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7 July 2019; revised ?; accepted ?.
Abstract

A weak turbulence theory is derived for magnetohydrodynamics under rapid rotation and in the presence of a large-scale magnetic field. The angular velocity is assumed to be uniform and parallel to the constant Alfvén speed . Such a system exhibits left and right circularly polarized waves which can be obtained by introducing the magneto-inertial length . In the large-scale limit (; being the wave number), the left- and right-handed waves tend respectively to the inertial and magnetostrophic waves whereas in the small-scale limit () pure Alfvén waves are recovered. By using a complex helicity decomposition, the asymptotic weak turbulence equations are derived which describe the long-time behavior of weakly dispersive interacting waves via three-wave interaction processes. It is shown that the nonlinear dynamics is mainly anisotropic with a stronger transfer perpendicular () than parallel () to the rotating axis. The general theory may converge to pure weak inertial/magnetostrophic or Alfvén wave turbulence when the large or small-scales limits are taken respectively. Inertial wave turbulence is asymptotically dominated by the kinetic energy/helicity whereas the magnetostrophic wave turbulence is dominated by the magnetic energy/helicity. For both regimes a family of exact solutions are found for the spectra which do not correspond necessarily to a maximal helicity state. It is shown that the hybrid helicity exhibits a cascade whose direction may vary according to the scale at which the helicity flux is injected with an inverse cascade if and a direct cascade otherwise. The theory is relevant for the magnetostrophic dynamo whose main applications are the Earth and giant planets for which a small () Rossby number is expected.

Weak turbulence theory for rotating MHD and planetary dynamos] Weak turbulence theory for rotating magnetohydrodynamics and planetary dynamos Sébastien Galtier]Sébastien Galtier thanks: Email address for correspondence: sebastien.galtier@lpp.polytechnique.fr

Key words: Dynamo theory, MHD turbulence, wave-turbulence interactions

 

1 Introduction

Rotation is a commonly observed phenomenon in astronomy: planets, stars and galaxies all spin around their axis. The rotation rate of planets in the solar system was first measured by tracking visual features whereas stellar rotation is generally measured through Doppler shift or by following the magnetic activity. One consequence of the Sun rotation is the formation of the Parker interplanetary magnetic field spiral well detected by space crafts, whereas the Earth rotation has a strong impact on the turbulent dynamics of large-scale geophysical flows. These few examples show that the study of rotating flows interests a wide range of problems, ranging from engineering (turbomachinery) to geophysics (oceans, earth’s atmosphere, gaseous planets), weather prediction and turbulence (Davidson 2004). Rotation is often coupled with other dynamical factors, it is therefore important to isolate the effect of the Coriolis force to understand precisely its impact. The importance of rotation can be measured with the Rossby number:

(1.0)

where , and are respectively typical velocity, length-scale and rotation rate. This dimensionless number measures the ratio of the advection term on the Coriolis force in the Navier-Stokes equations, also a small value of means a dynamics mainly driven by rotation. Typical large-scale planetary flows are characterized by (Shirley & Fairbridge 1997) whereas the liquid metals (mainly iron) in the Earth’s outer core are much more affected by rotation with (Roberts & King 2013). Note that for a giant planet like Jupiter in which liquid metallic hydrogen is present in most of the volume, it is believed that the Rossby number may even be smaller (see e.g.  Jones 2011). These situations contrast with the solar convective region where the magnetic field is believed to be magnified and for which .

