"The Solar Wind as a Turbulence Laboratory"
Roberto Bruno and Vincenzo Carbone 
1 Introduction
1.1 What does turbulence stand for?
1.2 Dynamics vs. statistics
2 Equations and Phenomenology
2.1 The Navier–Stokes equation and the Reynolds number
2.2 The coupling between a charged fluid and the magnetic field
2.3 Scaling features of the equations
2.4 The non-linear energy cascade
2.5 The inhomogeneous case
2.6 Dynamical system approach to turbulence
2.7 Shell models for turbulence cascade
2.8 The phenomenology of fully developed turbulence: Fluid-like case
2.9 The phenomenology of fully developed turbulence: Magnetically-dominated case
2.10 Some exact relationships
2.11 Yaglom’s law for MHD turbulence
2.12 Density-mediated Elsässer variables and Yaglom’s law
2.13 Yaglom’s law in the shell model for MHD turbulence
3 Early Observations of MHD Turbulence in the Ecliptic
3.1 Turbulence in the ecliptic
3.2 Turbulence studied via Elsässer variables
4 Observations of MHD Turbulence in the Polar Wind
4.1 Evolving turbulence in the polar wind
4.2 Polar turbulence studied via Elsässer variables
5 Numerical Simulations
5.1 Local production of Alfvénic turbulence in the ecliptic
5.2 Local production of Alfvénic turbulence at high latitude
6 Compressive Turbulence
6.1 On the nature of compressive turbulence
6.2 Compressive turbulence in the polar wind
6.3 The effect of compressive phenomena on Alfvénic correlations
7 A Natural Wind Tunnel
7.1 Scaling exponents of structure functions
7.2 Probability distribution functions and self-similarity of fluctuations
7.3 What is intermittent in the solar wind turbulence? The multifractal approach
7.4 Fragmentation models for the energy transfer rate
7.5 A model for the departure from self-similarity
7.6 Intermittency properties recovered via a shell model
8 Observations of Yaglom’s Law in Solar Wind Turbulence
9 Intermittency Properties in the 3D Heliosphere: Taking a Look at the Data
9.1 Structure functions
9.2 Probability distribution functions
10 Turbulent Structures
10.1 On the statistics of magnetic field directional fluctuations
10.2 Radial evolution of intermittency in the ecliptic
10.3 Radial evolution of intermittency at high latitude
11 Solar Wind Heating by the Turbulent Energy Cascade
11.1 Dissipative/dispersive range in the solar wind turbulence
12 The Origin of the High-Frequency Region
12.1 A dissipation range
12.2 A dispersive range
13 Two Further Questions About Small-Scale Turbulence
13.1 Whistler modes scenario
13.2 Kinetic Alfvén waves scenario
13.3 Where does the fluid-like behavior break down in solar wind turbulence?
13.4 What physical processes replace “dissipation” in a collisionless plasma?
14 Conclusions and Remarks
A Some Characteristic Solar Wind Parameters
B Tools to Analyze MHD Turbulence in Space Plasmas
B.1 Statistical description of MHD turbulence
B.2 Spectra of the invariants in homogeneous turbulence
B.3 Introducing the Elsässer variables
C Wavelets as a Tool to Study Intermittency
D Reference Systems
D.1 Minimum variance reference system
D.2 The mean field reference system
E On-board Plasma and Magnetic Field Instrumentation
E.1 Plasma instrument: The top-hat
E.2 Measuring the velocity distribution function
E.3 Computing the moments of the velocity distribution function
E.4 Field instrument: The flux-gate magnetometer
F Spacecraft and Datasets

2 Equations and Phenomenology

In this section, we present the basic equations that are used to describe charged fluid flows, and the basic phenomenology of low-frequency turbulence. Readers interested in examining closely this subject can refer to the very wide literature on the subject of turbulence in fluid flows, as for example the recent books by, e.g., Pope (2000Jump To The Next Citation Point); McComb (1990); Frisch (1995Jump To The Next Citation Point) or many others, and the less known literature on MHD flows (Biskamp, 1993Jump To The Next Citation Point; Boyd and Sanderson, 2003; Biskamp, 2003Jump To The Next Citation Point). In order to describe a plasma as a continuous medium it will be assumed collisional and, as a consequence, all quantities will be functions of space r and time t. Apart for the required quasi-neutrality, the basic assumption of MHD is that fields fluctuate on the same time and length scale as the plasma variables, say ω τ ≃ 1 H and kLH ≃ 1 (k and ω are, respectively, the wave number and the frequency of the fields, while τH and LH are the hydrodynamic time and length scale, respectively). Since the plasma is treated as a single fluid, we have to take the slow rates of ions. A simple analysis shows also that the electrostatic force and the displacement current can be neglected in the non-relativistic approximation. Then, MHD equations can be derived as shown in the following sections.

2.1 The Navier–Stokes equation and the Reynolds number

Equations which describe the dynamics of real incompressible fluid flows have been introduced by Claude-Louis Navier in 1823 and improved by George G. Stokes. They are nothing but the momentum equation based on Newton’s second law, which relates the acceleration of a fluid particle2 to the resulting volume and body forces acting on it. These equations have been introduced by Leonhard Euler, however, the main contribution by Navier was to add a friction forcing term due to the interactions between fluid layers which move with different speed. This term results to be proportional to the viscosity coefficients η and ξ and to the variation of speed. By defining the velocity field u (r,t) the kinetic pressure p and the density ρ, the equations describing a fluid flow are the continuity equation to describe the conservation of mass

∂ ρ --- + (u ⋅ ∇ )ρ = − ρ∇ ⋅ u, (1 ) ∂t
the equation for the conservation of momentum
[ ] ∂u- 2 ( η) ρ ∂t + (u ⋅ ∇) u = − ∇p + η∇ u + ξ + 3 ∇ (∇ ⋅ u ), (2 )
and an equation for the conservation of energy
[ ] ( )2 ∂s- η- ∂ui- ∂uk- 2- 2 ρT ∂t + (u ⋅ ∇ )s = ∇ ⋅ (χ ∇T ) + 2 ∂xk + ∂xi − 3 δik∇ ⋅ u + ξ (∇ ⋅ u) , (3 )
where s is the entropy per mass unit, T is the temperature, and χ is the coefficient of thermoconduction. An equation of state closes the system of fluid equations.

The above equations considerably simplify if we consider the incompressible fluid, where ρ = const. so that we obtain the Navier–Stokes (NS) equation

( ) ∂u ∇p 2 -∂t + (u ⋅ ∇ )u = − -ρ-- + ν ∇ u, (4 )
where the coefficient ν = η∕ρ is the kinematic viscosity. The incompressibility of the flow translates in a condition on the velocity field, namely the field is divergence-free, i.e., ∇ ⋅ u = 0. This condition eliminates all high-frequency sound waves and is called the incompressible limit. The non-linear term in equations represents the convective (or substantial) derivative. Of course, we can add on the right hand side of this equation all external forces, which eventually act on the fluid parcel.

