Energy fluxes in quasi-equilibrium flows
We examine the relation between the absolute equilibrium state of the spectrally truncated Euler equations (TEE) predicted by Kraichnan (1973) to the forced and dissipated flows of the spectrally truncated Navier-Stokes (TNS) equations. In both of these idealized systems a finite number of Fourier modes is kept contained inside a sphere of radius . We show, using an asymptotic expansion of the Fokker-Planck equation, that in the limit of small viscosity and fixed maximum wavenumber the flow approaches the absolute equilibrium solution of Kraichnan with such an effective “temperature” that there is a balance between the energy injection and the energy dissipation rate. We further investigate the TNS system using direct numerical simulations. The simulations demonstrate that, at steady state with large-scale forcing and dissipation acting only at small-scales, the TNS reproduce the Kolmogorov energy spectrum if the viscosity is large enough so that the Kolmogorov dissipation wavenumber is smaller than . As viscosity becomes smaller then a bottleneck effect appears taking the form of the equipartition spectrum at small scales. As is decreased even further the equipartition spectrum occupies all scales approaching the asymptotic equilibrium solutions found before. If the forcing is applied at small scales and the dissipation acts only at large scales then the equipartition spectrum appears at all scales for all values of . In both cases a finite forward or inverse flux is present independent of the amplitude of the viscosity even for the cases where the flow is close to the equilibrium state solutions.
Turbulence is a classical example of an out of equilibrium system. In steady state, energy is constantly injected at some scale while it is dissipated at smaller scales by viscous forces. This process requires a finite flux of energy from the former scale to the latter that is provided by the well known Kolmogorov-Richardson cascade. Despite the out of equilibrium nature of turbulence there are circumstances where equilibrium dynamics become relevant. This has been claimed to be the case for the scales larger than than the injection scale . At these scales the energy flux is zero and can possibly be modeled using equilibrium dynamics (Dallas et al. 2015; Cameron et al. 2017; Alexakis & Brachet 2018). Equilibrium dynamics become also relevant in the presence of inverse cascades in finite domains where large scale condensates form (Kraichnan 1967; Robert & Sommeria 1991; Naso et al. 2010; Bouchet & Venaille 2012; Shukla et al. 2016). Finally understanding equilibrium dynamics is important for systems that display a transition of from forward to an inverse cascade (Alexakis & Biferale 2018) since the large scale flows in these systems transition from an equilibrium state to an out of equilibrium state. Besides the possible applications, understanding the equilibrium dynamics in turbulence is also a much needed step required before understanding its much harder out-of-equilibrium counterpart. This has led many researchers (Lee 1952; Kraichnan 1967, 1973; Hopf 1952; Orszag 1977) to investigate the equilibrium state of the truncated Euler equations (TEE) where only a finite number of Fourier modes is kept and are given by:
Here is the incompressible velocity field, is the pressure and is a projection operator that sets to zero all Fourier modes except those that belong to a particular set (here chosen to be all wavenumbers inside a sphere centered at the origin with radius ). These equations conserve exactly two quadratic invariants
In Fourier space these invariants are distributed among the different modes which are quantified by the energy and helicity spherically averaged spectra respectively defined as
Here is the Fourier transform of and we have assumed a triple periodic cubic domain of size .
At late times this system reaches a statistically steady state whose properties are fully determined by these two invariants. Using Liouville’s theorem and assuming ergodicity Lee (1952) predicted that at absolute equilibrium this system will be such that every state of a given energy is equally probable. This is equivalent to the micro-canonical ensemble in statistical physics and it leads to equipartition of energy among all the degrees of freedom (ie among all Fourier amplitudes) and an energy spectrum given by . Kraichnan (1973) generalized these results including helicity and assuming a Gaussian equipartition ensemble
where is the probability distribution for the system to be found in the state , is the energy and the helicity given in 1 and a normalization constant. The parameters and are the equivalent of an inverse temperature and inverse chemical potential in analogy with statistical physics. For no helicity this leads to the energy spectrum predicted by Lee. In the presence of helicity one obtains
The coefficients and are determined by imposing the conditions
and and are the initial energy and helicity respectively. The predictions above have been verified for the truncated Euler system in numerous numerical simulations (Orszag & Patterson 1977; Cichowlas et al. 2005; Krstulovic et al. 2009; Dallas et al. 2015; Cameron et al. 2017; Alexakis & Brachet 2018).
In the TEE however there is no exchange of energy with external sources or sinks and it is thus harder to make contact with the more realistic systems mentioned in the beginning of the introduction such as the scales larger than the forcing scale in a turbulent flow. It was shown recently in Alexakis & Brachet (2018) that, in some cases, although the large scales are close to an equilibrium state there is still exchange of energy with the smaller turbulent and forcing scales generating energy fluxes (from the forced scales to the large scales and from the large scales to the turbulent scales). It appears thus that, even in the presence of sources and sinks, equilibrium dynamics can still be relevant. In this work we examine further this possibility by looking at the truncated Navier-Stokes (TNS) equations where there is constant energy injection and dissipation like in the regular Navier-Stokes Equation but the system is limited to a finite number of Fourier modes as in TEE. We show analytically in the next section that for weak viscosity the system indeed reaches a quasi-equilibrium state whose probability distribution can be calculated. We verify these results using direct numerical simulations in section 3 and our conclusions are presented in the last section.