Inertial waves are a ubiquitous feature of neutral fluids under rapid rotation (Greenspan 1968). Although much is known about their initial excitation, their nonlinear interactions is still a subject of intense research. Many papers have been devoted to pure rotating turbulence () but because of the different nature of the investigations (theoretical, numerical and experimental) it is difficult in general to compare directly the results obtained. From a theoretical point of view it is convenient to use a spectral description in terms of continuous wave vectors with the unbounded homogeneity assumption in order to derive the governing equations for the energy, kinetic helicity and polarization spectra (Cambon & Jacquin 1989). Although such equations introduce transfer terms which remain to be evaluated consistently, it is already possible to show with a weakly nonlinear resonant waves analysis (Waleffe 1993) the anisotropic nature of that turbulence with a nonlinear transfer preferentially in the perpendicular (to ) direction. For moderate Rossby numbers the eddy damped quasi-normal Markovian model may be used as a closure (Cambon et al. 1997), whereas in the small Rossby number limit the asymptotic weak turbulence theory can be derived rigorously (Galtier 2003). In the latter case, it was shown that the wave modes () are decoupled from the slow mode () which is not accessible by the theory, and the positive energy flux spectra were also obtained as exact power law solutions. The weak turbulence regime was also investigated numerically and it was shown in particular that the two-dimensional manifolds is an integrable singularity at , which is related to the scaling of the energy spectrum , and that the energy cascade goes forward (Bellet et al. 2006). Recently, the problem of confinement has been addressed explicitly in the inertial wave turbulence theory using discrete wave numbers (Scott 2014): three asymptotically distinct stages in the evolution of the turbulence are found with finally a regime dominated by resonant interactions. Pseudo-spectral codes are often used to investigate numerically homogeneous rotating turbulence (see e.g.  Mininni & Pouquet 2010a). Several questions have been investigated like the origin of the anisotropy or of the inverse cascade observed when a forcing is applied at intermediate scale . However, according to the question addressed the results may be affected by the discretization and by finite-box effects at too small Rossby number and too long elapsed time (Smith & Lee 2005; Bourouiba 2008). This seems to be the case in particular for the question of the inverse cascade mediated by the decoupling of the slow mode. For example, it was found that the one-dimensional isotropic energy spectrum may follow two different power laws with at small-scale () and at large-scale () (Smith & Waleffe 1999). But it was also shown that the scaling at large-scale was strongly influenced by the value of the aspect ratio between the parallel and the perpendicular (to ) resolution, a small aspect leading to a reduction of the number of available resonant triads, hence an alteration of the spectrum with the restoration of a spectrum for small enough vertical resolution. Several experiments have been devoted to rotating turbulence with different types of apparatus (Hopfinger et al. 1982; Jacquin et al. 1990; Baroud et al. 2002; Morize et al. 2005; van Bokhoven et al. 2009). Contrary to the theory and the simulation, it is very challenging to reproduce experimentally the conditions of homogeneous turbulence. Nevertheless, one of the main results reported is that the rotation leads to a bi-dimensionalisation of an initial homogeneous isotropic turbulence with anisotropic spectra where energy is preferentially accumulated in the perpendicular (to ) wave numbers . Energy spectra with were experimentally observed (Baroud et al. 2002; Morize et al. 2005; van Bokhoven et al. 2009) revealing a significant discrepancy with the isotropic Kolmogorov spectrum () for non-rotating fluids. Note that the wave number entering in the spectral measurements corresponds mainly to . Recently, direct measurements of energy transfer have been made in the physical space by using third-order structure function (Lamriben et al. 2011) and an increase of anisotropy at small scales has been found in agreement with some theoretical studies (Jacquin et al. 1990; Galtier 2003, 2009a; Bellet et al. 2006). The role of kinetic helicity – which quantifies departures from mirror symmetry (Moffatt 1969) – on rotating fluids has been the subject of few studies. One reason is that it is difficult to measure the helicity production from experimental studies. The other reason is probably linked to the negligible effect of helicity on energy in non-rotating turbulence. Indeed, in this case one observes a joint constant flux cascade of energy and helicity with a spectrum for both quantities (Chen et al. 2003a, b). But recently, several numerical simulations have demonstrated the surprising strong impact of helicity on fast rotating hydrodynamic turbulence (Mininni & Pouquet 2010a, b; Teitelbaum & Mininni 2009; Mininni et al. 2012) whose main properties can be summarized as follows. When the (large-scale) forcing applied to the system injects only negligible helicity, the dynamics is mainly governed by a direct energy cascade compatible with an energy spectrum which is precisely the weak turbulence prediction (Galtier 2003). However, when the helicity injection becomes so important that the dynamics is mainly governed by a direct helicity cascade, different scalings are found following the empirical law:

