitude to fit the axisymmetric mean-field formulation, but such measurements exist only from
1976. Photospheric magnetic flux and sunspot area averaged over several rotations correlate very
well (Dikpati et al 2006), so we are able to use the much longer sunspot area record to create
the forcing.
Starting from a fully converged calibrated model solution, we initialized the model from
the beginning of solar cycle 12, and ran it for the next 13 cycles extending through the
upcoming cycle 24. The results of our simulations and predictions for the peaks of cycles 12–24
are shown in figures 8(a) and (b), adapted from Dikpati et al (2006). In figure 8(a) we compare the
observed solar cycles (cycle periods all stretched or compressed to 10.75 years) with the sequence
of simulated peaks. It is clear from figure 8(a) that the correspondence between observed and
simulated cycles is very good, particularly beyond cycle 15. For the first few cycles the agreement
is not as good, because the model is still loading the meridional circulation ‘conveyer belt’ with
fields from previous cycles. The simulations show two curves for the upcoming cycle 24. The
solid curve represents the case for which the meridional circulation remains fixed at a value that
produces the 10.75 year dynamo period, and shows cycle 24 should have a ∼50% higher peak
than cycle 23. The dashed curve represents a simulation that takes account of time variations in
observed meridional circulation since 1996, as described above in section 3.7. For this case, cycle
24 is forecast to be ∼30% higher than cycle 23. Thus, even allowing for variations in meridional
circulation, we forecast that cycle 24 will be substantially more active than the current cycle.
There are many other forecasts for cycle 24, most saying it will be larger than cycle 23, but some
saying it will be smaller. But our forecast is the very first one made using a dynamo model with
real solar data.
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
17
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
The skill of our forecast model is demonstrated more clearly in the scatterplot shown in
figure 8(b) which shows the observed peaks for cycles 12–23 against the peaks of the toroidal flux
integral computed for the tachocline between the equator and 45◦ . The correlation coefficient for
these points is extremely high, 0.987 for cycles 16–23. This calculation was done for a particular
value of the magnetic diffusivity in the bulk of the convection zone, namely 5 × 1010 cm2 s−1 , but
in Dikpati and Gilman (2006) similar levels are reached for a wide range of diffusivity values,
illustrating the robustness of the model.
How the model works to achieve its forecast skill is discussed in detail in Dikpati and
Gilman (2006). A key element of its workings is shown in figure 10, in which is plotted the
latitudinal poloidal field and the toroidal field at similar phases in cycles 19 and 20. The frames
in figure 10 are snapshots from a continuous video animation for cycles 16–23, in which the path
followed by new poloidal field of a particular sign in Bθ can be tracked until it disappears near
the base of the convection zone near the equator 2–3 cycles later. The numbers within the plot
denote the cycle of origin of each field tracked in this way through all of cycles 16–23. Figure 10
then saves a sample from this tracking illustrating that there is a memory at high latitudes of
the poloidal field from previous cycles, with up to three successive cycles’ fields accumulated at
high latitudes by the meridional flow. The latitudinal differential rotation shearing these fields
produces the toroidal fields shown. The equatorward-flowing bottom branch of the meridional
circulation then sweeps both poloidal and toroidal fields toward the equator to put in place the
toroidal field that creates the peak of the next cycle.
3.9. Other dynamo-based forecast methods
Attempts to build dynamo-based schemes for forecasting upcoming solar cycle properties have
been made over more than 40 years. These schemes fall into two categories: (i) qualitative
and (ii) quantitative prediction schemes. Schatten (Schatten et al 1978) was the first to develop
a qualitative dynamo-based prediction scheme in which a ‘magnetic persistence’ between the
current sunspot cycle’s amplitude and previous cycle’s polar fields was considered. This scheme,
often called the ‘polar-field precursor method’, has recently been used to predict that the
upcoming cycle 24 will be very weak (Schatten 2005; Svalgaard et al 2005). Another dynamobased scheme, called the ‘geomagnetic precursor method’, respectively relates the ‘in-phase’ and
‘out-of-phase’components of the geomagnetic activity index to the Sun’s spot-producing toroidal
field and large-scale poloidal field, was first developed by Feynman (1982). But in contrast to
the polar field precursor method, recent implementation of this scheme has predicted a strong
cycle 24 (Hathaway and Wilson 2006).
