# Measuring Scale-dependent Shape Anisotropy by Coarse-Graining:
Application to Inhomogeneous Rayleigh-Taylor Turbulence

Dongxiao Zhao<EMAIL_ADDRESS>Department of Mechanical Engineering,
University of Rochester  School of Naval Architecture, Ocean and Civil
Engineering, Shanghai Jiao Tong University  Hussein Aluie  Department of
Mechanical Engineering, University of Rochester  Laboratory for Laser
Energetics, University of Rochester

###### Abstract

We generalize the ‘filtering spectrum’ (Sadek18, ) to probe scales along
different directions by spatial coarse-graining. This multi-dimensional
filtering spectrum quantifies the spectral content of flows that are not
necessarily homogeneous. From multi-dimensional spectral information, we
propose a simple metric for shape anisotropy at various scales. The method is
applied to simulations of 2D and 3D Rayleigh-Taylor (RT) turbulence, which is
inhomogeneous and anisotropic. We show that 3D RT has clear shape anisotropy
at large scales with approximately $4:3$ vertical to horizontal aspect ratio,
but tends toward isotropy at small scales as expected Chertkov03 ; Soulard12 ;
Zhou17-2 . In sharp contrast, we find that RT in 2D simulations, which are
still the main modeling framework for many applications, is isotropic at large
scales and its shape anisotropy increases at smaller scales where structures
tend to be horizontally elongated. While this may be surprising, it is
consistent with recent results in (Zhao22JFM, ); large-scale isotropy in 2D RT
is due to the generation of a large-scale overturning circulation via an
upscale cascade, while small scale anisotropy is due to the stable
stratification resultant from such overturning and the inefficient mixing in
2D.

##  I Introduction

Flows encountered in nature or engineering are often anisotropic. Anisotropy
can arise from the driving mechanisms, boundary conditions, or body forces
that break rotational symmetry such as those due to gravity, rotation, or
magnetic fields. Examples range from the motion of polymer fluids
graham2011fluid ; benzi2018polymers  and quantum superfluids Biferale19PRL ;
Yui20PRL , to flows in geophysics Manning86AAP ; Grechko92ASR ; Toselli11OSA ;
Cui15OSA ; Pouquet19POF  and astrophysics Shebalin83JPP ; Montgomery95APJ ;
Cho00APJ ; Horbury08PRL , including free shear flows such as jets and plumes
list1982turbulent , and bounded flows such as in a channel Antonia91JFM ;
Rasam11JOT .

In this work, we are concerned with _shape anisotropy_ and not with anisotropy
due to the vector components of a flow’s velocity. To distinguish the two
notions of anisotropy, we shall call the latter _vector anisotropy_. In the
turbulence literature, these two notions of anisotropy are related to so-
called _directional_ and _polarization_ anisotropy, respectively
Cambonetal97JFM ; Cambon01EJMB ; GodeferdChantal03JFM ; cambon2006anisotropic
; Burlotetal15POF . We avoid this terminology since it is suggestive of wave
phenomena. Here, we are motivated by more general flow structures that may be
spatially localized. Shape anisotropy and vector anisotropy are often
correlated, although the former can arise in scalar fields such as density or
temperature. Shape anisotropy of the velocity field may be diagnosed for each
velocity component separately or by analyzing the flow’s kinetic energy (KE)
as we do below.

In canonical turbulence that may be described by Kolmogorov’s theory
Kolmogorov41 , any anisotropy present at the large scales is expected to
decrease at smaller scales at sufficiently high Reynolds numbers, where the
flow is statistically isotropic Pope2001 . However, this is not always the
case, such as in magnetohydrodynamic turbulence
goldreich1997magnetohydrodynamic ; Boldyrev05APJL ; Bian19 ;
schekochihin2022mhd  where anisotropy is expected to be more pronounced at
smaller scales. For the objective testing of turbulence phenomenologies, and
for understanding and modeling complex flows such as those mentioned above,
the community has relied on a variety of metrics that quantify anisotropy at
different scales.

The choice of anisotropy metrics depends on the scales of interest. Bulk
turbulence (single-point) anisotropy is traditionally quantified using
invariants of the Reynolds stress tensor Lumley77JFM ; Livescu09 ; Zhou17-2 ;
Pouquet19POF . At the smallest scales, anisotropy is often measured from the
components of gradients Thiesset13JFM ; Carter17JFM ; Garanaik18POF . At
intermediate (possibly inertial) scales, anisotropy is often characterized
using two-point statistics such as autocorrelation functions, Fourier spectra,
or 2nd-order structure functions yeung1991response ; AradLvov99PRE ;
BiferaleProcaccia ; KurienSreenivasan00PRE ; BiferaleToschi01PRL ;
ishihara2002anisotropic ; Shen02POF ; cambon2006anisotropic ; Casciola07POF ;
burlot2015large ; Elsinga16POF ; Carter17JFM ; soulard2017influence . The
power-law scaling of wavelet spectra has also been used to gauge anisotropy
horbury2008anisotropic ; wicks2011anisotropy . Our motivation here is to
quantify anisotropy over all scales and in flows that may be inhomogeneous.

Fourier analysis is fraught with complications when applied to inhomogeneous
fields Sadek18 . Afterall, Fourier modes are not an eigenbasis for arbitrary
domains and boundary conditions champeney1987handbook ; krantz2019panorama .
Measuring the spectrum via a Fourier transform of the auto-correlation
function, sometimes known as the Wiener-Khinchin theorem
(champeney1987handbook, ), is not justified in the presence of boundaries or
if the field is statistically inhomogeneous such as with a spatially varying
mean or autocorrelation. In practical applications, Fourier analysis of
inhomogeneous fields (or non-stationary temporal signals) is often performed
oppenheim1999signalprocessing ; ThomsonEmery01  after removing the ensemble-
mean soulard2012inertial , detrending Savage+2017 ; ORourke2018 , and/or
tapering (i.e. windowing) Scott2005 ; khatri2018surface . Doing so removes
potentially important components of the dynamics. An emblematic example is the
global oceanic circulation, for which it had been asserted since the advent of
global satellite altimetry in the 1990s that its wavenumber spectrum’s peak is
at scales $O(100)~{}$km based on detrended and windowed Fourier analysis (e.g.
FerrariWunsch09 ; Torresetal2018jgr ; Kleinetal2019 ). It was recently shown
Storer2022NatComm ; buzzicotti2023spatio  that this is untrue and that the
spectral peak is in fact at $O(10^{4})~{}$km. The spectral peak and the
existence of a power-law scaling over scales $>10^{3}~{}$km in the oceanic
circulation could not have been detected from windowed Fourier analysis
because all scales larger than the window size (typically taken to be a few
hundred kilometers to avoid continental boundaries and curvature effects) are
implicitly removed. These limitations of Fourier analysis exist for many
realistic flows, including the Rayleigh-Taylor flows we consider here.

When removing the mean flow before using Fourier analysis, a starting
assumption is that identifying the statistical ensemble in an application is
well-posed. However, the choice of an ensemble is seldom unique and can have
important implications on the conclusions as discussed in the case of thermal
convection by Kraichnan kraichnan1964direct . For Earth systems applications,
such in the case of oceanic or atmospheric observations, we only have a single
realization. Appealing to ergodicity with respect to time to treat the
temporal record as an ensemble is formally unjustified (although it is often
employed) due to a lack of statistical stationarity: the system is variable
over a wide range of time-scales. Even if we overlook the difficulties in
choosing an ensemble, and if fluctuations about the ensemble-mean flow were
assumed to be statistically homogeneous, scales of the ensemble-mean flow
itself would remain inaccessible grea2013rapid . Often, there is a lack of
scale separation between the mean and fluctuating flow components. In such
systems, the ensemble-mean flow has significant spatial (or temporal)
variations at the same scales populated by the fluctuating field
buzzicotti2023spatio . This necessitates a self-consistent scale analysis of
both the mean and fluctuating contributions to the full non-linear dynamics.

Another tool for analyzing scales is the 2nd-order structure function. It has
been a valuable phenomenological tool in turbulence theory, but it requires
statistical averaging and is not a formal scale decomposition of a field
Frisch95 . Unlike a spectrum, which when integrated yields total energy
(Parseval’s relation), a 2nd-order structure function, $S_{2}(r)$, yields
total energy in the limit111While the sum of the 2nd-order structure function
and the autocorrelation yields total energy, the sum lacks scale information
and is not a scale decomposition. If Fourier analysis is justified, such as
for homogeneous flows, it is possible to relate the 2nd-order structure
function, $S_{2}(r)$, to the Fourier spectrum, $E(k)$, but this follows
directly from the Wiener-Khinchin relation and involves a weighted average of
$E(k)$ over the _entire_ $k$-space,
$S_{2}(r)=2\int_{0}^{\infty}dk\,(1-cos(k\,r))E(k)$ (e.g. babiano1985structure
; Pope2001 ). of length-scales $r\to 0$ (e.g. balwada2022direct ). At any
scale $r$, $S_{2}(r)$ can have significant contributions from all scales
larger or smaller than $r$. This is unsurprising since for a field such as
velocity $u(x)$, $S_{2}(r)=\langle|\delta u(x;r)|^{2}\rangle$ at scale $r$ is
constructed from increments $\delta u(x;r)=u(x+r)-u(x)$ of separation $r$
before spatial averaging, $\langle\dots\rangle$. Increments $\delta u(r)$ can
have contributions from all scales larger or smaller than $r$ depending on the
regularity (or smoothness) of the field $u(x)$ eyink2005locality ; Eyink09 ;
Aluie17  (see discussion following eq. (4) in aluie2010scale ). It is known
that the power-law scaling of a 2nd-order structure function, $S_{2}(r)\sim
r^{\alpha}$, is related to that of the Fourier spectrum, $E(k)\sim
k^{-\alpha-1}$, but only if $\alpha<2$, i.e. the scaling relation breaks down
if $E(k)$ is steeper than $k^{-3}$ as a function of wavenumber $k$ (e.g.
babiano1985structure ; biferale2001inverse ). Perhaps less well-known is that
the scaling relation between $S_{2}(r)$ and $E(k)$ also breaks down when
$\alpha<0$, i.e. $E(k)$ is shallower than $k^{-1}$ Eyink95 ; eyink2005locality
; aluie2010scale . Figs. A10-A11 in the Appendix provide a demonstration of
these facts. Another obvious limitation, shared with Fourier analysis, is that
structure functions do not provide spatial information about various scales.

Of relevance to results herein are KE spectra of variable density (VD) flows
livescu2008variable ; Livescu20 , characterized by significant variations in
mass density $\rho$. When using Fourier analysis, structure functions, or
wavelets to analyze KE, $\rho|{\bf u}|^{2}/2$, as a function of scale, KE is
often treated as quadratic to ensure the spectrum (i.e. power spectral
density) is positive. For example, the Fourier transform of an auxiliary field
${\bf w}\equiv\sqrt{\rho}{\bf u}$ is performed and then squared such that the
spectrum is positive and integrates to $\rho|{\bf u}|^{2}/2$ kida1990energy ;
CookZhou02 . This approach, while being mathematically sound on its own,
creates difficulties when analyzing the scale-dependent equations governing
such KE spectra as shown in zhao2018inviscid .

Recently, the so-called ‘filtering spectrum’ was proposed Sadek18  to measure
spectral content using straightforward coarse-graining in physical space,
which is closely related to the continuous wavelet transform daubechies1992ten
; perrier1995wavelet . This permits its application to inhomogeneous flows
with complex boundaries and allows us to probe scales of both the mean and
fluctuating fields concurrently buzzicotti2023spatio . The approach has been
recently adopted to measure the first global energy spectrum of the oceanic
general circulation Storer2022NatComm .

The filtering spectrum can be regarded as a generalization of the Fourier
spectrum to inhomogeneous fields. The filtering spectrum is an energy-
preserving scale decomposition Sadek18  and can represent the non-quadratic KE
content at different scales of VD flows as shown in Zhao22JFM . If the
filtering kernel has a sufficient number of vanishing moments, the filtering
spectrum follows any power-law scaling that the Fourier spectrum may have
(assuming Fourier analysis is possible). In fact, the filtering spectrum
converges to the Fourier spectrum when using a kernel with an infinite number
of vanishing moments (e.g. the Dirichlet kernel), which is justified only for
homogeneous fields given the highly non-local nature of such kernels in
$x$-space (Fig. A1 in Appendix).

In the appropriate domain, Fourier basis functions satisfy orthogonality,
which is a highly prized property. However, the sum of any finite number of
these basis functions222An equivalent statement can be made when Fourier modes
are continuous. suffers from significant spatial non-locality333Non-locality
of the Dirichlet kernel in $x$-space is a consequence of non-smoothness of its
Fourier transform (the sharp-spectral cutoff) in $k$-space, which is a
consequence of what is sometimes known as the Paley–Wiener theorem
sogge2017fourier ; krantz2019panorama .. Such spatial non-locality is not a
concern when working with homogeneous fields, but presents significant
challenges when working with inhomogeneous fields such as in the presence of
boundaries (see Fig. A1 and associated discussion in Appendix). What has been
shown in previous work Sadek18 ; Raietal2021 ; Zhao22JFM ; Storer2022NatComm ;
buzzicotti2023spatio  is the possibility of performing a meaningful scale
decomposition of inhomogeneous fields and determine their spectra, satisfying
both positive semi-definiteness and energy conservation, without the need for
orthogonality.

