Many fluid equations are expected to exhibit turbulence in their solutions, in which a significant portion of their energy ends up in high frequency modes. A typical example arises from the three-dimensional periodic Navier-Stokes equations

$\displaystyle \partial_t u + u \cdot \nabla u = \nu \Delta u + \nabla p + f$

$\displaystyle \nabla \cdot u = 0$

where ${u: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}^3}$ is the velocity field, ${f: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}^3}$ is a forcing term, ${p: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}}$ is a pressure field, and ${\nu > 0}$ is the viscosity. To study the dynamics of energy for this system, we first pass to the Fourier transform

$\displaystyle \hat u(t,k) := \int_{{\bf R}^3/{\bf Z}^3} u(t,x) e^{-2\pi i k \cdot x}$

so that the system becomes

$\displaystyle \partial_t \hat u(t,k) + 2\pi \sum_{k = k_1 + k_2} (\hat u(t,k_1) \cdot ik_2) \hat u(t,k_2) =$

$\displaystyle - 4\pi^2 \nu |k|^2 \hat u(t,k) + 2\pi ik \hat p(t,k) + \hat f(t,k) \ \ \ \ \ (1)$

$\displaystyle k \cdot \hat u(t,k) = 0.$

We may normalise ${u}$ (and ${f}$) to have mean zero, so that ${\hat u(t,0)=0}$. Then we introduce the dyadic energies

$\displaystyle E_N(t) := \sum_{|k| \sim N} |\hat u(t,k)|^2$

where ${N \geq 1}$ ranges over the powers of two, and ${|k| \sim N}$ is shorthand for ${N \leq |k| < 2N}$. Taking the inner product of (1) with ${\hat u(t,k)}$, we obtain the energy flow equation

$\displaystyle \partial_t E_N = \sum_{N_1,N_2} \Pi_{N,N_1,N_2} - D_N + F_N \ \ \ \ \ (2)$

where ${N_1,N_2}$ range over powers of two, ${\Pi_{N,N_1,N_2}}$ is the energy flow rate

$\displaystyle \Pi_{N,N_1,N_2} := -2\pi \sum_{k=k_1+k_2: |k| \sim N, |k_1| \sim N_1, |k_2| \sim N_2}$

$\displaystyle (\hat u(t,k_1) \cdot ik_2) (\hat u(t,k) \cdot \hat u(t,k_2)),$

${D_N}$ is the energy dissipation rate

$\displaystyle D_N := 4\pi^2 \nu \sum_{|k| \sim N} |k|^2 |\hat u(t,k)|^2$

and ${F_N}$ is the energy injection rate

$\displaystyle F_N := \sum_{|k| \sim N} \hat u(t,k) \cdot \hat f(t,k).$

The Navier-Stokes equations are notoriously difficult to solve in general. Despite this, Kolmogorov in 1941 was able to give a convincing heuristic argument for what the distribution of the dyadic energies ${E_N}$ should become over long times, assuming that some sort of distributional steady state is reached. It is common to present this argument in the form of dimensional analysis, but one can also give a more “first principles” form Kolmogorov’s argument, which I will do here. Heuristically, one can divide the frequency scales ${N}$ into three regimes:

• The injection regime in which the energy injection rate ${F_N}$ dominates the right-hand side of (2);
• The energy flow regime in which the flow rates ${\Pi_{N,N_1,N_2}}$ dominate the right-hand side of (2); and
• The dissipation regime in which the dissipation ${D_N}$ dominates the right-hand side of (2).

If we assume a fairly steady and smooth forcing term ${f}$, then ${\hat f}$ will be supported on the low frequency modes ${k=O(1)}$, and so we heuristically expect the injection regime to consist of the low scales ${N=O(1)}$. Conversely, if we take the viscosity ${\nu}$ to be small, we expect the dissipation regime to only occur for very large frequencies ${N}$, with the energy flow regime occupying the intermediate frequencies.

