Bidya Binay Karak
Jack Eddy Fellows, NASA LWS & UCAR VSP
Summary of selected recent works:
Manuscript under preparation: The increase of effective Prandtl number, a possible solution to the solar convection conundrum, causes anti-solar differential rotation
Observations suggest that the convective velocity at large scales in
global convection simulations might be over-estimated (convective
conundrum). One plausible solution to this could be that solar
convection is driven by the low entropy plumes, in a process known as
entropy rain. Another solution could be the magnetic field. The
Lorentz force of the magnetic field suppresses the convective motions
and also the turbulent mixing of entropy between upflows and
downflows, leading to a large effective Prandtl number (Pr). We
explore this idea in three-dimensional global rotating convection
simulations at different thermal conductivity (kappa), i.e., at
different Pr. In agreement with previous non-rotating simulations,
the convective velocity is systematically reduced with the increase of
Pr. However, in rotating simulations, the convective velocity
initially increases with the increase of Pr and when the thermal
conductive flux becomes negligible, the convective velocity
decreases. A subadiabatic layer is formed near the base of the
convection zone due to continuous deposition of low entropy plumes in
low-$\kappa$ simulations. The most interesting result of our
low-$\kappa$ simulations is that the convective motions are
accompanied by a change in the convection structure that is
increasingly influenced by small-scale plumes. These plumes tend to
transport angular momentum radially inward and thus establish an
anti-solar differential rotation, in striking contrast to the solar
rotation profile. If such low diffusive plumes, driven by the
radiative-surface cooling, are present in the Sun, then our results
cast doubt on the idea that a high effective Pr may be a viable
solution to the solar convective conundrum. Our study also emphasizes
that any resolution of the conundrum that relies on the downward
plumes must take into account angular momentum transport as well as
Figure caption: Entropy fluctuations at two different radial layers (top panels: near surface and bottom: lower part of the convection zone) from simulations at Pr = 1 (left panels), and Pr = 10 (right panels).
Can the Babcock-Leighton Dynamo Operate during Maunder-like Grand Minima?
The Sun occasionally goes through Maunder-like extended grand minima when its magnetic activity drops considerably from the normal activity level for several decades. Many possible theories have been proposed to explain the origin of these minima. However, how the Sun managed to recover from such inactive phases every time is even more enigmatic. The Babcock-Leighton type dynamos, which are successful in explaining many features of the solar cycle remarkably well, are not expected to operate during grand minima due to the lack of a sufficient number of sunspots. We explore the question of how the Sun could recover from grand minima through the Babcock-Leighton dynamo. In our 3D dynamo model, grand minima are produced spontaneously as a result of random variations in the tilt angle of emerging active regions. We find that the Babcock-Leighton process can still operate during grand minima with only a minimal number of sunspots and that the model can emerge from such phases without the need for an additional generation mechanism for the poloidal field. The essential ingredient in our model is a downward magnetic pumping which inhibits the diffusion of the magnetic flux across the solar surface.
Figure caption: (a) Temporal variation of the smoothed sunspot number obtained from 13,000-year simulation with random scatter in the sunspot tilt angle..
Blue shaded regions below the horizontal line
represent the grand minima.
(b) Monthly spot number (black/red: north/south) shown only for a selected 1600-year interval. (The manuscript is submitted in Phys. Rev. Lett.)
Scatter in the Sunspot Tilt around Joy's Law Produces Variable Solar Cycle
About 100 years back George Ellery Hale and his colleagues have shown that sunspots follow a certain polarity rule and the leading polarity of sunspot appears slightly closer to the equator, giving rise to a tilt which increases with the latitude of the sunspot. This is known as Joy's law.
About 50 years later, Babcock and Leighton independently have demonstrated that because of this tilt, when sunspots decay, they produce a poloidal field, which is observed on the solar surface. From the observations, we know that there is a large scatter in the sunspot tilt around Joy's law. Thus this scatter can produce a variation in the poloidal field. In a three-dimensional Babcock-Leighton dynamo model, we for the first time have shown that this observed scatter in the sunspot tilt can produce a significant variation in the amplitude and the period of the solar cycle. Thus our study has explicitly demonstrated that the observed tilt scatter is a cause of the variation in the solar cycle. See publication for details: Karak, & Miesch, ApJ.
