# Correlation and decomposition framework for identifying and disentangling
flow structures: canonical examples and application to isotropic turbulence

Siddhartha Mukherjee1  Merlijn Mascini1  Luis M. Portela1
<EMAIL_ADDRESS>1Section of Transport Phenomena, Department of Chemical
Engineering, Delft University of Technology, 2629 HZ, Delft, Netherlands

###### Abstract

Turbulence has long been held synonymous to structure. The description of its
phenomenology often invokes the concept of spatial “coherent structures”
arising in its vector fields. Despite advances in structure eduction
techniques, the organization of turbulence fields has resisted clear
description—primarily, due to a lack of tools to identify instantaneous
spatial organization, aggravated by an obfuscating scale superposition. We
present a generalized correlation framework, and introduce correlation
measures that identify structures as instantaneous patterns in vector fields;
coupled with a paradigm, using Helmholtz decomposition concepts, to
disentangle these structures. After testing the correlations using simple
canonical flows, we apply them to realizations from direct numerical
simulations of homogeneous isotropic turbulence. We find that regions of high
kinetic energy manifest as localized velocity jets, which intersperse the
velocity field, contrary to the prevalent view of high kinetic energy regions
as large swirling structures (eddies). We confirm that regions of high
enstrophy form small vorticity jets, invariably associated with a surrounding
region of swirling velocity. Correlation field statistics viz-a-vis turbulence
fields shows that the high kinetic energy jets and high enstrophy swirls are
mostly spatially exclusive. Decomposing the velocity field, using the Biot-
Savart law, into contributions from different levels and regions of the
vorticity field, reveals the organization of these structures. High kinetic
energy jets are neither self-inducing (due to low vorticity contents), nor
significantly induced by strong vorticity; they are almost entirely induced
by, non-local, intermediate range vorticity ($\sim\omega^{\prime}$, i.e. the
rms vorticity), which permeates the volume. High enstrophy swirls, on the
other hand, are a superposition of self-induced swirling motion along with a
background-induced flow. Intermediate vorticity, moreover, has the highest
contribution to the induction of the velocity field everywhere. This suggests
that turbulence organization could emerge from non-local and non-linear field
interactions, dominated by permeating intermediate vorticity, leading to an
alternative description of turbulence, contrary to the notion of a strict
hierarchy of coherent structures. The tools presented in this paper can be
readily applied to study generic vector and scalar fields associated with
diverse phenomena.

##  1 Introduction

“Structure” in a field can be defined as a certain distribution of the
properties of the field in a region, characterized by a small number of
parameters, which can be described (deterministically) in a “simple way”. For
instance, in a velocity field, swirling motion can be considered as a kind of
structure, which brings to mind examples such as a tornado, cyclone or a
simple bathtub vortex. The concept of structure in flow fields immediately
also invokes the notion of “coherent motion”, one interpretation of which is:
regions of the flow that have a certain spatial pattern (for instance a
swirling motion). This idea of structure can also be understood by considering
its opposite, i.e. a structure-less field, which mathematically may be defined
as random.

The structure in a general field, and in particular in a velocity field, can
be the result of (arbitrary) choices in constructing the field and of the
(intrinsic) dynamics of the field. For example, the addition of a translation
or a rotation generate a “coherent motion” that is not related with the
intrinsic dynamics of the velocity field. The pattern of the field at infinity
can be seen as the result of these arbitrary choices, and it can be “removed”,
in order to obtain patterns associated with the (intrinsic) dynamics of the
field. In classical Newtonian mechanics, this is equivalent to observing
motion with respect to the “distant stars”. The use of correlation and
Helmholtz decomposition concepts allows the generalization of these ideas,
which are essential for deciphering instrinsic field structures. For instance,
the spatial correlation of a field over a sphere with an infinite radius can
be made zero by performing an opposite transformation in the field (eg. a
translation or a rotation). From a Helmholtz decomposition perspective, this
is equivalent to making a transformation in the field such that the
generalized contribution of any region of the infinity-field (far-field
contribution from “large distances”) becomes equal to zero. We apply these
techniques to study turbulence, to investigate both the structures that arise
in turbulent flows, and the composition of these structures from a Helmholtz
decomposition perspective.

Turbulent flows have been found to be very rich in structure across different
representational spaces, so much so that turbulence has been held synonymous
to structure (Tsinober, 2014). Moreover, turbulent flow fields are intriguing
due to the superposition of structure and randomness across scales; uncovering
and characterizing which has garnered profound interest over the past decades.
In describing velocity field structures in turbulence, a key idea often used,
albeit ill-defined, is that of the “eddy”, which also refers to coherent
regions of swirling motion. The superposition of eddies (or coherent motion
across all scales) has served as the conceptual background upon which most of
turbulence theory has been built (Frisch, 1995; Dubrulle, 2019). How these
coherent structures arise across all scales, and what they look like, however,
is not fully known. In this paper, we are interested in finding out whether
the finite-sized spatial structures comprising turbulent flow fields can be
identified and isolated in the vector fields where they are believed to arise.
Further, we are interested in considering instantaneous structures which are
continuously produced and destroyed, and are not the result of an averaging
procedure. According to the conventional “cascade” perspective, these
structures may range from the largest scales that contain most of the kinetic
energy and “drive” the dynamics, to the small scales associated with the
dissipation of kinetic energy. In this framework, it should be noted that the
smallest scales are merely a consequence of the turbulence dynamics, and are
hence not dynamically significant in determining the overall flow (Tsinober,
2014).

There have been various approaches aimed at identifying coherent structures in
different contexts that are prevalent in the turbulence literature. Most
widely used are techniques based upon the velocity gradient tensor
$A_{ij}=\partial u_{i}/\partial x_{j}$ and its symmetric ($S_{ij}$) and skew-
symemtric ($\Omega_{ij}$) parts. For instance, Jeong & Hussain (1995) define a
criterion (called $\lambda_{2}$) based on the eigenvalues of the local
pressure Hessian, which is related to $S_{ij}$ and $\Omega_{ij}$. Dubief &
Delcayre (2000) used the second and third invariants ($Q$ and $R$) of
$A_{ij}$, originally used to characterize the topology of point flow patterns
(Chong et al., 1990), and Haller (2005) used the strain acceleration tensor
along fluid trajectories. Farge & Pellegrino (2001) used a wavelet
decomposition to identify coherent and incoherent vorticity structures,
Hussain (1986) and Sirovich (1987) studied statistically emerging lower
dimensional attractors, while others have extensively studied Lagrangian
structures crucial for material transport (Peacock & Dabiri, 2010; Peacock &
Haller, 2013; Haller, 2015). Non-linear equilibrium solutions have also been
classified as exact coherent structures (Waleffe, 1997, 2001; Deguchi & Hall,
2014). Lozano-Durán & Jiménez (2014) studied spatio-temporally coherent
vortical structures, while She et al. (1990) and Jiménez et al. (1993)
investigated the structure of strong vorticity (worms) in homogeneous
isotropic turbulence, and Moisy & Jiménez (2004) quantified the large-scale
spatial distribution of small, localized, intense vorticity worms.

These (and many other) studies and techniques have greatly informed our
understanding of coherent structures in turbulence. Several of these studies
use a “functional decomposition” approach (eg. wavelet and spectral
decomposition) to study the coherent structures, their relations and
“hierarchy”; eg. Argoul et al. (1989) and Alexakis & Biferale (2018) address
the “cascade” concept (Richardson, 1922) using wavelet and spectral
decompositions, respectively. However, when using a “functional decomposition”
approach the turbulence fields are separated into “classes” and this “class
perspective” does not represent individual coherent structures, and their
relations, occuring in the actual physical space.

Many basic concepts associated with coherent structures, like the existence of
a hierarchy of coherent structures (as invoked, for instance, in the
Richardson (1922) “cascade”), or the energetic interaction of eddies (Waleffe,
1992) and eddy breakups, have remained intractable in the physical space,
where these ideas were first envisioned. Part of this disconnect is due to the
lack of tools designed to identify instantaneous spatial structures, which may
be driving these processes. The other issue is extracting these structures
from their obfuscating scale superposition, in order to study their form and
dynamics. To address these issues we use correlation concepts and Helmholtz
decomposition concepts, which enable us to identify and extract individual
flow structures from turbulence fields. In this study, we deal with
incompressible, homogeneous isotropic turbulence, with a zero mean velocity,
hence the removal of a velocity pattern associated with an “artificial frame
of reference” is not an issue. We approach the concept of coherent structures
with a focus on the following key aspects:

  1. 1.

Finite structure size \- We consider a “coherent structure” to be a finite,
spatial structure, which represents a unit of coherent motion (eg. an “eddy”
is a “coherent structure” in which the coherent motion is a swirling veloity
field). It must, hence, have a spatial form, that is to say, it cannot be
completely irregular. “Coherence”, in this context, becomes almost synonymous
to “correlation”, as an ordered spatial structure must comprise of a
neighbourhood of vectors that are strongly correlated (either positively or
negatively). Here, it becomes important to highlight the distinction from
point-criteria used for educing structures, which are based on the velocity
gradient tensor (or derivatives thereof, like $Q$, $R$, etc.). These
techniques describe point-structures, reasoning from the Taylor expansion
perspective of the velocity field in the infinitesimal neighbourhood of each
point in the flow field. Structures in the flow field, however, are finite
regions of spatio-temporal order, and may not necessarily be related to local
velocity gradients, as the velocity at a point results from spatial integrals
of the velocity derivatives. We study coherent structures in velocity and
vorticity spaces by developing correlation measures designed to seek out
particular spatial order in these structures.

  2. 2.

Instantaneity \- The spatial structures described above exist instantaneously,
and are not consequences of averaging procedures. In fact, a structure will
have an entire lifecycle, from generation until destruction (driven by the
dynamics of the Navier-Stokes equations). While ignoring the temporal
evolution of the structures, in this work we limit ourselves to identifying
structures in instantaneous realizations (i.e. snapshots) of turbulence
fields. Hence, we consider only the geometry of structures and not their
kinematics or dynamics.

  3. 3.

Disentangling structures \- Part of the complexity of turbulence fields comes
from the superposition of structures, which makes it difficult to even define,
let alone extract and study an individual structure. The reason for this
superposition of (velocity field) structures can be understood from the Biot-
Savart law, which gives that the velocity at each point in space is generated
by the spatial integrals of quantities associated with the gradients of the
velocity. The integrals account for both near-field and far-field
contributions. The summative nature of the Biot-Savart law, hence, provides a
paradigm for disentangling the contributions that generate any given velocity
field pattern or structure, as different contributions may be isolated by
employing suitable conditional sampling criteria on the velocity
reconstruction. We use this method to identify regions of the velocity
gradient field which ‘generate’—in a Biot-Savart sense—a particular velocity
structure.

The tools developed in this study, namely a set of generalized correlation
measures, along with velocity reconstruction using the Biot-Savart law, allow
us to look at turbulence fields from a different perspective; for instance
they enable us to identify the structure of high kinetic energy and high
enstrophy regions. Reconstructing the velocity field using the Biot-Savart
law, further, reveals the distribution of the vorticity contributions in the
generation of these structures, along with the generation of the total
velocity field. This paves the way for studying turbulence as a dynamical
system of interacting structures that arise in its physical fields, the
interplay between which manifests as the dynamics. Moreover, our results give
novel insights into turbulence, in particular, regarding the emergence of flow
organization.

The layout of the paper is as follows. We begin by proposing different
instantaneous correlation measures in section 2, which are designed to
identify simple vector-field structures, based upon a generalization of the
correlation tensor, along with correlations associated with the Biot-Savart
law in section 3. These correlations are first applied to canonical flows in
section 4, where some of their features are highlighted. In section 5, the
correlations are applied to incompressible, homogeneous isotropic turbulence
flow fields, where the particular flow structures associated with high kinetic
energy and high enstrophy regions are identified. In section 6 we first show
instances of individual flow structures. These results, obtained using an in-
house code are shown to be essentially similar to those obtained upon using a
reference dataset in Appendix B. Further in section 6, we perform the velocity
field reconstruction using the Biot-Savart law, and both qualitatively show
and quantify, the vorticity composition of velocity field structures,
following which we end with the conclusions of this study where we describe
the picture of emergence of structures in turbulence.

##  2 Generalized correlation

Correlation, in its most general form, can be interpreted as the relation
between one region of a phase-space (or a field) with another; the two regions
and their relation being defined based upon certain rules, when viewed from
another phase-space region (the region of observation). This can be expressed
as the relation between $\mathcal{R}_{1}(\mathcal{S}_{1},\mathcal{T}_{1})$ and
$\mathcal{R}_{2}(\mathcal{S}_{2},\mathcal{T}_{2})$ as viewed from
$\mathcal{R}_{0}(\mathcal{S}_{0},\mathcal{T}_{0})$, as illustrated in figure
1. Here $\mathcal{S}$ denotes a space-set (eg. a bounded continuous region or
a set of points), $\mathcal{T}$ denotes a time-set (eg. a continuous time
interval or a set of time instances), and $\mathcal{R}$ denotes a phase-space-
set defined over $(\mathcal{S},\mathcal{T})$ (which could be defined, for
example, using the velocity field or the pressure field). Based upon a set of
rules given by any function $\psi$, defined over
$\mathcal{S}_{1},\mathcal{S}_{2},\mathcal{S}_{o},\mathcal{T}_{1},\mathcal{T}_{2},\mathcal{T}_{o}$,
the original phase-space-sets can be mapped to a correlation-set
$\mathcal{C}$, with appropriate dimensions, based upon the definition of
$\psi$.

