Lie Group Reconstruction of K41 and She-Leveque Scaling Laws in Incompressible Turbulence

1. Introduction

Incompressible turbulence is one of the most complex unsolved mysteries in classical physics[1]. At high Reynolds numbers, the flow field is filled with multi-scale chaotic vortex motions ranging from macroscopic to microscopic. Understanding the statistical laws of these multi-scale structures, especially the scaling laws of structure functions, is the core issue in turbulence research. In 1922, Richardson vividly depicted the physical picture of cross-scale energy cascade in his famous verse[2]: large vortices transfer energy to small vortices without dissipation, until molecular viscosity takes effect. This energy cascade picture laid the physical foundation for subsequent quantitative theories.

Guided by the Richardson cascade picture, in 1941, the Soviet mathematician Kolmogorov proposed the groundbreaking K41 theory[3,4]. The K41 theory is the first quantitative statistical theory with predictive power in the history of turbulence research. Its core lies in reducing complex multi-scale chaotic motions to a universal process controlled by a single physical quantity—the average energy dissipation rate $ϵ$. Based on the hypotheses of local isotropy and similarity, the K41 theory asserts that within the inertial subrange, the statistical properties are determined solely by $ϵ$ and the spatial scale r, thereby yielding the strictly linear law of the structure function scaling exponent (${\zeta }_p=p/3$) and the famous $-5/3$ scaling law of the turbulent kinetic energy spectrum.

However, with the improvement of observation precision, experiments and numerical simulations subsequently found that higher-order scaling exponents systematically deviate from the linear law of K41, a phenomenon known as anomalous scaling. Its physical root lies in intermittency—the energy dissipation is extremely uneven in space, and strong dissipation is concentrated in a very small number of active vortex tube structures[1]. To solve this problem, Kolmogorov himself proposed the refined similarity hypothesis (K62) in 1962, introducing the fluctuation of the dissipation rate, but failed to provide a closed-form solution for the scaling exponents. It was not until 1994 that She and Leveque proposed the famous SL scaling law[5], which, by introducing the hierarchical recursive assumption of singular structures, gave an explicit scaling formula for the first time that astonishingly matched experiments. However, the SL theory remains essentially a semi-phenomenological model, and its power-law assumption of hierarchical recursion lacks a more fundamental dynamical or geometric derivation.

From a deep mathematical physics perspective, the scaling laws of turbulence are essentially symmetry problems. The profound connection between continuous symmetry and conservation laws was first established by Lie’s pioneering work[6,7,8], and subsequently Noether proved their strict equivalence[9]. In the field of fluid mechanics, Ovsiannikov[10] and Barenblatt[11] developed the group analysis method and scaling law theory, respectively.

This paper aims to systematically introduce the infinitesimal generators of the Lie group and their prolongation theory into the algebraic reconstruction of turbulence scaling laws. It should be emphasized that the work of this paper is essentially an algebraic isomorphic mapping of existing phenomenological laws, rather than a derivation from the first principles of the Navier-Stokes equations. However, this reconstruction clarifies the nature of the parameters in the SL theory (such as $\beta$ and $C_0$) as algebraic structural constants, providing a clear algebraic framework for extending to other complex flow fields (such as 2D turbulence, magnetohydrodynamic turbulence, or compressible turbulence).

2. Mathematical Foundation: Scaling Lie Group and Infinitesimal Generator

Scale self-similarity is mathematically described by a one-parameter scaling Lie group. Define the scaling group $G_{\lambda }=\left\{g_{\lambda }\right|\lambda >0\}$ acting on the spatial scale r and the velocity field u:

$$g_{\lambda }:(r,u)\mapsto (\lambda r,{\lambda }^{h}u)$$

(1)

where $\lambda$ is the group parameter, h is the velocity scaling exponent, and the group multiplication satisfies $g_{{\lambda }_1}\circ g_{{\lambda }_2}=g_{{\lambda }_1{\lambda }_2}$.