We use the velocity scale U and the length scale L to define dimensionless independent variables, namely ′ r = r L (from which ′ ∇ = ∇ ∕L) and ′ t = t(L∕U ), and dependent variables ′ u = u U and ′ 2 p = p U ρ. Then, using these variables in Equation (4View Equation), we obtain

∂u-′ ′ ′ ′ ′′ − 1 ′2 ′ ∂t′ + (u ⋅ ∇ )u = − ∇ p + Re ∇ u . (5 )

The Reynolds number Re = UL ∕ν is evidently the only parameter of the fluid flow. This defines a Reynolds number similarity for fluid flows, namely fluids with the same value of the Reynolds number behaves in the same way. Looking at Equation (5View Equation) it can be realized that the Reynolds number represents a measure of the relative strength between the non-linear convective term and the viscous term in Equation (4View Equation). The higher Re, the more important the non-linear term is in the dynamics of the flow. Turbulence is a genuine result of the non-linear dynamics of fluid flows.

2.2 The coupling between a charged fluid and the magnetic field

Magnetic fields are ubiquitous in the Universe and are dynamically important. At high frequencies, kinetic effects are dominant, but at frequencies lower than the ion cyclotron frequency, the evolution of plasma can be modeled using the MHD approximation. Furthermore, dissipative phenomena can be neglected at large scales although their effects will be felt because of non-locality of non-linear interactions. In the presence of a magnetic field, the Lorentz force j × B, where j is the electric current density, must be added to the fluid equations, namely

[ ] ∂u- 2 ( η-) 1-- ρ ∂t + (u ⋅ ∇ ) u = − ∇p + η∇ u + ξ + 3 ∇ (∇ ⋅ u ) − 4πB × (∇ × B), (6 )
and the Joule heat must be added to the equation for energy
[ ] ∂s- ∂ui- 2 --c2-- 2 ρT ∂t + (u ⋅ ∇ )s = σik ∂xk + χ ∇ T + 16π2 σ(∇ × B ), (7 )
where σ is the conductivity of the medium, and we introduced the viscous stress tensor
( ∂ui ∂uk 2 ) σik = η ----+ ---- − --δik∇ ⋅ u + ξ δik∇ ⋅ u. (8 ) ∂xk ∂xi 3

An equation for the magnetic field stems from the Maxwell equations in which the displacement current is neglected under the assumption that the velocity of the fluid under consideration is much smaller than the speed of light. Then, using

∇ × B = μ0j

and the Ohm’s law for a conductor in motion with a speed u in a magnetic field

j = σ (E + u × B ),

we obtain the induction equation which describes the time evolution of the magnetic field

∂B--= ∇ × (u × B ) + (1∕σ μ0)∇2B, (9 ) ∂t
together with the constraint ∇ ⋅ B = 0 (no magnetic monopoles in the classical case).

In the incompressible case, where ∇ ⋅ u = 0, MHD equations can be reduced to

∂u --- + (u ⋅ ∇ )u = − ∇Ptot + ν ∇2u + (b ⋅ ∇ )b (10 ) ∂t
∂b- 2 ∂t + (u ⋅ ∇ ) b = − (b ⋅ ∇ )u + η ∇ b. (11 )
Here Ptot is the total kinetic Pk = nkT plus magnetic pressure 2 Pm = B ∕8π, divided by the constant mass density ρ. Moreover, we introduced the velocity variables √ ---- b = B ∕ 4πρ and the magnetic diffusivity η.

Similar to the usual Reynolds number, a magnetic Reynolds number Rm can be defined, namely

cAL0 Rm = -----, η

where √ ---- cA = B0 ∕ 4πρ is the Alfvén speed related to the large-scale L0 magnetic field B0. This number in most circumstances in astrophysics is very large, but the ratio of the two Reynolds numbers or, in other words, the magnetic Prandtl number Pm = ν ∕η can differ widely. In absence of dissipative terms, for each volume V MHD equations conserve the total energy E (t)

∫ E (t) = (v2 + b2)d3r , (12 ) V
the cross-helicity Hc (t), which represents a measure of the degree of correlations between velocity and magnetic fields
∫ 3 Hc(t) = v ⋅ b d r, (13 ) V
and the magnetic helicity H (t), which represents a measure of the degree of linkage among magnetic flux tubes
∫ 3 H (t) = a ⋅ bd r, (14 ) V
where b = ∇ × a.

The change of variable due to Elsässer (1950Jump To The Next Citation Point), say z± = u ± b′, where we explicitly use the background uniform magnetic field b ′ = b + cA (at variance with the bulk velocity, the largest scale magnetic field cannot be eliminated through a Galilean transformation), leads to the more symmetrical form of the MHD equations in the incompressible case

∂z-± ± ( ∓ ) ± ± 2 ± ∓ 2 ∓ ± ∂t ∓ (cA ⋅ ∇ )z + z ⋅ ∇ z = − ∇Ptot + ν ∇ z + ν ∇ z + F , (15 )
where ± 2ν = ν ± η are the dissipative coefficients, and ± F are eventual external forcing terms. The relations ∇ ⋅ z± = 0 complete the set of equations. On linearizing Equation (15View Equation) and neglecting both the viscous and the external forcing terms, we have
± ∂z--∓ (cA ⋅ ∇ ) z± ≃ 0, ∂t

which shows that − z (x − cAt) describes Alfvénic fluctuations propagating in the direction of B0, and z+ (x + cAt) describes Alfvénic fluctuations propagating opposite to B0. Note that MHD Equations (15View Equation) have the same structure as the Navier–Stokes equation, the main difference stems from the fact that non-linear coupling happens only between fluctuations propagating in opposite directions. As we will see, this has a deep influence on turbulence described by MHD equations.

It is worthwhile to remark that in the classical hydrodynamics, dissipative processes are defined through three coefficients, namely two viscosities and one thermoconduction coefficient. In the hydromagnetic case the number of coefficients increases considerably. Apart from few additional electrical coefficients, we have a large-scale (background) magnetic field B0. This makes the MHD equations intrinsically anisotropic. Furthermore, the stress tensor (8View Equation) is deeply modified by the presence of a magnetic field B0, in that kinetic viscous coefficients must depend on the magnitude and direction of the magnetic field (Braginskii, 1965). This has a strong influence on the determination of the Reynolds number.

2.3 Scaling features of the equations

The scaled Euler equations are the same as Equations (4View Equation and 5View Equation), but without the term proportional to R −1. The scaled variables obtained from the Euler equations are, then, the same. Thus, scaled variables exhibit scaling similarity, and the Euler equations are said to be invariant with respect to scale transformations. Said differently, this means that NS Equations (4View Equation) show scaling properties (Frisch, 1995Jump To The Next Citation Point), that is, there exists a class of solutions which are invariant under scaling transformations. Introducing a length scale ℓ, it is straightforward to verify that the scaling transformations ℓ → λ ℓ′ and h ′ u → λ u (λ is a scaling factor and h is a scaling index) leave invariant the inviscid NS equation for any scaling exponent h, providing 2h ′ P → λ P. When the dissipative term is taken into account, a characteristic length scale exists, say the dissipative scale ℓD. From a phenomenological point of view, this is the length scale where dissipative effects start to be experienced by the flow. Of course, since ν is in general very low, we expect that ℓ D is very small. Actually, there exists a simple relationship for the scaling of ℓD with the Reynolds number, namely −3∕4 ℓD ∼ LRe. The larger the Reynolds number, the smaller the dissipative length scale.

