Freely Decaying Turbulence Model

• Freely decaying turbulence model treats turbulence as an initial-value problem without external forcing, emphasizing nonlinear energy transfer and conservation of large-scale invariants.
• It applies to diverse systems such as homogeneous isotropic Navier–Stokes, MHD, 2D turbulence, and gyrokinetics, each exhibiting distinct decay laws and spectral scaling behaviors.
• The model highlights the impact of infrared invariants, self-similarity, and evolving measurement times on decay exponents and inverse transfer mechanisms.

A freely decaying turbulence model treats turbulence as an initial-value problem in which external forcing is absent after the initial condition is prescribed, so the subsequent evolution is controlled by nonlinear transfer, dissipation, symmetry constraints, and whatever large-scale invariants survive the decay. In the literature considered here, such models span homogeneous isotropic Navier–Stokes turbulence, non-helical and helical MHD, two-dimensional turbulence, surface semi-geostrophic dynamics, electrostatic gyrokinetics, force-free electrodynamics, and even stochastic-gravitational-wave source modeling. The common organizing variables are the kinetic or magnetic energy, the integral scale, the infrared form of the spectrum, and the characteristic time at which the cascade becomes dynamically developed (Yoffe et al., 2018, Olesen, 2015, Gorce et al., 2024).

1. Problem formulation and modeling scope

The canonical hydrodynamic setting is incompressible homogeneous isotropic turbulence (HIT) evolving without forcing in either a periodic box or a laboratory flow that has already become approximately homogeneous and shear-free. In DNS, this is commonly posed with a prescribed initial energy spectrum such as

$$E(k,0) = c \left(\frac{k}{k_0}\right)^4 \exp\!\left[-\left(\frac{k}{k_0}\right)^2\right], \qquad c=0.266,\quad k_0=3.536,$$

followed by pseudospectral evolution with dealiasing and viscous decay (Yoffe et al., 2018). In three-dimensional incompressible decay, the basic diagnostics are

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$

together with the integral scale, the Taylor microscale, and transfer spectra (Tsuzuki, 20 Jan 2026).

Several papers formulate the same unforced setting in more specialized languages. In MHD, the decay is described either with velocity and magnetic fields directly or with Elsässer variables $\mathbf z^\pm=\mathbf u\pm \mathbf b$, because the nonlinear transfer is naturally organized in that basis (Linkmann et al., 2015). In gyrokinetics, the evolving object is the perturbed gyrocenter distribution $g$, with electrostatic potential $\varphi$ recovered from quasineutrality and no parallel dynamics, so that nonlinear perpendicular phase mixing is isolated (Tatsuno et al., 2012). In force-free electrodynamics, the fluid inertia is eliminated and the magnetic field relaxes under force-free constraints, but the decay remains an unforced turbulent relaxation problem (Zrake et al., 2015).

A recurrent modeling assumption is self-preservation or self-similarity. In two-dimensional statistical theory, the Kármán–Howarth equation is reduced with a similarity variable $\eta=r/l(t)$ and a Sedov-type closure (Ran, 2010). In hydrodynamic decay theory, the same idea appears as power-law decay of energy and algebraic growth of the integral scale (Krogstad et al., 2011, Gorce et al., 2024). In more diagnostic formulations, the key issue is not only the asymptotic decay law but also the time at which the flow has become sufficiently developed for such a law to be meaningful (Yoffe et al., 2018).

2. Infrared invariants, decay exponents, and classical closures

The most persistent classification scheme is by infrared spectrum and large-scale invariant. Saffman-type decay is tied to

$E(k\to 0)\sim k^2,$

with invariant

$L\sim u^2 l^3,$

whereas Batchelor-type decay is tied to

$E(k)\sim k^4,$

with invariant

$I\sim u^2 l^5$

(Krogstad et al., 2011, Gorce et al., 2024). These hypotheses generate different self-similar decay exponents once combined with

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$0

or its equivalent dissipation closure (Krogstad et al., 2011).

