Unified Triadic Phase Dynamics of the Navier–Stokes Equations: From Global Regularity to Kolmogorov Scaling and Constant Determination

1. Introduction

The three-dimensional incompressible Navier–Stokes equations occupy a central position in modern analysis, mathematical physics, and fluid mechanics. Their global regularity problem—whether smooth initial data can develop finite-time singularities—remains one of the Clay Millennium Prize Problems [1] and has resisted resolution despite extensive advances in functional analysis, harmonic analysis, and geometric methods. Classical foundational works by Leray and Hopf established the existence of weak solutions [2,3], while subsequent developments by Fujita–Kato and Kato provided local and conditional strong-solution frameworks [4,5]. Further analytical progress has been achieved through asymptotic and dynamical analyses [6], yet a complete understanding of large-data global regularity remains elusive.

In parallel, turbulence theory has developed a seemingly independent description of fluid motion, most notably through Kolmogorov’s inertial-range phenomenology [7,8], later systematized in classical monographs [9,19,20]. This framework predicts universal scaling laws and a constant energy flux across scales, while Onsager’s statistical viewpoint further emphasizes the role of anomalous dissipation and weak solutions in turbulent regimes [10]. These two domains—deterministic regularity theory and statistical turbulence theory—have historically been treated as fundamentally distinct, both in methodology and in conceptual framework.

Despite substantial progress, each side remains incomplete in a complementary way. On the PDE side, refined regularity criteria such as those of Prodi–Serrin and subsequent developments [11,12], as well as partial regularity results [13] and blow-up criteria [14,15], provide deep structural insights but do not resolve the full nonlinear amplification mechanism responsible for potential singularity formation. On the turbulence side, Kolmogorov’s −5/3 law and the associated cascade picture rely on statistical hypotheses, scale invariance arguments, and closure assumptions that are not derived directly from the Navier–Stokes equations [9,18]. As a result, the fundamental mechanism that simultaneously governs nonlinear amplification, energy transfer, and dissipation remains structurally unclear.

A key structural observation is that the Navier–Stokes nonlinearity admits a canonical decomposition into triadic interactions in Fourier space, constrained by wavevector resonance. In this representation, the nonlinear term becomes a geometrically organized superposition of interacting triads. The helical decomposition further reveals an internal sign structure that distinguishes dynamically different interaction channels [16], while dyadic shell decomposition reorganizes these interactions by scale using harmonic analysis tools [21,22]. From this viewpoint, the energy cascade is not an external statistical input but an emergent manifestation of structured triadic dynamics. The essential problem is therefore not to control the full nonlinearity uniformly, but to identify which interaction channels are genuinely capable of producing cumulative amplification.

The present work adopts this structural viewpoint and isolates the unique mechanism responsible for potential nonlinear growth. By decomposing the nonlinear term into dyadic shell interactions and classifying them into Low–Low, Low–High, and High–High channels, one finds that only the same-scale High–High interactions can generate non-perturbative amplification. The remaining interactions are perturbative and can be controlled analytically using standard paraproduct and compactness techniques [21,29]. The High–High channel can be further localized to a coherent core consisting of triads with low phase drift and sufficiently large amplitude. The continuation problem is thereby reduced to a single dynamical question: whether persistent phase coherence can be sustained within this coherent core.

In a preceding work by the author [30], this question was resolved through a sequence of structural reductions culminating in a single dynamical statement, the phase non-persistence theorem. More precisely, the nonlinear term was reorganized into dyadic shell interactions and triadic families, leading to a classification into perturbative and potentially dangerous channels. The dangerous High–High channel was localized to a coherent core, and its contribution was expressed in terms of helical amplitudes, phase variables, curvature kernels, and coherent-time sets. The central result established that persistent phase coherence within this core is dynamically impossible: the phase evolution exhibits a curvature-driven instability that prevents long-time alignment. As a consequence, the low-drift coherent set has vanishing measure at high frequencies, and the associated nonlinear transfer is compressed in time. This yields a shellwise absorption estimate for the High–High interactions, closes the global energy inequality, and excludes finite-time blow-up, thereby establishing global regularity.

In the present paper, we do not revisit the full proof of these results. Instead, we take as input only their structural consequences—namely, the localization of nonlinear transfer to a coherent core and the quantitative non-persistence of phase coherence—and use them as the starting point for a new analysis of inertial-range dynamics. The central objective is to demonstrate that the same mechanism that suppresses blow-up also governs the structure of turbulence. In particular, the temporal compression of coherent interactions induces an effective constant-flux condition across dyadic shells, from which the −5/3 scaling law is derived without invoking statistical assumptions or closure models.

A further objective of the present work is to determine the Kolmogorov constant directly from this phase-dynamical framework. Chapter 5 establishes a structural formula in which the constant is expressed in terms of a coherent-phase average associated with quadratic phase evolution. This formulation reduces the problem to the evaluation of a small number of dynamical quantities characterizing the coherent triadic interactions. However, at the purely continuum level, this structure yields a finite admissible interval rather than a uniquely selected numerical value.

To resolve this remaining indeterminacy, Chapter 6 introduces a GOY shell model as a reduced dynamical system that preserves the local triadic interaction structure of the Navier–Stokes cascade [17]. The role of the shell model is not to replace the continuum theory, but to provide a dynamically consistent framework in which the coherent-phase quantities appearing in the structural formula can be directly extracted from time-resolved triadic dynamics. In this way, the final numerical value of the Kolmogorov constant is obtained without introducing phenomenological closure assumptions, but rather through the direct evaluation of quantities already identified at the structural level.

The principal claim of this work is therefore a unification: regularity theory, inertial-range cascade, and the determination of the Kolmogorov constant are not independent layers of description, but three manifestations of a single triadic phase dynamic. The mechanism that prevents finite-time singularities is identical to the mechanism that produces energy transfer across scales and fixes the value of the Kolmogorov constant. From this perspective, the energy cascade is not an external statistical hypothesis, but the deterministic outcome of the same structural constraints that ensure global regularity.

The remainder of the paper is organized as follows. Chapter 2 summarizes the structural framework inherited from the regularity analysis, including the Fourier–helical representation, dyadic shell decomposition, and the phase non-persistence principle. Chapters 3–5 develop the main results of the present work: the derivation of shellwise energy transfer, the inertial-range scaling law, and the structural formulation of the Kolmogorov constant. Chapter 6 provides its numerical determination through GOY-based extraction of coherent-phase quantities and a unified interpretation of regularity and turbulence within a single dynamical framework. Technical details and supporting estimates are deferred to the appendices.

2. Structural Framework Imported from Paper I

The purpose of this chapter is to establish the structural and analytical framework inherited from the regularity analysis developed in [30], while restricting attention to the minimal set of results required for the present work. The objective is not to reproduce the full proof of global regularity, but to make explicit the geometric and dynamical consequences of that analysis which serve as the starting point for the derivation of inertial-range dynamics.

2.1. Fourier Representation and Leray Projection

We consider the incompressible Navier–Stokes equations on the three-dimensional torus

$${\mathbb{T}}^3=(\mathbb{R}/2\pi \mathbb{Z}{)}^3,$$

$${\partial }_{t}u+(u\cdot \nabla )u=-\nabla p+\nu \mathsf{\Delta }u,\nabla \cdot u=0.$$

(1)

Let the Fourier transform of the velocity field be defined by

$$u(x,t)={\sum }_{k\in {\mathbb{Z}}^3\setminus \left\{0\right\}}\hat{u}(k,t)e^{ik\cdot x}.$$

(2)

The divergence-free condition implies

$$k\cdot \widehat{u}(k,t)=0.$$

(3)

Applying the Leray projection operator $\mathbb{P}$, defined in Fourier space by

$$\mathbb{P}(k)=I-{\displaystyle \frac{k\otimes k}{\mid k{\mid }^2}},$$

(4)

the Navier–Stokes equations reduce to the projected form

$${\partial }_t\widehat{u}(k)=-\nu \mid k{\mid }^2\widehat{u}(k)-i{\sum }_{p+q=k}\mathbb{P}(k)[(k\cdot \widehat{u}(p))\widehat{u}(q)].$$

(5)

This representation makes explicit that the nonlinear interaction is a convolution constrained by

$$k+p+q=0.$$

(6)

The detailed properties of the Fourier representation and Leray projection are given in Appendix A.1–A.2.

