Reynolds-number dependence of the dimensionless dissipation rate in homogeneous magnetohydrodynamic turbulence
This paper examines the behavior of the dimensionless dissipation rate for stationary and nonstationary magnetohydrodynamic (MHD) turbulence in presence of external forces. By combining with previous studies for freely decaying MHD turbulence, we obtain here both the most general model equation for applicable to homogeneous MHD turbulence and a comprehensive numerical study of the Reynolds number dependence of the dimensionless total energy dissipation rate at unity magnetic Prandtl number. We carry out a series of medium to high resolution direct numerical simulations of mechanically forced stationary MHD turbulence in order to verify the predictions of the model equation for the stationary case. Furthermore, questions of nonuniversality are discussed in terms of the effect of external forces as well as the level of cross- and magnetic helicity. The measured values of the asymptote lie between for free decay, where the value depends on the initial level of cross- and magnetic helicities. In the stationary case we measure .
The dynamics of conducting fluids is relevant to many areas in geo- and astrophysics as well as in engineering and industrial applications. Often the flow is turbulent, and the interaction of the turbulent flow with the magnetic field leads to considerable complexity. Being a multi-parameter problem, techniques that have been successfully applied to turbulence in nonconducting fluids sometimes fail to deliver unambiguous predictions in magnetohydrodynamic (MHD) turbulence. This concerns e.g. the prediction of inertial range scaling exponents by extension of Kolmogorov’s arguments Kolmogorov (1941) to MHD, and considerable effort has been put into the further understanding of inertial range cascade(s) in MHD turbulence Iroshnikov (1964); Kraichnan (1965); Goldreich and Sridhar (1995); Boldyrev (2005, 2006); Beresnyak and Lazarian (2006); Mason et al. (2006); Gogoberidze (2007). The difficulties are partly due to the many different configurations that can arise in MHD turbulence because of e.g. anisotropy, different levels of vector field correlations, different values of the dissipation coefficients and different types of external forces, and as such are connected to the question of universality in MHD turbulence Dallas and Alexakis (2013a, b); Wan et al. (2012); Schekochihin et al. (2008); Mininni (2011); Grappin et al. (1983); A. Pouquet and P. Mininni and D. Montgomery and A. Alexakis (2008); Beresnyak (2011); Boldyrev et al. (2011); Grappin and Müller (2010); Lee et al. (2010); Servidio et al. (2008). The behavior of the (dimensionless) dissipation rate is representative of this problem, in the sense that the aforementioned properties of MHD turbulence influence the energy transfer across the scales, i.e. the cascade dynamics Frisch et al. (1975); Pouquet et al. (1976); Pouquet and Patterson (1978); Biskamp (1993); Dallas and Alexakis (2013b); Alexakis (2013), and thus the amount of energy that is eventually dissipated at the small scales.
The behavior of the total dissipation rate in a turbulent non-conducting fluid is a well-studied problem. As such it has been known for a long time that the total dissipation rate in both stationary and freely decaying homogeneous isotropic turbulence tends to a constant value with increasing Reynolds number following a well-known characteristic curve Sreenivasan (1984, 1998); McComb (2014); McComb et al. (2015a); Yeung et al. (2015); Jagannathan and Donzis (2016). For statistically steady isotropic turbulence this curve can be approximated by the real-space stationary energy balance equation, where the asymptote is connected to the maximal inertial flux of kinetic energy McComb et al. (2015a). The corresponding problem in MHD has received much less attention, however, recent numerical results for freely decaying MHD turbulence at unity magnetic Prandtl number report similar behavior. Mininni and Pouquet Mininni and Pouquet (2009) carried out direct numerical simulations (DNSs) of freely decaying homogeneous MHD turbulence without a mean magnetic field, showing that the temporal maximum of the total dissipation rate became independent of Reynolds number at a Taylor-scale Reynolds number (measured at the peak of ) of about 200. Dallas and Alexakis Dallas and Alexakis (2014) measured the dimensionless dissipation rate also from DNS data for free decay for random initial fields with strong correlations between the velocity field and the current density. Again, it was found that with increasing Reynolds number. Interestingly, a comparison with the data of Ref. Mininni and Pouquet (2009) showed that the approach to the asymptote was slower than for the data of Ref. Mininni and Pouquet (2009), suggesting an influence of the level of certain vector field correlations on the approach to the asymptote. A theoretical model for dissipation rate scaling in freely decaying MHD turbulence was put forward recently Linkmann et al. (2015) based on the von Kármán-Howarth energy balance equations (vKHE) in terms of Elsässer fields Politano and Pouquet (1998). For unity magnetic Prandtl number it predicts the dependence of on a generalized Reynolds number , with denoting the root-mean-square value of one Elsässer field, the integral scale corresponding to the other Elsässer field, while and are the kinematic viscosity and the magnetic resistivity, respectively. The model equation has the following form
where and are time-dependent coefficients depending on several parameters, which themselves depend on the magnetic, cross- and kinetic helicities. The predictions of this equation were subsequently tested against data obtained from medium to high resolution DNSs of freely decaying homogeneous MHD turbulence leading to a very good agreement between theory and data.