(1.0)

where and are respectively the power law indices of the one-dimensional energy and helicity spectra. This law cannot be explained by a consistent phenomenology where anisotropy is used which renders the relation (1) highly non-trivial. As shown by Galtier (2014), an explanation can only be found when a rigorous analysis is made on the weak turbulence equations: the relation corresponds in fact to the finite helicity flux spectra which are exact solutions of the equations.

It has been long recognized that the Earth’s magnetic field is not steady (Finlay et al. 2010). Changes occur across a wide range of timescales from second – because of the interactions between the solar wind and the magnetosphere – to several tens of millions years which is the longest timespan between polarity reversals. To understand the generation and the maintain of a large-scale magnetic field, the most promising mechanism is the dynamo (Pouquet et al. 1976; Moffatt 1978; Brandenburg 2001). Dynamo is an active area of research where dramatic developments have been made in the past several years (Dormy et al. 2000). The subject concerns primary the Earth where a large amount of data is available which allows us to follow e.g. the geomagnetic polarity reversal occurrences over million years (Finlay & Jackson 2003; Roberts & King 2013). This chaotic behavior contrasts drastically with the surprisingly regularity of the Sun which changes its magnetic field lines polarity every years. It is believed that the three main ingredients for the geodynamo problem are the Coriolis, Lorentz-Laplace and buoyancy forces. The latter force may be seen as a source of turbulence for the conducting fluids described by incompressible magnetohydrodynamics (MHD), whereas the two others are more or less balanced (Elsässer number of order one). This balance leads to the strong-field regime – the so-called magnetostrophic dynamo – for which we may derive magnetostrophic waves (Lehnert 1954; Schmitt et al. 2008). This regime is thought to be relevant not only for Earth but also for giant planets like Jupiter or Saturn, and by extension probably to exoplanets (Stevenson 2003). In order to investigate the dynamo problem several experiments have been developed (Pétrélis et al. 2007). In one of them, the authors were able to successfully reproduce with liquid sodium reversals and excursions of a turbulent dynamo generated by two (counter) rotating disks (Berhanu et al. 2007). This result follows a three-dimensional numerical simulation of the Earth’s outer core where the reversal of the dipole moment was also obtained (Glatzmaier & Roberts 1995). In this model, however, the inertial/advection terms are simply discarded to mimic a very small Rossby number. This assumption is in apparent contradiction with any turbulent regime (Reynolds number is about for the Earth’s outer core) and in particular with the weak turbulence one in which the nonlinear interactions – although weak at short-time scales compared with the linear contributions – become important for the nonlinear dynamics at asymptotically large-time scales. As we will see below, it is basically the regime that we shall investigate theoretically in this paper: a sea of helical (magnetized) waves (Moffatt 1970) will be considered as the main ingredient for the triggering of dynamo through the nonlinear transfer of magnetic energy and helicity.