The infinitesimal generator (Lie algebra element) corresponding to this Lie group is:

$$X=\xi (r,u){\displaystyle \frac{\partial }{\partial r}}+\eta (r,u){\displaystyle \frac{\partial }{\partial u}}=r{\displaystyle \frac{\partial }{\partial r}}+hu{\displaystyle \frac{\partial }{\partial u}}$$

(2)

To study the scaling properties of statistical moments such as structure functions, we need to prolong the generator to the Jet space of the manifold[10]. For any scaling physical observable $S\left(r\right)$ that depends only on the spatial scale r studied subsequently in this paper, the action of the generator prolonged to the first-order Jet space degenerates into differentiation with respect to its independent variable. Since $S\left(r\right)$ has been ensemble-averaged over the velocity field u, it depends only explicitly on r, so the action of the $u{\partial }_u$ term in the generator is zero, and the prolonged generator simplifies to:

$$X^{\left(1\right)}S\left(r\right)=\xi \left(r\right){\displaystyle \frac{dS}{dr}}=r{\displaystyle \frac{dS}{dr}}$$

(3)

If $S\left(r\right)$ transforms according to a power law under the scaling group, i.e., $S\left(\lambda r\right)={\lambda }^{\zeta }S\left(r\right)$, then it must be an eigenfunction of the prolonged generator:

$$X^{\left(1\right)}S\left(r\right)=\zeta S\left(r\right)$$

(4)

where $\zeta$ is called the scaling exponent or the generator eigenvalue. Therefore, the determination of turbulence scaling exponents is mathematically equivalent to solving the eigenvalue problem of physical quantities under the action of the prolonged generator.

3. K41 Scaling Law: Mapping between Generator Invariance and Conservation Law

The K41 theory is built upon the physical foundation of the Richardson cascade[2]. Kolmogorov proposed in his classical papers[3,4] that within the inertial subrange (integral scale $L\gg r\gg \eta$ dissipation scale), the statistical properties are determined solely by the energy dissipation rate $ϵ$ and the scale r. Physically, $ϵ$ represents the energy flux, which remains constant during the cascade process.

3.1. Classical Dimensional Analysis and Derivation of K41 Scaling Law

In the classical derivation of K41, dimensional analysis is the core tool[1]. Define the velocity difference between two points separated by a spatial distance r in the flow field as $\delta u\left(r\right)$. In the inertial subrange, $\delta u\left(r\right)$ is determined solely by the average energy dissipation rate $ϵ$ (energy dissipated per unit mass of fluid per unit time) and the scale r. Assuming $\delta u\left(r\right)\propto {ϵ}^a{r}^b$, through dimensional consistency $\left[L{T}^{-1}\right]={\left[L^2{T}^{-3}\right]}^a{\left[L\right]}^b$, we solve for $a=1/3,b=1/3$, yielding $\delta u\left(r\right)\propto {\left(ϵr\right)}^{1/3}$.

Thus, defining the p-th order velocity structure function $S_p\left(r\right)\equiv {\langle \left|\delta u\left(r\right)\right|}^p\rangle$ (where $\langle ·\rangle$ denotes the ensemble average), it satisfies:

$$S_p\left(r\right)\propto {\left(ϵr\right)}^{p/3}\Rightarrow {\zeta }_p={\displaystyle \frac{p}{3}}$$

(5)

Similarly, define the turbulent kinetic energy spectrum $E\left(k\right)$ at wavenumber k (corresponding to the reciprocal of the spatial scale), which represents the turbulent kinetic energy contained per unit mass of fluid per unit wavenumber interval. $E\left(k\right)$ is determined solely by $ϵ$ and k. Assuming $E\left(k\right)\propto {ϵ}^c{k}^d$, from $\left[L^3{T}^{-2}\right]={\left[L^2{T}^{-3}\right]}^c{\left[L^{-1}\right]}^d$, we solve $c=2/3,d=-5/3$, yielding the famous Kolmogorov $-5/3$ scaling law:

$$E\left(k\right)\propto {ϵ}^{2/3}{k}^{-5/3}$$

(6)

In addition, at the dissipation scale $\eta$, the kinematic viscosity $\nu$ begins to dominate. Dimensional analysis of $\eta \propto {ϵ}^m{\nu }^n$ yields $m=-1/4,n=3/4$, giving the Kolmogorov microscale $\eta ={({\nu }^3/ϵ)}^{1/4}$.

