Forecasting asymptotic states of a Galerkin approximation of 2D MHD equations

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

2. 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 Basic Processes in Turbulent Plasmas

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

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

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

6. ky K subspaces: fixed points v,b kx Px and Py subspaces: fixed points the fields lie on defined parity wave vectors ky ky b v v kx b kx kx Basic Processes in Turbulent Plasmas

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

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

9. K(2,2) subspace ky (2,2) kx Selective dissipation for Eint , Eext Eint is dissipated faster then Eext R increases in time K(2,2) subspace is a repeller: the distance from subspace increases Basic Processes in Turbulent Plasmas

10. Predictability of the dynamical behaviour • 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. Statistical analysis • We need a rule which relates each initial condition to a particular subspace. • We associate with the nearest subspace. • We calculate the belonging probability to a given subspace with energy selection criterion, using both ideal and dissipative runs. • The criterion attributes a general Ψ(t) state to the α-th subspace which has the minimum external energy. Basic Processes in Turbulent Plasmas

11. We report the correlation coefficients for four different values of time, t1 e t2 : probability to have both Dissipative runs Idelal runs • The system selects, on ideal times, the subspace towards which it goes asymptotically.

12. Conclusions • There are some time-invariant subspaces important for the system dynamics. • 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. • Starting from a random initial condition, it’s statistically possible the solution forecasting. Basic Processes in Turbulent Plasmas