The third-order structure function in two dimensions: The Rashomon effect

Abstract

We study the third-order longitudinal structure function, S 3 (r), in two-dimensional turbulence. In three dimensions, there is considerable theoretical, experimental, and numerical consensus regarding the validity of Kolmogorov's arch-famous " 4 5 th law" for S 3 (r). By contrast, in two dimensions, two disparate cascades, changed dissipation anomalies, a large-scale drag, and other factors conspire to create several versions of the S 3 (r) "law." This single quantity can vary considerably when viewed from different perspectives, reminiscent of the "Rashomon effect" in anthropology. After reviewing the history and usage of S 3 (r) in two-dimensional turbulence, we show that S 3 (r) generically embodies a mixture of energy and enstrophy fluxes. Building on this result, we derive S 3 (r) laws for freely decaying and forced two-dimensional turbulent flows, where we also account for the effects of a large-scale drag, an inextricable feature of quasi two-dimensional turbulence in experimental and atmospheric flows. We draw attention to the caution needed in interpreting S 3 (r) in two-dimensional turbulence.

Advective structure functions in anisotropic two-dimensional turbulence

“…In particular, the success of advective structure functions to diagnose energy and enstrophy cascade rates, in anisotropic, forced 2-D turbulence with large- and small-scale dissipation expands the breadth of practical scenarios where these relations could be applied. Our results extend beyond previous studies which have shown that, in isotropic 2-D turbulence, external forcing at a specific scale does not affect third-order structure function relations within the inertial cascades (Lindborg 1999; Cerbus & Chakraborty 2017), and similar results for 3-D turbulence (Banerjee & Galtier 2016). Cerbus & Chakraborty (2017) suggested that large-scale dissipation would reduce the down-scale enstrophy cascade rate as some enstrophy will be removed by large-scale drag or hypo-viscosity.…”

“…Our results extend beyond previous studies which have shown that, in isotropic 2-D turbulence, external forcing at a specific scale does not affect third-order structure function relations within the inertial cascades (Lindborg 1999; Cerbus & Chakraborty 2017), and similar results for 3-D turbulence (Banerjee & Galtier 2016). Cerbus & Chakraborty (2017) suggested that large-scale dissipation would reduce the down-scale enstrophy cascade rate as some enstrophy will be removed by large-scale drag or hypo-viscosity. In the present simulations, approximately 99 % of enstrophy is dissipated at small scales, and approximately 90 % of energy is dissipated at large scales (table 2).…”

“…Third-order structure function laws have been proposed for isotropic 2-D turbulence that apply outside idealised inertial cascades (Xie & Bühler 2018, 2019). It is possible that the derivations presented here could also be modified to extend outside of inertial ranges, such as near the forcing scale or in regions of overlapping energy and enstrophy cascades (Lindborg 1999; Thompson & Young 2006; Cerbus & Chakraborty 2017). The relationship shown in (2.9) means that any extant laws containing can be converted into laws in terms of , and similarly the relation allows the conversion of extant laws with third-order velocity structure functions into laws with second-order structure functions.…”

Coarsening in the two-dimensional incompressible Toner-Tu equation: Signatures of turbulence

“…Here S 3 (r, t) ≡ [δ r u] 3 , δ r u ≡ [u(x + r, t) − u(x, t)] .r, and the angular brackets indicate spatial and ensemble averaging [48,49]. In the statistically steady turbulence, enstrophy flux Z k is constant in the inertial range and is equal to the enstrophy dissipation rate.…”

Sufficient Conditions for Dual Cascade Flux Laws in the Stochastic 2d Navier–Stokes Equations

“…In the range of scales associated with the direct cascade we expect the following to hold over a range of scales ℓ ν ≪ |h| ≪ ℓ I (viscous scale and injection scale respectively): 8) which is indicative of a direct cascade of enstrophy (see e.g. [18]). As predicted by Eyink [32], one also expects Yaglom's law [74] for the vorticity in the inertial range ℓ ν ≪ |h| ≪ ℓ I (Yaglom originally derived this prediction for passive scalar turbulence),…”