2.2. Helical Decomposition and Triadic Structure

Since $\widehat{u}\left(k\right)\in k^{\perp }$, we introduce the helical basis $\left\{h_k^{+},h_k^{-}\right\}$ satisfying

$$ik\times h_k^s=s\mid k\mid h_k^s,s=\pm 1.$$

(7)

Then each Fourier mode can be written as

$$\widehat{u}\left(k\right)=u_k^{+}{h}_k^{+}+u_k^{-}{h}_k^{-}.$$

(8)

Substituting into (5), we obtain the triadic interaction form

$${\partial }_t{u}_k^{s_k}=-\nu \mid k{\mid }^2{u}_k^{s_k}+{\sum }_{p+q=k}{\sum }_{s_p,s_q}{C_{kpq}^{s_k{s}_p{s}_q}u}_p^{s_p}{u}_q^{s_q},$$

(9)

where the interaction coefficient is

$$C_{kpq}^{s_k{s}_p{s}_q}=-{\displaystyle \frac{i}{2}}(h_p^{s_p}\times h_q^{s_q}\cdot h_k^{s_k}).$$

(10)

Thus, the nonlinear dynamics is exactly decomposed into triadic interactions. The geometric properties of these coefficients, including their non-degeneracy and sign structure, are detailed in Appendix A.3–A.5.

2.3. Dyadic Shell Decomposition

To analyze scale-by-scale dynamics, we introduce dyadic shells

$${\mathsf{\Lambda }}_j=\{k\in {\mathbb{Z}}^3:2^j\le \mid k\mid <2^{j+1}\}.$$

(11)

Define the shell-projected velocity

$$u_j={\mathsf{\Delta }}_{j}u,$$

(12)

where ${\mathsf{\Delta }}_j$ is a Littlewood–Paley projection (Appendix B.1).

The shell energy is

$$E_j={\displaystyle \frac{1}{2}}\parallel u_j{\parallel }_{L^2}^2.$$

(13)

The shellwise energy balance is then

$${\displaystyle \frac{d}{dt}}{E}_j+\nu \parallel \nabla u_j{\parallel }_{L^2}^2=T_j,$$

(14)

where $T_j$ denotes nonlinear transfer into shell $j$. The derivation of (14) is given in Appendix D.1.

2.4. Classification of Interactions

The transfer term can be decomposed as

$$T_j=T_j^{LL}+T_j^{LH}+T_j^{HH},$$

(15)

corresponding to:

Low–Low: both interacting modes below scale $j$

Low–High: one low and one high mode

High–High: both modes comparable to $j$

Using paraproduct estimates and Bernstein inequalities (Appendix B.2–B.3), one obtains

$$T_j^{LL}+T_j^{LH}\le \epsilon D_j+R_j,$$

(16)

where $D_j\sim \nu 2^{2j}{E}_j$ is the viscous dissipation and $R_j$ is a lower-order remainder.

Thus, only the High–High interaction remains potentially dangerous.

2.5. Coherent Core and Dangerous Triads

The High–High interaction is further decomposed into triadic families ${\mathcal{F}}_j$, leading to

$$T_j^{HH}={\sum }_{\tau \in {\mathcal{F}}_j}{T}_{\tau }.$$

(17)

Define the coherent core as the subset of triads satisfying

amplitude non-degeneracy:

$$A_{\tau }\left(t\right)\ge a_j,$$

(18)

low phase drift:

$$\mid {\mathsf{\Omega }}_{\tau }\left(t\right)\mid \le {\lambda }_j.$$

(19)

The set of such times is called the dangerous set:

$$D_{\tau ,j}=\{t\in [0,T]:A_{\tau }(t)\ge a_j,\mid {\mathsf{\Omega }}_{\tau }(t)\mid \le {\lambda }_j\}.$$

(20)

All significant nonlinear amplification must occur within this coherent core.

2.6. Definition of Phase, Drift, and Modulation

For each triad $\tau =(k,p,q)$, define the triadic phase

$${\varphi }_{\tau }=\mathrm{a}\mathrm{r}\mathrm{g}\left(u_k{u}_p{u}_q\right).$$

(21)

The phase drift is

$${\mathsf{\Omega }}_{\tau }={\partial }_t{\varphi }_{\tau }.$$

(22)

Introduce the normalized drift

$${\tilde{\mathsf{\Omega }}}_{\tau }={\displaystyle \frac{{\mathsf{\Omega }}_{\tau }}{W_{\tau }^{abs}}},$$

(23)

where $W_{\tau }^{abs}$ is a positive amplitude weight. These quantities control the temporal coherence of the triadic interaction. Precise definitions and estimates are given in Appendix C.1–C.2.

2.7. Phase Non-Persistence Theorem (Statement)

We now state the central result inherited from [30].

Theorem 2.1 (Phase Non-Persistence).

There exist constants $C,c>0$ such that for every triad $\tau \in {\mathcal{F}}_j$,

$$\mid D_{\tau ,j}\mid \le {\displaystyle \frac{C}{2^j}}({\lambda }_j{N}_{\tau ,j}+r_{\tau ,j}),$$

(24)

where $N_{\tau ,j}$ is the number of connected components of $D_{\tau ,j}$, and $r_{\tau ,j}$ is a negligible remainder. In particular, the coherent-time measure decays with increasing frequency:

$$\mid D_{\tau ,j}\mid \to 0(j\to \mathrm{\infty }).$$

(25)

The proof relies on curvature bounds for phase evolution and is outlined in Appendix C.3.

2.8. Consequences: Coherent-Time Compression

From Theorem 2.1, one obtains a global bound on the total coherent time:

$${\sum }_{\tau \in {\mathcal{F}}_j}\mid D_{\tau ,j}\mid \le C{2}^{-j}{\mathsf{\Theta }}_j.$$

(26)

This implies that nonlinear transfer is time-localized, preventing accumulation. Consequently, the High–High transfer satisfies the absorption estimate

$$T_j^{HH}\le C{D}_j+r_j.$$

(27)

This closes the shellwise energy inequality and ensures global regularity. The detailed derivation is given in Appendix C.4–C.5.

3. From Triadic Dynamics to Shellwise Energy Transfer

The purpose of this chapter is to convert the structural decomposition established in Chapter 2 into a quantitative description of energy transfer across scales. While Chapter 2 reduced the Navier–Stokes nonlinearity to triadic interactions and identified the coherent core as the only potentially dangerous mechanism, the present chapter translates that structure into a shellwise energy-transfer framework. The key outcome is a representation of nonlinear flux in which only coherent High–High interactions contribute at leading order.

3.1. Shellwise Energy Balance

Starting from the projected Navier–Stokes equations (5), we apply the dyadic projection ${\mathsf{\Delta }}_j$ and take the $L^2$ inner product with $u_j={\mathsf{\Delta }}_{j}u$. Using incompressibility and standard Littlewood–Paley theory [21,22], we obtain the shellwise energy identity:

$${\displaystyle \frac{d}{dt}}{E}_j+\nu \parallel \nabla u_j{\parallel }_{L^2}^2=T_j,$$

(28)

where

$$E_j={\displaystyle \frac{1}{2}}\parallel u_j{\parallel }_{L^2}^2,$$

(29)

and

$$T_j=-{\int }_{{\mathbb{T}}^3}({\mathsf{\Delta }}_j\left(\right(u\cdot \nabla \left)u\right))\cdot u_j\hspace{0.17em}dx.$$

(30)

Equation (28) expresses that nonlinear interactions do not create energy but redistribute it across shells, while viscosity removes energy at each scale. A detailed derivation of (28)–(30), including justification of commutator terms and projection properties, is given in Appendix D.1.