Figure 1: Correlation between phase-space regions $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$, as observed from phase-space region $\mathcal{R}_{0}$; each
region is defined over a space-set $\mathcal{S}$ and a time-set $\mathcal{T}$.
Using a function $\psi$, these regions are mapped onto a correlation space,
producing a region shown here as $\mathcal{C}$.

The usual definition of the two-point correlation tensor for turbulent flows
can be seen as a particular case of this generalized definition, where the
definition additionally also involves statistical (averaging) concepts, with
the phase-space-sets being composed of an ensemble of different realizations;
making it a statistical measure. We first frame the usual definition of the
two-point correlation tensor in the context of this generalized definition.
Then, via analogy, we will define a deterministic two-point correlation, in
order to characterize the structure of individual fields. Note that since the
usual definition of the two-point correlation tensor is a statistical-concept,
applied to an ensemble of different field realizations, it is not necessarily
a good representation of the structure of each individual field. On the other
hand, the deterministic two-point correlation that we will define contains a
(simplified) characterization of the structure of each individual field.

The usual two-point correlation tensor for turbulent flows is defined as

$\mathcal{R}_{ij}(\mathbf{x},\mathbf{r},t,\Delta t)=\left\langle
u_{i}(\mathbf{x},t)u_{j}(\mathbf{x}+\mathbf{r},t+\Delta t)\right\rangle$ (1)

where $\mathbf{x}$ denotes a position, $\mathbf{r}$ the separation between two
positions, $t$ a time, $\Delta t$ the difference between two times,
$\mathbf{u}=(u_{x},u_{y},u_{z})$ the velocity at a given position and time and
$\left\langle{\cdot}\right\rangle$ ensemble averaging. In the context of the
generalized correlation, this definition can be framed as:

  * •

$\mathcal{S}_{1}$: spatial region of interest, containing $\mathbf{x}$

  * •

$\mathcal{S}_{2}$: spatial region of interest, containing
$\mathbf{x}+\mathbf{r}$

  * •

$\mathcal{T}_{1}$: time interval of interest, containing $t$

  * •

$\mathcal{T}_{2}$: time interval of interest, containing $t+\Delta t$

  * •

$\mathcal{R}_{1}$: phase-space-set, composed of the ensemble of velocity
fields defined over $\mathcal{S}_{1}$ and $\mathcal{T}_{1}$, i.e.
$\mathcal{R}_{1}(\mathcal{S}_{1},\mathcal{T}_{1})=\cup_{n}\mathbf{u}_{n}(\mathcal{S}_{1},\mathcal{T}_{1})$

  * •

$\mathcal{R}_{2}$: phase-space-set, composed of the ensemble of velocity
fields defined over $\mathcal{S}_{2}$ and $\mathcal{T}_{2}$, i.e.
$\mathcal{R}_{2}(\mathcal{S}_{2},\mathcal{T}_{2})=\cup_{n}\mathbf{u}_{n}(\mathcal{S}_{2},\mathcal{T}_{2})$

  * •

$\mathcal{S}_{0}=\mathcal{S}_{1}$, $\mathcal{T}_{0}=\mathcal{T}_{1}$,
$\mathcal{R}_{0}=\mathcal{R}_{1}$

  * •

$\psi$, a function composed of a deterministic part, defined over the
individual velocity fields of the ensemble,
$\mathbf{u}_{n}=(u_{x,n},u_{y,n},u_{z,n})$, and a statistical part:

    * –

Deterministic part: $\psi_{n}(\mathbf{x},\mathbf{r},t,t+\Delta
t)=u_{i,n}(\mathbf{x},t)u_{j,n}(\mathbf{x}+\mathbf{r},t+\Delta t)$

    * –

Statistical part: $\overline{\psi}(\mathbf{x},\mathbf{r},t,t+\Delta
t)=\left\langle{\psi_{n}(\mathbf{x},\mathbf{r},t,t+\Delta t)}\right\rangle$
where $\left\langle{\cdot}\right\rangle$ denotes the ensemble averaging
operator, which is usually a linear operator (eg. arithmetic averaging);
however, different (non-linear) operations could also be used, leading to
definitions different from the usual two-point two-time correlation tensor.

Therefore, in the context of the generalized definition, it results that the
usual two-point two-time correlation can be defined as
$\mathcal{C}(\mathcal{R}_{1},\mathcal{R}_{2})$, whose elements are given by
the correlation tensor

$\mathcal{R}_{ij}(\mathbf{x},\mathbf{r},t,\Delta
t)=\overline{\psi}(\mathbf{x},\mathbf{r},t,\Delta t)$ (2)

By considering $\Delta t=0$, i.e. $\mathcal{T}_{1}=\mathcal{T}_{2}$, the two-
point two-time correlation tensor is reduced to the usual two-point
correlation tensor:

$\mathcal{R}_{ij}(\mathbf{x},\mathbf{r},t)=\left\langle{u_{i}(\mathbf{x},t)u_{j}(\mathbf{x}+\mathbf{r},t)}\right\rangle$
(3)

This is a tensor with 9 components, which for a generic turbulent flow is a
function of $\mathbf{x}$, $\mathbf{r}$ and $t$. It contains a lot of
information (from a statistical perspective), however, for some turbulent
flows, the information needed for its characterization can be significantly
reduced. In particular, for homogeneous isotropic turbulence, the dependence
on $\mathbf{x}$ disappears, and the dependence on $\mathbf{r}$ reduces to a
function of the radial distance $r=|\mathbf{r}|$ alone, regardless of the
orientation of the vector $\mathbf{r}$. As the choice of the orientation of
$\mathbf{r}$ is arbitrary and spans all directions of space, the velocity
components $u_{i}$ can be considered along three orthogonal directions, which
can be given as $\mathbf{e}_{1}\parallel\mathbf{r}$,
$\mathbf{e}_{2}\bot\mathbf{r}$ and
$\mathbf{e}_{3}\bot\left(\mathbf{r},\mathbf{e}_{2}\right)$. This yield three
correlation functions $f(r,t)$ (longitudinal), $g_{1}(r,t)$ and $g_{2}(r,t)$
(transverse), respectively. For homogeneous isotropic turbulence, these three
correlation functions completely characterize the usual two-point correlation
tensor; if, additionally, the turbulence is also statistically-steady, the
three correlations depend only on $r$. A simplified characterization of these
correlation functions can be given by using an integral measure of them, which
can be obtained by integrating over $r$, to get the integral lengths.

The usual two-point correlation is a statistical concept, which mixes measures
of the structure of individual fields with a measure of the structure of their
ensemble. In general, the structure of the ensemble does not represent the
structure of individual fields; actually, it can be completely different. For
example, homogeneous isotropic turbulence refers to the ensemble; the
individual fields are often far from being homogeneous and isotropic. The
characterization of the structure using the usual two-point correlation, even
of individual field realizations, only holds statistically (i.e. upon suitable
spatial averaging). We propose a deterministic characterization of the
individual (instantaneous) fields using a correlation definition similar to
the one employed for the usual two-point correlation, but considering only the
deterministic part of $\psi$. This will be supplemented by a simplified
characterization of the correlation, using integral measures, which, since the
individual fields are not homogeneous and isotropic, are different and more
elaborate that the usual integral measures for homogeneous isotropic
turbulence. This simplified characterization, even though incomplete, is more
manageable.

We propose $\mathcal{C}(\mathcal{R}_{1},\mathcal{R}_{2})$ as a definition for
the correlation of individual (instantaneous) vector fields, which, for the
sake of concreteness, is illustrated here for the velocity field. Here again
$\mathcal{S}_{1}$ is the spatial region of interest containing $\mathbf{x}$,
$\mathcal{S}_{2}$ is the spatial region of interest containing
$\mathbf{x}+\mathbf{r}$, $\mathcal{T}_{1}=\mathcal{T}_{2}$ is the time
interval of interest containing $t$. The elements of the phase-space-sets are
the velocity fields and the phase-space-set of observation, $\mathcal{R}_{0}$,
is equivalent to $\mathcal{R}_{1}$. The elements of
$\mathcal{C}(\mathcal{R}_{1},\mathcal{R}_{2})$ are the correlation tensor
fields

$\displaystyle\mathcal{R}_{ij}\left(\mathbf{x},\mathbf{r},t\right)$
$\displaystyle=\psi\left(\mathbf{x},\mathbf{r},t\right)$ (4)
$\displaystyle\psi\left(\mathbf{x},\mathbf{r},t\right)$
$\displaystyle=u_{i}\left(\mathbf{x},t\right)u_{j}\left(\mathbf{x}+\mathbf{r},t\right)$
(5)

The separation vector $\mathbf{r}$ can be represented by a scalar,
$r=|\mathbf{r}|$ denoting the separation distance, and a direction. In 3D
space, this direction can be specified by two angles (and with only one angle
in 2D) i.e. the azimuthal angle $\alpha$ and the elevation angle $\beta$. We
can define the separation vector $\mathbf{r}_{\alpha\beta}$ as a vector of
length $r$ which points along the direction specified by $\alpha$ and $\beta$,
while being placed at point $\mathbf{x}$. Hence, the correlation tensor can be
written as $\mathcal{R}_{ij}(\mathbf{x},r_{\alpha\beta},t)$, which is a
function of seven variables, namely $x,y,z,r,\alpha,\beta,t$, where $x,y,z$
are the Cartesian components of $\mathbf{x}$. Since in this work we limit
ourselves to identifying structures in instantaneous field realizations, for
the remainder of this paper we will omit the time dependence $t$. The
correlation tensor can then be expressed in matrix form as

$\mathcal{R}_{ij}(\mathbf{x},\mathbf{r}_{\alpha\beta})=\left[\begin{array}[]{ccc}u_{x}(\mathbf{x})u_{x}(\mathbf{x}+\mathbf{r}_{\alpha\beta})&u_{x}(\mathbf{x})u_{y}(\mathbf{x}+\mathbf{r}_{\alpha\beta})&u_{x}(\mathbf{x})u_{z}(\mathbf{x}+\mathbf{r}_{\alpha\beta})\\\
\displaystyle
u_{y}(\mathbf{x})u_{x}(\mathbf{x}+\mathbf{r}_{\alpha\beta})&u_{y}(\mathbf{x})u_{y}(\mathbf{x}+\mathbf{r}_{\alpha\beta})&u_{y}(\mathbf{x})u_{z}(\mathbf{x}+\mathbf{r}_{\alpha\beta})\\\
\displaystyle
u_{z}(\mathbf{x})u_{x}(\mathbf{x}+\mathbf{r}_{\alpha\beta})&u_{z}(\mathbf{x})u_{y}(\mathbf{x}+\mathbf{r}_{\alpha\beta})&u_{z}(\mathbf{x})u_{z}(\mathbf{x}+\mathbf{r}_{\alpha\beta})\\\
\end{array}\right]$ (6)

This contains a lot of information, which needs to be reduced for practical
reasons. So, instead of considering the entire $\mathcal{R}$ matrix, we work
with one of its invariants, the trace, which is written as

$\mathrm{tr}\left(\mathcal{R}_{ij}(\mathbf{x},\mathbf{r}_{\alpha\beta})\right)=u_{x}(\mathbf{x})u_{x}(\mathbf{x}+\mathbf{r}_{\alpha\beta})+u_{y}(\mathbf{x})u_{y}(\mathbf{x}+\mathbf{r}_{\alpha\beta})+u_{z}(\mathbf{x})u_{z}(\mathbf{x}+\mathbf{r}_{\alpha\beta})$
(7)

This is easier to conceptualize, as the trace is also the dot product between
the velocities at points $\mathbf{x}$ and
$\mathbf{x}+\mathbf{r}_{\alpha\beta}$

$\mathrm{tr}\left(\mathcal{R}_{ij}(\mathbf{x},\mathbf{r}_{\alpha\beta})\right)=\mathbf{u}(\mathbf{x})\cdot\mathbf{u}(\mathbf{x}+\mathbf{r}_{\alpha\beta})$
(8)

A further reduction can be performed by integrating this quantity along
directions specified by $\mathbf{r}_{\alpha\beta}$, to associate an integral
measure $L_{\alpha\beta}(\mathbf{x},\Lambda)$ along each direction as

$L_{\alpha\beta}(\mathbf{x},\Lambda)=\int_{0}^{\Lambda}\mathbf{u}({\mathbf{x}})\cdot\mathbf{u}({\mathbf{x}+\mathbf{r}_{\alpha\beta})}\mathrm{d}r_{\alpha\beta}$
(9)

The correlation tensor field is hence reduced to a two-dimensional manifold
around each point, as illustrated in figure 2. This 2D manifold gives a
simplified characterization of the structure of the field around $\mathbf{x}$.
For a given $\Lambda$, its shape gives an integral measure of how this
structure depends on the direction. For instance, the direction of maximum
$L_{\alpha\beta}$ could be determined as a function of position; however, even
though interesting in itself, this will not be explored here. Further, this
manifold is invariant under translation and rotation of the original
coordinate axes, along with being invariant under reflection (similarly to an
axial vector).

