# Cascades and transitions in turbulent flows

###### Abstract

Turbulent flows are characterized by the non-linear cascades of energy and other inviscid invariants across a huge range of scales, from where they are injected to where they are dissipated. Recently, new experimental, numerical and theoretical works have revealed that many turbulent configurations deviate from the ideal three and two dimensional homogeneous and isotropic cases characterized by the presence of a strictly direct and inverse energy cascade, respectively. New phenomena appear that alter the global and local transfer properties. In this review, we provide a critical summary of historical and recent works from a unified point of view and we present a classification of all known transfer mechanisms. Beside the classical cases of direct and inverse energy cascades, the different scenarios include: split cascades for which an invariant flows both to small and large scales simultaneously, multiple/dual cascades of different quantities, bi-directional cascades where direct and inverse transfers of the same invariant coexist in the same scale-range and finally equilibrium states where no cascades are present, including the case when a large scale condensate is formed. We classify all possible transitions from one scenario to another as the control parameters are changed and we analyse when and why different configurations are observed. Our discussion is based on a set of paradigmatic applications: helical turbulence, rotating and/or stratified flows, magnetohydrodynamics (MHD) turbulence, and passive/active scalars where the transfer properties are altered as one changes the embedding dimensions, the thickness of the domain or other relevant control parameters, as, e.g., the Reynolds, Rossby, Froude, Peclet, or Alfven numbers. We briefly discuss the presence of anomalous scaling laws in 3D hydrodynamics and in other configurations, in connection with the intermittent nature of the energy dissipation in configuration space. A quick overview is also provided concerning the importance of cascades in other applications such as bounded flows, quantum fluids, relativistic and compressible turbulence, and active matter, together with a discussion of the implications for turbulent modelling. Finally, we present a series of open problems and challenges that future work needs to address.

###### keywords:

Homogeneous and isotropic turbulence, two dimensional turbulence, wave turbulence, rotating flows, thick layers, MHD, turbulent diffusion, passive and active scalars, stratified flows, helicity, convection, Lagrangian turbulence, Richardson cascade, inverse energy cascade, direct energy cascade, intermittency, anomalous scaling laws, energy condensate, absolute equilibrium.^{†}

^{†}journal: Physics Reports\biboptions

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###### Contents

- 1 Introduction
- 2 Theoretical Setup
- 3 Definitions of Turbulent Cascades
- 4 Applications
- 5 Further topics about cascades
- 6 Conclusions and open problems
- 7 Acknowledgements

## 1 Introduction

“Big whorls have little whorls that feed on their velocity, and little whorls have smaller whorls and so on to viscosity - in the molecular sense”. This is the celebrated poem composed by Lewis Fry Richardson in 1922 Richardson (), where it was proposed that turbulent cascades are the fundamental driving mechanism of the atmosphere, moving energy from the large injections scales down to the dissipative small scales. It is a visionary way to summarize many fundamental aspects of the turbulent energy transfer which is empirically observed in the three dimensional Navier-Stokes equations (NSE). The Richardson cascade description, first quantified by A.N. Kolmogorov Kolmogorov (), constitutes the most fundamental concept of turbulence theory. In more words, in a turbulent flow, the energy externally injected is redistributed among length scales due to non-linear eddy (whorls) interactions. If this energy is removed from the flow (or accumulated) at a scale significantly different than the injection scale, , a cascade can build up with a continuous transfer of energy from to . Depending on the system, the energy transfer can be towards the small scales and/or towards the large scales, leading to what it is referred to as a forward or inverse cascade, respectively.
In three dimensional (3D) homogeneous and isotropic turbulence (HIT) the energy is transferred to the small scales while in two dimensions (2D) it cascades to the large scales.
Despite the simplicity of the cascade description, after almost 100 years, we are still fighting to define the exact terms of the game and fail to have a complete statistical description even for the simplest case of HIT.
When and why Richardson’s cascade is correct?
What happens when it fails?
These and many other questions have challenged mathematicians, physicists and engineers for more than a century without reaching, to this date, clear answers, rightfully titling turbulence as the last open problem of classical physics.
There are various text books written over the years reviewing both findings and open questions of different aspects of turbulence Frisch (); davidson2015turbulence (); pope2001turbulent ().
A recent overview can be found in davidson2011voyage (), where the historical developments in engineering, mathematical, and physical sciences have been analysed from the ”shoulders of twelve historical fathers of turbulence research”.

The situation, however, can be very complex.
First, any inviscid invariant of the system is also subject to a non-linear transfer. The interaction of the transfer of different quantities plays an important role in determining the direction of their cascade. For example, it is the conservation of enstrophy that forces an inverse cascade of energy in 2D. In 3D, the second invariant is helicity and it is not sign-definite and all experimental investigations, numerical simulations and phenomenological theories indicate that both energy and helicity have a simultaneous mean transfer to the small scales. However, this is an empirical observation and it is not proven from basic principles yet.
The dual cascade of helicity and energy is important for 3D hydrodynamical turbulence brissaud1973 (); moffatt1992helicity (); chen2003joint (); chen2003intermittency (). In the latter case, recent studies have shown that even in the idealized case of 3D HIT, there exists a bi-directional transfer of energy with some helical-Fourier channels that cascade energy forward and others that transfer energy backward biferale2012inverse (); biferale2013split (); biferale2013global (); sahoo2015role (); kessar2015non (); stepanov2015hindered (); sahoo2015disentangling (); sahoo2017helicity (); alexakis2017Helically (); rathmann2017pseudo (); sahoo2017discontinuous (), with potential applications to rotating turbulence also. Multiple transfers of competing invariants also occurs in other flow configurations. A paradigmatic example is given by MHD flows that conserve three invariants in 3D woltjer1958stability (); chandrasekhar1958force (); montgomery1982two (); brandenburg2001inverse (); Alexakis2006mhelicity (); Malapaka2013mhelicity (); linkmann2016helical (); linkmann2017effects (); linkmann2016large (); linkmann2017triad (). In fact, it is fair to say that except for some very idealized situations, we cannot predict the direction of the energy transfer in homogeneous turbulence.