We can heuristically predict the dividing line between the energy flow regime. Of all the flow rates ${\Pi_{N,N_1,N_2}}$, it turns out in practice that the terms in which ${N_1,N_2 = N+O(1)}$ (i.e., interactions between comparable scales, rather than widely separated scales) will dominate the other flow rates, so we will focus just on these terms. It is convenient to return back to physical space, decomposing the velocity field ${u}$ into Littlewood-Paley components

$\displaystyle u_N(t,x) := \sum_{|k| \sim N} \hat u(t,k) e^{2\pi i k \cdot x}$

of the velocity field ${u(t,x)}$ at frequency ${N}$. By Plancherel’s theorem, this field will have an ${L^2}$ norm of ${E_N(t)^{1/2}}$, and as a naive model of turbulence we expect this field to be spread out more or less uniformly on the torus, so we have the heuristic

$\displaystyle |u_N(t,x)| = O( E_N(t)^{1/2} ),$

and a similar heuristic applied to ${\nabla u_N}$ gives

$\displaystyle |\nabla u_N(t,x)| = O( N E_N(t)^{1/2} ).$

(One can consider modifications of the Kolmogorov model in which ${u_N}$ is concentrated on a lower-dimensional subset of the three-dimensional torus, leading to some changes in the numerology below, but we will not consider such variants here.) Since

$\displaystyle \Pi_{N,N_1,N_2} = - \int_{{\bf R}^3/{\bf Z}^3} u_N \cdot ( (u_{N_1} \cdot \nabla) u_{N_2} )\ dx$

we thus arrive at the heuristic

$\displaystyle \Pi_{N,N_1,N_2} = O( N_2 E_N^{1/2} E_{N_1}^{1/2} E_{N_2}^{1/2} ).$

Of course, there is the possibility that due to significant cancellation, the energy flow is significantly less than ${O( N E_N(t)^{3/2} )}$, but we will assume that cancellation effects are not that significant, so that we typically have

$\displaystyle \Pi_{N,N_1,N_2} \sim N_2 E_N^{1/2} E_{N_1}^{1/2} E_{N_2}^{1/2} \ \ \ \ \ (3)$

or (assuming that ${E_N}$ does not oscillate too much in ${N}$, and ${N_1,N_2}$ are close to ${N}$)

$\displaystyle \Pi_{N,N_1,N_2} \sim N E_N^{3/2}.$

On the other hand, we clearly have

$\displaystyle D_N \sim \nu N^2 E_N.$

We thus expect to be in the dissipation regime when

$\displaystyle N \gtrsim \nu^{-1} E_N^{1/2} \ \ \ \ \ (4)$

and in the energy flow regime when

$\displaystyle 1 \lesssim N \lesssim \nu^{-1} E_N^{1/2}. \ \ \ \ \ (5)$

Now we study the energy flow regime further. We assume a “statistically scale-invariant” dynamics in this regime, in particular assuming a power law

$\displaystyle E_N \sim A N^{-\alpha} \ \ \ \ \ (6)$

for some ${A,\alpha > 0}$. From (3), we then expect an average asymptotic of the form

$\displaystyle \Pi_{N,N_1,N_2} \approx A^{3/2} c_{N,N_1,N_2} (N N_1 N_2)^{1/3 - \alpha/2} \ \ \ \ \ (7)$

for some structure constants ${c_{N,N_1,N_2} \sim 1}$ that depend on the exact nature of the turbulence; here we have replaced the factor ${N_2}$ by the comparable term ${(N N_1 N_2)^{1/3}}$ to make things more symmetric. In order to attain a steady state in the energy flow regime, we thus need a cancellation in the structure constants:

$\displaystyle \sum_{N_1,N_2} c_{N,N_1,N_2} (N N_1 N_2)^{1/3 - \alpha/2} \approx 0. \ \ \ \ \ (8)$

On the other hand, if one is assuming statistical scale invariance, we expect the structure constants to be scale-invariant (in the energy flow regime), in that