In summary, there is compelling numerical and theoretical evidence for finite dissipation in freely decaying MHD turbulence at least for unity magnetic Prandtl number , while so far no systematic results for the stationary case have been reported. In this paper we extend the derivation carried out in Ref. Linkmann et al. (2015) to include the effects of external forces and we present the first systematic study of dissipation rate scaling for stationary MHD turbulence. In order to be able to test the model equation against DNS data for a large range of generalized Reynolds numbers, we concentrate on the case . The most general form of Eq. (1) for nonstationary flows with large-scale external forcing is derived, which can be applied to freely decaying and stationary flows by setting the corresponding terms to zero. This generalization of Eq. (1) is the first main result of the paper, it is applicable to both freely decaying and stationary MHD turbulence. It implies that the dissipation rate of total energy is finite in the limit in analogy to hydrodynamics, and highlights the dependence of the coefficients and on the external forces. As such, Eq. (1) predicts nonuniversal values of the asymptotic value of the dimensionless dissipation rate in the infinite Reynolds number limit and of the approach to the asymptote for a variety of MHD flows. The resulting theoretical predictions for the stationary case are compared to DNS data for stationary MHD turbulence for three different types of mechanical forcing while the results for the freely decaying case Linkmann et al. (2015) are reviewed for completeness. The DNS data shows good agreement with Eq. (1) and the different forcing schemes have no measurable effect on the values of the coefficients in Eq. (1). The measured values of lie between for free decay, where the value depends on the initial level of cross- and magnetic helicities. In the stationary case we measure .
This paper is organized as follows. We begin by reviewing the formulation of
the MHD equations in terms of Elsässer fields in Sec. II
where we introduce the basic quantities we aim to study in both formulations of
the MHD equations. In Section III we extend the derivation
put forward in Ref. Linkmann et al. (2015) to nonstationary MHD turbulence. The
model equation is verified against DNS data for statistically steady MHD
turbulence and the comparison to data for freely decaying MHD turbulence
presented in Ref. Linkmann et al. (2015) is reviewed in Sec. IV,
where special emphasis is given to the question of nonuniversality of MHD
turbulence in the context of external forces and the level of cross- and
magnetic helicities. Our results are summarized and discussed in the context
of related work in hydrodynamic and MHD turbulence in
Sec. V, where we also outline suggestions for further work.
Ii The total dissipation in terms of Elsässer fields
In this paper we consider statistically homogeneous MHD turbulence in the absence of a background magnetic field. The flow is taken to be incompressible, leading to the following set of coupled partial differential equations
where denotes the velocity field, the magnetic induction expressed in Alfvén units, the kinematic viscosity, the magnetic resistivity, the thermodynamic pressure, and are external mechanical and electromagnetic forces, which may be present, and denotes the density which is set to unity for convenience. Equations (2)-(4) are considered on a three-dimensional domain , which due to homogeneity can either be the full space or a subdomain with periodic boundary conditions. The MHD equations (2)-(4) can be formulated more symmetrically using Elsässer variables Elsässer (1950)
where and the pressure consists of the sum of the thermodynamic pressure and the magnetic pressure . Which formulation of the MHD equations is chosen often depends on the physical problem, for some problems the Elsässer formalism is technically convenient, while the formulation using the primary fields and facilitates physical understanding. The ideal invariants total energy , cross-helicity and magnetic helicity are given in the respective formulations of the MHD equation by
with , and denoting the respective Fourier transforms of the magnetic,
velocity and Elsässer fields, while is the Fourier transform of the magnetic vector potential .