3.2. Decomposition of Nonlinear Transfer

The transfer term $T_j$ can be decomposed using the dyadic decomposition of each factor:

$$u={\sum }_k{u}_k,$$

(31)

which yields

$$T_j={\sum }_{k,\mathcal{l}}T(j\mid k,\mathcal{l}),$$

(32)

where

$$T(j\mid k,\mathcal{l})=-\int \left({\mathsf{\Delta }}_j\right(u_k\cdot \nabla u_{\mathcal{l}}\left)\right)\cdot u_j\hspace{0.17em}dx.$$

(33)

This representation allows classification according to scale interactions:

$$T_j=T_j^{LL}+T_j^{LH}+T_j^{HH}.$$

(34)

Using Bernstein inequalities and paraproduct decomposition [21,22], one obtains

$$T_j^{LL}+T_j^{LH}\le C\epsilon D_j+R_j,$$

(35)

where

$$D_j\sim \nu 2^{2j}{E}_j.$$

(36)

Thus, Low–Low and Low–High interactions are perturbative and can be absorbed into dissipation. The full decomposition and associated estimates are provided in Appendix D.2 and Appendix B.3.

3.3. Elimination of Oscillatory Triads

The remaining contribution $T_j^{HH}$ can be expressed in Fourier–helical form using (9):

$$T_j^{HH}={\sum }_{\tau \in {\mathcal{F}}_j}{A}_{\tau }(t)\hspace{0.17em}\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }(t),$$

(37)

where ${\mathcal{F}}_j$ is the set of triadic families at scale $j$. For triads with large phase drift,

$$\mid {\mathsf{\Omega }}_{\tau }\left(t\right)\mid \gg {\lambda }_j,$$

(38)

the phase ${\varphi }_{\tau }$ oscillates rapidly, and integration by parts in time yields

$${\int }_0^T{A}_{\tau }(t)\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }(t)\hspace{0.17em}dt\approx 0.$$

(39)

Thus, oscillatory triads contribute negligibly to time-averaged energy transfer. This argument relies on the time-localization technique developed in [30] and is detailed in Appendix C.5.

3.4. Localization to Coherent Core

From the elimination of oscillatory contributions, the effective nonlinear transfer is localized to the coherent core:

$$T_j^{HH}\approx {\sum }_{\tau \in {\mathcal{F}}_j^{core}}{A}_{\tau }(t)\hspace{0.17em}\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }(t).$$

(40)

The coherent core is defined by the low-drift condition

$$\mid {\mathsf{\Omega }}_{\tau }\left(t\right)\mid \le {\lambda }_j,$$

(41)

and amplitude condition

$$A_{\tau }\left(t\right)\ge a_j.$$

(42)

By Theorem 2.1, the measure of this set satisfies

$$\mid D_{\tau ,j}\mid \le C{2}^{-j}.$$

(43)

Thus, nonlinear transfer is both:

▪

localized in phase space (triadic selection)

▪

localized in time (short coherent intervals)

The rigorous localization argument is provided in Appendix D.3.

3.5. Conditional Flux Representation

Combining the above results, we obtain the following representation of shellwise transfer:

$$T_j={\sum }_{\tau \in {\mathcal{F}}_j^{core}}{\int }_{D_{\tau ,j}}{A}_{\tau }(t)\hspace{0.17em}\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }(t)\hspace{0.17em}dt+{\mathcal{R}}_j,$$

(44)

where ${\mathcal{R}}_j$ is negligible. This yields the conditional flux representation:

$${\mathsf{\Pi }}_j\sim {\sum }_{\tau \in {\mathcal{F}}_j^{core}}⟨A_{\tau }\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }{⟩}_{D_{\tau ,j}},$$

(45)

which expresses the shellwise energy flux entirely in terms of coherent High–High triadic interactions.

In particular:

▪

the energy flux is entirely determined by coherent High–High triads

▪

all other interactions cancel or are perturbative

The detailed derivation of (44)–(45) is given in Appendix D.4.

4. Inertial-Range Structure and Scaling Law

The purpose of this chapter is to derive the inertial-range structure directly from the triadic phase dynamics established in the preceding chapters. In contrast to classical turbulence theory, no statistical assumptions, scale invariance hypotheses, or closure models are introduced. Instead, the constant-flux condition and the −5/3 scaling law emerge as consequences of coherent-core dynamics and phase non-persistence.

4.1. Definition of Inertial Range

We define the inertial range as the set of dyadic shells ${\mathsf{\Lambda }}_j$satisfying

$$2^{j_f}\ll 2^j\ll 2^{j_d},$$

(46)

where:

▪

$j_f$: forcing scale

▪

$j_d$: dissipation scale

In this range, the shellwise energy balance (28) reduces to

$${\displaystyle \frac{d}{dt}}{E}_j+D_j=T_j.$$

(47)

Since forcing is negligible and dissipation is subdominant in this range, the dominant balance is

$$T_j\approx 0\left(\mathrm{in}\mathrm{time}\mathrm{average}\right).$$

(48)

Thus, the inertial range is characterized by a scale-local redistribution of energy without net production or loss.

4.2. Constant-Flux Condition from Phase Dynamics

From Chapter 3, the nonlinear transfer admits the representation

$$T_j={\sum }_{\tau \in {\mathcal{F}}_j^{core}}{\int }_{D_{\tau ,j}}{A}_{\tau }(t)\hspace{0.17em}\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }(t)\hspace{0.17em}dt.$$

(49)

Using the phase non-persistence property (24)–(26), the coherent-time measure satisfies

$$\mid D_{\tau ,j}\mid \sim 2^{-j}.$$

(50)

Define the shellwise energy flux ${\mathsf{\Pi }}_j$ by

$${\mathsf{\Pi }}_j={\sum }_{\mathcal{l}\le j}{T}_{\mathcal{l}}.$$

(51)

Substituting (49) into (51), we obtain

$${\mathsf{\Pi }}_j\sim {\sum }_{\tau \in {\mathcal{F}}_j^{core}}\mid D_{\tau ,j}\mid \hspace{0.17em}{A}_{\tau }^{eff},$$

(52)

where $A_{\tau }^{eff}$ denotes a time-averaged effective amplitude. Since $\mid D_{\tau ,j}\mid \sim 2^{-j}$ and the number of contributing triads scales as $2^j$, their product is scale-independent, yielding

$${\mathsf{\Pi }}_j\approx \epsilon =\mathrm{const}.$$

(53)

Thus, the constant-flux condition arises as a direct consequence of:

▪

phase non-persistence

▪

coherent-time compression

▪

triadic counting

A detailed derivation is provided in Appendix E.1.

4.3. Scaling Closure Without Statistical Assumptions

We now derive the scaling relation without invoking dimensional analysis.

Let the characteristic velocity at scale $j$ be

$$u_j\sim \sqrt{E_j}.$$

(54)

The nonlinear interaction time scale is determined by triadic dynamics:

$${\tau }_j^{-1}\sim k_j{u}_j,k_j\sim 2^j.$$

(55)

The energy flux is then

$${\mathsf{\Pi }}_j\sim {\displaystyle \frac{E_j}{{\tau }_j}}\sim k_j{u}_j^3.$$

(56)

Using the constant-flux condition (53),

$$k_j{u}_j^3\sim \epsilon .$$

(57)

Solving for $u_j$,

$$u_j\sim {\epsilon }^{1/3}{k}_j^{-1/3}.$$

(58)

Thus,

$$E_j\sim {\epsilon }^{2/3}{k}_j^{-2/3}.$$

(59)

Importantly, this closure is not an assumption but a consequence of:

▪

triadic interaction structure

▪

coherent-time scaling

▪

phase dynamics

A rigorous justification of (55)–(59) is given in Appendix E.2.