3.2. Invariance under the Generator and Conservation Law

Although dimensional analysis is concise and effective, the profound physical mechanism behind it can be reinterpreted through the invariance of the Lie group. Inspired by the profound connection between symmetry and conservation laws revealed by Noether’s theorem[9], if a system possesses a certain continuous symmetry, it must correspond to an invariant. It should be pointed out that since the Navier-Stokes equations contain dissipation terms, the traditional Noether theorem cannot be directly applied to yield energy flux conservation; however, in the inertial subrange, the phenomenological physical picture of the Richardson cascade provides the empirical fact of energy flux conservation. In the language of Lie groups, this phenomenological conservation law means that $ϵ$ is an invariant of the scaling group.

We apply the infinitesimal generator to $ϵ$. Physically, since $ϵ\sim \delta u^3/r$, using the linearity of the generator:

$$Xϵ=X\left({\displaystyle \frac{\delta u^3}{r}}\right)={\displaystyle \frac{3\delta u^2\left(X\delta u\right)}{r}}-{\displaystyle \frac{\delta u^3\left(Xr\right)}{r^2}}={\displaystyle \frac{3\delta u^2\left(h\delta u\right)}{r}}-{\displaystyle \frac{\delta u^3\left(r\right)}{r^2}}=(3h-1)ϵ$$

(7)

Invariance requires $Xϵ=0$ (i.e., $ϵ$ is a zero eigenvalue state), which strictly yields the velocity scaling exponent:

$$3h-1=0\Rightarrow h={\displaystyle \frac{1}{3}}$$

(8)

This derivation profoundly reveals the algebraic origin of the K41 scaling exponent $1/3$: it is the unique eigenvalue choice that maintains the existence of an invariant under the action of the generator.

3.3. Prolongation of Structure Functions and Linear Character

We apply the infinitesimal generator directly to the p-th order structure function $S_p\left(r\right)$. Here we assume that under the stationary statistical ensemble of locally isotropic turbulence, the generator action and the ensemble average are commutative (this assumption is usually reasonable because the scaling transformation does not break the ergodicity hypothesis of the ensemble), thus obtaining:

$$X{S}_p=X\langle \delta u^p\rangle =\langle p\delta u^{p-1}X\delta u\rangle =hp\langle \delta u^p\rangle =hp{S}_p$$

(9)

On the other hand, according to Ovsiannikov’s prolongation theory[10], if $S_p\left(r\right)$ satisfies self-similarity, it must be an eigenfunction of the prolonged generator $X^{\left(1\right)}$, let the eigenvalue be ${\zeta }_p$:

$$X^{\left(1\right)}{S}_p\left(r\right)=r{\displaystyle \frac{d{S}_p}{dr}}={\zeta }_p{S}_p\left(r\right)$$

(10)

Since the finite group action on the manifold must be consistent with the infinitesimal generator action, comparing the above two equations, we must have:

$${\zeta }_p=hp={\displaystyle \frac{p}{3}}$$

(11)

The linear scaling law of K41 is thus proven. Its physical essence is that the statistical laws in the inertial subrange are strictly invariant under the scaling Lie group $g_{\lambda }:(r,u)\mapsto (\lambda r,{\lambda }^{1/3}u)$, and structure functions of all orders share the same generator eigenvalue determined by the invariant, exhibiting a perfect linear character. However, this perfect symmetry is broken by intermittency in real turbulence[1].

4. SL Scaling Law: Symmetry Breaking and Algebraic Recursion of Generator Eigenvalues

Experiments find that for higher-order moments ${\zeta }_p<p/3$ (anomalous scaling), which means the strict single-parameter generator symmetry of K41 is broken. The energy dissipation rate $ϵ$ is intermittently distributed in space and is no longer a global invariant. Statistical moments of different orders are no longer linearly determined by a single eigenvalue h, but correspond to a series of complex eigenvalues ${\tau }_p$. She and Leveque in 1994[5] successfully characterized this symmetry reconstruction after breaking by introducing a hierarchy of singular structures. Below we will elaborate on the physical derivation process of the classical SL scaling law and reveal its equivalent mapping relationship with the extraction of Lie group generator eigenvalues.

