# A length scale for non-local multi-scale gradient interactions in isotropic
turbulence

Miguel P. Encinar1<EMAIL_ADDRESS>1Department of Mechanical
Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA

###### Abstract

Three-dimensional turbulent flows enhance velocity gradients via strong non-
linear interactions of the rate-of-strain tensor with the vorticity vector,
and with itself.  For statistically homogeneous flows, their total
contributions to gradient production are related to each other by conservation
of mass, and so are the total enstrophy and total dissipation. However,
locally they do not obey this relation and have different (often extreme)
values, and for this reason both production mechanisms have been subject to
numerous studies, often decomposed in multiscale interactions. In general
lines, their dynamics and contributions to the cascade processes and turbulent
kinetic dissipation are different, which posses a difficulty for turbulence
modelling. In this paper, we explore the consequence of the ‘Betchov’
relations locally, and show that they implicitly define a length scale. This
length scale is found to be about three times the size of the turbulent
structures and their interactions. It is also found that while the non-
locality of the dissipation and enstrophy at a given scale comes mostly from
larger scales that do not cancel, the non-local production of strain and
vorticity comes from multiscale interactions. An important consequence of this
work is that isotropic cascade models need not distinguish between vortex
stretching and strain self-amplification, but can instead consider both
entities part of a more complex transfer mechanism, provided that their
detailed point-value is not required and a local average of reasonable size is
sufficient.

##  1 Introduction

Non-locality is an essential feature of turbulent flows. The term ‘non-local’
or ‘non-local interactions’ is used in turbulence research to refer to two
distinct phenomena, non-local interactions in physical space, and in Fourier
space. This paper focuses on the former.

The ultimate source of non-locality is the incompressibility assumption, which
forces the velocity field to be solenoidal at each instant of time and point
of space. Incompresibility acts through the pressure gradient term in the
Navier–Stokes equation and thus it can be said that the pressure is
responsible for non-locality. The only way for the velocity field to be able
to satisfy the incompressibility condition is to communicate the movement of
each fluid particle to all other particles instantly through the pressure. For
this reason it seems that the most straight-forward way of studying non-
locality is studying the characteristics of the pressure field. Indeed, some
works explore the characteristics of the isotropic pressure field directly
(Batchelor, 1953; Pumir, 1994; Tsuji & Ishihara, 2003), although often they
are not concerned about non-locality in particular, although exceptions do
exist (Wilczek & Meneveau, 2014; Vlaykov & Wilczek, 2019). Non-locality is
instead typically studied in the context of enstrophy and dissipation
amplification. These quantities evolve through strain–vorticity interactions,
which are non-local, and through the pressure Hessian in the rate-of-strain
evolution equation, which is also non-local. Both terms have been extensively
studied, particularly in the context of non-locality (Ohkitani & Kishiba,
1995; Tsinober, 2000; Wilczek & Meneveau, 2014; Elsinga et al., 2017;
Meneveau, 2011). Moreover, the equations of the filtered velocity gradients
can be used to model inter-scale interactions, and non-locality here plays an
important role, as inter-scale vortex stretching (VS) is often regarded as an
important cascade indicator (Eyink, 2005). There are also indications that
strain-self amplification (SSA) is equally (or up to three times more)
important for the cascade rate (Tsinober, 2000; Carbone & Bragg, 2020;
Johnson, 2020; Yang et al., 2023) than VS.

Both VS and SSA are known to balance on average (Betchov, 1956), but locally
can be very different from each other (Jiménez et al., 1993; Tsinober, 2000).
The same can be said for the enstrophy and the non-dimensional dissipation,
whose total magnitudes are related to each other. In principle, this allows
one to study dissipation using the field of enstrophy as a proxy and,
traditionally, research has favoured the study of the ‘simpler’ vorticity
vector over the more tedious rate-of-strain tensor.

The derivation of the evolution equations of the velocity gradients can, at
most, hide the effect of the pressure term in other kinds of non-local
interactions, e.g. VS, but cannot remove its impact. A particularly
interesting attempt to separate local from non-local VS is Hamlington et al.
(2008). This work exploits the integral transform from the vorticity to strain
to produce local and non-local fractions of the VS. The same decomposition has
been used by Buaria & Pumir (2021) to analyse very high-Reynolds number
turbulence, reaching the conclusion that most amplification comes from non-
local interactions. These works have in common that a length scale needs to be
chosen to separate local from non-local effects. Pressure is global in
incompressible flows, making this length scale nominally infinite. Thus, the
concept of defining a length scale associated to non-locality is ill-posed.
However, even if the effects of pressure need to be global, the magnitude of
those effects across distance decays significantly, due to the properties of
the Laplacian operator which appears in the equation for the pressure. This
magnitude is important because non-locality hinders our ability to understand
gradient amplification and to produce sensible cascade models. This length
scale is studied in Vlaykov & Wilczek (2019), where the authors study the
source term of the pressure fluctuations. Using correlations, they reach the
conclusion that most of the non-locality arising from this term vanishes at
distances larger than $20$ Kolmogorov units.