The filtering spectrum as a method is especially valuable in permitting us to
visualize (in physical space) the flow at different scales in a self-
consistent manner buzzicotti2021 , along with any associated anisotropy as we
shall see below. A disadvantage of the filtering spectrum compared to the
Fourier spectrum is that it involves smoothing as a function of scale Sadek18
. This is the price paid for gaining spatially local information at different
scales and generalizing the notion of a spectrum to non-homogeneous fields.
Concurrently exact spatial and scale localization is forbidden by the
uncertainty principle sogge2017fourier ; krantz2019panorama .

The following section II is a brief review of the 1D filtering spectrum
proposed by Sadek18 . In section III, we generalize the filtering spectrum to
multiple dimensions and define simple metrics to quantify scale anisotropy. In
section IV, we demonstrate our approach using illustrative examples and then
apply it to anisotropic inhomogeneous turbulence generated from the Rayleigh-
Taylor (RT) instability in 2D and in 3D. The paper closes with a brief summary
and practical comments about the approach’s usage.

##  II One-dimensional filtering spectrum

For any field ${\bf a}({\bf x})$, a coarse-grained or (low-pass) filtered
version of this field, which contains spatial variations at scales $>\ell$, is
defined in $n$-dimensional Euclidean space444Coarse-graining on curved
manifolds is more involved aluie2019convolutions . as Leonard75 ; Germano92 ;
MeneveauKatz00 ; eyink2005locality

$\overline{\bf a}_{\ell}({\bf x})=\int\mathrm{d}^{n}{\bf r}~{}G_{\ell}({\bf
x}-{\bf r})\,{\bf a}({\bf r}).$ (1)

Kernel $G_{\ell}({\bf r})=\ell^{-n}G({\bf r}/\ell)$ is the dilated version of
the “parent kernel” $G({\bf r})$, which is normalized. $G_{\ell}({\bf r})$ has
its main support over a region of diameter $\ell$. Operation (1) may be
interpreted as a local space average in a region of size $\ell$ centered at
point ${\bf x}$. It is, therefore, a scale decomposition performed in x-space
that partitions length scales in the system into large ($\gtrsim\ell$),
captured by $\overline{\bf a}_{\ell}$, and small ($\lesssim\ell$), captured by
the residual

${\bf a}^{\prime}_{\ell}={\bf a}-\overline{\bf a}_{\ell}.$ (2)

We assume $G_{\ell}$ is an even function such that, $\int{\bf
r}\,G_{\ell}({\bf r})\,\mathrm{d}^{n}{\bf r}=0$, which ensures that local
averaging is symmetric and operation (1) can be rewritten as

$\overline{\bf a}_{\ell}({\bf x})=\int\mathrm{d}^{n}{\bf r}~{}G_{\ell}({\bf
r})\,{\bf a}({\bf x}+{\bf r}).$ (3)

Coarse-graining is a very general scale-analysis framework and includes
Fourier analysis (e.g. Krantz99 ; Aluie09 ) and wavelet analysis (e.g.
meneveau1991analysis ; meneveau1991dual ) as special cases with the
appropriate choice of kernel $G_{\ell}$. See Sadek18  for further discussion.
Coarse-graining also lays the foundational framework of large-eddy simulation
(LES) (Leonard75, ; Germano92, ; MeneveauKatz00, ; GhateLeLe20JFM, ;
Johnson22JFM, ). For inhomogeneous flows such as in the presence of
boundaries, it is traditional in LES to filter only along the homogeneous
directions (e.g. Piomellietal91 ; yang2015integral ). This is to prevent
commutation errors that arise when deforming the kernel to avoid the boundary
SagautLES . However, at least for the purpose of diagnosing scales, it was
shown recently that these considerations are moot if regions beyond boundaries
are treated in a manner that is consistent with the boundary conditions
satisfied by the dynamics zhao2018inviscid ; Aluieetal2018_jpo ; rai2021scale
; buzzicotti2023spatio . We will discuss this point in greater detail when
applying the method to data from Rayleigh-Taylor simulations.

Since spatial coarse-graining allows for extracting spatial information at
varying length scales $\ell$, it can be used for calculating the one-
dimensional filtering spectrum Sadek18 . As an example, the filtering spectrum
for the velocity field ${\bf u}$ is

$\displaystyle\overline{E}(k_{\ell})=\frac{d}{dk_{\ell}}\left\\{\frac{|\overline{{\bf
u}}_{\ell}({\bf x})|^{2}}{2}\right\\},$ (4)

where $k_{\ell}=L/\ell$ is a ‘filtering wavenumber’, $L$ is a characteristic
length-scale (e.g. domain size), $\ell$ is the scale being probed, and
$\\{\dots\\}\equiv(\mbox{Volume})^{-1}\int(\dots)d^{n}{\bf r}$ is spatial
averaging in $n$-dimensions. Eq. (4) measures the energy density (per
wavenumber) at scale $\ell$ by varying it and probing the associated
variations in coarse KE, $\\{|\overline{{\bf u}}_{\ell}({\bf x})|^{2}\\}/2$,
which is the cumulative spectrum at _all_ scales larger than $\ell$. The main
advantage of this method is that it does not rely on Fourier transforms and,
therefore, can be easily applied to non-periodic or homogeneous data. In a
periodic domain, Fourier and filtering spectra agree if $G_{\ell}$ has
sufficient vanishing moments. In fact, the two spectra have an explicit
relationship expressed by eq. (16) in Sadek18 . Another advantage to the
filtering spectrum is its ease of generalization to VD flows Sadek18 ;
Zhao22JFM  where KE, $\rho|{\bf u}|^{2}/2$, is non-quadratic due to variations
in the mass density field, $\rho$. For VD flows, such as Rayleigh-Taylor flows
considered in this paper, the KE filtering spectrum is Sadek18

$\displaystyle\overline{E}(k_{\ell})\equiv\frac{d}{dk_{\ell}}\left\\{\frac{1}{2}\frac{|\overline{\rho{\bf
u}}_{\ell}({\bf x})|^{2}}{\overline{\rho}_{\ell}}\right\\},$ (5)

which reduces to eq. (4) when $\rho$ is constant. Energy spectra based on
Fourier and wavelet transforms, on the other hand, are constrained to
quadratic quantities to ensure that the spectrum is positive while also
satisfying Parseval’s relation. Therefore, a Fourier or wavelet decomposition
usually treats kinetic energy as quadratic by transforming the auxillary field
$\sqrt{\rho}{\bf u}$, even when this treatment obscures the inertial-range
dynamics (zhao2018inviscid, ). In contrast, the filtering spectrum conserves
energy due to the fundamental theorem of calculus and is not constrained by
Parseval’s relation. Figure 5 in reference Zhao22JFM  shows the density,
velocity, and kinetic energy filtering spectra of 2D and 3D RT turbulence
obtained with this new approach.

##  III Multi-dimensional filtering spectra and scale anisotropy

While the filtering spectrum in eqs. (4),(5) can be applied to anisotropic
fields as was done in Zhao22JFM , it is a one-dimensional spectrum since it
does not distinguish between scales along different directions. The
contribution from the most energetic scale direction dominates the value of
$\overline{E}(k_{\ell})$. In order to distinguish the spectral content in
different directions, we generalize the above definition to multiple
dimensions by using anisotropic kernels.

This is most easily done when considering multi-dimensional filtering kernels
that are separable. For example in 3D with Cartesian coordinates,

$G_{\ell}({\bf r})=G_{\ell}(r_{x})\,G_{\ell}(r_{y})\,G_{\ell}(r_{z}),$ (6)

where ${\bf r}=(r_{x},r_{y},r_{z})$, which is the case for a Gaussian or
Boxcar kernel. For separable kernels, isotropic filtering in eq. (3) reduces
to filtering in each direction separately:

$\overline{{\bf u}}_{\ell}({\bf x})=\int d{\bf r}\,G_{\ell}({\bf r})\ {\bf
u}({\bf x}+{\bf r})=\int dr_{x}\,G_{\ell}(r_{x})\int
dr_{y}\,G_{\ell}(r_{y})\int dr_{z}\,G_{\ell}(r_{z})\ {\bf u}({\bf x}+{\bf
r})~{}.$ (7)

Therefore, a natural generalization to anisotropic filtering is

$\overline{{\bf u}}_{{\vec{\ell}}}\,({\bf x})=\int
dr_{x}\,G_{\ell_{x}}(r_{x})\int dr_{y}\,G_{\ell_{y}}(r_{y})\int
dr_{z}\,G_{\ell_{z}}(r_{z})\ {\bf u}({\bf x}+{\bf r})~{}.$ (8)

This definition makes it clear how we can probe length scales along different
directions by utilizing a vector of filtering scales,
${\vec{\ell}}=(\ell_{x},\ell_{y},\ell_{z})$. The “cumulative spectrum”

$\displaystyle\mathcal{E}({\bf
k})\equiv\frac{1}{2}\left\\{\frac{|\overline{\rho{\bf
u}}_{{\vec{\ell}}}|^{2}}{\overline{\rho}_{{\vec{\ell}}}}\right\\}$ (9)

at filtering wavevector

${\bf k}=(k_{x},k_{y},k_{z})\equiv(L/\ell_{x},L/\ell_{y},L/\ell_{z})$ (10)

yields the energy content at all scales larger than $\ell_{i}$ in the $i$th-
direction. Here, $L$ is a reference scale common to all directions. Fig. 1(a)
illustrates the cumulative spectrum in 2-dimensions,
$\mathcal{E}(k_{1},k_{2})$, which is shaded in blue and accounts for all
energy at filtering wavevectors $|k_{x}|<k_{1}$ and $|k_{y}|<k_{2}$. Note that
${\bf k}$ in eq. (10) occupies only the first quadrant, $k_{x},k_{y},k_{z}\geq
0$. This is because $\mathcal{E}({\bf k})$ in eq. (9) is invariant under rigid
(Euclidean isometry) transformations of the fields $\rho({\bf x})$ and ${\bf
u}({\bf x})$, which involve a sequence of rotations, translations, and
reflections. Phase information, which may seem lost in the definition of
$\mathcal{E}({\bf k})$, is easily retrieved from the spatial field
$\frac{1}{2}|\overline{\rho{\bf u}}_{{\vec{\ell}}}|^{2}({\bf
x})/\overline{\rho}_{{\vec{\ell}}}({\bf x})$ in eq. (9) without spatial
averaging, $\\{\dots\\}$.

From $\mathcal{E}({\bf k})$, we can quantify the spectral energy density at
any scale ${\vec{\ell}}$ in $n$-dimensions by defining the $n$-dimensional
filtering spectrum as

$\displaystyle\overline{E}^{n\mathrm{D}}({\bf
k})\equiv\frac{\partial^{n}}{\partial k_{1}\dots\partial
k_{n}}\mathcal{E}({\bf k})~{}.$ (11)

It satisfies energy conservation in a straightforward manner by the
fundamental theorem of calculus,

$\frac{1}{2}\left\\{\frac{|\overline{\rho{\bf
u}}_{{\vec{\ell}}_{0}}|^{2}}{\overline{\rho}_{{\vec{\ell}}_{0}}}\right\\}+\int_{k_{0}}^{\infty}{\mathrm{d}}^{n}{\bf
k}~{}\overline{E}^{n\mathrm{D}}({\bf k})=\frac{1}{2}\left\\{\rho|{\bf
u}|^{2}\right\\}~{},$ (12)

where ${\vec{\ell}}_{0}=(L/k_{0},L/k_{0},L/k_{0})$ is the largest length scale
at which the flow is filtered.

(a) Multi-D filtering spectrum in eq. (11) (b) Anisotropy based on shell
centroid

Figure 1: Panel (a): Schematic of extracting the multi-dimensional “filtering
spectrum”. Filtering is performed in physical space to disentangle scales,
without resorting to Fourier transforms. Filtering wavenumbers
$(k_{x},k_{z})=(L/\ell_{x},L/\ell_{z})$ are only a proxy for length-scales,
but may be thought of as Fourier wavenumbers conceptually. The cumulative
spectrum $\mathcal{E}(k_{1},k_{2})$ (shaded in blue) yields the energy content
at _all_ scales larger than $\ell_{1}$ in the x-direction and larger than
$\ell_{2}$ in the z-direction, such that
$\partial_{k_{z}}\partial_{k_{x}}\mathcal{E}$ evaluated at $(k_{1},k_{2})$
yields the spectral energy (density) content at exactly those scales
$(\ell_{1},\ell_{2})$. It is a generalization of the traditional multi-
dimensional Fourier spectrum, $\frac{1}{2}|\widehat{{\bf
u}}|^{2}(k_{1},k_{2})$, and reduces to it when using a sharp-spectral
filtering kernel. Panel (b): Shell (blue) in filtering wavenumber space,
$|{\bf k}|\in[k/\sqrt{2},\sqrt{2}k]$. It has logarithmic thickness $\times 2$
(Aluie09, ). Its centroid (yellow star) based on the distribution of
$\overline{E}^{n\mathrm{D}}({\bf k})$ within the shell (${\bf
K}^{\mathrm{shell}}(k)$ in eq. (13)) gives a measure of anisotropy at scale
$\ell=L/k$ from its proximity to the diagonal line (dashed orange).