4.1. Definition of Dissipation Rate Moments and Hierarchical Structure

Define the local energy dissipation rate at spatial local scale r as ${ϵ}_r$. Its p-th order statistical moment is denoted as ${\Pi }_p\left(r\right)\equiv \langle {ϵ}_r^p\rangle$. Under the broken symmetry, ${\Pi }_p\left(r\right)$ is no longer a zero eigenvalue state. Within the inertial subrange, ${\Pi }_p\left(r\right)$ approximately satisfies a power-law scaling, so it can be locally written as the eigenvalue equation of the prolonged generator:

$$X^{\left(1\right)}{\Pi }_p\left(r\right)=r{\displaystyle \frac{d{\Pi }_p}{dr}}={\tau }_p{\Pi }_p\left(r\right)$$

(12)

Here ${\tau }_p$ is the generator eigenvalue (anomalous scaling exponent) of the local dissipation rate moment under the broken symmetry. According to the refined similarity hypothesis of K62, the scaling exponent of the velocity structure function is:

$${\zeta }_p={\displaystyle \frac{p}{3}}+{\tau }_{p/3}$$

(13)

Therefore, the core of solving anomalous scaling lies in solving the eigenvalue sequence ${\tau }_p$.

To characterize the intermittent distribution of dissipation, the SL theory introduces a hierarchical structure. Define the p-th order hierarchical structure as the ratio of two adjacent moments:

$${ϵ}_r^{\left(p\right)}\equiv {\displaystyle \frac{{\Pi }_{p+1}\left(r\right)}{{\Pi }_p\left(r\right)}}={\displaystyle \frac{\langle {ϵ}_r^{p+1}\rangle }{\langle {ϵ}_r^p\rangle }}$$

(14)

Physically, ${ϵ}_r^{\left(p\right)}$ reflects the typical dissipation rate of those structures in the flow field with stronger dissipation intensity. As p increases, ${ϵ}_r^{\left(p\right)}$ increasingly focuses on characterizing extremely strong dissipation events (i.e., the most singular structures).

4.2. Three Physical Hypotheses of the Classical SL Theory

She and Leveque pointed out in [5] that there exists a hierarchy of singular structures with different intensities in turbulence, and proposed three key hypotheses:

Hypothesis 1: Dominance of the highest singularity (asymptotic scaling). When the order $p\to \infty$, the most singular dissipation structure dominates the statistical behavior, and the hierarchical structure ${ϵ}_r^{\left(p\right)}$ converges to the dissipation rate of the most singular structure ${ϵ}_r^{(\infty )}\propto r^{-\gamma }$. This means that the scaling exponent sequence $X_p$ of the hierarchical structure (defined as ${ϵ}_r^{\left(p\right)}\propto r^{X_p}$) has a limit $X_{\infty }\equiv -\gamma$.

Hypothesis 2: Geometric constraint of the most singular structure. In 3D incompressible turbulence, the most singular structure is identified as quasi-one-dimensional filamentary vortex tubes. Under the contraction of the scaling group on the 3D spatial volume, the probability of encountering such a 1D structure scales as $P\propto r^{D_c}$, where $D_c$ is the codimension of this singular structure (for a 1D vortex tube, $D_c=3-1=2$). The SL theory demonstrated that this geometric constraint determines the scaling exponent of the most singular structure $\gamma =2/3$, and defined the parameter reflecting the intermittency intensity $C_0=D_c=2$.

Hypothesis 3: Self-similar recursive relation of the hierarchy. This is the core of the SL theory. Between the hierarchies of dissipation structures of different intensities, an invariant self-similar relation is satisfied. Define the relative fluctuation $R_p\equiv {ϵ}_r^{\left(p\right)}/{ϵ}_r^{(\infty )}$. SL assumed a power-law relation between higher-order and lower-order relative fluctuations: $R_{p+1}\propto R_p^{\beta }$, where $\beta \in (0,1)$ is the hierarchical recursion parameter. This hypothesis indicates that the closer to the most singular structure, the slower the relative deviation between hierarchies decays, forming a memory effect with fractal characteristics.