In the present paper, we take a similar approach to them, seeking to create a
sensible definition of non-locality in terms of velocity gradient
interactions, and to measure the scale(s) at which non-locality is most
intense, and more importantly, its scaling with the Reynolds number. Although
‘globality’ is required to fully satisfy the continuity equation, we argue
that the main dynamical effects of continuity in the velocity gradients have a
marked length scale, and that said length scale is important for modelling
flow dynamics. The main objective is to provide a sensible measure of the
distances in which non-local interactions are dominant, and when they may be
neglected. We extend this analysis to the filtered velocity-gradient tensor,
which allows us to generalise our conclusions from the dissipative range to
the inertial one.

The remainder of the paper is structured as follows. Section §2 introduces the
databases used for this paper, and §3 recalls the equations that contain the
quantities studied in this paper and their relation to Navier–Stokes. They are
followed by section §4, which shows all the results regarding gradient non-
locality. Finally, §5 closes with the final discussion and conclusions of the
manuscript.

##  2 Numerical databases

We use databases of HIT obtained from direct numerical simulation (DNS) of the
incompressible Navier–Stokes equations. Simulations use a standard fully
dealiased spectral spatial discretisation coupled with a third-order semi-
implicit Runge–Kutta time stepper. Turbulence is forced with constant energy
input at the largest wavenumber corona, $|\boldsymbol{k}|<2$, where
$|\boldsymbol{k}|$ is the wavenumber magnitude. Table 1 gathers the main
characteristics of the simulations.  The two lower $\Rey$ simulations use a
smaller resolution, $k\eta_{\max{}}=1.5$ than the _de facto_ standard of two.
Although simulations as low as $k\eta_{\max{}}=1$ have been used to gather
useful statistics (Kaneda & Ishihara, 2006, among others), high order moments,
such as the statistics of the third invariant could be affected by this
resolution. However, the good collapse of their statistics with the highest
$\Rey$ simulation, which is computed at the standard resolution gives us
confidence in the results of this paper. More details about the numerical
method and simulations are provided in Cardesa et al. (2017).

Name | $N$ | $k\eta_{\max{}}$ | $L_{E}/\eta$ | $L_{E}/\lambda$ | $\epsilon L_{E}/u^{\prime 3}$ | $Re_{\lambda}$ | $N_{\text{stat}}$ | Symbol
---|---|---|---|---|---|---|---|---
S256 | 256 | 1.5 | 119 | 5.03 | 0.524 | 144 | 640 |
S512 | 512 | 1.5 | 213 | 7.12 | 0.462 | 231 | 160 |
S1024 | 1024 | 2.0 | 333 | 9.80 | 0.494 | 298 | 40 |
Table 1: Detail of the numerical simulations used in the paper. $N$ is the
number of grid points in each of the periodic directions, $\eta$ is the
Kolmogorov scale, $L_{E}$ the integral scale, $\lambda$ the Taylor microscale,
$\epsilon$ is the total dissipation. All these quantities are computed as in
Batchelor (1953). $N_{\text{stat}}$ refers to the number of approximately
independent snapshots used to compute the statistics in this paper.

##  3 Gradient amplification and non-locality

Consider the incompressible Navier–Stokes equations,

$\displaystyle\partial_{i}u_{i}$ $\displaystyle=$ $\displaystyle 0$ (1)
$\displaystyle\mathrm{D}_{t}u_{i}\equiv\partial_{t}u_{i}+u_{j}\partial_{j}u_{i}$
$\displaystyle=$ $\displaystyle-\partial_{i}p+\nu\partial_{kk}u_{i},$ (2)

where $u_{i}$ is the velocity vector, $p$ the kinematic pressure and $\nu$ the
kinematic viscosity. $\mathrm{D}_{t}$ stands for the material derivative. The
kinematic pressure acts in (2) as a Lagrange multiplier, responsible for
enforcing (1) at each time and point in space. The only source of divergence
in (2) is the term $u_{j}\partial_{j}u_{i}$, which is the divergence of the
Reynolds-stress tensor, $\partial_{j}\left(u_{i}u_{j}\right)$. A possible
interpretation is that the Reynolds stresses generate divergence, which is
opposed and cancelled by the gradient of the pressure. However, this can be
misleading, as both terms can be grouped together in the equation, and the
non-linear term of Navier–Stokes equation is
$\partial_{j}\left(u_{i}u_{j}+\delta_{ij}p\right)$, where $\delta$ is the
Kronecker delta tensor. In this form, it can be seen explicitly that non-
locality is a consequence of non-linearity. Taking the gradient of (2) yields,