Figure 2: A correlation surface around point $\mathbf{x}$, defined by the
measure $L_{\alpha\beta}$ found by integrating
$\mathrm{tr}(\mathcal{R}_{ij}(\mathbf{x},\mathbf{r}_{\alpha\beta}))$ upto
distance $\Lambda$, along a direction specified by the angles $\alpha$
(azimuth) and $\beta$ (elevation).

In principle, such a manifold can be calculated for any vector field, leading,
for a given $\Lambda$, to a correlation surface for each point in space. This
still contains a lot of information, which, in general, can pose difficulties
to represent in a compact form and interpret. Also, since we shall utilize
numerical datasets, the calculation of the manifold requires binning the
angles $\alpha$ and $\beta$ into discrete increments. The resolution of these
angles will depend significantly upon the resolution of the data, where high
resolution simulations will be required to acurately describe even a small
subset of angles, along with demanding computational requirement to calculate
the correlation manifold at each point in space.

Therefore, we perform a final simplification, where instead of the entire
manifold $L_{\alpha\beta}(\mathbf{x},\Lambda)$, we represent it by a three-
tuple $\mathbf{L}(\mathbf{x},\Lambda)$, along three arbitrary orthogonal
directions $x,y,z$ (forming an orthonormal base), which can be summarized as

$\mathbf{L}(\mathbf{x},\Lambda)=\left[\begin{array}[]{c}\displaystyle
L_{x}(\mathbf{x},\Lambda)\\\ \displaystyle L_{y}(\mathbf{x},\Lambda)\\\
\displaystyle L_{z}(\mathbf{x},\Lambda)\\\
\end{array}\right],\quad\mathrm{where}\quad
L_{i}(\mathbf{x},\Lambda)=\int_{-\Lambda}^{\Lambda}\mathbf{u}(\mathbf{x})\cdot\mathbf{u}(\mathbf{x}+\mathbf{r}_{i})\mathrm{d}r_{i}$
(10)

where $i\in\left\\{x,y,z\right\\}$ represents the three spatial directions.
Note that this three-tuple, being a simplifcation of the manifold, is also
invariant to rotation, translation and reflection. Hence, $\mathbf{L}$ at each
point $\mathbf{x}$ depends only on the choice of the arbitrary $x,y,z$
directions used to “sample” the manifold.

This correlation measure, $\mathbf{L}$, can be expected to yield large
absolute values at points $\mathbf{x}$ that are surrounded by large regions in
which (i) the local flow streamlines are well-aligned (such that the
directions of $\mathbf{u}(\mathbf{x})$ and $\mathbf{u}(\mathbf{x}+\mathbf{r})$
are similar) and (ii) the magnitude of these vectors is high. The definition
of $\mathbf{L}$ provides a combined measure of the organization and size of
the structure, and of the magnitude of the field in that region. The
definition does not include an implicit normalization, which, for instance,
could be achieved by dividing $L_{i}(\mathbf{x},\Lambda)$ by the integral of
the kinetic energy along the $i$ direction, within the limits
$-\Lambda<\mathbf{r}_{i}<\Lambda$. The current definition is expected to
identify regions of the flow which contain both structural organization (in
the manner of well-aligned streamlines), and a large field magnitude.
Normalizing the correlation can allow identifying regions with structural
organization alone, while disregarding the field magnitude. Note that, since
the current definition also includes the size of the structure in the
correlation measure, a (very) small region with a high field magnitude and a
high structural organization will not have a large absolute value of
$\mathbf{L}$. A separate consideration of the size of the structure could be
achieved by analyzing the influence of $\Lambda$ on $\mathbf{L}$. The value of
$\Lambda$ limits the size of the structures, and the limit $\Lambda\to 0$
leads to a point-criterion. For “large” values of $\Lambda$, structures of all
sizes can contribute to $\mathbf{L}$, with the larger ones being associated
with larger absolute values of $\mathbf{L}$, hence, the analysis of the
variation of $\mathbf{L}$ with $\Lambda$ allows to separate the effect of the
size from the structural organization and field magnitude effects. Different
forms of the correlation measures can be defined, to educe different aspects
of structural organization, including, or not, in different ways, the size of
the structure and field magnitude effects. For the present study, we do not
normalize the correlation measures. Also, we limit ourselves to a value of
$\Lambda$ that is “large enough” to include all the “relevant” structure
sizes. In section 5.3 we perform a limited study on the influence of the
choice of $\Lambda$ and show that for the situation considered in this study
(homogeneous isotropic turbulence) the Taylor microscale is a good choice for
$\Lambda$.

A different way of constructing the $\mathbf{L}$ correlation measure can be

$\mathbf{L^{s}}(\mathbf{x},\Lambda)=\left[\begin{array}[]{c}\displaystyle
L^{s}_{x}(\mathbf{x},\Lambda)\\\ \displaystyle
L^{s}_{y}(\mathbf{x},\Lambda)\\\ \displaystyle
L^{s}_{z}(\mathbf{x},\Lambda)\\\ \end{array}\right],\quad\mathrm{where}\quad
L^{s}_{i}(\mathbf{x},\Lambda)=\int_{0}^{\Lambda}\mathbf{u}(\mathbf{x}-\mathbf{r}_{i})\cdot\mathbf{u}(\mathbf{x}+\mathbf{r}_{i})\mathrm{d}r_{i}$
(11)

Here, the correlation measure along an axis is constructed using the dot
product between velocity pairs equidistant from $\mathbf{x}$, symmetrically,
along a given direction (hence the notation $\mathbf{L^{s}}$ for
$\mathbf{L}$-symmetric). This measure is also expected to yield high values
when the flow streamlines are parallel (or anti-parallel) in the
$\Lambda-$neighbourhood of $\mathbf{x}$, and when the magnitude of the vectors
is high. Moreover, this measure will be more sensitive to the larger
symmetries and anti-symmetries in the field. These two correlation measures
are illustrated in figure 3.

(a) In the $\mathbf{L}$ correlation measure, the velocity at point
$\mathbf{x}$ is correlated to the velocities between points
$-\Lambda<\mathbf{r}_{i}<\Lambda$.

(b) In the $\mathbf{L^{s}}$ correlation measure, velocity pairs equidistant
from the point $\mathbf{x}$ are correlated, between
$0<|\mathbf{r}_{i}|<\Lambda$. These pairs are marked by the same symbol.

Figure 3: Schematic of the $\mathbf{L}$ and $\mathbf{L^{s}}$ correlation
measures, shown along the $x$ direction.

These measures can be applied to any vector field. We define correlation
measures $\mathbf{G}(\mathbf{x},\Lambda)$ and
$\mathbf{G^{s}}(\mathbf{x},\Lambda)$ for the vorticity field, which by analogy
are given as

$\mathbf{G}(\mathbf{x},\Lambda),\quad\mathrm{where}\quad
G_{i}(\mathbf{x},\Lambda)=\int_{-\Lambda}^{\Lambda}\boldsymbol{\omega}(\mathbf{x})\cdot\boldsymbol{\omega}(\mathbf{x}+\mathbf{r}_{i})\mathrm{d}r_{i}$
(12)

and

$\mathbf{G^{s}}(\mathbf{x},\Lambda),\quad\mathrm{where}\quad
G^{s}_{i}(\mathbf{x},\Lambda)=\int_{0}^{\Lambda}\boldsymbol{\omega}(\mathbf{x}-\mathbf{r}_{i})\cdot\boldsymbol{\omega}(\mathbf{x}+\mathbf{r}_{i})\mathrm{d}r_{i}$
(13)

The correlation measure $\mathbf{G}$ is expected to yield large absolute
values at points $\mathbf{x}$ that are surrounded by large regions in which
(i) the vorticity streamlines are well-aligned and (ii) the magnitude of the
vorticity is high. The correlation measure $\mathbf{G^{s}}$ is the vorticity
field equivalent of $\mathbf{L^{s}}$, and is expected to be more sensitive to
the symmetries and anti-symmetries in the vorticity field, along with being
sensitive to the vorticity magnitude.

Note that, in general, structures in the velocity and vorticity fields can be
very different, with very different sizes and magnitudes, hence, the concept
of “large regions” and “large values” are relative and need to be interpreted
in the individual context of the different correlation measures. In
homogeneous isotropic turbulence, regions of high vorticity magnitude are
related to the smaller scales of turbulence; they correspond to the long tails
of the statistical distribution of the vorticity (i.e.
$\omega\gg\omega^{\prime}$ where
$\omega^{\prime}=\left\langle\omega^{2}\right\rangle^{1/2}$) and occur
intermittently. The correlation measures $\mathbf{L}$ (and $\mathbf{L^{s}}$)
and $\mathbf{G}$ (and $\mathbf{G^{s}}$) give a sepearate (simplified)
characterization of the structure of the individual (instantaneous) velocity
and vorticity fields. For simplicity of language, here onward in the paper we
will refer to them, and other correlation measures, simply, as correlations.

The correlations defined so far consider the velocity and vorticity fields
separately, however, other correlations can be defined, which use both these
fields, exploiting the relation between the velocity and vorticity. The
vorticity $\boldsymbol{\omega}$ is defined as
$\boldsymbol{\omega}=\nabla\times\mathbf{u}$. The velocity field, in turn, can
be reconstructed from the vorticity field using the Biot-Savart law. This
serves as an important tool to identify, as well as disentangle, structures,
and is briefly described below.

##  3 Biot-Savart reconstruction and associated correlations

###  3.1 Biot-Savart reconstruction

We start with the Helmholtz decomposition, which states that a sufficiently
smooth (twice continuously differentiable) vector field, defined on a bounded
or an unbounded domain, can be uniquely decomposed into three components : (i)
an irrotational vector field $\mathbf{D}$, (ii) a solenoidal vector field
$\mathbf{C}$ and (iii) a harmonic vector field $\mathbf{B}$. Applied to the
velocity field $\mathbf{u}$, this can be written as

$\mathbf{u}=\mathbf{D}+\mathbf{C}+\mathbf{B}$ (14)

where

$\displaystyle\mathbf{D}$ $\displaystyle=\nabla\phi$ (15a)
$\displaystyle\mathbf{C}$ $\displaystyle=\nabla\times\boldsymbol{\varphi}$
(15b) $\displaystyle\mathbf{B}$ $\displaystyle=\nabla\psi$ (15c)

and

$\displaystyle\nabla^{2}\phi$ $\displaystyle=\nabla\cdot\mathbf{u}$ (16a)
$\displaystyle\nabla^{2}\boldsymbol{\varphi}$
$\displaystyle=-\nabla\times\mathbf{u}=-\boldsymbol{\omega}$ (16b)
$\displaystyle\nabla^{2}\psi$ $\displaystyle=0$ (16c)

In a bounded domain with a volume $V$ and a bounding surface $S$, the three
components can be written as a generalized Biot-Savart law (see for instance
Wu et al. (2007))

$\displaystyle\mathbf{D}$
$\displaystyle=\frac{1}{4\pi}\int_{V}\frac{\nabla\cdot\mathbf{u}}{|\mathbf{r}|^{3}}\mathop{}\\!\mathrm{d}V^{\prime}$
(17a) $\displaystyle\mathbf{C}$
$\displaystyle=\frac{1}{4\pi}\int_{V}\frac{\boldsymbol{\omega}\times\mathbf{r}}{|\mathbf{r}|^{3}}\mathop{}\\!\mathrm{d}V^{\prime}$
(17b)
$\displaystyle\mathbf{B}=-\frac{1}{4\pi}\int_{S}\frac{\left(\mathbf{n}\cdot\mathbf{u}\right)\mathbf{r}+\left(\mathbf{n}\times\mathbf{u}\right)\times\mathbf{r}}{|\mathbf{r}|^{3}}\mathop{}\\!\mathrm{d}S^{\prime}$
$\displaystyle=-\frac{1}{4\pi}\int_{S}\frac{\left(\mathbf{n}\cdot\mathbf{r}\right)\mathbf{u}+\left(\mathbf{n}\times\mathbf{r}\right)\times\mathbf{u}}{|\mathbf{r}|^{3}}\mathop{}\\!\mathrm{d}S^{\prime}$
(17c)

where $\mathbf{r}$ is the position vector, from a point in the volume $V$ (or
the surface $S$) to the point $\mathbf{x}$ where the integrals are being
evaluated. The integrals over the volume $V$ can be considered the “near-
field” contribution whereas the surface integral over the bounding surface $S$
can be considered the “far-field” contribution. Any region within the domain
has a bounding surface that separates it from the rest of the domain.
Therefore, the integral over this bounding surface can be seen as a
representing the sum of the contributions of (i) the integral over the volume
surrounding the region and (ii) the integral over the bounding surface of the
whole domain.

In an unbounded (infinite) domain, the Helmholtz decomposition reduces to

$\mathbf{u}=\mathbf{D}+\mathbf{C}$ (18)

provided that $\mathbf{B}$ goes to zero when $|\mathbf{r}|$ goes to infinity.
This happens if the distance over which the velocity field is correlated along
the surface grows slower than $|\mathbf{r}|$ when $|\mathbf{r}|$ goes to
infinity.

