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Functional renormalisation group approach to shell models of turbulence
by Côme Fontaine, Malo Tarpin, Freddy Bouchet, Léonie Canet
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Léonie Canet |
| Submission information | |
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| Preprint Link: | scipost_202310_00022v1 (pdf) |
| Code repository: | https://gitlab.com/CFon10/sabra_frg_solver |
| Date accepted: | 2023-11-10 |
| Date submitted: | 2023-10-20 08:35 |
| Submitted by: | Canet, Léonie |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Shell models are simplified models of hydrodynamic turbulence, retaining only some essential features of the original equations, such as the non-linearity, symmetries and quadratic invariants. Yet, they were shown to reproduce the most salient properties of developed turbulence, in particular universal statistics and multi-scaling. We set up the functional renormalisation group (RG) formalism to study generic shell models. In particular, we formulate an inverse RG flow, which consists in integrating out fluctuation modes from the large scales (small wavenumbers) to the small scales (large wavenumbers), which is physically grounded and has long been advocated in the context of turbulence. Focusing on the Sabra shell model, we study the effect of both a large-scale forcing, and a power-law forcing exerted at all scales. We show that these two types of forcing yield different fixed points, and thus correspond to distinct universality classes, characterised by different scaling exponents. We find that the power-law forcing leads to dimensional (K41-like) scaling, while the large-scale forcing entails anomalous scaling. }
Author comments upon resubmission
In fact, independently of the referee’s suggestions, the paper has also changed quite significantly. This is because we detected a small mistake in the numerics to solve the FRG flow equations. The final conclusions are similar, regarding in particular the interplay between the long-range forcing and the large-scale one, and the existence of anomalous scaling at the large-scale fixed point. However, the way to obtain them has required more efforts. In particular, we have implemented both a direct RG, as was done in the previous version, and an inverse RG, which is a new aspect. We believe that this result is interesting on its own, and this has substantially increased the interest of the manuscript. We have also improved on the level of approximation, by pushing further the vertex expansion.
We think that the overall changes, both on technical aspects, as well as on the discussion of the analogy with hydrodynamical turbulence, have been very beneficial to the paper.
List of changes
* An inverse RG flow has been implemented, in addition to the direct RG flow.
Results from both RG flows are now presented and compared.
* The discussion on the analogies with Navier-Stokes turbulence has been extended.
* The approximation scheme has been pushed further, up to bi-quadratic order in the fields.
Published as SciPost Phys. 15, 212 (2023)