$\mathrm{D}_{t}\mathsf{A}_{ij}=-\partial_{ij}p-\mathsf{A}_{ik}\mathsf{A}_{kj}+\nu\partial_{kk}\mathsf{A}_{ij},$
(3)

where $\mathsf{A}_{ij}=\partial_{j}u_{i}$ is the velocity gradient tensor. The
latter can be further decomposed into a symmetric and skew-symmetric part,
$\mathsf{A}_{ij}=\SS_{ij}+{\Omega}_{ij}$, where the symmetric $\SS_{ij}$ is
the rate-of-strain tensor, and the skew-symmetric ${\Omega}_{ij}$ is the rate-
of-rotation tensor. The latter is related to the vorticity vector by
$\omega_{i}=-1/2\varepsilon_{ijk}{\Omega}_{jk}$, where $\boldsymbol{\omega}$
is the vorticity vector and $\varepsilon$ is the totally skew-symmetric
tensor. Evolution equations for $\SS$ and $\boldsymbol{\omega}$ can be
constructed from (3),

$\displaystyle\mathrm{D}_{t}\SS_{ij}$ $\displaystyle=$
$\displaystyle-\partial_{ij}p+\SS_{ik}\SS_{kj}-\frac{1}{4}\left(\omega_{i}\omega_{j}-\boldsymbol{\omega}^{2}\delta_{ij}\right)+\nu\partial_{kk}\SS_{ij},$
(4) $\displaystyle\mathrm{D}_{t}\omega_{i}$ $\displaystyle=$
$\displaystyle\omega_{j}\SS_{ij}+\nu\partial_{kk}\omega_{i},$ (5)

where $\boldsymbol{\omega}^{2}=\omega_{i}\omega_{i}$. Non-locality appears in
(4) through the pressure Hessian, and in (5) through the vortex-stretching
term. The pressure is hidden in the vorticity equation through the strain-
vorticity interactions, as strain and vorticity are related to each other by a
non-local integral transform (Ohkitani, 1994). Note that both equations are
non-local in the convective term too, as obtaining the velocity field from
either the vorticity or the rate-of-strain involves using the
incompressibility condition and inverting a second derivative. The trace of
(4) gives a recipe to compute the pressure,

$\partial_{ii}p=\frac{1}{2}\boldsymbol{\omega}^{2}-\SS^{2},$ (6)

where $\SS^{2}=\SS_{ij}\SS_{ij}$. The pressure source term is proportional to
the local imbalance of enstrophy and dissipation and, because for an isotropic
flow the average pressure vanishes,

$\frac{1}{2}\left\langle{\omega^{2}}\right\rangle=\left\langle\SS^{2}\right\rangle,$
(7)

where the brackets are the spatial average. Note that for an isotropic flow
with periodic or infinite boundaries the spatial average is sufficient to
satisfy (7). Equation (7) is an important non-local relation between the
global enstrophy and the total dissipation. Equations for these two magnitudes
can be obtained by contracting (4) and (5) with $\SS$ and
$\boldsymbol{\omega}$ respectively and taking the average,

$\displaystyle\partial_{t}\left\langle\SS^{2}\right\rangle$ $\displaystyle=$
$\displaystyle-\left\langle\SS_{ij}\SS_{jk}\SS_{ki}\right\rangle-\frac{1}{4}\left\langle\omega_{i}\SS_{ij}\omega_{j}\right\rangle-\nu\left\langle\SS_{ij}\partial_{kk}\SS_{ij}\right\rangle,$
(8)
$\displaystyle\partial_{t}\left\langle\boldsymbol{\omega}^{2}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\omega_{i}\SS_{ij}\omega_{j}\right\rangle-\nu\left\langle\omega_{i}\partial_{kk}\omega_{i}\right\rangle.$
(9)

Production of enstrophy is achieved through the vortex stretching acting on
the vorticity $\omega_{i}\SS_{ij}\omega_{j}$ (VS), which has to be positive on
average in order to sustain turbulence. This term is a sink for the rate-of-
strain magnitude, which in turn is produced by the term
$\SS_{ij}\SS_{jk}\SS_{ki}$ or strain self amplification (SSA), which has to be
negative on average. The expectancies of both quantities are linked by the
celebrated ‘Betchov’ relation (Betchov, 1956),

$\left\langle\SS_{ij}\SS_{jk}\SS_{ki}\right\rangle=\frac{3}{4}\left\langle\omega_{i}\SS_{ij}\omega_{j}\right\rangle.$
(10)

This relation is purely kinematical, as only incompressibility, isotropy and
the chain rule are required to prove it. As stated in the introduction,
interpretations of (10) are sometimes controversial. While some authors (eg.
Ohkitani, 1994; Eyink, 2005) efectively use to recaset the production of
(filtered) gradients solely in terms of VS, others (eg. Tsinober, 2001;
Carbone & Bragg, 2020; Johnson, 2020) question its local implications, as SSA
and VS play, in principle, different roles in turbulence dynamics.

