1 Introduction

The interplanetary medium is permeated by a supersonic, highly turbulent plasma flowing out from the solar corona, the solar wind (Tu et al. 1995; Bruno et al. 2005). The turbulent character of the flow, at frequencies below the ion gyrofrequency \(f_{ci}\simeq 1\) Hz, has been invoked since the first Mariner mission (Coleman et al. 1968). In fact, velocity and magnetic fluctuations power spectra are close to the Kolmogorov’s −5/3 law (Bruno et al. 2005; Frisch 1995). However, even though fields fluctuations are usually considered within the framework of magnetohydrodynamic (MHD) turbulence (Bruno et al. 2005), a firm established proof of the existence of an energy cascade, the main characteristic of turbulence, remains a conjecture so far (Dobrowolny et al. 1980). Here we show, comparing data analysis to new theoretical results, that solar wind fields are in a state of fully developed turbulence. It is possible to derive, for the MHD case, the only exact, nontrivial theoretical result on turbulence, namely a relation between the third order moment of the longitudinal increments of the fields and the increment scale (Kolmogorov 1941). Using Ulysses spacecraft measurements, we have observed the existence of such relation in solar wind. This observation firmly puts low frequency solar wind fluctuations within the framework of MHD turbulence. Since solar wind is the only natural plasma accessible for in situ measurements, the importance of our finding stands beyond the understanding of the basic physics of solar wind turbulence.

Incompressible MHD equations are more complicated than the standard neutral fluid mechanics equations because the velocity of the charged fluid is coupled with the magnetic field generated by the motion of the fluid itself. However, written in terms of the Elsässer variables defined as \({\textbf{\textit{z}}}^\pm = {\textbf{\textit{v}}} \pm (4\pi\rho)^{-1/2} {\textbf{\textit{b}}}\) (v and b are the velocity and magnetic field, respectively, and ρ the mass density), they have the same structure as the Navier-Stokes equations (Dobrowolny et al. 1980)

$$ \begin{aligned} \partial_t {{\textbf{\textit{z}}}}^\pm+{{\textbf{\textit{z}}}}^\mp \cdot {\varvec \nabla}{{\textbf{\textit{z}}}}^\pm =& -{\varvec \nabla} P/\rho+ \left({\nu+\kappa \over 2}\right) \nabla^2 {{\textbf{\textit{z}}}}^\pm \\ +&\,\left({\nu-\kappa \over 2}\right) \nabla^2 {{\textbf{\textit{z}}}}^\mp,\\ \end{aligned} $$
(1)

where P is the total hydromagnetic pressure, ν is the viscosity and κ the magnetic diffusivity. In particular, the nonlinear term appears as \({\bf z}^\mp \cdot {\varvec \nabla}{\textbf{\textit{z}}}^\pm\) , suggesting the form of a transport process, in which Alfvénic MHD fluctuations z ± propagating along the background magnetic field are transported by fluctuations z propagating in the opposite direction. This transport is active as z ± and z are clearly not independent. Still, following the same procedure as in (Antonia et al. 1997; Danaila et al. 2001), and assuming local homogeneity, a relation similar to the Yaglom equation for the transport of a passive quantity (Monin and Yaglom 1975) can be obtained in the stationary state11

$$ \begin{aligned} \partial_\parallel Y^\pm(r)&=-{4 \over 3} \epsilon^\pm + 2\nu \nabla^2 \left\langle |\Updelta {\textbf{\textit{z}}}^\pm|^2 \right\rangle\\ & -2\left\langle \Updelta {\textbf{\textit{z}}}^\pm \cdot ({\varvec \nabla}+{\varvec \nabla}')\Updelta P/\rho \right\rangle \\ & + \left\langle {\textbf{\textit{z}}}^\mp \cdot ({\varvec \nabla}+{\varvec \nabla}')|\Updelta {\textbf{\textit{z}}}^\pm|^2 \right\rangle. \end{aligned} $$

Here, \(\Updelta {\textbf{\textit{z}}}^\pm \equiv {\textbf{\textit{z}}}^\pm({\textbf{\textit{x}}}')-{\textbf{\textit{z}}}^\pm({\textbf{\textit{x}}})\) are the (vector) increments of the fluctuations between two points \({\textbf{\textit{x}}}\) and \({\textbf{\textit{x}}}' \,{\equiv}\,{\textbf{\textit{x}}}+{\textbf{\textit{r}}},{\varvec \nabla}\) and \({\varvec \nabla}'\) are the gradients at the corresponding two points, ∂ is the longitudinal derivative along the separation r, while Y ±(r) are the mixed third order structure function \(\langle |\Updelta {\textbf{\textit{z}}}^\pm|^2\, \Updelta z^\mp_\parallel \rangle\) and \(\epsilon^{\pm} \equiv \nu\, \left\langle |{\varvec\nabla}{\textbf{\textit{z}}}^\pm|^2 \right\rangle \stackrel{\rm hom}{=} 3\nu\, \left\langle |\partial_\parallel\, {\textbf{\textit{z}}}^\pm|^2 \right\rangle\) are the pseudo-energy average dissipation rates, namely the dissipation rates of both \(\left\langle |{\textbf{\textit{z}}}^\pm|^2 \right\rangle /2,\) respectively. Finally, ΔP represent the increment of the total pressure fluctuations and the kinematic viscosity ν is here assumed to be equal to the magnetic diffusivity κ (this last assumption is in fact not necessary if we concentrate on the inertial range, as we will do from now on). The last term on the r.h.s. of equation (2) is related to large-scale inhomogeneities and disappears if the flow is globally homogeneous. Also, assuming local isotropy, the term containing pressure correlation vanishes, so that after longitudinal integration of (2) and in the inertial range of MHD turbulence (i.e. when ν → 0), a linear scaling law

$$ Y^\pm(r) = -{4 \over 3} \epsilon^\pm r $$

is obtained, characterizing a turbulent cascade with a well-defined finite energy flux ε±. An alternative derivation of this result using correlators instead of structure functions was also obtained in (Politano and Pouquet 1998), and observed in numerical simulations (Sorriso-Valvo et al. 2002).