As it is easily verified, ideal MHD equations display similar scaling features. Say the following scaling transformations u → λhu ′ and B → λ βB′ (β here is a new scaling index different from h), leave the inviscid MHD equations unchanged, providing 2β ′ P → λ P, 2h ′ T → λ T, and 2(β− h) ′ ρ → λ ρ. This means that velocity and magnetic variables have different scalings, say h ⁄= β, only when the scaling for the density is taken into account. In the incompressible case, we cannot distinguish between scaling laws for velocity and magnetic variables.

2.4 The non-linear energy cascade

The basic properties of turbulence, as derived both from the Navier–Stokes equation and from phenomenological considerations, is the legacy of A. N. Kolmogorov (Frisch, 1995Jump To The Next Citation Point).3 Phenomenology is based on the old picture by Richardson who realized that turbulence is made by a collection of eddies at all scales. Energy, injected at a length scale L, is transferred by non-linear interactions to small scales where it is dissipated at a characteristic scale ℓD, the length scale where dissipation takes place. The main idea is that at very large Reynolds numbers, the injection scale L and the dissipative scale ℓD are completely separated. In a stationary situation, the energy injection rate must be balanced by the energy dissipation rate and must also be the same as the energy transfer rate 𝜀 measured at any scale ℓ within the inertial range ℓD ≪ ℓ ≪ L. From a phenomenological point of view, the energy injection rate at the scale L is given by 2 𝜖L ∼ U ∕τL, where τL is a characteristic time for the injection energy process, which results to be τL ∼ L ∕U. At the same scale L the energy dissipation rate is due to 𝜖D ∼ U 2∕ τD, where τD is the characteristic dissipation time which, from Equation (4View Equation), can be estimated to be of the order of 2 τD ∼ L ∕ν. As a result, the ratio between the energy injection rate and dissipation rate is

𝜖L τD --- ∼ ---∼ Re , 𝜖D τL
that is, the energy injection rate at the largest scale L is Re-times the energy dissipation rate. In other words, in the case of large Reynolds numbers, the fluid system is unable to dissipate the whole energy injected at the scale L. The excess energy must be dissipated at small scales where the dissipation process is much more efficient. This is the physical reason for the energy cascade.

Fully developed turbulence involves a hierarchical process, in which many scales of motion are involved. To look at this phenomenon it is often useful to investigate the behavior of the Fourier coefficients of the fields. Assuming periodic boundary conditions the α-th component of velocity field can be Fourier decomposed as

∑ uα(r,t) = u α(k,t)exp(ik ⋅ r), k

where k = 2πn ∕L and n is a vector of integers. When used in the Navier–Stokes equation, it is a simple matter to show that the non-linear term becomes the convolution sum

∂u α(k,t) ∑ ---∂t---- = M αβγ(k ) u γ(k − q,t)uβ(q,t), (16 ) q
where M (k) = − ik (δ − k k ∕k2) αβγ β αγ α β (for the moment we disregard the linear dissipative term).

MHD equations can be written in the same way, say by introducing the Fourier decomposition for Elsässer variables

∑ z±α(r,t) = z±α (k,t)exp (ik ⋅ r), k

and using this expression in the MHD equations we obtain an equation which describes the time evolution of each Fourier mode. However, the divergence-less condition means that not all Fourier modes are independent, rather ± k ⋅ z (k,t) = 0 means that we can project the Fourier coefficients on two directions which are mutually orthogonal and orthogonal to the direction of k, that is,

∑2 z±(k,t) = z±a (k,t)e(a)(k ), (17 ) a=1
with the constraint that (a) k ⋅ e (k) = 0. In presence of a background magnetic field we can use the well defined direction B0, so that
ik × B0 ik e (1)(k) = ---------; e (2)(k) = --- × e(1)(k). |k × B0 | |k|

Note that in the linear approximation where the Elsässer variables represent the usual MHD modes, z± (k,t) 1 represent the amplitude of the Alfvén mode while z± (k,t) 2 represent the amplitude of the incompressible limit of the magnetosonic mode. From MHD Equations (15View Equation) we obtain the following set of equations:

[ ] ( )3 δ 2 ∂-- ± L-- ∑ ∑ ± ∓ ∂t ∓ i(k ⋅ cA) za (k,t) = 2π Aabc(− k, p,q )z b (p,t)zc (q, t). (18 ) p+q=kb,c=1
The coupling coefficients, which satisfy the symmetry condition Aabc(k,p, q) = − Abac(p,k,q ), are defined as
A (− k, p,q) = [(ik)⋆ ⋅ e(c)(q)][e (a)∗(k ) ⋅ e(b)(p )], abc

and the sum in Equation (18View Equation) is defined as

δ ( )3 ∑ 2π- ∑ ∑ ≡ L δk,p+q, p+q=k p q

where δk,p+q is the Kronecher’s symbol. Quadratic non-linearities of the original equations correspond to a convolution term involving wave vectors k, p and q related by the triangular relation p = k − q. Fourier coefficients locally couple to generate an energy transfer from any pair of modes p and q to a mode k = p + q.

The pseudo-energies ± E (t) are defined as

1 1 ∫ 1∑ ∑2 E ±(t) = ---- |z±(r,t)|2d3r = -- |za±(k,t)|2 2L3 L3 2 k a=1

and, after some algebra, it can be shown that the non-linear term of Equation (18View Equation) conserves separately E ±(t). This means that both the total energy E(t) = E+ + E − and the cross-helicity + − Ec (t) = E − E, say the correlation between velocity and magnetic field, are conserved in absence of dissipation and external forcing terms.

In the idealized homogeneous and isotropic situation we can define the pseudo-energy tensor, which using the incompressibility condition can be written as

( L )3 ⟨ ⟩ ( kakb ) U ±ab(k, t) ≡ --- z±a (k,t)z±b (k,t) = δab −---2- q±(k), 2 π k

brackets being ensemble averages, where ± q (k ) is an arbitrary odd function of the wave vector k and represents the pseudo-energies spectral density. When integrated over all wave vectors under the assumption of isotropy

[∫ ] ∫ ∞ 3 ± ± Tr d k Uab(k,t) = 2 0 E (k,t)dk,

where we introduce the spectral pseudo-energy E± (k,t) = 4πk2q± (k, t). This last quantity can be measured, and it is shown that it satisfies the equations

∂E-±(k,t)- ± 2 ± ± ∂t = T (k,t) − 2νk E (k, t) + F (k,t). (19 )
We use ν = η in order not to worry about coupling between + and − modes in the dissipative range. Since the non-linear term conserves total pseudo-energies we have
∫ ∞ dk T ±(k,t) = 0, 0

so that, when integrated over all wave vectors, we obtain the energy balance equation for the total pseudo-energies

dE ±(t) ∫ ∞ ∫ ∞ -------= dkF ±(k, t) − 2ν dk k2E ±(k,t). (20 ) dt 0 0
This last equation simply means that the time variations of pseudo-energies are due to the difference between the injected power and the dissipated power, so that in a stationary state
∫ ∫ ∞ ± ∞ 2 ± ± dk F (k,t) = 2ν dk k E (k,t) = 𝜖 . 0 0

Looking at Equation (19View Equation), we see that the role played by the non-linear term is that of a redistribution of energy among the various wave vectors. This is the physical meaning of the non-linear energy cascade of turbulence.

