The turbulent cascade in the solar wind Luca Sorriso-Valvo LICRYL – IPCF/CNR, Rende, Italy R. Marino, V. Carbone, R. Bruno, P. Veltri, A. Noullez, B. Bavassano

Reynolds Number Turbulence L v0v0 l Analysis of longitudinal velocity differences

Since dissipation is efficient only at very small scales, the system dissipates energy by transferring it to small scales Nonlinear energy cascade (Richardson picture) Energy injection (  ) Non-linear energy transfer (  ) Energy dissipation (  ) Integral scale L Inertial range Dissipative scale l d eddies Energy cascade

R e = Non-linear Dissipative = v L E in E ou t E nl Energy balance power-law (observed universal exponent: -5/3) Navier-Stokes MHD Elsasser fields Energy injection (  ) Non-linear energy transfer (  ) Energy dissipation (  ) Integral scale L Inertial range Dissipative scale l d

This leads to the Kolmogorov scaling law Under the Kolmogorov hypothesis (K41) of constant energy transfer rate, the scaling parameter is h = 1/3 Phenomenology of fluid turbulence Kolmogorov spectrum Introducing the energy dissipation rate The characteristic time to realize the cascade (eddy- turnover time) is the lifetime of turbulent eddies

Phenomenological arguments for magnetically dominated MHD turbulence Since the Alfvén time might be shorter than the eddy-turnover time, nonlinear interactions are reduced and the cascade is realized in a time T When the flow is dominated by a (large-scale) magnetic field, there is one more characteristic time, the Alfvén time  A, related to the sweeping of Alfvénic fluctuations A different scaling relation for the pseudo-energies transfer rates Kraichnan spectrum

Belcher and Davis, JGR, 1971 Energy cascade needs both z+ and z- fluctuations for the non-linear term to exist. Is the turbulent spectrum compatible with the observed Alfvénic fluctuations ? Observations indicate that one of the Elsasser fluctuations is approximately zero (Alfvénic turbulence), thus the turbulent non-linear Energy cascade should be inhibited. A puzzle for MHD turbulence z+z+ z-z- Elsasser variables fluctuations The energy cascade is due to the nonlinear term of MHD equations Nonlinear interactions occur between fluctuations propagating in opposite direction with respect to the magnetic field.

Evidences of power spectrum in the solar wind attributed to fully developed MHD turbulence Coleman, ApJ, 1968 Bavassano et al., JGR 1982 Open question about the existence of a MHD turbulent energy cascade DYNAMIC ALIGNMENT: Dobrowolny et al. (PRL, 1980) proposed a possible “solution” of the puzzle: if the two Elsasser fields have the same energy transfer rate (same spectral slope), an initial small unbalance at injection scale (meaning quasi-correlated Alfvénic fluctuations) is maintained along a nonlinear cascade toward smaller scales. This is enough to explain the simultaneous observation of a turbulent spectrum and the presence of one single Alfvénic “mode”.

An exact law for incompressible MHD turbulent cascade: Politano & Pouquet Mixed third-order moment Large-scale inhomogeneities Pressure term (anisotropy) Dissipative term (vanishing in the inertial range) term including the pseudo- energy dissipation rate tensor From incompressible MHD equations, an exact relation can be derived for the mixed third-order moment assuming stationarity, homogeneity, isotropy (Kolmogorov 4/5) Politano & Pouquet, PRE 1998 Politano, Pouquet, Carbone,EPL 1998 Sorriso-Valvo et al, PRL 2007 IF THE YAGLOM RELATION IS VALID, A NONLINEAR ENERGY CASCADE MUST EXIST. THE RELATION IS THE ONLY EXACT AND NONTRIVIAL RESULT IN TURBULENCE

Numerical evidences Sorriso-Valvo et al, Phys. of Plasmas 2002 From 2-dimensional numerical simulation of MHD equations (1024X1024) A snapshot of the current j from the simulation in the statistically steady state

Ulysses data 1996: results High latitude (  > 35°) Running windows of 10 days (2000 data points each) have been used to avoid radial distance and latitudinal variations, as well as non-stationariety effects. 8 minutes averages of velocity, magnetic field and density are used to build the Elsasser fields Z ± Low solar activity ( )

