1. A low order dynamo model and possible applications
Dário Passos, Ilídio Lopes
CENTRA
Instituto Superior
Técnico, Lisboa
Nordita Dynamo School, Stockholm, 2010

2. Monthly Mean Sunspot Number since 1750 Sunspots are a signature of Bf
How ?
Monthly Mean Sunspot Number since 1750
Sunspots are a signature of Bf
Assumption: Sunspot number is proportional to the magnetic energy 
Create a proxy for the toroidal field Bf
Calculate Phase Space of Bf
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3. Reducing the order of the dynamical system
Axisymmetric Mean Field Dynamo Theory coupled with Dynamical Systems Analysis
What?
Magnetic field averaged and divided into toroidal and poloidal components
Axisymmetric dynamo model equations (Charbonneau (2005) Liv. Rev. Sol. Phys. 2)
Reducing the order of the dynamical system
Truncated Dynamo equations by dimensional analysis, variable separation(!)
( Mininni et al. 2001, Sol Phys. 201, Pontieri et al. 2003, Sol.Phys. 203, Passos & Lopes 2008, ApJ 686 )
Simplest way, truncate the dynamo equations by assuming:
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4. The low order dynamo (toy) model
What ?
The low order dynamo (toy) model
Tipical van der Pol – Duffing
oscillator phase space
Example of possible solutions for the Spatial (or Structural) coefficients
dB/dt
R – Solar Radius
vp – Meridional Circulation Amplitude
a – The alpha effect
g – magnetic buoyancy coefficient
r – plasma density
h – Magnetic diffusivity
l0 – Characteristic magnetic interaction length
W – Differential rotation B
Different models can give different
spatial coefficients dependencies
Passos, Lopes (2010) Jastp     
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5. How ?
Fit individual magnetic cycles and retrieve spatial coefficients for each one
Phase space for magnetic cycles
number 7, 8, 9, 10 and 11 (gray) and corresponding theoretical fit (black)
For each fit we get a set of coefficients mn, xn, wn, ln
Assumption: As first approximation, we consider that each cycle corresponds to an equilibrium solution.
Compute evolution of the coefficients from cycle to cycle  infer evolution of physical quantities
To calculate the evolution of vp use:
assuming that h does not change and that all variations in m is due to vp
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6. Validation of inferred results using a numerical dynamo model
Example of application: Deriving amplitude variation for vp
Validation of inferred results using a numerical dynamo model
Modified version of the Surya Dynamo Code (Nandy, Choudhuri (2002), Sience 296)
Input the derived profile for vp
Look for the number of eruptions at the surface (sunspot proxy)
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7. Results for three simulated scenarios
SSN1 – Changing vp every sunspot minima (full)
SSN2 – Changing vp every sunspot maxima (full)
SSN3 – Changing vp every sunspot minima (half)
Lopes, I. and Passos, D., (2009) Sol. Phys, 1, 257
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8. Improve the theoretical model behind this method
C o n c l u s i o n s
We developed a method that can extract new information about the evolution of the solar magnetic field from sunspot time series
We can use it to determine the evolution of certain background structures of the solar dynamo (assuming that they evolve in a time frame longer than the mag. field)
Within the theoretical frame work used , meridional circulation variations can account, at least, for part of the variability observed in the Sun
C u r r e n t a n d f u t u r e w o r k
Improve the theoretical model behind this method
Introduce an adaptative fitting method
Use the method to impose new constraints to dynamo theory
Study long term trends in dynamo action or background structures
Dário Passos CENTRA / IST