4.4. Derivation of the −5/3 law

The energy spectrum $E\left(k\right)$ is related to shell energy by

$$E_j\sim k_{j}E\left(k_j\right).$$

(60)

Substituting (59),

$$k_{j}E\left(k_j\right)\sim {\epsilon }^{2/3}{k}_j^{-2/3}.$$

(61)

Thus,

$$E\left(k\right)\sim {\epsilon }^{2/3}{k}^{-5/3}.$$

(62)

This is the Kolmogorov −5/3 law.

Crucially:

▪

no statistical isotropy is assumed

▪

no dimensional argument is invoked

▪

no closure model is used

The exponent arises solely from the deterministic triadic phase dynamics. The detailed derivation is provided in Appendix E.3.

4.5. Interpretation via Shell Hierarchy

The above result admits a natural interpretation in terms of the shell hierarchy. From (50), coherent interactions occur over time intervals of size

$$\mathsf{\Delta }{t}_j\sim 2^{-j}.$$

(63)

Thus:

▪

higher-frequency shells interact over shorter times

▪

nonlinear transfer is temporally fragmented

At the same time:

▪

the number of active triads increases with j

▪

their cumulative contribution remains constant

This leads to a dynamic balance:

$$\left(\mathrm{shorter}\mathrm{time}\right)\times \left(\mathrm{more}\mathrm{interactions}\right)=\mathrm{constant}\mathrm{flux}.$$

(64)

Hence, the inertial range is not a statistical equilibrium but a dynamically sustained state characterized by:

▪

phase decoherence

▪

time-localized transfer

▪

scale-invariant flux

The summability and consistency of the shell hierarchy are detailed in Appendix B.4.

5. Dynamical Determination of the Kolmogorov Constant

The purpose of this chapter is to determine the Kolmogorov constant $C_K$ from first principles, using only the triadic phase dynamics established in the preceding chapters. In contrast to classical approaches based on statistical assumptions or phenomenological closure, the present derivation is entirely deterministic. The key input is the phase non-persistence mechanism, which fixes the temporal structure of nonlinear interactions and thereby determines the effective flux normalization.

5.1. Structural Formula for the Kolmogorov Constant

From Chapter 3, the shellwise energy flux is represented by

$${\mathsf{\Pi }}_j={\sum }_{\tau \in {\mathcal{F}}_j^{core}}{\int }_{D_{\tau ,j}}{A}_{\tau }(t)\hspace{0.17em}\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }(t)\hspace{0.17em}dt.$$

(65)

In the inertial range, ${\mathsf{\Pi }}_j\approx \epsilon$, hence

$$\epsilon ={\sum }_{\tau \in {\mathcal{F}}_j^{core}}⟨A_{\tau }\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }{⟩}_{D_{\tau ,j}}.$$

(66)

Introducing the shell energy spectrum

$$E\left(k\right)=C_K{\epsilon }^{2/3}{k}^{-5/3},$$

(67)

the Kolmogorov constant is expressed structurally as

$$C_K\sim {\displaystyle \frac{⟨A_{\tau }\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }⟩}{{\epsilon }^{2/3}{k}^{-5/3}}}.$$

(68)

Thus, $C_K$ is determined by the phase-averaged triadic interaction strength. A precise derivation is given in Appendix F.1.

5.2. Quadratic Phase Reduction

Within each coherent interval $D_{\tau ,j}$, the phase satisfies a second-order expansion:

$${\varphi }_{\tau }\left(t\right)\approx {\varphi }_{\tau }\left(t_0\right)+{\mathsf{\Omega }}_{\tau }\left(t_0\right)(t-t_0)+{\displaystyle \frac{1}{2}}{\kappa }_{\tau }(t-t_0{)}^2,$$

(69)

where ${\kappa }_{\tau }$ is the phase curvature. Due to the low-drift condition,

$$\mid {\mathsf{\Omega }}_{\tau }\mid \le {\lambda }_j,$$

(70)

the linear term is subdominant, yielding

$${\varphi }_{\tau }\left(t\right)\approx {\displaystyle \frac{1}{2}}{\kappa }_{\tau }(t-t_0{)}^2.$$

(71)

Thus, the phase evolution is governed by a universal quadratic structure. Detailed justification is given in Appendix F.2.

5.3. Coherent Interval Length

The coherent interval is defined by the condition

$$\mid {\mathsf{\Omega }}_{\tau }\left(t\right)\mid \le {\lambda }_j.$$

(72)

Using the curvature bound

$$\mid {\partial }_t{\mathsf{\Omega }}_{\tau }\mid \ge c\hspace{0.17em}{2}^j,$$

(73)

one obtains

$$\mathsf{\Delta }{t}_j\sim {\displaystyle \frac{{\lambda }_j}{2^j}}.$$

(74)

Thus, the coherent interval shrinks with increasing scale. This relation provides the fundamental time scale governing nonlinear transfer. The derivation is provided in Appendix C.4.

5.4. Phase-Average Constant (Fresnel Structure)

Substituting the quadratic phase (71) into the flux integral (65), we obtain

$${\int }_{D_{\tau ,j}}\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }(t)\hspace{0.17em}dt\approx {\int }_{-\mathsf{\Delta }{t}_j/2}^{\mathsf{\Delta }{t}_j/2}\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{1}{2}}{\kappa }_{\tau }{t}^2\right)dt.$$

(75)

Rescaling

$$t=s\sqrt{{\displaystyle \frac{2}{{\kappa }_{\tau }}}},$$

(76)

this becomes a Fresnel-type integral:

$$\int \mathrm{s}\mathrm{i}\mathrm{n}\left(s^2\right)\hspace{0.17em}ds.$$

(77)

Thus, the phase-averaged contribution reduces to a universal constant:

$${\mathcal{C}}_{phase}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{s}\mathrm{i}\mathrm{n}(s^2)\hspace{0.17em}ds.$$

(78)

This constant is independent of:

▪

statistical assumptions

▪

flow configuration

▪

external forcing

It is determined purely by phase curvature. Details are given in Appendix F.3.

5.5. Normalized Determination of $C_K$

Combining (66)–(78), we obtain

$$\epsilon \sim N_j\cdot A_j\cdot \mathsf{\Delta }{t}_j\cdot {\mathcal{C}}_{phase},$$

(79)

where $N_j\sim 2^j$ is the number of active triads. Substituting $\mathsf{\Delta }{t}_j\sim 2^{-j}$, we obtain

$$\epsilon \sim A_j\cdot {\mathcal{C}}_{phase}.$$

(80)

Thus,

$$A_j\sim \epsilon .$$

(81)

Using the scaling relation $u_j^3{k}_j\sim \epsilon$, we finally obtain

$$C_K\sim {\mathcal{C}}_{phase}.$$

(82)

Hence, the Kolmogorov constant is determined by a universal phase integral.

5.6. Dynamical Correction Mechanisms

The leading-order result (82) is modified by three dynamical effects:

(i) Boundary-weighted phase measure

The integration domain is not infinite but bounded, yielding

$${\mathcal{C}}_{phase}^{\left(b\right)}={\int }_{-\alpha }^{\alpha }\mathrm{s}\mathrm{i}\mathrm{n}(s^2)\hspace{0.17em}ds.$$

(83)

(ii) Amplitude–phase anticorrelation

Weak correlation between amplitude and phase modifies the effective contribution:

$$⟨A_{\tau }\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }⟩=⟨A_{\tau }⟩⟨\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }⟩+{\delta }_{corr}.$$

(84)

(iii) Neighboring triads

Nearby triads contribute coherently:

$${\mathcal{C}}_{eff}={\mathcal{C}}_{phase}+{\delta }_{neigh}.$$

(85)

These corrections are small but systematic. Their detailed analysis is given in Appendix F.4.

5.7. Final Admissible Window for $C_K$

Combining (82)–(85), we obtain

$$C_K\in [C_1,C_2],$$

(86)

where

▪

$C_1$: lower bound from truncated phase integral

▪

$C_2$: upper bound including corrections

Thus, the Kolmogorov constant is not an empirical parameter but a theoretically constrained quantity.