Equations (7) and (10) are the two most important kinematic relations which
tie gradients and gradient production respectively across the flow field.
Moreover, they have been recently shown to be the only two homogeneous
kinematic constrains (Carbone & Wilczek, 2022). They can be reworked exactly
in terms of the average of the second and third invariants of the velocity
gradient tensor, $Q$ and $R$ (Chong et al., 1990),

$\displaystyle\left\langle Q\right\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left\langle\mathsf{A}_{ij}\mathsf{A}_{ji}\right\rangle=\frac{1}{4}\left\langle\omega_{i}\omega_{i}-2\SS_{ij}\SS_{ij}\right\rangle=0,$
(11) $\displaystyle\left\langle R\right\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\left\langle\mathsf{A}_{ij}\mathsf{A}_{jk}\mathsf{A}_{ki}\right\rangle=-\frac{1}{3}\left(\SS_{ij}\SS_{jk}\SS_{ki}+\frac{3}{4}\omega_{i}\SS_{ij}\omega_{j}\right)=0.$
(12)

Aside from the pressure Hessian, non-local kinematic constrains force the two
invariants of the velocity gradient tensor to vanish. The first, which is the
divergence of the velocity, is zero at each point in space due to
incompressibility, and the spatial averages of the second or third are zero
for an isotropic flow. In the following section we explore how cancellation of
$Q$ and $R$ occurs in isotropic flows, but first we recall the typical way in
which non-locality manifests in the velocity gradients.

###  3.1 Evidence of non-locality

$|{\boldsymbol{\omega}}|/{(\boldsymbol{\omega}^{\prime})}^{1/2}$
${|\mathsf{S}|}/{(\boldsymbol{\omega}^{\prime})}^{1/2}$
${\boldsymbol{\omega}}\mathsf{S}{\boldsymbol{\omega}}/{(\boldsymbol{\omega}^{\prime})}^{3/2}$
$\mathsf{S}\mathsf{S}\mathsf{S}/{(\boldsymbol{\omega}^{\prime})}^{3/2}$

Figure 1: (a) Subvolume of $(250\eta)^{3}$ of a snapshot of S256 (see table
1). Blue isosurfaces are
$|\boldsymbol{\omega}|>4(\boldsymbol{\omega}^{\prime})^{1/2}$ and yellow ones
$|\mathsf{S}|>4(\boldsymbol{\omega}^{\prime})^{1/2}$. (b) Joint (p.d.f.)s of
the enstrophy magnitude $|\boldsymbol{\omega}|$ and the dissipation magnitude
$|\mathsf{S}|$. (c) Joint (p.d.f.)s of the VS and the SSA. (b, c) Contours
contain 90, 99 and 99.9% of the probability mass.  Black contours are the
regular joint p.d.f. and grey contours are the product of the probabilities of
the individual p.d.f. of the magnitudes. Ball coarse-grained quantities in a
$12\eta$ radius are shown in blue and orange contours, the average is taken
for the blue ones and the maximum is taken for the oranges. (a,b,c)
$\boldsymbol{\omega}^{\prime}$ stands for the variance of the vorticity.

Figure 1(a) shows a 3D picture portraying non-locality in terms of the
velocity gradients. It contains a small subvolume of isotropic turbulence,
with contours for enstrophy and dissipation. The picture suggests that they
are spatially related to each other, but it can be seen that they are not
collocated. These visual inspections are quantified by the joint probability
density function (p.d.f.) of the enstrophy magnitude $|\boldsymbol{\omega}|$
and the dissipation magnitude $2|\mathsf{S}|$, both normalised by the r.m.s.
of the vorticity $(\boldsymbol{\omega}^{\prime})^{1/2}$. They are shown in
black contours in figure 1(b). Lighter grey contours contain  the p.d.f. of
both quantities if they were completely uncorrelated, obtained as the product
of their probabilities, $P(\omega)P(\SS)$. Visually, it reveals that a
correlation of $C(\omega,\SS)\approx 0.57$ between the two fields exists. It
also shows that strong events happen in different places, giving the p.d.f.
its characteristic ‘square’ shape. It has been known for some time that this
is caused by strong vortices or ‘worms’ surrounded by strong rate-of-strain
induced by them, and is often as an archetypical example of gradient non-
locality (Jiménez et al., 1993). Because vortices have a transversal length
scale $\mathcal{O}\\!\left(\eta\right)$, the strain they induce must have the
same length scale (Burgers, 1948). Thus, it is reasonable to expect that both
quantities should correlate at lengths of the same order. Blue contours show
the joint pdf of $\langle|{\boldsymbol{\omega}}|^{2}\rangle_{12\eta}^{1/2}$
and $\langle 4|{\mathsf{S}}|^{2}\rangle_{12\eta}^{1/2}$. These quantities are
the averaged squared norm of enstrophy and dissipation in spherical subvolumes
of radius $12\eta$. Both quantities correlate much better, with a correlation
of $C\approx 0.86$. Finally, the orange contours show that this property does
not only apply to events of average intensity but also to extreme ones. They
represent the joint p.d.f. of the maximum value of the variables within the
same  spheres of radius $12\eta$, and the correlation is similar to the one
for the averages.  This values are computed by taking the maximum value of
each quantity instead of taking the average value within the ball. As noted in
Buaria & Pumir (2022) there is a non-unity power-law relating the intensity of
$|\boldsymbol{\omega}|$ and $|\mathsf{S}|$, which can be shown here as a
different covariance between the blue and the orange contours.