Fig. 1(a) illustrates the motivation for our definition (11) in 2D. The
filtering spectrum $\overline{E}^{2\mathrm{D}}(k_{1},k_{2})$ at filtering
wavevector ${\bf k}=(k_{1},k_{2})$ quantifies the spectral energy density by
measuring the cumulative spectrum’s response,
$\partial_{k_{z}}\partial_{k_{x}}\mathcal{E}(k_{1},k_{2})$, to concurrent
scale variations in all directions, $\Delta k_{x}$ and $\Delta k_{z}$, at
$(k_{1},k_{2})$. Here, the filtering scale vector is
$(\ell_{1},\ell_{2})=(L/k_{1},L/k_{2})$. The one-dimensional spectrum
$\overline{E}(k_{\ell})$ in eq. (5) is essentially an integral of
$\overline{E}^{2\mathrm{D}}({\bf k})$ at all filtering wavevectors ${\bf
k}=(k_{x},k_{y})$ within a thin shell of width $\Delta k$,
$\left|\sqrt{k_{x}^{2}+k_{y}^{2}}-k_{\ell}\right|<\Delta k$.

When mass density $\rho$ is constant, $\overline{E}^{n\mathrm{D}}({\bf k})$ is
a generalization of the traditional Fourier spectrum,
$\frac{1}{2}|\widehat{{\bf u}}|^{2}({\bf k})$, where $\widehat{{\bf u}}({\bf
k})$ is the Fourier transform of ${\bf u}({\bf x})$. The filtering spectrum
reduces to the Fourier spectrum when using a sharp-spectral filtering
(Dirichlet or Bessel) kernel in eq. (8). However, a main advantage of our
multi-dimensional filtering spectrum in equation (11) is that it can diagnose
bounded or inhomogeneous flows in a straightforward manner. Moreover, as
discussed in section II, it allows for measuring the spectrum of KE in VD
flows while respecting its nonquadratic nature.

###  III.1 Scale-dependent Metrics for Shape Anisotropy

The multi-D filtering spectrum allows us to quantify anisotropy as a function
of length-scale as follows. For any filtering wavenumber $k$ associated with
length scale $\ell$, consider the logarithmic shell $|{\bf
k}|\in[k/\sqrt{2},\sqrt{2}k]$ in filtering wavevector space shown in Fig. 1b.
Within the shell, the normalized “scale-dependent centroid” (or first moment)
of $\overline{E}^{n\mathrm{D}}$ is

$\displaystyle{\bf
K}^{\mathrm{shell}}(k)\equiv\begin{cases}\left.\displaystyle\frac{1}{k}\displaystyle\int\limits_{|{\bf
k}|\in\left[\nicefrac{{k}}{{\sqrt{2}}},\sqrt{2}k\right]}{\mathrm{d}}^{n}{\bf
k}\,{\bf k}\,\overline{E}^{n\mathrm{D}}({\bf
k})~{}\middle/\displaystyle\int\limits_{|{\bf
k}|\in\left[\nicefrac{{k}}{{\sqrt{2}}},\sqrt{2}k\right]}{\mathrm{d}}^{n}{\bf
k}\,\overline{E}^{n\mathrm{D}}({\bf k})~{}\right.,\\\ \\\ \hskip
14.22636pt{\mbox{\boldmath$0$}}\hskip 28.45274pt\text{if
$\displaystyle\int\limits_{|{\bf
k}|\in\left[\nicefrac{{k}}{{\sqrt{2}}},\sqrt{2}k\right]}{\mathrm{d}}^{n}{\bf
k}\,\overline{E}^{n\mathrm{D}}({\bf
k})<\epsilon^{\textrm{numeric}}$.}\end{cases}$ (13)

Integration is performed in the first quadrant of ${\bf k}$-space. Definition
(13) is dimensionless and is zero if the shell is devoid of energy. We set
$\epsilon^{\textrm{numeric}}=10^{-14}$ in this work to avoid numerical
overflow errors. When multiplied by wavenumber, $k\,{\bf
K}^{\mathrm{shell}}(k)$ is the centroid for the shell’s energy as shown in
Fig. 1b. Note the shells $\left[\nicefrac{{k}}{{\sqrt{2}}},\sqrt{2}k\right]$
in definition (13) have a width that is constant on a logarithmic (not linear)
scale. This is important to capture spatially localized structures at
different scales, sometimes called a Littlewood-Payley decomposition Aluie09 .

Figure 2: Anisotropic shapes and their corresponding anisotropy metric
$\mathrm{AR}$ from eqs. (14),(A-3).

The centroid’s distance from the diagonal in Fig. 1b is one measure of
anisotropy at each $k$, which is equivalent to the contrast between different
components, ${K}_{i}^{\mathrm{shell}}$, of ${\bf K}^{\mathrm{shell}}$. A
convenient measure for such contrast is what we call the scale-dependent
anisotropy metric (AR) tensor,

$\displaystyle\mathrm{AR}_{ij}^{\mathrm{shell}}(k)\equiv\frac{({K}_{i}^{\mathrm{shell}})^{2}-({K}_{j}^{\mathrm{shell}})^{2}}{({K}_{i}^{\mathrm{shell}})^{2}+({K}_{j}^{\mathrm{shell}})^{2}}~{},\hskip
28.45274pt\mathrm{for~{}}i,j=1,\dots,n.$ (14)

If ${K}_{i}^{\mathrm{shell}}={K}_{j}^{\mathrm{shell}}=0$, we define
$\mathrm{AR}_{ij}^{\mathrm{shell}}\equiv 0$. For example,
$\mathrm{AR}_{zx}^{\mathrm{shell}}(k)>0$, indicates a dominant concentration
of energy within shell $k$ at small $k_{x}$ and large $k_{z}$. Therefore,
$\mathrm{AR}_{zx}^{\mathrm{shell}}(k)>0$ indicates structures that are
horizontally elongated (along the $x$-direction) at scale $\ell\sim k^{-1}$ as
sketched in Fig. 2. The converse is also true, where
$\mathrm{AR}_{zx}^{\mathrm{shell}}(k)<0$ indicates vertically elongated
structures at scale $\ell\sim k^{-1}$. The anisotropy metric tensor is
antisymmeteric,
$\mathrm{AR}_{ij}^{\mathrm{shell}}=-\mathrm{AR}_{ji}^{\mathrm{shell}}$, and
each of its components lies within
$-1\leq\mathrm{AR}_{ij}^{\mathrm{shell}}\leq 1$. We shall demonstrate these
definitions numerically using simple examples in the following section. In
Appendix section B, we present a reduced (and less expensive) scale analysis
based on what we call marginal spectra and discuss their limitations.

##  IV Numerical Implementation

In this section, we measure scale anisotropy using illustrative examples as
well as data from Rayleigh-Taylor (RT) turbulence simulations in 3D and 2D. RT
turbulence is markedly anisotropic and inhomogeneous, which makes it a good
application for our method.

###  IV.1 Illustrative Examples

separation

Figure 3: Illustrative examples of anisotropic fields. Left panel shows the
Gaussian scalar field in eq. (15). Right panel shows the magnitude of the
Taylor-Green velocity in eq. (16), where vortices are elongated along the
$x$-direction.

For illustrative purposes, we consider two examples in a 2D periodic domain: a
Gaussian scalar field and a Taylor-Green velocity field. Defining these in a
periodic domain allows us to use a sharp spectral filtering kernel in Fourier
space for comparison. The Gaussian scalar field $\phi(x,z)$ is of the form

$\displaystyle\begin{split}\phi(x,z)=&\frac{1}{\sqrt{2\pi}\sigma_{x}}\frac{1}{\sqrt{2\pi}\sigma_{z}}e^{-\frac{(x-\pi)^{2}}{2\sigma_{x}^{2}}-\frac{(z-\pi)^{2}}{2\sigma_{z}^{2}}}~{},\\\
\mathrm{where}\hskip 28.45274pt&(\sigma_{x},\sigma_{z})=(\pi/32,\pi/8)\hskip
28.45274pt\mathrm{and}\hskip 28.45274ptx,z\in[0,2\pi]~{}.\end{split}$ (15)

This Gaussian field is anisotropic, elongated in the $z$-direction as shown in
figure 3(a). It populates a broad band of Fourier wavenumbers (see Appendix
Fig. A12).

The Taylor-Green (TG) velocity field

$\displaystyle\begin{split}(u_{x},u_{z})=&\left(p_{z}\sin\left(p_{x}x\right)\cos\left(p_{z}z\right),-p_{x}\cos\left(p_{x}x\right)\sin\left(p_{z}z\right)\right),\\\
\mathrm{where}\hskip 28.45274pt&(p_{x},p_{z})=(8,32)\hskip
28.45274pt\mathrm{and}\hskip 28.45274ptx,z\in[0,2\pi]~{},\end{split}$ (16)

is anisotropic and elongated in the $x$-direction as shown in figure 3(b). It
consists of only a single Fourier wavevector $(p_{x},p_{z})$. Both examples
are discretized on a $512\times 512$ grid.

(a) cumulative spectrum ${\mathcal{E}}(k_{x},k_{z})$ (b) 2D filtering spectrum
$\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ (c) scale-dependent centroid (d)
scale-dependent anisotropy metric

Figure 4: Filtering spectra of the Gaussian scalar field (eq. (15)) using
Gaussian filters (eq. (17)). Panel (a) is the cumulative spectrum in 2D
filtering wavenumber space. Panel (b) is the 2D filtering spectrum obtained
via eq. (11) (note the logarithmic color bar). Panel (c) shows the scale-
dependent centroid obtained from integrating $\overline{E}^{2\textrm{D}}$
within a shell (eq. (13)) . Panel (d) shows the anisotropy metric from eqs.
(14).

(a) $\overline{\phi}_{L/2}$

(b) $\overline{\phi}_{L/8}$

(c) $\phi^{\prime}_{L/8}$

(d) $\phi^{\prime}_{L/32}$

Figure 5: Scale-dependent anisotropy demonstrated through the low-pass
($\overline{\phi}_{\ell}$) and high-pass ($\phi^{\prime}_{\ell}$) filtered
field in eq. (15) (see Fig. 3a). We use an isotropic Gaussian filtering kernel
defined in eq. (17), with $\ell_{x}=\ell_{z}=\ell$. Note that $\phi$ is
isotropic at very large scales and becomes increasingly anisotropic at smaller
scales, with the smallest scales in panel (d) exhibiting the most anisotropy
among those shown here. These panels highlight the advantages of coarse-
graining over Fourier space analysis or structure functions.

####  IV.1.1 Gaussian scalar field

Figure 4 demonstrates the filtering spectrum,
$\overline{E}^{2D}(k_{x},k_{z})$, and how to quantify anisotropy for the
Gaussian scalar field, $\phi$. The cumulative spectrum,
$\mathcal{E}(k_{x},k_{z})$, is calculated from eq. (9) by setting $\rho=1$ and
replacing ${\bf u}$ with $\phi$. In defining filtering wavenumber, $k=L/\ell$,
we use the domain size $L=2\pi$. The filtering kernel in eq. (8) is itself a
normalized Gaussian of the form

$G_{\vec{\ell}}(r_{x},r_{z})=\left(\frac{6}{\pi\ell_{x}^{2}}\right)^{1/2}e^{-6|r_{x}/\ell_{x}|^{2}}\left(\frac{6}{\pi\ell_{z}^{2}}\right)^{1/2}e^{-6|r_{z}/\ell_{z}|^{2}}~{}.$
(17)

Fig. 4(a) shows the cumulative spectrum, $\mathcal{E}(k_{x},k_{z})$, as a
function of filtering wavevector $(k_{x},k_{z})=(L/\ell_{x},L/\ell_{z})$. It
is clear from Fig. 4(a) that $\mathcal{E}(k_{x},k_{z})$ is asymmetric with
respect to the diagonal line, $k_{x}=k_{z}$. $\mathcal{E}(k_{x},k_{z})$
saturates more quickly with increasing $k_{z}$ than with increasing $k_{x}$.
Equivalent, albeit more intuitive information is conveyed by the filtering
spectrum, $\overline{E}^{2D}(k_{x},k_{z})$, in Fig. 4(b). It shows that
$\overline{E}^{2D}(k_{x},k_{z})$ is asymmetric with respect to the diagonal,
$k_{x}=k_{z}$, with more weight present at smaller $k_{z}$ (i.e. larger
vertical length scales $\ell_{z}\sim k_{z}^{-1}$). This is consistent with the
elongated profile of $\phi$ along the $z$-direction relative to the
$x$-direction. Note that the asymmetry in Fig. 4(b) is only over an
intermediate range of scales. Symmetry seems to be recovered in the lower-left
and upper-right corners of panel 4(b), which we shall now discuss.

