# Comparing local energy cascade rates in isotropic turbulence using structure
function and filtering formulations

H. Yao1<EMAIL_ADDRESS>M. Schnaubelt2  A. Szalay2  T. Zaki1  C. Meneveau1
1Department of Mechanical Engineering & IDIES, Johns Hopkins University
2Department of Physics and Astronomy & IDIES, Johns Hopkins University

###### Abstract

Two common definitions of the spatially local rate of kinetic energy cascade
at some scale $\ell$ in turbulent flows are (i) the cubic velocity difference
term appearing in the generalized Kolmogorov-Hill equation (GKHE) (structure
function approach), and (ii) the subfilter-scale energy flux term in the
transport equation for subgrid-scale kinetic energy (filtering approach). We
perform a comparative study of both quantities based on direct numerical
simulation data of isotropic turbulence at Taylor-scale Reynolds number of
1250. While observations of negative subfilter-scale energy flux (backscatter)
have in the past led to debates regarding interpretation and relevance of such
observations, we argue that the interpretation of the local structure
function-based cascade rate definition is unambiguous since it arises from a
divergence term in scale space. Conditional averaging is used to explore the
relationship between the local cascade rate and the local filtered viscous
dissipation rate as well as filtered velocity gradient tensor properties such
as its invariants. We find statistically robust evidence of inverse cascade
when both the large-scale rotation rate is strong and the large-scale strain
rate is weak. Even stronger net inverse cascading is observed in the “vortex
compression” $R>0$, $Q>0$ quadrant where $R$ and $Q$ are velocity gradient
invariants. Qualitatively similar, but quantitatively much weaker trends are
observed for the conditionally averaged subfilter scale energy flux. Flow
visualizations show consistent trends, namely that spatially the inverse
cascade events appear to be located within large-scale vortices, specifically
in subregions when $R$ is large.

###### keywords:

##  1 Introduction

The classic description of the energy cascade in turbulence postulates that
kinetic energy originates from forcing large-scale eddies, is subsequently
transferred to smaller-scale eddies (forward cascade), and is eventually
dissipated due to viscous effects (Richardson, 1922; Kolmogorov, 1941). In a
statistical sense, the sign and magnitude of third-order moments of velocity
increments confirm this general direction of the energy cascade, as described
by the $4/5$ law governing the global average of the third-order longitudinal
velocity increment (Kolmogorov, 1941; Frisch, 1995), $\langle\delta
u_{L}(\ell)^{3}\rangle\equiv\langle([{\bf u}({\bf x}+{\bm{\ell}})-{\bf u}({\bf
x})]\cdot{{\bm{\ell}}/\ell})^{3}\rangle=-\frac{4}{5}\,\ell\,\langle\epsilon\rangle$,
where $\langle...\rangle$ denotes global averaging, $\delta u_{L}(\ell)$ is
the longitudinal velocity increment and $\epsilon$ is the viscous dissipation
rate, while the displacement $\ell=|{\bm{\ell}}|$ is assumed to be well inside
the inertial range of turbulence. In this sense, the quantity
$-\frac{5}{4}\langle\delta u_{L}(\ell)^{3}\rangle/\ell$ is often interpreted
as a measure of the energy flux going from scales larger than $\ell$ to all
smaller scales. Because turbulence is known to be highly intermittent in space
and time (Kolmogorov, 1962; Frisch, 1995; Meneveau & Sreenivasan, 1991) there
has also been much interest in characterizing the local properties of the
energy cascade, i.e. the fluctuations of the energy flux before averaging.
However, without statistical averaging, the $4/5$-law is less meaningful,
e.g., the quantity $-\frac{5}{4}\delta u_{L}^{3}/\ell$ cannot simply be
interpreted as an energy flux locally in space and time. To enable such
interpretation, it is necessary to consider explicit angular averaging over
all possible directions of the vector $\bm{\ell}$. Such formulations have been
developed in prior works by Duchon & Robert (2000), Eyink (2002) and Hill
(2001, 2002a). Duchon & Robert (2000) and Eyink (2002) use such equations to
study the energy cascade and energy dissipation in the limit of zero
viscosity. A review about extensions to the classic Kolmogorov equation is
presented by Dubrulle (2019), specifically focusing on the Duchon & Robert
(2000) local formulation.

Hill (2001, 2002a) developed a local version of the Kolmogorov equation in
which the reference position ${\bf x}$ is symmetrically located halfway
between the two points ${\bf x}+{\bf r}/2$ and ${\bf x}-{\bf r}/2$ separated
by ${\bf r}$ over which the velocity increment is computed. This equation,
which we shall denote as the generalized Kolmogorov-Hill equation (GKHE),
describes the evolution of the second-order (squared) velocity difference, a
measure of energy content of all scales smaller than $|\bm{r}|$ at a specific
physical position $\bf x$. As will be reviewed in §2, scale-space integration
over ${\bf r}$ of the GKHE up to some scale $\ell$ in the inertial range and
without additional statistical averaging provides a localized description of
the energy cascade process. The GKHE also includes effects of viscous
dissipation, viscous diffusion, advection, and pressure. A number of prior
works have studied various versions of the GKH equation. For isotropic
turbulence, Yasuda & Vassilicos (2018) quantified the variability of the
energy flux that arises in this equation, while Carbone & Bragg (2020)
considered a definition of mean energy flux approximated based on a solenoidal
filtered velocity increments and examined its connections to average vortex
and strain stretching rates. Besides applications to isotropic homogeneous
flow, numerous studies have investigated the application of the statistically
averaged GKHE to spatially non-homogeneous flows. For instance, in wall
boundedf lows, researchers have explored the energy cascade using a Reynolds
decomposition to isolate effects of mean shear and non-homogeneity (Antonia et
al., 2000; Danaila et al., 2001, 2004, 2012; Marati et al., 2004; Cimarelli et
al., 2013). Investigations have also studied the energy cascade rates in
boundary layer bypass transition (Yao et al., 2022), and flow separation
(Mollicone et al., 2018). Furthermore, specific attention has been given to
the study of inverse cascade in wake flows (Gomes-Fernandes et al., 2015;
Portela et al., 2017) and at turbulent/non-turbulent interfaces (Zhou &
Vassilicos, 2020; Cimarelli et al., 2021; Yao & Papadakis, 2023).

The notion of transfer, or flux, of kinetic energy across length-scales is of
particular practical interest also in the context of large eddy simulation
(LES). There the rate of energy cascade is commonly referred to as the subgrid
or subfilter-scale (SGS, SFS) rate of dissipation. It is defined as the
contraction between the subgrid stress tensor and the filtered strain-rate
tensor and arises as a source term in the transport equation for
subgrid/subfilter-scale kinetic energy (Piomelli et al., 1991; Meneveau &
Katz, 2000). This quantity characterizes the energy transfers between the
resolved scale and the residual scale within the inertial range, which is also
a local property (Eyink & Aluie, 2009). The SGS dissipation is highly
intermittent (Cerutti & Meneveau, 1998), and can be both positive and negative
locally, but on average, energy is known to be transferred from large scales
to the residual scales (forward cascade). There is considerable literature on
the subject starting from the seminal papers by Lilly (1967), Leonard (1975)
and Piomelli et al. (1991). Some reviews include Meneveau & Katz (2000);
Meneveau (2010); Moser et al. (2021).

Without averaging, it has been a common observation that the SGS/SFS
dissipation can be negative which has often been interpreted as indicative of
local inverse cascading of kinetic energy, i.e., energy transfering from small
to large scales of motion (“backscatter” (Piomelli et al., 1991)). Borue &
Orszag (1998) noted that the forward cascade occurs predominantly in regions
characterized by strong straining, where the magnitude of negative skewness of
the strain tensor and vortex stretching are large. Conversely, backscatter was
observed in regions with strong rotation. The relationship between SGS
dissipation and stress topology and stress strain alignment geometry is
discussed and measured based on 3D PIV measurements by Tao et al. (2002). In a
more recent study, Ballouz & Ouellette (2018) investigated the SGS tensor by
considering the relative alignment of the filtered shear stress and strain
tensors. They found that the energy cascade efficiency is quite low, a trend
they attributed to energy being transfered largely between positions in
physical space. Quantitatively, in expressing the subgrid-stress tensor as a
superposition of all smaller scale Gaussian-filtered velocity gradients,
Johnson (2020, 2021) was able to isolate the relative contributions of small-
scale strain self-stretching and vortex stretching finding both to be
important.

It has been questioned whether it is the local quantity
$-\tau_{ij}\tilde{S}_{ij}$ (where $\tau_{ij}$ and $\tilde{S}_{ij}$ are the
subgrid-scale stress and resolved strain-rate tensors, respectively), or the
work done by the SGS/SFS force, $\tilde{u}_{i}\partial_{j}\tau_{ij}$ (where
$\tilde{u}_{i}$ is the resolved velocity), that should be the genuine
definition of local energy cascade rate. For instance Kerr et al. (1996) use
the latter in their study of correlations of cascade rate and vorticity, and
more recently Vela-Martín & Jiménez (2021) uses both quantities in their
analysis. Moreover the SGS force plays a central role for optimal LES modeling
(Langford & Moser, 1999). The SGS force is invariant to divergence-free tensor
fields which therefore do not affect the large-scale dynamics but addition of
such a tensor field to $\tau_{ij}$ can certainly affect the usual definition
of subgrid-scale dissipation $-\tau_{ij}\tilde{S}_{ij}$. By re-expressing the
SGS stress and dissipation terms using an optimization procedure, Vela-Martín
(2022) provided arguments that the often observed backscatter does not
actually contribute to the energy cascade between scales but rather to the
energy flux in the physical space, also suggesting that backscatter does not
need to be explicitly modeled in LES.