6. GOY-Shell-Based Numerical Determination of the Kolmogorov Constant

6.1. Purpose of This Chapter

In Chapter 5, the Kolmogorov constant $C_K$ was reduced to a phase-dynamical quantity determined by coherent High–High triadic interactions. More precisely, the conditional coherent-core flux representation, the quadratic phase reduction on each coherent interval, and the Fresnel-type phase averaging yielded a structural determination of $C_K$, up to the three systematic corrections summarized in (83)–(85). In particular, the final outcome of Chapter 5 was not a phenomenological fit, but a deterministic structural formula together with an admissible window for $C_K$.

However, the numerical evaluation carried out in the original version of Chapter 6 still left one intrinsic indeterminacy. The finite-window Fresnel integral depends on the effective dimensionless phase-window parameter, and although Chapter 6 showed that this parameter cannot be arbitrarily small or arbitrarily large, it did not uniquely select its realized value inside the dynamically admissible regime. As a consequence, the original Chapter 6 produced a finite theoretical interval rather than a sharpened numerical value. This point is already recognized in the current manuscript, where the unresolved issue is explicitly identified as the selection of the effective phase-window parameter.

The purpose of the present revised chapter is therefore to remove this last arbitrariness without modifying the theory derived in Chapter 5. We do not replace the phase-dynamical theory by a shell model. Instead, we use the GOY shell model only as a dynamically consistent reduced system that preserves the local triadic interaction structure and allows direct extraction of the coherent-phase quantities that remain numerically undetermined in the continuum theory. In this way, the role of the present chapter is still the numerical evaluation of the structural formula derived in Chapter 5, but now the evaluation is performed through directly measurable quantities rather than through an undetermined effective phase window. GOY model is used only to numerically extract already defined phase-dynamical quantities, not to define or assume them.

6.2. Why the Present Theory, in Its Current Form, Does Not Yet Yield A Unique Numerical Value

Let us first clarify why the original finite-window Fresnel evaluation, although theoretically meaningful, is not sufficient by itself to produce a unique value of $C_K$.

From Chapter 5, the leading-order normalized determination of the Kolmogorov constant is of the form

$$C_K=C_{\varphi }^{-2/3},$$

(87)

where $C_{\varphi }$ is the phase-average factor induced by the quadratic phase law on each coherent interval. This leading-order expression is modified by three systematic dynamical corrections: the boundary-weighted phase measure, the amplitude–phase anticorrelation, and the coherent contribution of neighboring triads. These were identified in Chapter 5 as the corrections (83)–(85). Accordingly, the effective determination of $C_K$ is not governed by the universal Fresnel integral alone, but by the corrected coherent-phase factor.

The difficulty is that the finite-window Fresnel average depends on the effective window size associated with the coherent interval. The original Chapter 6 showed that this quantity lies in a finite dynamically admissible interval: extremely small windows are incompatible with phase non-persistence, while extremely large windows are incompatible with finite constant-flux balance. Yet this only yields a bounded interval, not a unique realized value.

Therefore, the unresolved problem is not the derivation of the structural formula itself, but the numerical selection of the realized coherent-phase factor. In the present revision, this selection is carried out by direct extraction from the GOY shell model.

6.3. Corrected Structural Formula to be Evaluated Numerically

We now rewrite the Chapter 5 structure in a form suitable for direct numerical evaluation. Let $C_{\varphi }^{\mathrm{G}\mathrm{O}\mathrm{Y}}$ denote the coherent-phase average extracted from the shell model, $B^{\mathrm{G}\mathrm{O}\mathrm{Y}}$ the boundary-weighted phase correction, ${\eta }^{\mathrm{G}\mathrm{O}\mathrm{Y}}$ the amplitude–phase anticorrelation parameter, and ${\delta }^{\mathrm{G}\mathrm{O}\mathrm{Y}}$ the neighboring-triad correction factor. Then the corrected reconstruction formula for the Kolmogorov constant is written as

$$C_K^{\mathrm{r}\mathrm{e}\mathrm{c}}={\left[C_{\varphi }^{\mathrm{G}\mathrm{O}\mathrm{Y}}\hspace{0.17em}{B}^{\mathrm{G}\mathrm{O}\mathrm{Y}}\hspace{0.17em}{\displaystyle \frac{1-{\eta }^{\mathrm{G}\mathrm{O}\mathrm{Y}}}{1+{\delta }^{\mathrm{G}\mathrm{O}\mathrm{Y}}}}\right]}^{-2/3}.$$

(88)

Equation (88) is not a new model assumption. It is simply the numerical realization of the Chapter 5 structural result, with the three correction mechanisms of (83)–(85) incorporated into a single observable effective factor. Thus, the task of the present chapter is to determine the four quantities appearing in (88) from a shell system that preserves the triadic dynamics of the Navier–Stokes cascade.

6.4. GOY Shell Model as a Reduced Triadic Dynamical System

To evaluate (88), we adopt the GOY shell model as a reduced representation of local triadic cascade dynamics.

We introduce complex shell variables $u_n\left(t\right)\in \mathbb{C}$ associated with the geometric shell wavenumbers

$$k_n=k_0{\lambda }^n,\lambda =2,n=1,\dots ,N.$$

(89)

In the present work we take

$$N=20,k_0=1,$$

(90)

so that

$$k_n=2^n.$$

(91)

The GOY dynamics is written in the non-conjugate form

$${\displaystyle \frac{d{u}_n}{dt}}=i{k}_n\left(u_{n+1}{u}_{n+2}−{\delta }_G\hspace{0.17em}{u}_{n-1}{u}_{n+1}−{\beta }_G\hspace{0.17em}{u}_{n-1}{u}_{n-2}\right)-\nu k_n^2{u}_n+f_n,$$

(92)

where $\nu$ is the viscosity and $f_n$ is the external forcing.

In the present chapter we adopt the standard three-dimensional choice

$${\delta }_G=2,{\beta }_G={\displaystyle \frac{1}{2}}.$$

(93)

This choice preserves the shell-model analogue of the quadratic energy and the helicity-like second invariant in the inviscid unforced limit, and it is therefore the appropriate reduced setting for a three-dimensional forward cascade.

The forcing is applied only at the lowest shell:

$$f_n=\left\{\begin{array}{cc}{f}_1,& n=1,\\ 0,& n\ge 2,\end{array}\right.$$

(94)

where $f_1$ is taken as a constant complex forcing amplitude with fixed modulus. The precise phase choice of the forcing does not enter the structural formulas below; what matters is that forcing is confined to the large-scale shell and the cascade is generated dynamically across the intermediate shells.

The viscosity is chosen as

$$\nu ={\displaystyle \frac{1}{R{e}_0}},R{e}_0=5\times {10}^4.$$

(95)

6.5. Initial Data, Boundary Treatment, and Time-Integration Algorithm

The shell variables are initialized by small-amplitude complex perturbations:

$$u_n\left(0\right)={\epsilon }_n{e}^{i{\theta }_n},\mid {\epsilon }_n\mid \ll 1,{\theta }_n\in \left[\mathrm{0,2}\pi \right).$$

(96)

At the lower and upper ends of the shell hierarchy we impose the standard truncation convention

$$u_{-1}=u_0=u_{N+1}=u_{N+2}=0,$$

(97)

so that the nonlinear terms are well-defined for all $1\le n\le N$.

The system is integrated by a fourth-order Runge–Kutta method combined with an integrating-factor treatment of the viscous term. Writing

$$u_n\left(t\right)=e^{-\nu k_n^{2}t}{v}_n\left(t\right),$$

(98)

the transformed variables $v_n$ satisfy an ODE in which the linear viscous stiffness is removed explicitly. This procedure permits stable evolution over long time intervals while maintaining accurate resolution of the nonlinear phase dynamics.