Symmetry in the lower-left corner of Fig. 4(b) is consistent with isotropy at
the largest scales, comparable to the domain size, where filtering in eq. (8)
approaches a domain average, $\overline{\phi}_{\vec{\ell}}\to\\{\phi\\}$ as
${\vec{\ell}}\to(\infty,\infty)$. The spatial mean, $\\{\phi\\}$, is uniform
and, thus, trivially isotropic. Fig. 5 demonstrates this in physical space,
which also highlights the advantages of coarse-graining. It may be helpful to
think of a length scale decomposition as an operation analogous to removing
one’s eyeglasses, blurring an image isotropically. If the blurring is only
slight (small $\ell$ in filtering kernel $G_{\ell}$) as in Fig. 5b, one can
still detect any anisotropy present at the large scales. However, such
anisotropy is undetectable if the blurring is severe ($\ell\to\infty$) as in
Fig. 5a. Fig. 5c,d show how anisotropy of the field $\phi$ in eq. (5) persists
at smaller scales, which is explained by simple analysis in Appendix sectionI.
How can this be reconciled with the seeming symmetry in the upper-right corner
of Fig. 4(b)?

Symmetry in the upper-right corner of Fig. 4(b) is merely due to the finite
grid resolution, which precludes fully resolving shells beyond a certain
wavenumber. Indeed, the closer we approach the upper-right corner along the
diagonal $k_{x}=k_{z}$, the less of the shell is resolved on either side of
the diagonal. This is related to having isotropic numerical grid cells at the
smallest scales. The analytical (non-discretized) $\phi$ exhibits persistent
anisotropy to arbitrarily small scales as explained in Appendix sectionI.

Fig. 4c evaluates centroid components $K^{\textrm{shell}}_{x}(k)$ and
$K^{\textrm{shell}}_{z}(k)$ from eq. (13). It shows that
$K^{\textrm{shell}}_{z}\leq K^{\textrm{shell}}_{x}$ over the entire range of
$k$, which implies that the field is elongated along the $z$-direction at all
scales. This is also expressed by the anisotropy metric in Fig. 4d, where
$AR^{\textrm{shell}}_{zx}\leq 0$ at all scales. As we discussed above, at the
largest length scales ($k\to 0$), we see that $K^{\textrm{shell}}_{z}\approx
K^{\textrm{shell}}_{x}$ and $AR^{\textrm{shell}}_{zx}\approx 0$, indicating
isotropy at very large scales. A similar behavior occurs as $k\to\infty$, but
this is due to numerical resolution limits.

(a) cumulative spectrum ${\mathcal{E}}(k_{x},k_{z})$ (b) 2D filtering spectrum
$\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ (c) scale-dependent centroid (d)
scale-dependent anisotropy metric

Figure 6: Same as in Fig. 4 but filtering is done with a sharp-spectral kernel
in Fourier space.

Fig. 6 displays similar results as in Fig. 4 but using a sharp spectral
filter. This is defined in Fourier space as

$\displaystyle\widehat{\mbox{Sinc}}_{\vec{k_{c}}}(k_{x},k_{z})=\begin{cases}1,&\text{if
$|k_{x}|<k_{c,x}$ and $|k_{z}|<k_{c,z}$},\\\ 0,&\text{otherwise},\end{cases}$
(18)

which is straightforward to use due the domain’s periodicity in our example.
In eq. (18), $\vec{k_{c}}$ is the wavevector filtering cutoff.
$\widehat{\mbox{Sinc}}_{\vec{k_{c}}}$ can be written in $x$-space as a product
of Dirichlet kernels Riveraetal14  (see Appendix Fig. A1).

Comparing Figs. 4,6 shows that the results are broadly consistent. Some
notable differences are that plots in Fig. 6 are (i) sharper as a function of
$k$ and (ii) decay more rapidly at large $k$.

Differences in (i) sharpness arise from the filtering spectrum
$\overline{E}(k_{\ell})$ being essentially a weighted average of the Fourier
spectrum $E(k)$ over a range of Fourier wavenumbers $k$ centered around
$k_{\ell}$ (Sadek18, ),

$\overline{E}(k_{\ell})=\int_{0}^{\infty}{\mathrm{d}}p~{}\left[\frac{d}{dk_{\ell}}\left|\widehat{G}\left(\frac{p}{k_{\ell}}\right)\right|^{2}\right]E(p).$
(19)

Here, $\widehat{G}$ is the Fourier transform of filtering kernel $G$. For a
sharp spectral filtering kernel, the factor in square brackets in eq. (19) is
a delta function, yielding $\overline{E}=E$. Therefore, a comparison between
Fig. 6 and Fig. 4 highlights that spatially localized filtering kernels such
as the Gaussian are not strictly local in $k$-space compared to a sharp-
spectral filter, which can lead to additional smoothing as a function of
scale. This is a necessary price for gaining spatially local information (e.g.
Fig. 5, showing the field in $x$-space at different scales), since
concurrently exact spatial and scale localization is forbidden by the
uncertainty principle.

The difference in (ii) the decay rate at large $k$ between Fig. 6 and Fig. 4
was discussed in detail in Sadek18  and is a consequence of eq. (19). It is
due to using the first-order kernel in eq. (17) (using the terminology of
Sadek18 ) to calculate the filtering spectrum, which precludes
$\overline{E}(k_{\ell})$ decaying faster than $k_{\ell}^{-3}$ as
$k_{\ell}\to\infty$. In contrast, the Fourier spectrum decays rapidly,
$E(k)\sim e^{-k^{2}\sigma^{2}}$, for the Gaussian field $\phi$ in eq. (15)
(see Appendix sectionI). The slower $k_{\ell}^{-3}$ decay is not a limitation
of the filtering spectrum method, but of the filtering kernel itself. Faster
decay rates can be captured by using higher-order kernels Sadek18  (Appendix
Fig. A7). Further discussion of this issue along with a comparison between
Fourier and filtering spectra are provided in Appendix sectionI. Despite
differences in the decay rate, we see that anisotropy inferred from spatial
filtering in Fig. 4 is consistent with that in Fig. 6 from Fourier analysis.
Therefore, even when using a first-order kernel, the underlying anisotropy is
still detected from the filtering spectrum
$\overline{E}^{n\textrm{D}}(k_{x},k_{z})$, although it may be underestimated.

####  IV.1.2 Taylor-Green

Figure 7 shows the filtering spectrum, $\overline{E}^{2D}(k_{x},k_{z})$, and
anisotropy for the Taylor-Green flow (eq. (16)). For clarity, we use the sharp
spectral kernel (eq. (18)) for filtering since the flow is localized at a
single Fourier wavevector. We provide a corresponding Fig. A15 in the Appendix
using the Gaussian filter for completeness.

In Fig. 7a, the cumulative spectrum, $\mathcal{E}(k_{x},k_{z})$ from eq. (9),
is calculated by setting $\rho=1$. In defining filtering wavenumber,
$k=L/\ell$, we use the domain size $L=2\pi$. It is clear from Fig. 7(a),(b)
that $\mathcal{E}(k_{x},k_{z})$ and $\overline{E}^{2\textrm{D}}(k_{x},k_{z})$
are asymmetric with respect to the diagonal line, $k_{x}=k_{z}$, with more
weight toward smaller $k_{x}$ and larger $k_{z}$. In fact,
$\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ is zero everywhere except at
$(k_{x},k_{z})=(8,32)$ as defined in eq. (16).

Fig. 7(c) evaluates the normalized centroid ${\bf K}^{\textrm{shell}}$ from
eq.(13). Note that unlike in eq.(13), here we use thin bands $|{\bf
k}|\in(k,k+1]$ for clarity, since our field comprises of nonlocalized plane
waves. For completeness, Fig. A14 in the Appendix uses the dyadic bands as in
eq.(13). Fig. 7(c) shows a spike in ${\bf K}^{\textrm{shell}}$ at $k=31$ with
$K^{\textrm{shell}}_{x}=0.23K^{\textrm{shell}}_{z}$ as expected, where the
field is elongated along the $x$-direction. This is also expressed by the
anisotropy metric in Fig. 7d, where
$\textrm{AR}^{\textrm{shell}}_{zx}(31)=0.90$.

(a) cumulative spectrum ${\mathcal{E}}(k_{x},k_{z})$ (b) 2D filtering spectrum
$\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ (c) scale-dependent centroid (d)
scale-dependent anisotropy metric

Figure 7: Similar to Fig. 4 but analyzing the Taylor-Green velocity (eq. (16))
using a sharp-spectral filter in Fourier space. The only Fourier mode present
is $(k_{x},k_{z})=(8,32)$, seen in panel (b). Panels (c) and (d) show the
centroid (eqs. (13),(A-2)), but with the integrals in eqs. (13),(A-2) done
over thin bands $k<|{\bf k}|\leq k+1$ to highlight the precise wavenumber
locations. A complementary figure A14 in the appendix shows integrals over
dyadic bands $(k/\sqrt{2},\sqrt{2}k]$ as in eqs. (13),(A-2).

###  IV.2 Rayleigh-Taylor Turbulence

Figure 8: Visualizations of the density field from 2D (left) and 3D (right)
Rayleigh-Taylor turbulence. Adapted from Figure 2 of reference Zhao22JFM ,
with permission from Cambridge University Press.

separate

Our method is now applied to a fluid dynamics problem, that of Rayleigh-Taylor
(RT) turbulence simulated in both 2D and 3D. RT turbulence occurs when a heavy
fluid is supported from below by a light fluid in the presence of gravity or
an equivalent acceleration field. An initial small perturbation at the fluid
interface grows due to the buoyancy-driven RT instability. It is a fluid
dynamics problem that is of both fundamental interest and practical importance
in many fields. For example, RT instability plays a leading role in the
propagation of the thermonuclear flame front in supernova explosions
Supernova00 , and is a main obstacle degrading energy yields in inertial
confinement fusion (ICF) Betti16 .

It may be apparent to many fluid dynamics experts that modeling RT unstable 3D
systems using 2D simulations is inferior to using 3D simulations. Yet, such
propositions remain vague and speculative without clearly identifying the
precise aspects of the dynamics that are misrepresented in a 2D simulation.
This is necessary to convince and guide practitioners who are focused on
global system modeling while facing finite computational resources. For
example, ICF modelers aim to predict or postdict results from several shots
(i.e. experiments) per day in the laboratory. Conducting a single 3D
simulation can take several weeks on today’s high-end supercomputers, even
without incorporating important system components such as the hohlraum and
laser physics Haines2020 ; GatuJOHNSON2020 . The trade-off ICF modelers face
is between including more physical processes in simulations at the expense of
conducting them in 2D, despite the recognition that 3D modeling is preferable.
To many, trade-offs between 2D and 3D flow physics are not as clear as trade-
offs between whether or not to include components that are critical to the
system’s evolution such laser deposition, hohlraum physics, or resolving
small-scale effects such as from the tent or stalk-mount Clarketal16 . This is
why in ICF, 2D simulations are still the main “work horse” for experimental
design Schlossberg2021 ; Kritcher2022-1 ; Kritcher2022-2  as routine 3D
simulations are prohibitively expensive. In the hope of improving 2D modeling,
we shall present clear evidence of the stark differences in the fundamental RT
scale physics between 2D and 3D models.

At high Reynolds numbers and at sufficiently late times, the system will
develop into turbulence Zhou17-1 ; Zhou17-2 ; Livescu20 . RT turbulence is
inhomogeneous, anisotropic, and unsteady. Since RT turbulence is inhomogeneous
in the vertical $z$-direction (along gravity), it is a good representative of
the challenges faced when analyzing scales in inhomogeneous flows. Previous
work had been restricted to analyzing scales in the horizontal directions,
which are homogeneous Zhou17-1 ; Zhou17-2 . An important exception is the work
of Soulard and Griffond soulard2012inertial  who developed a successful theory
for the anisotropic spectral scaling in the inertial range of RT flows. They
employed Fourier analysis on the fluctuations in all directions, including the
inhomogeneous direction, after removing Reynolds averages. The Soulard-
Griffond RT theory applies to scales that are sufficiently small such that the
nonlinear term dominates over buoyancy and anisotropy can be regarded as a
perturbation around a Kolmogorov-Obukhov equilibrium spectrum
ishihara2002anisotropic ; Chertkov03 . Soulard and Griffond
soulard2012inertial  tested their theory using direct numerical simulations of
RT and calculated Fourier transforms of the fluctuating fields along all
directions, including the inhomogeneous vertical direction grea2013rapid .