The angled brackets indicate an ensemble average.
Equation (9) is gauge-independent as shown in Appendix A.
We now motivate the use of the Elsässer formulation for the study of the dimensionless dissipation coefficient in MHD. In hydrodynamics, the dimensionless dissipation coefficient is defined in terms of the Taylor surrogate expression for the total dissipation rate, , where denotes the root-mean-square (rms) value of the velocity field and the integral scale defined with respect to the velocity field, as
However, in MHD there are several quantities that may be used to define an MHD analogue to the Taylor surrogate expression, such as the rms value of the magnetic field, one of the different length scales defined with respect to either or , or the total energy.
Since the total dissipation in MHD turbulence should be related to the flux of total energy through different scales, one may think of defining a dimensionless dissipation coefficient for MHD in terms of the total energy. However, this would lead to a nondimensionalization of the hydrodynamic transfer term with a magnetic quantity. This can be seen by considering the analog of the von Kármán-Howarth energy balance equation in real space Chandrasekhar (1951) stated here for the case of free decay
where , and are the longitudinal structure functions, the longitudinal correlation function and another correlation function. The longitudinal structure and correlation functions are given by
where and denotes the longitudinal component of a vector field , that is its component parallel to the displacement vector , and
its longitudinal increment. The function is defined through the third-order correlation tensor
As can be seen from their respective definitions, the functions
and scale with while the function scales with
. If Eq. (11) were to be nondimensionalized with
respect to the total energy
then the purely hydrodynamic term would be scaled partially
by a magnetic quantity.
This problem of inconsistent nondimensionalization can be avoided by working with Elsässer fields, which requires an expression for the total dissipation rate in terms of Elsässer fields. The total rate of energy dissipation in MHD turbulence is given by the sum of Ohmic and viscous dissipation
Similarly, the total dissipation rate can be decomposed into its respective contributions from the Elsässer dissipation rates
where the Elsässer dissipation rates are defined as
with . The total dissipation rate relates to the sum of the Elsässer dissipation rates
where the cross-helicity dissipation rate is given by
Since this paper is concerned with both stationary and nonstationary flows, the total energy input rate must also be considered. Similar to the dissipation rate, the input rate can be split up into either kinetic and magnetic contributions or the Elsässer contributions
The latter equation can be rewritten as
where denotes the input rate of the cross-helicity.
Iii Derivation of the equation
Since the total dissipation rate can be expressed either in terms of the Elsässer fields or the primary fields and , it should be possible to describe it also by the vKHE for Politano and Pouquet (1998). For the freely decaying case no further complication arises as the rate of change of total energy, which figures on the left-hand side of the energy balance, equals the total dissipation rate. However, in the more general case the rate of change of the total energy is given by the difference of energy input and dissipation. That is, in the most general case the total energy dissipation rate is given by
For the stationary case and one obtains . For the freely decaying case and the change in total energy is due to dissipation only, that is . In terms of Elsässer variables can also be expressed as
where denote the Elsässer energies. Since we have related the total dissipation rate to the rate of change of the Elsässer energies, we are now in a position to consider the energy balance equations for , which are stated here for the most general case of homogeneous forced nonstationary MHD flows without a mean magnetic field
where are (scale-dependent) energy input terms and
are the third-order longitudinal correlation function and the second-order structure functions of the Elsässer fields, respectively. As can be seen from the definition, the third-order correlation function scales with , where denote the respective rms values of the Elsässer fields. This permits a consistent nondimensionalization of the Elsässer vKHE using the appropriate quantities defined in terms of Elsässer variables. As such the complication that arose if the energy balance was written in terms of and can be circumvented. This motivates the definition of the dimensionless Elsässer dissipation rates as
are the integral scales defined with respect to 111The scaling is ill-defined for the (measure zero) cases , which correspond to exact solutions to the MHD equations where the nonlinear terms vanish. Thus no turbulent transfer is possible, and these cases are not amenable to an analysis which assumes nonzero energy transfer Politano and Pouquet (1998). . For balanced MHD turbulence, i.e. , one should expect , since
Therefore all quantities defined with respect to the rms fields and should be the same in this case. Finally, the dimensionless dissipation rate is defined as
Using the definition given in Eq. (34), the Elsässer energy balance equations (30) can now be consistently nondimensionalized. For conciseness the explicit time and spatial dependences are from now on omitted, unless there is a particular point to make.