As can be seen from this partial summary of the literature on backscatter and
inverse cascade in the LES filtering approach, no consensus has been reached
regarding the possible importance and physical interpretation of local
backscatter using the definition based on the inner product of the subgrid
stress and filtered strain-rate tensors. Also, the question of inverse cascade
has not received much attention from the point of view of the local versions
of the Kolmogorov equation in the structure function approach. Therefore, in
this paper we first revisit the generalized local structure function
formulation (§2.1). We argue that in this formulation the term responsible for
the energy cascade can be unambiguously interpreted as a flux of kinetic
energy between scales since it appears inside a divergence in scale space. In
this sense it differs from the filtering formulation used in LES (reviewed in
§2.2) in which typically a fixed filter scale is used and no change in scales
is considered, thus making the concept of a “flux in scale space” less clearly
defined and open to various interpretations.

With the definition of local cascade rate or energy flux clarified for the
structure function approach, we perform a comparative study of both the
structure function and the filtering approaches’ energy flux terms in a
relatively high Reynolds number DNS database of forced isotropic turbulence at
a Taylor-scale Reynolds number of 1,250. The data analysis is greatly
facilitated by the availability of these data in a new version of the Johns
Hopkins Turbulence Database (JHTDB) System, in which python notebooks access
the data directly (see appendix A). The comparisons involve various
statistical properties of the energy flux. First, in §3 we provide comparisons
of both quantities by means of simple statistical measures such as their mean
values, joint probability density distributions and correlation coefficients,
comparing both the two definitions of kinetic energy and kinetic energy
cascade rate or flux. We then comparatively examine conditional averages based
on the local molecular dissipation rate averaged over a ball of size $\ell$,
specifically re-examining the Kolmogorov refined similarity hypothesis (KRSH)
in §4. Then, in §5, we present comparative conditional averages of kinetic
energy flux based on properties of the large-scale velocity gradient field
such as the strain and rotation magnitudes, and the $Q$ and $R$ invariants.
Particular attention is placed on events of local negative energy flux and
whether or not such events can be considered to be of statistical
significance. Overall conclusions are presented in §6.

##  2 Local energy flux in the structure function and filtering approaches

In this section, both the structure function based (GKHE) and filtering (LES)
energy equations are reviewed. We focus on the term representing energy
cascade (energy flux) in each equation, and describe some of the prior efforts
in the literature relating the structure function and filtering approaches.

###  2.1 Energy cascade rate/flux in the generalized Kolmogorov-Hill equation

The GKHE is a generalized Karman-Horwath equation that is directly derived
from the incompressible Navier-Stokes equations without any modeling. Before
averaging, the instantaneous GKHE with no mean flow and neglecting the forcing
term reads (Hill, 2001, 2002b):

$\displaystyle\frac{\partial\delta u_{i}^{2}}{\partial
t}+u^{*}_{j}\frac{\partial\delta u_{i}^{2}}{\partial
x_{j}}=-\frac{\partial\delta u_{j}\delta u_{i}^{2}}{\partial
r_{j}}-\frac{8}{\rho}\frac{\partial p^{*}\delta u_{i}}{\partial
r_{i}}+\nu\frac{1}{2}\frac{\partial^{2}\delta u_{i}\delta u_{i}}{\partial
x_{j}\partial x_{j}}+2\nu\frac{\partial^{2}\delta u_{i}\delta u_{i}}{\partial
r_{j}\partial r_{j}}-4\epsilon^{*},$ (1)

where $\delta u_{i}=\delta u_{i}({\bf x},{\bf r})=u_{i}^{+}-u_{i}^{-}$ is the
velocity increment vector in the $i^{\textrm{th}}$ Cartesian direction over
displacement vector ${\bm{r}}$. The superscripts $+$ and $-$ represent two
points ${\bf x}+{\bf r}/2$ and ${\bf x}-{\bf r}/2$ in the physical domain that
have a separation vector $r_{i}=x^{+}_{i}-x^{-}_{i}$ and middle point
$x_{i}=(x^{+}_{i}+x^{-}_{i})/2$ (see Fig. 1 (a)). The superscript $*$ denotes
the average value between two points, e.g., the two-point average dissipation
is defined as $\epsilon^{*}({\bf x},{\bf r})=(\epsilon^{+}+\epsilon^{-})/2$,
and $\epsilon^{\pm}$ here is the “pseudo-dissipation” defined at every point
as $\epsilon=\nu({\partial u_{i}}/{\partial x_{j}})^{2}$. Also, we will use
${\bf r}_{s}={\bf r}/2$ to denote the radial coordinate vector from the local
“origin” ${\bf x}$.

As remarked by Hill (2001, 2002b) it is then instructive to apply integration
over a sphere in ${\bf r}_{s}$-space up to a radius $\ell/2$, i.e. over a
sphere of diameter $\ell$. The resulting equation is divided by the sphere
volume $V_{\ell}=\frac{4}{3}\pi({\ell}/{2})^{3}$ and a factor 4, and Gauss’
theorem is used for the ${\bf r}$-divergence terms (recalling that
$\partial{\bf r}=2\partial{\bf r}_{s}$), yielding

$\displaystyle\frac{1}{2\,V_{\ell}}\iiint\limits_{V_{\ell}}\left(\frac{\partial\delta
u_{i}^{2}/2}{\partial t}+u^{*}_{j}\frac{\partial\delta u_{i}^{2}/2}{\partial
x_{j}}\right)d^{3}{\bf
r}_{s}=-\frac{3}{4\ell}\frac{1}{S_{\ell}}\oint\limits_{S_{\ell}}\delta
u_{i}^{2}\delta
u_{j}\hat{n}_{j}dS-\frac{6}{\rho\,\ell}\frac{1}{S_{\ell}}\oint\limits_{S_{\ell}}p^{*}\delta
u_{j}\hat{n}_{j}dS$ (2)
$\displaystyle+\frac{\nu}{4}\frac{1}{V_{\ell}}\iiint\limits_{V_{\ell}}\left(\frac{1}{2}\frac{\partial^{2}\delta
u_{i}\delta u_{i}}{\partial x_{j}\partial x_{j}}+2\frac{\partial^{2}\delta
u_{i}\delta u_{i}}{\partial r_{j}\partial r_{j}}\right)d^{3}{\bf
r}_{s}-\frac{1}{V_{\ell}}\iiint\limits_{V_{\ell}}\epsilon^{*}d^{3}{\bf
r}_{s},$

where $S_{\ell}$ represents the bounding sphere’s surface of area
$S_{\ell}=4\pi(\ell/2)^{2}$ and $\hat{n}_{j}$ is the unit normal vector. Eq. 2
suggests defining a structure-function based kinetic energy at scale $\ell$
according to

$k_{{\rm sf},\ell}({\bf
x},t)=\frac{1}{2\,V_{\ell}}\iiint\limits_{V_{\ell}}\frac{1}{2}\delta
u_{i}^{2}({\bf x},{\bf r})\,d^{3}{\bf r}_{s},$ (3)

so that the first term in Eq. 2 corresponds to $\partial\,k_{{\rm
sf},\ell}/\partial t$. The 1/2 factor in front of the integral is justified
since the volume integration over the entire sphere will double count the
energy contained in $\delta u_{i}^{2}=u_{i}^{+}-u_{i}^{-}$. Equation 2 thus
describes the transport of two-point, structure function energy $k_{{\rm
sf},\ell}$, which represents energy within eddies with length scales up to
$\ell$ (Davidson, 2015) in both the length scale $\ell$ and physical position
$\bf x$ spaces. The last term in equation 2 represents $r$-averaged rate of
dissipation with the radius vector ${\bf r}_{s}={\bf r}/2$ being integrated up
to magnitude $\ell/2$,

Figure 1: (a): Sketch showing local domain of integration over a ball of
diameter $\ell$ used in the symmetric Hill (2002b) structure function approach
in which pairs of points separated by distances $r=2r_{s}$ up to $\ell$ are
used. (b) shows integration up to a ball of radius $\ell$ in which pairs of
points separated by distances $r$ up to $\ell$ are used as in the approach of
Duchon & Robert (2000). For volume averaging, in (a) 3D integration over the
vector ${\bf r}_{s}$ is performed at fixed ${\bf x}$ while in (a) 3D
integration over the vector ${\bf r}$ is performed at fixed ${\bf x}$. For
surface integrations, in (a) integration is done over the spherical surface of
radius $\ell/2$ while in (b) is is done over a spherical surface of radius
$\ell$.

$\epsilon_{\ell}({\bf
x},t)\equiv\frac{1}{V_{\ell}}\iiint\limits_{V_{\ell}}\epsilon^{*}({\bf x},{\bf
r})\,d^{3}{\bf r}_{s}.$ (4)

As remarked by Hill (2001, 2002b) this quantity corresponds directly to the
spherical average of local dissipation at scale $\ell$ and plays a central
role in the celebrated Kolmogorov Refined Similarity Hypothesis, KRSH
(Kolmogorov, 1962).