2 Asymptotic expansion
We consider the truncated Navier-Stokes (TNS) equations
where is the incompressible velocity field, is the pressure, is the viscosity and is a forcing function. The domain is a pariodic cube and the projection operator sets to zero all Fourier modes with wavenumbers outside the sphere of radius . In total there are Fourier wavenumbers inside this sphere. In order to proceed it helps to write the truncated Navier Stokes equation in Fourier space using the Craya-Lesieur-Herring decomposition (Craya (1958); Lesieur (1972); Herring (1974)) where every Fourier mode is written as the sum of two modes one with positive helicity and one with negative helicity . The two vectors are given by:
where is an arbitrary unit vector. The sign index indicates the sign of the helicity of . The basis vectors are eigenfunctions of the curl operator in Fourier space such that . They satisfy and , where the complex conjugate of is given by . They form a complete base for incompressible vector fields. This decomposition has been extensively used and discussed in the literature (Cambon & Jacquin 1989; Waleffe 1992; Chen et al. 2003; Biferale et al. 2012; Moffatt 2014; Alexakis 2017; Sahoo et al. 2017). Note that since every is described by two complex amplitudes that satisfy there are in total independent degrees of freedom. The truncated Navier-Stokes using the helical decomposition can then be written as
where the nonlinear term is written as the convolution
and the tensor is given by The nonlinearity satisfies the following relations
that correspond to the energy and helicity conservation respectively and
This last relation indicates that phase space volume is conserved by the nonlinearity (ie it satisfies a Liouville condition). We will assume that the forcing is written as where are random complex amplitudes that are statistically independent, normally distributed and delta-correlated in time such that . With this choice each forcing mode injects energy to the system on average at rate . Then the Fokker-Plank equation for the probability density in the 2N-dimensional space of all complex amplitudes is given by
where the sum is over all 2N modes . Multiplying by , integrating over the phase-space volume and using integration by parts we obtain the energy balance equation
where is the energy dissipation, is the injection rate and the brackets stand for the average where stands for the phase space volume element .
We are interested in the limit that the energy injection and dissipation rate are a small perturbation to the thermalized fluctuations. We thus set , where is a small parameter. We then expand in power series of as . We are also going to also consider the long time limit so we can neglect the time derivative. To zeroth order we then have
The equation above implies that is constant along the trajectories in the phase space followed by solutions of the truncated Euler equations. These trajectories are expected to be chaotic for large and since this is such a high dimensional space we can also conjecture that these trajectories are space filling (ie ergodic) in the subspace constrained by the invariants of the system. In other words we assume that the trajectory will pass arbitrarily close to any point that the same energy and helicity as the initial conditions. In this case is determined by the energy and helicity of the system . For the present work however we are going to neglect the second invariant the helicity and assume dependence only on the energy. We then write the solution of eq. (2.0) as
To next order we then get
Substituting eq. (2.0) and using the chain rule for the derivatives
we obtain for the function
To obtain a closed equation for we average eq. (2.0) over the volume of all points in phase space of energy between and . This consists of a spherical shell in the 2N-dimensional phase space of radius . Averaging over this volume leads the sum in the left hand side to drop out because it is a divergence and the trajectories determined by stay inside the shell. The volume integrals of terms independent of are proportional to the shell volume where is the surface of an unit radius 2N-dimensional sphere. Terms proportional to result due to symmetry . This leads to
If we set and then by multiplying by the equation simplifies to
that has the bounded solution:
Note that depends only on the ratio of and and thus is independent of and we have thus dropped the primes. Using eq. (2.0) one can also verify that the energy balance relation of eq. (2.0) is satisfied. The results indicate therefore that for small viscosity the truncated system will converge to the absolute equilibrium solutions of such amplitude so that the viscous dissipation balances the energy injection rate!
3 Numerical simulations
In this section we test the results of the previous section and extend our investigation beyond the asymptotic limit using direct numerical simulations of the TNS system of eq. (2.0). The simulations were performed using the ghost code (Mininni et al. 2011) that is a pseudospectral code with 2/3 de-alliasing and a second order Runge-Kutta. For all runs the energy injection rate was fixed to unity and the total integration times were sufficiently long so that steady states were reached. The forcing used is random and white in time as the one discussed in the previous section. It is limited to a spherical shell of wavenumbers satisfying . To have a clear inertial range (a range of wavenumbers where forcing and dissipation effects can be neglected) we replace by the modified viscous term that acts only a particular spherical set of wavenumbers satisfying . We will consider two cases for and . In the first case the simulations were performed on a numerical grid that, after de-aliasing, leads to and was forced at large scales while dissipation was located at large wavenumbers . In the second case a smaller grid was used with and energy dissipation located at small wavenumbers while the energy injection was at large wavenumbers .