iii.1 Dimensionless von Kármán-Howarth equations
Before proceeding further, the scale-dependent forcing term on the left-hand side of this equation needs to be analyzed in some detail in order to clarify its relation to the energy input rates and . The Elsässer energy input is given by
Since the energy input rate is given by , the correlation function can be expressed as
where are dimensionless even functions of satisfying and the characteristic scale of the forcing. At scales much smaller than the forcing scale, i.e. for , for suitable types of forces can be expanded in a Taylor series Novikov (1965), leading to the following expression for the energy input
In the limit of infinite Reynolds number the inertial range extends through all wavenumbers, formally implying that , where Eq. (41) implies . Therefore it should be possible to split the term into a constant, , and a scale-dependent term , which encodes the additional scale dependence introduced by realistic, finite Reynolds number forcing. For consistency, this scale-dependent term must vanish in the formal limit . This can be achieved by writing in terms of the correlation of the force and Elsässer field increments
Therefore we define
where . Hence the energy input can be expressed as the sum of the scale-independent energy input rate and a scale-dependent term which vanishes in the formal limit
with . Substitution of Eq. (44) into the nondimensionalized energy balance Eq. (III.1) leads to the dimensionless version of the Elsässer vKHE for homogeneous MHD turbulence in the most general case for nonstationary flows at any magnetic Prandtl number
where and denote generalized large-scale Reynolds numbers given by
In order to express Eq. (45) more concisely, the following dimensionless functions are defined
such that Eq. (45) can be written as
This equation can be applied to the two simpler cases of freely decaying and stationary MHD turbulence by setting the corresponding terms to zero. For the case of free decay there are no external forces therefore , while for the stationary case the terms and vanish. A further simplification concerns the case , that is , where the inverse of the generalized Reynolds numbers vanish. In this case the evolution of depends only on , and an approximate analysis using asymptotic series is possible. Most numerical results are concerned with this case due to computational constraints, hence it would be very difficult to test an approximate equation against DNS data if not only but also needs to be varied. From now on the magnetic Prandtl number is therefore set to unity, keeping in mind that the analysis could be extended to provided the approximate equation derived in the following section is consistent with DNS data.
iii.2 Asymptotic analysis for the case
Equation (54) suggests a dependence of on , however, the structure and correlation functions also have a dependence on Reynolds number, which describes their deviation from their respective inertial-range forms. The highest derivative in Eq. (54) is multiplied by the small parameter , which suggests that this equation may be viewed as singular perturbation problem amenable to asymptotic analysis Lundgren (2002). The Elsässer vKHE was rescaled by the rms values of the Elsässer fields and the corresponding integral length scales, where the integral scales are by definition the large-scale quantities, the interpretation in hydrodynamics usually being that they represent the size of the largest eddies. As such, the nondimensionalization was carried out with respect to quantities describing the large scales, that is, with respect to ‘outer’ variables. Hence outer asymptotic expansions of the nondimensional structure and correlation functions are considered with respect to the inverse of the (large-scale) generalized Reynolds numbers . We point out that the case would require expansions in two parameters, where the cases and must be treated separately due to a sign change in between the two cases.
The formal asymptotic series of a generic function [used for conciseness in place of the functions on the right-hand side of Eq. (54)] up to second order in reads
After substitution of the expansions into Eq. (54), collecting terms of the same order in , one arrives at equations describing the behavior of and
up to second order in , using the coefficients , and defined as
in order to write Eq. (54) in a more concise way. The zero-order term in the expansion of the function vanishes, since corresponds to the scale-dependent part of the energy input which vanishes in the limit , hence . According to the definition of in Eq. (37), the asymptote is given by
and using the definition of the generalized Reynolds numbers, which implies one can define
( is defined analogously), resulting in the following expression for the dimensionless dissipation rate
Since the time dependence of the various quantities in this problem has been suppressed for conciseness, it has to be emphasized that Eq. (62) is time dependent, including the Reynolds number . Equation (62) in conjunction with eqs. (57)-(59) is the most general asymptotic expression for the Reynolds number dependence of developed so far. It is applicable for freely decaying, stationary and non-stationary MHD turbulence in the presence of external forces, and it may be applied to the corresponding problem in non-conducting fluids by setting . As such it extends previous results for freely decaying MHD turbulence Linkmann et al. (2015), as well as for the stationary case in homogeneous isotropic turbulence of non-conducting fluids McComb et al. (2015a).
