The workshop consisted in the cross-collaboration of the participants on the 11 projects listed below. The subjects range from numerical methods (6) to properties of real and simulated turbulence (5,8,9), with, in between, turbulence generation mechanisms, large scales (1,2) and small scales (11), and the relaxation of collisionless/dispersive turbulence in (3,7,10). Some subjects attracted more collaborators than others. However, everyone took profit of knowing in some detail what the others were working on.
 

1. List of projects

(1) MHD within shear flows - F. Lignieres and J. Leorat
(2) Forced helical turbulence in spherical geometry - Wolfgang Dobler
(3) Kinetic effects in the expanding solar wind: Petr Hellinger, Roland Grappin, Pavel Travnicek, Andre Mangeney
(4) Parametric instability of Alfven waves in the expanding wind.: L.Primavera and R.Grappin
(5) Origin of the magnetic excess in 3D incompressible MHD turbulence (continuation of last CIAS workshop): W. Mueller and R. Grappin
(6) High Order Schemes for turbulence: Tony Arber and Wolfgang Dobler
(7) Filamentation of dispersive Alfven waves: Andre Mangeney, Petr Hellinger, Pavel Travnicek
(8) Finite size scaling of the magnetic field magnitude from the simulated MHD turbulence: Bogdan Hnat and Tony Arber.
(9) The Lyapunov spectrum of WIND--MFI data: B. Hnat and L. Nocera
(10) Turbulence in the DNLS-Burgers equation: L. Nocera
(11) Electron acceleration in a turbulent electrostatic wave field: Filippo Pantellini and Petr Hellinger
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2. Reports on projects

(1) MHD within shear flows
Collaborators: F. Lignieres and J. Leorat

Our objective is to study the non linear dynamo action by shear turbulence in stably stratified atmosphere like stellar radiative zone. Such shear
flows occur within a flow at larger (stellar) scale, not represented in the simulations. Some modelisation of the effects of the large scale flow,
considered as given, on the simulated scales of the shear is thus needed. This may be done for example, by applying a uniform rotation or
by imposing the mean flow to keep a given uniform gradient.
During this week, we considered two types of shear flows in slab geometry and started to experiment numerically with a code solving the non linear
MHD equations.
a) In the first configuration a shear layer anticyclonic with respect to the local rotation may be unstable at a rather low Reynolds number (cf Lezius & Johnston, 1996, JFM, 77,153-175). We have checked the occurence of the linear instability (the resulting flow is 2D). We have now to verify if the flow in non linearly saturated regime is an efficient kinematic dynamo (for a related example, see P.C. Matthews' work)
b) In the second configuration , turbulence is maintained by an uniform shear (see for example Schumacher JFM,441,109-118, 2001). The
numerical study has just started, and stratification effects will be examined next.
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(2) Forced helical turbulence in spherical geometry
Collaborators: Wolfgang Dobler

Magnetic helicity conservation is considered one of the big problems for cosmic turbulent dynamos, which are operating at very large magnetic
Reynolds numbers. The magnetic helicity can only evolve on the diffusive time scale, which is determined by the molecular magnetic diffusivity, not a turbulent
diffusivity, and the corresponding timescales are extremely long for stars or galaxies.
The project is dedicated to investigating the helicity balance (production + flux) for spherical turbulent dynamos. Currently the dynamo is realised
by imposing, within the sphere, a stochastic helical forcing at a given small wavenumber. The velocity field quickly evolves into a helical state
with positive sign of helicity in one hemisphere and negative in the other (just as the helicity of the imposed forcing).
The magnetic field generated by this flow is quite asymmetrical and has prominent small structures even after the (statistically) steady state is
achieved.
A comparison with a simulation where the forcing has the same sign of helicity in the whole sphere shows that in this latter case (after an
almost identical phase of linear growth) the field grows faster and attains a considerably higher steady-state value than in the case
discussed before. This difference is most probably due to two effects, the different efficiency of magnetic field generation (which manifests itself in the
different critical magnetic Reynolds numbers for the corresponding mean-field dynamo models with symmetric or antisymmetric alpha-effect),
and the exchange of magnetic flux and magnetic helicity through the equatorial plane. The next step is to disentangle these two effects, which requires us to adapt
tools from slab simulations for calculating magnetic helicity and its flux.
In the long perspective,, the addition of gravity, rotation and an energy equation with central heating and surface cooling will allow us to drop the
forcing altogether and to generate helical turbulence and the accompanying magnetic fields in a much more realistic fashion.
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(3) Kinetic effects in the expanding solar wind.
Collaborators: Petr Hellinger, Roland Grappin, Pavel Travnicek and Andre Mangeney