2.5 The inhomogeneous case

Equations (19View Equation) refer to the standard homogeneous and incompressible MHD. Of course, the solar wind is inhomogeneous and compressible and the energy transfer equations can be as complicated as we want by modeling all possible physical effects like, for example, the wind expansion or the inhomogeneous large-scale magnetic field. Of course, simulations of all turbulent scales requires a computational effort which is beyond the actual possibilities. A way to overcome this limitation is to introduce some turbulence modeling of the various physical effects. For example, a set of equations for the cross-correlation functions of both Elsässer fluctuations have been developed independently by Marsch and Tu (1989Jump To The Next Citation Point), Zhou and Matthaeus (1990Jump To The Next Citation Point), Oughton and Matthaeus (1992), and Tu and Marsch (1990aJump To The Next Citation Point), following Marsch and Mangeney (1987Jump To The Next Citation Point) (see review by Tu and Marsch, 1996Jump To The Next Citation Point), and are based on some rather strong assumptions: i) a two-scale separation, and ii) small-scale fluctuations are represented as a kind of stochastic process (Tu and Marsch, 1996Jump To The Next Citation Point). These equations look quite complicated, and just a comparison based on order-of-magnitude estimates can be made between them and solar wind observations (Tu and Marsch, 1996).

A different approach, introduced by Grappin et al. (1993), is based on the so-called “expanding-box model” (Grappin and Velli, 1996Jump To The Next Citation Point; Liewer et al., 2001Jump To The Next Citation Point; Hellinger et al., 2005). The model uses transformation of variables to the moving solar wind frame that expands together with the size of the parcel of plasma as it propagates outward from the Sun. Despite the model requires several simplifying assumptions, like for example lateral expansion only for the wave-packets and constant solar wind speed, as well as a second-order approximation for coordinate transformation Liewer et al. (2001) to remain tractable, it provides qualitatively good description of the solar wind expansions, thus connecting the disparate scales of the plasma in the various parts of the heliosphere.

2.6 Dynamical system approach to turbulence

In the limit of fully developed turbulence, when dissipation goes to zero, an infinite range of scales are excited, that is, energy lies over all available wave vectors. Dissipation takes place at a typical dissipation length scale which depends on the Reynolds number Re through −3∕4 ℓD ∼ LRe (for a Kolmogorov spectrum −5∕3 E (k) ∼ k). In 3D numerical simulations the minimum number of grid points necessary to obtain information on the fields at these scales is given by N ∼ (L ∕ℓD)3 ∼ Re9 ∕4. This rough estimate shows that a considerable amount of memory is required when we want to perform numerical simulations with high Re. At present, typical values of Reynolds numbers reached in 2D and 3D numerical simulations are of the order of 4 10 and 3 10, respectively. At these values the inertial range spans approximately one decade or a little more.

Given the situation described above, the question of the best description of dynamics which results from original equations, using only a small amount of degree of freedom, becomes a very important issue. This can be achieved by introducing turbulence models which are investigated using tools of dynamical system theory (Bohr et al., 1998). Dynamical systems, then, are solutions of minimal sets of ordinary differential equations that can mimic the gross features of energy cascade in turbulence. These studies are motivated by the famous Lorenz’s model (Lorenz, 1963) which, containing only three degrees of freedom, simulates the complex chaotic behavior of turbulent atmospheric flows, becoming a paradigm for the study of chaotic systems.

The Lorenz’s model has been used as a paradigm as far as the transition to turbulence is concerned. Actually, since the solar wind is in a state of fully developed turbulence, the topic of the transition to turbulence is not so close to the main goal of this review. However, since their importance in the theory of dynamical systems, we spend few sentences abut this central topic. Up to the Lorenz’s chaotic model, studies on the birth of turbulence dealt with linear and, very rarely, with weak non-linear evolution of external disturbances. The first physical model of laminar-turbulent transition is due to Landau and it is reported in the fourth volume of the course on Theoretical Physics (Landau and Lifshitz, 1971Jump To The Next Citation Point). According to this model, as the Reynolds number is increased, the transition is due to a infinite series of Hopf bifurcations at fixed values of the Reynolds number. Each subsequent bifurcation adds a new incommensurate frequency to the flow whose dynamics become rapidly quasi-periodic. Due to the infinite number of degree of freedom involved, the quasi-periodic dynamics resembles that of a turbulent flow.

The Landau transition scenario is, however, untenable because incommensurate frequencies cannot exist without coupling between them. Ruelle and Takens (1971Jump To The Next Citation Point) proposed a new mathematical model, according to which after few, usually three, Hopf bifurcations the flow becomes suddenly chaotic. In the phase space this state is characterized by a very intricate attracting subset, a strange attractor. The flow corresponding to this state is highly irregular and strongly dependent on initial conditions. This characteristic feature is now known as the butterfly effect and represents the true definition of deterministic chaos. These authors indicated as an example for the occurrence of a strange attractor the old strange time behavior of the Lorenz’s model. The model is a paradigm for the occurrence of turbulence in a deterministic system, it reads

dx- dy- dz- dt = Pr(y − x ), dt = Rx − y − xz , dt = xy − bz , (21 )
where x(t), y(t), and z(t) represent the first three modes of a Fourier expansion of fluid convective equations in the Boussinesq approximation, Pr is the Prandtl number, b is a geometrical parameter, and R is the ratio between the Rayleigh number and the critical Rayleigh number for convective motion. The time evolution of the variables x(t), y(t), and z(t) is reported in Figure 12View Image. A reproduction of the Lorenz butterfly attractor, namely the projection of the variables on the plane (x,z) is shown in Figure 13View Image. A few years later, Gollub and Swinney (1975) performed very sophisticated experiments,4 concluding that the transition to turbulence in a flow between co-rotating cylinders is described by the Ruelle and Takens (1971) model rather than by the Landau scenario.
View Image

Figure 12: Time evolution of the variables x(t), y (t), and z (t) in the Lorenz’s model (see Equation (21View Equation)). This figure has been obtained by using the parameters Pr = 10, b = 8∕3, and R = 28.
View Image

Figure 13: The Lorenz butterfly attractor, namely the time behavior of the variables z(t) vs. x(t) as obtained from the Lorenz’s model (see Equation (21View Equation)). This figure has been obtained by using the parameters Pr = 10, b = 8∕3, and R = 28.