In this work, we show that relation (3) is satisfied in solar wind turbulence. In order to avoid variations of the solar activity and ecliptic disturbances (like slow wind sources, Coronal Mass Ejections, ecliptic current sheet, and so on), we use high speed polar wind data measured by the Ulysses spacecraft (Smith et al. 1995; Balogh et al. 1995). In particular, we analyse here the first seven months of 1996, when the heliocentric distance slowly increased from 3 AU to 4 AU, while the heliolatitude decreased from about 55–30°. Since the wind speed in the spacecraft frame is much larger than the typical velocity fluctuations, and is nearly aligned with the R radial direction, time fluctuations are in fact spatial fluctuations with time and space scales (τ and r, respectively) related through the Taylor hypothesis, so that r = −〈v R 〉τ. From the 8 min averaged time series z ±(t), we compute the time increments Δz ±(τ;t) = z ±(t + τ)−z ±(t), and obtain the mixed third order structure function \(Y^\pm(-\langle v_R \rangle_t\, \tau) = \left\langle |\Updelta {\textbf{\textit{z}}}^\pm(\tau;t)|^2\, \Updelta z^\mp_R(\tau;t) \right\rangle_t\) using moving averages 〈•〉 t on the time t over periods spanning around 10 days, during which the fields can be considered stationary.

A linear scaling \(Y^\pm(\tau) = 4/3 \epsilon^\pm \langle v_R \rangle_t \tau\) is indeed observed in a significant fraction of the periods we examined, with an inertial range spanning up to two decades, indicating the existence of an energy cascade in plasma turbulence. This is the first experimental validation of the turbulence MHD theorem discussed above. Figure 1 shows two examples of scaling and the extension of the inertial range, for Y ±(τ). The linear scaling law generally extends from a few minutes to one day or more. This happens in 15 periods of a few days each in the 7 months considered. Several other periods are found in which the scaling range is considerably reduced. The sign of Y ±(τ) is observed to be either positive or negative. Since pseudo-energies dissipation rates are positive defined, a positive sign for Y ±(τ) (negative for Y ±(r)) indicates a (standard) forward cascade with pseudo-energies flowing towards the small scales to be dissipated. On the contrary, a negative Y ±(τ) suggests an inverse cascade where the energy flux is being transferred on average toward larger scales. Figure 2 shows the location of the most evident scaling intervals, together with the values of the flux rate ε± estimated through a fit of the scaling law (3), typically of the order of a few hundreds in J,kg−1s−1.

Fig. 1
figure 1

The scaling behaviour of Y ±(τ) as a function of the time scale τ for one of the periods we examined. Different curves refer to positive and negative values of the mixed structure functions Y ±(τ) and thus of ε±. The full black line correspond to a linear scaling.

Fig. 2
figure 2

Hourly averaged quantities are represented as a function of the flight time of Ulysses. The top panels represent, respectively, the solar wind speed, the magnitude of the magnetic field, the particle density, the distance from the sun and the heliolatitude angle. In the bottom panel the values of ε±, calculated through a fit with the function (3) during the periods where a clear linear scaling exists, are reported.

Scaling is not observed all the time within the solar wind. As already stated, Eq. 2 reduces to the linear law (3) only when local homogeneity, incompressiblity and isotropy conditions are satisfied. In general, solar wind inhomogeneities play a major role at large scales so that local homogeneity is generally fulfilled within the range of interest. Regarding incompressibility, it has been shown that compressive phenomena mainly affect shocked regions and dynamical interaction regions like stream-stream interface (Tu et al. 1995; Bruno et al. 2005). However, the time interval we analyze, because of Ulysses high latitude location, is not affected by these compressive phenomena (Gosling et al. 1995). On the other hand, it has also been shown (Bruno et al. 2005) that magnetic field compressibility increases mainly at very small scales within the fast wind regime. It follows that the incompressibility assumption can be considered valid to a large extent for the analyzed interval and at intermediate scales. The large scale anisotropy, mainly due to the average magnetic field, is only partially lost at smaller scales, and a residual anisotropy is always present (Horbury et al. 2001; Sorriso-Volvo et al. 2006), generally breaking the local isotropy assumption.

In conclusion, we observed, for the first time in the solar wind evidence of Yaglom MHD scaling law indicating the existence of a local energy cascade in hydromagnetic turbulence. The scaling holds in a number of relatively long periods of about 10 days, and also provides the first estimation of the pseudo-energy dissipation rate. Although our data might not fully satisfy requirements of homogeneity, incompressibility and isotropy everywhere, the observed linear scaling extends on a wide range of scales and appears very robust. The unexpected existence of the scaling law in anisotropic, weakly compressible and inhomogeneous turbulence still needs to be fully understood. Our result estabilishes a firm point within solar wind phenomenology, and, more generally, provides a better knowledge of plasma turbulence, carrying along a wide range of practical implications on both laboratory fusion plasmas and space physics.