We consider hybrid simulation taking into account solar wind expansion. For different plasma parameters we consider how the ideal plasma system
could stop adiabatic perpendicular cooling that results from a radial expansion. In the case of cold plasma there is no way to stop the
cooling while for a hot plasma fire hose instabilities are able to transfer energy from parallel to perpendicular direction and to stop
perpendicular cooling. The situation is strongly modified when we add to the plasma a population of alpha particles with a drift velocity
with respect to the protons. In this case the ratio of the drift velocity to the local Alfven velocity increases during the expansion.
When the ratio exceeds a certain treshhold (between 1 and 2) various instabilities appear that slow down the alphas and
heat alphas and protons. The generated waves are able to stop the adiabatic cooling of protons even for the
cold plasma. Such instabilities may explain the observation that show that solar wind protons are not adiabatic.
Another process we considered during the workshop is bringing the energy from large to small resonant scales: we considered a shear (parallel
to the mean field). We currently are considering two cases:
a) almost all energy is within the shear
b) the shear and the wave have comparable energies.
The first case corresponds to waves within large-scale stream shear; the second to parallel Alfven waves interacting with perpendicular
(non propagating) Alfven waves, which might lead to regimes resembling turbulence.
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(4) Parametric instability of Alfven waves in the expanding wind.
Collaborators: L.Primavera and R.Grappin

A steady rise of z-/z+ (the ratio of reflected over outward propagating Alfven wave energy) with increasing heliocentric distance is observed
by Ulysses, with a saturation at about 2.2 AU. A comparable evolution has also been obtained via simulations starting with
circularly polarized Alfven waves propagating in a homogeneous medium, subject to parametric instability. The question is whether this mechanism
still works within the expanding solar wind. This question is related to the theme of the relative importance of the compressible and incompressible
components in this turbulent medium. It is clear that even a circularly polarized Alfven wave propagating within an expanding medium leads to
compressive fluctuations. However, at the same time, expansion slows down significantly nonlinear coupling, as has been shown in the early work by
Grappin Mangeney Velli 1993 and Grappin & Velli 1994.
We used the expanding box model (coordinates are comobile with a given uniform radial wind) to study first an all-radial case (mean field, as well as
wavevector are radial). For reasonable parameter values (beta=0.1, expansion parameter 0.1, corresponding to 6 hours waveperiods), the
numerical simulations lead to a growth rate (measured in wave frequency) comparable to the unexpanding case. This is not unexpected in the radial
case, as both the relative wave amplitude and plasma beta increase with distance almost at the same rate.
We plan to study soon the more realistic non-radial case where both the wavevector and mean field rotate (in opposite directions) when distance
increases, so that the wave no longer remains a circularly polarized wave.
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(5) Origin of the magnetic excess in 3D incompressible MHD turbulence (continuation of last CIAS workshop)
Collaborators: W. Mueller and R. Grappin