Figure caption: Variable sunspot cycle obtained from 3D Babcock-Leighton dynamo model (STABLE) with the observed scatter in sunspot tilt around Joy's law which folows a Gaussian distribution with sigma of 15 degrees. (a) 19 magnetic cycles are highlighted with red and blue representing the northern and southern hemispheres respectively. Red shaded areas indicate timess when the sunspot number (SSN) in the north exceeds that in the south and blue shaded areas indicate the opposite. (b) Long-term variability in the same simulation, exhibiting extended periods of low and high activity. Black and red indicate northern and southern hemispheres and the dotted line shows the observed SSN for the Sun, averaged over the last 13 cycles, for comparison.
Babcock-Leighton Solar Dynamo: Flux Transport vs. Dynamo Wave?
The generation of poloidal field through the decay and dispersal of
tilted bipolar active regions on the solar surface is observed.
Parameterizing this process for the generation of poloidal field
and by including an observed differential rotation for the generation of toroidal field
we have developed a kinematic dynamo model.
When a meridional circulation--poleward near the surface and
equatorward near the bottom--is included in this model,
most of the basic features of the solar cycle, including the equatorward migration of the toroidal field, are reproduced. This is consistent with
the previous flux transport dynamo models.
However, when a radially downward magnetic pumping near the surface is included,
the poloidal field becomes predominantly radial
which allows the negative radial shear in the near-surface layer to act on the radial field
to produce a toroidal field. This consequently causes the dynamo wave with an
equatorward migration of the toroidal field.
In this scenario, we observe a clear equatorward migration of the toroidal field at low latitudes
even when there is no meridional flow near the bottom
or when there is a shallow meridional circulation with
no flow underneath.
Furthermore, the magnetic pumping suppresses the diffusion of fields at the surface
which helps to achieve the 11-year dynamo cycle at a moderately larger value of
Figure caption: Left panel: downward pumping speed needed to get 11-year cycle
for different values of the bulk diffusivities at a fixed value of
the surface diffusivity of 3*10^12 cm^2/s. Black points and red squares represent two sets of simulations in which the Babcock-Leighton source is related to the mean toroidal flux in the CZ and the toroidal field in the tachocline, respectively. Right panel: corresponding values of critical alpha needed to get stable solutions.
See the paper for details
: Karak, & Cameron, ApJ.
Is the Small-scale Magnetic Field Correlated with the Dynamo Cycle?
The small-scale magnetic field is ubiquitous at the solar surface - even at high latitudes. From observations we know that this field is uncorrelated (or perhaps even weakly anticorrelated) with the global sunspot cycle. Our aim is to explore the origin, and particularly the cycle dependence of this field using three-dimensional dynamo simulations. We use a simple model of a turbulent dynamo in a shearing box driven by helically forced turbulence. Depending on the dynamo parameters, large-scale (global) and small-scale (local) dynamos can be excited independently in this model.
Based on simulations in different parameter regimes, we find that, when only the large-scale dynamo is operating in the system, the small-scale magnetic field generated through shredding and tangling of the large-scale magnetic field is positively correlated with the global magnetic cycle.
However, when both dynamos are operating, the small-scale field is produced from both the small-scale dynamo and the tangling of the large-scale field. In this situation, when the large-scale field is weaker than the equipartition value of the turbulence, the small-scale field is almost uncorrelated with the large-scale magnetic cycle.
On the other hand, when the large-scale field is stronger than the equipartition value, we observe a clear anticorrelation between the small-scale field and the large-scale magnetic cycle. This anticorrelation can be interpreted as a suppression of both the small-scale dynamo and the tangling of the large-scale field. Based on our studies we conclude that the observed small-scale magnetic field in the Sun is generated by the combined mechanisms of small-scale dynamo and tangling of the large-scale field. The observed cyclic variation of the small-scale field is produced by the interaction between the large-scale field and the flow.
Figure caption: butterfly diagrams of x and y components of large-scale field, energies
of large-scale, and small-scale fields (top to bottom) as functions of z and t,
normalized by the diffusive time scale.