The Politano & Pouquet relation is satisfied in several Ulysses samples (polar, fast wind  high Alfvénic correlations!) Although the data may be affected by inhomogeneity, local anisotropy and compressible effects, the observed P&P scaling law is robust in most periods of Ulysses dataset. The first REAL evidence that (low frequency) solar wind can be described in the framework of MHD turbulence Sorriso-Valvo et al., PRL (2007) 1. Observation of inertial range scaling

The estimated values of energy transfer rates are about 100J/Kg sec. For comparison, energy transfer rates per unit mass in usual fluid flows are 1  50 J/Kg s 2. The energy transfer rates

3. The inertial range of SW turbulence EvEv EBEB f -5/3 The velocity spectrum extends to large scales, while magnetic spectral break is often (but not always) observed around a few hours. Radial velocity and magnetic field spectra for one sample of wind with Yaglom scaling 1h In literature: up to 6-12 hours. Our data: up to 1-2 days. What is the actual extent of the inertial range in SW? Velocity inertial range can locally extend up to 1-2 days! Seen from Yaglom law and from spectral properties. 1day

Magnetic contribution to energy transport Velocity contribution to energy transport 4. MHD or Navier-Stokes? The role of magnetic field It is possible to separate the contributions to the energy cascade: terms advected by velocity and terms advected by magnetic field. In this example, Yaglom scaling is dominated by the magnetic field at small scales (from 10 mins to 3h), but at large scale only the velocity advects energy Total energy transport Separation of scales around the Alfvén time

 NON UNIVERSALITY OF SOLAR WIND TURBULENCE! SMALL SCALES (< hours): Magnetic filed dominates or equilibrium: MHD cascade LARGE SCALES (> hours): Velocity dominates (Navier-Stokes cascade), or non- negligible magnetic field contribution (MHD cascade) Different behaviour in different samples

5. Compressive turbulence in solar wind: phenomenological scaling law Energy-transfer rates per unit volume in compressible MHD Density-weighted Elsasser variables: what scaling law for third-order mixed moment? Kowal & Lazarian, ApJ 2007, Kritsuk et al., ApJ 2007, Carbone et al., PRL 2009 Introducing phenomenological variables, dimensionally including density Low-amplitude density fluctuations could play a role in the scaling law

5. Compressive turbulence in solar wind: observation from data Phenomenological compressible P&P enhanced scaling is observed, even in samples where the incompressible law is not verified.

Estimate of the heating rate needed for the solar wind: models for turbulence Vasquez et al., JGR 2007 Solar wind models  Adiabatic expansion, temperature should decrease with helioscentric distance Spacecraft measurements  Temperature decay is slower than expected from adiabatic expansion Is the measured turbulent energy flux enough to explain the observed non-adiabatic expansion of the solar wind? 6. Solar wind heating

Compressive and incompressive dissipation rates, compared with model wind heating (2 temperatures) estimated energy transfer rate: compressible case estimated energy transfer rate: incompressible cascade energy transfer rate required for the observed T

7. The role of cross-helicity Fast streams (high cross-helicity) have a weak turbulent cascade on only one mode. Slow streams, in which the two modes coexist and can exchange energy more rapidly, the energy cascade is more efficient (giving larger transfer rates) and is observable on both modes. This observation reinforces the scenario proposed by Dobrowolny et al. suggesting that MHD cascade is favoured in low cross-helicity samples. Cross-helicity plays a relevant role in the MHD turbulent cascade in solar wind. Fast: only one mode fast slow Slow: both modes

Conclusions 1.The MHD cascade exists in the solar wind. 2.First measurements of energy dissipation rate (statistical analysis in progress). 3.The extent of the inertial range is larger than previously estimated. 4.The role of magnetic field is relevant (but not always) for the cascade, especially at small scales. However, no universal phenomenology is observed (work in progress). 5.Density fluctuations, despite their low-amplitude, enhance the turbulent cascade (more theoretical results are needed). 6.Heating of solar wind could be entirely due to the compressible turbulent cascade. 7.Cross-helicity plays an important role in the scaling: Alfvénic fluctuations inhibit the cascade, in the spirit of dynamic alignment (work in progress). The turbulent cascade was observed in Ulysses solar wind data through the Politano & Poquet law, providing several results: