High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR) The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Research under sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer. SOLAR DYNAMO MODELING AND PREDICTION Mausumi Dikpati High Altitude Observatory, NCAR

Observational signature for global evolution of solar magnetic fields From url of D. Hathaway

What is a dynamo? A dynamo is a process by which the magnetic field in an electrically conducting fluid is maintained against Ohmic dissipation In astrophysical object, there can always be a dynamo whenever the plasma consists of seed magnetic fields and flow fields All these magnetic fields are maintained by dynamo action

Flux-transport Dynamo (i) Generation of toroidal (azimuthal) field by shearing a pre-existing poloidal field (component in meridional plane) by differential rotation (Ω-effect ) (ii) Re-generation of poloidal field by lifting and twisting a toroidal flux tube by helical turbulence (α-effect) (iii) Flux transport by meridional circulation <

Fixing dynamo ingredients While Ω -effect and meridional circulation can be fixed from observations, the α–effect could be of different types as suggested theoretically. One directly observed α–effect can arise from decay of tilted, bipolar active regions Babcock 1961, ApJ, 133, 572

How a Babcock-Leighton Flux-transport dynamo works Shearing of poloidal fields by differential rotation to produce new toroidal fields, followed by eruption of sunspots. Spot-decay and spreading to produce new surface global poloidal fields. Transport of poloidal fields by meridional circulation (conveyor belt) toward the pole and down to the bottom, followed by regeneration of new toroidal fields of opposite sign.

Mathematical Formulation Under MHD approximation (i.e. electromagnetic variations are nonrelativistic), Maxwell’s equations + generalized Ohm’s law lead to induction equation : Applying mean-field theory to (1), we obtain the dynamo equation as, Differential rotation and meridional circulation from helioseismic data Poloidal field source from active region decay Turbulent magnetic diffusivity (1) (2) Toroidal fieldPoloidal field Meridional circulation Differential rotation Assume axisymmetry, decompose into toroidal and poloidal components:

Poloidal and Toroidal Equations and Boundary Conditions (3a) (3b) (i) Both poloidal and toroidal fields are zero at bottom boundary (ii) Toroidal field is zero at poles, whereas poloidal field is parallel to polar axis (iii) Toroidal field zero at surface; poloidal fields from interior match potential field above surface (iv) Both poloidal and toroidal fields are antisymmetric about the equator

Evolution of Magnetic Fields In a Babcock-Leighton Flux-Transport Dynamo Dikpati & Charbonneau 1999, ApJ, 518, 508 Dynamo cycle period ( T ) primarily governed by meridional flow speed

Refining a Babcock_Leighton flux-transport dynamo A full-spherical-shell Babcock-Leighton dynamo relaxes to a quadrupole parity, violating the observed Hale’s polarity rule which implies dipole parity about the equator Remedy: a tachocline α-effect Dikpati & Gilman, 2001, ApJ, 559, 428;Bonanno et al, 2002, A&A, 390, 673

Calibrated Flux-transport Dynamo Model Near-surface diffusivity same as used by Wang, Shelley & Lean, 2002; Schrijver 2002 in their surface flux-transport models. Zita is exploring in details the sensitivity of diffusivity profiles to flux-transport dynamo N-Pole S-Pole Red: α -effect location Green: rotation contours Blue: meridional flow Magnetic diffusivity usedFlows derived from observations

Contours: toroidal fields at CZ base Gray-shades: surface radial fields Observed NSO map of longitude-averaged photospheric fields Validity test of calibration Dikpati, de Toma, Gilman, Arge & White, 2004, ApJ, 601, 1136

Why is solar cycle prediction important? Qian, Solomon & Roble; GRL, 2006  High atmosphere density varies as function of solar cycle  Density variation at 400 km depth is 2-3 times that of cycle amplitude variation  Satellites are placed at that altitude, and so drag due to density variation affects their lifetime

Issues with polar field precursor techniques Q1. How can the 5.5 year-old polar fields from previous cycle determine the next cycle’s amplitude? Q2. Do they remain radial down to shear layer? Q3. Are stronger radial fields associated with stronger or weaker latitudinal fields? It depends on field geometry inside convection zone: see 3 possible cases < < < 1. Weak radial; strong latitudinal 3. Strong radial; weak latitudinal 2. Weak radial; weak latitudinal