4.3. Algebraic Solution of the Classical SL Scaling Law

Based on the above three phenomenological hypotheses, we can rigorously solve for ${\tau }_p$ through classical algebraic recursion.

First, calculate the scaling exponent $X_p$ of the hierarchical structure ${ϵ}_r^{\left(p\right)}$. According to the definition ${\Pi }_p\propto r^{{\tau }_p}$, we have:

$${ϵ}_r^{\left(p\right)}={\displaystyle \frac{{\Pi }_{p+1}}{{\Pi }_p}}\propto {\displaystyle \frac{r^{{\tau }_{p+1}}}{r^{{\tau }_p}}}=r^{{\tau }_{p+1}-{\tau }_p}$$

(15)

Therefore, $X_p={\tau }_{p+1}-{\tau }_p$.

Next, taking the logarithm of the recursive relation in Hypothesis 3, $R_{p+1}\propto R_p^{\beta }$, we obtain a linear relation:

$$ln{R}_{p+1}=\beta ln{R}_p+\mathrm{const}$$

(16)

Extract the scaling exponent of the above equation. Since the scaling exponent of the relative fluctuation $R_p$ is $X_p-X_{\infty }$, the scaling behavior of $ln{R}_p$ is proportional to $(X_p-X_{\infty })lnr$. Matching the coefficients of $lnr$ on both sides of the equation, we obtain the difference equation for the scaling exponent:

$$X_{p+1}-X_{\infty }=\beta (X_p-X_{\infty })$$

(17)

Defining $Y_p\equiv X_p-X_{\infty }$, the above equation transforms into a simple geometric recursion $Y_{p+1}=\beta Y_p$, whose solution is:

$$Y_p=Y_0{\beta }^p$$

(18)

Calculate the initial value $Y_0=X_0-X_{\infty }$. Because ${\Pi }_0=\langle {ϵ}_r^0\rangle =1$ is a constant, ${\tau }_0=0$; since the average dissipation rate is conserved $\langle {ϵ}_r\rangle =ϵ$ being a constant, ${\tau }_1=0$. Therefore $X_0={\tau }_1-{\tau }_0=0$. Substituting this yields $Y_0=-X_{\infty }=\gamma$. Thus:

$$X_p=X_{\infty }+\gamma {\beta }^p=-\gamma (1-{\beta }^p)$$

(19)

Now solve for ${\tau }_p$ from $X_p={\tau }_{p+1}-{\tau }_p$ and ${\tau }_0=0$. Summing $X_k$ from $k=0$ to $p-1$:

$$\begin{array}{cc}\hfill {\tau }_p& ={\displaystyle \sum _{k=0}^{p-1}}{X}_k={\displaystyle \sum _{k=0}^{p-1}}\left[-\gamma (1-{\beta }^k)\right]\hfill \\ \hfill & =-\gamma p+\gamma {\displaystyle \sum _{k=0}^{p-1}}{\beta }^k=-\gamma p+\gamma {\displaystyle \frac{1-{\beta }^p}{1-\beta }}.\hfill \end{array}$$

(20)

Introducing the geometric invariant $C_0\equiv {\displaystyle \frac{\gamma }{1-\beta }}$ from Hypothesis 2 (its physical meaning is codimension), the above equation simplifies to:

$${\tau }_p=-\gamma p+C_0(1-{\beta }^p)$$

(21)

For vortex tubes in 3D incompressible turbulence, substituting $C_0=2$ and $\gamma =2/3$, we inversely solve $\beta =2/3$. Substituting ${\tau }_p$ into ${\zeta }_p={\displaystyle \frac{p}{3}}+{\tau }_{p/3}$, we get:

$$\begin{array}{cc}\hfill {\zeta }_p& ={\displaystyle \frac{p}{3}}+\left[-{\displaystyle \frac{2}{3}}\left({\displaystyle \frac{p}{3}}\right)+2\left(1-{\left({\displaystyle \frac{2}{3}}\right)}^{p/3}\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{p}{9}}+2\left[1-{\left({\displaystyle \frac{2}{3}}\right)}^{p/3}\right].\hfill \end{array}$$

(22)

This is the famous She-Leveque scaling law.