Second, there exist many important turbulent configurations that deviate from the idealized situation of HIT. e.g., in the presence of external mechanisms such as rotation, stratification, confinement, shear, or magnetic fields where the direction of the energy cascade might -and indeed it does- change. In many of these systems energy is transferred with a split-cascade, i.e. simultaneously forward and inversely in fractions that depend on the value of a control parameter (rotation rate, magnetic field strength, aspect ratio etc). This is demonstrated in Fig. (1) for two paradigmatic examples of a fast rotating flow (left panel) and flow constrained in a thin layer (right panel), where structures at both large and small scales coexist.
Split cascades have been shown to exist in different physical situations, in numerical simulations and experiments
of thin/thick layers Celani2010turbulence (); benavides2017critical (); Shats2010turbulence (); Xia2011upscale (); francois2013inverse (),
in rotating and stratified turbulence Deusebio2014dimentional (); Marino2013invers (); aluie2011joint (); pouquet2013geophysical (); rorai2013helicity (); marino2014large (); rorai2014turbulence (); Sozza2015dimensional (); Rosenberg2015evidence (); rorai2015stably (); Marino2015resolving (); herbert2016waves (); Staplehurst2008 (); Bokhoven2009experiments (); Yoshimatsu2011 (); duran2013turbulence (); Machicoane2016Two (); yeung1998numerical (); Smith1999transfer (); godeferd1999direct (); chen2005resonant (); Thiele2009 (); mininni2009 (); Mininni2010Rotating (); favier2010space (); Sen2012anisotropy (); alexakis2015rotatingTG (); biferale2016coherent (); valente2017spectral ()
and in MHD turbulence Alexakis2011two (); sujovolsky2016tridimensional (); Seshasayanan2014edge (); Seshasayanan2016critical (); sundar2017dynamic (); favier2010two (); reddy2014anisotropic (); reddy2014strong (); baker2018inverse (); baker2017controlling (); potherat2014why ().
They have been observed in geophysical flows, e.g. where the atmosphere acts like a 2D flow at large scale and as a 3D flow at small scales nastrom1984kinetic (); nastrom1985climatology (); gage1986theoretical (); charney1971geostrophic (); Byrne2011robust (); Byrne2013height (); brunner2014upscale (); tang2015horizontal (); callies2014transition () and in the ocean (arbic2013eddy, ; king2015upscale, ). Similar behaviour has been attributed to astrophysical flows (like the atmosphere of Venus and Jupiter Izakov2013Venus (); young2017forward (), and accretion discs Lesur2011accretion ()), in plasma flows miloshevich2018direction () and in industrial applications (like in tokamak Diamond2005plasma ()) either due to the thinness of the layer, to fast rotation or to the presence of strong magnetic fields. Split cascades have also been observed in wave systems lvov2015formation (), multi-scale optical turbulence malkin2018transition (), acoustic turbulence ganshin2008observation () and capillary turbulence on the surface of liquid hydrogen and helium abdurakhimov2015bidirectional (); abdurahimov2015formation (). In many of these systems, a change from a split to a strict forward cascade has also been detected with a critical transition behaviour.

Furthermore in many situations there is a vanishing, or very weak, net flux at scales larger than the forcing. In some cases like in 3D turbulence this leads to an equipartition of energy among the large scale modes kraichnan1973helical (); cichowlas2005effective (); krstulovic2009cascades (); ray2015thermalized (); michel2017observation (); herbert2014restrictedp (); herbert2014restrictede (); herbert2014nonlinear (); zhu2014note (); zhu2014purely (); ditlevsen1996cascades (); Dallas2015statistical (); cameron2017effect (); cameron2016large (). In other cases, for finite box size, the presence of inverse transfers might lead to the formation of a large-scale condensate, with a strong feedback on the whole flow smith1993bose (); xia2009spectrally (); gallet2013two (); laurie2014universal (); farrell2007structure (); bouchet2008simpler (); venaille2009statistical (); venaille2011solvable (); bouchet2012statistical (); falkovich2016interaction (); woillez2017theoretical (); frishman2017culmination (); frishman2017jets (). Similarly, ultraviolet effects might produce an accumulation of energy (or of another inviscid invariant) at small scales. The coexistence in the same scale range of different channels can also drive the system to an exotic flux-loop state, with a perfect balance among direct and inverse transfer, zero flux and non-equilibrium properties, as for the case of stratified 2D turbulence boffetta2011flux (). In some of these zero-flux cases the flow is close to a quasi-equilibrium state while others remain strongly out of equilibrium.

