Cascade and scaling

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Gregory Falkovich (2008), Scholarpedia, 3(8):6088. doi:10.4249/scholarpedia.6088 revision #137326 [link to/cite this article]
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Figure 1: Vorticity clusters in the inverse cascade

Turbulence is a state of a continuous medium (or a system with many degrees of freedom) deviated far from thermal equilibrium. That state is accompanied by dissipation and needs an external pumping to sustain it. Developed turbulence corresponds to the case when the scales of externally excited and effectively dissipated motions are vastly different. For example, a moving car leaves behind meter-size vortices while viscous friction is only effective for eddies smaller than a millimeter. Instabilities of large vortices, their breakdown and fragmentation bring energy from input to dissipation scales by a cascade:

Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls, and so on to viscosity.
Richardson, (1922)

Cascade must be a natural state of any nonlinear system where input and output are far away as long as the interaction is effectively local. Locality here means that effective energy exchange between different modes goes to zero with the ratio of their scales. Apart from energy, other quantities conserved by interaction can cascade too. For example, during ore pulverization (when colliding stones are broken) mass cascades towards smaller sizes, while in water droplet coagulation (say, in clouds) mass cascades towards larger sizes. The cascade towards small scales is usually called direct while that towards large scales is called inverse. If a system has more than one conservation law in the absence of input and dissipation, then input at some scale can generate both direct and inverse cascades simultaneously, as happens for two-dimensional vortex turbulence or wave turbulence on water surface (see e.g. Zakharov et al. 1992, Frisch 1995). The interval of scales between input and output is called inertial interval (or transparency window).

Developed turbulence contains many excited degrees of freedom and requires a statistical description. Since most cases of turbulence involve many strongly interacting degrees of freedom, they can be neither described theoretically nor satisfactory modelled on computer. Therefore, a general symmetry aspects of turbulence statistics are of prime importance. Pumping and dissipation usually break symmetries (isotropy, scale invariance and time reversibility) and one asks if the symmetries are restored in the inertial interval (so that some of the information on pumping is forgotten). Scale invariance (or scaling) is particularly important (to predict the properties of the scales unresolved by modelling). One calls probability density function of the velocity difference \(v\) measured at the distance \(r\) scale invariant if it is actually a function of a single variable (rather than two)\[P(v,r)=f(v/r^a)/v\ .\] One can ask even more fundamental question of universality in the inertial interval: are there any statistical properties that are independent of the statistics of input (say, whether we excite turbulence by uniformly rotating blades or random kicks) and the precise form of dissipation. In other words, is it enough to know just the flux i.e. the input rate of energy (or other quantity) in a statistical steady state? The answer on scale invariance and universality is "definitely no for direct cascades" and "probably yes for inverse cascades", as discussed in more details below.


Burgers turbulence

Consider arguably the simplest case of turbulence. In the reference frame moving with the sound velocity, one-dimensional weakly compressible flows are described by the Burgers equation \(u_t+uu_x-\nu u_{xx}=f\ .\) Without the external force \(f\ ,\) it has a propagating shock-wave solution \(u=2v\{1+\exp[v(x-vt)/\nu]\}^{-1}\) with the energy dissipation rate \(\nu\int u_x^2\,dx\) independent of the kinematic viscosity \(\nu\ .\) The shock width \(\nu/v\) is a dissipative scale. A force correlated on a much larger scale \(L\) produces a direct cascade of acoustic turbulence which is a set of shocks at random positions. Consider the single-time statistics of the velocity difference \(w(x,t)=u(x,t)-u(0,t)\) whose moments are called the structure functions \(S_n(x,t)=\langle w^n\rangle\ .\) Angular brackets denote averages over time. Quadratic non-linearity relates the time derivative of the second moment to the third one\[{\partial S_2\over\partial t}=-{\partial S_3\over3\partial x}-4\epsilon+ \nu{\partial^2 S_2\over\partial x^2} \ .\] Here \(\epsilon=\nu \langle u_x^2\rangle\) is the mean energy dissipation rate which in a steady state must be equal to the input rate \(\langle fu\rangle\ .\) Consider now a steady state at the limit \(\nu\to0\) at fixed \(x\ .\) Shock dissipation provides for a finite limit of \(\epsilon\) at \(\nu \to0\ ,\) then \(S_3=-12\epsilon x\ .\) Flux constancy fixes \(S_3\ ,\) which is thus universal that is determined solely by \(\epsilon\ .\) On the contrary, other structure functions \(S_n\) cannot be (even approximately) estimated by \((\epsilon x)^{n/3}\ .\) Indeed, the scaling of the structure functions can be readily understood for any dilute set of shocks (that do not cluster in space) which seems to be the case for a general large-scale pumping. In this case, \(S_n(x)=A_n|x|^n+B_n|x|\ ,\) where the first term comes from the regular (smooth) parts of the velocity (the right \(x\)-interval in the Figure 2) while the second term comes from \(O(x)\) probability to have a shock in the \(x\)-interval.