Very recently quantitative dynamo-based forecast schemes are being developed. All of them
so far have relied on flux-transport type solar dynamo models operating either purely at the surface
or in the meridional plane of the Sun’s convection zone. We have already discussed in detail in
the previous section the predictive tool of the latter type. These schemes use surface magnetic
data from many past cycles and integrate the induction equation extending into the future to
make quantitative estimation for the upcoming cycle’s dynamo-generated flux at the base of the
convection zone. A predictive tool based on surface flux-transport model has been constructed by
Cameron and Schüussler (2007) who postulated that the cross-equatorial transport of magnetic
flux in the late phase of a cycle is an accurate precursor for the strength of the next cycle. By
assimilating observed surface magnetic data into their ID model and integrating forward in time,
they found a very high correlation (r = 0.90) between the observed amplitude of the next cycle
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
18
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
and their predicted cross-equatorial transport at the end of the previous cycle, when they represent
the latitude information of the surface poloidal source with equatorward migrating Gaussian as
in Dikpati and Gilman (2006). They also predict a high cycle 24. By contrast, the correlation
between the polar fields computed from this model and the observed cycle amplitude is poor.
They also noted that the predictive skill goes down significantly with the correlation coefficient
declining to 0.45, when the actual latitude information for the spot eruption is incorporated in the
model. It is yet to be seen how the results from our flux-transport dynamo-based scheme change
when the observed daily latitude profile of the emerged flux is incorporated into the model.
Finally, we note that certain theoretical results of flux-transport type solar dynamos have led
to the development of another quantitative dynamo-based scheme that is based on the fact that
the latitudinal drift speed of centroid of sunspot zone is positively correlated with the strength of
second following cycle (Hathaway and Wilson 2004). This scheme has also predicted a strong
cycle 24.
3.10. Criticisms, future calculations and new results
Certain criticisms have been put forward concerning forecasting solar cycle peaks using
flux-transport dynamo models (Tobias et al 2006). This correspondence did not undergo scientific
review, and we were not allowed by Nature editors to respond to it. So we respond here and
elsewhere.
The authors of this paper state a priori that because they contain parameterizations,
flux-transport dynamos have no predictive power. But all weather and climate forecasting models
also have many parameterizations, and they have long been demonstrated to have skill. It is a
logical certainty that predictive skill of all models, not just flux transport dynamos, can only
be judged a posteriori by the results they produce. Furthermore the skill of our model cannot
be tested by using some other model with other physics, such as one that omits meridional
circulation and the time–latitude data assimilation. Also, much of the uncertainty of the effect
of poorly known parameterizations of physical processes is in effect removed by calibrating the
model to solar observations, and then forecasting only departures from the calibrated solutions.
This is common practice and used with great success in forecast models in other fields, such as
meteorology.
Tobias et al also claim that since the dynamo equations are extremely nonlinear, chaos
intrinsic to the system makes prediction much more difficult, particularly for longer time periods.
Within a solar cycle, there are short-term turbulent features which are probably governed by a
variety of nonlinear processes including the buoyant rise of initially toroidal fluxtubes. But we
are not attempting to forecast them with our forced linear predictive tool. Obviously there are
aggregate effects of many short-term nonlinear events which lead to, for example, the average
surface poloidal fields that have a well-defined observed pattern. Our predictive tool captures
this pattern. By analogy, mean flow in a river can be forecast without forecasting accurately the
various eddies that occur in the flow.
Many additional calculations can and should be done with our solar cycle prediction model
to test its robustness with respect to changes and fluctuations with time and space in various
parameters currently included in it, such as the meridional circulation, turbulent magnetic
diffusivity, α-effects, as well as the form and time dependence of the surface magnetic data
from previous cycles. Since the differential rotation is observed to vary very little (torsional
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
19
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
oscillations are only a few of the mean differential rotation) variations in that input are likely to
be less important, at least in kinematic models.
Doing such tests systematically probably would be best done employing so-called
‘ensemble-forecast’ methods, such as described in a meteorological context by Kalnay (2003),
as well as more sophisticated methods of assimilating observed data and iterating with updated
forecast models, such as described in Kalnay (2003) and Talagrand (1997).