After this discovery, the strange attractor model gained a lot of popularity, thus stimulating a large number of further studies on the time evolution of non-linear dynamical systems. An enormous number of papers on chaos rapidly appeared in literature, quite in all fields of physics, and transition to chaos became a new topic. Of course, further studies on chaos rapidly lost touch with turbulence studies and turbulence, as reported by Feynman et al. (1977), still remains …the last great unsolved problem of the classical physics. Furthermore, we like to cite recent theoretical efforts made by Chian and coworkers (Chian et al., 1998, 2003) related to the onset of Alfvénic turbulence. These authors, numerically solved the derivative non-linear Schrödinger equation (Mjølhus, 1976; Ghosh and Papadopoulos, 1987) which governs the spatio-temporal dynamics of non-linear Alfvén waves, and found that Alfvénic intermittent turbulence is characterized by strange attractors. Note that, the physics involved in the derivative non-linear Schrödinger equation, and in particular the spatio-temporal dynamics of non-linear Alfvén waves, cannot be described by the usual incompressible MHD equations. Rather dispersive effects are required. At variance with the usual MHD, this can be satisfied by requiring that the effect of ion inertia be taken into account. This results in a generalized Ohm’s law by including a (j × B¯) ¯-term, which represents the compressible Hall correction to MHD, say the so-called compressible Hall-MHD model.

In this context turbulence can evolve via two distinct routes: Pomeau–Manneville intermittency (Pomeau and Manneville, 1980) and crisis-induced intermittency (Ott and Sommerer, 1994). Both types of chaotic transitions follow episodic switching between different temporal behaviors. In one case (Pomeau–Manneville) the behavior of the magnetic fluctuations evolve from nearly periodic to chaotic while, in the other case the behavior intermittently assumes weakly chaotic or strongly chaotic features.

2.7 Shell models for turbulence cascade

Since numerical simulations, in some cases, cannot be used, simple dynamical systems can be introduced to investigate, for example, statistical properties of turbulent flows which can be compared with observations. These models, which try to mimic the gross features of the time evolution of spectral Navier–Stokes or MHD equations, are often called “shell models” or “discrete cascade models”. Starting from the old papers by Siggia (1977) different shell models have been introduced in literature for 3D fluid turbulence (Biferale, 2003). MHD shell models have been introduced to describe the MHD turbulent cascade (Plunian et al., 2012Jump To The Next Citation Point), starting from the paper by Gloaguen et al. (1985Jump To The Next Citation Point).

The most used shell model is usually quoted in literature as the GOY model, and has been introduced some time ago by Gledzer (1973) and by Ohkitani and Yamada (1989). Apart from the first MHD shell model (Gloaguen et al., 1985Jump To The Next Citation Point), further models, like those by Frick and Sokoloff (1998) and Giuliani and Carbone (1998Jump To The Next Citation Point) have been introduced and investigated in detail. In particular, the latter ones represent the counterpart of the hydrodynamic GOY model, that is they coincide with the usual GOY model when the magnetic variables are set to zero.

In the following, we will refer to the MHD shell model as the FSGC model. The shell model can be built up through four different steps:

a) Introduce discrete wave vectors:
As a first step we divide the wave vector space in a discrete number of shells whose radii grow according to a power kn = k0λn, where λ > 1 is the inter-shell ratio, k0 is the fundamental wave vector related to the largest available length scale L, and n = 1,2,...,N.

b) Assign to each shell discrete scalar variables:
Each shell is assigned two or more complex scalar variables un(t) and bn(t), or Elsässer variables Z ±(t) = un ± bn(t) n. These variables describe the chaotic dynamics of modes in the shell of wave vectors between k n and k n+1. It is worth noting that the discrete variable, mimicking the average behavior of Fourier modes within each shell, represents characteristic fluctuations across eddies at the scale ℓn ∼ k−n1. That is, the fields have the same scalings as field differences, for example Z ±n ∼ |Z ±(x + ℓn) − Z± (x)| ∼ ℓhn in fully developed turbulence. In this way, the possibility to describe spatial behavior within the model is ruled out. We can only get, from a dynamical shell model, time series for shell variables at a given kn, and we loose the fact that turbulence is a typical temporal and spatial complex phenomenon.

c) Introduce a dynamical model which describes non-linear evolution:
Looking at Equation (18View Equation) a model must have quadratic non-linearities among opposite variables Z ±(t) n and ∓ Zn (t), and must couple different shells with free coupling coefficients.

d) Fix as much as possible the coupling coefficients:
This last step is not standard. A numerical investigation of the model might require the scanning of the properties of the system when all coefficients are varied. Coupling coefficients can be fixed by imposing the conservation laws of the original equations, namely the total pseudo-energies

± 1-∑ || ±||2 E (t) = 2 Z n , n

that means the conservation of both the total energy and the cross-helicity:

1-∑ 2 2 ∑ ∗ E(t) = 2 |un | + |bn| ; Hc (t) = 2ℜe (unb n), n n

where ℜe indicates the real part of the product unb∗n. As we said before, shell models cannot describe spatial geometry of non-linear interactions in turbulence, so that we loose the possibility of distinguishing between two-dimensional and three-dimensional turbulent behavior. The distinction is, however, of primary importance, for example as far as the dynamo effect is concerned in MHD. However, there is a third invariant which we can impose, namely

2 ∑ n |bn|-- H (t) = (− 1 ) kα , (22 ) n n
which can be dimensionally identified as the magnetic helicity when α = 1, so that the shell model so obtained is able to mimic a kind of 3D MHD turbulence (Giuliani and Carbone, 1998Jump To The Next Citation Point).

After some algebra, taking into account both the dissipative and forcing terms, FSGC model can be written as

dZ±n-- ±∗ ν-±-μ- 2 + ν ∓-μ- 2 − ± dt = iknΦ n + 2 k nZn + 2 k nZn + F n , (23 )
( ) ( ) Φ ± = 2-−-a-−-c Z± Z ∓ + a +-c Z± Z ∓ + n 2 n+2 n+1 2 n+1 n+2 ( c − a ) ( a + c) + ----- Z±n−1Z ∓n+1 − ----- Zn∓− 1Z±n+1 + ( 2λ ) ( 2λ ) c − a ∓ ± 2 − a − c ∓ ± − ----2 Zn−2Z n−1 − -----2--- Zn− 1Z n−2, (24 ) 2λ 2 λ
where5 λ = 2, a = 1 ∕2, and c = 1∕3. In the following, we will consider only the case where the dissipative coefficients are the same, i.e., ν = μ.

2.8 The phenomenology of fully developed turbulence: Fluid-like case

Here we present the phenomenology of fully developed turbulence, as far as the scaling properties are concerned. In this way we are able to recover a universal form for the spectral pseudo-energy in the stationary case. In real space a common tool to investigate statistical properties of turbulence is represented by field increments ± ± ± Δz ℓ (r ) = [z (r + ℓ) − z (r)] ⋅ e, being e the longitudinal direction. These stochastic quantities represent fluctuations6 across eddies at the scale ℓ. The scaling invariance of MHD equations (cf. Section 2.3), from a phenomenological point of view, implies that we expect solutions where Δz ±ℓ ∼ ℓh. All the statistical properties of the field depend only on the scale ℓ, on the mean pseudo-energy dissipation rates 𝜀±, and on the viscosity ν. Also, ± 𝜀 is supposed to be the common value of the injection, transfer and dissipation rates. Moreover, the dependence on the viscosity only arises at small scales, near the bottom of the inertial range. Under these assumptions the typical pseudo-energy dissipation rate per unit mass scales as 𝜀± ∼ ( Δz ±)2∕t± ℓ ℓ. The time t± ℓ associated with the scale ℓ is the typical time needed for the energy to be transferred on a smaller scale, say the eddy turnover time ± ∓ tℓ ∼ ℓ∕Δz ℓ, so that

𝜀± ∼ (Δz ±)2 Δz ∓∕ℓ. ℓ

When we conjecture that both ± Δz fluctuations have the same scaling laws, namely ± h Δz ∼ ℓ, we recover the Kolmogorov scaling for the field increments

Δz ± ∼ (𝜀± )1∕3ℓ1∕3. (25 ) ℓ
Usually, we refer to this scaling as the K41 model (Kolmogorov, 1941Jump To The Next Citation Point, 1991Jump To The Next Citation Point; Frisch, 1995Jump To The Next Citation Point). Note that, since from dimensional considerations the scaling of the energy transfer rate should be 𝜀± ∼ ℓ1−3h, h = 1∕3 is the choice to guarantee the absence of scaling for 𝜀±.