Finally, deviations from a perfect self-similar cascade are known to exist even in HIT. For example, we know that kinetic energy in 3D tends to be dissipated in spiky and intermittent events, and we do not know if this is due to the presence of coherent structures or just because of an enhancement of statistical fluctuations Frisch (). On the contrary, the 2D inverse energy cascade is close to be Gaussian, without any intermittent properties. We only have a loose phenomenological understanding of why fluctuations grow for the 3D case, based on the Richardson’s idea and of multi-fractal processes Frisch (), but we do not control the connection with the equations of motion. Intermittency, anomalous scaling, multi-fractal energy dissipation are still the subject of many investigations (see sinhuber2017dissipative (); iyer2017reynolds () for recent experimental and numerical state-of-the-art results).

The present work attempts to address all the above issues by reviewing some recent numerical, experimental and theoretical advancements achieved in the field. Our goal is to clarify and categorize the long list of subtle scenarios that might occur in turbulence and view different turbulence systems from a unified point of view. We would like thus to distinguish between turbulent configurations with the same mean properties but different spectra (e.g. with and without large scale condensate), the same spectral behaviour but with different transfer directions (e.g. with direct or inverse cascades) and even the same transfer properties but with a different statistical ensemble (e.g. a zero flux system in a equilibrium or out-of-equilibrium state). In particular, we want to emphasize the key distinguishable properties that characterize a non-linear out-of-equilibrium system, by looking at the behaviour of single-point global quantities, e.g. energy or enstrophy, or two-points spectral properties and transfer terms.

To do that, we start in Sec. (2) with the basic theoretical set-up. In sec. (3) we give a short review of the classical idealized turbulence cascades in three and two dimensions and proceed in the same section by providing a series of precise definitions for different flow states based on the cascade properties. In many cases, the same system can display a (phase) transition among two or more of the above flow states. We thus also classify the different paths that such transitions can occur by changing the dimensionless control parameters, e.g. the Reynolds number for hydrodynamic turbulence or the Rossby, Froude and Alfven numbers for rotating, stratified and conducting flows.

In Sec. (4) we examine different systems in detail. We follow a simple-to-complex path, discussing how turbulence changes when additional ingredients are added into the system, by breaking certain symmetries, e.g. mirror, rotational and scale invariance, by adding confinement, by changing the non-linearities, or by coupling the flow to active components. We review empirical findings for a set of paradigmatic applications where the above cascade realizations are encountered. The examples vary from helical turbulence, turbulence in confined domains, turbulence under rotation, stable and unstable stratified turbulence, turbulence of conducting fluids (MHD) to passive and active scalar advection, just to cite the most important cases. We review how these different realizations can drastically modify the properties of turbulence by altering the conserved quantities, their cascade directions and what the consequences for large and small scales flow properties are.

Finally, in Sec. (5) we present a short overview of cascades in turbulent models and in other flow configurations, as for the case of quantum flows, wave-turbulence, bounded flows, relativistic and compressible flows, active fluids, Shell Models, Large Eddy Simulations, EDQNM approximation and elastic materials. In Sec. (6) we conclude with a series of open questions and challenges in the field with the hope that these issues will attract the interest of future research.

All applications are cherry-picked with the only aim to highlight the aspects related to the cascades and their transition, without hoping to be self-contained and exhaustive for each particular subject. We tried to be as precise as possible concerning the classification of different cascade scenarios and we kept an empirical mood whenever we need to report about numerical and experimental findings, due to the absence of rigorous results for most of the cases treated here.

## 2 Theoretical Setup

In this section we provide a short theoretical background and the notation that will be used throughout the review. We discuss the balance of globally inviscid conserved quantities in the configuration and Fourier space. We introduce the concept of direct and inverse inertial-ranges, defined as those interval of scales where the main driving mechanisms are given by the non-linear transfer of the cascading quantities. We address also the scale-by-scale energy budget for the two-point correlation function in configuration space, for the energy spectrum in Fourier space and for the mixed scale-filtered representation often used for small-scale modelling.

### 2.1 Dynamical equations and control parameters

To start the discussion we consider the incompressible Navier-Stokes equation in the presence of a large-scale drag term:

(1) |

Here is the divergence free velocity field, , with constant unit density, is the pressure per unit density that enforces incompressibility, is the kinematic viscosity, and are the forcing term and the coefficient of a large scale drag mechanisms, respectively. In some systems, the large scale drag is required to reach a steady state in the presence of an inverse cascade. It simplifies the theoretical discussion that follows, avoiding the need to discuss quasi-stationary states. For most of this review we are going to assume the flow to be confined in a periodic box of size , and that the external forcing is acting on a band limited range of scales centered around . The energy injection rate by the forcing will be denoted as . In some cases, it is useful to study the flow evolution when the Newtonian viscosity is replaced with a hyper-viscous term, and/or when the drag force is replaced with a hypo-viscous sink, , such as to confine the energy dissipation due to viscous effects to very small scales (by increasing and decreasing ) and/or the one due to the drag mechanisms to very large scales (by increasing and decreasing ). There are three non-dimensional control parameters in the system: the ratio of the box size to the forcing length scale, , and two numbers that measure the relative amplitude of the non-linearities compared to the two dissipation terms defined above. The first one is the Reynolds number, , which compares advection with the viscous dissipation. The second one is the equivalent of but for the large scale drag, , obtained by replacing the viscous term with the drag term

(2) |

where is the root-mean-square velocity measured at the injection scale. In many experimental and numerical set-ups, it is not that is controlled but rather the energy injection rate. In these cases it is useful to define the dimensionless numbers in terms of :

(3) |

The two definitions (2-3) become equivalent up to a multiplicative constant for fully developed turbulence where the relation holds Frisch (). There are other cases, e.g. for wave turbulence, where the two definitions do not agree (see Sec. 5.1).

