Multifractals, free turbulence decay laws and the Scaling Gyroscopes Cascade model
Authors: Li Xin , Schertzer Daniel , Roustan Yelva , Tchiguirinskaia Ioulia 
Published: March 2021

&lt;p&gt;Turbulence being a dissipative system decays when being &quot;free&quot;, i.e. without any force. The law of this decay has been intriguing for quite a while. Assuming that for vanishing viscosity, the whole spectrum is self-similar, as well as stationary for low wave numbers/large eddies&amp;#160; (E(k,t) &amp;#8776;C&lt;sub&gt;S&lt;/sub&gt; k&lt;sup&gt;S&lt;/sup&gt;, k &amp;#8594; 0) , it was shown [1] that the total energy of turbulence has a power-law decay: E(t) = &amp;#8747; E(k,t) dk &amp;#8776; t&lt;sup&gt;-a(s)&lt;/sup&gt;: a(s) =2(s+1)/(s+3) . This was particularly thought to be relevant for s=4, C&lt;sub&gt;4&lt;/sub&gt; being proportional to the Loitsianski integral, assumed to be time-invariant [2]. However, it was shown with the help of the eddy-damped quasi-normal Markovian (EDQNM) [3] that there is an energy backscatter term transferring energy from energy-containing eddies by nonlocal triads interactions to large eddies, which behaves like T&lt;sub&gt;NL&lt;/sub&gt;&amp;#8776; k&lt;sup&gt;4&lt;/sup&gt; and therefore prevents the invariance of the Loitsianski integral. This implies that the theoretical exponent a(s) = 2(s+1)/(s+3)&amp;#160; is only valid for s&lt;4 and that a(s) =a(4)=-(10-2&amp;#947;)/7 for s&amp;#8805; 4 with C&lt;sub&gt;4&lt;/sub&gt;(t) &amp;#8776; t &lt;sup&gt;&amp;#947;&lt;/sup&gt;, &amp;#947;&gt;0. The turbulence decay is therefore slower than previously expected for s &amp;#8805; 4 due to the backscatter term that progressively stores energy in large eddies.&amp;#160;&lt;br&gt;EDQNM provides the estimate &amp;#947; &amp;#8776; 0.16. However, a strong limitation of EDQNM and similar models (e.g. Direct Interaction Approximation, Test Field Model) is that these models are not able to represent intermittency, which is a fundamental phenomenon of turbulence [4] and this could bring into questions the previous results. We, therefore, investigate this question with the Scaling Gyroscopes Cascade (SGC) model [5], which is based on nonlocal interactions and display multifractal intermittency [6]. We first theoretically argue that SGC confirms the existence of the backscatter term, but the turbulence decay is no longer smooth but occurs by puffs and we provide numerical evidence of this.&lt;/p&gt;&lt;p&gt;Keywords: Loitsianski integral; intermittency; infrared spectrum; SGC model; energy decay&lt;/p&gt;&lt;p&gt;[1]M. Lesieur and D. Schertzer, &amp;#8216;&amp;#8216;Amortissement auto-similaire d&amp;#8217;une turbulence a&amp;#8216; grand nombre de Reynolds,&amp;#8217;&amp;#8217; J. Mec. 17, 609 1978 .&lt;/p&gt;&lt;p&gt;[2]Davidson, P. A. (2000). Was Loitsyansky correct? A review of the arguments. &lt;em&gt;Journal of Turbulence&lt;/em&gt;, &lt;em&gt;1&lt;/em&gt;(1), 006-006.&lt;/p&gt;&lt;p&gt;[3]Frisch, U., Lesieur, M.,Schertzer, D. (1980). Comments on the quasi- normal Markovian approximation for fully-developed turbulence. Jour- nal of Fluid Mechanics, 97(1), 181-192.&lt;/p&gt;&lt;p&gt;[4]Morf, R. H., Orszag, S. A., Frisch, U. (1980). Spontaneous singularity in three-dimensional inviscid, incompressible ow. Physical Review Letters, 44(9), 572.&lt;/p&gt;&lt;p&gt;[5]Chigirinskaya, Y., Schertzer, D.,&amp;#160; Lovejoy, S. (1997). Scaling gyroscopes cascade: universal multifractal features of 2-D and 3-D turbulence. &lt;em&gt;Fractals and Chaos in Chemical Engineering. World Scientific, Singapore&lt;/em&gt;, 371-384.&lt;/p&gt;&lt;p&gt;[6]Chigirinskaya, Y.,&amp;#160; Schertzer, D. (1997). Cascade of scaling gyroscopes: Lie structure, universal multifractals and self-organized criticality in turbulence. In &lt;em&gt;Stochastic Models in Geosystems&lt;/em&gt; (pp. 57-81). Springer, New York, NY.&lt;/p&gt;