In the real space turbulence properties can be described using either the probability distribution functions (PDFs hereafter) of increments, or the longitudinal structure functions, which represents nothing but the higher order moments of the field. Disregarding the magnetic field, in a purely fully developed fluid turbulence, this is defined as S(ℓp)= ⟨Δup ℓ⟩. These quantities, in the inertial range, behave as a power law S (p) ∼ ℓξp ℓ, so that it is interesting to compute the set of scaling exponent ξ p. Using, from a phenomenological point of view, the scaling for field increments (see Equation (25View Equation)), it is straightforward to compute the scaling laws S (ℓp) ∼ ℓp∕3. Then ξp = p∕3 results to be a linear function of the order p.

When we assume the scaling law Δz ± ∼ ℓh ℓ, we can compute the high-order moments of the structure functions for increments of the Elsässer variables, namely ⟨ ± p⟩ ξp (Δz ℓ ) ∼ ℓ, thus obtaining a linear scaling ξp = p∕3, similar to usual fluid flows. For Gaussianly distributed fields, a particular role is played by the second-order moment, because all moments can be computed from S (2) ℓ. It is straightforward to translate the dimensional analysis results to Fourier spectra. The spectral property of the field can be recovered from (2) Sℓ, say in the homogeneous and isotropic case

∫ ( ) (2) ∞ sinkℓ Sℓ = 4 E (k) 1 − --kℓ-- dk, 0

where k ∼ 1∕ℓ is the wave vector, so that in the inertial range where Equation (41View Equation) is verified

E(k) ∼ 𝜀2∕3k−5∕3. (26 )
The Kolmogorov spectrum (see Equation (26View Equation)) is largely observed in all experimental investigations of turbulence, and is considered as the main result of the K41 phenomenology of turbulence (Frisch, 1995Jump To The Next Citation Point). However, spectral analysis does not provide a complete description of the statistical properties of the field, unless this has Gaussian properties. The same considerations can be made for the spectral pseudo-energies E± (k), which are related to the 2nd order structure functions ⟨ [ ±]2⟩ Δz ℓ.

2.9 The phenomenology of fully developed turbulence: Magnetically-dominated case

The phenomenology of the magnetically-dominated case has been investigated by Iroshnikov (1963Jump To The Next Citation Point) and Kraichnan (1965Jump To The Next Citation Point), then developed by Dobrowolny et al. (1980bJump To The Next Citation Point) to tentatively explain the occurrence of the observed Alfvénic turbulence, and finally by Carbone (1993Jump To The Next Citation Point) and Biskamp (1993Jump To The Next Citation Point) to get scaling laws for structure functions. It is based on the Alfvén effect, that is, the decorrelation of interacting eddies, which can be explained phenomenologically as follows. Since non-linear interactions happen only between opposite propagating fluctuations, they are slowed down (with respect to the fluid-like case) by the sweeping of the fluctuations across each other. This means that ± ( ±)2 ± 𝜀 ∼ Δz ℓ ∕T ℓ but the characteristic time ± T ℓ required to efficiently transfer energy from an eddy to another eddy at smaller scales cannot be the eddy-turnover time, rather it is increased by a factor t±ℓ ∕tA (tA ∼ ℓ∕cA < t±ℓ is the Alfvén time), so that Tℓ± ∼ (t±ℓ )2∕tA. Then, immediately

± 2 ∓ 2 𝜀± ∼ [Δz-ℓ-][Δz-ℓ ]-. ℓcA

This means that both ± modes are transferred at the same rate to small scales, namely 𝜖+ ∼ 𝜖− ∼ 𝜖, and this is the conclusion drawn by Dobrowolny et al. (1980bJump To The Next Citation Point). In reality, this is not fully correct, namely the Alfvén effect yields to the fact that energy transfer rates have the same scaling laws for ± modes but, we cannot say anything about the amplitudes of + 𝜀 and − 𝜀 (Carbone, 1993Jump To The Next Citation Point). Using the usual scaling law for fluctuations, it can be shown that the scaling behavior holds 1− 4h ′ 𝜀 → λ 𝜀. Then, when the energy transfer rate is constant, we found a scaling law different from that of Kolmogorov and, in particular,

Δz ± ∼ (𝜀cA)1∕4ℓ1∕4. (27 ) ℓ
Using this phenomenology the high-order moments of fluctuations are given by (p) p∕4 Sℓ ∼ ℓ. Even in this case, ξp = p ∕4 results to be a linear function of the order p. The pseudo-energy spectrum can be easily found to be
E ±(k) ∼ (𝜀c )1∕2k−3∕2. (28 ) A
This is the Iroshnikov–Kraichnan spectrum. However, in a situation in which there is a balance between the linear Alfvén time scale or wave period, and the non-linear time scale needed to transfer energy to smaller scales, the energy cascade is indicated as critically balanced (Goldreich and Sridhar, 1995Jump To The Next Citation Point). In these conditions, it can be shown that the power spectrum P(k) would scale as −5∕3 f when the angle 𝜃B between the mean field direction and the flow direction is 90∘ while, the same scaling would follow f− 2 in case 𝜃 = 0∘ B and the spectrum would also have a smaller energy content than in the other case.

2.10 Some exact relationships

So far, we have been discussing about the inertial range of turbulence. What this means from a heuristic point of view is somewhat clear, but when we try to identify the inertial range from the spectral properties of turbulence, in general the best we can do is to identify the inertial range with the intermediate range of scales where a Kolmogorov’s spectrum is observed. The often used identity inertial range ≃ intermediate range, is somewhat arbitrary. In this regard, a very important result on turbulence, due to Kolmogorov (1941Jump To The Next Citation Point, 1991Jump To The Next Citation Point), is the so-called “4/5-law” which, being obtained from the Navier–Stokes equation, is “…one of the most important results in fully developed turbulence because it is both exact and nontrivial” (cf. Frisch, 1995Jump To The Next Citation Point). As a matter of fact, Kolmogorov analytically derived the following exact relation for the third order structure function of velocity fluctuations:

⟨(Δv (r,ℓ))3⟩ = − 4𝜖ℓ , (29 ) ∥ 5
where r is the sampling direction, ℓ is the corresponding scale, and 𝜖 is the mean energy dissipation per unit mass, assumed to be finite and nonvanishing.