A major goal of such efforts would be to achieve significant skill in simulating and
forecasting simultaneously both the amplitude and the duration of solar cycles. How do we
achieve that goal, starting from our current model and approaches? The current model contains
fixed meridional circulation that is chosen, along with the amplitude of the turbulent magnetic
diffusivity in the bulk of the convection zone, to produce a cycle period equal to the average
period of cycles 12–24, 10.75 years. The quenching of all α-effects is picked to produce a
sequence of cycle peaks of equal amplitude, an amplitude typical of a solar cycle, with a butterfly
diagram from tachocline-level toroidal and surface radial fields that is also well calibrated to the
Sun. This forms the starting point in time for simulations and predictions from observed surface
magnetic data for cycles 12–24, which data has been stretched or compressed also to the observed
average cycle period. This is done to ensure that over several cycles the period intrinsic to the
model for that meridional circulation, and that of the surface observations, do not lead to a time
phase mismatch that seriously degrades the simulation and forecast of cycle amplitudes. The
high skill of the amplitude prediction results reviewed above indicate that such degradation has
been avoided.
We have recently found that, even with the constraints described above, our model does
produce cycles with variable period. Since both the meridional flow and diffusivity are fixed
with time, these variations must come from some other source. We find that they are caused
by the amplitude variations from cycle to cycle in the input data, as well as the stretching and
compression of that data that we have introduced. Despite these variations, we achieve the high
skill reported in section 3.8; evidently this effect does not degrade the amplitude forecast. But
with our expanded goal of forecasting cycle amplitude and period simultaneously, we need to
know how even the current model behaves and why.
The basic result we have found concerning cycle periods in the model is shown in
figure 10(a)–(c). Data for these plots was recovered from the prediction runs shown in Dikpati
and Gilman (2006) with the turbulent magnetic diffusivity η = 5 × 1010 cm2 s−1 . Figure 10(a)
shows the correlation between the cycle period from the model and the observed smoothed peak
sunspot area. The grey dots are for cycles 12–15, the first four cycles in the simulation, during
which the model is loading the conveyor belt with the observed data, while the black dots are for
cycles 16–23, produced after the conveyor belt is loaded. We see that the scatter of the periods is
much larger for cycles 12–15 than for the later cycles, indicating that cycle amplitude can have
a large effect on the period during an initial transient. During the later cycles the variation in
period is ∼1.8 years, compared to 4.5 years for the earlier ones. The straight line is fitted only
to the later cycles and is associated with a positive correlation coefficient of 0.45. This indicates
that bigger cycles tend to have longer periods in the model, but obviously many more cases
are needed to pin down the degree of correlation. We could have correlated the model period
with the model peak toroidal flux, but since our prediction results show (figure 8(b)) such a high
correlation between observed spot area and predicted toroidal flux, we should get the same result
from such a correlation.
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
20
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
With the method of handling the observed data we used to force the dynamo from the top,
the observed peaks are unchanged. However, the stretching and compression in time of the data
results in changes in the total surface magnetic flux the dynamo is forced with, in linear proportion
to the amount of stretching or compression. In figure 10(b) we show the correlation between
the model produced cycle periods and the ‘excess’ spot area that results from the stretching
or compression. We estimate this amount by calculating the change in the ‘full width at half
maximum’ of the sunspot area curve with time for each cycle (see figure 7(a)). Here again we
see much more scatter in the early cycles, and again a variation of about 1.8 years in the periods
with the late cycles. Here too, the correlation is positive (r = 0.52), indicating a tendency for
cycles for which some flux has been added to be longer.
Thus our handling of the data has introduced an artifact into the determination of the period.
We are not using this model to make a forecast of the period, so no harm is done, but clearly when
we want to forecast both amplitude and period, we must replace the stretching and compression
with some other technique for conditioning the initial data to give an optimum result. Presumably
even if we remove this artifact, the cycle period variation with cycle amplitude would remain,
though there may be some ‘crosstalk’ between these two effects in the model. Also, since in the
forecast model the forecast of one cycle depends on information from more than one previous
cycle (see figure 9), the effects of stretching and compression of previous cycles must be playing
some role.
Even though we know that there is a weak anti-correlation between the amplitude and the
duration of a solar/stellar cycle (Soon et al 1994; Charbonneau and Dikpati 2000), in the model it
is opposite (see figure 10(a)) because the next cycle takes more time to overcome a strong cycle.
This positive correlation between cycle amplitude and period was also seen in Charbonneau and
Dikpati (2000). In figure 10(b) the tendency for a cycle that is stretched to the average period
is to be even longer in the simulation, because that cycle has more flux than it would otherwise
have had, and therefore it should end late. We must await an ensemble of sensitivity tests such
as that outlined previously in this section to investigate whether fluctuations in other quantities
lead to the amplitude-duration anti-correlation.
