1. 23-28 September 2003 Basic Processes in Turbulent Plasmas Forecasting asymptotic states of a Galerkin approximation of 2D MHD equations Forecasting asymptotic states of a Galerkin approximation of 2D MHD equations Rosita De Bartolo, Vincenzo Carbone Rosita De Bartolo, Vincenzo Carbone Dipartimento di Fisica Dipartimento di Fisica Università della Calabria Università della Calabria

2. 23-28 September 2003Basic Processes in Turbulent Plasmas Mhd equations *, plasma model with large characteristic length (L →∞ ) and low frequency ( ω→0 ), approximate a conducting fluid motion as in plasma devices and solar corona. Mhd equations *, plasma model with large characteristic length (L →∞ ) and low frequency ( ω→0 ), approximate a conducting fluid motion as in plasma devices and solar corona. *homogeneous and incompressible magnetic fluid equations

3. 23-28 September 2003Basic Processes in Turbulent Plasmas Infinite set of velocity and magnetic fields equations ∙ Quadratic invariants Total energy Cross helicity Mean square of vector potential → Magnetic field alignment towards velocity field → Linking degree of magnetic field force lines For each triad of interacting wave vectors such that

4. Numerical simulations Numerical simulations MHD usually uses spectral methods (Fast Fourier Transform) Physical space → FFT→ Spectral space Physical space → FFT→ Spectral space For all infinite values of play a role. For all infinite values of play a role. ↓ It is impossible to realize simulations containing infinite modes It is impossible to realize simulations containing infinite modes Galerkin model all interactions Galerkin model all interactions with with outside the domain outside the domain (-N,N) are set to zero (-N,N) are set to zero → N -N Yes No kxkx kyky

5. 23-28 September 2003Basic Processes in Turbulent Plasmas kyky kxkx v Fluid subspace: trivial subspace v=±b kyky kxkx Alfvénic subspaces: fixed points kyky kxkx b Magnetic subspaces: fixed points The simplified system displays an interesting property: time invariant subspaces; they play a crucial role in the dynamical behaviour of solutions.

6. 23-28 September 2003Basic Processes in Turbulent Plasmas v,b kyky kxkx K subspaces: fixed points kyky kxkx kyky kxkx kxkx v b v b P x and P y subspaces: fixed points the fields lie on defined parity wave vectors

7. Dynamical behaviour of the system near the invariant subspace Ideally stable subspaces Ideally stable subspaces Ideally unstable subspaces Ideally unstable subspaces We want to discuss what happens when we add a small perturbation to a solution which belongs to a particular invariant subspace I α. subspace t=0 subspace t=0

8. Subspace trivially stable: no dynamo effect in 2D; a seed of magnetic field cannot rise Fluid subspace R decreases in time Fluid subspace is an attractor: the distance from subspace decreases Selective dissipation for E int, E ext E ext is dissipated faster then E int

9. 23-28 September 2003Basic Processes in Turbulent Plasmas K(2,2) subspace Selective dissipation for E int, E ext E int is dissipated faster then E ext R increases in time K(2,2) subspace is a repeller: the distance from subspace increases kyky kxkx (2,2)

10. 23-28 September 2003Basic Processes in Turbulent Plasmas Statistical analysis Predictability of the dynamical behaviour It’s possible to predict the final stages of the free decay solutions which start near the invariant subspace. It’s possible to predict the final stages of the free decay solutions which start near the invariant subspace. We wonder now if it is possible to extend this kind of predictability to arbitrary starting conditions. We wonder now if it is possible to extend this kind of predictability to arbitrary starting conditions. We need a rule which relates each initial condition to a particular subspace. We associate with the nearest subspace. The criterion attributes a general Ψ(t) state to the α-th subspace which has the minimum external energy. The criterion attributes a general Ψ(t) state to the α-th subspace which has the minimum external energy. We calculate the belonging probability to a given subspace with energy selection criterion, using both ideal and dissipative runs. We calculate the belonging probability to a given subspace with energy selection criterion, using both ideal and dissipative runs.

11. The system selects, on ideal times, the subspace towards which it goes asymptotically. The system selects, on ideal times, the subspace towards which it goes asymptotically. t 1 =0 t 1 =1 t 1 =2 t 1 =20 t 2 =0 0.260.250.22 t 2 =1 0.260.840.87 t 2 =2 0.250.840.85 t 2 =20 0.220.870.85 t 1 =0 t 1 =1 t 1 =2 t 1 =20 t 2 =0 0.340.270.25 t 2 =1 0.340.720.62 t 2 =2 0.270.720.73 t 2 =20 0.250.620.73 Idelal runs Dissipative runs We report the correlation coefficients for four different values of time, t 1 e t 2 : probability to have both

12. 23-28 September 2003Basic Processes in Turbulent Plasmas Conclusions Conclusions There are some time-invariant subspaces important for the system dynamics. There are some time-invariant subspaces important for the system dynamics. Subspaces can be ideally stable (attactors) or ideally unstable (repellers). Subspaces can be ideally stable (attactors) or ideally unstable (repellers). The time evolution of the system can be investigated on the basis of stability properties of subspace. The time evolution of the system can be investigated on the basis of stability properties of subspace. Starting from a random initial condition, it’s statistically possible the solution forecasting. Starting from a random initial condition, it’s statistically possible the solution forecasting.