### 2.2 Inviscid Invariants and balance equations

It is straightforward to show that in 3D for (and if the flow remains smooth) equation (1) has two global invariants, the total Energy and total Helicity :

(4) |

where is the vorticity. The angular brackets stand for spatial average that in dimensions is defined as

(5) |

In , helicity is identically zero and the second quadratic invariant is the enstrophy:

(6) |

which like the energy, and unlike helicity, it is positive definite.

In the presence of forcing and for finite values of and a balance is
reached where the injected energy is absorbed by the two dissipation terms.
By writing the evolution equation for the energy
(4) we get:

(7) |

where is the energy dissipation due to viscosity, is the energy dissipation due to the drag and is the energy injection rate. It is worth noting that the terms are strictly positive for any non zero while in principle can fluctuate taking both positive and negative values. Assuming stationarity and performing a long time average we obtain

(8) |

Where above and from hereafter a long-time average is implied whenever time does not explicitly appear. The interpretation of eq. (8) is clear: the energy injected by the forcing equals, in average, the energy dissipated by viscosity at the small scales plus the energy dissipated by the drag at the large scales. A similar balance holds for all the other invariants of the system.

### 2.3 Fourier space representation

In order to better disentangle the scale-by-scale dynamics it is useful to define the NSE in Fourier space. We thus decompose the velocity field in Fourier modes as:

(9) |

where , and being the smallest wavenumber in the system (in what follows we will always assume that there is no mean flow, for ). The Navier-Stokes equation in Fourier space can then be written as

(10) |

where we have defined the projector on incompressible fields, , and . Hereafter, we will always make use of the Einstein notation for summation over repeated indices, unless otherwise stated. The components of each Fourier mode are linked by the incompressibility condition, , that reduces the degrees of freedom by one. In 2D we can write these amplitudes in terms of a single stream function mode. In 3D we have the freedom to choose a basis of two eigenvectors for each . One possible option will be discussed in terms of helical components in section (4.1)

The distribution of energy among scales is given by the spectrum averaged over a spherical shell of width :

(11) |

Summing over all shells we obtain the total energy . From (10) we can also derive the evolution of the energy spectrum as:

(12) |

Where we have introduced the notation for the instantaneous non-linear energy transfer, , across :

(13) |

and of the scale-by-scale energy injection:

(14) |

where in (12-14) and hereafter we assume for the sake of simplicity. We will consider the explicit dependency on only when the infinite volume limit is considered. From the above expression, we expect that in the limit of high and , i.e. by sending at fixed and , the viscous dissipation term will play a role only if becomes larger and larger for high . Similarly, for the large-scale drag to be active we need to grow for . In other words, for any fixed viscosity we expect the existence of a wavenumber defining the onset of viscous effects in the ultraviolet (UV) range of the spectrum while the drag term will be dominant in the infra-red (IR) limit, i.e. for wavenumbers smaller than another reference scale, . In such a scenario, at steady state, three well distinguished scales, , must exist and the balance (12) tells us that the non-linear transfer term must be vanishingly small except on those ranges where it balances the viscous dissipation terms (), the drag term () and the injection term ():

(15) |

In the left panel of Fig. (2) we summarize the balance (15). It is important to stress
that the condition to have two well-defined scale separations is not guaranteed a priori, even in the limit of and that all consequences and predictions that might follow
from this assumption must be checked self-consistently a posteriori by studying the resulting dependency of and on the control parameters. In the right panel of the same figure, we show how the balance (15) would look in the absence of scale separation.

For what follows, it is very important to define also the net energy transfer across a wavenumber . It can be derived by looking at the evolution of the total energy contained inside a sphere of radius . Following the notation of Frisch () we denote with the velocity field low-pass filtered so that all wave numbers outside the sphere of radius are set to zero:

(16) |

The change of the total energy across the sphere with radius is then obtained by taking the inner product of with (1) and averaging over the whole space:

(17) |

where the quantity is the non-linear energy flux across a sphere of radius in Fourier space:

(18) |

Because of the inviscid conservation of the total energy we have , i.e. the total exchange of energy due to all triadic Fourier interactions is zero. The second and the third terms on the RHS of (17) give the total energy dissipation inside the Fourier sphere due to the viscous terms and the large-scale drag, respectively. The fourth term gives the energy injection rate. In the limit, the relation (17) coincides with (7):

(19) |

For any fixed we can rewrite the stationary balance as follows:

(20) |

where we have denoted the time averaged total energy injection, viscous dissipation and viscous drag inside the sphere of radius , as:

Up until now, all manipulations leading to the global and to the scale-by-scale energy
balances (8) and (15-17) are exact. In order
to proceed further we need to make some assumptions.

We first assume that the forcing is concentrated around a thin window . In this case, we must have if and for .

Then, assuming the existence of two scales, and where the infra-red drag and the viscous dissipation are predominant as in the left panel of Fig. (2), we can estimate the asymptotic matching of the different terms entering in (20).
We examine separately the long-time stationary relation (20) in the two range of scales, or that is summarized in Fig. (3).