This important relation can be obtained in a more general framework from MHD equations. A Yaglom’s relation for MHD can be obtained using the analogy of MHD equations with a transport equation, so that we can obtain a relation similar to the Yaglom’s equation for the transport of a passive quantity (Monin and Yaglom, 1975Jump To The Next Citation Point). Using the above analogy, the Yaglom’s relation has been extended some time ago to MHD turbulence by Chandrasekhar (1967Jump To The Next Citation Point), and recently it has been revised by Politano et al. (1998Jump To The Next Citation Point) and Politano and Pouquet (1998Jump To The Next Citation Point) in the framework of solar wind turbulence. In the following section we report an alternative and more general derivation of the Yaglom’s law using structure functions (Sorriso-Valvo et al., 2007Jump To The Next Citation Point; Carbone et al., 2009c).

2.11 Yaglom’s law for MHD turbulence

To obtain a general law we start from the incompressible MHD equations. If we write twice the MHD equations for two different and independent points xi and ′ xi = xi + ℓi, by substraction we obtain an equation for the vector differences ± ± ′ ± Δz i = (zi ) − zi. Using the hypothesis of independence of points ′ xi and xi with respect to derivatives, namely ∂i(z±j )′ = ∂′iz±j = 0 (where ∂′i represents derivative with respect to x′i), we get

± ∓ ′ ± ∓ ′ ± ′ ∂tΔz i + Δz α ∂αΔz i + z α(∂α + ∂α[)Δz i = − (∂i + ∂i])ΔP + + (∂2′α + ∂2α) ν ±Δz+i + ν∓ Δz −i (30 )
(′ ΔP = Ptot − Ptot). We look for an equation for the second-order correlation tensor ± ± ⟨Δz i Δz j ⟩ related to pseudo-energies. Actually the more general thing should be to look for a mixed tensor, namely ⟨Δz ±i Δz ∓j ⟩, taking into account not only both pseudo-energies but also the time evolution of the mixed correlations ⟨z+ z−⟩ i j and ⟨z− z+⟩ i j. However, using the DIA closure by Kraichnan, it is possible to show that these elements are in general poorly correlated (Veltri, 1980Jump To The Next Citation Point). Since we are interested in the energy cascade, we limit ourselves to the most interesting equation that describes correlations about Alfvénic fluctuations of the same sign. To obtain the equations for pseudo-energies we multiply Equations (30View Equation) by Δz ± j, then by averaging we get
± ± -∂-- ∓ ± ± ∂t⟨Δz i Δz j ⟩ + ∂ℓ ⟨ΔZ α(Δz i Δzj )⟩ = α 2 = − Λ − Π + 2ν ∂--⟨Δz ±Δz ± ⟩ − 4-∂--(𝜖±ℓ ), (31 ) ij ij ∂ℓ2α i j 3∂ ℓα ij α
where we used the hypothesis of local homogeneity and incompressibility. In Equation (31View Equation) we defined the average dissipation tensor
𝜖±ij = ν⟨(∂αZ ±i )(∂αZ ±j )⟩. (32 )
The first and second term on the r.h.s. of the Equation (31View Equation) represent respectively a tensor related to large-scales inhomogeneities
Λij = ⟨z∓α(∂ ′α + ∂ α)(Δz ±i Δz ±j )⟩ (33 )
and the tensor related to the pressure term
Π = ⟨Δz ±(∂ ′+ ∂ )ΔP + Δz ±(∂′ + ∂ )ΔP ⟩. (34 ) ij j i i i j j
Furthermore, In order not to worry about couplings between Elsässer variables in the dissipative terms, we make the usual simplifying assumption that kinematic viscosity is equal to magnetic diffusivity, that is ± ∓ ν = ν = ν. Equation (31View Equation) is an exact equation for anisotropic MHD equations that links the second-order complete tensor to the third-order mixed tensor via the average dissipation rate tensor. Using the hypothesis of global homogeneity the term Λij = 0, while assuming local isotropy Πij = 0. The equation for the trace of the tensor can be written as
∂ ∂2 4 ∂ ∂t⟨|Δz ±i |2⟩ +---⟨ΔZ ∓α|Δz ±i |2⟩ = 2ν---⟨|Δz ±i |2⟩ − -----(𝜖±iiℓα ), (35 ) ∂ℓα ∂ ℓα 3 ∂ℓα
where the various quantities depends on the vector ℓα. Moreover, by considering only the trace we ruled out the possibility to investigate anisotropies related to different orientations of vectors within the second-order moment. It is worthwhile to remark here that only the diagonal elements of the dissipation rate tensor, namely ± 𝜖ii are positive defined while, in general, the off-diagonal elements 𝜖±ij are not positive. For a stationary state the Equation (35View Equation) can be written as the divergenceless condition of a quantity involving the third-order correlations and the dissipation rates
∂ [ ∂ 4 ] ---- ⟨Δz ∓α |Δz ±i |2⟩ − 2ν----⟨|Δzi±|2⟩ − --(𝜖±iiℓα) = 0, (36 ) ∂ℓα ∂ ℓα 3
from which we can obtain the Yaglom’s relation by projecting Equation (36View Equation) along the longitudinal ℓα = ℓer direction. This operation involves the assumption that the flow is locally isotropic, that is fields depends locally only on the separation ℓ, so that
( ) [ ] 2- -∂- ∓ ± 2 ∂-- ± 2 4-± ℓ + ∂ℓ ⟨Δz ℓ |Δz i | ⟩ − 2ν ∂ℓ⟨|Δz i | ⟩ + 3𝜖iiℓ = 0. (37 )
The only solution that is compatible with the absence of singularity in the limit ℓ → 0 is
⟨Δz ∓|Δz ±|2⟩ = 2ν-∂-⟨|Δz ± |2⟩ − 4-𝜖±ℓ, (38 ) ℓ i ∂ ℓ i 3 ii
which reduces to the Yaglom’s law for MHD turbulence as obtained by Politano and Pouquet (1998Jump To The Next Citation Point) in the inertial range when ν → 0
± ∓ ± 2 4- ± Y ℓ ≡ ⟨Δzℓ |Δzi |⟩ = − 3 𝜖iiℓ. (39 )
Finally, in the fluid-like case where + − zi = zi = vi we obtain the usual Yaglom’s law for fluid flows
⟨Δv ℓ|Δvi|2⟩ = − 4-(𝜖ℓ) , (40 ) 3
which in the isotropic case, where ⟨Δv3ℓ⟩ = 3⟨Δv ℓΔv2y⟩ = 3⟨Δv ℓΔv2z⟩ (Monin and Yaglom, 1975Jump To The Next Citation Point), immediately reduces to the Kolmogorov’s law
3 4- ⟨Δv ℓ⟩ = − 5𝜖ℓ (41 )
(the separation ℓ has been taken along the streamwise x-direction).

