Fundamentals of magnetohydrodynamics

Transcription

1 Fndamentals of magnetohydrodynamics Part II Daniel Gómez 1, Webpage: astro.df.ba.ar (1 Institto de Astronomía y Física del Espacio, CONICET, Argentina ( Departamento de Física, Universidad de Benos Aires, Argentina ICTP-SAIFR, Sao Palo, Brazil, October 14, 013

2 The MHD eqations are: These eqations describe a large nmber of important plasma processes, sch as instabilities and wave propagación (Alfven and magnetosonic waves dynamo mechanisms to generate magnetic fields MHD trblence magnetic reconnection Note that even thogh the electric field is not present, it does not mean that it is not relevant πσ η σ 4, 1 1 c J B E c = = + 0, ( ( 4 1 ( ( ( 0 0 = + = = = = B B B t B F B B p t p p t visc ext η σ π ρ ρ ρ ρ ρ ρ γ MHD eqations

3 injection Flid trblence Energy cascade - energy flx toward high - vortex breadown Scale invariance - energy flx in : - energy power spectrm: ε τ E inertial range dissipation 1 τ, ε = τ const. Therefore E = ε Kolmogorov spectrm (K41

4 Simlations We integrate the MHD eqations nmerically, sing a spectral scheme in all three spatial directions (Gomez, Milano and Dmitr 000; also Dmitr, Gomez & Matthaes 003 We show reslts from 56x56x56 rns performed in (CAPS, or linx clster with 80 cores For the spatial derivatives, we se a psedo-spectral scheme with /3-dealiasing. Spectral codes are well sited for trblence stdies, since they provide exponentially fast convergence. Time integration is performed with a second order Rnge-Ktta scheme.the time step is chosen to satisfy the CFL condition.

5 Simlations: spatial integration We focs on Forier-Galerin methods. Let s ilstrate on Brgers eqation t + x = ν xx for (x,t on the interval 0 x < π the initial condition ( x,0 = ( x 0 assming periodic bondary conditions and We expand in a trncated Forier expansion N / N ( x, t = ( t = N / e ix Demanding zero projection of the soltion (x,t on the trncated Forier space t = ( x ν, ( x = im l+ m= l m This trncated expansion soltion as N N ( x, t converges exponentially fast to the exact However, it is comptationally very demanding, it involves O( N operations.

6 Simlations: spatial integration The FFT algorithm yields the discrete set { ˆ } from the set { ( x } j after O( N log N floating point operations. (x The strategy of compting spatial derivatives in Forier space and nonlinear terms in physical space, is nown as psedo-spectral, i.e. t j, x j = ( π = j, j = 0,,N 1 FFT N x ν, ( x = { ˆ, = N / + 1,,N / } FFT ( FFT 1 ( FFT 1 ( i The relation between discrete Forier coefficients ˆ = + m 0 + Nm { ˆ } and the continos ones is This sm cases a sprios effect nown as aliasing when compting nonlinear terms. Aliasing effects can be sppressed by applying the two-thirds rle, i.e. ˆ = 0, N 3

7 Simlations: temporal integration We advance the soltion throgh discrete time steps t i = i t In compact notation, if where F is a nonlinear and spatial differential operator, we se a second order Rnge-Ktta scheme. du = dt F( U, t We first advance half a step U i+ 1 = U i + t F( U i, t i and se U 1 i+ to jmp the whole step U i+ 1 i+ i = U + t F( U, ti This is second order accrate (i.e. O(( t. The size of the step is limited by the CFL condition, i.e t x / 0 for t = 0 x

8 Stretch-twist-fold dynamo Stretch-twist-fold (Vainshtein-Zeldovich 197 A.L=const. A.B=const..L.B L.B (a Magnetic flx tbe (b Stretching (frozen to inhomogeneos flows (c Twisting (helicoidal flows (d Folding (with reconnection bac to (a with twice as mch magnetic flx!!

