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Functional renormalisation group approach to shell models of turbulence
by Côme Fontaine, Malo Tarpin, Freddy Bouchet, Léonie Canet
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Submission summary
Authors (as registered SciPost users):  Léonie Canet 
Submission information  

Preprint Link:  https://arxiv.org/abs/2208.00225v1 (pdf) 
Code repository:  https://gitlab.com/CFon10/LORGSabra 
Date submitted:  20220802 14:24 
Submitted by:  Canet, Léonie 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Shell models are simplified models of hydrodynamic turbulence, retaining only some essential features of the original equations, such as the nonlinearity, symmetries and quadratic invariants. Yet, they were shown to reproduce the most salient properties of developed turbulence, in particular universal statistics and multiscaling. We set up the functional renormalisation group formalism to study generic shell models. Focusing on the Sabra shell model, we obtain the renormalisation group fixedpoint describing its universal properties for a forcing exerted at large scales. We find that the structure functions exhibit universal scaling, and show that they acquire anomalous scaling exponents, with $\zeta_2\simeq 0.74\pm 0.02$, which is in good agreement with results from numerical simulations. Moreover, we study the influence of a longrange powerlaw forcing, and resolve its interplay with the Sabra fixedpoint corresponding to large scale forcing.
Current status:
Reports on this Submission
Anonymous Report 2 on 2023520 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2208.00225v1, delivered 20230520, doi: 10.21468/SciPost.Report.7220
Strengths
Introduction of a powerful theoretical framework to describe hydrodynamic turbulence.
Captures anomalous scaling in shell model.
Able to contrast longrange and shortrange forcing with an important consequence for existing attempts at describing turbulence within the perturbative renormalization group.
Manuscript is clearly presented.
Weaknesses
Lowest order approximation does not describe multifractal scaling
Report
In this manuscript the authors apply the functional Renormalization Group to simplified models of hydrodynamic turbulence known as shell models. They more specifically focus on one representative of the class of models, the Sabra shell model. The authors show for the first time the existence of a fixed point controlling the universal properties in the case of a forcing at large scale. They do so by approximating the exact functional RG flow equations through a truncation at the lowest order of the vertex expansion. The full functional dependence on wavenumber is retained, which allows the authors to calculate the scaling dimensions of the structure functions. The anomalous character of the scaling that originates from intermittency is well captured and the exponent for the secondorder structure function is is good agreement with computer simulation results. However, the present approximation fails at describing the expected multifractal scaling of the model.
An interesting property of the functional RG method is that it allows a study of the nature of forcing on the fixed point. Longrange forcing is defined as scale free while shortrange forcing is concentrated at the large scale only. The authors show that the two types of forcing lead to distinct fixed points and, more importantly, that the two fixed points do not coincide when the exponent characterizing the scalefree (power law) forcing goes to zero. This casts doubts on previous attempts to access the shortrange forcing case as a limit of the longrange one within the perturbative RG.
This work is a valuable step for the development of a functional RG approach to turbulence and it provides the framework to improve the results for accessing multiscaling and for describing the NavierStokes turbulence. The manuscript is also clearly presented. I therefore recommend its publication in SciPost Physics.
Anonymous Report 1 on 20221019 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2208.00225v1, delivered 20221018, doi: 10.21468/SciPost.Report.5925
Strengths
1 clearly written
2scientifically valid, state of the art application of functional renormalization methods to scaling analysis.
Weaknesses
functional renormalization scaling predictions rely on closures. This is however not a weakness of the paper in itself rather of the method so far.
Report
The manuscript describes the application of functional RG to the ``Sabra'' shell model of turbulence. Shell models are stylized models of fluid turbulence. Before the proof of anomalous scaling in the Kraichnan methods, shell models possibly provided convincing evidences
though only numerical that anomalous scaling in passive scalar turbulence is not a finite effect and that the same may well be true for the nonlinear advecting field.
The present paper introduces a closure, the so called LO approximations, of the functional renormalization flow which is then numerically solved.
The results are consistent within numerical accuracy with the prediction of anomalous scaling of the shell model counterparts of structure functions
by direct numerical simulations. The authors took also care to verify that whereas the anomalous exponents are universal with respect to the bare forcing amplitude and molecular dissipation they may not be Fig 2 right panel with respect to the shell ratio and the shell model parameter.
Another interesting observation regards the dependence of the RG fixed point upon the type of forcing. The authors' finding is that there exist two distinct fixed points depending on the nature of the forcing. If the forcing is concentrated on one infrared shell the fixed point describes anomalous scaling. If instead the forcing is a powerlaw on all shells the corresponding fixed point is associated to dimensional scaling. The authors speculate that the same phenomenon may occur in realistic models of fluid turbulence. This conclusion maybe, however, too hasty or perhaps not clearly formulated. Numerical simulations of the stochastic NavierStokes equation in two dimensions by A Mazzino et al PRL 99 (14), 144502, and JSTAT 2009 (10), P10012 give, so far undisputed evidence, that power law forcing may bring about both type of behaviors. Specifically, dimensional scaling dominates in parametric regions where it is expected to be so based on the analysis of the KarmanHowarthMonin equation. Conversely inverse energy cascade for ($ d=2 $ and $ \varepsilon<4 $) and or direct enstrophy cascade ($ \varepsilon>6 $) become the dominant contributions to scaling in complementary parametric regions again in agreement with predictions based on the asymptotic analysis of the KHM equation. In my understanding, the benchmark for identifying a ``turbulence'' fixed point for functional RG in NavierStokes is the 5/3 law and consistency with Galilean invariance (Ward identities). As both types of forcing (e.g. power law at $ \epsilon=4 $ and large scale in $d=3$) satisfy these requirements, the interpretation of the fixed points they bring about in what should be the fully developed turbulence regime remains, in my opinion, inconclusive. In the case of shell models the absence of spatial structure maybe ultimately responsible for the observed phenomenon. A more careful discussion by the authors of this point would be desirable.
Overall the paper is well written and comes from a collaboration including one of the leading experts (Canet) on FRG methods so I have no remarks on the technical implementation. Hence my recommendation is to publish with minor amendments in relation to my comments above.