The local energy cascade rate in the inertial range at position ${\bf x}$ and
time $t$ is defined as

$\Phi_{\ell}({\bf
x},t)\equiv-\frac{3}{4\,\ell}\frac{1}{S_{\ell}}\oint\limits_{S_{\ell}}\delta
u_{i}^{2}\,\delta u_{j}\,\hat{n}_{j}\,dS=-\frac{3}{4\,\ell}\,[\delta
u_{i}^{2}\delta u_{j}\hat{n}_{j}]_{S_{\ell}},$ (5)

where $[..]_{S_{\ell}}$ indicates area averaging over the sphere of diameter
$\ell$. We note that in this definition, $\Phi_{\ell}({\bf x},t)$ represents
the surface average of a flux that is defined positive if energy is flowing
into the sphere in the ${\bf r}$-scale space. The position is fixed at ${\bf
x}$ and thus the quantity $\Phi_{\ell}({\bf x},t)$ does not contain possible
confounding spatial transport effects.

In terms of the overall average of Eq. 2, under the assumptions of homogeneous
isotropic flow and statistical steady-state conditions, and for $\ell$ in the
inertial range of turbulence, the unsteady, transport and viscous terms
vanish. The pressure term is also zero due to isotropy and incompressibility.
Therefore, Eq. 2 can be simplified and yields as expected

$\langle\Phi_{\ell}\rangle=\langle\epsilon_{\ell}\rangle=\langle\epsilon\rangle,$
(6)

or equivalently $[\delta u_{i}^{2}\delta
u_{j}\hat{n}_{j}]_{S_{\ell}}=-4/3\,\ell\,\langle\epsilon\rangle$, the 4/3-law
Frisch (1995).

In this paper the focus will be mainly on the flux term $\Phi_{\ell}$ with
some attention also on the dissipation term $\epsilon_{\ell}$. Analysis of the
time derivative, spatial advection terms and pressure terms is left for other
ongoing studies. The viscous flux terms (in both spatial and scale spaces) are
also not considered, since our present interest concerns the inertial range.

###  2.2 Energy cascade rate/flux in the filtering approach

In this section, we review the transport equation of the subgrid-scale kinetic
energy (Germano, 1992) for $k_{{\rm sgs},\ell}\equiv\frac{1}{2}\tau_{ii}$,
where $\tau_{ij}=\widetilde{u_{i}u_{j}}-\tilde{u}_{i}\tilde{u}_{j}$ is the
subgrid-scale stress tensor, where the tilde symbol $(\sim)$ denotes spatial
filtering of variables. The transort equation for $k_{{\rm sgs},\ell}$ reads
(Germano, 1992)

$\displaystyle\frac{\partial k_{{\rm sgs},\ell}}{\partial
t}+\tilde{u}_{j}\frac{\partial k_{{\rm sgs},\ell}}{\partial x_{j}}=$
$\displaystyle-\frac{1}{2}\frac{\partial}{\partial
x_{j}}\left(\widetilde{u_{i}u_{i}u_{j}}-2\tilde{u}_{i}\widetilde{u_{i}u_{j}}-\tilde{u}_{j}\widetilde{u_{i}u_{i}}-\tilde{u}_{i}\tilde{u}_{i}\tilde{u}_{j}\right)-\frac{\partial}{\partial
x_{j}}\left(\widetilde{pu_{j}}-\tilde{p}\tilde{u}_{j}\right)$ (7)
$\displaystyle+\frac{\partial}{\partial x_{j}}\left(\nu\frac{\partial k_{{\rm
sgs},\ell}}{\partial x_{j}}\right)+\nu\frac{\partial\tilde{u}_{i}}{\partial
x_{j}}\frac{\partial\tilde{u}_{i}}{\partial
x_{j}}-\nu\widetilde{\frac{\partial u_{i}}{\partial x_{j}}\frac{\partial
u_{i}}{\partial x_{j}}}-\tau_{ij}\widetilde{S}_{ij}.$

The last term is called the subgrid-scale rate of dissipation at position
$({\bf x})$, and is often denoted as

$\Pi_{\ell}({\bf x},t)\equiv-\tau_{ij}\widetilde{S}_{ij}.$ (8)

For filtering, in the present work we consider a spherical-shaped sharp top-
hat filter in physical space with a diameter equal to $\ell$. Therefore, for
any field variable $A({\bf x})$, we define the filtered variable as
$\tilde{A}({\bf x})={V_{\ell}}^{-1}\iiint\limits_{V_{\ell}}A({\bf x}+{\bf
r}_{s})\,d^{3}{\bf r}_{s}$. Note that each term in Equation 2 and in Equation
7 are thus evaluated at the same length scale. Terms in Equation 7 can be
compared directly to terms in Equation 2, in particular the local dissipation
terms are exactly the same, i.e,

$-\nu\widetilde{\frac{\partial u_{i}}{\partial x_{j}}\frac{\partial
u_{i}}{\partial
x_{j}}}=-\frac{1}{V_{\ell}}\iiint\limits_{V_{\ell}}\nu\frac{\partial
u_{i}}{\partial x_{j}}\frac{\partial u_{i}}{\partial x_{j}}d^{3}{\bf
r}_{s}=\epsilon_{\ell}({\bf x},t).$ (9)

Again, for homogeneous steady state turbulence in the inertial range
(neglecting viscous diffusion and resolved dissipation terms), upon averaging
Eq. 7 simplifies to

$\langle\Pi_{\ell}\rangle=\langle\epsilon_{\ell}\rangle=\langle\epsilon\rangle$
(10)

which is similar to Equation 6 and thus on average certainly both definitions
of energy cascade rate/flux agree with each other, i.e.,
$\langle\Pi_{\ell}\rangle=\langle\Phi_{\ell}\rangle$.

It is also of interest to compare the average value of the two definitions of
kinetic energy used in both definitions of energy cascade rate/flux. In the
inertial range of high Reynolds number turbulence, both $\langle k_{{\rm
sf},\ell}\rangle$ and $\langle k_{{\rm sgs},\ell}\rangle$ can be evaluated
based on the Kolmogorov $r^{2/3}$ law and $k^{-5/3}$ spectrum, respectively.
The result is (see Appendix B for details), $\langle k_{{\rm
sf},\ell}\rangle=1.575\,\langle\epsilon\rangle^{2/3}\ell^{2/3}$ and $\langle
k_{{\rm sgs},\ell}\rangle=1.217\,\langle\epsilon\rangle^{2/3}\ell^{2/3}$. In
other words, they are of similar order of magnitude but the SGS kinetic energy
is slightly smaller.

###  2.3 Other relationships between structure function and filtering
approaches

In the present paper, we shall perform the data analysis and comparisons using
the two approaches mentioned above (GKHE and filtering). However, it is useful
at this stage to include some remarks regarding other structure function and
energy definitions used in earlier works by Vreman et al. (1994), Constantin
et al. (1994), Duchon & Robert (2000), Eyink (2002) and Dubrulle (2019). Those
approaches typically focus on the structure function written at one of the
endpoints instead of the midpoint. Duchon & Robert (2000) and Dubrulle (2019)
focus on the two-point correlation quantity $C({\bf x},{\bf r})=u_{i}({\bf
x})u_{i}({\bf x}+{\bf r})$ (see Fig. 1 (b)). Local averaging over all values
of ${\bf r}$ from ${\bf r}=0$ up to scale $|{\bf r}|=\ell$ at any given ${\bf
x}$ then corresponds to the “mixed” energy quantity $u_{i}\tilde{u}_{i}/2$
(denoted as $E^{\ell}$ in Dubrulle (2019)), and where the filtering is over a
sphere of diameter $2\ell$ so as to combine two points with separation
distances up to $\ell$. The quantity $C({\bf x},{\bf r})$ combines filtered
and unfiltered velocities and hence it is more difficult to interpret for
comparisons of structure function and LES filtering approaches. In its
transport equation, Duchon & Robert (2000) show that a term similar to the
third-order structure function term of Eq. 5 arises. However, in order for the
structure function to correspond to scale $\ell$, one has to choose to
integrate over a sphere of diameter $2\ell$ (the locally integrated
dissipation rate would then be $\epsilon_{2\ell}$). In a spherical integration
over ${\bf r}$ of powers of the velocity difference $[u_{i}({\bf x}+{\bf
r})-u_{i}({\bf x})]$, only the first term is affected by filtering or
averaging over the spherical shell, while the center velocity $u_{i}({\bf x})$
remains fully local. Note that in the GKHE formalism, the averaging affect
both end-point velocities in the same way, and both become averaged at scale
$\ell$ in a formally symmetric way.

An early connection between structure functions and filtering approaches was
developed by Vreman et al. (1994). In the Vreman analysis, the structure
function is defined based on the difference of velocity $u_{i}({\bf x}+{\bf
r})$ and the locally filtered velocity $\tilde{u}_{i}$ centered at ${\bf x}$.
Spherical integration of $(u_{i}({\bf x}+{\bf r})-\tilde{u}_{i})^{2}$ over a
sphere of radius $\ell/2$ then yields equivalence with the SGS kinetic energy
at scale $\ell$. But $(u_{i}({\bf x}+{\bf r})-\tilde{u}_{i})^{2}$ does not
equal the usual structure function definition, now due to a mixture of
filtered and unfiltered quantities at two points even before local filtering.