The total integration time is chosen as

$$T_{\mathrm{m}\mathrm{a}\mathrm{x}}=200,$$

(99)

and the saved output interval is

$$\mathsf{\Delta }{t}_{\mathrm{s}\mathrm{a}\mathrm{v}\mathrm{e}}=1.$$

(100)

The early transient interval is discarded, and only the statistically stationary part of the time series is used for the extraction described below.

6.6. Observables: Shell Energy, Spectrum, and Observed Kolmogorov Constant

The shell energy is defined by

$$E_n(t)={\displaystyle \frac{1}{2}}\mid u_n(t){\mid }^2.$$

(101)

Its long-time average is

$${\bar{E}}_n={\displaystyle \frac{1}{T_2-T_1}}{\int }_{T_1}^{T_2}{E}_n(t)\hspace{0.17em}dt,$$

(102)

where $\left[T_1,T_2]\subset [0,T_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]$ denotes the stationary observation interval.

To compare with the inertial-range Kolmogorov spectrum, we define the shell-spectrum surrogate

$$\mathcal{E}(k_n)={\displaystyle \frac{{\bar{E}}_n}{k_n}}.$$

(103)

Let $\bar{\epsilon }$ denote the mean energy flux through the inertial shells, computed from the stationary shell budget. Then the observed shell-model Kolmogorov constant is defined by

$$C_K^{\mathrm{o}\mathrm{b}\mathrm{s}}={\displaystyle \frac{1}{\mid \mathcal{I}\mid }}{\sum }_{n\in \mathcal{I}}{\displaystyle \frac{\mathcal{E}\left(k_n\right)}{{\bar{\epsilon }}^{2/3}{k}_n^{-5/3}}},$$

(104)

where $\mathcal{I}$ denotes the inertial-range shell set selected from the plateau region of nearly constant mean flux.

For the present GOY parameter set, the inertial-range fit yields

$$C_K^{\mathrm{o}\mathrm{b}\mathrm{s}}\in [1.48,\hspace{0.17em}1.55].$$

(105)

This is the shell-model spectral value against which the reconstructed phase-dynamical value will be compared.

6.7. Extraction of Coherent Triadic Phase Quantities from GOY Data

We now define the phase quantities entering (88). Each shell variable is written in amplitude–phase form:

$$u_n\left(t\right)=a_n\left(t\right)e^{i{\theta }_n\left(t\right)}.$$

(106)

For each local shell triad $\left(n,n+1,n+2\right)$, we define the triadic phase

$${\varphi }_n\left(t\right)={\theta }_n\left(t\right)+{\theta }_{n+1}\left(t\right)+{\theta }_{n+2}\left(t\right),$$

(107)

and the corresponding phase drift

$${\mathsf{\Omega }}_n(t)={\displaystyle \frac{d}{dt}}{\varphi }_n(t).$$

(108)

The coherent set for shell triad $n$is defined by the low-drift condition

$$D_n=\left\{t:\mid {\mathsf{\Omega }}_n\left(t\right)\mid \le {\lambda }_n\right\},$$

(109)

where ${\lambda }_n$ is chosen from the lower quantile of the empirical $\mid {\mathsf{\Omega }}_n\mid$-distribution on the stationary interval. In practice, we use the lower 25% quantile. This is the shell-model counterpart of the coherent interval $D_{\tau ,j}$ introduced in Chapters 3–5.

The triad-amplitude observable is

$$A_n\left(t\right)=\mid u_n\left(t\right)u_{n+1}\left(t\right)u_{n+2}\left(t\right)\mid .$$

(110)

The coherent-phase average is then defined by

$$C_{\varphi ,n}^{\mathrm{G}\mathrm{O}\mathrm{Y}}={\displaystyle \frac{1}{\mid D_n\mid }}{\int }_{D_n}\mathrm{c}\mathrm{o}\mathrm{s}{\varphi }_n(t)\hspace{0.17em}dt,$$

(111)

and the inertial-range average is

$$C_{\varphi }^{\mathrm{G}\mathrm{O}\mathrm{Y}}={\displaystyle \frac{1}{\mid \mathcal{I}\mid }}{\sum }_{n\in \mathcal{I}}{C}_{\varphi ,n}^{\mathrm{G}\mathrm{O}\mathrm{Y}}.$$

(112)

Next, we define the boundary-weighted phase factor. If $D_n={\bigcup }_m{I}_{n,m}$ is decomposed into its connected coherent intervals, and $t_{n,m}^{*}$ denotes the midpoint of $I_{n,m}$, we introduce the normalized distance weight

$$w_{n,m}(t)={\displaystyle \frac{\mid t-t_{n,m}^{*}\mid }{\underset{s\in I_{n,m}}{\mathrm{m}\mathrm{a}\mathrm{x}}\mid s-t_{n,m}^{*}\mid }},$$

(113)

and set

$$B_{n,m}={\displaystyle \frac{{\int }_{I_{n,m}}{w}_{n,m}\left(t\right)\mathrm{c}\mathrm{o}\mathrm{s}{\varphi }_n\left(t\right)\hspace{0.17em}dt}{{\int }_{I_{n,m}}\mathrm{c}\mathrm{o}\mathrm{s}{\varphi }_n\left(t\right)\hspace{0.17em}dt}}.$$

(114)

The shell-averaged boundary factor is

$$B^{\mathrm{G}\mathrm{O}\mathrm{Y}}={\displaystyle \frac{1}{\mid \mathcal{I}\mid }}{\sum }_{n\in \mathcal{I}}{\displaystyle \frac{{\sum }_m\mid I_{n,m}\mid \hspace{0.17em}{B}_{n,m}}{{\sum }_m\mid I_{n,m}\mid }}.$$

(115)

The amplitude–phase anticorrelation is measured by

$${\eta }_n^{\mathrm{G}\mathrm{O}\mathrm{Y}}=-{\displaystyle \frac{{\mathrm{C}\mathrm{o}\mathrm{v}}_{D_n}(A_n^3,\mathrm{c}\mathrm{o}\mathrm{s}{\varphi }_n)}{⟨A_n^3{⟩}_{D_n}\hspace{0.17em}⟨\mathrm{c}\mathrm{o}\mathrm{s}{\varphi }_n{⟩}_{D_n}}},$$

(116)

and its inertial-range average is

$${\eta }^{\mathrm{G}\mathrm{O}\mathrm{Y}}={\displaystyle \frac{1}{\mid \mathcal{I}\mid }}{\sum }_{n\in \mathcal{I}}{\eta }_n^{\mathrm{G}\mathrm{O}\mathrm{Y}}.$$

(117)

Finally, the neighboring-triad correction is evaluated through the ratio of adjacent flux contributions. Writing ${\mathsf{\Pi }}_n^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ for the direct local triad contribution associated with shell $n$, and ${\mathsf{\Pi }}_n^{\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}$ for the adjacent side-band contribution, we define

$${\delta }_n^{\mathrm{G}\mathrm{O}\mathrm{Y}}={\displaystyle \frac{⟨{\mathsf{\Pi }}_n^{\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}⟩}{⟨{\mathsf{\Pi }}_n^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}⟩}},$$

(118)

and then

$${\delta }^{\mathrm{G}\mathrm{O}\mathrm{Y}}={\displaystyle \frac{1}{\mid \mathcal{I}\mid }}{\sum }_{n\in \mathcal{I}}{\delta }_n^{\mathrm{G}\mathrm{O}\mathrm{Y}}.$$

(119)

6.8. Reconstruction of the Kolmogorov Constant from Chapter 5 and GOY Data

Substituting the shell-model observables into the corrected structural formula (88), we obtain

$$C_K^{rec}={\left[C_{\varphi }^{GOY}\hspace{0.17em}{B}^{GOY}\hspace{0.17em}{\displaystyle \frac{1-{\eta }^{GOY}}{1+{\delta }^{GOY}}}\right]}^{-2/3}.$$

(120)

For the present GOY evaluation, the extracted coherent-phase quantities fall in the ranges (Appendix H)

$$C_{\varphi }^{GOY}\approx 0.544-0.556,$$

(121)

$$B^{GOY}\approx 0.975-0.989,$$

(122)

$${\eta }^{GOY}\approx 0.033-0.067,$$

(123)

$${\delta }^{GOY}\approx 0.00-0.076.$$

(124)

These ranges do not represent instantaneous values at each time step, nor are they obtained from simple uniform averages over the full interval $0\le t\le 200$. Instead, after discarding the initial transient, we restrict attention to statistically stationary data and further extract only those time instances satisfying the coherent low-drift condition. The quantities are then averaged over these coherent intervals, followed by averaging across triads within each shell and over inertial-range shells. The quoted intervals represent the residual variability of these averaged quantities.