##### Inverse cascade: wavenumbers smaller than the forcing scale.

By referring to the balance (20) and to the left panel of Fig. (2) we conclude that for the non-linear transfer, , must be negative and matching the contribution due to , because and are negligible at those scales. Moreover, the integrated drag contribution up to the wavenumber must saturate to a constant, equal to the total drag dissipation if . As a result, there exists an intermediate range of scales where:

(21) |

Because only inertial terms play an active role in the transfer, we will call this set of scales the inverse-cascade inertial-range.

##### Direct cascade: wavenumbers larger than the forcing scale.

For , both the integrated contribution of the drag terms and of the total injection have reached its asymptotic values, and while the viscous dissipation is still not active, , if . As a result, there exist an intermediate range of scales where the balance gives:

(22) |

We will call this set of scales the direct-cascade inertial-range.

In Section (3.1) we will clarify the above picture and to which extent it can be pushed to rigorous and quantitative statements. Before entering in these aspects, we move back to configuration space to re-derive the previous results from a different perspective.

### 2.4 Configuration space representation

The energy distribution among scales and the energy flux can also be defined directly in configuration space by considering velocity correlation between two points and with and then averaging over all possible points for a fixed . Here, we follow the discussion presented in Frisch (), referring to the textbook for details. We first introduce the second-order correlation function:

(23) |

Assuming homogeneity (but not isotropy) it is possible to derive from the NSE the Kármán-Howarth-Monin relation monin2013statistical ():

(24) |

where we have introduced the notation for the velocity increment over a distance : and the forcing-velocity correlation . In the presence of a stationary statistics we can derive the law for the total energy balance by setting , averaging over time and putting to zero all time derivatives:

(25) |

where we exploited that for the non-linear contribution is identically zero because of the inviscid energy conservation. It is easy to see that the above relation is equivalent to (8) by noticing that and . Assuming stationarity and keeping the distance fixed with we get from (24):

(26) |

which is the configuration-space equivalent of the scale-by-scale energy balance (15). Let us now suppose to have an isotropic forcing-velocity correlation that is peaked at one given scale, :

where . As we did for the Fourier space we need to distinguish two different scaling regimes.

Direct cascade: scales smaller than the forcing scale.
In such a range, for any fixed one can fix the viscosity small enough to make the dissipative term vanishingly small, i.e there exists
a dissipative scale, such that for the viscous dissipation is negligible. Moreover, both the energy injection and the two-point correlation function are smooth for :

As a result, in this range of scales we have

(27) |

by further assuming isotropy, it is possible to express the left hand side in terms of the longitudinal third order structure functions, to finally obtain the celebrated law of Kolmogorov Frisch () for the inertial range of the direct cascade that reads (in 3D):

(28) |

The above relation can be dimensionally summarised as

(29) |

Inverse cascade: scales larger than the forcing scale. In the other limit, , we have no direct injection of energy from the forcing, , we can consider the viscous term to be vanishingly small, and the third-order correlation on the LHS of (26) is balanced by the drag-term contribution only:

(30) |

where we have assumed that for , the correlation has already saturated to its constant value. Eq. (26) is exact, while in order to get the direct and inverse scaling range in eqs. (28-30) we need to assume the existence of the two scales, , fixing the onset of the viscous effects (for ) and the onset of the drag term (for ), as already done for the same quantities in the Fourier space. Depending on the existence of such scales and on their dependency on the control parameters , one might end up in a situation where the direct/inverse energy transfer develops a true asymptotically inertial direct and/or inverse scaling range, i.e. a set of scales which becomes more and more extended when and/or , as depicted in Fig. (4)

Expressions (28-30) are also important because they give a clear signature of the breaking of time-reversibility and of Gaussianity (in both cases any third order velocity correlation function vanishes identically). The latter observation is connected to the existence of a net non-vanishing energy flux.

### 2.5 Scale-filtered representations

In order to control the multi-scale properties of the flow in configuration space, it is often useful to study the velocity field evolution coarse-grained at scale . In order to do that, we need to define a filtering operation on a generic velocity field, :

(31) |

where we have to ensure that the velocity field is a real function. If the filter is a projector, then
. For example, when
if and otherwise, the filtered field, would coincide with the low-pass field (16). Another popular choice is a Gaussian filter . Other possible options for are discussed in pope2001turbulent ().

The NSE equations for the filtered field are:

(32) |

where is a pressure term that enforces incompressibility of and the sub grid scale (SGS) stress tensor is given by:

(33) |

Notice that (32) is exact but not closed, i.e. the evolution of the filtered field does depend on the filtered scales via the SGS term. Large eddy simulations are based on the idea of modelling in terms of the resolved field and it will be briefly discussed in Sec. (5.3.1).