For nonstationary MHD turbulence at the peak of dissipation the term in Eq. (57) vanishes for constant flux of cross-helicity (that is, ), since in the infinite Reynolds number limit the second-order structure function will have its inertial range form at all scales. By self-similarity the spatial and temporal dependences of e.g. should be separable in the inertial range, that is
for some value , and
At the peak of dissipation
which implies . Equation (57) taken for nonstationary flows at the peak of dissipation is thus identical to Eq. (57) for stationary flows, which suggests that at this point in time a nonstationary flow may behave similarly to a stationary flow. We will come back to this point in Sec. IV. Due to selective decay, that is the faster decay of the total energy compared to and Biskamp (1993), in most situations one could expect to be small compared to in the infinite Reynolds number limit. In this case and
which recovers the inertial-range scaling results of Ref. Politano and Pouquet (1998) and reduces to Kolmogorov’s four-fifth law for .
iii.3 Relation of to energy and cross-helicity fluxes
In analogy to hydrodynamics, the asymptotes should describe the total energy flux, that is the contribution of the cross-helicity flux to the Elsässer flux should be canceled by the respective terms and in Eq. (57). However, since this is not immediately obvious from the derivation, further details are given here. For nonstationary turbulence at the peak of dissipation, Eq. (57) for the asymptotes reduces to
The dimensional version of this equation is
where it is assumed that the function has its inertial range form corresponding to . The function can also be expressed through the Elsässer increments Politano and Pouquet (1998)
which can be written in terms of the primary fields and as
(see e.g. Ref. Politano and Pouquet (1998)). The two terms on the first line of Eq. (III.3) are the flux terms in the evolution equation of the total energy, while the two terms on last line correspond to the flux terms in the evolution equation of the cross-helicity Politano and Pouquet (1998). Now Eq. (68) can be expressed in terms of the primary fields
where is the flux of total energy and the cross-helicity flux, which must equal for nonstationary MHD turbulence. Thus the contribution from the third-order correlator resulting in is canceled by , or, after nondimensionalization, the cross-helicity flux is canceled by . The two simpler cases of freely decaying and stationary MHD turbulence are recovered by setting either (free decay) or (stationary case).
Since is a measure of the flux of total energy across different scales in the inertial range, differences for the value of this asymptote should be expected for systems with different initial values for the ideal invariants and . The flux of total energy and thus the asymptote is an averaged quantity. This implies that cancellations between forward and inverse fluxes may take place leading on average to a positive value of the flux, that is, forward transfer from the large scales to the small scales. In case of , the value of should therefore be less than for due to a more pronounced inverse energy transfer in the helical case, the result of which is less average forward transfer and thus a smaller value of the (average) flux of total energy. For the asymptote is expected to be smaller than for , since alignment of and weakens the coupling of the two fields in the induction equation, leading to less transfer of magnetic energy across different scales and presumably also less transfer of kinetic to magnetic energy.
Furthermore, from an analysis of helical triadic interactions in ideal MHD carried out in Ref. Linkmann et al. (2016) it may be expected that high values of cross-helicity have a different effect on the asymptote , depending on the level of magnetic helicity. The analytical results suggested that the cross-helicity may have an asymmetric effect on the nonlinear transfers in the sense that the self-ordering inverse triadic transfers are less quenched by high levels of compared to the forward transfers. The triads contributing to inverse transfers were mainly those where magnetic field modes of like-signed helicity interact, and so for simulations with maximal initial magnetic helicity the dynamics will be dominated by these triads. If the inverse fluxes are less affected by the cross-helicity than the forward fluxes, then the expectation is that for a comparison of the value of between systems with (i) high and , (ii) high and , (iii) and high and finally (iv) and