Simulations by Biskamp and Mueller (2000) found that decaying turbulence starting with equipartition between magnetic and kinetic energies and
with no mean magnetic field give rise to a Kolmogorov slope for the total (magnetic+kinetic) energy spectrum. Our aim is to use a simple approach to obtain a phenomenology of the process leading to the magnetic excess. We find that the case with no magnetic helicity is well explained by a balance between the Alfven relaxation (linear propagation within the local mean field, that is basically a non-local coupling in Fourier space) and the dynamo effect, estimated as a local coupling.
A problem remains for the case with large-scale magnetic helicity. In that case, the magnetic excess is much enhanced, and not only in the large scales where the magnetic helicity is confined, but also at medium scales. Hence, classical (EDQNM closure) estimations of the effects of helical terms cannot explain the large magnetic excess at medium scales. We tested an hypothesis of non WKB propagation in which the large scale helical structures are fixed, but a wave propagating in this structure shows no more magnetic excess than without helicity. We are thus left with the conclusion that a large-scale magnetic excess leads to a cascade with reduced Alfven effect, even though the (isotropic) Alfven time is significantly smaller than the nonlinear time. The only explanation is that the average Alfven time is larger than the
isotropic average, due to anisotropies in spectral space, most structures having their wavevectors perpendicular to the local mean field. This anisotropy would be larger when large scales have a large magnetic helicity. This idea has to be tested in detail.
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(6) High Order Schemes for turbulence
Collaborators: Tony Arber and Wolfgang Dobler

High order finite difference schemes have the advantage over spectral codes of being easy to code and parallelise. For compressible simulations in which shocks may form spectral techniques are also unsuitable. The aim was to evaluate the effectiveness of a 6th order in space, third order in time finite difference code for strongly driven turbulence. This has high accuracy for smooth regions but is capable of employing localised shock viscosity where needed. Results were compared directly with those from a 2nd order shock code based on Lagrangian remap techniques. By testing the finite difference code on simple tests such as the Brio-Wu shock we determined that upwind biasing, shock resistivity and some thermal conduction were required to remove all false oscillations. With these parameters set the code was tested on a volume forced turbulence simulation where the forcing was through delta-correlated, large amplitude Beltrami waves. The code does work for this problem. Hence, the same high order scheme can be used for problems ranging from linear waves, weakly non-linear interactions and strongly driven turbulence. However, for the strongly driven turbulence test the results were no more accurate than those from the shock capturing code which suggests that the optiminal strategy for strongly non-linear, compressible turbulence is probably a PPM or ENO based scheme. The high order finite difference scheme is freely availably to all participants at the workshop for future work.
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(7) Filamentation of dispersive Alfven waves
Collaborators: Andre Mangeney, Petr Hellinger and Pavel Travnicek

We consider the filamentation of a dispersive Alfven wave in low beta plasma. This work is motivated by Cluster II observations behind a low Mach number quasi-perpendicular shock. This study is done in two steps. First, we consider a nearly perpendicular shock with a relatively low beta and we study the formation of the Alfven wave behind the shock with a two-dimensional (2-D) hybrid code. Second, we have started a detailed study of the filamentation process using 2-D (and in the future 3-D) hybrid code: we start with a homogeneous plasma with a single, left handed, circularly polarised Alfven wave. The Alfven wave is unstable in the transverse direction: The filamentation process leads to magnetosonic waves perpendicular to the Alfven wave propagation.
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(8) Finite size scaling of the magnetic field magnitude from the simulated MHD turbulence.
Collaborators: Bogdan Hnat and Tony Arber.

Statistical properties of the interplanetary magnetic field fluctuations can provide an important insight into the solar wind turbulent cascade. Analysis of the Probability Density Functions (PDF) of the velocity and magnetic field fluctuations has shown that these exhibit non-Gaussian properties on small time scales while large scale features appear to be uncorrelated. Recently we have applied the finite size scaling technique to explore the scaling of the magnetic field energy density fluctuations as seen
by WIND. We have found a single scaling sufficient to collapse the curves over the entire investigated range. The rescaled PDF followed a non Gaussian distribution with asymptotic behavior well described by the Gamma distribution arising from a finite range Levy walk. Such mono scaling suggests that a Fokker-Planck approach can be applied to study the PDF dynamics. This finite size scaling technique has never been applied to the simulated MHD turbulence before. During the workshop we applied this method to the magnetic field magnitude time series and compared it with the results obtained from the real observations. Interestingly, we identified a dual scaling (in time) which, in general, resembles that found in the observed data from the real driven systems such as the magnetosphere. The PDFs for small scales are consistent with the Levy distribution with alpha index of about 1.32 while the large scale PDFs are subdiffusive with the scaling index of about 0.3. The time scale where this break occurs can be related to the turnover time of the smallest driven structure in the simulation. This result will be investigated further by studying different driving methods in the simulation. Finally, we will attempt to apply the results of this analysis to the physical turbulent systems where the dual scaling has been identified.
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(9) The Lyapunov spectrum of WIND--MFI data
Collaborators: B. Hnat and L. Nocera

