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

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 Dário Passos CENTRA / IST

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: Dário Passos CENTRA / IST

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.2009.12.019     Dário Passos CENTRA / IST

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 Dário Passos CENTRA / IST

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) Dário Passos CENTRA / IST

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 Dário Passos CENTRA / IST

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