The principal formulations summarized below are explicitly given in the cited literature (Krogstad et al., 2011, Gorce et al., 2024, McComb et al., 2014, Baumert, 2012, Olesen, 2015).

Formulation	Infrared or invariant statement	Decay prediction
Saffman	$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$1, $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$2	$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$3
Batchelor	$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$4, $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$5	$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$6
Isotropic infrared expansion	$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$7, $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$8, $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$9	leading infrared term is $\mathbf z^\pm=\mathbf u\pm \mathbf b$0
Two-fluid dipole model	quadratic sink $\mathbf z^\pm=\mathbf u\pm \mathbf b$1 from pair collisions	$\mathbf z^\pm=\mathbf u\pm \mathbf b$2
Non-helical MHD self-similarity	dimension-independent spectral similarity	$\mathbf z^\pm=\mathbf u\pm \mathbf b$3

The Saffman branch is supported experimentally in two distinct settings. Grid-generated turbulence in a large recirculating wind tunnel, after the near-grid inhomogeneous region is excluded, decays with $\mathbf z^\pm=\mathbf u\pm \mathbf b$4 very close to the classical Saffman exponent $\mathbf z^\pm=\mathbf u\pm \mathbf b$5, and the spectra collapse in the classical manner on integral scales at low $\mathbf z^\pm=\mathbf u\pm \mathbf b$6 and Kolmogorov microscales at high $\mathbf z^\pm=\mathbf u\pm \mathbf b$7 (Krogstad et al., 2011). Magnetic-stirrer experiments in a closed container similarly report early-time conservation of $\mathbf z^\pm=\mathbf u\pm \mathbf b$8, a large-scale $\mathbf z^\pm=\mathbf u\pm \mathbf b$9 spectrum, and decay laws closer to Saffman than Batchelor, with $g$0 during the early decay (Gorce et al., 2024).

A separate infrared analysis reaches a different conclusion for incompressible isotropic turbulence. By writing

$g$1

and expressing the correlation through the longitudinal function $g$2, the $g$3 term cancels identically under the condition that $g$4 decays faster than $g$5, so $g$6 and

$g$7

without assuming exponential large-distance decay (McComb et al., 2014). This produces a genuine controversy rather than a mere notational difference: one line of work treats freely decaying HIT as Saffman-type, while another derives a Loitsyansky-type $g$8 infrared structure from isotropic incompressibility itself (Krogstad et al., 2011, Gorce et al., 2024, McComb et al., 2014).

The two-fluid quasi-particle theory offers a distinct asymptotic closure. Turbulence is modeled as a dense ensemble of quasi-rigid vortex-dipole tubes in an inviscid background fluid; pair collisions either scatter diffusively or annihilate into unstable von Kármán couples. The resulting free-decay balances

$g$9

yield

$\varphi$0

with constant asymptotic eddy viscosity

$\varphi$1

(Baumert, 2012).

3. Self-similarity, inverse transfer, and dimensional reduction in MHD

For freely decaying turbulent non-helical magnetic fields, the central claim of the dimensional-reduction analysis is that the self-similarity law is the same in all spatial dimensions. Starting from the scaling symmetry

$\varphi$2

the spectral energy densities satisfy

$\varphi$3

and, with $\varphi$4,

$\varphi$5

The resulting decay laws are

$\varphi$6

so the characteristic wavenumber decreases and energy shifts toward larger scales, i.e. an inverse transfer occurs in all dimensions (Olesen, 2015).

The same paper gives a gauge-specific mechanism for that inverse transfer. With

$\varphi$7

and the Lorenz gauge

$\varphi$8

the scaling of $\varphi$9 implies scale invariance of $\eta=r/l(t)$0. The spectral density $\eta=r/l(t)$1 then obeys the same self-similarity as the energy spectrum, and

$\eta=r/l(t)$2

is independent of $\eta=r/l(t)$3. The inverse transfer can therefore be assigned, in this gauge, to a time-independent squared vector potential, directly analogous to the conventional two-dimensional MHD inverse-cascade mechanism (Olesen, 2015).