$\displaystyle c_{\lambda N, \lambda N_1, \lambda N_2} = c_{N,N_1,N_2} \ \ \ \ \ (9)$

for dyadic ${\lambda > 0}$. Also, since the Euler equations conserve energy, the energy flows ${\Pi_{N,N_1,N_2}}$ symmetrise to zero,

$\displaystyle \Pi_{N,N_1,N_2} + \Pi_{N,N_2,N_1} + \Pi_{N_1,N,N_2} + \Pi_{N_1,N_2,N} + \Pi_{N_2,N,N_1} + \Pi_{N_2,N_1,N} = 0,$

which from (7) suggests a similar cancellation among the structure constants

$\displaystyle c_{N,N_1,N_2} + c_{N,N_2,N_1} + c_{N_1,N,N_2} + c_{N_1,N_2,N} + c_{N_2,N,N_1} + c_{N_2,N_1,N} \approx 0.$

Combining this with the scale-invariance (9), we see that for fixed ${N}$, we may organise the structure constants ${c_{N,N_1,N_2}}$ for dyadic ${N_1,N_2}$ into sextuples which sum to zero (including some degenerate tuples of order less than six). This will automatically guarantee the cancellation (8) required for a steady state energy distribution, provided that

$\displaystyle \frac{1}{3} - \frac{\alpha}{2} = 0$

or in other words

$\displaystyle \alpha = \frac{2}{3};$

for any other value of ${\alpha}$, there is no particular reason to expect this cancellation (8) to hold. Thus we are led to the heuristic conclusion that the most stable power law distribution for the energies ${E_N}$ is the ${2/3}$ law

$\displaystyle E_N \sim A N^{-2/3} \ \ \ \ \ (10)$

or in terms of shell energies, we have the famous Kolmogorov 5/3 law

$\displaystyle \sum_{|k| = k_0 + O(1)} |\hat u(t,k)|^2 \sim A k_0^{-5/3}.$

Given that frequency interactions tend to cascade from low frequencies to high (if only because there are so many more high frequencies than low ones), the above analysis predicts a stablising effect around this power law: scales at which a law (6) holds for some ${\alpha > 2/3}$ are likely to lose energy in the near-term, while scales at which a law (6) hold for some ${\alpha< 2/3}$ are conversely expected to gain energy, thus nudging the exponent of power law towards ${2/3}$.

We can solve for ${A}$ in terms of energy dissipation as follows. If we let ${N_*}$ be the frequency scale demarcating the transition from the energy flow regime (5) to the dissipation regime (4), we have

$\displaystyle N_* \sim \nu^{-1} E_{N_*}$

and hence by (10)

$\displaystyle N_* \sim \nu^{-1} A N_*^{-2/3}.$

On the other hand, if we let ${\epsilon := D_{N_*}}$ be the energy dissipation at this scale ${N_*}$ (which we expect to be the dominant scale of energy dissipation), we have

$\displaystyle \epsilon \sim \nu N_*^2 E_N \sim \nu N_*^2 A N_*^{-2/3}.$

Some simple algebra then lets us solve for ${A}$ and ${N_*}$ as

$\displaystyle N_* \sim (\frac{\epsilon}{\nu^3})^{1/4}$

and

$\displaystyle A \sim \epsilon^{2/3}.$

Thus, we have the Kolmogorov prediction

$\displaystyle \sum_{|k| = k_0 + O(1)} |\hat u(t,k)|^2 \sim \epsilon^{2/3} k_0^{-5/3}$

for

$\displaystyle 1 \lesssim k_0 \lesssim (\frac{\epsilon}{\nu^3})^{1/4}$

with energy dissipation occuring at the high end ${k_0 \sim (\frac{\epsilon}{\nu^3})^{1/4}}$ of this scale, which is counterbalanced by the energy injection at the low end ${k_0 \sim 1}$ of the scale.