Another interesting approach was presented in Constantin et al. (1994) and
connected to the LES filtering approach by Eyink (1995), Eyink (2006)
(equations 2.12-2.14). In fact as recounted in the review by Eyink &
Sreenivasan (2006), early unpublished work by Onsager anticipated such
expressions half a century prior. Written in terms of the sharp spherical
filter we use here, the expression for the trace of the SGS stress reads

$\tau_{ii}({\bf x})=\frac{1}{V_{\ell}}\iiint_{V_{\ell}}[u_{i}({\bf x}+{\bf
r})-u_{i}({\bf x})]^{2}\,d^{3}{\bf
r}-\left(\frac{1}{V_{\ell}}\iiint_{V_{\ell}}[u_{i}({\bf x}+{\bf r})-u_{i}({\bf
x})]\,d^{3}{\bf r}\right)^{2}.$ (11)

This equation represents an exact relationship between two-point structure
functions and the subgrid-scale kinetic energy. But for the RHS to correspond
to structure functions up to scale $\ell$, the integration must be done over a
sphere of radius $\ell$ and thus a filtering scale of $2\ell$ for the stress
tensor in the filtering formulation. The suggested relationship then appears
to be between subgrid-scale stress kinetic energy at scale $2\ell$ and
structure functions up to two-point separations $\ell$ but averaged over a
local domain of size $2\ell$, similarly as in the Duchon & Robert (2000)
approach. Note that while each of the terms in Eq. 11 is also a mixture of
filtered and unfiltered velocities, the subtraction cancels the local term and
restores the fully filtered property inherent in the definition of
$\tau_{ii}$.

While not expecting qualitatively different results (except perhaps using the
diameter instead of the radius as a name for “scale”), we here continue our
focus on the more “symmetric” formulation by Hill, with fixed position ${\bf
x}$ specified at the midpoint between two points separated by vector ${\bf r}$
whose magnitude then spans up to scale $\ell$ (or integration radius ${\bf
r}_{s}$ up to radius $\ell/2$).

##  3 Comparisons between kinetic energies and cascade rates/fluxes

In this section, we provide comparisons of local kinetic energies in the
structure function formalism, $k_{{\rm sf},\ell}$, with that in the filtering
formalism, $k_{{\rm sgs},\ell}$. We also compare the local energy cascade
rates $\Phi_{\ell}$ and $\Pi_{\ell}$. We consider data from a direct numerical
simulation (DNS) of forced isotropic turbulence at $R_{\lambda}=1{,}250$ (the
Taylor-scale Reynolds number) that used 8,1923 grid points (Yeung et al.,
2012). The analysis is performed at three length-scales in the inertial range,
$\ell=30\eta,\,45\eta,\,60\eta$ where
$\eta=(\nu^{3}/\langle\epsilon\rangle)^{1/4}$ is the (average) Komogorov
scale. Quantities are evaluated directly from their definitions using
spherical integration. To compute volume spherically filtered quantities such
as $\epsilon_{\ell}$, $k_{{\rm sgs},\ell}$, $k_{{\rm sf},\ell}$ or $\tau_{ij}$
from data, we fix the middle point coordinate ${\bf x}$ in the physical
domain. Subsequently, we download data in a cubic domain using the JHTDB’s
cutout service in a cube of size equal to $\ell$. The data are then multiplied
by a spherical mask (filter) to evaluate local filtered quantities. The
velocity components are obtained by utilizing pre-computed Getfunctions from
the Johns Hopkins Turbulence database (JHTDB) including spatial interpolation,
as explained in more detail in Appendix A. For surface averages, we discretize
the outer surface of diameter $\ell$ into 500 points pairs ($+$ and $-$
points) that are approximately uniformly distributed on the sphere. The
accuracy of this method of integration has been tested by increasing the
number of points used in the discretization. Data on the specified points are
obtained from the database using 8th-order Lagrange spatial interpolation.

Overall mean values are obtained at the three scales and are plotted in Fig. 2
(a). The results for kinetic energy for the structure function approach are
consistent with the analytical evaluation (see Appendix B). For the SGS
kinetic energy, the numerical results fall below the theoretical inertial
range prediction, due to effects of the viscous range that reduces the amount
of SGS kinetic energy even at scales much larger than the Kolmogorov scale.
The average values of the spectral fluxes agree very well with the averaged
rate of viscous dissipation.

Figure 2:  Panel (a) shows normalized mean kinetic energies and mean cascade
rates as function of three filter scales for the $R_{\lambda}=1250$ DNS
isotropic turbulence dataset. Specifically, closed squares show $\langle
k_{{\rm sf},\ell}\rangle/(\langle\epsilon\rangle\ell)^{2/3}$ while closed
circles show $\langle k_{{\rm
sgs},\ell}\rangle/(\langle\epsilon\rangle\ell)^{2/3}$. Open squares show
$\langle\Phi_{\ell}\rangle/\langle\epsilon\rangle$ while open circles show
$\langle\Pi_{\ell}\rangle/\langle\epsilon\rangle$. The horizontal lines show
the expected asymptotic values in the inertial range for mean kinetic energies
in the structure function formulation (1.575) and in the filtering formulation
(1.217), while the expected energy cascade rates equal unity. Panel (b) shows
the correlation coefficients between kinetic energies ($\rho_{kk}$, downward
triangles) and between cascade rates ($\rho_{\Phi\Pi}$,upward triangles).

Figure 3(a) shows the joint PDF of $k_{{\rm sf},\ell}$ and $k_{{\rm
sgs},\ell}$ at scale $\ell=45\eta$. The correlation coefficient between both
quantities is $\rho_{kk}=0.97$ (Fig. 2 (b)). Similarly, Figure 3(b) shows the
joint PDF of $\Pi_{\ell}$ and $\Phi_{\ell}$, also at scale $\ell=45\eta$ for
the same dataset. The correlation coefficient between both quantities is
measured to be $\rho_{\Phi\Pi}=0.58$ (Fig. 2 (b)), significantly lower than
for the energies but still appreciable. It can be seen that negative values
occur for both $\Pi_{\ell}$ and $\Phi_{\ell}$, although it appears that
$\Phi_{\ell}$ has more variability and larger negative excursions than
$\Pi_{\ell}$. As summarized in §1, the relevance of locally negative values of
$\Pi_{\ell}$ to the flow physics remains unclear, especially given the fact
that upon averaging, the quantity becomes positive. Conversely, the quantity
$\Phi_{\ell}$ has a clearer local interpretation, in the sense that locally
negative values can clearly be interpreted as kinetic energy (local $\delta
u_{i}^{2}/2)$) showing a net flux into a sphere of diameter $\ell$ in scale
space, i.e. becoming associated with energy at smaller scales. Its overall
average also is positive. An interesting question is whether negative values
of $\Pi_{\ell}$ or $\Phi_{\ell}$ survive under some type of statistical
averaging. In the next sections we use conditional averaging to quantify the
importance of negative values (inverse local cascade, or backscatter).

Figure 3: Joint PDFs of (a) $k_{{\rm sgs},\ell}$ and $k_{{\rm sf},\ell}$ with
values $0.01,0.03,0.1,0.3,1,3$ (b) $\Pi_{\ell}$ and $\Phi_{\ell}$ with values
$0.001,0.003,0.01,0.3,0.1,0.3$, at scale $\ell=45\eta$ measured in DNS of
isotropic turbulence at $R_{\lambda}$=1250. The red dash line represents a
$45$ degree slop line. The data and the editable notebook can be found at:
https://cocalc.com/…/Figure3.

##  4 Conditional averaging based on local dissipation

In this section, we compare conditionally averaged cascade rates/fluxes for
both the structure function and filtering formulations, conditioned on
$\epsilon_{\ell}$, i.e, $\langle\Phi_{\ell}|\epsilon_{\ell}\rangle$ and
$\langle\Pi_{\ell}|\epsilon_{\ell}\rangle$. According to KRSH (Kolmogorov,
1962), the statistical properties of velocity increments depend on the local
average dissipation within a sphere of scale $\ell$, rather than being
determined by the globally averaged dissipation. Written in terms of the
quantities of present interest, the KRSH would read
$\langle\Phi_{\ell}|\epsilon_{\ell}\rangle=\epsilon_{\ell}$ since
$\Phi_{\ell}$ is fully determined by the velocity increments envisioned in the
KRSH. Loosely extending the KRSH arguments to the filtering formalism would
suggest $\langle\Pi_{\ell}|\epsilon_{\ell}\rangle=\epsilon_{\ell}$.

In order to assess this hypothesis, we evaluate the conditional averages based
on the same dataset described before. Results for
$\langle\Phi_{\ell}|\epsilon_{\ell}\rangle$ and
$\langle\Pi_{\ell}|\epsilon_{\ell}\rangle$ are shown in Figure 4. Results for
the three scales considered are included. As can be seen, the plot shows a
close agreement between both $\langle\Phi_{\ell}|\epsilon_{\ell}\rangle$ and
$\langle\Pi_{\ell}|\epsilon_{\ell}\rangle$, with $\epsilon_{\ell}$. It is
important to note that $\Phi_{\ell}$ and $\Pi_{\ell}$ are conditioned on the
exact same values of $\epsilon_{\ell}$. The similarities and differences
observed in Figure 4 indicate that $\Phi_{\ell}$ and $\Pi_{\ell}$ share many
properties (same conditional averages) but they are not identical. For
instance, it is clear from Fig. 3 (b) that the variance of $\Phi_{\ell}$
exceeds that of $\Pi_{\ell}$, even though their mean values are the same.