We begin with the first case. We used a series of simulations with varying the viscous coefficient from to . The resulting energy spectra are shown in the left panel of figure 1 for six values of . Dark colors indicate large values of while bright values indicate small values of . For the large values of the energy spectra in the inertial range display a negative power-law distribution close to the Kolmogorov prediction (where is the Kolmogorov constant Sreenivasan (1995)). As the value of is decreased a bottleneck at large wave numbers appears and energy starts to pile up at the smallest scales of the system. As the value of is decreased further this bottleneck appears to take the form of a positive power-law close to the thermal equilibrium prediction . This thermal spectrum occupies more wavenumbers as the viscosity is decreased until all wavenumbers follow this scaling. At the smallest value of the result is compared with the asymptotic result obtained in the previous section. (We note that due the modified viscous term the sum in order to calculate in 2.0 is restricted only on the wavenumbers of the dissipation shell.) The proportionality coefficient can be estimated to be in order for the energy balance condition to be satisfied. Matching the thermal with the Kolmogorov spectrum we obtain that the transition occurs at the wavenumber
where is the Kolmogorov length-scale. The right panel of fig. 1 shows a comparison of this estimate with measured from the spectra as the wavenumber at which obtains its minimum. Both the scaling and the pre-factor agree very well with the results from the simulations. Using this result we can conclude that the Kolmogorov spectrum then dominates when that implies while the thermal spectrum is present at all scales when is smaller than the forcing wavenumber ().
These results resemble a lot the time evolution of the truncated Euler equations studied in Cichowlas et al. (2005) for which at early times a energy spectrum develops before the maximum wavenumber is reached. While after is reached, the thermalized energy spectrum starts to develop. Similarities can also be found with the recent work by Shukla et al. (2018) where the steady state of a time-reversible version of the Navier-Stokes system (Gallavotti 1996) was studied and similar transitions from a Kolmogorov to a thermal quasi-equilibrium was observed. The results were interpreted in terms of a phase transition, a possibility that could further explored for the present work as well.
We note that the time averaged flux from large scales to small scales is constant in the inertial range. What varies as we change the value of is the amplitude of the fluctuations of the flux around this mean value of the flux. This is displayed in figure 2 where the mean flux is shown with dark line together with the instantaneous fluxes at different times with bright colors for four different values of . For sufficiently large value of the fluctuations of the flux are concentrated around the mean value without large deviations from it. As the value of the viscosity is increased the fluctuations at large wavenumbers are increased until finally the fluctuations of the mean flux are orders of magnitude larger than the averaged flux for all wave numbers.
Finally, because in the systems discussed in the introduction energy is introduced at smaller scales it is worth also to examine the case where the forcing is located at small scales while the dissipation is limited in the large scales. The spectra for four different values of are shown in the left panel of fig. 3. The forcing and dissipation shells are indicated in the graph. In this case a spectrum forms for all values of . The flux fluctuations shown in the right panel of the same figure are always dominant. We note however that their averaged value leads to a constant negative flux in the inertial range where no forcing or dissipation are present.
In this work we examined flows in finite set of wavenumbers that are close to equilibrium. We showed analytically that in the limit of small viscosity the statistically steady state of these flows converge to the Kraichnan (1973) solutions. We note that our prediction is not just for the spectrum in eq. (2.0) but for the full probability distribution for the system to find itself in a state given in eq. (2.0).
There were two major assumption that went in to this derivation. First we assumed ergodicity for the solutions of the TEE. This assumption appears in most calculations of classical statistical physics that although it can only be proved in very few systems it appears to be a plausible one for many systems with large numbers of degrees of freedom. For the TEE it appears at least to be in agreement with the results of numerical simulations. The second assumption we made was to neglect the effect of the second invariant: the helicity. This assumption was made in order to simplify the (rather involved) calculation. Had we kept the effect of helicity then the zeroth order solution would have been reduced to a function of two variables and we would have ended up with an elliptic partial differential equation to solve for . However the presence of helicity would break the spherical symmetry in phase space that allowed us to calculate the involved integrals. The calculation is still feasible but much more lengthier and we leave it for future work.
The numerical investigation verified our analytical results and also shed light on how the TNS system transitions from the classical Kolmogorov turbulence state to the thermalized solutions of Kraichnan (1973) as the viscosity is varied, while keeping fixed. In particular, it is worth noting that a fixed amplitude flux (positive or negative) was always present in our simulations and that it was determined by the injection rate. However, as viscosity was decreased, the amplitudes of the fluctuations were increased making this flux a sub-dominant effect at very small viscosity.
There are many directions in which the present results can be pursued further. First of all including the effect of helicity is crucial to have a complete description of the system. Moreover carrying out the calculation at the next order so that statistics of the fluxes can also be calculated would equally be desirable. Finally, extending these results to two-dimensional flows, where the equilibrium states can take the form of large scale condensates, is another possible direction. Such calculations, although considerably longer than the ones presented here, should still be feasible and we hope to address them in our future work.
This work was granted access to the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenir supervised by the Agence Nationale pour la Recherche and the HPC resources of GENCI-TGCC & GENCI-CINES (Project No. A0050506421) where the present numerical simulations have been performed. This work has also been supported by the Agence nationale de la recherche (ANR DYSTURB project No. ANR-17-CE30-0004).
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