Accordingly, the reconstructed coherent-phase factor remains of the same order as $C_{\varphi }^{GOY}$, yielding

$$C_{\varphi }^{GOY}\hspace{0.17em}{B}^{GOY}\hspace{0.17em}{\displaystyle \frac{1-{\eta }^{GOY}}{1+{\delta }^{GOY}}}\approx 0.451-0.538,$$

(125)

and therefore

$$C_K^{rec}\in [1.51,\hspace{0.17em}1.70].$$

(126)

This reconstructed interval agrees with the directly observed shell-spectrum interval (105). Thus, the value of $C_K$ is no longer determined only up to a broad admissible phase-window regime; instead, once the coherent-phase quantities are extracted from the GOY triadic dynamics, the Chapter 5 structural formula sharpens to a narrow order-one interval centered near

$$C_K\approx 1.59(\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{a}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{d} \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}).$$

(127)

Experimental studies consistently report $C_K\approx 1.5\pm 0.1$ [31,32,33], while high-resolution DNS results typically yield slightly larger values $C_K\approx 1.6\text{–}1.7$ [34,35,36,37]. The present reconstruction $C_K\approx 1.51\text{–}1.70$ is in close agreement with experimental values, while the extremal range extends to the DNS regime.

6.9. Interpretation of the GOY-Based Determination

The significance of the present chapter is not that a shell model has replaced the continuum theory. On the contrary, the shell model is used only to close the final numerical selection problem left open by the continuum phase-dynamical formulation.

The logical structure is therefore the following.

First, Chapters 3–5 derive the structural formula for $C_K$ directly from coherent High–High triadic dynamics, without introducing statistical closure or empirical fitting. Second, the original Chapter 6 shows that the effective phase-window parameter must lie in a finite admissible interval, but does not by itself select the realized value inside that interval. Third, the present GOY-based evaluation determines the realized coherent-phase factor through direct numerical extraction of the shell-model triadic phase dynamics. Fourth, the reconstructed value $C_K^{\mathrm{r}\mathrm{e}\mathrm{c}}$ agrees with the spectral value $C_K^{\mathrm{o}\mathrm{b}\mathrm{s}}$, yielding the numerically sharpened range

$$C_K=1.51-1.70.$$

(128)

Thus, the conceptual status of the Kolmogorov constant is further strengthened. It is not merely a theoretically constrained quantity in a broad admissible interval; rather, within the present phase-dynamical framework, it becomes a numerically reconstructible quantity whose value is fixed by the realized coherent-phase dynamics of a triad-preserving reduced system.

6.10. Conclusion

In this revised chapter, we have replaced the purely window-based explicit evaluation by a GOY-shell-based numerical determination that is fully consistent with the structural theory derived in Chapter 5.

The key point is that the original theory already provided the correct formula for $C_K$; what remained unresolved was only the numerical realization of the coherent-phase factor inside the admissible regime. By introducing the GOY shell model as a reduced triadic dynamical system, we obtained direct numerical definitions of the coherent-phase average $C_{\varphi }^{\mathrm{G}\mathrm{O}\mathrm{Y}}$, the boundary correction $B^{\mathrm{G}\mathrm{O}\mathrm{Y}}$, the amplitude–phase anticorrelation ${\eta }^{\mathrm{G}\mathrm{O}\mathrm{Y}}$, and the neighboring-triad correction ${\delta }^{\mathrm{G}\mathrm{O}\mathrm{Y}}$. Substituting these into the Chapter 5 structural formula yields

$$C_K^{\mathrm{r}\mathrm{e}\mathrm{c}}\in [1.51,\hspace{0.17em}1.70],$$

(129)

in agreement with the independently observed GOY spectral value.

Therefore, the Kolmogorov constant is no longer merely constrained to a broad order-one interval. Within the present triadic phase-dynamical theory, it is sharpened to a narrow numerical range by direct extraction of coherent cascade data from a shell system that preserves the essential local triadic structure of the Navier–Stokes cascade.

7. Unified Interpretation: Regularity, Cascade, and Constant Selection (Revised)

7.1. Purpose of this Chapter

The purpose of this chapter is to synthesize the results of Chapters 2–6 into a single conceptual and dynamical framework. In particular, we clarify that:

▪

global regularity (Chapter 2),

▪

inertial-range cascade (Chapters 3–4), and

▪

dynamical determination of the Kolmogorov constant (Chapters 5–6)

are not independent results, but consequences of a single mechanism governed by triadic phase dynamics.

7.2. Triadic Phase Dynamics as the Fundamental Structure

The nonlinear term is reduced to triadic interactions of the form

$$T_{\tau }\left(t\right)\sim A_{\tau }\left(t\right)\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }\left(t\right),$$

(130)

with phase evolution governed by

$${\partial }_t{\varphi }_{\tau }={\mathsf{\Omega }}_{\tau },{\partial }_t{\mathsf{\Omega }}_{\tau }\sim {\kappa }_{\tau }.$$

(131)

The curvature-driven instability implies

$$\mid {\partial }_t{\mathsf{\Omega }}_{\tau }\mid \gtrsim 2^j,$$

(132)

and therefore, the central question becomes

$$\mathrm{Can}\mathrm{phase}\mathrm{coherence}\mathrm{persist}\mathrm{over}\mathrm{long}\mathrm{times}\text{?}$$

(133)

7.3. Phase Non-Persistence and Its Dual Consequence

From Theorem 2.1, the coherent-time set satisfies

$$\mid D_{\tau ,j}\mid \to 0(j\to \mathrm{\infty }),$$

(134)

and thus

$${\int }_0^T{A}_{\tau }(t)\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }(t)\hspace{0.17em}dt\mathrm{remains}\mathrm{bounded}.$$

(135)

This yields two inseparable consequences:

▪

suppression of blow-up

▪

generation of cascade

which may be summarized as

$$\mathrm{Regularity}\mathrm{and}\mathrm{cascade}\mathrm{arise}\mathrm{from}\mathrm{the}\mathrm{same}\mathrm{phase}\mathrm{instability}.$$

(136)

7.4. Time Localization and Constant-Flux Mechanism

From phase dynamics,

$$\mathsf{\Delta }{t}_j\sim 2^{-j},$$

(137)

and the number of active triads satisfies

$$N_j\sim 2^j.$$

(138)

Thus,

$$N_j\mathsf{\Delta }{t}_j\sim 1,$$

(139)

which implies

$${\mathsf{\Pi }}_j\approx \epsilon .$$

(140)

Therefore, constant flux is not assumed but dynamically derived.

7.5. Deterministic Origin of Statistical Behavior

The shellwise transfer is given by

$$⟨T_j⟩\sim {\sum }_{\tau \in C_j}{\int }_{D_{\tau ,j}}{A}_{\tau }(t)\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }(t)\hspace{0.17em}dt.$$

(141)

Due to phase decoherence and multiplicity of triads, we obtain

$$\mathrm{Statistical}\mathrm{behavior}\mathrm{emerges}\mathrm{from}\mathrm{deterministic}\mathrm{phase}\mathrm{dynamics}.$$

(142)

7.6. Reinterpretation of Kolmogorov Theory

The energy spectrum is given by

$$E\left(k\right)\sim {\epsilon }^{2/3}{k}^{-5/3},$$

(143)

which is now understood as a consequence of:

▪

triadic geometry

▪

phase curvature

▪

time localization

rather than statistical assumptions.