Figure 2: Velocity profile in Burgers turbulence

The scaling exponents, \(\xi_n=d\ln S_n/d\ln x\ ,\) thus behave as follows\[\xi_n=n\] for \(n\leq1\) and \(\xi_n=1\) for \(n>1\ .\) That means that the probability density function (PDF) of the velocity difference in the inertial interval \(P(w,x)\) is not scale-invariant, that is the function of the re-scaled velocity difference \(w/x^a\) cannot be made scale-independent for any \(a\ .\) Simple bi-modal nature of Burgers turbulence (shocks and smooth parts) means that the PDF is actually determined by two (non-universal) functions, each depending of a single argument\[P(w,x)=w^{-1}f_1(w L/u_\mathrm{rms}x)+xf_2(w/u_\mathrm{rms})\ .\] Breakdown of scale invariance means that the low-order moments decrease faster than the high-order ones as one goes to smaller scales, i.e. the smaller the scale the more probable are large fluctuations. In other words, the level of fluctuations increases with the resolution. The moments with \(n>3\) are higher while those with \(n<3\) are lower than the estimate \((\epsilon x)^{n/3}\) suggested by the scaling hypothesis. The second moment (whose Fourier transform gives a spectral energy density) is by a factor \((x/L)^{1/3}\) smaller. When the scaling exponents \(\xi_n\) do not lie on a straight line, this is called an anomalous scaling. The term "anomaly" in theoretical physics means that the effect of symmetry breaking stays finite when the symmetry-breaking factor goes to zero. Here we have two anomalies:

  • finite third moment signals time-irreversibility even when viscosity goes to zero,
  • scale invariance is broken by pumping and not restored even when \(x/L\to0\ .\)

As an alternative to the description in terms of structures (shocks), one can relate the anomalous scaling in Burgers turbulence to the additional integrals of motion (see e.g. Polyakov 1995). Indeed, the integrals \(E_n=\int u^{2n}\,dx/2\) are all conserved by the inviscid Burgers equation. Any shock dissipates a finite amount of \(E_n\) at the limit \(\nu\to0\) so that one can express \(S_{2n+1}\) via these dissipation rates for integer \(n\ .\) For example, \(S_{5}=-40 x [\langle u^{2}u_{x}^2\rangle+\langle u^{2}\rangle\langle u_{x}^2\rangle]\).

We conclude that the statistics of velocity differences in the inertial interval depends on the infinitely many pumping-related parameters, the fluxes of all dynamical integrals of motion.

Passive scalar

Consider a scalar quantity \(\theta(r,t)\) which is subject to molecular diffusion and advection by the fluid flow but has no back influence on the velocity (i.e. passive)\[\partial_t\theta+(v\cdot\nabla)\theta=\kappa\nabla^2\theta+\varphi\ .\] If the random source \(\varphi\) produces the fluctuations of \(\theta\) on some scale \(L\) then the inhomogeneous velocity field stretches, contracts and folds the field \(\theta\) producing progressively smaller and smaller scales — this is the mechanism of the scalar cascade, which is eventually dissipated by diffusion. To describe the correlation functions of the scalar in the inertial interval, one employs the Lagrangian description that follows fluid trajectories. Indeed, if we neglect diffusion, then the equation can be solved along the characteristics \(\mathbf{R}(t)\) which are called Lagrangian trajectories and satisfy \(\mathrm{d}\mathbf{R}/\mathrm{d}t=v(\mathbf{R}, t)\ .\) Presuming zero initial conditions for \(\theta\) at \(t\to-\infty\) we write \(\theta(\mathbf{R}(t),t)=\int^t_{-\infty}\varphi(\mathbf{R}(t'),t')\mathrm{d}t'\ .\) In that way, the correlation functions of the scalar can be obtained by integrating the correlation functions of the pumping along the trajectories that satisfy the final conditions \(\mathbf{R}(t)=r_i\ .\) We consider a pumping which is Gaussian, statistically homogeneous and isotropic in space and white in time\[\langle\varphi(r_1,t_1)\varphi(r_2,t_2)\rangle= \Phi(|r_1-r_2|)\delta(t_1-t_2)\] where the function \(\Phi\) is constant at \(r\ll L\) and goes to zero at \(r\gg L\ .\) Assuming zero initial conditions at \(t=0\ ,\) the scalar pair correlation function is as follows\[ \langle\theta({r}_1)\theta({r}_2)\rangle= \int_{0}^t\Phi\left(R_{12}\left(t'\right)\right)\,\mathrm{d}t'\ .\] (*)