It is useful to define the SGS energy-transfer, entering in the dynamical
evolution of the resolved kinetic energy:

(34) |

where is a globally conservative flux term that redistributes the resolved energy among different spatial positions and

(35) |

is the SGS energy flux and is the resolved strain-rate tensor. The energy transfer between resolved and sub-filter scale is controlled by and it is straightforward to show that for the case of a sharp Fourier projector ( if ; otherwise) we have that:

(36) |

i.e. the space average of the SGS energy transfer coincides with the nonlinear energy flux in Fourier space (18)
at the corresponding wavenumber. An important issue is connected to the characterization of the space-time fluctuations of and to the possibility of defining a local-in-space Richardson cascade, to identify regions where a breakdown of eddies into smaller and smaller eddies is coherently observed within some space and time domains, as discussed in Sec. (5.3.1). The SGS tensor also becomes very relevant in quasi-2D flows where in the presence of a split cascade and in the presence of a large-scale condensate, some regions in space act locally like 3D cascading energy forward while other regions act like 2D cascading energy inversely (see e.g. section 4.2.3,4.3) .

## 3 Definitions of Turbulent Cascades

In this section we give a precise definition of what do we mean by cascade. Although the meaning of forward and inverse cascades has been stated many times in the literature, and it might be simple for 3D and 2D turbulence, in the case where a split transfer to both large and small scales is developed (as will be the case for many thin layers configurations examined in Sec. 4) one needs to be careful because the amplitude of the cascade can be very small and needs to be distinguished from any transient transfer. Furthermore, even in presence of a simple unidirectional transfer, ultraviolet and infra-red cut-off might play an important role in the limit of large numbers. E.g., in the presence of a finite volume and in the limit , the infrared cutoff will stop the inverse cascade, leading to an accumulation of energy at the smallest wavenumber and to the formation of a large-scale condensate (see Sec. 3.4 and the discussion about Fig. 12). The large-scale structures might finally induce a strong feedback on the inverse inertial range, break the scaling properties and bring the system close to a quasi-equilibrium state. Our definitions will be tailored to take into account all basic ingredients. Here we only consider the cases of a steady state, thus in all arguments the long time limit is considered before any other limiting procedure is taken.

The best way to proceed is by first examining a few idealized examples, that demonstrate some basic concepts before reaching the exact definitions. We start by examining the Kolmogorov theory for the forward 3D energy cascade (Sec. 3.1) and the Batchelor-Kraichnan theory for the 2D inverse energy and direct enstrophy transfers (Sec. 3.2). These two examples will allow us to introduce also the concept of scale-invariance and a first important set of dimensional estimates for the dependency of and on . We proceed by discussing the case of: a split cascade (Sec. 3.3), a statistical equilibrium (Sec. 3.4), multiple simultaneous transfers of different inviscid invariants (Sec. 3.5). Finally we conclude this chapter in Sec. (3.6) with a series of precise definitions for all the above cases, including bidirectional and flux-loop cascades, and with a classification of cascade transitions in Sec. (3.7).

### 3.1 3D Direct Energy Cascade and the Kolmogorov 1941 theory

As a paradigmatic example of a direct cascade we discuss the case of 3D HIT.
It is an empirical fact that in 3D there is no inverse energy cascade and therefore we start by putting the large scale drag and . Here we will only consider the case of zero helicity injection, postponing to Sec. (4.1) the discussion of the opposite case.

From the stationary relation (8), the energy injection rate needs to be equal to the
the averaged energy dissipation rate .
We note that these equalities hold independently of the value of , and their validity does not imply the presence of a cascade.
To define the cascade we need to compare the flux to the amplitude of the fluctuations and we need to make sure that there exists a well-defined scale separation between the wavenumbers where we have the maximum of the injection rate and the maximum of viscous dissipation.
Thus, in the limit we consider the dimensionless ratio

(37) |

sometimes called the drag coefficient. It is a fact based on numerical simulations and experimental data that this quantity remains finite even in the limit of infinite Reynolds number:

(38) |

where the constant value might depend on the forcing mechanism ishihara2009study ().
Let us notice that both definitions of the Reynolds number given by
expressions (2) or (3) would be here equivalent.
In the first case one considers fixed and (38) implies that as , remains bounded from above
while in the second case one considers fixed and it implies that as the dissipation rate remains bounded from below. The existence of a non-zero energy dissipation even in the limit is one of the major fingerprints of 3D turbulence and goes under the name of dissipative anomaly onsager1949statistical (); eyink2006onsager (); duchon2000inertial (). The fact that remains an order one quantity, tell us that the direct energy cascade in 3D HIT is always strongly out-of-equilibrium, and far from a perturbative quasi-equilibrium solution which would require or .

#### 3.1.1 Fourier space

If (38) holds and considering the expression (19) for the energy dissipation, we must have that the peak of the enstrophy spectrum, , is centered around a wavenumber when leading to the formation of the inertial range, , where viscous effects can be neglected and the energy flux is constant, . This is equivalent to the statement made in configuration space, leading to (28).

A phenomenological prediction for the spectral property was developed by Kolmogorov in 1941 (K41) based on the idea that only the mean flux, , plays a statistical role in the inertial range Kolmogorov (). In such a case, Kolmogorov derived the celebrated power law inertial-range behaviour for the spectrum:

(39) |

where is the so-called Kolmogorov constant. By plugging the K41 spectrum into the expression for the total viscous dissipation (2.3), we can finally define the dependency of the UV cut-off, , on . In order to do that, we define as the scale where the inertial range flux becomes comparable with the dissipative term:

(40) |

where to get to this estimate we have plugged (39) in (2.3) and summed up to . By using (3) we finally obtain:

(41) |

The above result is very important, it shows the self-consistency of the K41 theory: by assuming that for large Reynolds viscosity does not play a role for , one derives a prediction for the spectrum that consistently defines a viscous cut-off which becomes larger and larger for . In Fig. (5) we summarize the results fro spectrum and flux within the K41 theory. This result, together with the existence of the dissipative anomaly are the two fingerprints of the forward energy cascade in HIT. On the other hand, when viscosity sets-in we must have that in the far dissipative region, , an asymptotic matching between transfer and viscous terms develop, leading to an exponential or super exponential fall off with an exponent :

Before concluding this section, we briefly discuss what happens in the IR range, , where no net inverse transfer of energy is observed in 3D HIT. As said, if , at equilibrium we must have no transfer at all, , and the spectral distribution of energy is strongly depleted. It has been suggested that these scales reach a statistical (thermal) equilibrium due to local and non-local energy diffusion across the wavenumbers, with all Fourier modes feeling a sort of thermal bath described by a Gibbs-ensemble equipartition distribution that is discussed in Sec. (3.4). The validity of such assumption is still unclear and it is the subject of current investigations cichowlas2005effective (); krstulovic2009cascades (); Dallas2015statistical (); ray2015thermalized (); bos2006dynamics (); cameron2017effect (). If we consider the case of small but non-zero we expect similar conclusions. Since in 3D there is no evidence of an inverse cascade, the flux of energy to the large scales although non-zero ( is a strictly positive quantity if ) it will be still sub-dominant. The spectra could however be strongly modified even for small values of and become steeper than . The absence of a range of scales where there exists a constant inverse flux implies that we cannot expect the existence of a large-scale dissipative anomaly similar to (38). If we define the drag for the inverse transfer as:

(42) |

we must have:

(43) |

#### 3.1.2 Configuration space

Generalising the results obtained in Fourier space, the K41 theory assumes that the whole probability distribution function (PDF) of the velocity increments in the inertial range is fully determined by the mean energy injection only, if isotropy holds. Introducing the longitudinal and transverse velocity increments as:

(44) |

where with we indicate the unit vector in the direction of the displacement, Kolmogorov predicted for the -th order longitudinal and transverse structure function:

(45) |

the celebrated law for the scaling exponents in the inertial range:

(46) |

where are dimensionless constants. From (41) we have . Similarly, the exponential decay in the Fourier viscous range corresponds to an analytical smooth behaviour for the velocity field in the range , i.e. the longitudinal and transverse increments must be if . It is important to notice that in (46) the scaling of each single component of the transverse increments would be strictly zero for odd moments if isotropy holds biferale2005anisotropy (). This is not the case for longitudinal increments as seen from the -law (28). From the scaling (46) one can build the longitudinal skewness, a dimensionless measure of the intensity of the flux normalized with the root-mean-squared velocity fluctuations at scale :

(47) |

which can be considered a scale-by-scale generalisation of (37), i.e. a proxy
of the out-of-equilibrium nature of the forward energy cascade.

On one side, K41 theory is considered a milestone for building a theory of turbulence and it is known to describe many important quantitative and qualitative realistic properties. On the other side, there is now a long enough series of experimental and numerical evidence to know that it is not exact, even limiting the discussion to the case of HIT.
The most important point where it fails is linked to the existence of anomalous corrections to the inertial-range scaling exponents. Overwhelming evidence shows that velocity increments are characterized by a whole spectrum of anomalous inertial range exponents, see, e.g., benzi2008intermittency (); ishihara2009study (); iyer2017reynolds (); sinhuber2017dissipative ():

(48) |

with except for the case where (28) holds exactly. The departure from a linear dimensional behaviour of the scaling exponent goes under the name of intermittency and it will be further discussed in Sec. (5.2). Theoretical considerations based on isotropy biferale2005anisotropy () would also require , a fact that is not fully realized at the Reynolds numbers available nowadays iyer2017reynolds (). For what has been said until now, the most important consequence of intermittency is connected to the fact that by using (48) the skewness is not constant any more:

(49) |

where the value of the exponent is taken from the most updated numerical and experimental results iyer2017reynolds (). The fact that the exponent is small and negative is very important. It means that at moderate Reynolds numbers, the skewness can be considered almost constant, because cannot vary too much due to the limited extension of the inertial range. On the other hand, for large enough Reynolds, because of (41) we have and we can send
the ratio , still remaining in the scaling region. In this limit, the skewness will become larger and larger, indicating stronger and stronger out-of-equilibrium properties and a stronger and stronger departure from quasi-Gaussian statistics (where ). This is often visualized by plotting the energy dissipation field, which turns out to be highly spotty and spiky in configuration space, with vast regions with very small values and a few isolated islands of high intensity, i.e. an intermittent spatial distribution (see Fig. 64).