This project is aimed at applying modern concepts and techniques of signal data analysis to the time series provided by the observations
of magnetic field fluctuations in the solar wind as performed by the MFI probe on WIND; it was conceived entirely during the last week of
the workshop and it is part and complement to the project on the computation of the PDF of the same data.
Specifically, our task is to compute the whole spectrum of Lyapunov exponents of the time series using the technique of phase space
reconstruction and to estimate the information dimension of the series: this latter point is a contribution to the debate on whether
solar wind fluctuations are to be regarded as deterministic chaos (finite fractal dimension) or rather as infinite dimensional stochastic noise.
The large amount of records in the series (N>10^6, sampled at an interval of 92 sec for more than three years) provides an excellent
opportunity also to test several aspects of the phase space reconstruction technique in situations of large embedding dimensions
D, viz Eckman's 'N>10^D' conjecture, Kaplan and Yorke's information dimension formula and Hardy's and linear reconstruction algorithms.
Preliminary results with embedding dimensions as large as 7 show that all but one of the Lyapunov exponents are positive and that the information dimension of the time series should be larger than 6.
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(10) Turbulence in the DNLS-Burgers equation
Collaborators: L. Nocera

This project is motivated by the numerical simulations of quasi-parallel propagating Alfven waves in the solar wind which can be described by the Derivative Nonlinear Schrodinger Burgers (DNLS-Burgers) Equation. Specifically, during these simulations, a turbulent state appears to converge to a state where the magnetic field is mostly concentrated into isolated peaks. These peaks, or 'structures', look strikingly similar to the solitons of the DNLS equation with zero dissipation, though of course, these solitons cannot solve the complete DNLS-Burgers equation.
To explain this state of affairs, we conjecture that those structures appear due to a relaxation process much in the same way as it happens for two dimensional turbulence, according to Taylor's hypothesis. We verified that the properties hold true for the DNLS-Burgers equation:
1. existence of nonlinear invariants; 2. existence of selective dissipation of invariants; 3. existence of an inverse cascade.
In particular we verified that hypothesis 2 is actually met by applying the general perturbation theory of solitons to the DNLS-Burgers equation; we also verified numerically that this property holds true by
actually computing the invariants for the specific case in which an initial non vanishing oblique algebraic soliton state was evolved according to the DNLS-Burgers equation. Though proving  hypothesis 3 was not possible on general grounds, we took space spectra out of the simulations of the DNLSB equation and found that the inverse cascade process indeed exists in the specific situation of turbulence evolving out of the same algebraic soliton state mentioned above.
Last, using techniques of variational calculus, we found that the distribution of the magnetic field is given by the 'hyperbolic vanishing soliton', which was finally observed in our direct numerical simulations of the DNLS-Burgers equation.In conclusion, we proved that turbulence in the DNLS-Burgers equation can evolve into self-organized states.
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(11) Electron acceleration in a turbulent electrostatic wave field
Collaborators: Filippo Pantellini and Petr Hellinger

The plasma of the upper solar chromosphere is characterized by the fact that the collisional mean free path is large compared to the typical wave length of the electrostatic plasma waves. Under such conditions the waves may stochastically accelerate (Fermi mechanism) a fraction of the electrons to high energy enabling them to escape upward in the solar atmosphere. If the acceleration mechanism is efficient enough to overcome the thermalizing effect of collisions a non thermal electron population is generated. The presence of non thermal electron populations in the solar atmosphere can substantially affect the transport properties of the atmospheric plasma. For example, transport properties are of fundamental importance in the problem of the heating of the solar corona.
During this workshop we could define what essential ingredients a numerical code must have in order to simulate the acceleration. For example, what local characteristics and what kind of height dependence shall one assume for the spectrum. A working fortran version of the code has been written during the workshop. The code still needs some testing, but the first, and most important steps, have been successfully accomplished during the workshop.
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