4.4. Equivalent Mapping between Classical Recursion and Lie Group Generator Eigenvalues and Algebraic Reconstruction of the SL Scaling Law

The derivation of the aforementioned classical SL scaling law is based on intuitive physical hypotheses and algebraic matching of scaling exponents, but once the framework of the infinitesimal generator of the Lie group is introduced, the profound algebraic essence of these physical hypotheses becomes manifestly clear. The classical recursion process and the extraction of Lie group generator eigenvalues are not only numerically equal in result but also strictly isomorphic in mathematical structure. It must be clear that this equivalent mapping transforms the SL phenomenological hypotheses into a Lie algebra structure, thereby endowing the latter with a deeper mathematical connotation.

First, consider the eigenvalue problem of the hierarchical structure ${ϵ}_r^{\left(p\right)}$ under the action of the generator. Applying the prolonged infinitesimal generator $X^{\left(1\right)}=r{\displaystyle \frac{d}{dr}}$ to the hierarchical structure ${ϵ}_r^{\left(p\right)}$ and using the Leibniz rule:

$$X^{\left(1\right)}{ϵ}_r^{\left(p\right)}=X^{\left(1\right)}\left({\displaystyle \frac{{\Pi }_{p+1}}{{\Pi }_p}}\right)={\displaystyle \frac{{\Pi }_p\left(X^{\left(1\right)}{\Pi }_{p+1}\right)-{\Pi }_{p+1}\left(X^{\left(1\right)}{\Pi }_p\right)}{{\Pi }_p^2}}$$

(23)

Substituting the eigenvalue equation $X^{\left(1\right)}{\Pi }_p={\tau }_p{\Pi }_p$ into the above equation, the numerator becomes ${\Pi }_p\left({\tau }_{p+1}{\Pi }_{p+1}\right)-{\Pi }_{p+1}\left({\tau }_p{\Pi }_p\right)=({\tau }_{p+1}-{\tau }_p){\Pi }_p{\Pi }_{p+1}$. Thus we obtain:

$$X^{\left(1\right)}{ϵ}_r^{\left(p\right)}=({\tau }_{p+1}-{\tau }_p){ϵ}_r^{\left(p\right)}\equiv X_p{ϵ}_r^{\left(p\right)}$$

(24)

This result has profound algebraic and physical implications. In the unbroken symmetry of K41, ${\tau }_p=0$, hence $X_p=0$, and all hierarchical structures are “zero modes” (invariants) under the action of the generator. However, in intermittent turbulence, $X_p\ne 0$. Physically, $X_p$ quantifies the nonlinear sensitivity of the p-th order hierarchical structure to scale variations r. It is no longer a single a priori constant, but is determined by the difference of the eigenvalues of two adjacent moments, which means that as the order p increases, the scaling behavior of the dissipation rate moment is not a simple linear superposition, but there exists a “nonlinear increment” caused by intermittency. From the perspective of continuous group representation, in the infinite-dimensional space spanned by ${\Pi }_p$, the hierarchical structure ${ϵ}_r^{\left(p\right)}$ constitutes an eigenstate of the difference operator representation. $X_p$ is the “transition” effect produced by the operator acting on adjacent basis vectors, similar to the energy level difference produced by the raising and lowering operators of the angular momentum operator in quantum mechanics, or the weight difference in representation theory. It directly characterizes the nonlinear bifurcation structure of the spectrum after symmetry breaking: the larger $X_p$, the more severely the higher-order moments deviate from the linear scaling, and the more significant the intermittency effect.