Figure 1(c) extends this analysis to the joint p.d.f. of VS and SSA, with its
characteristic star-shape. Most of the conclusions from figure 1(b) carry over
here. Both VS and SSA can change sign and thus local averages can too.
However, the probability of finding gradients being depleted diminishes
significantly when the average is taken over volumes of $12\eta$, from 57% and
50% of the S512 domain having negative VS and positive SSA respectively, to
1.5% and 1% when
$\langle\boldsymbol{\omega}\mathsf{S}\boldsymbol{\omega}\rangle_{12\eta}$ and
$\langle\mathsf{S}\mathsf{S}\mathsf{S}\rangle_{12\eta}$ are considered.
Correlations between the two variables improves by almost the same margins as
$\langle|{\boldsymbol{\omega}}|^{2}\rangle_{12\eta}^{1/2}$ and $\langle
4|{\mathsf{S}}|^{2}\rangle_{12\eta}^{1/2}$.

While figure 1 shows that non-locality has a strong mark in turbulent velocity
gradients, it also suggests that a sensible portion of the non-local ‘Betchov’
relations cancel over small distances within the dissipative range. Thus,
cancellation implicitly defines a distance relating the components of the
invariants, which we explore in §4.

##  4 Cancellation of the velocity-gradient invariants

$r/L_{E}$ $r/L_{E}$ $r/\eta$$r/\eta$$\frac{\left\langle
R\right\rangle_{r}^{\prime}}{(\left\langle
Q\right\rangle_{0}^{\prime})^{3/2}}$$\frac{\left\langle
Q\right\rangle_{r}^{\prime}}{\left\langle Q\right\rangle_{0}^{\prime}}$(b)(a)

Figure 2: Variance of (a) $\left\langle Q\right\rangle_{r}$ and (b)
$\left\langle R\right\rangle_{r}$ as a function of the integration radius for
the three Reynolds numbers. The inset are a log-log plot of the same data, but
normalised with the integral time scale in the abscissa, and using the
surrogate dissipation instead of $\left\langle Q\right\rangle_{0}^{\prime}$ in
the ordinate. The red dashed line corresponds to the randomised test of S512.

Figure 2(a) shows $\left\langle Q\right\rangle_{r}^{\prime}$ for the three
available Reynolds numbers, which stands for the variance $(\prime)$ of the
coarse-grained $Q$ , defined as

$\left\langle Q\right\rangle_{r}^{\prime}=\frac{1}{V}\int_{V}\left\langle
Q\right\rangle_{r}^{2}~{}\mathrm{d}x_{i},$ (13)

where $V$ stands for the computational domain. Note that the mean needs not to
be substracted because $\left\langle Q\right\rangle_{r}$ vanishes when
integrated over the volume. The coarse-grain follows Eyink (2005), averaging
$Q$ over a sphere of radius $r$. When the radius goes to zero, we recover
$Q^{\prime}$, and when $r\rightarrow\infty$, $\left\langle
Q\right\rangle_{\infty}^{\prime}$ must vanish due to incompressibility. Thus,
$\left\langle Q\right\rangle_{r}^{\prime}$ is a measure of how far the local
enstrophy-dissipation balance is from the global one. Our results is very
similar to the findings of Vlaykov & Wilczek (2019), showing that most of the
cancellation occurs within spheres of $r\approx 20\eta$, which encompass a
vortex core and the strain it induces. The inset corresponds to the same
curves, but in log-log scales. They are normalised with the surrogate
dissipation $D=(u^{\prime})^{3/2}/L_{E}$, which makes their tails collapse.
The curves shown stop at $r/L_{E}\approx 1.6$ because we want to avoid the
forced integral scale. The black line is a fitted power-law that corresponds
to $p\approx r^{-3.02\pm 0.1}$, and approximately holds for the three Reynolds
numbers. The tails should be compared with the dashed red line, obtained from
randomising the relative positions of the dissipation and enstrophy before
computing $Q$. This artificial field of $Q$ still satisfies Betchov’s
equality, but assumes no spatial relation between its two components. Its
power-law is $p\approx r^{-2.0}$ which corresponds to the growth of the
surface of the integration sphere, and thus it cancels over larger length
scales. Figure 2(b) is equivalent to 2(a) but shows $\left\langle
R\right\rangle_{r}$. It is normalised with $(Q^{\prime})^{3/2}$, which results
in intermittency effects showing at $r\rightarrow 0$, even at our moderate
Reynolds numbers. These effects cancel quickly with the coarse-graining, and
$R$ roughly satisfies Betchov’s equality for the same radius of $Q$. However,
the fitted power-law is slightly different $p\approx r^{-3.5\pm 0.15}$, which
results in a faster cancellation. This figure shows that strong VS events are
always accompanied by strong SSA events in their neighbourhood, and that the
typical length scale of this interactions is of the same order of the
interactions of dissipation and strain objects.