This can be illustrated by considering an integral correlation measure
$L_{\alpha}$ defined on the surface of a sphere, similarly to what was done
for 3D, which is given as

$L_{\alpha}\left(\mathbf{x},\Lambda\right)=\int_{0}^{\Lambda}\mathbf{u}({\mathbf{x}})\cdot\mathbf{u}({\mathbf{x}+\mathbf{r}_{\alpha})}\mathrm{d}r_{\alpha}$
(19)

where $\mathbf{x}$ is a point on the surface of the sphere. Since this
calculation is confined to the surface, the correlation around each point
$\mathbf{x}$ is simply a function of an angle $\alpha$ and an integration
length $\Lambda$, measured along the direction (on the surface) specified by
$\alpha$. In the Helmholtz decomposition, the contribution of the boundary to
the velocity field (i.e. the contribution of $\mathbf{B}$ to $\mathbf{u}$)
goes to zero if

$\lim_{|\mathbf{r}|\to\infty}\frac{L_{\alpha}\left(\mathbf{x},\Lambda\right)}{|\mathbf{r}|}=0$
(20)

for any $\mathbf{x}$, $\alpha$ and $\Lambda<\pi|\mathbf{r}|$ (see, for
example, Phillips (1933)). For an infinite domain, the contribution of
$\mathbf{B}$ to the velocity field could be the result of an “organized motion
over an infinite distance”, which can be seen as the result of (arbitrary)
choices in constructing the field (e.g. the choice of a particular frame of
reference). This arbitrary artificial “coherent motion”, which is the
“extrinsic motion” can be removed by making a transformation in the velocity
field such that the condition given by eq. 20 becomes true for any
$\mathbf{x}$, $\alpha$ and $\Lambda<\pi|\mathbf{r}|$; this condition will
result in the integrals given by eq. 17c becoming equal to zero for any
surface $S$ in the sphere, when the radius of the sphere goes to infinity.

When doing turbulent flow simulations with periodic boundary conditions, any
arbitrary artificial “coherent motion” can be removed by not considering
$\mathbf{B}$, provided that the domain is large enough to take into account
any relevant “intrinsic motion”; i.e. provided that the domain is
“significantly” larger than the distance over which the velocity field is
correlated; this requires that the domain needs to be “significantly” larger
than the integral length scale of the turbulence, which is our case. If this
happens, for $|\mathbf{r}|$ larger than half the domain size the contribution
of $\mathbf{B}$ to the velocity approaches a constant, which can be made zero
by neglecting $\mathbf{B}$ (which is equivalent to “choosing the appropriate
frame of reference”). Actually, in our case, as usually true for turbulence
simulations in triperiodic domains, this constant is already zero, since the
mean velocity is equal to zero; as it will be shown, in this case $\mathbf{C}$
(i.e. eq. 17b) gives a good approximation of $\mathbf{u}$ (since we deal with
an incompressible flow, $\mathbf{D}=0$) for $|\mathbf{r}|$ approaching the
domain size.

With these considerations, the generalized Helmholtz decomposition reduces to
the simplified Biot-Savart law, applicable for incompressible flows over
periodic (or infinite) domains, which is given as

$\mathbf{u}(\mathbf{x})=\frac{1}{4\pi}\int_{V}\frac{\boldsymbol{\omega}\times\mathbf{r}}{|\mathbf{r}|^{3}}\mathrm{d}V^{\prime}$
(21)

The Biot-Savart law provides a way to disentangle flow structures by isolating
the contributions from different vorticity regions to a local velocity
structure. For instance, local and non-local vorticity contributions can be
separated using this paradigm, or the vorticity field can be conditionally
sampled to identify the contribution of different vorticity levels in
generating velocity field structures.

Note that, even though far-field vorticity contribution, i.e. $\mathbf{B}$,
may be absent, the volumetric region can also be split into an isolated local
region ($V_{\mathrm{L}}$), surrounded by the non-local region
($V_{\mathrm{NL}}$), which essentially behaves as a far-field for the region
$V_{\mathrm{L}}$. This leads to the consequence that, if the local region has
negligible vorticity within $V_{\mathrm{L}}$, while the non-local region
$V_{\mathrm{NL}}$ induces a velocity field within $V_{\mathrm{L}}$, then this
velocity field must be a potential flow. This means that the local flow in
$V_{\mathrm{L}}$ can be described by the gradient of a harmonic function
$\psi$, i.e. $\mathbf{u}_{\mathrm{L}}=\nabla\psi$, while $\nabla^{2}\psi=0$.
The non-local contributions from $V_{\mathrm{NL}}$ cannot generate vorticity
within the local region $V_{\mathrm{L}}$ (as illustrated in figure 4). A last
feature to note regarding the Biot-Savart law is the rapid decay (of $1/r^{2}$
over a distance $r$) of the vorticity contribution, which means that a small,
isolated, vorticity region cannot extend its influence over a large distance
beyond its immediate neighbourhood.

Figure 4: Schematic of a Biot-Savart reconstruction of the velocity field in a
region $V_{\mathrm{L}}$, from three isolated non-local regions $V_{1},V_{2}$
and $V_{3}$. When the region $V_{\mathrm{L}}$ has negligible vorticity of its
own, the flow generated within the region by non-local vorticity contributions
is a potential flow which can be written as the gradient of a harmonic
function $\psi$.

###  3.2 Correlations related to the Biot-Savart law

Ideas associated with the Biot-Savart law can be used to define correlation
measures in order to identify, extract and disentangle structures associated
with the relation between the velocity and vorticity fields. In regions of
strong vorticity associated with swirling-flow in the orthogonal plane, the
Lamb vector, i.e. $\boldsymbol{\omega}\times\mathbf{u}$, yields high values.
Although, this is again a local quantity. We propose a correlation which
utilizes this idea and extends it to a non-local form, where the vorticity at
point $\mathbf{x}$ is correlated with the velocity at point
$\mathbf{x}+\mathbf{r}_{i}$, leading to a three-tuple
$\mathbf{H}(\mathbf{x},\Lambda)$, which, similarly to
$\mathbf{L}(\mathbf{x},\Lambda)$, can be written as follows

$\mathbf{H}(\mathbf{x},\Lambda)\quad\mathrm{where}\quad
H_{i}(\mathbf{x},\Lambda)=\int_{-\Lambda}^{\Lambda}\left(\frac{\boldsymbol{\omega}(\mathbf{x})\times\mathbf{r}_{i}}{|\mathbf{r}_{i}|}\right)\cdot\mathbf{u}(\mathbf{x}+\mathbf{r}_{i})\mathrm{d}r_{i}$
(22)

The above correlation has a flavour of the Biot-Savart law and it allows
correlating the contribution of the local vorticity
$\boldsymbol{\omega}(\mathbf{x})$ with the global vorticity contribution to
its neighbouring velocity field, since the velocity
$\mathbf{u}(\mathbf{x}+\mathbf{r}_{i})$ can be seen as an integral result of
the global vorticity field. Note that, with the above definition,
$\mathbf{H}(\mathbf{x},\Lambda)$ will tend to be orthogonal to
$\boldsymbol{\omega}(\mathbf{x})$, as $H_{i}$ (i.e. $H$ along the
$i-$direction) will have a high magnitude when the vorticity is large and
orthogonal to the $i-$direction.

Since the vorticity at a point generates flow, in a Biot-Savart sense, in the
plane orthogonal to the vorticity vector, a more natural correlation
definition is proposed, which takes into account this fact. This is done by
correlating the vorticity along a particular direction, say $x$, to the flow
in the orthogonal plane, i.e. $yz$. At each point, a velocity field is
generated using the $x-$vorticity (i.e. $\omega_{x}$) in the orthogonal
$yz-$plane within a circular region (along perimeters of circles of radius
$0<r_{yz}\leq\Lambda$). This local velocity field is calculated with a
simplification of the Biot-Savart law, by taking the cross product of the
vorticity with unit vectors in the orthogonal plane
($\mathbf{r}_{yz}/|\mathbf{r}_{yz}|$), as done for the $\mathbf{H}$
correlation. This is illustrated in figure 5, where the $\omega_{x}$ vorticity
component generates the velocity field shown in blue (solid lines), while the
real velocity field generated from the global vorticity contributions is shown
in red (dashed lines). The $\mathbf{H^{p}}$ correlation (for
$\mathbf{H}-$planar) is calculated as the integral of the dot product between
the vorticity-generated velocity vectors (blue) and the real velocity (red),
over rings of radius $0<r_{yz}<\Lambda$. Since the length of the rings
increases proportionally with the radius $r$, the integral over each ring is
further divided by $r$ (i.e. $|\mathbf{r}_{yz}|$), to give an average
correlation at a distance $r$, though other definitions can be used. This
correlation is given by

$\mathbf{H^{p}}(\mathbf{x},\Lambda)\quad\mathrm{where}\
H^{p}_{x}(\mathbf{x},\Lambda)=\int_{0}^{\Lambda}\frac{1}{|\mathbf{r}_{yz}|}\oint_{\mathcal{L}}\left(\frac{{\omega_{x}}(\mathbf{x})\hat{i}\times\mathbf{r}_{yz}}{|\mathbf{r}_{yz}|}\right)\cdot\mathbf{u}(\mathbf{x}+\mathbf{r}_{yz})\
\mathrm{d}l\ \mathrm{d}r_{yz}$ (23)

where $\hat{i}$ is the unit vector in the $x$-direction. $H^{p}_{y}$ and
$H^{p}_{z}$ are defined in a similar way. Evidently, this correlation is more
computationally expensive to calculate than $\mathbf{H}$, as three planar
regions need to be considered for each point in space. This requirement can be
relaxed by sampling the rings $0<r\leq\Lambda$ with a chosen frequency, i.e.
using every $n$-th ring such that $r\in\left\\{n,2n,3n,\dotsc\right\\}$.

Figure 5: A schematic of the $\mathbf{H^{p}}$ correlation. The $\omega_{x}$
vorticity is used to generate a velocity field in the $yz-$plane (shown in
solid-blue lines), which is correlated with the real velocity field (dashed-
red lines) within a circular region of radius $\Lambda$. The correlation is
integrated around rings ($\mathrm{d}l$ elements) and then along the radial
direction ($\mathrm{d}r$). The correlation will be strong when the local
vorticity dominates in producing, in a Biot-Savart sense, the velocity field
in its neighbourhood (left), and low when the local velocity field is not
associated with the central vorticity (right).

$\mathbf{H^{p}}(\mathbf{x},\Lambda)$ can be seen as an area-integral
correlation-measure of the planar (2D) version of $\mathbf{C}$ in eq. 17b
(see, for e.g., Wu et al. (2007)) with $\mathbf{u}$. In other words, it
correlates the contribution of $\omega_{x}$ (the vorticity at $\mathbf{x}$) to
$\mathbf{C}$ at $\mathbf{x}+\mathbf{r}$ with $\mathbf{u}$ at
$\mathbf{x}+\mathbf{r}$; i.e. it correlates the value of
$\mathbf{u}(\mathbf{x}+\mathbf{r})$ due to $\omega_{x}(\mathbf{x})$ with the
actual value of $\mathbf{u}(\mathbf{x}+\mathbf{r})$. However, contrary to,
e.g., $L_{\alpha\beta}(\mathbf{x},\Lambda)$, which is a line-integral
correlation-measure over a distance $\Lambda$,
$\mathbf{H^{p}}(\mathbf{x},\Lambda)$ is an area-integral correlation-measure
over a disc of radius $\Lambda$. Similarly, different combinations of the 2D
and 3D versions of $\mathbf{C}$, and line, area and volume integral
correlation-measures can be constructed, providing different “Biot-Savart
perspectives” on the correlation between the vorticity field and the velocity
field; however, we do not further explore these possibilities here.

##  4 Correlations applied to simplified canonical flows

In this section we test the correlations developed in the two previous
sections using simplified flows, which are constructed using simplified
canonical flows. In order to illustrate some key features of the different
correlations, we consider simplified 1D, 2D and 3D velocity fields.

###  4.1 One-dimensional fields

We construct a 1D velocity field along a line, starting with Oseen vortices,
which can be defined by a tangential velocity field and a vorticity field,
given as

$\displaystyle u_{\theta}(r)$ $\displaystyle=\frac{\Gamma}{2\pi
r}\left[1-\exp\left(-\frac{r^{2}}{4\nu t}\right)\right]$ (24)
$\displaystyle\omega_{z}(r)$ $\displaystyle=\frac{\Gamma}{4\pi\nu
t}\exp\left(-\frac{r^{2}}{4\nu t}\right)$ (25)

where $r$ is the radial distance from the vortex center, $\Gamma$ is the
circulation, $\nu$ is the fluid kinematic viscosity and $t$ is the time. The
Oseen vortex comprises a small core region in (near) solid body rotation,
within which the velocity increases (approximately) radially as
$u_{\theta}\propto r$ to its maximum value. Beyond this, there exists a (near)
potential flow region (where $\omega_{z}$ is nearly zero) and
$u_{\theta}\propto 1/r$. We add a noise $\zeta$ to the velocity field given by
eq. 24, to generate a “structure immersed in noise”. The vortex has a certain
‘reach’, which depends on the amplitude of $\zeta$, and is defined as the
distance beyond which $\zeta>u_{\theta}$. The vorticity of the Oseen vortex is
calculated in Cartesian coordinates as
$\omega_{z}=\nabla\times\mathbf{u}=\partial u_{y}/\partial x-\partial
u_{x}/\partial y$, and not using eq. 25, due to the addition of the noise to
the velocity field.