Weak turbulence is the study of the long time statistical behavior of a sea of weakly nonlinear dispersive waves (Nazarenko 2011). The energy transfer between waves occurs mostly among resonant sets of waves and the resulting energy distribution, far from a thermodynamic equilibrium, is characterized by a wide power law spectrum and a high Reynolds number. This range of wavenumbers – the inertial range – is generally localized between large-scales at which energy is injected in the system (sources) and small-scales at which waves break or dissipate (sinks). Pioneering works on weak turbulence date back to the sixties when it was established that the stochastic initial value problem for weakly coupled wave systems has a natural asymptotic closure induced by the dispersive nature of the waves and the large separation of linear and nonlinear time scales (Benney & Saffman 1966; Benney & Newell 1967, 1969). In the meantime, Zakharov & Filonenko (1966) showed that the kinetic equations derived from the weak turbulence analysis have exact equilibrium solutions which are the thermodynamic zero flux solutions but also – and more importantly – finite flux solutions which describe the transfer of conserved quantities between sources and sinks. The solutions, first published for isotropic turbulence (Zakharov 1965; Zakharov & Filonenko 1966) were then extended to anisotropic turbulence (Kuznetsov 1972). Weak turbulence is a very common natural regime with applications, for example, to capillary waves (Kolmakov et al. 2004), gravity waves (Falcon et al. 2007), superfluid helium and processes of Bose-Einstein condensation (Lvov et al. 2003), nonlinear optics (Dyachenko et al. 1992), inertial waves (Galtier 2003), Alfvén waves (Galtier et al. 2000, 2002; Galtier & Chandran 2006) or whistler/kinetic Alfvén waves (Galtier 2006b).

In this paper, the weak turbulence theory will be established for rotating MHD in the limit of small Rossby and Ekman numbers, the latter measuring the ratio of the viscous on Coriolis terms. We shall assume the existence of a strong uniform magnetic field parallel to the fast and constant rotating rate. The combination of the Coriolis and Lorentz-Laplace forces leads to the appearance of two types of circularly polarized waves and a possible non equipartition between the kinetic and magnetic energies (Moffatt 1972; Favier et al. 2012). After a general introduction to rotating MHD in §2, a weak helical turbulence formalism is developed in §3 by using a technique developed in Galtier (2006b). The phenomenology of weak turbulence dynamo is given in §4, the general properties of the weak turbulence equations are discussed in §5, whereas the exact spectral solutions are derived in §6. We conclude with a discussion in §7. Generally speaking, it is believed that the present work can be useful for better understanding the nonlinear magnetostrophic dynamo (Roberts & King 2013) with in background an application to Earth but also to giant planets like Jupiter or Saturn for which the intensity of the Coriolis force is relatively strong, the range of available length scales wide, and the magnetic field mainly dipolar with a weak tilt () of the dipole relative to the rotation axis. By extension we may even think that the analysis is relevant for exoplanets and some magnetized stars (Morin et al. 2011).

2 Rotating magnetohydrodynamics

2.1 Governing equations

The basic equations governing incompressible MHD under solid rotation and in the presence of a uniform background magnetic field are:

(2.0)
(2.0)
(2.0)
(2.0)

with the velocity, the total pressure (including the magnetic pressure and the centrifugal term), the magnetic field normalized to a velocity (, with the constant density), the uniform normalized magnetic field, the rotating rate, the kinematic viscosity and the magnetic diffusivity. Note the presence of the Coriolis force in the first equation (second term in the left hand side). Turbulence can only be maintained if a source is added to balance the small-scale dissipation. For example, in the geodynamo problem we may think that this external forcing is played by the convection (since the Rayleigh number ) with the buoyancy force (Braginsky & Roberts 1995). In our case, we shall perform a pure nonlinear analysis, therefore the source and dissipation terms will be discarded. The weak turbulence equations that will be derived may describe, however, any magnetic Prandtl limit since the (linear) dissipative terms may be added to the equations after having made the nonlinear asymptotic analysis. In the rest of the paper, we shall assume that:

(2.0)

with a unit vector (). We introduce the magneto-inertial length defined as:

(2.0)

This length scale will be useful to characterize the main properties of rotating MHD.

2.2 Three-dimensional inviscid invariants

The two inviscid () quadratic invariants of incompressible rotating MHD in the presence of a background magnetic field parallel to the rotating axis are the total energy:

(2.0)

and the hybrid helicity:

(2.0)

where is the vector potential () and is the volume over which the average is made. The second invariant is a mixture of cross-helicity, , and magnetic helicity, , which are not conserved in the present situation (Matthaeus & Goldstein 1982). Indeed, it is straightforward to show from (2.1)–(2.1) that (see also Shebalin 2006):

(2.0)
(2.0)
(2.0)

where is the vorticity and is the normalized current density. Therefore, the previous equations demonstrate that a second invariant may emerge if and only if . Below, we will verify that for the weak turbulence equations these two inviscid invariants are conserved for each triad of wave vectors.