See the published paper for details
: Karak, & Brandenburg, ApJ 816, 9 (2016).
Hysteresis between Distinct Modes of Turbulent Dynamos:
We perform three-dimensional simulations of large-scale dynamos in a shearing box with helically forced turbulence. As an initial condition, we either take a weak random magnetic field or we start from a snapshot of an earlier simulation. Two quasi-stable states are found to coexist in a certain range of parameters close to the onset of the large-scale dynamo. The simulations converge to one of these states in dependence on the initial conditions. When either the fractional helicity or the magnetic Prandtl number is increased between successive runs above the critical value for the onset of the dynamo, the field strength jumps to a finite value. However, when the fractional helicity or the magnetic Prandtl number is then decreased again, the field strength stays at a similar value even below the original onset. We also observe intermittent decaying phases away from the strong field branch close to the point where large-scale dynamo action is just possible.
From this study, we argue that stars just before the dynamo activity ceased show grand minima and probably our Sun is just slightly above the dynamo transition. Stars rotating much faster do not show grand minima but show irregular cycles. Also, we show that if the dynamo is only slightly supercritical we get grand minima. This is in agreement with the flux transport dynamo simulations which produce grand minima only when they are not highly supercritical.
Figure caption: Dynamo hysteresis, as seen in the rms value of the large-scale magnetic field as a function of helicity (measure of rotation) of the fluid. The filled circles and the red diamonds are from simulations that started with weak random seed fields and strong oscillatory fields of the previous simulation, respectively. Runs E5-E11 show intermittent behavior.
See the published paper for details: Karak et al. ApJ 803, 2 (2015).
Stellar Differential Rotation: Transition from Solar to Anti-solar Profile and Dynamo Activities in global MHD simulations:
We have studied the behaviour of solar differential rotations near the transition from solar to anti-solar profiles using global compressible MHD simulations in spherical geometry. By taking different radiative conductivities, the convective velocities and hence the rotational influence on the convection is varied in a set of simulations. When we decrease the Coriolis number, differential rotation changes from solar-like to anti-solar. We find that the magnetic field helps to produce solar-like differential rotation. In our simulations we do not find any evidence of the bistable states of differential rotation which has been previously observed in hydrodynamic simulations. In anti-solar differential rotation cases we get coherent single cell meridional circulations, whereas in solar-like rotation we get multi-cellular circulations. In all cases, the poleward propagating speed near the surface is close to the observed value. The large-scale flows show significant temporal variations which are also in observational ranges.
In the slowly rotating cases, we find activity cycles, but no clear polarity reversals, whereas in the more rapidly rotating cases irregular variations are obtained.
See the published paper for details: Karak et al. A&A, 576, A26 (2015).
Results from two different global MHD convection simulations: Run A (upper left panel) and Run E
(upper right). These two simulations are basically same except Run E is more rotation dominated than Run A. Top:
Angular velocity in the meridional plan. Note that Run A produced anti-solar differential rotation whereas
Run E produced solar-like rotation. Lower panels: the toroidal
component of the magnetic field at the bottom of the convection zone from these two simulations as a
function of latitide and time.
Quenching of Turbulent Transport Coefficients:
Turbulent transport in hydromagnetic turbulence determines the evolution of
large-scale magnetic fields in astrophysical objects.
These transport effects are usually suppressed, and becomes anisotropic.
Using different variants of the test-field method, we determine the quenching of the
turbulent transport coefficients for different flows.
We observe significant quenching only when the mean magnetic field is larger than
the equipartition value of the turbulence. Expressing the magnetic field in terms of the equipartition value of the quenched flows,
we obtain for the quenching exponents of the turbulent magnetic diffusivity for convection about 1.3 convection.
However, when the magnetic field is expressed in terms of the equipartition value of the unquenched flows,
this quenching exponents become about 2.3. For the alpha effect, the exponent is about 2 in the first case,
but 3 in the second.
The quenching of turbulent pumping follows the same power law as turbulent diffusion,
while for the coefficient describing the Omega*J effect nearly the same quenching exponent
is obtained as for alpha.