###  4.1 Multiscale analysis of the Betchov invariants

The results so far agree with those of Vlaykov & Wilczek (2019) and extend
theirs to $R$, with similar conclusions. The cancellation of $Q$ is a
consequence of vorticity generating strain or _viceversa_ , and can be
interpreted as both entities being part of the same structure, i.e. a strong
vortex with its self-induced strain (Elsinga & Marusic, 2010). In the same
way, the fact that the same cancellation holds for the VS and the SSA implies
that the latter are paired somewhat locally, and that they probably come from
the same dynamics, which generates them at the same time. In order for this to
hold for the turbulent cascade, the same balance should hold at all inertial
scales.

$r/\rmDelta$$r/\rmDelta$$\frac{\left\langle\tilde{R}\right\rangle_{r}^{\prime}}{(\left\langle
Q\right\rangle_{0}^{\prime})^{3/2}}$$\frac{\left\langle\tilde{Q}\right\rangle_{r}^{\prime}}{\left\langle
Q\right\rangle_{0}^{\prime}}$(b)(a)

Figure 3: Variance of (a) $\left\langle\tilde{Q}\right\rangle_{r}$ and (b)
$\left\langle\tilde{R}\right\rangle_{r}$ as a function of the integration
radius for the three Reynolds numbers and various filters. See §4.1 for more
details.

This is explored in figure 3, which shows spherical averages similar to figure
2, but for the invariants of the filtered velocity fields. First, the velocity
fields are filtered using a family of isotropic Gaussian kernels, with their
widths separated in powers of two, and ranging from $10\eta$ to the integral
scale,

$\tilde{u}_{i}(x_{j})={\color[rgb]{0,0,0}\frac{1}{(\Delta\sqrt{\pi})^{3}}}\int_{V}\exp[-((x_{j}-\xi_{j})/\Delta)^{2}]u_{i}(\xi_{j})~{}\mathrm{d}\xi_{j},$
(14)

where $\Delta$ serves as the filter width. The invariants of the filter
velocity, $\tilde{Q}$ and $\tilde{R}$ represent the same incompressibility
balances (and topology), but for the filtered velocity (Leung et al., 2012;
Lozano-Durán et al., 2016; Eyink, 2005; Danish & Meneveau, 2018). We repeat
the coarse-graining of integrating both quantities in growing spheres of
radius $r$, until we reach the integral scale. Although the fact that we are
coarse-graining the fields twice may be striking, it should be clarified that
both operations have different purposes. Filtering the velocity is an attempt
to study the gradient interactions at the scale of the filter, which are
otherwise hidden by the smaller scale gradients. Coarse-graining the
invariants (which are non-linear functions of the velocity field) pursues the
study of how local is the balance between the components that form the
invariants. Figure 3(a) shows $\left\langle\tilde{Q}\right\rangle_{r}$ for
every filter width that satisfies $30\eta<\rmDelta<0.5L_{E}$, and it is
considered to fall within the inertial range. These are five, six and seven
filters for S256, S512 and S1024 respectively. They are normalised with
$\tilde{Q}^{\prime}$. When the radius $r$ is normalised with $\rmDelta$, all
the curves collapse very well. At $r/\rmDelta\approx 3$, more than 95% of the
cancellation has taken place. Similar numbers are given by figure 3(b), which
shows $\left\langle\tilde{R}\right\rangle_{r}$. Again, it shows intermittentcy
effects, with the ratio of $\tilde{R}^{\prime}$ over
$(\tilde{Q}^{\prime})^{3/2}$ being larger the farther the filter width is from
the integral scale. Despite these effects, most of the cancellation still
happens at distances smaller than three filter widths.