In general, the behaviors of both $\langle\Phi_{\ell}|\epsilon_{\ell}\rangle$
and $\langle\Pi_{\ell}|\epsilon_{\ell}\rangle$ confirm the validity of the
KRSH in the present context. More detailed analysis of the KRSH for
$\Phi_{\ell}$ and connections to Eq. 2 are reported in Yao et al. (2023)

Figure 4: Conditional averages of $\Phi_{\ell}$ and $\Pi_{\ell}$ based on
local dissipation $\epsilon_{\ell}$, i.e.,
$\langle\Phi_{\ell}|\epsilon_{\ell}\rangle$ (black symbols and lines),
$\langle\Pi_{\ell}|\epsilon_{\ell}\rangle$ (open symbols and lines). The red
line indicates the value of $\epsilon_{\ell}$. Different symbols denote
different scales $\ell/\eta=30$ (squares), $45$ (triangles) and $60$
(circles). All values are normalized with the globally averaged rate of
dissipation $\langle\epsilon\rangle$.

##  5 Conditional averaging based on large-scale velocity gradients

In this section, we explore correlations between properties of the velocity
gradient tensor filtered at scale $\ell$ and the two definitions of energy
cascade rate/flux. The velocity gradient tensor encapsulates information about
fluid deformation and rotation and connections to the energy cascade have been
studied extensively. Already Bardina et al. (1985) examined the impact of
rotation on homogeneous isotropic turbulence (HIT) and observed that rotation
decreases the dissipation (cascade) rate while increasing the length scales,
suggestive of inverse energy cascade effects. Goto (2008) investigated
physical mechanisms underlying forward energy cascade and argued that forward
cascade can be triggered in regions characterized by strong strain between two
large-scale tubular vortices. The role of the filtered gradient tensor for
energy cascade was first explored numerically in Borue & Orszag (1998) and
experimentally in Van der Bos et al. (2002) building on the “Clark model” that
approximates features of the subgrid-scale tensor using Taylor-series
expansion. Recent studies by Johnson (2020, 2021) and Carbone & Bragg (2020)
have significantly expanded on such analyses and examined the roles of self-
strain amplification and vortex stretching driving the forward energy cascade
process. For inverse cascade, a vortex thinning mechanism may be at play
(Johnson, 2021).

A first level of characterization of the properties of the velocity gradient
tensor are its invariants. To characterize deformation and rotation, we
evaluate the strain and rotation invariants from data, defined according to

$S_{\ell}^{2}({\bf x},t)=\tilde{S}_{ij}\tilde{S}_{ij},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Omega_{\ell}^{2}({\bf
x},t)=\tilde{\Omega}_{ij}\tilde{\Omega}_{ij},$ (12)

where $S_{ij}$ and $\Omega_{ij}$ are the symmetric and antisymmetric parts of
the velocity gradient tensor $A_{ij}=\partial u_{i}/\partial x_{j}$ and the
tilde denotes, as before, spherical tophat filtering over a ball of diameter
$\ell$. For consistency with prior literature, these values will be normalized
by the overall average $\langle
Q_{w}\rangle=\frac{1}{2}\langle\Omega_{\ell}^{2}\rangle$ (equal to
$\frac{1}{2}\langle S_{\ell}^{2}\rangle$ in isotropic turbulence).

A more detailed characterization of the statistics of velocity gradients
involves the invariants $Q$ and $R$ (Vieillefosse, 1982). It is well-known
that the joint probability density function (JPDF) of $Q$-$R$ exhibits a
characteristic tear-drop shape (Chong et al., 1990; Meneveau, 2011), from
which flow topology information such as vortex stretching and compression can
be inferred (Chong et al., 1990; Borue & Orszag, 1998; Lüthi et al., 2009;
Danish & Meneveau, 2018). These two invariants (at scale $\ell$) are defined
as usual according to

$Q_{\ell}({\bf
x},t)=-\frac{1}{2}\tilde{A}_{ij}\tilde{A}_{ji},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ R_{\ell}({\bf
x},t)=-\frac{1}{3}\tilde{A}_{ij}\tilde{A}_{jk}\tilde{A}_{ki}.$ (13)

Under the assumption of restricted Euler dynamics (Meneveau, 2011), the
transport equation for the velocity gradient tensor leads to
${dQ_{\ell}}/{dt}=-3R_{\ell}$ and ${dR_{\ell}}/{dt}=-\frac{2}{3}Q_{\ell}^{2}$
(Cantwell, 1992). The quantity $R_{\ell}$ can thus be considered as the
(negative) rate of change of $Q_{\ell}$ and contains both vortex stretching
and strain self-stretching mechanisms (Johnson, 2021). In our comparative
investigation of energy cascade rates, conditional averaging based on the four
invariant quantities $S^{2}_{\ell}$, $\Omega^{2}_{\ell}$, $Q_{\ell}$ and
$R_{\ell}$ will be undertaken.

We begin with qualitative visualizations of the fields in small subsets of the
domains analzyed. Panel (a) and (b) of Figure 5 depict a sample instantaneous
field of $\Phi_{\ell}$ and $\Pi_{\ell}$ respectively, highlighting regions of
strong local forward cascade (indicated by solid red circles) and strong
inverse cascade (indicated by dashed circles). The correlation between these
two variables is evident, the computed correlation coefficient between the
snapshots is 0.64. On both panels (a) and (b) the fluxes are normalized by the
global averaged dissipation $\langle\epsilon\rangle$. As already noted based
on the joint PDFs, there are differences between $\Phi_{\ell}$ and
$\Pi_{\ell}$. The maximum magnitude of the positive cascade rate in
$\Phi_{\ell}$ is about twice that of $\Pi_{\ell}$, while the magnitude of the
negative cascade rate in $\Phi_{\ell}$ is about 3 to 4 times larger. Since
$\langle\Phi_{\ell}\rangle\sim\langle\Pi_{\ell}\rangle\sim\langle\epsilon_{\ell}\rangle\sim\langle\epsilon\rangle$,
the significant different maximum values indicates $\Phi_{\ell}$ is more
variable and intermittent than $\Pi_{\ell}$. Also, $\Phi_{\ell}$ exhibits
somewhat finer-scale spatial features.

###  5.1 Conditional statistics based on strain rate $(S^{2}_{\ell})$ and
rotation rate $(\Omega^{2}_{\ell})$

Figure 5: Panels (a), (b), (c), (d) are instantaneous $\Phi_{\ell}$,
$\Pi_{\ell}$, $S_{\ell}^{2}$ and $\Omega_{\ell}^{2}$ field with $\ell=45\eta$
in a $750\eta\times 750\eta$ domain ($500\times 500$ points of the DNS grid.
The black solid circles in (a) and (b) are located at strong local forward
cascade region, which is correlated to the strong local strain rate marked in
the black circle in panel (c). The red dashed circles in (a) and (b) are
located at strong local inverse cascade regions (negative energy fluxes),
which appear qualitatively correlated to relatively strong local rotation
rates marked in the red dashed circles in panel (d).

Panels (a) and (b) in figure 5 show distinct regions including both local
forward (red area) and inverse (blue area) cascade rates. It is visually
apparent that the presence of a strong local forward energy cascade is
associated with increased local strain rate, as indicated by the solid red
circle in both panels (a) and (b) of Figure 5 and the corresponding black
solid circle in panel (c). Similarly, a strong local inverse energy cascade is
observed alongside a significant local rotation rate, depicted by the dashed
red circle in panels (a) and (b) of Figure 5 and the corresponding red dashed
circle in panel (d). The strong correlation between forward cascade and local
straining is consistent with multiple earlier observations and prior works in
the literature (e.g. Borue & Orszag (1998) and recently Johnson (2021);
Carbone & Bragg (2020)). We focus attention on the regions with negative
energy cascade rates. Conditional averaging can elucidate the statistical
significance of these regions. Specifically, we inquire whether there are
large-scale flow local features as characterized by the filtered velocity
gradient invariants that are systematically accompanied by inverse cascade,
i.e., negative $\Phi_{\ell}$. Thus we perform conditional averaging of
$\Phi_{\ell}$ based on the invariants $S_{\ell}^{2}$ and $\Omega_{\ell}^{2}$
and repeat the analysis for the SGS energy flux quantity $\Pi_{\ell}$.

Figure 6:  Panels (a), (b), (c) are
$\langle\Phi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell}\rangle$; panels (d), (e),
(f) are $\langle\Pi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell}\rangle$ at
$\ell=30\eta,45\eta,60\eta$. The black dashed line in panel (c) represent iso-
line of $Q_{\ell}\equiv-\frac{1}{2}(S_{\ell}^{2}-\Omega_{\ell}^{2})$.
$\Phi_{\ell}$ and $Pi_{\ell}$ are normalized by $\langle\epsilon\rangle$;
$S^{2}_{\ell}$ and $\Omega^{2}_{\ell}$ are normalized by $\langle
Q_{w}\rangle=\frac{1}{2}\langle\Omega_{\ell}^{2}\rangle$.