2.3 Helical MHD waves

One of the main effects produced by the Coriolis force is to modify the polarization of the linearly polarized Alfvén waves – solutions of the standard MHD equations – which become circularly polarized and dispersive (Lehnert 1954). Indeed, if we linearize equations (2.1)–(2.1) such that:

(2.0)

with a small parameter () and a three-dimensional displacement vector, then we obtain the following inviscid () and ideal () equations in Fourier space:

(2.0)
(2.0)
(2.0)
(2.0)

where the wave vector (, , ) and . The index denotes the Fourier transform, defined by the relation:

(2.0)

where (the same notation is used for the other fields). The linear dispersion relation () reads:

(2.0)

with:

(2.0)

We obtain the general solution:

(2.0)

where the value () of defines the directional wave polarity such that we always have ; then is a positive definite pulsation. The wave polarization tells us if the wave is right () or left () circularly polarized. In the first case, we are dealing with the magnetostrophic branch, whereas in the latter case with the inertial branch (see figure 1). We see that the transverse circularly polarized waves are dispersive and that we recover the two well-known limits, i.e. the pure inertial waves () in the large-scale limit (), and the standard Alfvén waves () in the small-scale limit (). For the pure magnetostrophic waves we find the pulsation, . Note that the Alfvén waves become linearly polarized only when the Coriolis force vanishes: when it is present, whatever its magnitude is, the modified Alfvén waves are circularly polarized. This property is also found in MHD when the Hall term is added (Sahraoui et al. 2007).

Figure 1: Dispersion relation for rotating MHD permeated by a background magnetic field in linear (top) and logarithmic (bottom) coordinates, with . The upper and lower branches correspond respectively to left- and right-handed polarized waves. The Alfvén wave dispersion relation is also given (dotted line).

2.4 Polarization

The polarizations and can be related to two well-known quantities, the reduced magnetic helicity and the reduced cross-helicity . The reduced magnetic helicity is defined as:

(2.0)

where denotes the complex conjugate. For circularly polarized waves, we can use relation (2.3) which gives . On the other hand, the reduced cross-helicity is defined as:

(2.0)

The linear solution implies, , which leads to . The use of both relations gives eventually:

(2.0)

This result is only valid for the linear solutions but may be generalized to any fluctuations in order to find the properties of helical turbulence (Meyrand & Galtier 2012).

2.5 Magnetostrophic equation

The governing equations of rotating MHD can also be written in the following form:

(2.0)
(2.0)

where the relation has been introduced. The magnetostrophic regime corresponds to a balance between the Coriolis and the Lorentz-Laplace forces (Finlay 2008). If we balance such terms in the linear case, we obtain the relation:

(2.0)

which can be introduced in equation (2.5) to give:

(2.0)

Expression (2.5) is the magnetostrophic equation which describes the nonlinear evolution of the magnetic field when both the rotation and the uniform magnetic field are relatively strong. It is asymptotically true in the sense that it only corresponds to the lower part of the magnetostrophic branch shown in figure 1. We may note immediately the similarity with the electron MHD equation introduced in plasma physics (Kingsep et al. 1990) to describe the small space-time evolution of a magnetized plasma. The difference resides in the coefficient which is the ion skin depth in electron MHD. Then, it is not surprising that the linear solution gives the same (up to a factor ) dispersion relation as for whistler waves which are also right circularly polarized. We will see in section 5.6 that indeed the general weak turbulence equations gives in the large-scale right-polarization limit the same equation (up to a factor) as in the electron MHD case (Galtier & Bhattacharjee 2003).

