SOLAR DYNAMO MODELING AND PREDICTION

High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR)

The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Researchunder sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer.

SOLAR DYNAMO MODELING AND PREDICTION

Mausumi DikpatiHigh 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 FormulationUnder 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 rotationand meridional circulation

from helioseismic data

Poloidal field source from active region

decay

Turbulent magnetic diffusivity

(1)

(2)

. BBUB ηt

, BBBUB ηαt

Toroidal field Poloidal field Meridionalcirculation

Differentialrotation

, ˆ ,, ˆ ,, φφφ tθrAtθrB eeB ,ˆ ,Ωsin, φθrθrθr euU

Assume axisymmetry, decompose into toroidal and poloidal components:

Poloidal and Toroidal Equations and Boundary Conditions(3a)

(3b)

, 222 ,,

sin1sin

sin1

φφ BBθrSAθr

ηAθrθrt

A

u

φθφr Buθ

Brurrt

φB 1 , 222

sin1ˆΩ sin φφφp Bθr

ηBηθr

eB

(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 FieldsIn 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-Po

leS-

Pole

Red: α -effect locationGreen: rotation contoursBlue: meridional flow

Magnetic diffusivity used Flows 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 techniquesQ1. 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

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Timing Prediction For Cycle 24 Onset

Dikpati, 2004, ESA-SP, 559, 233

Recent Support For Delayed Onset Of Cycle 24Cycle 23

onsetPred.

cycle 24 onset

Recent Support For Delayed Minimum At End of Cycle 23

Mar. 29, 2006

Early 1996Nov. 1994

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

Latitudinalfield

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