Figure 6 shows the joint conditionally-averaged $\Phi_{\ell}$ and $\Pi_{\ell}$
based on $S^{2}_{\ell}$ and $\Omega^{2}_{\ell}$, denoted as
$\langle\Phi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell}\rangle$ and
$\langle\Pi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell}\rangle$, respectively. The
analysis is performed by computing averages over two million randomly
distributed points ${\bf x}$. In the presented results, $\Phi_{\ell}$ is
normalized by $\langle\epsilon\rangle$, while $S^{2}_{\ell}$ and
$\Omega^{2}_{\ell}$ are normalized by $\langle
Q_{w}\rangle=\frac{1}{2}\langle\Omega^{2}_{\ell}\rangle$. Panels (a), (b), and
(c) of figure 6 present the joint conditionally-averaged
$\langle\Phi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell}\rangle$ at three different
length scales, namely $r=30\eta,45\eta,60\eta$. Panels (a), (b), and (c)
highlight the dominance of the forward cascade by the extensive red region.
This magnitude is manytimes larger than the maximum magnitude observed in the
blue region, representing the inverse cascade. The red region covers a wide
range of $S^{2}_{\ell}$ and $\Omega^{2}_{\ell}$ values, consistent with the
expectation that the global average would favor a forward cascade
($\langle\Phi_{\ell}>0\rangle$). The highest positive values of
$\langle\Phi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell}\rangle$ correspond to high
strain rates and low rotation rates, and they decrease as the strain rate
decreases. Interestingly, the inverse cascade appears explicitly in the lower-
right corner of the plots, where the rotation rate is strong but the strain
rate is weak. It is worth noting that the conditionally averaged values shown
in Figure 6 reflect the combined outcome of the forward and inverse cascades.
Consequently, in specific regions characterized by distinct strain and
rotation rates, events with forward and inverse cascades can cancel each other
out. Only in the lower right corner is there an indication of net inverse
cascade when the cascade rate is defined using the structure function
approach.

In panel (c), we superimpose dashed lines representing the isolines of
$Q_{\ell}$, with the $Q_{\ell}=0$ line indicating the condition of equal
strain and rotation rates. The parallel dashed lines correspond to
$Q_{\ell}=-10$, $-5$, $0$, $5$, $10$, and $15$, respectively. The
$Q_{\ell}=15$ line appears near the boundary separating the red and blue
regions. However, the boundary of the blue region does not appear to align
well with the $Q_{\ell}$ isoline. This observation suggests that $Q_{\ell}$
might be not enough to provide an adequate threshold for distinguishing the
net forward and inverse cascade regions.

Panels (d), (e), and (f) in Figure 6 present results for the joint
conditionally-averaged $\Pi_{\ell}$ based on $S^{2}_{\ell}$ and
$\Omega^{2}_{\ell}$, corresponding to the same filter scales as panels (a),
(b), and (c). It is evident that trends for the positive cascade rate (red
region) for $\Pi_{\ell}$ closely resemble those of $\Phi_{\ell}$, with the
peak of the forward cascade occurring at a high strain rate and low rotation
rate. The magnitude of the maximum forward cascade rate for $\Pi_{\ell}$ is
slightly weaker compared to that of $\Phi_{\ell}$. The most significant
difference is that only a few instances of blue squares are observed in
regions characterized by strong rotation and weak strain, indicating that the
overall predominance of the forward cascade persists regardless of the local
values of $S^{2}_{\ell}$ and $\Omega^{2}_{\ell}$. These results highlight some
important statistical differences between $\Phi_{\ell}$ and $\Pi_{\ell}$.

Figure 7:  Panels (a) and (b) present the conditional averages of cascade
rates obtained by sampling positive and negative signs, i.e,
$\langle\Phi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell},\Phi_{\ell}>0\rangle$ and
$\langle\Phi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell},\Phi_{\ell}<0\rangle$,
respectively. Panel (c) and (d) display the logarithm base 10 of the number of
samples on the $S^{2}_{\ell},\Omega^{2}_{\ell}$ map (out of a total of
$2\times 10^{6}$ samples). The isolines in panels (c) and (d) are the values
corresponding to the contour.

To develop a more detailed understanding of the inverse cascade region within
$\langle\Phi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell}\rangle$, we performe a
further analysis by dividing the samples based on $\Phi_{\ell}>0$ and
$\Phi_{\ell}<0$ for $r=45\eta$ (results for $r=\\{30,60\\}\eta$, not shown,
are similar). We then calculate the conditional average of these separated
samples considering $S^{2}_{\ell}$ and $\Omega^{2}_{\ell}$, denoted as
$\langle\Phi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell},\Phi_{\ell}>0\rangle$ and
$\langle\Phi_{\ell}|S^{2}_{\ell},\Omega^{2}_{\ell},\Phi_{\ell}<0\rangle$. The
results are presented in Figure 7. From panel (a), it can be observed that the
forward cascade clearly increases with $S^{2}_{\ell}$, with the highest values
of $\Phi_{\ell}$ concentrated in the upper-left corner. It increases also with
$\Omega^{2}_{\ell}$ but less rapidly. Combined, the trend seems to be an
increase roughly proportional to $\sim S^{2}_{\ell}+0.75\,\Omega^{2}_{\ell}$.
Differently, panel (b) illustrates that the inverse cascade is roughly
proportional to $\sim S^{2}_{\ell}+0.5\,\Omega^{2}_{\ell}$, i.e. shallower
isolines extending more in the horizontal direction than in the vertical
compared to the forward cascade case shown in (a). This observation elucidates
why the strongest red region in panel (b) of Figure 6 emerges at the largest
$S^{2}_{\ell}$, while below this threshold, the forward cascade events
progressively weaken and are gradually canceled out by the inverse cascade.
Finally, in regions characterized by a weak strain rate and strong rotation
rate, the inverse cascade becomes the dominant contribution.

Panel (c) and (d) display the distribution of the number of samples
corresponding to positive and negative cascade rates in logarithmic scale (out
of the 2 million samples (balls) considered). Our focus is specifically
directed towards the bottom-right corner of the plots, which corresponds to
the region where the inverse cascade is observed in panel (b) of Figure 6.
Interestingly, we observe that at $\Omega^{2}_{\ell}/\langle
Q_{w}\rangle\approx 40$ and $S^{2}_{\ell}/\langle Q_{w}\rangle<10$, the number
of samples representing both inverse and forward cascade rates is roughly
equivalent, falling within the range of $10^{1}$ to $10^{1.5}$. This implies
that within this region, the magnitude of the inverse cascade must be
significant to achieve net negative values for the conditional average. Still,
for the conditions with net inverse cascade, the number of occurrences for
both forward and inverse cascade rates is quite small, on the order of only
$1/10^{5}$ of the total samples, indicating a very low frequency. This
observation suggests that the inverse cascade region depicted in Figure 6 must
be primarily attributed to very rare but intense events.

###  5.2 Conditional statistics based on $Q_{\ell}$ and $R_{\ell}$ invariants

In the context of the $\Omega^{2}_{\ell}$ and $S^{2}_{2}$ map shown in Figure
6, we observe the presence of a distinct inverse energy cascade in the region
characterized by strong rotation but weak strain (corresponding to large $Q$
values) for $\Phi_{\ell}$. However, such observations did not hold for
$\Pi_{\ell}$. But these results do not preclude the possibility that net
forward and inverse cascade may be associated with other invariants of the
filtered velocity gradient tensor $\tilde{A}_{ij}$.

Figure 8 shows the joint conditional averaged
$\langle\Phi_{\ell}|Q_{\ell},R_{\ell}\rangle$ and
$\langle\Pi_{\ell}|Q_{\ell},R_{\ell}\rangle$ at three different scales,
namely, $\ell=\\{30,45,60\\}\eta$. Across all figures, we can observe the
distinctive teardrop shape pattern on the Q-R map, as reported in previous
studies (Chong et al., 1990; Meneveau, 2011). The black solid lines are the
boundaries, separating the four quadrants based on the sign of $Q$ and $R$.
Notably, it becomes evident that both $\Phi_{\ell}$ and $\Pi_{\ell}$ exhibit a
strong and dominant inverse cascade in the quadrant characterized by $Q>0$ and
$R>0$, commonly referred to as the “vortex compression” region. This
observation is particularly interesting considering the absence of an
observable inverse cascade for $\Pi_{\ell}$ in the $S^{2}_{\ell}$ and
$\Omega^{2}_{\ell}$ map. Hence, these results indicate that the variables $Q$
and $R$ provide a more effective characterization to identify inverse cascade
using conditional averaging.

The region characterized by $Q<0$ and $R>0$, which corresponds to the strain-
dominated region, exhibits the most pronounced forward cascade. This
observation aligns with the findings of many prior analyses in the literature
Borue & Orszag (1998); Van der Bos et al. (2002); Johnson (2021); Carbone &
Bragg (2020) as well as those depicted in Figure 8, further emphasizing that
the strong local strain rate plays a crucial role in driving the forward
energy cascade. We note that Borue & Orszag (1998); Van der Bos et al. (2002)
display conditional averages weighted by the joint PDF of $R$ and $Q$. In
their results, there was hardly any indication of backscatter/inverse cascade
in the $Q>0$ and $R>0$ “vortex compression” region, because the overall
probability density of that region is smaller than the other regions. However,
the unweighted conditional averaging represents the relevant values if the
large-scale flow is in that particular state ($Q>0$ and $R>0$), and is
therefore relevant to our analysis.