Furthermore, the core of the SL theory—the self-similar recursive relation of Hypothesis 3—corresponds in the Lie algebra to an internal algebraic constraint on the sequence of generator eigenvalues. Taking the logarithm of the power-law relation in Hypothesis 3 actually maps the multiplicative structure on the manifold to the additive structure of the tangent space (where the Lie algebra resides): $ln{R}_{p+1}=\beta ln{R}_p+\mathrm{const}$. At this point, we apply the prolonged generator $X^{\left(1\right)}$ to this linear equation of the tangent space. Since the action of the generator on the logarithmic function is equivalent to extracting the scaling exponent of this function, i.e., $X^{\left(1\right)}lnR={\displaystyle \frac{X^{\left(1\right)}R}{R}}$, and according to the previous derivation, $X^{\left(1\right)}{R}_p=(X_p-X_{\infty })R_p$, we have:

$$X^{\left(1\right)}(ln{R}_{p+1})={\displaystyle \frac{X^{\left(1\right)}{R}_{p+1}}{R_{p+1}}}=X_{p+1}-X_{\infty }$$

(25)

$$X^{\left(1\right)}(\beta ln{R}_p+\mathrm{const})=\beta {\displaystyle \frac{X^{\left(1\right)}{R}_p}{R_p}}=\beta (X_p-X_{\infty })$$

(26)

Since $ln{R}_{p+1}$ and $\beta ln{R}_p+\mathrm{const}$ describe the same scaling field, the generator acting on them must extract the same eigenvalue deviation, thereby strictly obtaining the linear difference equation for the infinitesimal generator eigenvalues:

$$X_{p+1}-X_{\infty }=\beta (X_p-X_{\infty })$$

(27)

This proves that the hierarchical recursion hypothesis of SL is essentially an a priori restriction on the action law of the Lie group infinitesimal generator on the tangent space. The parameter $\beta$ is no longer merely an empirical fitting constant, but a structural constant in the Lie algebra recursion of the eigenvalue deviation sequence, which determines the decay rate of the nonlinear bifurcation of eigenvalues after symmetry breaking.

Having established this core Lie algebraic difference equation, we no longer need to rely on any phenomenological scaling matching; we only need pure algebraic solution and the injection of geometric boundary conditions to reconstruct the final SL scaling law in a closed manner. Solving the difference equation yields $X_p=X_{\infty }+\gamma {\beta }^p=-\gamma (1-{\beta }^p)$; restoring ${\tau }_p$ from the summation of eigenvalue differences:

$${\tau }_p=-\gamma p+\gamma {\displaystyle \frac{1-{\beta }^p}{1-\beta }}$$

(28)

Finally, introduce geometric boundary conditions to determine the algebraic structural constants. In 3D incompressible turbulence, the most singular structure is a 1D vortex tube, and its codimension is $C_0=2$. In the Lie algebra framework, the codimension directly determines the proportional relationship of the asymptotic behavior of the eigenvalues, i.e., $C_0\equiv {\displaystyle \frac{\gamma }{1-\beta }}=2$. Meanwhile, the energy dissipation rate scaling constraint of vortex tube dynamics gives the most singular eigenvalue $X_{\infty }=-\gamma =-2/3$. Substituting $\gamma =2/3$ into the proportional relationship immediately solves the structural constant $\beta =1-{\displaystyle \frac{2/3}{2}}=2/3$. Substituting these strictly determined geometric and dynamical parameters into the general solution of ${\tau }_p$, and utilizing ${\zeta }_p={\displaystyle \frac{p}{3}}+{\tau }_{p/3}$, we finally rigorously derive: ${\zeta }_p={\displaystyle \frac{p}{3}}+\left[-{\displaystyle \frac{2}{3}}\left({\displaystyle \frac{p}{3}}\right)+2\left(1-{\left({\displaystyle \frac{2}{3}}\right)}^{p/3}\right)\right]={\displaystyle \frac{p}{9}}+2\left[1-{\left({\displaystyle \frac{2}{3}}\right)}^{p/3}\right]$, which is exactly same as Eq.(22).