The number of sunspots observed on the face of the Sun varies roughly periodically with a period of 11 years. However, the strength and the duration of the cycle vary in an irregular manner. One puzzling aspect of this 11-year sunspot cycle is the Maunder minimum (1645-1715) when sunspots were very scarce. Indirect studies suggest that there were several other such events in the past. There is evidence that some other solar-like stars also show solar (stellar) cycle and some of them show irregular cycle with frequent maunder-like grand minima. We may also mention that the sunspot cycle might have an effect on the space environment and the Earth climate system. It has been shown that in the pre-industrial era the Sun was the main driver for the Earth's global temperature. Therefore understanding the solar cycle is not only important to solar and stellar physics community but also to the space weather and Earth climate community.
We applied a dynamo model to understand the origin of this sunspot cycle. Basically, the dynamo is the cyclic conversion between the poloidal field (which is in the poloidal direction and responsible for giving Sun's dipole field) and the toroidal field (which is in the azimuthal direction and responsible for giving sunspot eruptions). Presently, the flux transport dynamo model is a promising model for studying the solar cycle. In this model, the toroidal field is generated by the strong differential rotation near the base of the convection zone and the poloidal field is generated near the solar surface from the decay of the sunspots. The turbulent diffusion and the meridional circulation are the two important transport agents in this model which communicate these two spatially segregated source regions of magnetic fields.
Modeling the Maunder minimum
Maunder minimum is an episode during 1645-1715 when very few sunspot appeared on the surface of the Sun. This is not an artifact of a few observations but a real phenomenon. Other indicators of solar
activity (e.g., auroral activity) were also equally muted. We have explored whether the present dynamo models can explain this Maunder minimum.
We have shown that if the poloidal field at the end of a cycle falls to a very low value due to the strong fluctuations in its generation process, then this can produce a Maunder minimum. To know more about this work see: Choudhuri & Karak RAA, 9, 953 (2009).
We have also shown that if the meridional circulation at the end of a cycle falls to a very low value, then this can produce a Maunder minimum. To know more about this work see: Karak, ApJ 724, 1021 (2010).
Modeling the irregular solar cycle
The solar cycle is not very regular and periodic. Some cycles are weaker and some are stronger. In the flux transport dynamo model, the poloidal field is generated from the decay of the tilted bipolar sunspots near the solar surface through Babcock-Leighton process. We believe that this Babcock-Leighton process of generating poloidal field involves randomness and due to this, the net poloidal field at the end of a cycle becomes different than others. This randomness in the poloidal field gives rise to the irregularity in the solar cycle. However, we have shown that the fluctuations in the meridional circulation can introduce one more important source of irregularities in the solar cycle. For details of this work see: Karak, ApJ 724, 1021
We have applied dynamo models to study one important aspect of the solar cycle,
known as the Waldmeier effect.
Basically, this says that the stronger cycles are having shorter rise time and vice-versa. We have successfully reproduced this effect theoretically using flux transport dynamo model. For details of this work see: Karak & Choudhuri MNRAS, 410, 1503, (2011).
Origin of grand minima
One of the most striking aspects of this sunspot cycle is that there have been times in the past when sunspots did not appear for several years and a few cycles went missing. A most well-known example of this is the Maunder minimum during 1645-17153. Although reliable sunspot data did not exist before 1610, occurrences of such grand minima in the past can be inferred from the study of cosmogenic isotopes 14C in old tree rings and 10Be in polar ice. When sunspots are absent, the magnetic field in the solar wind becomes weak and more galactic cosmic rays can reach the Earth, producing more of such radioactive isotopes. Analyses of these isotopes indicate that there have been about 27 grand minima in the last 11,000 years. Since there were about 1,000 solar cycles during this period, the occurrence of 27 grand minima implies that about 2.7% cycles had conditions appropriate for forcing the Sun into grand minima. We address the question how grand minima are produced and specifically calculate the frequency of occurrences of grand minima from a theoretical model. With reasonable guidance from the observational data of last 28 solar cycles, we show that a flux transport solar dynamo model leads to the conclusion that about 2-4% of the sunspot cycles may have conditions suitable for inducing grand minima. For details of this work see: Choudhuri & Karak (2012) Phys. Rev. Lett., 109, 171103 (2012).