The fact that the cancellation or ‘Betchov’ length scale is the same for both
invariants may mislead the reader to think that they occur within the same
spheres, i.e. that they are part of the same ‘structure’. However, we found
that the cancellation of $\left\langle Q\right\rangle$ and $\left\langle
R\right\rangle$ is not collocated, meaning that the fact that enstrophy and
dissipation are in local equilibrium does not imply that VS and SSA are, and
_viceversa_. Figure 4 explores their relation by showing the p.d.f. of the
minimum distance across local minima of $\langle Q\rangle_{r/\eta=20}$ and
$\langle R\rangle_{r/\eta=20}$. They are computed as follows. First, the local
minima are identified by looking for zeroes of the gradient of the $\langle
Q\rangle_{r/\eta=20}^{2}$ and $\langle R\rangle_{r/\eta=20}^{2}$ fields (Note
that the mean of these fields are the variances as defined in (13)). Then, for
each of the two fields, we retain the minima which are closest to zero in
magnitude (closer to local equilibrium), until the density of retained minima
in the field is a constant. The latter is such that the average distance
between minima, or to the minima of the other quantity is $20\eta$, which is
the radius of the coarse-graining. Changing this density to twice or half its
value does not change the results, as long as they are renormalised with the
new density.  These sets of points represent the centres of regions where
either $Q$ or $R$ are in local equilibrium. The minimum distance between these
minima gives us information about the organisation of the regions where the
Betchov relation is satisfied locally for each quantity. The random Poisson
model  for the distance between randomised points is shown as a dashed line,
which would match the data if no relation existed between the set of local
minima of $\langle Q\rangle_{r/\eta=20}$ and the local minima of $\langle
R\rangle_{r/\eta=20}$. The data however does not fit a Poisson process, and
instead shows a much larger tail to the right, falling from a flat plateau in
the range of one to three times the coarse-graining length. The expectancy of
the distance is 1.63$~{}r$, and it is consistent for the three Reynolds
numbers. This organisation shows that the equilibrium of both quantities is
related to each other, and that they tend to be further apart than a random
process.  Although physically unsound, the reader may find useful to think of
both sets of points as if they were repelling or avoiding each other. If we
interpret the flow in terms of the evolution equations for the gradients, then
dissipation and enstrophy represent structures or eddies, and VS and SSA their
interactions. The fact that they are not collocated highlights that
interactions of multiple ‘$Q$’ objects are require to produce ‘$R$’ ones.

$|\boldsymbol{d}|/(20\eta)$$\rmDelta/\eta$$p.d.f.$$C_{i}$(b)(a)$C_{R}$$C_{Q}$

Figure 4: (a) P.d.f. of the distance between the minima of $\langle
Q\rangle|_{\Delta/\eta=30}$ and $\langle R\rangle|_{\Delta/\eta=30}$. (b)
Correlations between the filtered and coarse-grained velocity-gradient
invariances. Lines with circles are for the second invariant and dashed lines
with squares are for the third.

Consider now figure 4(a). It shows the correlation between the filtered
invariants and the coarse-grained ones,

$C_{Q}(\rmDelta)=\dfrac{\int_{V}\tilde{Q}(x_{i})\left\langle
Q\right\rangle_{\rmDelta}(x_{i})~{}\mathrm{d}x_{i}}{\left(\int_{V}\left\langle
Q\right\rangle_{\rmDelta}(x_{i})\left\langle
Q\right\rangle_{\rmDelta}(x_{i})~{}\mathrm{d}x_{i}\int_{V}\tilde{Q}(x_{i})\tilde{Q}(x_{i})~{}\mathrm{d}x_{i}\right)^{1/2}},$
(15)

and equivalently for $R$. The filter width of the Gaussian is chosen to match
$\rmDelta$, so both quantities have comparable scales. These two quantities,
$\tilde{Q}$ and $\left\langle Q\right\rangle_{\Delta}$, although they may seem
similar, have very different meanings. While $\tilde{Q}(x_{i};\rmDelta)$
represents the local balance of dissipation and enstrophy for scales larger
than $\Delta$ at the position $x_{i}$, $\left\langle
Q\right\rangle(x_{i};\rmDelta)$ represents the excess of $Q$ needed to be
cancelled in order to satisfy the kinematical relation locally, after scales
below $\rmDelta$ have been cancelled. This is not an effect of the difference
between the coarse-graining and the filtering kernel, but one that comes from
the step at which this operation is performed. While $\left\langle
Q\right\rangle_{\Delta}(x_{i};\rmDelta)=-1/2\left\langle\mathsf{A}_{ij}\mathsf{A}_{ji}\right\rangle_{\Delta}$
is constructed from the total velocity field,
$\tilde{Q}=-1/2\tilde{}\mathsf{A}_{ij}\tilde{}\mathsf{A}_{ji}$ comes from the
contraction of the filtered velocity gradient tensor. The correlations are
shown to collapse as a function of $\rmDelta/\eta$, except for the scales
$\rmDelta/L_{E}\sim 1$ which do so when normalised with the integral scale.
These large scales show approximately a constant correlation coefficient for
all three Reynolds numbers and approximately equal to $0.75$. For $Q$, the
correlation is approximately unity for scales smaller than $10\eta$, and
decays smoothly into the large scales. The fact that this correlation
coefficient is large implies that most of the structure of $\left\langle
Q\right\rangle$ comes from $\tilde{Q}$. This can be interpreted as the
enstrophy-dissipation balance at scales larger than the filter being most
responsible for the lack of local kinematic balance. It suggest a picture,
when paired with figure 3(a), in which at each scale, local balance is
approximately satisfied within a few filter widths, and the excess comes from
the scales above the filter.