We generate the velocity field by placing the centers of two counter-rotating
Oseen vortices along the $x-$axis, separated by a distance greater than the
typical reach of either vortex. Along the $x-$axis the velocity only has a
$y-$component, $u_{y}$, and it is this one-dimensional velocity field that we
consider. Furthermore, we impose a periodicity in $u_{y}$, over a length of
$N_{x}$. This field, hence, consists of a large periodic structure, with
smaller sub-structures (which are associated with each of the Oseen vortices
and their interaction). This creates a pattern of symmetries and anti-
symmetries in the velocity and vorticity fields.

The two vortices are generated using the same parameters, which are given (in
arbitrary units) as: $\Gamma=10$, $\nu=2.0$ and $t=2.0$. The length of the
field is $N_{x}=500$ and the centers of the vortices are placed at $150$ and
$350$, with a core solid body rotation region extending over roughly $5$
units, and an Oseen velocity pattern extending roughly $50$ units, on either
side of the centers. Uniformly distributed random noise ($-1<\zeta<1$), scaled
to an amplitude of $1\%$ of the maximum velocity magnitude, was added to the
velocity. The velocity reduces to within $2.5\%$ of the maximum value in the
range of $x<50$ and $x>450$. As can be seen in the top panel of figure 6, the
velocity field consists of a “larger structure” (approximately in the range
$50<x<450$), which is symmetric, except for the noise, with respect to its
middle ($x\approx 250$). This larger structure contains the following sub-
structures: (i) two Oseen velocity patterns around the center of each vortex
(within approximately $100<x<200$ and $300<x<400$), containing a solid body
rotation region and a potential flow region, and (ii) an almost uniform
velocity region (between roughly $200<x<300$) due to the interaction between
the two vortices.

Figure 6: (a) A one-dimensional velocity field comprising a larger structure
composed of two smaller sub-structures (i.e. two counter-rotating Oseen
vortices, with their centers marked with circles). Correlations calculated for
this field are shown in panels (b) $L_{x}$ and (c) $L^{s}_{x}$, obtained by
integrating over $\Lambda=N_{x}/2$, which spans the entire length of the
field, which is periodic over $N_{x}=500$.

The middle and bottom panels of figure 6 show $\mathbf{L}$ and
$\mathbf{L^{s}}$, respectively (which, here, only have a $x-$component, i.e.
$L_{x}$ and $L_{x}^{s}$), with the integration length spanning the entire
length of the velocity field, i.e. $\Lambda=N_{x}/2$. A few features of the
correlation profiles point at the nature of the correlation definitions, as
well as the importance of the choice of $\Lambda$.

First, $L_{x}$ is found to have a shape similar to the function itself. If we
consider the definition of $L_{x}$, it is the product between the value of the
function at a point and its integral over a length, hence, if either of the
two is zero, the correlation becomes zero. Therefore, the correlation will
have significant, or large, values where the function itself has significant,
or large, values, and the structure of the function (i.e. its symmetries
and/or anti-symmetries) does not make its integral small.

When the function is the sum of several “basis functions” (in this case two
“pure” Oseen vortices and a “noise”), the $L_{x}$ correlation at a point
involves the product between the local values of the basis functions and the
integrals of the basis functions. So, even if the structure of each of the
“basis functions” makes their integral small (i.e. if $L_{x}$ of a basis
function itself is small), $L_{x}$ of the total function is not necessarily
small. In the particular case presented here, the function $u_{y}$ is the sum
of a “pure function”, $u_{yp}$, and a “noise function”, $\zeta$. The pure
function $u_{yp}$ is the sum of two “basis functions”, the Oseen vortices; the
interaction between the velocity fields of these two Oseen vortices results in
a larger structure with a finite integral. The integral of the noise function,
$\zeta$, is itself a “noise”. Hence, the $L_{x}$ correlation of $u_{y}$ has
the same shape as $u_{y}$ itself. Mathematically, this can be seen as

$\displaystyle L_{x}(x)$
$\displaystyle=\int_{-\Lambda}^{\Lambda}[u_{yp}(x)+\zeta(x)]\cdot[u_{yp}(x+r_{x})+\zeta(x+r_{x})]\mathrm{d}r_{x}$
$\displaystyle=[u_{yp}(x)+\zeta(x)]\cdot\int_{-\Lambda}^{\Lambda}[u_{yp}(x+r_{x})+\zeta(x+r_{x})]\mathrm{d}r_{x}$
$\displaystyle=[u_{yp}(x)+\zeta(x)]\cdot\left(\int_{-\Lambda}^{\Lambda}u_{yp}(x+r_{x})\mathrm{d}r_{x}+\int_{-\Lambda}^{\Lambda}\zeta(x+r_{x})\mathrm{d}r_{x}\right)$
(26)

If either the integral of $u_{yp}$ or the integral of $\zeta$ are non-zero,
$L_{x}$ will be similar to $u_{yp}$. Interestingly, even for a velocity
profile which leads to the integral of $u_{yp}$ becoming zero (i.e. $L_{x}$ of
the pure function is zero), the additional noise breaks the overall symmetries
and anti-symmetries, such that $L_{x}$ becomes non-zero, and the shape of the
pure velocity field $u_{yp}$ can be extracted from $L_{x}$. If the noise is
removed from the velocity field, and the vortices are placed sufficiently far
from each other such that the integral of $u_{yp}$ becomes zero, $L_{x}$,
indeed, goes to zero for $\Lambda=N_{x}/2$ (not shown here).

$L^{s}_{x}$, in figure 6(c), shows different features, starting with a central
peak around $x=250$, which is exactly between the two counter-rotating
vortices. Although the velocity around this position is small, $L^{s}_{x}$
attains a large value since the velocity field is essentially mirrored around
this point, hence being perfectly correlated (with only the noise values being
different between the mirrored halves), i.e. the large peak around $x=250$ is
associated with the symmetry of the larger structure around this point. At the
core of the two vortices ($x=150$ and $x=350$), where $L_{x}$ becomes zero,
$L^{s}_{x}$ shows a large negative peak, since the velocity field on the left
and right of these points is anti-correlated up to the reach of each vortex,
i.e. the large magnitude of $L_{x}^{s}$ at $x=150$ and $x=350$ is associated
with the anti-symmetry of the “local sub-structures” around these points. The
part of $L^{s}_{x}$ in the region outside of the larger structure (i.e.
approximately $x<100$ and $x>400$), is a repetition of the $L^{s}_{x}$ profile
in the center (i.e. $150<x<350$) due to the periodicity of the velocity field
and the integration length spanning the entire length of the field
($\Lambda=N_{x}/2$).

Figure 7 shows the correlations for a similar velocity field, now integrated
over a length of $\Lambda=35$, which corresponds approximately to the size of
the sub-structures (i.e. the individual vortices). This is because within a
distance of $35$ units from the center of each vortex, the velocity reduces to
roughly $10\%$ of its maximum value (note that a slightly lower or higher
$\Lambda$ does not change the results significantly). The $L_{x}$ and
$L^{s}_{x}$ profiles show a few similarities and differences. First, at the
vortex cores, where the velocity is close to zero, $L_{x}$ goes to zero, while
$L_{x}^{s}$ yields strong negative peaks because the velocity is strongly
anti-correlated across the core region. In the potential flow regions of the
two vortices and in the region between the vortices, the general shape of the
two correlations is similar, with both $L_{x}$ and $L^{s}_{x}$ yielding
positive correlation values, reflecting that the velocity is mostly uniform in
these regions. This shows that in the core regions of vortices $\mathbf{L}$
and $\mathbf{L^{s}}$ correlations behave very differently while in regions of
roughly uniform flow they have a similar behaviour. $L^{s}_{x}$ does not
identify the ‘larger structure’, which in this case has a lengthscale larger
than the integration length of the correlation. This shows the importance of
the choice of $\Lambda$ in identifying larger or smaller symmetries and anti-
symmetries of the vector fields. We shall revisit the importance of the choice
of $\Lambda$, in the context of turbulence, in section 5.3.

Figure 7: (a) One-dimensional velocity field and correlations (b) $L_{x}$ and
(c) $L^{s}_{x}$ calculated for an integration length of $\Lambda=35$, which
spans the approximate size of the individual ‘sub-structures’, i.e the two
counter-rotating Oseen vortices. Here, at the vortex cores $L_{x}$ goes to
zero while $L^{s}_{x}$ yields strong negative peaks. In the potential flow
regions of the two vortices and in the region between the vortices, $L_{x}$
and $L^{s}_{x}$ have a similar shape overall, with both correlations yielding
positive values.

Note that it is not only the length of integration $\Lambda$, but also the
lower and upper limits of integration which determine the symmetries and
structures being identified. For instance, eq. 9, can be changed to, instead,
find non-local structures between lengths
$\Lambda_{1}<r_{\alpha\beta}<\Lambda_{2}$, while looking around from point
$\mathbf{x}$. We show an example using the $\mathbf{L^{s}}$ correlation, as
follows

$L^{s}_{i}(\mathbf{x},\Lambda_{1},\Lambda_{2})=\int_{\Lambda_{1}}^{\Lambda_{2}}\mathbf{u}(\mathbf{x}-\mathbf{r}_{i})\cdot\mathbf{u}(\mathbf{x}+\mathbf{r}_{i})\mathrm{d}r_{i}$
(27)

Recalling the generalized correlation definition, this change in the limits of
$\Lambda$ is essentially defining $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ as
finite regions going from $x\in[x_{0}+\Lambda_{1},x_{0}+\Lambda_{2}]$ and
$x\in[x_{0}-\Lambda_{2},x_{0}-\Lambda_{1}]$ respectively, while $x_{0}$ is a
point of the region of observation $\mathcal{S}_{o}$. One example of this is
shown in figure 8, with $\Lambda_{1}=75$ and $\Lambda_{2}=125$. These
integration limits are such that, when $x_{0}$ corresponds to the middle of
the larger structure in the velocity field (i.e. at $x=250$), $\Lambda_{1}$
and $\Lambda_{2}$ span across most of the vortex regions. The $L^{s}_{x}$
profile, consequently, shows a strong peak in between the two vortices,
resulting from the larger structure comprising the two counter-rotating
vortices. These particular limits of integration do not identify any
significant large symmetries, as observed from other $x$ locations.

Figure 8: $L^{s}_{x}$ correlation integrated from $\Lambda_{1}=75$ to
$\Lambda_{2}=125$ shows how non-local symmetries can be identified by varying
the limits of integration.

Figure 9 shows correlations $G_{x}$ and $G^{s}_{x}$, which are the vorticity
field equivalents of $L_{x}$ and $L^{s}_{x}$, integrated over
$\Lambda=N_{x}/2$. The $G_{x}$ correlation remains mostly zero throughout,
with small, noisy fluctuations. This is because, unlike the velocity field,
which yields a finite value for the intergal of $u_{y}(x)$ (over the length
$\Lambda=N_{x}/2$) due to the interaction between the two vortices and the
non-zero contribution from the integral of the noise term $\zeta$, the
integral of $\omega_{z}$ remains nearly zero, since the vorticity is localized
at the core of the two vortices and has the same magnitude, but opposite sign
at the two cores; the only contribution here is from the non-zero integral of
the noise term, which breaks the anti-symmetry of the vorticity field.
Similarly for $L_{x}$ with $u_{y}$, the break of the anti-symmetry due to the
noise, leads to the shape of $G_{x}$ being similar to the shape of
$\omega_{z}$; however, since here the only contribution is from the non-zero
integral of the noise term, the magnitude of $G_{x}$ is much smaller than the
magnitude of $\omega_{z}$.

The $G^{s}_{x}$ correlation has a very similar behaviour to $L^{s}_{x}$, just
with the opposite sign. It yields a large negative peak at the middle of the
larger structure ($x=250$), associated with the anti-symmetry of the vorticity
of the larger structure around this point (while $L^{s}_{x}$ has a large
positive peak, associated with the symmetry of the velocity field around this
point). It also shows smaller positive peaks at the core of the vortices
($x=150$ and $x=350$), around which the vorticity is high and symmetric (while
$L^{s}_{x}$ has a negative peak, associated the anti-symmetry of the velocity
field around the vortex core). The $G^{s}_{x}$ profile is also repeated due to
the periodicity of the velocity field and the large integration length,
similarly to the $L^{s}_{x}$ profile in figure 6.

Figure 9: $G_{x}$ and $G^{s}_{x}$ correlations (integrated up to
$\Lambda=N_{x}/2$) are shown for the Oseen vortex-pair. Analogously for
$L_{x}$ with $u_{y}$, the shape of $G_{x}$ is similar to the shape of
$\omega_{z}$. $G^{s}_{x}$ has a profile similar to $L^{s}_{x}$ in figure 6,
but with the opposite sign.