A complementary MHD model focuses not on the spectral decay law but on the dimensionless dissipation rate. Using Elsässer variables and the exact real-space energy balance, the asymptotic expansion gives

$\eta=r/l(t)$4

where $\eta=r/l(t)$5 is a generalized Reynolds number built from $\eta=r/l(t)$6 and $\eta=r/l(t)$7. Here $\eta=r/l(t)$8 is identified with the asymptotic total energy transfer flux. The key point is nonuniversality: magnetic helicity and cross helicity change the nonlinear transfer efficiency, so the asymptotic constant depends on the vector-field correlations rather than only on Reynolds number (Linkmann et al., 2015).

The helical and non-helical inverse-transfer problem sharpens that distinction. In helical decay, a mean-field $\eta=r/l(t)$9-effect closure,

$E(k\to 0)\sim k^2,$0

together with coupled equations for $E(k\to 0)\sim k^2,$1 and $E(k\to 0)\sim k^2,$2, reproduces the large-scale growth semi-analytically (Park, 2017). In non-helical decay, the same paper concludes that the $E(k\to 0)\sim k^2,$3-effect is not suitable; instead, the inverse transfer is attributed to the induction equation in the presence of inhomogeneities, the curl of the EMF, and EDQNM-type triadic transfer. This suggests that the phrase “inverse transfer” covers at least two physically distinct mechanisms in freely decaying MHD (Park, 2017).

4. Symmetry competition, attractors, and the choice of evolved time

One model of freely decaying turbulence treats the spectrum as distributed chaos controlled by competing attractors. The $E(k\to 0)\sim k^2,$4-attractor is associated with translational symmetry and the Birkhoff–Saffman integral $E(k\to 0)\sim k^2,$5, while the $E(k\to 0)\sim k^2,$6-attractor is associated with rotational symmetry and the Loitsyanskii integral $E(k\to 0)\sim k^2,$7: $E(k\to 0)\sim k^2,$8 From

$E(k\to 0)\sim k^2,$9

one obtains a stretched-exponential spectrum

$L\sim u^2 l^3,$0

hence $L\sim u^2 l^3,$1 for the $L\sim u^2 l^3,$2-attractor and $L\sim u^2 l^3,$3 for the $L\sim u^2 l^3,$4-attractor (Bershadskii, 2016).

The central result of that model is temporal competition. In statistically stationary homogeneous isotropic turbulence, the $L\sim u^2 l^3,$5-attractor usually dominates because its basin of attraction is larger. In freely decaying turbulence, however, the $L\sim u^2 l^3,$6-attractor can dominate during an intermediate stage because its basin is small and thin but it becomes effective earlier, while the $L\sim u^2 l^3,$7-attractor is still not fully developed. DNS from Wray’s decaying turbulence dataset supports this interpretation: at $L\sim u^2 l^3,$8 and $L\sim u^2 l^3,$9, plotting $E(k)\sim k^4,$0 against $E(k)\sim k^4,$1 gives a near-linear relation, whereas by $E(k)\sim k^4,$2 the $E(k)\sim k^4,$3 scaling becomes competitive, suggesting a transition toward $E(k)\sim k^4,$4-attractor dominance (Bershadskii, 2016).

A different but related issue is the definition of the “evolved” time $E(k)\sim k^4,$5. DNS of freely decaying HIT over $E(k)\sim k^4,$6 shows that the curve of dimensionless dissipation versus Reynolds number depends strongly on when it is measured. The paper studies onset of power-law decay, onset of $E(k)\sim k^4,$7, peak dissipation time $E(k)\sim k^4,$8, peak skewness time $E(k)\sim k^4,$9, and peak inertial-transfer time $I\sim u^2 l^5$0, and then proposes the composite criterion

$I\sim u^2 l^5$1

With this choice, the decaying-turbulence $I\sim u^2 l^5$2 curve becomes virtually identical to the forced stationary case (Yoffe et al., 2018).