While we agree with soulard2012inertial  that Fourier analysis can be well-
posed for a component of the RT flow (at times when mixing zone is
sufficiently small such that the fluctuating fields are at sufficiently small
scales relative to the vertical domain size), our goal here is to demonstrate
an approach that generalizes Fourier analysis and can be used to diagnose
scales of the entire RT flow, including the Reynolds mean fields. As mentioned
in the introduction, there is often a lack of scale separation between the
mean and fluctuating fields Zhao22JFM ; buzzicotti2023spatio , necessitating a
characterization of spectral information for both. Moreover, Fourier analysis,
due to its associated Dirichlet kernel that is highly nonlocal in $x$-space
(Fig. A1 in Appendix) presents additional difficulties to scale analysis of
inhomogeneous flows. Scales have to be limited to sufficiently large Fourier
wavenumbers $k$ in the vertical and to sufficiently early times in the RT
mixing layer growth to avoid domain boundary artifacts soulard2012inertial .

Recently, reference (Zhao22JFM, ) measured the 1-dimensional filtering spectra
(see Fig. A9 below) for RT turbulence, including scales in the vertical
inhomogeneous direction, but did not distinguish the spectral contribution
from different directions. Here, we examine shape anisotropy as a function of
scale. In addition, reference (Zhao22JFM, ) showed the limitation of 2D RT
simulations as surrogates for 3D configurations from the perspective of energy
transfer across scales. Here we shall extend the comparison between 2D and 3D
RT from the perspective of scale-dependent anisotropy.

Visualizations of the simulated flow in 2D and 3D are shown in figure 8. They
convey qualitatively the turbulent nature of these flows (see Zhao22JFM ). The
simulations are of the fully compressible Navier-Stokes equation with an ideal
gas equation of state. The governing equations are solved in 2D and 3D
rectangular domains in Cartesian coordinates. The boundary conditions are
periodic in the horizontal directions and no-slip rigid walls in the vertical
$z$-direction. The equations are solved using a hybrid solver, which uses the
pseudo-spectral scheme in the horizontal and a 6th order compact finite
difference scheme in the vertical. Here, we use data from simulations 3D1024
and 2D4096 described in Table 1 in Zhao22JFM , with the Atwood number 0.5 and
the domain-size based Reynolds number 13854 and 44562 for 3D1024 and 2D4096,
respectively. More details can be found in Zhao22JFM .

As discussed in Appendix sectionB, extracting the 3-dimensional cumulative
spectrum, $\mathcal{E}({\bf k})$ in eq. (9), can be computationally expensive.
For expediency when analyzing the 3D RT data, we probe the following
2-dimensional scale sub-spaces, $(\ell_{x},0,\ell_{z})$ and
$(\ell_{x},\ell_{y},0)$, separately. For example, analyzing
$(\ell_{x},0,\ell_{z})$ is accomplished by convolving with kernel
$G_{{\vec{\ell}}}({\bf x})=G_{\ell_{x}}(x)\,\delta(y)\,G_{\ell_{z}}(z)$. In
this case, no scale decomposition is performed in the $y$-direction, but the
convolution is done at every location $(x,y,z)$ in 3D physical space.

In this section, we use the Gaussian kernel in eq. (17). For the definition of
filtering wavenumbers, $k=L/\ell$, we use $L=3.2$, which is the domain size
along the horizontal direction. In simulations with non-periodic boundary
conditions, such as our RT flows with rigid walls at the top and bottom,
filtering near the boundary requires a choice for the fields beyond the
boundary. Following past work (Aluieetal2018_jpo, ; zhao2018inviscid, ;
Zhao22JFM, ), we extend the domain beyond the physical boundaries in a manner
consistent with the boundary conditions. For our RT problem, velocity is kept
zero beyond the walls, density field is kept constant (zero normal gradient),
and the extended pressure field satisfies the hydrostatic condition
$dP/dz=-\rho g$. The coarse-graining operation (3) can then be performed at
every point in the flow domain.

####  IV.2.1 3D RT

To quantify scale anisotropy in 3D RT, we first calculate
$\mathcal{E}(k_{x},\infty,k_{z})$ and
$\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ in filtering wavenumber subspace
$(k_{x},k_{z})$. These are shown in Fig. 9. The asymmetry relative to the
diagonal $k_{z}=k_{x}$ is especially clear from $\overline{E}^{2\textrm{D}}$
in Fig. 9b at wavenumbers smaller than $\approx 20$, where
$\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ is skewed toward smaller $k_{z}$
(i.e. larger vertical length scales $\ell_{z}\sim k_{z}^{-1}$). This indicates
that the flow is elongated in the vertical, which is consistent with the
notion of large scale vertically inter-penetrating bubbles and spikes. Further
evidence for large scale anisotropy is derived from Fig. 10, where vertically
elongated shapes are visually obvious in panels (a)-(c) showing the flow at
scales larger than $\ell=L/10$. Fig. 10 underscores the utility of the coarse-
graining approach to analyzing anisotropy in a natural and intuitive manner.

The asymmetry relative to the diagonal is less clear at higher wavenumbers in
Fig. 9b. Indeed, visualizing the corresponding small-scales in physical space
in Fig. 10d suggests that they are in approximate isotropy. The quantity shown
in Fig. 10d is energy at scales smaller than $\ell$ and defined as

${\mathcal{E}}_{<{\vec{\ell}}}=\frac{1}{2}\left\\{\overline{\left(\rho|{\bf
u}|^{2}\right)}_{{\vec{\ell}}}\,-\frac{|\overline{\rho{\bf
u}}_{{\vec{\ell}}}|^{2}}{\overline{\rho}_{{\vec{\ell}}}}\right\\}~{},$ (20)

which is positive semi-definite as proved in Sadek18 .

Panels (c) and (d) in Fig. 9 provide a more quantitative measure of
anisotropy. Discrepancy between $K_{x}^{\textrm{shell}}$ and
$K_{z}^{\textrm{shell}}$ is clear around $k=5$, where their ratio reaches
$\approx 4/3$. This discrepancy is better captured by the anisotropy metric
$AR_{zx}^{\textrm{shell}}(k)$ in Fig. 9(d), which has an extremum of
$AR_{zx}^{\textrm{shell}}(5)=-0.27$. $AR_{zx}^{\textrm{shell}}(k)$ trends
toward zero at higher $k$, which indicates approximate isotropy field at
smaller length scales, although it never reaches zero over the dynamic range
of scales we resolve in the simulation. This is consistent with the presence
of residual small-scale _vector_ anisotropy observed in Livescu et al.
Livescu09  (Fig. 14 in Livescu09 ). It is also consistent with anisotropic (or
“directional”) spectra of the velocity fluctuations in Soulard and Griffond
soulard2012inertial  (Figs. 6-7 in soulard2012inertial ). We are unsure if
this residual anisotropy at high $k$ in Fig. 9(c),(d) would persist in the
limit of higher Reynolds numbers. Appendix Figs. A7-A8 repeat the anisotropy
analysis using a Boxcar and a 3rd-order filtering kernel, which in some sense
is more similar to a truncated Dirichlet kernel spyropoulos1994evaluation ,
showing that our results are insensitive to the Gaussian kernel we are using.

Panels 9(c),(d) show that at scales $\ell$ approaching the horizontal domain
size $L$ ($k=1$), we have
$K_{x}^{\textrm{shell}}(1)=K_{z}^{\textrm{shell}}(1)$ and
$AR_{zx}^{\textrm{shell}}(1)\approx 0$ indicating isotropy. At first glance,
this may seem surprising since we expect RT flows to be highly anisotropic at
large scales Livescu09 ; soulard2012inertial ; Zhou17-1 . However, note that
scale $\ell=L$ ($k=1$) is much larger than the largest bubble or spike in Fig.
8. As the coarse-graining scale ${\vec{\ell}}$ approaches that of the domain
size, spatially local KE, $\nicefrac{{1}}{{2}}|\overline{\rho{\bf
u}}_{\vec{\ell}}|^{2}/\overline{\rho}_{\vec{\ell}}$, approaches that of a
domain average, $\nicefrac{{1}}{{2}}|\\{\rho{\bf u}\\}|^{2}/\\{\rho\\}$. A
domain average has trivial shape isotropy because it is spatially constant. We
saw this in our illustrative example of an anisotropic Gaussian in Fig. 3(a),
which is isotropic at scales comparable to the domain size as shown in Fig. 5.
For 3D-RT, we have a similar situation in Fig. 9(d) where
$AR^{\mathrm{shell}}(k=1)\approx 0$. This can be seen from the visualization
of coarse KE at $k=1$ in Fig. A6 of the Appendix.

In contrast to Fig. 9, comparing scales in the two horizontal ($x$-$y$)
directions in Fig. 11 shows almost perfect isotropy, consistent with physical
expectations. We see that $\mathcal{E}(k_{x},k_{y},\infty)$ and
$\overline{E}^{2\textrm{D}}(k_{x},k_{y})$ are symmetric relative to the
diagonal $k_{y}=k_{x}$, and $AR_{yx}^{\textrm{shell}}=0$ for all $k$.

(a) cumulative spectrum ${\mathcal{E}}(k_{x},\infty,k_{z})$ (b) 2D filtering
spectrum $\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ (c) scale-dependent
centroid (d) scale-dependent anisotropy metric

Figure 9: 3D-RT filtering spectra of KE along the $x$-$z$ directions using
Gaussian kernels (eq. (17)). Panel (a) is the cumulative spectrum (eq. (9)) in
filtering wavenumber subspace $(k_{x},k_{z})$. Panel (b) is the associated 2D
filtering spectrum obtained via eq. (11) (note the logarithmic color bar).
Panel (c) shows the scale-dependent centroid obtained from integrating
$\overline{E}^{2\textrm{D}}$ within a shell (eq. (13)). Panel (d) shows the
anisotropy metric from eq. (14). Note the asymmetry relative to the diagonal
in panels (a),(b), associated with vertically elongated flow structures at
large scales (small $k$). The asymmetry seems to become less at smaller scales
(larger $k$), which is made clear in panel (c) where the centroid moves closer
to the diagonal at high $k$ ($K_{x}\approx K_{z}$). This is also reflected in
the anisotropy metric in (d), where $AR\to 0$ as $k\to\infty$.

(a) coarse density $\overline{\rho}_{\ell}$

(b) coarse KE ${\mathcal{E}}$

(c) coarse streamlines

(d) fine KE ${\mathcal{E}}_{<\ell}$

Figure 10: 3D-RT at different scales. Visualizations show a 2D slice at the
mid-plane $y=L/2$, where $L$ is the domain extent in the horizontal direction.
Panels (a),(b) show the coarse-grained density, $\overline{\rho}_{\ell}$ in
(a), and kinetic energy, ${\mathcal{E}}(k_{x},k_{z})$ in (b), at scale
$k_{x}=k_{z}=L/\ell=10$. The filtering kernel is a Gaussian (eq. (17)) with
$\ell_{x}=\ell_{z}=\ell$. Panel (c) shows $\overline{\rho}_{\ell}$ from panel
(a) in the background and the streamlines based on the coarse-grained velocity
field
$\left(\overline{\left(\rho\,u_{x}\right)}_{\ell}/\overline{\rho}_{\ell},\overline{\left(\rho\,u_{z}\right)}_{\ell}/\overline{\rho}_{\ell}\right)$
with filtering as in panels (a),(b) at $k=10$. Panel (d) shows the small-scale
(or fine) KE, ${\mathcal{E}}_{<{\vec{\ell}}}$ defined in eq. (20), with
$k_{x}=k_{z}=100$. Anisotropy at large scales is evident in panels (a)-(c)
while small scales seem to regain a semblance of isotropy.

(a) cumulative spectrum ${\mathcal{E}}(k_{x},k_{y},\infty)$ (b) 2D filtering
spectrum $\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ (c) scale-dependent
centroid (d) scale-dependent anisotropy metric

Figure 11: Same as in Fig. 9 but in the horizontal $x$-$y$ directions for
3D-RT. The flow is isotropic at all scales as expected.

####  IV.2.2 2D RT

To quantify scale anisotropy in 2D RT, we calculate
$\mathcal{E}(k_{x},\infty,k_{z})$ and
$\overline{E}^{2\textrm{D}}(k_{x},k_{z})$, shown in Fig. 12. From panels (a)
and (b), the asymmetry relative to the diagonal $k_{z}=k_{x}$ seems absent at
small filtering wavenumbers but becomes discernible at $k\gtrsim 50$, where
$\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ is skewed slightly toward smaller
$k_{x}$ (i.e. larger horizontal length scales $\ell_{x}\sim k_{x}^{-1}$). This
indicates that the flow is elongated in the horizontal, which may be
surprising for Rayleigh-Taylor flows where it is expected that the flow would
be elongated in the vertical due to buoyancy forcing Livescu08 ; Soulard12 ;
Zhou17-2 , similar to what we observed in section IV.2.1 in 3D-RT.

Panels (c) and (d) in Fig. 12 provide a more quantitative measure of
anisotropy. We see that $K_{x}^{\textrm{shell}}(1)=K_{z}^{\textrm{shell}}(1)$
at the smallest wavenumbers. The discrepancy between $K_{x}^{\textrm{shell}}$
and $K_{z}^{\textrm{shell}}$ becomes discernible at $k>10$. This is also
captured by $AR_{zx}^{\textrm{shell}}(k)$ in Fig. 12(d), which is positive and
increases at larger $k$, indicating that the flow is elongated in the
horizontal and becomes increasingly anisotropic at those smaller length
scales.