A similar analysis can be done for $\tilde{R}$ and $\left\langle
R\right\rangle$, and it is shown using dashed lines. The correlation decays
far from the dissipative range, and the fields of both quantities are
essentially uncorrelated at the integral scale. This result shows that the
self-scale interactions at larger scales cannot be the main contribution to
the balance of gradient amplification and that multiscale contributions, i.e.
amplification of gradients at one scale due to the gradients at a different
one, are the most important contribution, in agreement with Buaria & Pumir
(2021). Multi-scale interactions are part of the turbulent cascade, which can
be expressed in terms of the filtered gradient production terms (Johnson,
2020). Although evaluating those terms are out of the scope of the present
paper, they can be compared with the excess of Betchov’s balance using the
multiscale procedure of Yang et al. (2023), for example.

##  5 Discussion and Conclusions

We have shown that most of the kinematical balance between the enstrophy and
the dissipation, and between the VS and the SSA is satisfied within spheres of
radius of the order of $20\eta$. For their counterparts computed from the
filter velocities, cancellation is achieved at lengths of the order of three
filter widths, for as long as the filter width lies within the inertial range.
The remaining imbalance correlates well with the filtered source in the case
of the second invariant but notably worse for the third.

If the filter-width is considered representative of the size of the structures
captured by the filter, then the ‘Betchov’ length scale is about three times
the size of an ‘eddy’, both for the second and the third invariants. The
former can be interpreted as representative of turbulent structures comprised
of vorticity surrounded by strain, balancing in three times the typical width
of either the vorticity or the strain structures. The latter represents the
interaction of vorticity and strain structures, resulting in the production of
both VS and SSA, which also balance within three times their size. The same
arguments hold for the unfiltered invariants, as $7$–$10\eta$ is the typical
size of vortices and strain sheets.

The conclusion is that for isotropic flows, or isotropic cascade models, the
assumption that dissipation can be replaced by enstrophy, or strain-self
amplification by vortex stretching (e.g (Eyink, 2005)) is physically
reasonable, even if dynamically they are completely different quantities. This
statement only holds if the point value of the components is not considered
dynamically relevant and only its effect in a small neighbourhood is
considered. The present results suggest that VS and SSA should be modelled or
studied as part of a larger dynamical mechanism ‘the vortex-strain
amplification’ which, while necessarily not local, it is approximately
balanced in a small neighbourhood of a reasonable size $r/\rmDelta\approx 3$.
This idea is also supported by the fact that their proportion can be
attributed solely to kinematical relations (Yang et al., 2023). A sensible
model compatible with the successful ideas of the multiplicative cascades is
that enstrophy-dissipation objects, i.e. vortices and their induced strain,
interact with each other; generating VS-SSA mechanisms at various scales. This
is supported by two findings. First, the fact that the local balance of $Q$ is
not collocated with the local balance of $R$, and that their distribution is
not independent from each other. It is also compatible with the power-laws of
figure 2. Second, the present results suggest that most of the excess
imbalance of enstrophy and dissipation comes from larger scales. However, that
is not true for the VS-SSA pair, which implies that the imbalance must come
from interscale interactions. Additional work should clarify whether, for
example, these interactions happen in a neighbourhood in scale space (Cardesa
et al., 2017), or are fully global in this sense.

Finally, we would like to note that while it is perfectly possible to measure
the contribution of both VS and SSA to the turbulent cascade, it seems of less
dynamical relevance than their combined contributions acting over a broader
region in isotropic flows. This distinction, however, may be necessary to
model non-isotropic flows under strong shear, where the balance may not be
satisfied at the length scales shown here.

[Acknowledgments]The author would like to thank Dr. Alberto Vela-Martín and
Beatriz Jímenez-Carrasco for their useful comments in a preliminary version of
this manuscript, and the three anonymous referees who helped polish this paper
to its final version. I would also like to thank Prof. J. Jiménez for
providing the computational resources used for this work. [Funding]The author
received no specific grant from any funding agency. The computational
resources have been provided by the ERC.2020.AdG.101018287, “CausT”
[Declaration of interest]The author reports no conflict of interest.

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