A low order dynamo model and possible applications

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

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

What? Axisymmetric Mean Field Dynamo Theory coupled with Dynamical Systems Analysis 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, Ap. J 686 ) Simplest way, truncate the dynamo equations by assuming: Dário Passos CENTRA / IST

What ? The low order dynamo (toy) model Example of possible solutions for the Spatial (or Structural) coefficients d. B/dt Tipical van der Pol – Duffing oscillator phase space R – Solar Radius vp – Meridional Circulation Amplitude a – The alpha effect g – magnetic buoyancy coefficient r – plasma density h – Magnetic diffusivity l 0 – Characteristic magnetic interaction length W – Differential rotation Passos, Lopes (2010) Jastp. 2009. 12. 019 B Different models can give different spatial coefficients dependencies 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 Compute evolution of the coefficients from cycle to cycle infer evolution of physical quantities Assumption: As first approximation, we consider that each cycle corresponds to an equilibrium solution. 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

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 SSN 1 – Changing vp every sunspot minima (full) SSN 2 – Changing vp every sunspot maxima (full) SSN 3 – Changing vp every sunspot minima (half) Lopes, I. and Passos, D. , (2009) Sol. Phys, 1, 257 Dário Passos CENTRA / IST

Conclusions 1. We developed a method that can extract new information about the evolution of the solar magnetic field from sunspot time series 2. 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) 3. Within theoretical frame work used , meridional circulation variations can account, at least, for part of the variability observed in the Sun Current and future work 1. Improve theoretical model behind this method 2. Introduce an adaptative fitting method 3. Use the method to impose new constraints to dynamo theory 4. Study long term trends in dynamo action or background structures Dário Passos CENTRA / IST