Isotropy at large scales and increasing anisotropy at small scales in 2D RT is
consistent with recent results in Zhao22JFM . In that work, it was shown that
2D RT is characterized by the emergence of a large scale overturning
circulation, which isotropizes the 2D flow. This is in contrast to the picture
of vertically rising bubbles and sinking spikes often invoked for RT flows,
which nevertheless seems consistent with 3D-RT. Direct evidence for the
existence of a large scale overturning circulation in 2D RT is provided in
Figs. 13,14. The overturning circulation in Fig. 13c is quasi-isotropic and is
absent at the same large scales ($>\ell=L/10$) in 3D RT shown in Fig. 10c,
which are clearly anisotropic. Fig. 14 shows even larger scales ($>\ell=L/3$)
in 2D RT, where the large-scale isotropic overturning circulation is clearer.
In the Appendix Fig. A5, we calculate the anisotropy metrics at different
locations along the vertical direction.

It was shown in Zhao22JFM  that 2D RT does not cascade energy to smaller
scales, resulting in weaker small inertial scales relative to 3D-RT. Weaker
inertial scales and an associated suppressed molecular mixing was also
discussed in several previous studies Cabot06POF ; Zhou17-1 ; Huasen19 ;
Bian20 .

Therefore, our results in Fig. 12 indicate that the combination of a large-
scale overturning and suppressed small-scale mixing in 2D-RT yields strong
stable stratification at small-scales. Such stratification leads to a flow
that is elongated in the horizontal as quantified by the anisotropy metrics in
Fig. 12c,d. Direct visual evidence for the anisotropic flow at small scales is
provided in Fig. 13d, which should be contrasted to the corresponding Fig. 10d
in 3D RT.

(a) cumulative spectrum ${\mathcal{E}}(k_{x},k_{z})$ (b) 2D filtering spectrum
$\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ (c) scale-dependent centroid (d)
scale-dependent anisotropy metric

Figure 12: 2D-RT filtering spectra of KE as in Fig. 9. Note in panels (a),(b)
the symmetry relative to the diagonal at large-scales (small $k$), associated
with a quasi-isotropic overturning circulation. However, asymmetry emerges at
smaller scales (larger $k$), which is made clear in panel (c) where the
centroid moves away from the diagonal at high $k$ ($K_{x}<K_{z}$), associated
with horizontal layering due to stable stratification. This is also reflected
in the anisotropy metric in (d), where $AR>0$ as $k\to\infty$.

(a) coarse density $\overline{\rho}_{\ell}$

(b) coarse KE ${\mathcal{E}}$

(c) coarse streamlines

(d) fine KE ${\mathcal{E}}_{<\ell}$

Figure 13: 2D-RT at different scales, as in Fig. 10. Panels (a)-(c) are
filtered at scale $k_{x}=k_{z}=L/\ell=10$ using a Gaussian kernel. Panel (d)
shows small-scale KE at $k_{x}=k_{z}=100$. Large-scale flow in panels (a)-(c)
is approximately isotropic, which is especially clear in the large-scale
overturning circulation in panel (c). In contrast, the small-scale flow in (d)
shows a tendency for horizontal layering due to the stable stratification that
results from overturning.

(a) coarse density $\overline{\rho}_{\ell}$

(b) coarse KE ${\mathcal{E}}$ (c) coarse streamlines

pp

Figure 14: 2D-RT has a quasi-isotropic large-scale overturning circulation due
to an upscale cascade Zhao22JFM , which is absent in 3D-RT. Panels (a)-(b) are
similar to those in Fig. 13 but filtered at scale $k_{x}=k_{z}=L/\ell=3$.
Panel (c) demonstrates clearly the large-scale overturning circulation.

##  V Conclusions

This work establishes how spatial coarse-graining can be used to quantify
anisotropy at different length scales. It is based on generalization of ideas
from Sadek18 , which we demonstrated via simple illustrative examples and an
application to Rayleigh-Taylor turbulence.

In addition to the method, a main result of this work showed that anisotropy
of Rayleigh-Taylor turbulence in 2D is fundamentally different from that of
3D-RT. We showed that 2D-RT is characterized by enhanced anisotropy at small
scales in a manner that was perhaps unexpected, but in hindsight is consistent
with recent work Zhao22JFM . A clear identification of how 2D modeling of
applications involving RT can misrepresent the actual 3D hydrodynamics is a
necessary step to develop new modeling strategies without having to conduct
fully 3D simulations. This is because 3D hydrodynamics modeling remains
prohibitively expensive in many applications such as ICF as discussed above.

We envisage that our method for quantifying shape anisotropy would be
especially valuable in flows that are inhomogeneous, where established methods
such as Fourier analysis cannot be used in a straightforward manner. In this
regard, it is important to be aware of limitations of filtering spectra. In
particular, it was shown in Sadek18  that when using first-order kernels such
as a Gaussian, filtering spectra do not decay faster than $k^{-3}$ as
$k\to\infty$, even if there is little or no energy at those small scales when
probed with more complex (higher-order) kernels. Therefore, determining
whether or not energy conveyed by filtering spectra is physical or a spurious
artifact of the kernel being used requires using higher order kernels if a
$k^{-3}$ scaling is observed. Regarding anisotropy, the potential for
‘locking’ in at a $k^{-3}$ scaling when using first-order filtering kernels
would still allow us to detect anisotropy using the metrics, ${\bf
K}^{\mathrm{shell}}(k)$ and $\mathrm{AR}_{ij}^{\mathrm{shell}}(k)$, although
the level of anisotropy may be underestimated.

###### Acknowledgements.

We thank Olivier Soulard and two anonymous reviewers for their valuable
comments that helped improve our paper. This research was funded by US DOE
grant DE-SC0020229 and NSF grant PHY-2206380. Partial support from US NSF
grants PHY-2020249 is acknowledged. HA was also supported by US DOE grants DE-
SC0014318, DE-SC0019329, US NSF grant OCE-2123496, US NASA grant
80NSSC18K0772, and US NNSA grants DE-NA0003856, DE-NA0003914, DE-NA0004134. DZ
also acknowledges financial support from the National Natural Science
Foundation of China (No. 12202270). Computing time was provided by NERSC under
Contract No. DE-AC02-05CH11231.

## APPENDIX

###  A Filtering kernels and Fourier analysis

Fig. A1 shows three example kernels that can be used to calculate the
filtering spectrum. When using the Dirichlet kernel, the filtering spectrum is
exactly the same as the Fourier spectrum. The Dirichlet is highly non-local
and decays as $1/|x-x_{0}|$ away from the center at $x_{0}$, where the
filtered field $\overline{u}(x_{0})$ is being evaluated. Therefore, the
filtered field at location $x_{0}$ has contributions from spatial locations
$x$ far away from $x_{0}$. This is in comparison to filtering with a more
localized kernel, such as a Gaussian or Boxcar, which is practically (or
exactly) zero beyond a distance $\sim\ell/2$ away from $x_{0}$.

Figure A1: (left) Comparing the Gaussian ($G_{\ell}$), Boxcar ($H_{\ell}$),
and Dirichlet ($\mbox{Sinc}_{\ell}$) kernels in physical space, with the
filter width $\ell=L_{x}/8=0.4$. The Boxcar is exactly zero beyond a distance
$\ell/2$ from the center, the Gaussian decays faster than exponentially, while
the Dirichlet is highly non-local and decays as $1/|x-x_{0}|$ away from the
center $x_{0}=0$. (right) The same kernels shown in Fourier $k$-space.

Therefore, in a bounded domain, regions outside the boundaries can make non-
negligible (possibly spurious) contributions to the Fourier scale analysis
(equivalent to filtering with a Dirichlet kernel) even in regions that are
presumably far from the boundaries. When using a more compact kernel, such as
the Gaussian or Boxcar, the “domain of dependence” is localized to be within
$\sim\ell/2$ from the location being analyzed. This makes it feasible to
extend the fields beyond the domain boundaries (in a manner satisfying the
boundary conditions) to analyze scales near those boundaries.

###  B Marginal Filtering Spectra

The cumulative spectrum, ${\mathcal{E}}$, or equivalently, the filtering
spectrum, $\overline{E}^{nD}$, evaluated over the entire scale-grid (Fig.
1(a)) contains all information about the energy content of spatial scales in
different directions. However, ${\mathcal{E}}$ can be computationally
expensive to calculate using data on $N^{n}$ grid-points in $n$-dimensions
since it relies on spatial filtering via convolutions. Scanning the entire
range of scales in each direction requires nominally $O(N^{n})$ different
scale vectors $(\ell_{1},\dots,\ell_{n})$, each of which can cost of
$O(N^{2n})$ operations for the convolution. Therefore, in 3-dimensions, the
cost can be as high as $O(N^{9})$. While there are several algorithmic
optimizations to reduce the cost (StorerAluieJOSS, ; Storer2022NatComm, ;
buzzicotti2023spatio, ), such as subsampling the scale-grid (especially at
high $k$), the computational cost can still be considerable.

(a) Marginal spectrum in eq. (A-1) (b) Anisotropy based on marginal centroid

ppp

Figure A2:  Panel (a): similar to Fig. 1(a) but for “marginal spectra”,
$\partial_{k_{x}}{\mathcal{E}}$ and $\partial_{k_{z}}{\mathcal{E}}$. We see
that $\partial_{k_{x}}{\mathcal{E}}(k_{1},\infty)$ (in red) accounts for field
variations at scale $\ell_{1}=L/k_{1}$ in the $x$-direction, but contains all
scales in the $z$-direction. Similarly, $\partial_{k_{z}}{\mathcal{E}}$ (in
green) accounts for variations in the $z$-direction. Panel (b): (b) similar to
Fig. 1(b) but for the marginal centroid (${\bf K}^{\mathrm{mar}}(k)$ in eq.
(A-2)), depicted as a star. It is based on $\overline{E}^{\mathrm{mar}}(k)$,
which is more easily computed than $\overline{E}^{\mathrm{shell}}(k)$ but at
the expense of accuracy since it incorporates _all_ scales in other
directions. The restriction on ${\bf K}^{\mathrm{mar}}(k)$ to reside where the
two (red and green) strips overlap also makes it a less sensitive gauge of
anisotropy.

It is possible to quantify anisotropy by sampling a much smaller subset of the
scale-grid but at the cost of accuracy and sensitivity. Fig. A2(a) sketches
what we call the “marginal spectra,” in analogy to the marginal probability
distribution. A scale decomposition via filtering is performed along each
direction separately rather than concurrently. The marginal spectra along
different directions are defined as

$\displaystyle\overline{E}_{x}^{\mathrm{mar}}(k)$
$\displaystyle\equiv\frac{\partial}{\partial
k_{x}}\mathcal{E}(k_{x},\infty,\infty)\Big{\rvert}_{k_{x}=k},$ (A-1a)
$\displaystyle\overline{E}_{y}^{\mathrm{mar}}(k)$
$\displaystyle\equiv\frac{\partial}{\partial
k_{y}}\mathcal{E}(\infty,k_{y},\infty)\Big{\rvert}_{k_{x}=k},$ (A-1b)
$\displaystyle\overline{E}_{z}^{\mathrm{mar}}(k)$
$\displaystyle\equiv\frac{\partial}{\partial
k_{z}}\mathcal{E}(\infty,\infty,k_{z})\Big{\rvert}_{k_{z}=k}~{}.$ (A-1c)

Here, $\infty$ denotes the absence of filtering ($\ell=L/k=0$) along the
corresponding direction, thereby retaining the contribution from all scales
along that direction. The marginal spectrum
$\overline{E}_{x}^{\mathrm{mar}}(k)$, for example, is obtained from
$\overline{{\bf u}}_{\ell_{x},0,0}({\bf x})$ by convolving with kernel
$G_{\ell_{x}}(r_{x})\delta(r_{y})\delta(r_{z})$ in eq. (8), where the delta
function $\delta(r)=G_{\ell=0}(r)$.

Comparing $E_{x}^{\mathrm{mar}}(k)$, $E_{y}^{\mathrm{mar}}(k)$, and
$E_{z}^{\mathrm{mar}}(k)$ gives us a gauge of anisotropy at different scales.
However, it is important to be mindful of what information is conveyed in eq.
(A-1) and its limitations. When integrated over $k$, all three quantities
yield the same total energy. Therefore, if
$E_{z}^{\mathrm{mar}}(k_{1})>E_{x}^{\mathrm{mar}}(k_{1})$ at $k_{1}$, it is
guaranteed that $E_{x}^{\mathrm{mar}}(k_{2})>E_{z}^{\mathrm{mar}}(k_{2})$ for
some other $k_{2}$. In comparison, $\overline{E}^{n\mathrm{D}}$ has no similar
constraint.