2.6 Complex helicity decomposition

Given the incompressibility constraints (2.3) and (2.3), it is convenient to project the rotating MHD equations in a plane orthogonal to . We will use the complex helicity decomposition technique which has been shown to be effective in providing a compact description of the dynamics of three-dimensional incompressible fluids (Craya 1954; Kraichnan 1973; Waleffe 1992; Lesieur 1997; Turner 2000; Galtier 2003, 2006b). The complex helicity basis is also particularly useful since it allows to diagonalize systems dealing with circularly polarized waves. We introduce the complex helicity decomposition:

(2.0)

where:

(2.0)

and ==. We note that (, , ) form a complex basis with the following properties:

(2.0)
(2.0)
(2.0)
(2.0)

We project the Fourier transform of the original vectors and on the helicity basis (see also Appendix B):

(2.0)
(2.0)

and in particular, we note that:

(2.0)
(2.0)

We introduce the expressions of the new fields into the rotating MHD equations written in Fourier space and we multiply it by the vector . First, we will focus on the linear dispersion relation () which reads:

(2.0)

with:

(2.0)
(2.0)

Equation (2.6) shows that are the canonical variables for our system. These eigenvectors combine the velocity and the magnetic field in a non trivial way by a factor (with ). In the small-scale limit (), we see that : the Elsässer variables used in standard MHD are then recovered. In the large-scale limit (), we have for (inertial waves), or for (magnetostrophic wave). Therefore, can be seen as a generalization of the Elsässer variables to rotating MHD. In the rest of the paper, we shall use the relation:

(2.0)

where is the wave amplitude in the interaction representation for which we have, in the linear approximation, . In particular, that means that weak nonlinearities will modify only slowly in time the helical MHD wave amplitudes. The coefficient in front of the wave amplitude is introduced in advance to simplify the algebra that we are going to develop.

3 Helical weak turbulence formalism

3.1 Fundamental equations

We decompose the inviscid nonlinear MHD equations (2.3)–(2.3) on the complex helicity basis introduced in the previous section. Then, we project the equations on the vector . After simplifications we obtain:

(3.0)

and:

(3.0)

where . The delta distributions come from the Fourier transforms of the nonlinear terms. We introduce the generalized Elsässer variables in the following way:

(3.0)
(3.0)

Then, we obtain in the interaction representation (variable ):

(3.0)

where:

(3.0)

and:

(3.0)

Equation (3.0) is the wave amplitude equation from which it is possible to extract some information. As expected we see that the nonlinear terms are of order . This means that weak nonlinearities will modify only slowly in time the helical MHD wave amplitude. They contain an exponentially oscillating term which is essential for the asymptotic closure. Indeed, weak turbulence deals with variations of spectral densities at very large time, i.e. for a nonlinear transfer time much greater than the wave period. As a consequence, most of the nonlinear terms are destroyed by phase mixing and only a few of them, the resonance terms, survive (see e.g. Newell et al. (2001)). The expression obtained for the fundamental equation (3.0) is classical in weak turbulence. The main difference between different problems is localized in the matrix which is interpreted as a complex geometric coefficient. We will see below that the local decomposition allows to get a polar form for such a coefficient which is much easier to manipulate. From equation (3.0) we see eventually that, contrary to incompressible MHD, there is no exact solutions to the nonlinear problem in incompressible rotating MHD. The origin of such a difference is that in MHD the nonlinear term involves Alfvén waves traveling only in opposite directions whereas in rotating MHD this constrain does not exist (we have a summation over and ). In other words, if one type of wave is not present in incompressible MHD then the nonlinear term cancels whereas in the present problem it is not the case (see e.g.  Galtier et al. 2000).

3.2 Local decomposition

In order to evaluate the scalar products of complex helical vectors found in the geometric coefficient (3.0), it is convenient to introduce a vector basis local to each particular triad (Waleffe 1992; Turner 2000; Galtier 2003). For example, for a given vector , we define the orthonormal basis vectors:

(3.0)
(3.0)
(3.0)

where and:

(3.0)

We see that the vector is normal to any vector of the triad (,,) and changes sign if and are interchanged, i.e. . Note that does not change by cyclic permutation, i.e. , . A sketch of the local decomposition is given in figure 2.