Figure 8:  Panels (a), (b), (c) are
$\langle\Phi_{\ell}|Q_{\ell},R_{\ell}\rangle$; panels (d), (e), (f) are
$\langle\Pi_{\ell}|Q_{\ell},R_{\ell}\rangle$ at $\ell=30\eta,45\eta,60\eta$.
The black dotted lines separate four quadrants and the two lines
$Q=-(\frac{27}{4}R^{2})^{1/3}$ (Viellefosse lines) are also shown.
$\Phi_{\ell}$ and $\Pi_{\ell}$ are normalized by $\langle\epsilon\rangle$;
$Q_{\ell}$ are normalized by $\langle Q_{w}\rangle$ and $R_{\ell}$ are
normalized by $\langle Q_{w}\rangle^{(3/2)}$. The data and editable analysis
code that generated these joint PDFs (for the case at $\ell=45\eta$) can be
found at: https://cocalc.com/…/Figure8.

In a similar manner to Figure 7, we perform further conditional averaging also
distinguishing positive and negative cascade rates. Figure 9 (a) and (b)
present $\langle\Phi_{\ell}|Q_{\ell},R_{\ell},\Phi_{\ell}>0\rangle$ and
$\langle\Phi_{\ell}|Q_{\ell},R_{\ell},\Phi_{\ell}<0\rangle$ at $\ell=45\eta$.
In the case of the inverse cascade, it is observed to occur in all four
quadrants (panel (b)), with a more evenly distributed and symmetric presence
in the upper two quadrants associated with $Q>0$, i.e, the rotation-dominated
regions. The characteristic teardrop shape is less prominent and exhibits a
shorter tail compared to the forward cascade (panel (a)). Regarding the
forward cascade, it is evident that it is most dominant in the $Q<0,R>0$
quadrant, consistent with Figure 8. However, in the $Q>0,R>0$ quadrant, the
forward cascade is weaker and is overall canceled out by the stronger inverse
cascade in that particular region.

Panels (c) and (d) display the distribution of number of samples of forward
and inverse cascade rates, respectively. The shapes of the distributions align
with Figure 9 (a) and (b), but a majority of the samples are concentrated at
the center, corresponding to small values of Q and R. This observation
confirms that the strong instances of inverse cascade and forward cascade
observed in Figure 8 are primarily determined by infrequent but extreme events
(intermittency).

Figure 9: Panels (a) and (b) show
$\langle\Phi_{\ell}|Q_{\ell},R_{\ell},\Phi_{\ell}>0\rangle$ and
$\langle\Phi_{\ell}|Q_{\ell},R_{\ell},\Phi_{\ell}<0\rangle$. (c) and (d)
display the logarithm base 10 of the number of samples on the $Q,R$ map. The
isolines in panels (c) and (d) are the values corresponding to the contour.
Figure 10: Panel (a) and (b) show isosurface of local
$\Phi_{\ell}/\langle\epsilon\rangle=-60$ (light blue) and $Q_{\ell}/\langle
Q_{w}\rangle=20$ (yellow), and $R/\langle Q_{w}\rangle^{(3/2)}=30$ (black) in
two different 3D subdomains. Panel (c) and (d) show isosurface of local
$\Pi_{\ell}/\langle\epsilon\rangle=-20$ (green-blue) in the same subdomians
and isosurface of $Q_{\ell}/\langle Q_{w}\rangle$ and $R/\langle
Q_{w}\rangle^{(3/2)}$ as panel (a) and (b). Interactive visualizations are
available for each panel at: Panel(a), Panel(b), Panel(c), Panel(d). The link
to the directory containing the visualization code and the 3D fields with
these data can be found at: https://cocalc.com/…/.

Finally, to provide a visual impression of the spatial distribution of regions
of negative $\Phi_{\ell}$ in the flow, in Fig. 10 we provide a 3D
visualization of two instances in specific small subdomains of size $150^{3}$
gridpoints (i.e. $(225\eta)^{3}$ out of the overall $8192^{3}$ DNS domain. The
selection of these subdomains was based on the condition that
$Q_{\ell}/\langle Q_{w}\rangle>15$ and $R_{\ell}/\langle
Q_{w}\rangle^{(3/2)}>15$ at the center of each subdomain such that the center
is at a strong vortex compression region. We then calculate the values of
$\Phi_{\ell}$, $\Pi_{\ell}$, $Q_{\ell}$ and $R_{\ell}$ at every second grid
point. In panel (a) and (b) of Figure 10, the light blue regions correspond to
isosurface of a large negative value of
$\Phi_{\ell}/\langle\epsilon\rangle=-60$, indicating the presence of an
inverse cascade with significant magnitude. Clearly, we can see that the
occurrence of strong inverse cascade is closely associated with the presence
of the vortices. Panel (a) depicts that large negative $\Phi_{\ell}$ appears
near the center and not at the core of the yellow tube, although one should
recall that $\Phi_{\ell}({\bf x},t)$ is defined locally as centered at ${\bf
x}$ but represents the energy cascade into balls of diameter $45\eta$, i.e.
comparable to the diameter of the vortex (yellow region) shown. The blue
region is also largely connected with the black isosurface,
($R_{\ell}=20\langle Q_{w}\rangle^{3/2}$) indicating a strong “vortex
compression” region within the yellow tube. Interactive 3D versions of the
figure that can be accessed following the links in the figure caption help
elucidate the spatial structure. Panel (b) is an entirely different instance
of similar conditions, showing a more broken up vortex and showing that
$\Phi_{\ell}$ can also peak near the sides, and appear more scattered within
the vortex. Coupled with the results shown in Fig. 8, the visualizations
suggest that the strong inverse cascade occurs along the large scale vortices,
in regions of these vortices in which $R>0$, i.e. the vortex compression
regions. We can also observe some yellow tubes, within which inverse cascade
and compression are both absent. This is consistent to the statistics such
that, when conditionally averaging in terms of $Q_{\ell}$ but irrespective of
$R$, the inverse cascade becomes very week and almost non-existent. However,
once one only considers $R>0$ regions, inverse cascade can be clearly observed
in high vortical regions.

We also show $\Pi_{\ell}/\langle\epsilon\rangle=-20$ (green-blueish
isosurface) in the corresponding 3D subdomains shown as panels (c) and (d) of
Figure 10. Clearly we can see that the green-blueish and black regions largely
overlap within the yellow region in panel (c). In panel (d), the overlapping
between green-blueish, yellow and black region occurs at the center and right-
top region of the subdomain, indicating strong negative $\Pi_{\ell}$ is also
associated with strong vortex compression within high vortical region,
consistent with Figure 8. However, the patterns of the green SGS flux regions
are smoother, consistent to the 2D visualisation in Figure 5.

Caution must be expressed that visualizations only provide qualitative
impressions and more quantitative analysis requires structure-based
conditional averaging such as recently undertaken in Park & Lozano-Duran
(2023). While such analysis is beyond the scope of the present paper, the
conditional statistics presented in Fig. 8 already by themselves provide the
strong statistically robust connection between cascade rate measures and
features of the large scale velocity gradient tensor.

##  6 Conclusions

In this paper we explore, based on a DNS dataset of isotropic forced
turbulence at a relatively high Reynolds number ($R_{\lambda}=1250$), local
features of the energy cascade. We compare two common definitions of the
spatially local rate of kinetic energy cascade at some scale $\ell$. The first
is based on the cubic velocity difference term appearing in the Generalized
Kolmogorov-Hill equation (GKHE), in the structure function approach. The
second is the subfilter-scale energy flux term in the transport equation for
subgrid-scale kinetic energy, i.e., as used in filtering approach often
invoked in LES. Particular attention is placed on interpretation and
statistical robustness of observations of local negative structure-function
energy flux and subfilter-scale energy flux. The notion and relevance of local
inverse cascade or “backscatter” has been open to debates in the literature.
We argue that the interpretation of $\Phi_{\ell}({\bf x},t)$ as a spatially
local energy flux appears unambiguous because it arises naturally from a
divergence term in scale space. And, the symmetric formulation of Hill (2001,
2002b) leads to the spherically averaged third-order structure function-based
definition of a local cascade rate involving velocities at two points that are
treated equally via angular averaging over the sphere.

The data confirm the presence of local instances where $\Phi_{\ell}({\bf
x},t)$ is negative, i.e. indicative of local inverse cascade events in 3D
turbulence. Flow visualizations show that spatially the inverse cascade events
are often located near the core of large-scale vortex structures. Comparable
observations for the LES-based energy flux $\Pi_{\ell}({\bf x},t)$ (which also
displays negative values at many locations in the flow as is well-known in the
LES literature on “backscatter”) show that $\Pi_{\ell}({\bf x},t)$ displays
smoother and more blob-like features. Regarding the statistical significance
of such observations, local observations from single realizations are extended
using conditional averaging. Attention is placed first on relationships
between the local cascade rate and the local filtered viscous dissipation rate
$\epsilon_{\ell}({\bf x},t)$ that plays a central role in the classic KRSH
Kolmogorov (1962). Results show that conditional averaging of both
$\Phi_{\ell}({\bf x},t)$ and $\Pi_{\ell}({\bf x},t)$ eliminates negative
values and that the conditional averages in fact equate $\epsilon_{\ell}({\bf
x},t)$ to very good approximation, entirely consistent with KRSH predictions.