Similar to eq. (13), the normalized scale-dependent centroid of marginal
spectra within 1D shells are

$\displaystyle{K}^{\mathrm{mar}}_{i}(k)\equiv\begin{cases}\left.\displaystyle\frac{1}{k}\int\limits_{k/\sqrt{2}}^{\sqrt{2}k}dq\,q\,\overline{E}_{i}^{\mathrm{mar}}(q)\middle/\displaystyle\int\limits_{k/\sqrt{2}}^{\sqrt{2}k}dq\,\overline{E}_{i}^{\mathrm{mar}}(q),\right.\hskip
28.45274pt\mathrm{for~{}}i=1,\dots,n,\\\ \\\ \hskip 14.22636pt0\hskip
28.45274pt\text{if
$\displaystyle\int\limits_{k/\sqrt{2}}^{\sqrt{2}k}dq\,\overline{E}_{i}^{\mathrm{mar}}(q)<\epsilon^{\textrm{numeric}}$.}\end{cases}$
(A-2)

We set $\epsilon^{\textrm{numeric}}=10^{-14}$ below. Since
$\overline{E}_{i}^{\mathrm{mar}}$ incorporates _all_ scales in other
directions as shown in Fig. A2(a), the marginal centroid, ${{\bf
K}}^{\mathrm{mar}}$, is a less accurate gauge of anisotropy at any one scale.
Moreover, its restriction to reside where the two (red and green) strips
overlap in Fig. A2(b) also makes it a less sensitive gauge of anisotropy.

Similar to eq. (14), we define the anisotropy metric associated with marginal
spectra as

$\displaystyle\mathrm{AR}_{ij}^{\mathrm{mar}}(k)\equiv\frac{({K}_{i}^{\mathrm{mar}})^{2}-({K}_{j}^{\mathrm{mar}})^{2}}{({K}_{i}^{\mathrm{mar}})^{2}+({K}_{j}^{\mathrm{mar}})^{2}},\hskip
28.45274pt\mathrm{for~{}}i,j=1,\dots,n.$ (A-3)

It has an interpretation similar to that in eq. (14), as sketched in Fig. 2.
However, the anisotropy metrics $\mathrm{AR}^{\mathrm{shell}}$ and
$\mathrm{AR}^{\mathrm{mar}}$ can be differ quantitatively due to restrictions
on ${{\bf K}}^{\mathrm{mar}}$ explained in Fig. A2.

Below, we provide numerical examples to demonstrate how these metrics quantify
anisotropy and to highlight their limitations relative to shell-based metrics
presented in the main text.

###  C Centroids and anisotropic metric associated with marginal spectra

The anisotropy metric based on the marginal spectra is shown in Fig. A3 for
the two illustrative examples, and in Fig. A4 for the 2D and 3D Rayleigh-
Taylor fields.

(a) Gauss centroid, Gaussian filter (b) Gauss anisotropy metric, Gaussian
filter (c) Gauss centroid, sharp spectral filter

(d) Gauss anisotropy metric, sharp spectral filter (e) TG vortex centroid,
sharp spectral filter (f) TG vortex anisotropy metric, sharp spectral filter

Figure A3: Marginal spectra averaged centroid (eq. (A-2)) and anisotropic
metric (eq. (A-3)) of the Gaussian scalar field using Gaussian filters in
panels (a),(b), and of the Gaussian scalar using the sharp spectral filter in
panels (c), (d), and of the Taylor-Green vortex using the sharp spectral
filter in panels (e),(f).

Fig. A3(a) plots $K^{\textrm{mar}}_{x}(k)$ and $K^{\textrm{mar}}_{z}(k)$ from
eq. (A-2) for the Gaussian data using Gaussian filters. Consistent with their
${\bf K}^{\textrm{shell}}$ analogue shown in fig. 4c, it shows that
$K^{\textrm{mar}}_{z}\leq K^{\textrm{mar}}_{x}$ over the entire range of $k$,
which implies that the field is elongated along the $z$-direction at all
scales. We see that $K^{\textrm{mar}}_{z}\approx K^{\textrm{mar}}_{x}$ in the
limits of $k\to 0$ and $k\to\infty$. The anisotropy metric
$\mathrm{AR}_{zx}^{\mathrm{mar}}(k)$ in panel (b) from eq. (A-3) reflects this
behavior and is qualitatively consistent with its analogue
$\mathrm{AR}_{zx}^{\mathrm{shell}}(k)$ in fig. 4d, although with seemingly
lower sensitivity. This is not surprising given that ${\bf
K}^{\textrm{mar}}(k)$ is derived from a highly reduced spectral representation
as discussed in section B (see also Fig. 1). The sharp spectral filtering
results shown in panels (c) and (d) are similar to the Gaussian kernel results
in panels (a) and (b).

Fig. A3 (e) and (f) show for the Taylor-Green data the centroid ${\bf
K}^{\textrm{mar}}$ based on marginal spectra, which exhibits a spike in
$K_{x}^{\textrm{mar}}$ at $k=7$ and a spike in $K_{z}^{\textrm{mar}}$ at
$k=31$. This is because $\overline{E}_{x}^{\textrm{mar}}$ and
$\overline{E}_{z}^{\textrm{mar}}$ spike at those respective wavenumbers.
However, they highlight a limitation of marginal spectra we mentioned; since
$\overline{E}_{x}^{\textrm{mar}}$ and $\overline{E}_{z}^{\textrm{mar}}$ have
to yield the same total energy when integrated, if
$\overline{E}_{x}^{\textrm{mar}}(k_{1})>\overline{E}_{z}^{\textrm{mar}}(k_{1})$
at $k_{1}$, then we necessarily have
$\overline{E}_{x}^{\textrm{mar}}(k_{2})<\overline{E}_{z}^{\textrm{mar}}(k_{2})$
at another wavenumber $k_{2}$. This is also expressed by the anisotropy metric
$\mathrm{AR}_{ij}^{\mathrm{mar}}(k)$ in panel (f), where
$\textrm{AR}^{\textrm{mar}}_{zx}(7)=-1$ and
$\textrm{AR}^{\textrm{mar}}_{zx}(31)=1$.

(a) centroid of 3D RT KE, x-z (b) anisotropy metric of 3D RT KE, x-z (c)
centroid of 3D RT KE, x-y

(d) anisotropy metric of 3D RT KE, x-y (e) centroid of 2D RT KE, x-z (f)
anisotropy metric of 2D RT KE, x-z

Figure A4: Marginal spectra based centroid (eq. (A-2)) and anisotropic metric
(eq. (A-3)) of the 3D Rayleigh-Taylor kinetic energy along the $x$-$z$
directions in panels (a),(b), and of 3D RT KE along $x$-$y$ directions in
panels (c), (d), and of kinetic energy of 2D RT in panels (e),(f).

The marginal spectra based centroids and anisotropy metrics for the 2D and 3D
RT data are shown in fig. A4. For the shape anisotropy along the $x$-$z$
directions for 3D RT, metrics based on marginal spectra in Fig. A4(a),(b) show
qualitatively consistent results as the shell-based results in fig. 9(c),(d),
albeit with a much weaker anisotropic signal. This is consistent with our
earlier remarks about their lower sensitivity to anisotropy (Fig. 1). After
all, $\overline{E}^{\textrm{mar}}$ has significantly less scale information
than $\overline{E}^{2\textrm{D}}$.

The marginal spectra based shape anisotropy for 3D RT along the $x$-$y$
direction in fig. A4(d) is zero, indicating the flow is isotropy among the
horizontal directions as expected. For 2D RT data in panels (e), (f), metrics
based on marginal spectra fail to capture the small-scale anisotropy compared
to the shell-based results in fig. 12(c),(d), due to their lower sensitivity
discussed in the main text.

###  D Local measurement of shape anisotropy

One advantage of our filtering approach is that we can measure the anisotropy
associated with local sub-regions of the flow in addition to the whole domain.
Fig. A5 shows the centroids of the 2D RT corresponding to three different sub-
regions along vertical direction. The results indicate that the three sub-
regions are qualitatively similar but exhibit slight quantitative differences
in their anisotropy. We defer further analysis to future work.

(a) lower third of the domain (b) middle third of the domain (c) upper third
of the domain

(d) lower third of the domain (e) middle third of the domain (f) upper third
of the domain

Figure A5: The centroids (a)-(c) and anisotropy metrics (d)-(f) of 2D RT
associated with different vertical regions within $z\in[0,L_{z}]$. Panels
(a,d),(b,e),(c,f) are associated with regions $0<z\leq L_{z}/3$,
$L_{z}/3<z\leq 2L_{z}/3$, and $2L_{z}/3<z\leq L_{z}$, respectively.

###  E Isotropy of 3D-RT at the domain-size scales

As we discuss in the main text, at the largest possible scales, coarse-grained
fields approach the spatial mean, which is trivially isotropic. Specifically,
$\overline{\phi}_{\vec{\ell}}\to\\{\phi\\}$ as
${\vec{\ell}}\to(\infty,\infty)$. The spatial mean, $\\{\phi\\}$, is uniform
and, thus, trivially isotropic. This can be easily seen from the illustrative
example of an anisotropic Gaussian in Fig. 3(a), which is isotropic at scales
comparable to the domain size as shown in Fig. 5. This is why the anisotropic
metric $AR^{\textrm{shell}}_{zx}\approx 0$ at $k=1$ in Fig. 4(d).

For 3D-RT, we have a similar situation in Fig. 9(d) where
$AR^{\mathrm{shell}}\approx 0$ at the horizontal domain scale $k=1$ (i.e.
$\ell=L$, where $L=3.2$ is the domain size along the $x$-direction). This can
be seen from the visualization of coarse KE at $k=1$ in Fig. A6. Coarse KE is
isotropic because the filter width is comparable to the horizontal domain
size, such that the coarse-grained fields approache the spatial mean, which is
trivially isotropic. Note that these large scales ($k=1$) where isotropy
exists are larger than the size of biggest bubble or spike in 3D-RT seen in
Fig. 8 and Fig. 10.

Figure A6: Visualization from 3D-RT of coarse KE
$\frac{1}{2}\overline{\rho}|\widetilde{{\bf u}}|^{2}$ filtered at
$\ell=L_{x}$, corresponding to $k=1$.

###  F Insensitivity of our results to filtering kernels

To verify that our results are not sensitive to the particular filtering
kernel adopted in the paper, which is a first-order Gaussian Sadek18 , we
repeat our analysis with a Boxcar kernel, and with a third-order modified
Gaussian filter. The expression of the third-order kernel $G^{I}_{\ell}(x)$
with filtering size $\ell$ is

$\displaystyle
G^{I}_{\ell}(x)=c\,G_{\ell}(x,\ell)-c^{\prime}\,G_{\ell^{\prime}}(x-x_{0})-c^{\prime}\,G_{\ell^{\prime}}(x+x_{0})$
(A-4)

where $x$ is the spatial variable,
$G_{\ell}(x,\ell)=\sqrt{\frac{6}{\pi\ell^{2}}}e^{-\frac{6x^{2}}{\ell^{2}}}$ is
the first-order Gaussian kernel as in eq. 17,
$c=1.1,c^{\prime}=0.05,\ell^{\prime}=0.5\ell,x_{0}=\sqrt{\frac{43}{48}}\ell$
are parameters to satisfy the third-order kernel constraints $\int
G^{I}_{\ell}\,dx=0$ and $\int x^{3}G^{I}_{\ell}\,dx=0$. This higher-order
kernel is, in some sense, more similar to a truncated Dirichlet kernel
spyropoulos1994evaluation  than a Guassian. The shapes of the first- and the
third-order Gaussian, as well as the Boxcar, kernels are shown in fig. A7.

Figure A7: The Boxcar, first- and third-order Gaussian filtering kernels of
width $\ell=L_{x}/8=0.4$.

For convenience, we sample an $x$-$z$ slice from the 3D Rayleigh-Taylor data
and perform the anisotropy metrics using the above three kernels. Fig. A8 and
fig. A9(b) show the centroids, the anisotropic metrics, and the shell-averaged
1-D filtering spectra of KE, in which both the qualitative trends and the
quantitative values are similar. Thus our shape anisotropy results for RT are
not sensitive to the particular filtering kernel we have chosen.

(a) centroid, first-order Gaussian (b) anisotropy metric first-order Gaussian
(c) centroid, boxcar kernel

(d) anisotropy metric boxcar kernel (e) centroid, third-order Gaussian (f)
anisotropy metric third-order Gaussian

Figure A8: Centroid (eq. (A-2)) and anisotropic metric (eq. (A-3)) of the 3D
Rayleigh-Taylor kinetic energy within a $x$-$z$ slice. Panels (a),(b) are for
first-order kernels adopted in the main paper, panels (c),(d) are the results
of top-hat or boxcar filter, while panels (c),(d) are obtained with third-
order kernel of eq. A-4.

Figure A9: Left panel: The shell-integrated (or 1D spetrum in eq. (5)) kinetic
energy spectra associated with the 2D and 3D RT data used in the paper. Right
panel: from 3D-RT data, we show the shell-integrated kinetic energy spectra
using three different filtering kernels: first- and third-order Gaussian and a
boxcar filter. A $k^{-5/3}$ and a $k^{-3}$ scaling is included in each figure.