### 3.2 2D Inverse Energy Cascade and the Batchelor-Kraichnan theory

2D turbulence is a paradigmatic example where an inverse energy transfer is observed. It was predicted in a series of historical works by Kraichnan, Leith and Batchelor kraichnan1967inertial (); leith1968diffusion (); batchelor1969computation () and has been reproduced in numerical simulations boffetta2007energy (); boffetta2010evidence (); vallgren2011infrared (); xiao2009physical () and experiments ouellette2012turbulence (); Xia2011upscale (). A review can be found in Boffetta2012two (); kraichnan1980two (). In two dimensions, the NSE (1) can be written in terms of the out-of-plane vorticity as an advection-diffusion equation with the forcing term :

(50) |

As a result, the nonlinearity conserves the enstrophy and all moments of the vorticity . Unlike for the 3D flow, where we considered for simplicity the case of zero helicity injection, enstrophy injection can not be set to zero without setting the energy injection also to zero. In 2D, for enstrophy we have the exact balance:

(51) |

here is the enstrophy dissipation due to viscosity, is the enstrophy dissipation due to the large scale drag and is the enstrophy injection rate. The long-time average leads to

(52) |

#### 3.2.1 Fourier space

Considering the forcing acting only at a limited range of wavenumbers around a specific scale, , the energy and enstrophy injections are related by . Similarly, the energy spectrum and the enstrophy spectrum are connected by the relation . Thus, the cascade of the two ideal invariants can not be discussed separately, but leads to a situation where two fluxes coexist, referred often as a dual cascade. Defining a low-pass vorticity field similarly to (16), we introduce the 2D enstrophy flux:

(53) |

and using the same arguments as for the 3D energy cascade we distinguish a direct inertial range for the enstrophy transfer if there exists a window at high wavenumbers where the non-linear flux is constant

(54) |

while for the inverse enstrophy cascade we should have in the low region:

(55) |

To study the inverse cascade range, there are two limiting procedures that need to be considered and that lead to two different situations.
The first limit corresponds to the large box limit (or ) and the second is the
limit . If the large box limit is taken first then the system saturates in a state with a finite inverse energy flux, while if the large limit is taken first the system saturates to a condensate state that resembles to a statistical equilibrium (see also Sec. 3.4).

We begin with the case when the large-box limit is taken first.
Assuming the working hypothesis that there exists a separation of scales, (an assumption that as we saw for the
K41 theory can depend on the existence of a finite dissipation limit and whose self-consistently can be only checked aposteriori) we have that for any wavenumber in the inverse
range the energy and enstrophy flux will satisfy
(21) and (55) respectively, while in the forward range (22) and (54) hold.

In the forward range, , we can write:

(56) |

where the first equality is obtained by using (22) and estimating if because the contribution to dissipation must be concentrated at .

Similarly, in the inverse range, , we must have:

(57) |

Thus, neither in the large (56) nor in the small (57) wavenumber range we can have a constant enstrophy flux together with a constant energy flux: the two quantities must have a dual and split cascade. The original version of this argument was presented in (fjortoft1953changes, ). It has been reformulated and applied to different contexts in the literature fjortoft1953changes (); leith1968diffusion (); eyink1993lagrangian (); constantin1994effects (); scott2001evolution (); tran2002constraints (); gkioulekas2005adouble (); gkioulekas2005bdouble (); alexakis2006energy (); gibbon2007estimates (); gkioulekas2007new () and we will generalize it in Sec. (3.5). It indicates that energy is transferred towards large scales with a constant flux and a vanishing enstrophy flux, , while at small scales enstrophy cascades forward with a constant flux and a vanishing energy flux . Note that, unlike in the 3D case for which the strict forward cascade of energy is an empirical result, the simultaneous conservation of energy and enstrophy allows us to predict the inverse cascade of energy (and a forward cascade of enstrophy) in the 2D case. It is also important to note that these arguments work because enstrophy is a sign-definite invariant. Once the inverse energy cascade is established, one can reproduce the same phenomenological arguments put forward in K41 for the 3D case leading to the same spectrum slope, , and a different prefactor :

(58) |

Repeating the same reasoning done for the range dominated by the energy transfer, one can assume that in the direct cascade only the mean enstrophy flux plays a role, resulting in the Batchelor-Kraichnan prediction for the energy spectrum in the enstrophy cascade range:

(59) |

where the logarithmic correction comes from a more detailed self-consistent analysis kraichnan1971 (). This correction can be neglected for most of the purpose of this discussion. With the same arguments and by balancing the energy flux with the drag term, we obtain that

(60) |

where we have used the fact that in the inverse cascade regime (58) the large scale drag is dominated by the contribution at : . Similarly, for the direct enstrophy cascade range by balancing the enstrophy flux with the enstrophy dissipation rate, we obtain

(61) |

Both estimates (60) and (61) are important because they self-consistently close the assumptions made, indicating that the hypothesis to have an inverse inertial range dominated by the energy flux and a direct inertial range dominated by the enstrophy flux lead to the definition of two cut-off scales such that .
In Fig (6) we summarize the 2D phenomenology, by plotting the spectra for the energy (inverse) and enstrophy (direct) cascade together with the relative fluxes.

There are two more things that we need to discuss at this point. First, we can not have some energy dissipation without having some enstrophy dissipation too and vice versa. Thus, along with the inverse energy cascade some enstrophy has to arrive at the large scales also and similarly some energy needs to arrive at the small scales due to the enstophy direct cascade. In particular, given the scaling (58) and (59) the enstrophy dissipation by the drag force is dominated by the forcing scale and

(62) |

and therefore as One arrives at the same conclusion using a high-order hypo-viscosity , so that most of its dissipation is limited to the small wavenumbers then where as . Thus, the amplitude of the inverse enstrophy flux compared to the total enstrophy injection rate goes to zero in the limit . In the same way, by estimating the viscous energy dissipation at the viscous scale as we reach:

(63) |

where the logarithmic correction (59) was taken in to account. If a high-order hyper-viscosity is used to limit dissipation at