Flux-transport dynamo-based prediction scheme Meridional circulation plays an important role in this class of model, by governing a) the dynamo cycle period b) the memory of the Sun’s past magnetic fields <

Timing Prediction For Cycle 24 Onset Dikpati, 2004, ESA-SP, 559, 233

Recent Support For Delayed Onset Of Cycle 24 Cycle 23 onset Pred. cycle 24 onset

Recent Support For Delayed Minimum At End of Cycle 23 Mar. 29, 2006 Early 1996 Nov This coronal structure not yet close to minimum; more like 18 months before minimum Corona at last solar minimum looked like this

Amplitude prediction: Data-assimilation In Solar Cycle Models Given the strong correlation between area and flux, we apply data-assimilation techniques to our calibrated dynamo Such techniques used in meteorology for 50 years, but just starting in solar physics Appropriate time for data-assimilation in solar physics: large new data-sets becoming available First example; predicting relative solar cycle peaks. Future goal: simultaneous predictions of cycle amplitude and timing, using “sequential” and “variational” data-assimilation techniques

Construction Of Surface Poloidal Source: 2D Data Assimilation Period adjusted to average cycle Original data (from Hathaway) Assumed pattern extending beyond present

Three techniques for treating surface poloidal source in simulating and forecasting cycles 1) Continuously update of observed surface source including cycle predicted (a form of 2D data assimilation) 2) Switch off observed surface source for cycle to be predicted 3) Substitute theoretical surface source, derived from dynamo- generated toroidal field at the bottom, for observed surface source Forecasted quantity : integrated toroidal magnetic flux at the bottom in latitude range of 0 to 45 degree (which is the sunspot-producing field) We use these three techniques in succession to simulate and forecast

Simulating Relative Peaks Of Cycles 12 Through 24  We reproduce the sequence of peaks of cycles 16 through 23  We predict cycle 24 will be 30-50% bigger than cycle 23 Dikpati, de Toma & Gilman, 2006, GRL, 33, L05102

Evolution of predictive solution Color shades denote latitudinal (left) and toroidal (right) field strengths; orange/red denotes positive fields, green/blue negative Latitudinal fields from past 3 cycles are lined-up in high-latitude part of conveyor belt These combine to form the poloidal seed for the new cycle toroidal field at the bottom (Dikpati & Gilman, 2006, ApJ, 649, 498) Latitudinal field Toroidal field

How Does The Model Work Color shades denote latitudinal (top) and toroidal (bottom) field strengths; orange/red denotes positive fields, green/blue negative Latitudinal fields from past 3 cycles are lined-up in high-latitude part of conveyor belt These combine to form the poloidal seed for the new cycle toroidal field at the bottom Dikpati & Gilman, 2006, ApJ, 649, 498 Latitudinal field Toroidal field

Results from separating North and South hemispheres Model reproduces:  N/S asymmetry when large  relative sequence of peaks in N & S separately  short time-scale (monthly) features within a cycle; high surface diffusivity and long traversal time of surface poloidal fields to shear layer smooths short-term features in the model Model cannot reproduce: Observations indicate N/S asymmetry, often persisting for several cycles, but no systematic switching in strength between N & S Dikpati, Gilman, de Toma & Ghosh 2007, Solar Physics (submitted)

How many cycles can we predict ? Surface poloidal source constructed from the predicted bottom toroidal field; BL flux- transport dynamo in self-excited mode

Summary  Meridional circulation is an essential ingredient for large-scale solar dynamo  Flux-transport dynamo with input of observed surface magnetic flux displays high skill in forecasting peak of the next solar cycle, as well as significant skill for 2 cycles ahead  High skill extends to input data separated into N & S hemispheres  High surface diffusivity and long transport time to the bottom together smooth out the short-term observational features; therefore we will not be able to forecast short-term solar cycle features by this model

Future Directions  Going beyond axisymmetry: simulating and predicting the Sun’s active-longitudes  Simulating Grand-minima  Predict amplitude and timing simultaneously by applying “sequential” assimilation technique