Quenching of meridional circulation in flux transport dynamo models
We have included a parametric quenching of the meridional circulation in solar dynamo models such that the meridional circulation becomes weaker when the magnetic field at the base of the convection zone is stronger. We find that a flux transport solar dynamo tends to become unstable on including this quenching of meridional circulation if the diffusivity in the convection zone is less than about a certain value. For dynamo models with high diffusivity, the quenching of meridional circulation does not produce a large effect and the dynamo remains stable. For details of this work see: Karak & Choudhuri Sol. Phys. (2012).
Effect of turbulent magnetic pumping on the solar cycle memory and the prediction of the solar activity
The Sun's magnetic cycle is not regular. The strength of the solar cycle varies cycle to cycle making the prediction of future cycle difficult. However the prediction of the Sun's magnetic activity is important because of its effects on space environmental conditions and climate. However, recent efforts to predict the amplitude of the solar cycle have resulted in diverging forecasts with no consensus. In the Babcock-Leighton dynamo model, the poloidal field is generated near the solar surface whereas the toroidal field is generated near the base of the convection zone. Therefore a finite time necessary to reach the poloidal field to the base of the convection zone introduces the memory in the dynamo model which allows the dynamo model to predict the future solar cycle. We have shown that is dynamical memory of solar cycle is strongly reduced by the inclusion of downward turbulent pumping of magnetic flux and therefore this affects the solar cycle prediction. A reliable predictions for the maximum of solar activity can be made only at the preceding minimum and for more accurate predictions, sequential data assimilation would be necessary in forecasting models to account for the Sun's short memory. For details of this work see: Karak & Nandy ApJL (2012).
On the compatibility of a flux transport dynamo with a fast tachocline scenario
Tachocline is a thin layer (radial extent about 15-20 Mm) near the base of the solar convection zone where rotation switches from differential in the convection zone to nearly uniform in the radiation zone below. The narrow radial extent of this layer implies a strongly anisotropic angular momentum transport mechanism, much more effective in the horizontal direction than in the radial direction. One most plausible candidate for such anisotropic momentum transfer is the Maxwell stress in a predominantly horizontal magnetic field configuration. As the above considerations suggest that the dynamo generated toroidal field resides in the tachocline, it is plausible to assume that this field is responsible for the confinement of the tachocline. The feasibility of this so-called fast tachocline scenario has been demonstrated by Petrovay and Forgacs-Dajka.
We employ a flux transport dynamo model coupled with simple feedback formula relating the thickness of the tachocline to the Maxwell stress. The dynamo model is found to be robust against the nonlinearity introduced by this simplified fast tachocline mechanism. Solar-like butterfly diagrams are found to persist and, even without any parameter fitting, the overall latitudinal and the temporal variation of the thickness of the tachocline is well within the range admitted by helioseismic constraints. For details of this work see: Karak & Petrovay (2012) Sol. Phys.
Search for Chaos in Neutron Star Systems: Is Cyg X-3 a Black Hole?
The accretion disk around a compact object is a nonlinear general relativistic system involving magnetohydrodynamics. Naturally, the question arises whether such a system is chaotic (deterministic) or stochastic (random) which might be related to the associated transport properties whose origin is still not confirmed. Earlier, the black hole system GRS 1915+105 was shown to be low-dimensional chaos in certain temporal classes. However, so far such nonlinear phenomena have not been studied fairly well for neutron stars which are unique for their magnetosphere and kHz quasi-periodic oscillation (QPO). On the other hand, it was argued that the QPO is a result of nonlinear magnetohydrodynamic effects in accretion disks. If a neutron star exhibits chaotic signature, then what is the chaotic/correlation dimension? We analyze RXTE/PCA data of neutron stars Sco X-1 and Cyg X-2, along with the black hole Cyg X-1 and the unknown source Cyg X-3, and show that while Sco X-1 and Cyg X-2 are low-dimensional chaotic systems, Cyg X-1 and Cyg X-3 are stochastic sources. Based on our analysis, we argue that Cyg X-3 may be a black hole. For details of this work see: Karak et al. ApJ, 708, 862, (2010).
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