Figure 10 shows the $G_{x}$ and $G^{s}_{x}$ correlations for an integration
length of $\Lambda=35$. Both $G_{x}$ and $G^{s}_{x}$ show a very similar
overall shape, yielding sharp positive peaks at the vortex cores, where the
vorticity is high and symmetric. The $G_{x}$ correlation decays with some
“noise”, in the potential flow regions corresponding to the two vortices. The
$G^{s}_{x}$ correlation, here, gives a sharper and less noisy profile. This is
because, at the vortex core, the vorticity values to the left and right are
perfectly symmetric (apart from the noise values), which gives a large
correlation $G^{s}_{x}$ value at the vortex core. Slightly moving away from
the core in either direction strongly disturbs this symmetry, leading to a
sharper profile for $G^{s}_{x}$ than for $G_{x}$.

Figure 10: $G_{x}$ and $G^{s}_{x}$ correlations, integrated up to
$\Lambda=35$, are shown for the Oseen vortex-pair. Both correlations are
similar and show a strong peak at the vortex cores, while the peak in
$G^{s}_{x}$ decays more rapidly, since it is more dependent on the symmetry of
the vorticity field.

Finally, figure 11 shows the $H_{x}$ correlation for the Oseen vortex-pair,
integrated over lengths of $\Lambda=N_{x}/2$ (panel b) and $\Lambda=35$ (panel
c). The shape of the $H_{x}$ correlation is found to be almost insensitive to
the choice of $\Lambda$, while a higher $\Lambda$ increases the amplitude of
the correlation. This can be understood from the construction of this
correlation, which, in a Biot-Savart sense, is designed to identify regions
where the angular velocity aligns with the vorticity. In this example, $H_{x}$
yields large positive values in the core region of the two vortices, since the
flow around the vortex cores is generated by the vorticity at the cores. The
independence of the choice of $\Lambda$ is because the influence of the local
vorticity at a point $x_{0}$ rapidly decays over distance, such that at larger
$\Lambda$ values, the velocity field $u_{y}(x_{0}\pm\Lambda)$ is not
influenced by $\omega_{z}(x_{0})$. The larger structure in this example is
induced by the sum of the two individual vortices, and is, hence, “externally
generated”, which is why $H_{x}$ remains zero in the middle of the two
structures. The results, for a ‘linear’ version of the $H_{x}^{p}$ correlation
(instead of its planar construction), are identical, and are not additionally
shown here.

Figure 11: $H_{x}$ correlation, integrated up to (b) $\Lambda=N_{x}/2$ and (c)
$\Lambda=35$, is shown for the Oseen vortex-pair. The correlation yields
strong positive peaks at the vortex cores, and is found to be independent of
the choice of $\Lambda$ (except for a change in the magnitude of the
correlation).

The $L_{x}$ and $L^{s}_{x}$ correlations help in identifying the structure of
high velocity regions, while their analogues for the vorticity field, the
$G_{x}$ and $G^{s}_{x}$ correlations, help in identifying the structure of
high vorticity regions. The $H_{x}$ (and $H^{p}_{x}$) correlation gives
information on the vorticity-velocity structure. The similarities and
differences between these correlation measures gives information on the
structure of the velocity and vorticity fields. In this example, where the
velocity field is “induced” by a pair of vortices, at the core of the
vortices, while $L_{x}$ is close to zero, there exists a large peak in the
values of $L^{s}_{x}$, $G_{x}$, $G^{s}_{x}$ and $H_{x}$. On the other hand, in
the region outside of the core of the vortices where the velocity is
approximately uniform, if the integration length does not allow the capture of
the larger structure periodicity, $L_{x}$ and $L^{s}_{x}$ have a similar
shape, while $G_{x}$, $G^{s}_{x}$ and $H_{x}$ remain close to zero.

###  4.2 Two-dimensional Taylor-Green flow

We next test the correlations on a two-dimensional Taylor-Green flow field,
which comprises a set of counter-rotating vortices. Interaction between
vortex-pairs generates regions of unidirection flow, seprated by regions of
swirling flow. A simplified Taylor-Green flow pattern can be generated by
creating a velocity field as follows

$\displaystyle u_{x}(x,y)$ $\displaystyle=\sin(x)\cos(y)$ $\displaystyle
u_{y}(x,y)$ $\displaystyle=-\cos(x)\sin(y)$ (28)

We generate such a velocity field with $0\leq x<2\pi$ and $0\leq y<2\pi$, in
arbitrary units. A uniform noise $\zeta$, of amplitude $1\%$ of the maximum
velocity magnitude (which is unity), is added to the velocity field. The
vorticity field, which has just one component, i.e. $\omega_{z}$, is
calculated using central differences. In figure 12(a), the kinetic energy
$E_{k}$ is shown, while panel (b) shows the vorticity field $\omega_{z}$,
along with the flow streamlines, where the counter-rotating vortex array can
be seen. We would like to highlight that the velocity field structure in
Taylor-Green flow is determined by the four large vortices, which are
generated by regions of diffused, relatively high vorticity (as opposed to
small, localized regions of high vorticity found in turbulence). These
vortices have a diameter of approximately $\pi$. Essentially, the velocity
field structure consists of: (i) four large vortices with a core region of
high vorticity and low kinetic energy, (ii) regions of well-aligned jet-like
velocity, with high kinetic energy and low vorticity, in between the large
vortices, and (iii) a stagnation flow in the center with low kinetic energy
and low vorticity.

Figure 12: A Taylor-Green velocity field. Panel (a) shows the kinetic energy
$E_{k}$ field, , while panel (b) shows the vorticity field $\omega_{z}$, along
with the flow streamlines (in black, with thickness varying with the velocity
amplitude).

In figure 13, the correlation fields for the Taylor-Green flow are shown. The
correlation two-tuples are for the $x$ and $y$ directions, with an integration
length of $\Lambda=\pi/2$. Panel (a) shows $\mathbf{L}$, which is found to
have a high magnitude in regions of high kinetic energy $E_{k}$ (refer to
figure 12a). This is because the $\mathbf{L}$ correlation identifies velocity
regions that have simultaneously a high velocity magnitude and a high degree
of local vector alignment. At the vortex core regions, $|\mathbf{L}|$ yields
low values. It can be noted that for this velocity field, the regions of high
$|\mathbf{L}|$ are separated from regions of high vorticity, showing that jet-
like coherent regions can be generated by non-local vorticity (since the
vorticity within high $|\mathbf{L}|$ regions is negligible, see figure 12b).

Figure 13(b) shows $|\mathbf{L^{s}}|$, which has a very different structure
than $|\mathbf{L}|$, since it identifies symmetries and anti-symmetries in the
velocity field, which are not just associated with jet-like parallel
streamlines. There are regions of high $|\mathbf{L^{s}}|$ associated with
different types of symmetries/anti-symmetries, other than the jet-like
parallel streamlines. First, the regions of highest $|\mathbf{L^{s}}|$ are
found at the vortex cores, as the vortices have a large swirling region with a
velocity that is perfectly anti-correlated across the vortex cores. High
$|\mathbf{L^{s}}|$ regions are also found coinciding with high $|\mathbf{L}|$
regions, where the flow has a symmetry associated with jet-like parallel
streamlines. These two regions of high $|\mathbf{L^{s}}|$ are separated by
thin regions where $|\mathbf{L^{s}}|$ goes to zero, due to the larger
symmetries in the velocity field which cancel out in the calculation of
$\mathbf{L^{s}}$. There is also a region of high $|\mathbf{L^{s}}|$ at the
center, which is a stagnation flow region, which reflects the anti-symmetries
in the velocity field of the four large vortices that produce the stagnation
region.

Panel (c) shows $|\mathbf{G}|$, which yields high values in the vortex core
regions (which also are the regions of high vorticity, see figure 12b). In the
regions of jet-like flow between vortex-pairs, $|\mathbf{G}|$ has values very
close to zero, which is due to the negligible vorticity in these regions.
Panel (d) shows $|\mathbf{G^{s}}|$, which has a very different strucure than
$|\mathbf{G}|$, since it identifies the symmetries and anti-symmetries in the
vorticity field across every point. First, smaller diffused region of high
$|\mathbf{G^{s}}|$ are found corresponding to the vortex cores, similarly to
$|\mathbf{G}|$, since the vorticity across the vortex cores is highly
correlated. In between counter-rotating vortex-pairs, both in the $x$ and $y$
directions, there appear slightly elongated regions of high
$|\mathbf{G^{s}}|$. This is because along lines connecting the centres of
counter-rotating vortex-pairs, the vorticity field varies uniformly from
positive to negative, and this sign transition coincides with the jet-like
velocity region, which itself has negligible vorticity; this strong anti-
symmetry in the vorticity field gives the high $|\mathbf{G^{s}}|$. Note that
at the center there exists also a strong symmetry in the vorticity field along
the diagonals of the square, however, this is not reflected in
$\mathbf{G^{s}}$, which is a two-tuple for the $x$ and $y$ directions, along
which the vorticity is zero. This indicates that the particular alignment of
the structures can lead to different values of the correlation, depending on
the directions chosen for calculating the correlations. However, in
homogeneous isotropic turbulence, with an arbitrary alignment of the
structures, this effect will not appear, in a statistical sense.

Lastly, panels (e) and (f) show the $|\mathbf{H}|$ and $|\mathbf{H^{p}}|$
fields. Both correlations have a pattern that closely resembles
$|\mathbf{G}|$, with large values at the vortex core regions, since the
vortices here are generated by strong vorticity values at the vortex cores.
Regions of jet-like flow, which are externally induced by the interaction
between the velocity fields of adjacent vortices, yield negligible values of
$|\mathbf{H}|$ and $|\mathbf{H^{p}}|$.

Figure 13: Correlation field amplitudes for the Taylor-Green flow field,
obtained upon integrating the correlations over $\Lambda=\pi/2$.

###  4.3 Three-dimensional Burgers vortices

As a final example of the correlations applied to canonical flows, we now
consider a three-dimensional velocity field generated by superposing two
Burgers vortices, which again generates a ‘large-scale’ structure comprising
two smaller sub-structures. The Burgers vortex is an exact solution of the
Navier-Stokes equation, consisting of a radial velocity component along with a
tangential velocity, and can be constructed as

$u_{z}=az,\quad u_{r}=-\frac{ar}{2},\quad u_{\theta}=\frac{\Gamma}{2\pi
r}\left[1-\exp\left(-\frac{ar^{2}}{4\nu}\right)\right]$ (29)

where $a$ represents the rate of strain, $\Gamma$ the circulation and $\nu$
the kinematic viscosity. Here $u_{z}$, $u_{r}$ and $u_{\theta}$ give velocity
components in the axial, radial and tangential directions, which are converted
from cylindrical to Cartesian coordinates. We isolate the vortex in space by
multiplying the velocity field with a three-dimensional Gaussian function
$\mathcal{G}$ to contain the Burgers vortex within a spherical region,

$\mathcal{G}(x,y,z)=\exp\left[-\left(\frac{x^{2}+y^{2}+z^{2}}{\sigma^{2}}\right)\right]$
(30)

where $x$, $y$ and $z$ are measured with the origin placed at the center of
the vortex. The value of $\sigma$ is chosen such that it creates a spherical
region circumscribing the axial length of the vortex. This suppresses the
strain regions generated by each vortex, far away from its center, such that
the velocity field comprises primarily of two swirling-flow regions. The
swirling-flow of the Burgers vortex resembles the one-dimensional Oseen vortex
(as was described in Section 4.1), with a core in solid-body rotation where
$u_{\theta}\propto r$, followed by a potential flow region with
$u_{\theta}\propto 1/r$. A low amplitude uniform noise, $\zeta$, is added to
the final velocity field, over which the correlations are subsequently
calculated. Lastly, the vortices are also rotated at arbitrary azimuthal and
elevation angles $(\alpha,\beta)$. This is done to change the orientation of
the velocity field symmetries with respect to the orthogonal bases along which
the correlations are calculated, to test the applicability of the correlation
definitions for arbitrarily aligned structures, as will be encountered in
turbulence fields.

We generate two vortices, on a grid of $100^{3}$, with $a=0.1$, $\nu=0.025$,
$\Gamma=15$ and $\zeta=0.004$ (i.e. $\sim 1\%$ of the maximum velocity
magnitude), all quantities being presented in arbitrary units. The actual
values being used here are not important, as we simply intend to generate a
velocity field with a Burgers vortex structure. The resulting vortex has a
core region with solid body rotation up to $5$ units (grid cells) from its
axis, and the velocity magnitude in the potential flow region (with swirling
motion) decays to approximately $40\%$ and $10\%$ of the maximum velocity
magnitude within $15$ and $30$ grid cells from the axis, respectively. Both
vortices are multiplied with the Gaussian function $\mathcal{G}$, generated
with $\sigma=5$. The vortices are rotated at arbitrary azimuthal and elevation
angles of $(0.6,0.25)$ and $(-0.45,-0.3)$, measured in radians. These vortices
are then superposed by adding their velocity fields, with their centres placed
at $(40,40,40)$ and $(70,70,70)$. The resulting velocity field is shown in
figure 14. Since the vortices are placed close to each other, their velocity
fields begin to entangle and interact. The larger structure of the Burgers
vortex-pair, however, is distinct from the Oseen vortex-pair in Section 4.1,
and it does not have the same kind of symmetries and anti-symmetries.