###  G Structure function scalings of 1D fields

Here, we present the scaling of structure functions associated with periodic
1D field data $\phi(x)$ whose Fourier spectra follows a power-law scaling
$E_{\phi}(k)\sim k^{-\alpha}$, for a few different $\alpha$ values. The field
$\phi(x)$ is defined in a periodic domain $x\in[0,2\pi)$, discretized on a
uniform numerical grid of 32768 points. $\phi(x)$ and its associated Fourier
spectra are shown in Fig. A10. The corresponding second-order structure
function $S_{2}(r)=\langle(u(x+r)-u(x))^{2}\rangle$, where the spatial average
$\langle\dots\rangle$ is over $x\in[0,2\pi)$, is shown in Fig. A11 for 1D
fields with Fourier spectra ranging from $k^{-4}$ to $k^{0}$. Fig. A11(a)
demonstrates that power-law scaling of a 2nd-order structure function,
$S_{2}(r)\sim r^{\alpha-1}$, is related to scaling of the Fourier spectrum
$E(k)\sim k^{-\alpha}$ (legend in Fig. A11), but only when $1<\alpha<3$. We
see from Fig. A11(a) that the scaling of $S_{2}(r)$ is no longer related to
that of the Fourier spectrum when $E(k)$ is steeper than $k^{-3}$ (purple) or
shallower than $k^{-1}$ (blue).

Fig. A11(b) shows filtering spectra $\overline{E}(k_{\ell})$ using a Gaussian
(1st-order) filtering kernel applied to the same data used in panel A11(a). We
see that the filtering spectrum has the same scaling as the Fourier spectrum,
$\overline{E}(k)\sim E(k)\sim k^{-\alpha}$, for $\alpha<3$. Specifically, it
can correctly capture power-law scaling that is shallower than $k^{-1}$ (blue)
but fails for power-law scaling steeper than $k^{-3}$ (purple) since the
Gaussian kernel we are using to calculate $\overline{E}(k_{\ell})$ is a 1st-
order kernel. It is possible for $\overline{E}(k_{\ell})$ to correctly capture
power-laws steeper than $k^{-3}$ by using a higher-order kernel Sadek18 .

(a) 1D field with a (white) spectrum, $E(k)\sim k^{0}$ (b) 1D field with
spectrum $E(k)\sim k^{-3}$

Figure A10: One-dimensional fields in a periodic domain $[0,2\pi)$ whose
Fourier spectra scale as $E(k)\sim k^{0}$ (top row) and $E(k)\sim k^{-3}$
(bottom row).

Figure A11: Left panel: power-law scaling of a 2nd-order structure function,
$S_{2}(r)\sim r^{\alpha-1}$, is related to scaling of the Fourier spectrum
$E(k)\sim k^{-\alpha}$ (legend), but only when $1<\alpha<3$. We see that the
scaling of $S_{2}(r)$ is no longer related to that of the Fourier spectrum
when $E(k)$ is steeper than $k^{-3}$ (purple) or shallower than $k^{-1}$
(blue). The fields analyzed here are 1D periodic data similar to those shown
in Fig. A10. Right panel: filtering spectra $\overline{E}(k_{\ell})$ using a
Gaussian kernel applied to the same data used in the left panel. We see that
the filtering spectrum has the same scaling as the Fourier spectrum,
$\overline{E}(k)\sim E(k)\sim k^{-\alpha}$, for $\alpha<3$. Specifically, it
can correctly capture power-law scaling that is shallower than $k^{-1}$ (blue)
but fails for power-law scaling steeper than $k^{-3}$ (purple) since the
Gaussian kernel we are using to calculate $\overline{E}(k_{\ell})$ is a first-
order kernel. It is possible for $\overline{E}(k_{\ell})$ to correctly capture
power-laws steeper than $k^{-3}$ by using a higher-order kernel Sadek18 .

###  H Fourier Spectrum of a Gaussian

For a 1-dimensional Gaussian function with zero mean and standard deviation
$\sigma$,

$\phi(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^{2}}{2\sigma^{2}}},$ (A-5)

defined over the periodic domain $x\in[-\pi,\pi]$, its Fourier transform when
$\sigma\ll\pi$ is

$\displaystyle\begin{split}\widehat{\phi}(k)&=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-ikx}\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^{2}}{2\sigma^{2}}}dx=\frac{1}{2\pi}e^{-\frac{1}{2}k^{2}\sigma^{2}}~{}.\end{split}$
(A-6)

The factor $1/2\pi$ in the Fourier transform definition ensures that
$\widehat{\phi}(0)$ yields the domain average. Eq. (A-6) is an elementary
textbook result. Thus, the Fourier spectrum of a Gaussian is

$\displaystyle
E(k)=|\widehat{\phi}|^{2}(k)+|\widehat{\phi}|^{2}(-k)=\frac{1}{2\pi^{2}}e^{-k^{2}\sigma^{2}}~{}.$
(A-7)

We drop the factor $\nicefrac{{1}}{{2}}$ in defining $E(k)$ in the Appendix
for convenience. It is straightforward to verify that Parseval’s theorem,
$\int_{0}^{\infty}E(k)dk=\frac{1}{2\pi}\int_{-\pi}^{\pi}\phi(x)^{2}dx$, is
satisfied for $\sigma\ll\pi$. Fig. A12 compares the Fourier spectra of
Gaussians with $\sigma=\pi/8$ and $\sigma=\pi/32$, showing that the narrower
Gaussian ($\sigma=\pi/32$) extends to higher $k$.

Figure A12: Fourier spectra of Gaussians with standard deviations
$\sigma=\pi/8$ and $\sigma=\pi/32$

In our eq. (15) example of the 2D anisotropic Gaussian, we had

$\displaystyle\begin{split}\phi(x,z)=&\frac{1}{\sqrt{2\pi}\sigma_{x}}\frac{1}{\sqrt{2\pi}\sigma_{z}}e^{-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{z^{2}}{2\sigma_{z}^{2}}}~{},\\\
\mathrm{where}\hskip 28.45274pt&(\sigma_{x},\sigma_{z})=(\pi/32,\pi/8)\hskip
28.45274pt\mathrm{and}\hskip 28.45274ptx,z\in[-\pi,\pi]~{}.\end{split}$ (A-8)

Its 2-dimensional Fourier transform follows from eq. (A-6),

$\displaystyle\begin{split}\widehat{\phi}(k_{x},k_{z})=\frac{1}{(2\pi)^{2}}e^{-\frac{1}{2}k_{x}^{2}\sigma_{x}^{2}}~{}e^{-\frac{1}{2}k_{z}^{2}\sigma_{z}^{2}}~{},\end{split}$
(A-9)

and its 2D Fourier spectrum is, therefore,

$\displaystyle\begin{split}E^{2\textrm{D}}(k_{x},k_{z})=\frac{4}{(2\pi)^{4}}e^{-k_{x}^{2}\sigma_{x}^{2}}~{}e^{-k_{z}^{2}\sigma_{z}^{2}}~{}.\end{split}$
(A-10)

###  I Filtering Spectrum of a Gaussian

To obtain the analytical expression for the filtering spectrum of a 1-D
Gaussian function
$\phi(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^{2}}{2\sigma^{2}}}$, using the
filtering kernel
$G_{\ell}(x)=\frac{1}{\sqrt{\pi}\ell}e^{-\frac{x^{2}}{\ell^{2}}}$, we first
evaluate the filtered quantity $\overline{\phi}_{\ell}$, then obtain
$\\{\overline{\phi}_{\ell}^{2}\\}$, before calculating the filtering spectrum
$\overline{E}(k)=\frac{d}{dk}\\{\overline{\phi}_{\ell}^{2}\\}$, with
$k=L/\ell$ and $L=2\pi$. We now show the details.

The filtered Gaussian $\overline{\phi}_{\ell}$ is

$\displaystyle\overline{\phi}_{\ell}=\frac{1}{\sqrt{\pi}\ell}\frac{1}{\sqrt{2\pi}\sigma}\int^{\infty}_{-\infty}{\mathrm{d}}\xi~{}e^{-\frac{(x-\xi)^{2}}{\ell^{2}}}e^{-\frac{1}{2}\left(\frac{\xi}{\sigma}\right)^{2}}~{}.$
(A-11)

Note that in a domain $[-L/2,L/2)$ that is periodic, $\overline{\phi}_{\ell}$
is calculated in Fourier space. This ensures that eq. (A-11) is evaluated
correctly when $\ell\to L$, since the filtering kernel $e^{-x^{2}/\ell^{2}}$
extends beyond $[-L/2,L/2)$ for large $\ell$. Doing so is equivalent to
extending the domain periodically in $x$-space beyond its boundaries.

The cumulative spectrum is

$\displaystyle\begin{split}\\{\overline{\phi}_{\ell}^{2}\\}&=\frac{1}{L}\int^{L/2}_{-L/2}{\mathrm{d}}x~{}\frac{1}{\pi\ell^{2}}\frac{1}{2\pi\sigma^{2}}\int^{\infty}_{-\infty}{\mathrm{d}}\xi_{1}\int^{\infty}_{-\infty}{\mathrm{d}}\xi_{2}~{}e^{-\frac{(x-\xi_{1})^{2}}{\ell^{2}}-\frac{(x-\xi_{2})^{2}}{\ell^{2}}}e^{-\frac{\xi_{1}^{2}+\xi_{2}^{2}}{2\sigma^{2}}}\\\
&=\frac{1}{L\sqrt{2\pi}\sqrt{\ell^{2}+2\sigma^{2}}}=\frac{1}{L\sqrt{2\pi}\sqrt{L^{2}k^{-2}+2\sigma^{2}}}~{},\end{split}$
(A-12)

where the second line assumes $\sigma\ll L$ when evaluating integrals
involving Gaussians. The assumption is only for the convenience of obtaining
an analytical expression. Finally, the filtering spectrum is

$\displaystyle\overline{E}(k)=\frac{d}{dk}\\{\overline{\phi}_{\ell}^{2}\\}=\frac{1}{\sqrt{2\pi}}\frac{k^{-3}}{L^{2}(k^{-2}+2\sigma^{2}/L^{2})^{3/2}}~{},$
(A-13)

which is plotted in Fig. A13 for two different values of $\sigma$. In the
limit $k\to\infty$, we see that $\overline{E}(k)\sim k^{-3}$. As discussed in
Sadek18 , this ‘locking’ at $k^{-3}$ scaling overestimates the energy at large
$k$ and occurs when calculating the filtering spectrum using a 1st order
kernel such as the Gaussian kernel used here. In comparison, the Fourier
spectrum $E(k)$ decays faster than $k^{-3}$ as shown in eq. (A-6) and Fig.
A12. For the filtering spectrum to be meaningful, one has to use a kernel with
at least $p$ vanishing moments when analyzing a function whose Fourier
spectrum decays as $E(k)\sim k^{-\alpha}$ with $\alpha<p+2$. In practice, when
Fourier transforms are difficult to perform, one calculates the filtering
spectrum using a 1st-order kernel then decides if a higher order kernel is
required based on whether or not $\overline{E}(k)$ may be exhibiting a locked
$k^{-3}$ scaling. See Sadek18  for a detailed discussion.

Note that even with a 1st-order kernel, anisotropy is still detected using
filtering spectra. This can be seen from the filtering spectra of the two
Gaussians in Fig. A13, which shows differences in energy at different $k$.
However, the differences are less than those in Fig. A12 for the Fourier
spectra, i.e. the level of anisotropy can be underestimated when using a 1st-
order kernel.

Figure A13: Filtering spectra of Gaussians with standard deviations
$\sigma=\pi/8$ and $\sigma=\pi/32$.

###  J Taylor-Green using the dyadic bands

Fig. A14 shows centroid and anisotropy metric for the Taylor-Green example
using sharp-spectral filter in Fourier space as in Fig. 7 but with dyadic
bands ${\bf k}\in[k/\sqrt{2},\sqrt{2}k]$.

(a) scale-dependent centroid (b) scale-dependent anisotropy metric

Figure A14: Similar to Fig. 7 for the Taylor-Green velocity (eq. (16)) with a
sharp-spectral filter in Fourier space but using the dyadic bands ${\bf
k}\in[k/\sqrt{2},\sqrt{2}k]$. Unlike Fig. 7, no shift of the lines is
performed.

###  K Taylor-Green using Gaussian filter

Fig. A15 shows filtering spectra for the Taylor-Green example using a Gaussian
kernel. In defining filtering wavenumber, $k=L/\ell$, we use the domain size
$L=2\pi$.

(a) cumulative spectrum ${\mathcal{E}}(k_{x},k_{z})$ (b) 2D filtering spectrum
$\overline{E}^{2\textrm{D}}(k_{x},k_{z})$ (c) scale-dependent centroid (d)
scale-dependent anisotropy metric

Figure A15: Similar to Fig. 7 for the Taylor-Green velocity field but using
the Gaussian filtering kernel. Panels (c) and (d) use bands $k<|{\bf k}|\leq
k+1$ as in Fig. 7.

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