This onset analysis also shows that decay exponents are protocol-dependent rather than uniquely universal. The measured power-law exponent in

$I\sim u^2 l^5$3

decreases with increasing Reynolds number and lies in the range

$I\sim u^2 l^5$4

while an evolved-field decay run at $I\sim u^2 l^5$5 suggests a cascade delay of about

$I\sim u^2 l^5$6

so dissipation lags inertial transfer by a finite time (Yoffe et al., 2018). A plausible implication is that any freely decaying turbulence model that quotes a single exponent without specifying its invariant class and measurement time is incomplete.

5. Two-dimensional decay, quasi-shocks, and reduced models

Freely decaying two-dimensional turbulence supplies several non-equivalent model classes. In the statistical Kármán–Howarth/Sedov construction, the longitudinal correlation is written as

$I\sim u^2 l^5$7

and the low-$I\sim u^2 l^5$8 spectrum depends on a similarity index $I\sim u^2 l^5$9. The asymptotics are

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$00

while

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$01

The familiar $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$02 law is therefore only one branch of a broader self-similar family (Ran, 2010).

High-resolution DNS gives a more structural picture. In freely decaying 2D incompressible turbulence, the angle-averaged spectrum develops the Kraichnan-type law

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$03

but spatial filtration shows that the dominant contribution comes from sharp vorticity gradients or quasi-shocks, equivalently large $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$04. The 2D spectrum is highly anisotropic in Fourier space and organized into narrow jets; along each jet the spectrum behaves as

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$05

and angular averaging yields the isotropic $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$06 spectrum. Higher-order structure functions then show that $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$07 grows more slowly than linearly in $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$08, i.e. intermittency is present (Kudryavtsev et al., 2013).

Inverse-cascade behavior can also appear in unforced 2D turbulence. An ensemble of 50 runs at $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$09 and 10 runs at $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$10, all started with the same initial integral scale, energy, enstrophy, and Reynolds number but different random phases, shows that ensemble- and time-averaging can produce

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$11

together with negative energy flux and positive enstrophy flux, even without forcing. The source of the inverse-transfer behavior is the modal energy initially concentrated around the energy-containing scale $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$12 (Mininni et al., 2013).

Dimensional analysis supplies yet another decay model. For freely decaying 2D turbulence with initial scales $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$13 and $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$14, one obtains a Batchelor-like but explicitly time-dependent inertial range,

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$15

and, if an inverse cascade develops, a Kolmogorov-like spectrum

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$16

with a decaying flux

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$17

The same paper allows a 2D Saffman infrared law $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$18, a Loitsyansky-type $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$19, and, in finite domains under special conditions, a $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$20 condensate-like spectrum (Campanelli, 2019).

A reduced shell-model/Burgers formulation reproduces several of these asymptotics. In the continuum limit the model becomes

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$21

and a scaling solution gives

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$22

in the inertial range, plus a pre-viscous damping regime

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$23

before the purely viscous tail

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$24

takes over (Campanelli, 2022).

6. Extensions to geophysical, kinetic, and force-free systems

In surface semi-geostrophic (SSG) turbulence, the freely decaying model is built from conservation of potential temperature and potential vorticity in geostrophic coordinates, together with a nonlinear Monge–Ampère inversion,

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$25

In a doubly periodic horizontal domain with rigid lids and no forcing, the decay differs qualitatively from SQG: fronts and filaments are more prominent, the PDF of surface $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$26 becomes skewed and shifted toward negative values near the active boundary, and the kinetic-energy spectra are flatter than in SQG, with more energy concentrated at small scales as $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$27 increases (Ragone et al., 2015).