(a) View 1

(b) View 2

Figure 14: Two arbitrarily aligned Burgers vortices are shown, with isolated
swirling-flow regions. The velocity streamlines are coloured according to the
velocity magnitude, normalized beween $0$ and $1$, which show that the two
vortices begin to “interact”. The edges of the cubes shown here run from
$\left\\{x,y,z\right\\}\in[10,90]$.

Figure 15 shows the amplitude of all the correlations, integrated over a
length of $\Lambda=12$ for the Burgers vortex-pair. The value of $\Lambda$ is
large enough to encompass most of the structure of each vortex, and the
results were found to remain qualitatively unchanged for $\Lambda=10$ and
$\Lambda=15$, which only causes a change in the magnitude of the correlations.
The features of the correlation fields strongly reflect the behaviour of their
one-dimensional analogues as was presented for the Oseen vortex-pair (see
figures 7 and 10, where $\Lambda$ also encompasses, roughly, the structure of
each vortex). First, panels (a) and (b) show the $\mathbf{L}$ and
$\mathbf{L^{s}}$ correlations. The correlation regions are well aligned with
the axes of the vortices, and have a size slightly smaller than the extent of
the swirling-flow region. $\mathbf{L}$ goes to zero at the vortex core where
the velocity is also zero. $\mathbf{L^{s}}$ yields a large value at the vortex
core, across which the flow is highly anti-correlated, surrounding which is a
thin region of zero correlation, and an outer region of finite correlation
corresponding to the potential flow region of the vortex. Both these
correlations identify the outer swirling-flow region where the streamlines are
well-aligned, and locally parallel; however, in the core region of the vortex
they are very different. Due to their definitions, near the center of a strong
swirling flow $\mathbf{L^{s}}$ has a large magnitude whereas $\mathbf{L}$ has
a magnitude close to zero. Panels (c) and (d) show the $\mathbf{G}$ and
$\mathbf{G^{s}}$ correlations, both of which yield thin, elongated correlation
profiles aligned with the axes of the vortices, while the $\mathbf{G^{s}}$
correlation is sharper. This again reflects that, at the core of the vortices,
the vorticity vectors are well aligned. Panels (e) and (f) show $\mathbf{H}$
and $\mathbf{H^{p}}$, which also yield strong correlation profiles at the
cores of the vortices, aligned with the axes of the vortices. This is because
the swirling velocity field is associated with the vorticity at the vortex
core regions.

We also explored larger integration lengths, assuming a periodic field,
similarly to what was done for the one-dimensional flow. We found that for
larger integration lengths, of $\Lambda\approx N_{x}/2$, the correlations also
begin to recognize non-local symmetries (due to periodicity) in the twin
Burgers vortex velocity field, similarly to the one-dimensional Oseen vortex
pair (see figures 6 and 9), which is not additionally shown here. This shows
that the correlations, depending on the integration length, can identify large
complex structures, containing smaller sub-structures; however, this will not
be further explored here.

We note that, similar to the Oseen vortex examples, in the vortex core there
exists a strong relation between $\mathbf{L^{s}}$, $\mathbf{G}$,
$\mathbf{G^{s}}$, $\mathbf{H}$ and $\mathbf{H^{p}}$, with all of them having
large magnitudes, whereas $\mathbf{L}$ has a magnitude close to zero. On the
contrary, in the outer regions of the vortex, there exists a strong relation
between $\mathbf{L^{s}}$ and $\mathbf{L}$, with both of them having high
magnitudes, whereas $\mathbf{G}$, $\mathbf{G^{s}}$, $\mathbf{H}$ and
$\mathbf{H^{p}}$ have a magnitude close to zero. Essentially, over the entire
vortex, where the swirling flow is strong, $\mathbf{L^{s}}$ has a large
magnitude, and this region of high $|\mathbf{L^{s}}|$ encloses the vortex core
where $|\mathbf{L}|$ is close to zero and $|\mathbf{G}|$, $|\mathbf{G^{s}}|$,
$|\mathbf{H}|$ and $|\mathbf{H^{p}}|$ are also high. This happens because the
swirling flow region is self-induced by the high vorticity at the core of the
vortices leading to a highly anti-correlated velocity field around the center
of the vortex, where the velocity is close to zero; moving away from the
center of the vortex, the vorticity decays rapidly but the swirling flow is
still strong, and the velocity field, locally, is highly correlated due to the
parallel streamlines. Note that this is unlike the example of the Taylor-Green
flow, where the jet-like high $|\mathbf{L}|$ regions are externally-induced by
the interaction of vortices, leading to symmetries and anti-symmetries in the
vorticity field in this region, which can result in high magnitudes of
$\mathbf{G^{s}}$ even with low magnitudes of $\mathbf{G}$. The similarities
and differences between the different correlation definitions play a key role
in the subsequent analysis of turbulence field structures.

Overall, we find that the correlation definitions are adept at identifying
typical velocity and vorticity field patterns, also in fields arbitrarily
aligned with respect to directions along which the correlations are evaluated.

(a) $\mathbf{L}$ correlation

(b) $\mathbf{L^{s}}$ correlation

(c) $\mathbf{G}$ correlation

(d) $\mathbf{G^{s}}$ correlation

(e) $\mathbf{H}$ correlation

(f) $\mathbf{H^{p}}$ correlation

Figure 15: Amplitude of all correlations calculated for the two Burgers
vortices are shown for view 1 in figure 14, which gives a simultaneous look at
the axis of one vortex and the core region of the other. The edges of the
cubes shown here run from $\left\\{x,y,z\right\\}\in[10,90]$.

##  5 Correlations applied to homogeneous isotropic turbulence

After applying the correlations to canonical flows, we now study how these
ideas fare for real turbulence vector fields. A turbulence velocity field is
considered, typically, to have structures across multiple scales, while the
vorticity field mainly comprises structures at the smaller scales. These
fields, further, are highly complex and irregular. Applying the correlations
to instantaneous snapshots of turbulence vector fields, as obtained from
direct numerical simulations of homogeneous isotropic turbulence, reveals
their potential in educing coherent structures. In this section, we begin with
a brief description of the simulation method used to generate the turbulence
data. Since the correlations yield three-tuple fields containing information
about the structure of the flow, we first describe how the correlation fields
look qualitatively in comparison to the vector fields they are based on, i.e.
the velocity and the vorticity fields. We then describe the statistics of the
correlation fields, like the PDFs, CDFs, spectral characteristics and their
spatial organization. In the next section, we focus on individual structures
and unravel their Biot-Savart composition.

###  5.1 Simulation details and dataset

For this study, we use a dataset from DNS simulations of homogeneous isotropic
turbulence, for which the Navier-Stokes equations with a body force
$\mathbf{F}$ (as given below) are solved numerically

$\frac{\partial\mathbf{u}}{\partial
t}+\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}=-\frac{\nabla
p}{\rho}+\nu\nabla^{2}\mathbf{u}+\frac{\mathbf{F}}{\rho}$ (31)
$\nabla\cdot\mathbf{u}=0$ (32)

Turbulence is generated in a periodic box by means of low wavenumber forcing,
which is divergence-free by construction and is concentrated over a range of
Fourier modes. It is of the form given by Biferale et al. (2011), and has
properties similar to that devised by Alvelius (1999) and Ten Cate et al.
(2006), which can be written as

$\displaystyle F_{x}$ $\displaystyle=\sum_{k={k_{a}}}^{{k_{b}}}\rho
A(k)\left[\sin(2\pi ky+\phi_{y}(k))+\sin(2\pi kz+\phi_{z}(k))\right]$
$\displaystyle F_{y}$ $\displaystyle=\sum_{k={k_{a}}}^{{k_{b}}}\rho
A(k)\left[\sin(2\pi kx+\phi_{x}(k))+\sin(2\pi kz+\phi_{z}(k))\right]$
$\displaystyle F_{z}$ $\displaystyle=\sum_{k={k_{a}}}^{{k_{b}}}\rho
A(k)\left[\sin(2\pi kx+\phi_{x}(k))+\sin(2\pi ky+\phi_{y}(k))\right]$ (33)

The forcing is stochastic (white noise) in time, which is achieved by varying
each $\phi(k)$ randomly, and the force is distributed over a small range of
wavenumers, given by $k_{a}\leq k\leq k_{b}$ (for this study we fix
$k_{a}=1,k_{b}=8$), and the amplitude $A(k)$ of each of these wavenumbers is a
Gaussian distribution in Fourier space, centered around a central forcing
wavenumber $k_{f}$, given as

$A(k)=A\exp\left(-\frac{\left(k-k_{f}\right)^{2}}{c}\right)$ (34)

where $c$ sets the width of the distribution ($c=1.25$ here), $k_{f}=2$, and
$A$ is the forcing amplitude. We solve equations 31 and 32 with a standard
lattice-Boltzmann (LB) solver, incorporating the turbulence forcing as per
equation 33. This method has been used before for simulating homogeneous
isotropic turbulent flows of various kinds (Ten Cate et al., 2004, 2006;
Biferale et al., 2011; Perlekar et al., 2012; Mukherjee et al., 2019).

The simulation is performed in a periodic box of size $(2\pi)^{3}$ resolved
over $N^{3}$ grid points along each direction, all units being dimensionless,
hence resolving a range of wavenumbers from $k=2\pi/N$ (i.e. the largest scale
has a length $N\ [lu]$) to $k=2\pi/2=\pi$ (i.e. the smallest scale has a
length $2\ [lu]$). Since we simulate homogeneous isotropic turbulence, by
definition, all physical quantities are fluctuating and do not have a mean
value, i.e. $\mathbf{u}=\mathbf{u}^{\prime}$ and
$\boldsymbol{\omega}=\boldsymbol{\omega}^{\prime}$. The Kolmogorov scale is
defined as $\eta\sim\left(\nu^{3}/\epsilon\right)^{1/4}$, where $\nu$ and
$\epsilon$ are the kinematic viscosity and energy dissipation rate,
respectively. We adhere to the criterion for a DNS, as given by Moin & Mahesh
(1998), i.e. $k_{\mathrm{max}}\eta>1$. The Taylor microscale is calculated as

$\lambda=\left(\frac{15\nu{u^{\prime}}^{2}}{\left\langle{\epsilon}\right\rangle}\right)^{1/2}$
(35)

where $u^{\prime}$ is the root-mean-square velocity. The average rate of
energy dissipation $\left\langle{\epsilon}\right\rangle$ is calculated as
$\left\langle{\epsilon}\right\rangle=\nu\left\langle{\omega^{2}}\right\rangle$,
where $\left\langle{\omega^{2}}\right\rangle$ is the average enstrophy. Note
that the enstrophy $\omega^{2}=\boldsymbol{\omega}\cdot\boldsymbol{\omega}$ is
analogous to the turbulence kinetic energy
$E_{k}=\mathbf{u}\cdot\mathbf{u}/2$. For homogeneous isotropic turbulence,
since $u^{\prime}=v^{\prime}=w^{\prime}$, we have $E_{k}=3{u^{\prime}}^{2}/2$
or $u^{\prime}=\sqrt{2E_{k}/3}$. The root-mean-square vorticity,
$\omega^{\prime}$, is obtained as
$\left\langle{\boldsymbol{\omega}\cdot\boldsymbol{\omega}}\right\rangle^{1/2}$.
In general, $E_{k}$ and $\epsilon$ (apart from $\nu$) are average measures of
$u^{\prime}$ and $\omega^{\prime}$, respectively. The large eddy turnover
timescale is given as $T^{\star}=\mathcal{L}/u^{\prime}$, where $\mathcal{L}$
is the forcing lengthscale, given as $\mathcal{L}=N/k_{f}$. Using $\lambda$,
the Taylor Reynolds number is calculated as

$Re_{\lambda}=\frac{u^{\prime}\lambda}{\nu}$ (36)

and the Kolmogorov timescale is given as

$\tau_{k}=\left(\frac{\epsilon}{\nu}\right)^{-1/2}$ (37)

The turbulence simulation (parameters given in table 1) is performed for a
fluid initially at rest, to which the turbulence force is applied. After a
brief transient duration, turbulence becomes well developed and attains a
statistical steady-state, i.e. with a balance of power input and energy
dissipation. The simulation is then run for several additional large eddy
timescales ($\sim 20-30T^{\star}$), during which around $\sim 20$ field
snapshots are retained for analysis, all separated by $50\tau_{k}$, to give
converged statistical results.

$N^{3}$ | $k_{f}$ | $\nu$ | $u^{\prime}$ | $\omega^{\prime}$ | $\left\langle{E_{k}}\right\rangle$ | $\left\langle{\epsilon}\right\rangle$ | $\lambda$ | $Re_{\lambda}$ | $\eta$ | $\tau_{k}$
---|---|---|---|---|---|---|---|---|---|---
$256^{3}$ | $2$ | 0.0047 | 0.034 | 0.0103 | $1.8\times 10^{-3}$ | $5.0\times 10^{-7}$ | 13 | 95 | 0.67 | 97
Table 1: Simulation details, with all quantities presented in dimensionless
lattice units $[lu]$, average kinetic energy
$\left\langle{E_{k}}\right\rangle=\left(\sum_{k}E(k)\right)/N$, and the
average rate of energy dissipation
$\left\langle{\epsilon}\right\rangle=(\sum_{k}2\nu k^{2}E(k))/N$.