The stretching and compression of the observed input data appears to create another
effect, seen in figure 10(c), which shows the correlation between observed and simulated cycle
period is modestly negative (r = −0.50). Since from figure 10(a) higher peaks are associated
with longer periods, and cycles with observed longer periods had flux removed from them
by the compression, their periods should get shorter than before, leading to the negative
correlation.
We have demonstrated a simple, first example of data assimilation techniques in solar cycle
models in which the timing has been taken out of the calculation by making all cycles the same
length in the forcing term. Thus the updating of the model by comparing various simulation
outputs with observations has been avoided, and the model focussed purely on amplitude
prediction. However, the ultimate goal is to predict many solar cycle features simultaneously,
such as the timing, amplitude and the shape of the cycle. Clearly the stretching and compression
of the observational data needs to be handled in other ways. For example, we could assume that
each cycle retains the same total magnetic flux as measured by the sunspot area curve, rather
than the same peak spot area. This is yet to be done. The result might be less artificial variation
in the simulated cycle period, but perhaps at the expense of reduced skill in the simulation and
forecast of the cycle peak. More importantly, the ‘sequential assimilation technique’ needs to
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
21
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Figure 9. (a), (b) Latitudinal component of simulated poloidal field (red and
blue opposite signs) at phase 0.3 of cycles 19 and 20; (c), (d) same for simulated
toroidal field.
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
22
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Figure 10. Three frames respectively show correlations between (a) simulated
cycle period and observed peak spot area, (b) simulated cycle period and excess
flux in each cycle introduced due to stretching and compressing the original data,
(c) simulated cycle period and observed cycle period. Grey dots, representing the
first four cycles, are not considered for calculation of correlation coefficients or
the straight line fits.
be implemented in the model so that some time-varying model ingredients are updated after a
certain time interval in order to optimally adjust the model output with observations.
4. Concluding remarks
We have shown above that flux-transport dynamos when applied to the Sun can answer many
important questions, from what determines the dynamo period to how the fields of previous
cycles determine the amplitude of the next one. Building on this success, we plan many sensitivity
tests of the model, and expect to generalize it ultimately to forecast both the period and
the amplitude, as well as to simulate and even forecast the appearance and evolution of
nonaxisymmetric features, such as active longitudes. Such a model would require different
parameterizations of some processes than in the axisymmetric case, since more effects would
be calculated explicitly. We already know that when we split the observed surface magnetic
data into north and south hemispheres, we are able to correctly simulate the large differences in
amplitude between hemispheres in later cycles when they occur. In future, we will implement
the sequential assimilation technique to simultaneously predict the timing, amplitude and the
shape of a cycle.
Acknowledgments
We thank Pablo Mininni for a thorough review of the paper and for his helpful comments, and
extend our thanks to two anonymous reviewers of this paper for their critical reviews, which
have helped us improve the paper significantly. This work is partially supported by NASA grants
NNH05AB521, NNH06AD51I and the NCAR Director’s opportunity fund. National Center for
Atmospheric Research is sponsored by the National Science Foundation.
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
23
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
References
Babcock H W 1961 The topology of the Sun’s magnetic field and the 22-year cycle Astrophys. J. 133 572
Basu S and Antia H M 2003 Changes in solar dynamics from 1995 to 2002 Astrophys. J. 585 553
Browning M, Miesch M S, Brun A S and Toomre J 2006 Dynamo action in the solar convection zone and tachocline:
pumping and organization of toroidal fields Astrophys. J. 648 L157
Cameron R and Schüssler M 2007 Solar cycle prediction using precursor and flux transport models Astrophys. J.
659 801
Charbonneau P and Dikpati M 2000 Stochastic fluctuations in a Babcock–Leighton model of the solar cycle
Astrophys. J. 543 1027
Cowling T G 1945 On the Sun’s general magnetic field Mon. Not. R. Astron. Soc. 105 166
Dikpati M 1996 The evolution of weak diffuse magnetic fields of the sun, and the heating of the quiet corona Thesis
Indian Institute of Science Bangalore
Dikpati M 2004 Global MHD Theory of Tachocline and the Current Status of Large-Scale Solar Dynamo Proc. the
SOHO 14 / GONG 2004 Workshop ESA SP-559 233
Dikpati M 2005 Solar magnetic fields and the dynamo theory Adv. Space Res. 35 322
Dikpati M and Charbonneau P 1999 A Babcock-Leighton flux transport dynamo with solar-like differential rotation
Astrophys. J. 518 508
Dikpati M and Gilman P A 2001 Flux-transport dynamos with α-effect from global instability of tachocline
differential rotation: a solution for magnetic parity selection in the Sun Astrophys. J. 559 428
Dikpati M and Gilman P A 2006 Simulating and predicting solar cycles using a flux-transport dynamo Astrophys. J.