9 Mean field theory It provides a qantitative expresion for the coefficient alpha. The first assmption is that there is a scale separation between the large scale magnetic field being generated and the small scale convective motions, i.e B B + b, U +, < b >= 0 =< > where <...> is an average over small scales. To compte the evoltion of the mean field, we average the indct ion eqation B t = ( U B + < b >, B = 0 The extra term can be interpreted as an electromotive force exerted by small scale motions ε EMF =< b > We still need to obtain an expresion for the electromotive force, and that reqires some assmptions ((Steenbec, Krase & Radler

10 b t Mean field theory Let s sbstract the averaged eqation from the general indction eqation = ( U b + ( B + ( b < b >, b = 0 [1] [] [1] Can be removed with a Galilean transformation. [] It s a departre from average of a second order qantity (FOSA. Let s frther assme that this system evolves in a typical correlation time of these small scale convective motions. Therefore ε = τ < ( B >= α B β B EMF where we neglected the gradient of the large scale magnetic field. For an isotropic state of these small scale flows, these tensors become τ α ij = < > δ 3 ij τ β ij = < > δ ij The inetic helicity of convective flows is important for dynamo activity.

11 MHD 3D dynamos From mean field theory (Krase & Radler 1980, we now that the trblent generation of 1 magnetic fields (the alpha effect is proportional to the inetic helicity of the flow. H = To stdy this mechanism throgh direct simlations, we externally drive the flow with a helical force at large scales (an ABC pattern, ntil a stationary trblent state is reached (Mininni, Gómez & Mahajan, 003, ApJ, 587, 47; Mininni, Gómez & Mahajan, 005, ApJ, 619, 1019 At that point, a magnetic seed is implanted at small scales and the 3D MHD eqations are evolved (Menegzzi, Frisch & Poqet E inetic positive negative magnetic F S ν The boxes show the intermittent spatial distribtion of positive and negative inetic helicity H, clearly displaying a net nbalance.

12 Energy power spectra t = 0 t = 0 The power spectrm of magnetic energy grows in time ntil it reaches eqipartition at each scale (Brandenbrg et al The Kolmogorov slope is also displayed for reference. The fll line is the inetic energy power spectrm and the dotted line is the total energy.

13 Energy power spectra t = 0 t = 0 The power spectrm of magnetic energy grows in time ntil it reaches eqipartition at each scale (Brandenbrg et al The Kolmogorov slope is also displayed for reference. The fll line is the inetic energy power spectrm and the dotted line is the total energy.

14 Trblent dynamos The image on the right shows the spatio-temporal distribtion of magnetic energy. The image below shows an initial exponential growth stage (inematic dynamo for the total magnetic energy. At later times it satrates when it reaches approximate eqipartition with the total inetic energy of the trblent flow. As predicted by MFT (Steenbec et al. 1966, inematic helicity (H at the microscale prodces magnetic field at macroscopic scales (large-scale dynamos.

15 Force-free eqilibria When the forcing is applied at intermediate scales, an accmlation of magnetic energy is observed at the largest scales. This behavior is cased by the inverse cascade of magnetic helicity. The magnetic field at large scales is approximately force-free, i.e. B // B Small scales, however, are consistent with a strongly trblent MHD regime. This configration can be representative of active regions of the solar corona, which are approximately force-free at large scales and at the same time are being heated by a strong MHD trblence at smaller scales (Gómez & F.Fontán 1988

16 Conclsions Using Mean Field Theory, we have seen that large-scale dynamos are generated by trblent helical flows, i.e. with a high concentration of inematic helicity. We briefly mentioned the main featres of the spectral codes sed in or nmerical simlations. In particlar, spectral schemes present exponential convergence as the nmber of grid points increases. We showed MHD 3D simlations of a trblent and helical flow. An initial magnetic seed is seen to exponentially grow and to satrate when it reaches approximate eqipartition with the inetic energy of the flow. When the flow is externally driven at intermediate scales, an accmlation of magnetic energy is observed at the largest scales in the box. Frthermore, the magnetic field at these large scales is well described by a force-free regime. This state is relevant to coronal active regions, which are observed to be close to force-free eqilibria. Yet, at smaller scales they might show a strongly trblent regime responsible for the heating of the plasma confined in these regions.