Figure 2: Local decomposition for a given wave vector .

We now introduce the vectors:

(3.0)

and define the rotation angle , so that:

(3.0)
(3.0)

The decomposition of the helicity vector in the local basis gives (similar forms are obtained for and ):

(3.0)

After some algebra we obtain the following polar form for the matrix :

(3.0)

The angle refers to the angle opposite to in the triangle defined by (). To obtain equation (3.0), we have also used the well-known triangle relations:

(3.0)

Further modifications have to be made before applying the spectral formalism. In particular, the fundamental equation has to be invariant under interchange of and . To do so, we shall introduce the symmetrized matrix:

(3.0)

Finally, by using the identities given in Appendix A, we obtain:

(3.0)

where:

(3.0)

The matrix possesses the following properties:

(3.0)
(3.0)
(3.0)
(3.0)

Equation (3.0) is the fundamental equation that describes the slow evolution of the wave amplitudes due to the nonlinear terms of the incompressible rotating MHD equations. It is the starting point for deriving the weak turbulence equations. The local decomposition used here allows us to represent concisely complex information in an exponential function (polar form). As we will see below, it will simplify significantly the derivation of the asymptotic equations.

From equation (3.0) we note that the nonlinear coupling between helicity states associated with wave vectors, and , vanishes when the wave vectors are collinear (since then, ). This property is similar to the one found for pure rotating hydrodynamics. It seems to be a general property for helical waves (Kraichnan 1973; Waleffe 1992; Turner 2000; Galtier 2003, 2006b). Additionally, we note that the nonlinear coupling between helicity states vanishes whenever the wave numbers and are equal if their associated wave and directional polarities, , , and , respectively, are also equal. In the case of inertial waves, for which we have (left-handed waves), this property was already observed (Galtier 2003). Here, this finding is generalized to right and left circularly polarized waves. Note that in the large-scale limit for which we recover the linearly polarized Alfvén waves, this property tends to disappear (see also section 5.4).

We are interested by the long-time behavior of the helical wave amplitudes. From the fundamental equation (3.0), we see that the nonlinear wave coupling will come from resonant terms such that:

(3.0)

The resonance condition may also be written:

(3.0)

As we shall see below, relations (3.0) are useful in simplifying the weak turbulence equations and demonstrating the conservation of inviscid invariants.

3.3 Asymptotic weak turbulence equations

Weak turbulence is a state of a system composed of many simultaneously excited and interacting nonlinear waves where the energy distribution, far from thermodynamic equilibrium, is characterized by a wide power law spectrum. This range of wave numbers – the inertial range – is generally localized between large-scales at which energy is injected in the system and small dissipative scales. The origin of weak turbulence dates back to the early sixties and since then many papers have been devoted to the subject (see e.g. Hasselmann (1962); Benney & Saffman (1966); Zakharov (1967); Sagdeev & Galeev (1969); Kuznetsov (1972); Zakharov et al. (1992); Galtier (2009b); Nazarenko (2011)). The essence of weak turbulence is the statistical study of large ensembles of weakly interacting dispersive waves via a systematic asymptotic expansion in powers of small nonlinearity. This technique leads finally to the derivation of kinetic equations for quantities like the energy and more generally for the (quadratic) invariants of the system under investigation. Here, we will follow the standard Eulerian formalism of weak turbulence (see e.g. Benney & Newell (1969)).

We define the density tensor for an homogeneous turbulence, such that:

(3.0)

for which we shall write an asymptotic closure equation. The presence of the deltas and means that correlations with opposite wave or directional polarities have no long-time influence in the wave turbulence regime; the third delta distribution is the consequence of the homogeneity assumption. Details of the derivation of the weak turbulence equations are given in Appendix C. After a lengthly calculation, we obtain the following result:

(3.0)