Here \(\mathbf{R}_{12}(t')=\left\|\mathbf{R}_1(t')-\mathbf{R}_2(t')\right\|\) is the distance between two trajectories and \(\mathbf{R}_{12}(t)=\mathbf{r}_{12}\ .\) The function \(\Phi\) essentially restricts the integration to the time interval when the distance \(\mathbf{R}_{12}<L\ ,\) i.e. the pair correlation function is proportional to the time of separation from \({r}_{12}\) to \( L\ .\) Simply speaking, the correlation functions of \(\theta\) are proportional to the times spent by the particles within the correlation scales of the pumping. The structure functions of \(\theta\) are differences of correlation functions with different initial particle configurations, for instance, \(S_3(r_{12})\equiv\langle[\theta({r}_1)-\theta({r}_2)]^3\rangle= 3\langle\theta^2({r}_1)\theta({r}_2)-\theta({r}_1)\theta^2({r}_2) \rangle\ .\) In calculating \(S_3\ ,\) we are thus comparing two histories: the first one with two particles initially close to the position \({r}_1\) and one particle at \({r}_2\ ,\) and the second one with one particle at \({r}_1\) and two particles at \({r}_2\ ,\) see Figure 3.

Figure 3: The difference between upper and lower configurations determines the third structure function.

That is, \(S_3\) is proportional to the time during which one can distinguish one history from another, or to the time needed for an elongated triangle to relax to the equilateral shape. That time grows with the distance (as it takes longer to forget more elongated triangle) by the power law \(r_{12}^{\sigma_3}\) that can be inferred from the law of the decrease of the shape fluctuations of a triangle. Similarly, one defines the multi-point exponents \(\sigma_n\ ,\) which all depend in a non-trivial way on the space dimensionality and velocity properties and aren't equal \(n\sigma_2/2\ ,\) which means again an anomalous scaling (Falkovich et al. 2001). As shown in the Figure 4, \(\sigma_n\) grow with \(n\) slower than linearly which can be interpreted as due to mutual correlations between different fluid particles. One can build for every \(n\) a statistically conserved quantity (called also zero mode or martingale) where the increase of the distances between particles is compensated by the decrease of their angular correlations, such zero modes have scaling exponents equal to \(\sigma_n\ .\) The necessity of infinite number of parameters and breakdown of scale invariance can thus be related to an infinite number of conservation laws, but statistical rather than dynamical (as it was for Burgers). At very large \(n\ ,\) the exponents \(\sigma_n\) tend to a constant (like in Burgers) for the same reason that the shocks in the scalar field determine high moments.

Navier-Stokes turbulence

Generally, one can relate statistical conservation laws of Lagrangian evolution to the scaling of single-time moments by the following consideration. Let us ask, for instance, how to build a two-particle correlation function \(\langle |v_1(t)-v_2(t)|^2f(R_{12}(t)/R_{12}(0))\rangle\), which is conserved because the decay of the function of distance \(R_{12}(t)\) offsets the growth of the velocity difference. At \(t=0\), it is proportional to \(R^{\zeta_2}_{12}(0) \). If we assume that \(R_{12}(0)\) is irrelevant at large t, then \(f(x)\propto x^{-\zeta_2}\) at \(x\to\infty\), thus linking the Eulerian scaling exponent to the long-time limit of the Lagrangian conservation law (Falkovich and Frishman 2013). It is then possible that a similar mechanism of statistical conservation laws is responsible for an anomalous scaling in the direct energy cascade of three-dimensional (3d) incompressible turbulence described by the Navier-Stokes equation (Falkovich et al. 2001). The exponents of the velocity structure functions for this case are denoted \(\zeta_n\) in the Figure.
Figure 4: The scaling exponents\[\xi_n\] for Burgers, \(\zeta_n\) for 3d Navier-Stokes and \(\sigma_n\) for the passive scalar. The dotted straight line is n/3.
Like for Burgers turbulence, energy flux constancy requires \(\zeta_3=1\ ,\) other exponents do not lie on a straight line. It is likely that \(\zeta_2\not=2/3\,\!\) and it is given by the law of decorrelation of two velocity vectors in three dimensions. The Fourier image of \(S_2\) is the energy spectral density which thus must be different from the Kolmogorov law \(k^{-5/3}\ .\)