Thus far, examining from the generalized perspective of Noether’s theorem, the symmetry of K41 corresponds to the strict conservation of energy flux ($Xϵ=0$), while the SL theory demonstrates how the system maintains a reduced-dimensional covariant symmetry through algebraic recursion when this conservation law fails due to the emergence of singular structures in a complex flow field. The most singular structure in Hypothesis 1 corresponds to the algebraic fixed point at infinity of the generator eigenvalue sequence $X_{\infty }=-\gamma$; the geometric codimension in Hypothesis 2 serves as the boundary condition establishing the relationship between this algebraic fixed point and the structural constant $\beta$. The physical picture of the classical SL theory—from background uniform dissipation to the cross-scale fractal memory of vortex tube singularity—is perfectly translated into the algebraic bifurcation process of the generator eigenvalues from a linear spectrum to a nonlinear spectrum in the Lie algebra. This isomorphic mapping from physical intuition to rigorous algebra, along with the closed-form derivation of the final analytical solution, is precisely the most elegant logical consistency exhibited by Lie group theory in turbulence research.

5. Discussions and Conclusions

Based on the infinitesimal generator of the Lie group and its prolongation theory, this paper performs a unified algebraic reconstruction and equivalent mapping of the K41 and SL scaling laws for incompressible turbulence. The research shows that Kolmogorov’s physical hypotheses[3,4] in the K41 theory are mathematically equivalent to the invariance of the energy flux under the action of the scaling Lie group generator. Inspired by the profound connotation of Noether’s theorem[9], this continuous symmetry naturally corresponds to the conservation of energy flux in the Richardson cascade[2], thereby leading to the strictly linear character ${\zeta }_p=p/3$. We detailed the consistency from dimensional analysis to the derivation of generator eigenvalues, revealing the algebraic topological origin of the K41 scaling law.

However, intermittency breaks this perfect symmetry, leading to anomalous scaling[1]. The outstanding contribution of She and Leveque[5] lies in revealing that the broken symmetry does not disappear, but is implicit in the hierarchical recursion of singular structures. This paper reproduces in detail the hierarchical recursive algebraic solution process of the classical SL scaling law, and emphatically demonstrates the strict equivalence between this classical phenomenological process and the extraction of Lie group generator eigenvalues. Within the Lie algebra framework, we transform the phenomenological hypotheses of SL into a difference equation of generator eigenvalues, and in particular, reveal the physical and algebraic connotation of the difference eigenvalue $X_p$ as “scaling sensitivity” and “weight difference in representation theory”, quantifying the degree of symmetry breaking caused by intermittency. Combined with the geometric boundary condition of the 3D spatial vortex tube codimension $C_0=2$, we reconstruct the SL character formula. The anomalous scaling exponents are essentially nonlinear bifurcations of the Lie algebra eigenvalues, and the fractal memory effect between hierarchies is equivalent to the geometric recursion of eigenvalue deviations on the tangent space.

From Lie’s creation of continuous transformation groups[6,7,8], to Ovsiannikov’s development of group analysis of differential equations[10], and to modern statistical theory of turbulence, the reconstruction system of this paper demonstrates a clear logical main thread: K41 is the flat spacetime of the turbulent mean field, possessing perfect scaling symmetry; while the SL theory characterizes the scaling geometry in a curved spacetime induced by the singularity of vortex tubes. The algebraic tools of the Lie group provide the most elegant mathematical language for this transition from symmetry to broken symmetry, and from linearity to nonlinearity. At the same time, it should be acknowledged that the work of this paper is essentially an algebraic isomorphic mapping of existing phenomenological laws, and the dynamical origin of the SL recursion hypothesis still remains to be discovered from the first principles of the N-S equations. However, this algebraic reconstruction is not circular reasoning; it clarifies the nature of the parameters (such as $\beta$ and $C_0$) in the SL theory as algebraic structural constants. This not only deepens the understanding of the scaling laws of 3D incompressible turbulence, but also provides clear guidance for extending them to other complex flow fields (such as 2D turbulence, magnetohydrodynamic turbulence, or compressible turbulence): different physical systems only need to replace the geometric boundary conditions (algebraic fixed points) and structural constants of their most singular structures within the Lie algebra framework to systematically derive the corresponding scaling laws.