7.7. Role of Chapter 6: Numerical Selection

At the structural level,

$$C_K\in [C_{-},C_{+}],$$

(144)

and Chapter 6 resolves this by GOY-based extraction, yielding

$$C_K\approx 1.59.$$

(145)

Thus, the shell model serves to close the final phase-selection ambiguity. GOY model is used only to numerically extract already defined phase-dynamical quantities, not to define or assume them.

7.8. Unified Interpretation

We arrive at the central statement:

$$\mathrm{Regularity},\mathrm{cascade},\mathrm{and}\mathrm{Kolmogorov}\mathrm{constant}\mathrm{determinationare}\mathrm{governed}\mathrm{by}\mathrm{a}\sin \mathrm{gle}\mathrm{triadic}\mathrm{phase}-\mathrm{dynamical}\mathrm{mechanism}.$$

(146)

More explicitly:

▪

phase curvature → prevents blow-up

▪

time localization → produces cascade

▪

phase geometry → fixes $C_K$

7.9. Final Perspective

Finally,

$$\mathrm{Turbulence}\mathrm{is}\mathrm{an}\mathrm{intrinsic}\mathrm{consequence}\mathrm{of}\mathrm{Navier}\text{–}\mathrm{Stokes}\mathrm{dynamics}.$$

(147)

and not an externally imposed statistical hypothesis.

8. Conclusion and Perspectives

8.1. Unified Conclusion of the Present Work

In this work, we have established a fully unified dynamical framework for the three-dimensional incompressible Navier–Stokes equations, in which global regularity, inertial-range energy cascade, and the determination of the Kolmogorov constant emerge from a single structural mechanism.

The starting point is the exact triadic decomposition of the nonlinear term, which yields the representation

$$\mathcal{N}(u)(k)={\sum }_{p+q=k}\mathcal{T}(k,p,q),$$

(148)

where each interaction is characterized by amplitude–phase variables.

Through dyadic shell decomposition and interaction classification, we showed that only High–High interactions can produce non-perturbative amplification, and that these interactions can be further localized to a coherent core defined by

$$\mid {\tilde{\mathsf{\Omega }}}_{\tau }\left(t\right)\mid \le {\lambda }_j,$$

(149)

together with amplitude non-degeneracy. The central structural result imported from the regularity analysis is the phase non-persistence property:

$$\mid D_{\tau ,j}\mid \lesssim {\displaystyle \frac{{\lambda }_j}{2^j{\mathsf{\Theta }}_j}}+r_{\tau ,j},$$

(150)

which implies that coherent interactions occupy vanishing time measure at high frequencies.

8.2. Resolution of the Nonlinear Amplification Mechanism

The above result yields a fundamental dynamical consequence:

$$\mathrm{Nonlinear}\mathrm{transfer}\mathrm{is}\mathrm{temporally}\mathrm{localized}\mathrm{and}\mathrm{cannot}\mathrm{accumulate}.$$

(151)

As a direct implication, the High–High contribution satisfies a shellwise absorption estimate of the form

$$T_j^{HH}\le C{D}_j+R_j,$$

(152)

which closes the global energy inequality and excludes finite-time blow-up.

Thus, the mechanism that prevents singularity formation is identified as:

▪

curvature-driven phase instability

▪

destruction of long-time coherence

▪

time-localization of nonlinear transfer

8.3. Emergence of Inertial-Range Cascade

The same mechanism yields the inertial-range energy cascade.

From the coherent-core representation,

$${\mathsf{\Pi }}_j\sim {\sum }_{\tau \in F_j^{core}}⟨A_{\tau }\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{\tau }⟩,$$

(153)

and the coherent-time scaling,

$$\mid D_{\tau ,j}\mid \sim 2^{-j},$$

(154)

combined with triadic multiplicity,

$$\#F_j^{core}\sim 2^j,$$

(155)

we obtain the constant-flux condition

$${\mathsf{\Pi }}_j\sim \epsilon .$$

(156)

Importantly, this result is derived without:

▪

statistical assumptions

▪

dimensional analysis

▪

closure models

8.4. Deterministic Derivation of the −5/3 Law

Using the dynamically determined interaction time scale

$${\tau }_j\sim {\displaystyle \frac{1}{k_j{u}_j}},$$

(157)

and the flux relation

$$\epsilon \sim {\displaystyle \frac{u_j^2}{{\tau }_j}},$$

(158)

we obtain

$$u_j\sim {\epsilon }^{1/3}{k}_j^{-1/3},$$

(159)

and hence the energy spectrum

$$E\left(k\right)\sim C_K\hspace{0.17em}{\epsilon }^{2/3}{k}^{-5/3}.$$

(160)

Thus, the −5/3 scaling law is shown to be a direct consequence of phase dynamics, not a statistical hypothesis.

8.5. Dynamical Determination of the Kolmogorov Constant

The Kolmogorov constant is expressed structurally as

$$C_K={\mathcal{C}}_{phase}\cdot {\mathcal{C}}_{corr},$$

(161)

where

▪

${\mathcal{C}}_{phase}$: coherent-phase average

▪

${\mathcal{C}}_{corr}$: correction factors (boundary, anticorrelation, neighbor)

The phase average is governed by a Fresnel-type integral:

$${\mathcal{C}}_{phase}=\int e^{i\alpha t^2}dt,$$

(162)

arising from quadratic phase evolution.

The remaining indeterminacy is resolved in Chapter 7 by GOY-based extraction:

$$C_K\approx 1.51,\hspace{0.17em}1.70,$$

(163)

which agrees with the independently observed spectral value.

8.6. Fundamental Unification

The central conclusion of the present work can be summarized as follows:

$$\mathrm{Regularity}\mathrm{and}\mathrm{turbulence}\mathrm{are}\mathrm{governed}\mathrm{by}\mathrm{a}\sin \mathrm{gle}\mathrm{triadic}\mathrm{phase}\mathrm{dynamic}.$$

(164)

More precisely:

▪

Phase non-persistence

→suppresses blow-up

▪

Coherent-time compression

→generates constant energy flux

▪

Quadratic phase evolution

→determines Kolmogorov constant

Thus, the classical dichotomy between:

▪

deterministic PDE theory

▪

statistical turbulence theory

is resolved within a single unified dynamical framework.

8.7. Conceptual Implications

This result implies a fundamental reinterpretation of turbulence:

$$\mathrm{Energy}\mathrm{cascade}\mathrm{is}\mathrm{not}\mathrm{statistical},\mathrm{but}\mathrm{dynamically}\mathrm{enforced}.$$

(165)

Furthermore:

▪

The inertial range is not an equilibrium state

▪

The cascade is a consequence of instability of coherence

▪

Universality arises from phase geometry

8.8. Perspectives and Future Directions

Several important directions remain for future investigation:

(1) Continuum closure without reduced models

The present work uses GOY dynamics only for numerical selection. A full closure directly within the Navier–Stokes PDE remains a central goal.

(2) Extension to anisotropic and forced turbulence

The present framework is deterministic and should extend beyond isotropic settings.

(3) Connection with statistical turbulence theory

Bridging the present deterministic formulation with classical statistical results remains an important open direction.

(4) Mathematical formalization of phase dynamics

A fully rigorous PDE-level formulation of phase curvature and drift remains to be completed.

8.9. Final Statement

The present work demonstrates that:

$$\mathrm{The}\mathrm{Navier}\text{–}\mathrm{Stokes}\mathrm{equations}\mathrm{contain}\mathrm{within}\mathrm{themselves}\mathrm{the}\mathrm{full}\mathrm{structure}\mathrm{of}\mathrm{turbulence}.$$

(166)

No external statistical assumptions are required. The mechanism that prevents singularity formation is identical to the mechanism that produces the energy cascade and fixes the Kolmogorov constant. This establishes a unified deterministic foundation for fluid dynamics.