649 498
Dikpati M, de Toma G, Gilman P A, Arge C N and White O R 2004 Diagnostics of polar field reversal in solar
cycle 23 using a flux transport dynamo model Astrophys. J. 601 1136
Dikpati M, Gilman P A and MacGregor K B 2005 Constraints on the applicability of an interface dynamo to the
Sun Astrophys. J. 631 647
Dikpati M, Gilman P A and MacGregor K B 2006 Penetration of dynamo-generated magnetic fields into the Sun’s
radiative interior Astrophys. J. 638 564
Dikpati M, de Toma G and Gilman P A 2006 Predicting the strength of solar cycle 24 using a flux-transport
dynamo-based tool Geophys. Res. Lett. 33 L05102
Feynman J 1982 Geomagnetic and solar wind cycles, 1900–1975 J. Geophys. Res. 87 6153
Garaud P 1999 Propagation of a dynamo field in the radiative interior of the Sun Mon. Not. R. Astron. Soc. 304 583
Gilman P A 1983 Dynamically consistent nonlinear dynamos driven by convection in a rotating spherical
shell. II—Dynamos with cycles and strong feedbacks Astrophys. J. Suppl. 53 243
Gilman P A 2000 Fluid dynamics and MHD of the solar convection zone and tachocline: Current understanding
and unsolved problems—(Invited Review) Sol. Phys. 192 27
Gough D O and McIntyre M 1998 Inevitability of a magnetic field in the Sun’s radiative interior Nature 394 755
Haber et al 2002 Evolving Submerged meridional circulation cells within the upper convection zone revealed by
ring-diagram analysis Astrophys. J. 570 855
Hathaway et al 1996 GONG Observations of solar surface flows Science 272 1306
Hathaway D H and Wilson R M 2004 What the Sunspot record tells us about space climate Sol. Phys.
224 5
Hathaway D H and Wilson R M 2006 Geomagnetic activity indicates large amplitude for sunspot cycle 24 Geophys.
Res. Lett. 33 L18101
Kalnay E 2003 Atmospheric Modelling, Data Assimilation and Predictability (Cambridge: Cambridge University
Press) 341 pp
Markiel A J and Thomas J H 1999 Solar interface dynamo models with a realistic rotation profile Astrophys. J.
523 827
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)
24
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Mestel L and Weiss N O 1987 Magnetic fields and non-uniform rotation in stellar radiative zones Mon. Not. R.
Astron. Soc. 226 123
Parker E N 1981 Residual fields from extinct dynamos Geophys. Astrophys. F 18 175
Parker E N 1993 A solar dynamo surface wave at the interface between convection and nonuniform rotation
Astrophys. J. 408 707
Schatten K et al 1978 Using dynamo theory to predict the sunspot number during solar cycle 21 Geophys. Res. Lett.
5 411–4
Schatten K 2005 Solar activity prediction: timing predictors and cycle 24 J. Geophys. Res. 107 SSH 15 1–7
Schüssler M 1975 Axisymmetric alpha-squared-dynamos in the Hayashi-phase Astron. Astrophys. 38 263
Soon W H, Balliunas S L and Zhang Q 1994 Variations in surface activity of the Sun and solar-type stars Sol. Phys.
154 385
Stix M 1973 Spherical alpha-omega-dynamos by a variational method Astron. Astrophys. 24 275
Svalgaard L, Cliver E W and Kamide Y 2005 Sunspot cycle 24: smallest cycle in 100 years? Geophys. Res. Lett.
32 L01104
Talagrand O 1997 Assimilation of observations, an introduction J. Meteor. Soc. Japan 75 (1B) 191
Tobias S M, Hughes D and Weiss N O 2006 Unpredictable sun leaves researchers in the dark Nature
442 26
Wang Y-M and Sheeley N R Jr 1991 Magnetic flux transport and the sun’s dipole moment—new twists to the
Babcock-Leighton model Astrophys. J. 375 761
Wang Y-M, Nash A and Sheeley N R Jr 1989 Evolution of the sun’s polar fields during sunspot cycle
21—poleward surges and long-term behavior Astrophys. J. 347 529
Wrubel M H 1952 On the decay of a primeval stellar magnetic field Astrophys. J. 116 291
New Journal of Physics 9 (2007) 297 (http://www.njp.org/)