The analysis then focuses on conditional averages of $\Phi_{\ell}$ and
$\Pi_{\ell}$ conditioned on properties of the filtered velocity gradient
tensor properties, in particular four of its most invariants (strain and
rotation rate square magnitude and the two $Q-R$ invariants). We find
statistically robust evidence of inverse cascade as measured with
$\Phi_{\ell}$ when both the large-scale rotation rate is strong and the large-
scale strain rate is weak. When defined using $\Pi_{\ell}$, the conditional
averaging based on large-scale strain and rotation rates does not lead to any
significant average backscatter. When conditioning based on the $R$ and $Q$
invariants, significant net inverse cascading is observed for $\Phi_{\ell}$ in
the “vortex compression” $R>0$, $Q>0$ quadrant. Qualitatively similar, but
quantitatively much weaker trends are observed for the conditionally averaged
subfilter scale energy flux $\Pi_{\ell}$. We recall that a multiscale
decomposition of $\Pi_{\ell}$ in terms of velocity gradients at multiple
scales Johnson (2020, 2021) shows that $\Pi_{\ell}<0$ appears associated with
a vortex-thinning mechanism occuring at smaller scales interacting with large-
scale strains.

In summary, present results show that locally negative values of kinetic
energy fluxes at scale $\ell$ are observed for both the structure function and
filtering approaches, and at least for the structure function approach, the
interpretation as a flux in scale space appears unambiguous. Regarding
statistical robustness and potential net impact of such local observations,
conditional averaging shows that such inverse cascade becomes the
statistically dominant mechanism in regions in which the turbulent motions at
scales larger than $\ell$ are of the “vortex compression” ($R>0$ and $Q>0$)
type.

Future work should extend conditional averaging to more accurately reflect
entire flow structures and their possible connections to local inverse cascade
mechanisms. Other pointwise quantities such as helicity can also be explored.
It would also be instructive to connect present results with the multiscale
decomposition of Johnson (2020, 2021) and thus be able to identify the small-
scale mechanisms associated to local backscatter/inverse cascade events. And,
further theoretically obtained exact relations between structure function and
filtering approaches may yet be found.

## Acknowledgements

We thank Prof. Gregory Eyink for fruitful discussions and comments on this
manuscript, Dr. Miguel Encinar for valuable suggestions, and the JHTDB/IDIES
staff for their assistance with the database and its maintenance. This work is
supported by the National Science Foundation (Grant # CSSI-2103874).

## Declaration of interests

The authors report no conflict of interest.

##  Appendix A Turbulence database access tools

The high-resolution isotropic DNS data are accessible via new python-based
tools built upon the data housed in JHTDB (the Johns Hopkins Turbulence
Database, (Li et al., 2008)). JHTDB has been operating for over a decade and
has led to hundreds of peer-reviewed articles on turbulence. A new set of data
access tools based on Jupyter Notebooks has been developed that enables direct
access to subsets of the data continuing the “virtual sensors” concept Li et
al. (2008). The new notebooks provide fast and stable operation on the
existing turbulence data sets while enabling user-programmable, server-side
computations. To date, the new data access tools have been implemented and
tested on the high Reynolds number, forced isotropic turbulence data set on
$8{,}192^{3}$ grid points (the isotropic8192 datasets) of which 5 snapshots
are at a Taylor microscope Reynolds number of $Re_{\lambda}=1250$ (Yeung et
al., 2012) and one with very high spatial resolution at $Re_{\lambda}=610$.

The isotropic8192 data set has been partitioned into 4096 Zarr database files,
each of which is a $512^{3}$ volume cubelet of the $8192^{3}$ data. Each Zarr
file stores the velocity and pressure variables in distinct Zarr groups, and
the data in each group is further broken down into chunks. The chunks are ijk-
ordered such that cutouts and interpolation buckets, the size of which are
dependent on the interpolation or differentiation method selected by the user,
can be cutout directly from the intersecting chunk(s).

The new Python-based data access tools, pyturb, are accessed via the SciServer
(sciserver.org) platform. Pyturb interfaces directly with the data files in
Zarr format, stored on volumes mounted to each user’s SciServer container. The
entirety of the isotropic8192 data set ($8192^{3}$ volume, 6 snapshots) in
Zarr format is publicly available through Python Notebook on SciServer. Users
can apply for a SciServer account freely and download the demo Notebook. In
the Notebook, users can get access to pre-coded “Get” functions for arbitrary
sets of points: GetPressure to retrieve and interpolate pressures,
GetPressureGradient to retrieve and interpolate pressure gradient, and
similarly GetPressureHessian, GetVelocity, GetVelocityGradient,
GetVelocityHessian, GetVelocityLaplacian, and GetCutout to read raw data for a
user-specified box.

Demo codes for accessing data at user-specified arrays of points (in various
sample geometrical configurations) are listed in the Notebook. The
isotropic8192 datasets can be also accessed via the web-portal cutout service
where the pyturb GetCutout function has replaced the legacy function for user
queries (see https://turbulence.pha.jhu.edu/newcutout.aspx). JHTDB still
provides and maintains other datasets
(https://turbulence.pha.jhu.edu/datasets.aspx) through legacy SQL systems with
C, Fortran, Matlab, Python, and .Net interface. However, the aim is to
transfer the existing datasets and any new coming datasets to the pyturb
system in the future for faster and more stable services.

##  Appendix B Evaluating and comparing $\langle k_{{\rm sf},\ell}\rangle$ and
$\langle k_{{\rm sgs},\ell}\rangle$

The average values $\langle k_{{\rm sf},\ell}\rangle$ and $\langle k_{{\rm
sgs},\ell}\rangle$ can be obtained from classical turbulence theory and the
Kolmogorov spectrum. To evaluate $\langle k_{{\rm sf},\ell}\rangle$ we use the
general expression for the structure function tensor in isotropic turbulence
in the inertial range

$\langle\delta u_{i}({\bf r})\delta u_{j}({\bf
r})\rangle=C_{2}\,\langle\epsilon\rangle^{2/3}\,r^{2/3}\left(\frac{4}{3}\delta_{ij}-\frac{1}{3}\frac{r_{i}r_{j}}{r^{2}}.\right),$
(14)

and write

$\langle k_{{\rm
sf},\ell}\rangle=\frac{1}{2\,V_{\ell}}\iiint\limits_{V_{\ell}}\frac{1}{2}\langle\delta
u_{i}^{2}({\bf r})\rangle d^{3}{\bf
r}_{s}=\frac{1}{4}\frac{1}{V_{\ell}}\iiint\limits_{V_{\ell}}C_{2}\,\langle\epsilon\,\rangle^{2/3}\,r^{2/3}\left(\frac{4}{3}\delta_{ii}-\frac{1}{3}\frac{r_{i}r_{i}}{r^{2}}\right)d^{3}{\bm{r}}_{s}$
(15)

where ${\bf r}_{s}={\bf r}/2$. The integration yields

$\langle k_{{\rm
sf},\ell}\rangle=\frac{C_{2}\,\langle\epsilon\rangle^{2/3}\ell^{2/3}}{4\times
3\pi(\ell/2)^{3}/4}\int\limits_{0}^{\ell/2}4\pi
r_{s}^{2}\,(2r_{s})^{2/3}\,dr_{s}=\frac{3}{4}\,C_{2}\,\langle\epsilon\rangle^{2/3}\ell^{2/3}\approx
1.575\,\langle\epsilon\rangle^{2/3}\ell^{2/3}$ (16)

when using the usual empirical Kolmogorov structure function constant
$C_{2}\approx 2.1$.

In order to evaluate $\langle k_{{\rm sgs},\ell}\rangle$ we use (Pope, 2000;
Li & Meneveau, 2004)

$\langle\tau_{ij}\rangle=\iiint\left(1-\hat{G}^{2}_{\ell}({\bf
k})\right)\,\Phi_{ij}({\bf k})\,d^{3}{\bf
k}\,\,\,\,\stackrel{{\scriptstyle\mathclap{\tiny(i=j)}}}{{=}}\,\,\,\,C_{K}\langle\epsilon\rangle^{2/3}\int_{0}^{\infty}\left(1-\hat{G}^{2}_{\ell}({k})\right)\,2\,k^{-5/3}dk,$
(17)

where $\hat{G}_{\ell}({\bf k})=\hat{G}_{\ell}({k})$ is the Fourier transform
of the filter function at scale $\ell$ and $\Phi_{ij}({\bf k})=E(k)/(4\pi
k^{2})(\delta_{ij}-k_{i}k_{j}/k^{2})$ is the spectral tensor for isotropic
turbulence, while $E(k)=C_{K}\langle\epsilon\rangle^{2/3}k^{-5/3}$ is the
radial 3D energy spectrum of turbulence.

For the spherical top-hat filter, its Fourier transform can be shown to be

$\hat{G}_{\ell}({k})=\frac{3}{(k\ell/2)^{3}}\,\left(\sin\frac{k\ell}{2}-\frac{k\ell}{2}\cos\frac{k\ell}{2}\right)$
(18)

where $k=|{\bf k}|$. The definite integral needed to evaluate the RHS of Eq.
17 exists (using WolframAlpha online) and is given by

$\int\limits_{0}^{\infty}\left(1-\left[\frac{3}{(\kappa/2)^{3}}\,\left(\sin\frac{\kappa}{2}-\frac{\kappa}{2}\cos\frac{\kappa}{2}\right)\right]^{2}\right)\,\kappa^{-5/3}\,d\kappa=-544\,\,\Gamma(-20/3),$
(19)

with $\kappa=k\ell$ and where $\Gamma(..)$ is the Gamma function. Evaluating
and using $C_{K}\approx 1.6$, the result is

$\frac{1}{2}\tau_{ii}=0.76\,C_{K}\langle\epsilon\rangle^{2/3}\,\ell^{2/3}\approx
1.217\,\langle\epsilon\rangle^{2/3}\,\ell^{2/3}.$ (20)

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