However, in two-dimensional (2d) turbulence the energy cascade is inverse. In considering fluctuations in the inertial interval, one effectively averages (both spatially and temporally) over the pumping. That makes it natural that the statistics in the inverse cascades was found empirically to be scale-invariant. Moreover, in some cases, the scale invariance can be promoted to conformal invariance, which is essentially a local scale invariance. In a global re-scaling transformation one uniformly enlarges the picture (simultaneously decreasing resolution) while in conformal transformations one distorts the picture non-uniformly yet in a smooth way (keeping the angles between any vectors). Finding conformal invariance in turbulence links it with 2d critical phenomena whose universality classes were classified by representations of the conformal group. Specifically, different inverse cascades of 2d turbulence were analyzed by describing the statistics of the isolines of different fields (vorticity, temperature, streamfunction). An example of vorticity cluster boundaries is shown in Figure 1. Those isolines were found to belong to a remarkable class of curves that can be mapped into a Brownian walk (called Schramm Loewner evolution or SLE curves). Since Brownian walk is ultimately local (forgets itself on every step), such curves are conformally invariant. In this way, it was possible to establish that the boundaries of vorticity clusters in 2d inverse energy cascade are statistically equivalent to critical percolation and related to self-avoiding random walk (Bernard et al. 2006,2007).

Direct cascade in 2d turbulence is that of vorticity which is an inviscid Lagrangian invariant like passive scalar. In this case, fluid particles separate exponentially and the time of separation determined by (*) grows as \(r_{12}\to0\ .\) In other words, the cascade time is large at large Re which may mean that an effective averaging precludes an anomalous scaling in this case.

Note that in a compressible flow fluid particle cluster rather than separate (Falkovich et al. 2001). By virtue of (*) it means that the correlation functions grow with time at larger and larger scales which can be interpreted as an inverse cascade of a passive scalar. The correlation functions are scale-invariant but not conformally invariant in that case.

We thus conclude that despite the fact that both direct and inverse cascades carry a constant flux, the latter are scale invariant while the former are generally not. The difference between systems with normal and anomalous scaling in turbulence is like difference between renormalizable and non-renormalizable quantum field theories. Exactly when scale invariance can be promoted to conformal invariance is unclear at the moment. Generally, there is more to turbulence than just cascade.


  • M. Bauer and D. Bernard, 2D growth process: SLE and Loewner chains. Physics Reports 432, 115 (2006).
  • D. Bernard, G. Boffetta, A. Celani and G. Falkovich, Conformal invariance in two-dimensional turbulence, Nature Physics 2, 124 (2006); Inverse turbulent cascades and conformally invariant curves, Phys. Rev. Let. 98, 024501 (2007).
  • G. Falkovich, Conformal invariance in hydrodynamic turbulence, Russian Math Surv. 62(3)497(2007)
  • G. Falkovich, K. Gawedzki and M. Vergassola, Particles and fields in turbulence, Rev. Mod. Phys. 73, 913 (2001)
  • G. Falkovich and K. Sreenivasan, Lessons of hydrodynamic turbulence, Physics Today 59, 43 (2006)
  • U. Frisch, Turbulence. The Legacy of A.N. Kolmogorov (Cambridge University Press, 1995)
  • L. F. Richardson, Weather prediction by numerical process (Cambridge University Press, 1922)
  • A. M. Polyakov, Turbulence without pressure, Physical Review E, 52, 6183 (1995)
  • V. Zakharov, V. L'vov and G. Falkovich, Kolmogorov Spectra of Turbulence[1] (Springer-Verlag, 1992)
  • G. Falkovich and A. Frishman, Single velocity snapshot reveals future and past of pair of fluid particles in turbulence, Physical Review Letters, (2013)

Internal references

  • Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.
  • Roman W. Jackiw (2008) Axial anomaly. Scholarpedia, 3(10):7302.

Further reading

J. Cardy, G. Falkovich, K. Gawedzki, Non-equilibrium Statistical Mechanics and Turbulence, London Math Soc Lect Note Series 355, Cambridge Univ. Press. 2008

G. Falkovich Fluid mechanics (a short course for physicists). Cambridge University Press 2011

External links

See also

Turbulence, Dynamical systems

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