The relations we obtained can be used, or better, in a certain sense they might be used, as a formal definition of inertial range. Since they are exact relationships derived from Navier–Stokes and MHD equations under usual hypotheses, they represent a kind of “zeroth-order” conditions on experimental and theoretical analysis of the inertial range properties of turbulence. It is worthwhile to remark the two main properties of the Yaglom’s laws. The first one is the fact that, as it clearly appears from the Kolmogorov’s relation (Kolmogorov, 1941Jump To The Next Citation Point), the third-order moment of the velocity fluctuations is different from zero. This means that some non-Gaussian features must be at work, or, which is the same, some hidden phase correlations. Turbulence is something more complicated than random fluctuations with a certain slope for the spectral density. The second feature is the minus sign which appears in the various relations. This is essential when the sign of the energy cascade must be inferred from the Yaglom relations, the negative asymmetry being a signature of a direct cascade towards smaller scales. Note that, Equation (39View Equation) has been obtained in the limit of zero viscosity assuming that the pseudo-energy dissipation rates ± 𝜖ii remain finite in this limit. In usual fluid flows the analogous hypothesis, namely 𝜖 remains finite in the limit ν → 0, is an experimental evidence, confirmed by experiments in different conditions (Frisch, 1995Jump To The Next Citation Point). In MHD turbulent flows this remains a conjecture, confirmed only by high resolution numerical simulations (Mininni and Pouquet, 2009).

From Equation (36View Equation), by defining ΔZ ±i = Δvi ± Δbi we immediately obtain the two equations

[ ] ∂ ∂ 4 ---- ⟨Δv αΔE ⟩ − 2⟨ΔbαΔC ⟩ − 2ν----⟨ΔE ⟩ −--(𝜖Eℓα) = 0 (42 ) ∂ℓα[ ∂ ℓα 3 ] -∂-- -∂-- 4- ∂ ℓα − ⟨Δb αΔE ⟩ + 2 ⟨Δv α ΔC ⟩ − 4ν∂ ℓα⟨ΔC ⟩ − 3 (𝜖Cℓα) = 0, (43 )
where we defined the energy fluctuations ΔE = |Δvi |2 + |Δbi|2 and the correlation fluctuations ΔC = ΔviΔbi. In the same way the quantities ( ) 𝜖E = 𝜖+ii + 𝜖−ii ∕2 and ( ) 𝜖C = 𝜖+ii − 𝜖−ii ∕2 represent the energy and correlation dissipation rate, respectively. By projecting once more on the longitudinal direction, and assuming vanishing viscosity, we obtain the Yaglom’s law written in terms of velocity and magnetic fluctuations
4- ⟨Δv ℓΔE ⟩ − 2⟨Δb ℓΔC ⟩ = − 3 𝜖Eℓ (44 ) 4 − ⟨Δb ℓΔE ⟩ + 2⟨Δv ℓΔC ⟩ = −--𝜖Cℓ. (45 ) 3

2.12 Density-mediated Elsässer variables and Yaglom’s law

Relation (39View Equation), which is of general validity within MHD turbulence, requires local characteristics of the turbulent fluid flow which can be not always satisfied in the solar wind flow, namely, large-scale homogeneity, isotropy, and incompressibility. Density fluctuations in solar wind have a low amplitude, so that nearly incompressible MHD framework is usually considered (Montgomery et al., 1987Jump To The Next Citation Point; Matthaeus and Brown, 1988Jump To The Next Citation Point; Zank and Matthaeus, 1993Jump To The Next Citation Point; Matthaeus et al., 1991Jump To The Next Citation Point; Bavassano and Bruno, 1995Jump To The Next Citation Point). However, compressible fluctuations are observed, typically convected structures characterized by anticorrelation between kinetic pressure and magnetic pressure (Tu and Marsch, 1994Jump To The Next Citation Point). Properties and interaction of the basic MHD modes in the compressive case have also been considered (Goldreich and Sridhar, 1995Jump To The Next Citation Point; Cho and Lazarian, 2002).

A first attempt to include density fluctuations in the framework of fluid turbulence was due to Lighthill (1955). He pointed out that, in a compressible energy cascade, the mean energy transfer rate per unit volume 𝜖V ∼ ρv3 ∕ℓ should be constant in a statistical sense (v being the characteristic velocity fluctuations at the scale ℓ), thus obtaining the scaling relation v ∼ (ℓ∕ρ )1∕3. Fluctuations of a density-weighted velocity field 1∕3 u ≡ ρ v should thus follow the usual Kolmogorov scaling 3 u ∼ ℓ. The same phenomenological arguments can be introduced in MHD turbulence Carbone et al. (2009aJump To The Next Citation Point) by considering the pseudoenergy dissipation rates per unit volume 𝜖±V = ρ𝜖±ii and introducing density-weighted Elsässer fields, defined as w ± ≡ ρ1∕3z±. A relation equivalent to the Yaglom-type relation (39View Equation)

± −1 ∓ ± 2 ± W ℓ ≡ ⟨ρ⟩ ⟨Δw ℓ |Δw i | ⟩ = C 𝜖iiℓ (46 )
(C is some constant assumed to be of the order of unit) should then hold for the density-weighted increments ± Δw. Relation ± W ℓ reduces to ± Yℓ in the case of constant density, allowing for comparison between the Yaglom’s law for incompressible MHD flows and their compressible counterpart. Despite its simple phenomenological derivation, the introduction of the density fluctuations in the Yaglom-type scaling (46View Equation) should describe the turbulent cascade for compressible fluid (or magnetofluid) turbulence. Even if the modified Yaglom’s law (46View Equation) is not an exact relation as (39View Equation), being obtained from phenomenological considerations, the law for the velocity field in a compressible fluid flow has been observed in numerical simulations, the value of the constant C results negative and of the order of unity (Padoan et al., 2007; Kowal and Lazarian, 2007).

2.13 Yaglom’s law in the shell model for MHD turbulence

As far as the shell model is concerned, the existence of a cascade towards small scales is expressed by an exact relation, which is equivalent to Equation (40View Equation). Using Equations (23View Equation), the scale-by-scale pseudo-energy budget is given by

d-∑ ± 2 [ ±] ∑ 2 ± 2 ∑ [ ± ±∗] dt |Zn | = knIm Tn − 2 νkn|Zn | + 2ℜe Zn Fn . n n n

The second and third terms on the right hand side represent, respectively, the rate of pseudo-energy dissipation and the rate of pseudo-energy injection. The first term represents the flux of pseudo-energy along the wave vectors, responsible for the redistribution of pseudo-energies on the wave vectors, and is given by

( ) ± ± ± ∓ 2 − a − c ± ± ∓ Tn = (a + c)Z n Z n+1Zn+2 + ----λ---- Z n−1Z n+1Zn + ( ) ± ± ∓ c-−-a ± ± ∓ + (2 − a − c)Z n Z n+2Zn+1 + λ Z )Zn Zn+1Z n−1. (47 )

Using the same assumptions as before, namely: i) the forcing terms act only on the largest scales, ii) the system can reach a statistically stationary state, and iii) in the limit of fully developed turbulence, ν → 0, the mean pseudo-energy dissipation rates tend to finite positive limits 𝜖±, it can be found that

± ± −1 ⟨Tn ⟩ = − 𝜖 kn . (48 )
This is an exact relation which is valid in the inertial range of turbulence. Even in this case it can be used as an operative definition of the inertial range in the shell model, that is, the inertial range of the energy cascade in the shell model is defined as the range of scales kn, where the law from Equation (48View Equation) is verified.

  Go to previous page Scroll to top Go to next page