The two-dimensional electrostatic gyrokinetic case is governed by two positive-definite collisionless invariants,

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$28

and therefore exhibits a phase-space dual cascade: $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$29 cascades forward to smaller scales in position and velocity space, while $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$30 cascades inversely along the diagonal $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$31. The late-time state is assumed to be dominated by a single evolving scale $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$32, and the dynamics are classified by

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$33

a gyrokinetic Reynolds-number analogue. The theory predicts three regimes: weakly collisional, marginal, and strongly collisional, with the marginal state marked by a critical $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$34 preserved in time. In the weakly collisional regime,

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$35

whereas in the marginal regime

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$36

(Tatsuno et al., 2012).

Force-free electrodynamics provides a magnetically dominated extreme. Freely decaying force-free turbulence on periodic 2D and 3D domains exhibits an inverse cascade in all helical and non-helical settings. In 3D, helical runs obey Taylor relaxation and settle into the lowest-energy linear force-free equilibrium allowed by helicity conservation,

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$37

while 3D non-helical runs decay toward zero magnetic energy. In 2D, by contrast, helical runs do not reach the Taylor minimum but instead relax into nonlinear force-free equilibria with persistent current layers and isolated magnetic bubbles, which the paper attributes to additional topological invariants associated with level sets of the magnetic potential (Zrake et al., 2015).

A cosmological application turns freely decaying non-helical vortical turbulence into a stochastic-gravitational-wave source. The model assumes self-similar one-scale decay,

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$38

with a Gaussian unequal-time velocity correlator predicted by sweeping decorrelation. A Gibbs-kernel construction is then used to enforce positive definiteness of the UETC, and Monte Carlo integration of the anisotropic-stress UETC reproduces the SGWB seen in hydrodynamical simulations (Auclair et al., 2022).

7. Diagnostics, precursors, and unresolved questions

Recent work emphasizes spectral diagnostics that precede traditional global markers such as the dissipation peak. In freely decaying three-dimensional incompressible turbulence, the curl-of-vorticity spectrum

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$39

acts as a curvature-weighted precursor to the dissipation maximum. Defining

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$40

the observed ordering at baseline viscosity is

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$41

across multi-mode ABC flows, randomized low-$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$42 ABC initial data, Taylor–Green vortex, and Kida–Pelz flow (Tsuzuki, 20 Jan 2026).

The exact timings reported in that study make the ordering concrete rather than schematic. For example,

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$43

for Taylor–Green turbulence, while Kida–Pelz at $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$44 gives

$$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$45

The precursor persists at lower viscosity when adequate resolution is used, but weakens or breaks at higher viscosity, and inadequate resolution can produce cutoff-proximate locking of the $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$46-weighted peak (Tsuzuki, 20 Jan 2026).

Across the broader literature, three unresolved issues recur. First, infrared universality remains contested: Saffman-type $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$47 decay is supported by some experiments, while a separate isotropic analysis eliminates the $$K(t)=\frac12\langle |\boldsymbol{u}|^2\rangle, \qquad \varepsilon(t) = -\frac{dK}{dt} = \nu \langle |\nabla \boldsymbol{u}|^2\rangle = 2\nu \sum_k k^2 E(k,t),$$48 term altogether (Gorce et al., 2024, McComb et al., 2014). Second, nonuniversality is intrinsic in MHD because helicity and cross helicity alter forward transfer and hence the asymptotic dissipation constant (Linkmann et al., 2015). Third, measurement time is constitutive, not incidental: the apparent decay law, dissipation coefficient, and even attractor dominance depend on whether one samples the transient, the intermediate stage, or a later self-similar regime (Yoffe et al., 2018, Bershadskii, 2016).

A plausible synthesis is that “freely decaying turbulence model” does not designate a single closure but a family of invariant-based, symmetry-based, and diagnostic-based formulations. Their differences are not merely technical; they reflect distinct assumptions about infrared structure, admissible conserved quantities, dimensionality, gauge choice, and the stage of decay at which the model is expected to apply.