Figure 16 shows the evolution of $\left\langle{E_{k}}\right\rangle$ and
$\left\langle{\omega^{2}}\right\rangle$. Both quantities attain their steady-
state values within a short transient phase, $\sim 100\tau_{k}$, after which
they continue to oscillate around their temporal mean values. Beyond
$100\tau_{k}$, turbulence is well developed, with a sufficient separation of
scales. The small temporal oscillation of $\left\langle{E_{k}}\right\rangle$,
due to the finite volume of the simulation, further manifests in the temporal
oscillation of $\left\langle{\omega^{2}}\right\rangle$, due to the turbulence
dynamics (Pearson et al., 2004; Biferale et al., 2011).

Figure 16: Time evolution of the volume averaged turbulence kinetic energy
$\left\langle{E_{k}}\right\rangle$ and enstrophy
$\left\langle{\omega^{2}}\right\rangle$. Both quantities attain a steady-state
value, reflecting a developed turbulence state.

Figure 17 shows a snapshot of the turbulence kinetic energy $E_{k}$ and
enstrophy $\omega^{2}$ fields, as 3D volume renderings and planar cross-
sections, at a simulation time of $500\tau_{k}$. Typical features of the
kinetic energy and enstrophy can be seen, where the kinetic energy is
distributed over a range of length scales, and forms diffused, small and
intermediate sized, irregular structures. Enstrophy (and vorticity in general)
is concentrated at the smaller scales, in spatially intermittent tube-like
structures, also called “worms”.

(a)

(b)

(c)

(d)

Figure 17: Snapshots of turbulence fields, when the flow is fully developed
(at $t=500\tau_{k}$), show qualitative features of the kinetic energy $E_{k}$,
and the enstrophy, $\omega^{2}$. Panel (a) shows a 3D volume rendering (at a
resolution of $256^{3}$) of the $E_{k}$ field, which contains, small to
intermediate, irregular structures that are distributed throughout the volume.
Panel (b) shows a 2D cross-section of the $E_{k}$ field (at a resolution of
$256^{2}$) at an arbitrary plane of the 3D simulation domain. Panel (c) shows
a volume rendering of the $\omega^{2}$ field, which is markedly different from
the $E_{k}$ field, as enstrophy is concentrated at the smaller scales, forming
tube-like structures (also called “worms”). Panel (d) shows a planar cross-
section of $\omega^{2}$, which shows that most of the field has low values,
interspersed with small regions of concentrated enstrophy. Both the fields
have been normalized with their respective volume-averaged quantity, i.e.
$\left\langle{E_{k}}\right\rangle$ and
$\left\langle{\omega^{2}}\right\rangle$.

Figure 18 shows the probability and cumulative distribution functions (PDFs
and CDFs, respectively), of the three velocity and vorticity components. These
profiles have been obtained using $20$ field snapshots, all separated by
$50\tau_{k}$. Figure 18(a) shows that the velocity components follow a
Gaussian distribution (shown as the dashed line), and that the velocity
fluctuations are not extreme (here they range from
$-4<u_{i}/u_{i}^{\prime}<4$). Figure 18(b) shows the CDFs of the velocity
components, where $65\%$ and $97\%$ of the velocity has a magnitude below
$u_{i}^{\prime}$ and $2u_{i}^{\prime}$, respectively. Extreme values of the
velocity, around $|u_{i}|>3u_{i}^{\prime}$ occupy a very small fraction of the
total velocity field. Similarly, figure 18(c) and figure 18(d) show the PDFs
and CDFs of the vorticity components. The PDFs show the typical long-tail
distribution of vorticity, which is highly non-Gaussian. The extent of these
tails gives a measure of the intermittency in the vorticity field, where
increasingly extreme values can occur with a low probability. The CDFs of the
vorticity show that most of the vorticity field has a low value, with $70\%$
and $95\%$ of the field below $\omega_{i}^{\prime}$ and
$2\omega_{i}^{\prime}$, respectively. In this regard, the vorticity field has
a similar composition as the velocity field; the difference being that the
vorticity can also assume much more extreme values (even $\sim
18\omega_{i}^{\prime}$ in this case).

We note that the vorticity field is often classified into a few “ranges”. She
et al. (1990, 1991) proposed a classification where, based upon the amplitude
and structure of the vorticity streamlines, the vorticity field is divided
into “low-vorticity”, “moderate-vorticity” and “high-vorticity” ranges.
According to their classification, “high-vorticity”
($\omega\gg\omega^{\prime}$), which occupies a very small fraction of the
volume and forms the long tails of the vorticity PDF, forms vorticity
streamlines that are well-aligned, while the velocity field in the vicinity of
these structures has a spiral, swirling motion. “Moderate-vorticity”
($\omega>\omega^{\prime}$), on the other hand, was found to be less organized,
the structure of which was described as “sheet-like” and “ribbon-like”. “Low-
vorticity”, at the level of the root-mean-square value
($\omega\sim\omega^{\prime}$ and $\omega<\omega^{\prime}$), which occupies
most of the volume, was found to form random vorticity streamlines with no
apparent structure. In this classification, the well-structured regions of the
vorticity field, associated with high $\omega$ values, are highlighted, though
they occupy a small fraction of the volume. For our subsequent Biot-Savart
analysis in section 6.4, where we disentangle the various vorticity
contributions to the generation of velocity field structures, we use a similar
classification. Since vorticity at the level of the $\omega^{\prime}$ occupies
most of the volume, it is expected to have a significant influence on the
Biot-Savart velocity generation. Hence we term this range of vorticity, based
upon its amplitude, as “weak” and “intermediate” background vorticity, since
it permeates the volume. We use this classification only as a guideline to
interpret our results, as we do not investigate the range of structures of the
vorticity field.

(a)

(b)

(c)

(d)

Figure 18: Probability and Cumulative Distribution Functions (PDF and CDF),
shown for the three velocity and vorticity components. The dashed line in
panels (a) and (c) show a typical Gaussian distribution.

The spectra of kinetic energy and enstrophy are calculated using the three-
dimensional Fourier transform $\hat{\phi}_{\mathbf{k}}$ of the velocity and
vorticity fields, respectively. The three-dimensional spectra are spherically
averaged over wavenumber shells $k\in\left[k-1/2,k+1/2\right]$, where
$k=\sqrt{\mathbf{k}\cdot\mathbf{k}}$, to give one-dimensional spectra over the
scalar wavenumber $k$ as follows

$\phi(k)=4\pi k^{2}\frac{\sum_{k}|\hat{\phi}_{\mathbf{k}}|^{2}}{\sum_{k}1}$
(38)

These one-dimensional spectra are further time averaged over 20 samples
separated by $50\tau_{k}$, to give time-averaged spectral characteristics.
Lastly, the spectrum $\phi(k)$ is normalized as
$\overline{\phi(k)}=\phi_{k}/\sum_{k}\phi(k)$ to facilitate comparison of
different quantities, as we are mainly interested in the relative distribution
of energy over wavenumber. Figure 19 shows the $E_{k}$ and $\omega^{2}$
spectra, where $E_{k}$ exhibits a well developed inertial range, which follows
the $k^{-5/3}$ spectral scaling, while the enstrophy spectra has a small,
positive slope, with a broad peak at higher wavenumbers.

Figure 19: Time-averaged, one-dimensional power spectra are shown for the
kinetic energy and enstrophy fields.

###  5.2 Qualitative and statistical features of the correlation fields

The various correlations are calculated for the snapshot of the data presented
in figure 17, for an integration length of $\Lambda=\lambda$. Each correlation
field, say $\mathbf{L}$, is normalized by its respective root-mean-square
(rms) value, $L^{\prime}$, which is calculated as
$L^{\prime}=\left\langle{\mathbf{L}\cdot\mathbf{L}}\right\rangle^{1/2}$. In
general, $\left\langle{\cdot}\right\rangle$ denotes ensemble averaging,
however, in the snapshots of the correlation fields a volume-average was used,
in order to normalize the field snapshot with its root-mean-square value at
the same instance of time. The amplitude of a correlation field, say
$|\mathbf{L}|$, is simply referred to as $L$. Each of the correlation fields
have been shown separately, to highlight their qualitative features, at an
arbitrary cross-sectional slice and as a three-dimensional volume rendering.
The PDF and CDF of the three components of each correlation have been shown as
well, which have been averaged over $20$ field realizations, each separated by
$50\tau_{k}$.

Figure 20 shows the $\mathbf{L}$ correlation. The volume rendering in panel
(a) shows the spatial distribution of the correlation field, which shows
features across various lengthscales, with diffused regions of high magnitude
ranging from intermediate to small sizes. These features, further, are very
similar to the features in the $E_{k}$ field (as seen in figure 17). Note that
small, isolated, regions of the correlation field with typically a high
magnitude, which we refer to as correlation kernels, are measures of the
correlation in larger regions of the velocity and vorticity field surrounding
them. Panel (b) shows a cross-sectional view of the correlation field, at the
sample plane as that shown in figure 17. The PDF of the components of
$\mathbf{L}$, in panel (c), shows that the correlation is highly positively
skewed. This follows from the definition of $\mathbf{L}$, which identifies
regions of well-aligned streamlines, i.e. the local velocity
$\mathbf{u}({\mathbf{x}})$ is expected to be aligned with the velocity in the
neighbourhood $-\Lambda<x_{i}<\Lambda$, and the product of the two is
positive. The strong coincidence of high $L$ with regions of high $E_{k}$
(which is a point quantitiy), reflects that high $E_{k}$ regions comprise
parallel streamlines of jet-like flow, hence, not only corresponding to
regions of high velocity magnitude, but also exhibiting a high degree of
alignment. This correspondence shall be quantified with the subsequent
analysis. Further, simlarly to $u_{i}$, the PDFs of $L_{i}$ do not extend over
a very large range of values, however, contrary to $u_{i}$, they are strongly
non-Gaussian. Panel (d) shows the CDFs of $L_{i}$, where approximately $70\%$
and $93\%$ of the $L_{i}$ fields are below $L_{i}^{\prime}$ and
$2L_{i}^{\prime}$, respectively.

(a)

(b)

(c)

(d)

Figure 20: The $\mathbf{L}$ correlation is shown for a single field snapshot,
at $500\tau_{k}$. Panel (a) shows the three-dimensional field as a volume
rendering, while panel (b) shows the cross-sectional view of
$L=\sqrt{\mathbf{L}\cdot\mathbf{L}}$ at the same plane shown in figure 17.
Panels (c) and (d) show the time-averaged PDFs and CDFs of the three
components of $\mathbf{L}$.

Figure 21 shows the $\mathbf{L^{s}}$ correlation. The volumetric distribution
in panel (a) and the planar crossection in panel (b) show a striking
similarity to the structures and distribution of $L$ (see figure 20). The
$L^{s}$ field, like $L$, also shows intermediate and small sized diffused
regions, found throughout the volume of the flow. While the $\mathbf{L^{s}}$
kernels are slightly smaller than the $\mathbf{L}$ kernels, the strong
correspondence between the two correlation fields is in stark contrast to the
canonical flows examples, particularly the Taylor-Green flow pattern (see
figure 13). This shows that in (homogeneous isotropic) turbulence velocity
fields, there are no large symmetries (within an integration length of
$\Lambda=\lambda$), which would, for instance, be associated with “large
eddies” with a swirling motion. High values of $L^{s}$, here, arise due to
parallel streamlines in jet-like flow regions, which in turn were found to
coincide strongly with regions of high $E_{k}$. This further corroborates that
high kinetic energy structures are jet-like. The PDFs of $L^{s}_{i}$, in panel
(c), point to the same fact, as the distribution is found to be highly
positively skewed (note that the $\mathbf{L^{s}}$ correlation, by definition,
yields large positive values for aligned streamlines, and large negative
values for anti-parallel streamlines). Negative values of $L^{s}$ are also
found, albeit with very small probability, which shows that a few regions have
anti-parallel streamlines, yielding mild $L^{s}$ values. Lastly, the CDFs in
panel (d) show that, similarly to the $L_{i}$ fields, approximately $70\%$ and
$93\%$ of the $L^{s}_{i}$ fields are below ${L^{s}_{i}}^{\prime}$ and
$2{L^{s}_{i}}^{\prime}$, respectively.

(a)

(b)

(c)

(d)

Figure 21: The $\mathbf{L^{s}}$ correlation is shown for a single field
snapshot, at $500\tau_{k}$. Panel (a) shows the three-dimensional field as a
volume rendering, while panel (b) shows the cross-sectional view of
$L^{s}=\sqrt{\mathbf{L^{s}}\cdot\mathbf{L^{s}}}$ at the same plane shown in
figure 17. Panels (c) and (d) show the time-averaged PDFs and CDFs of the
three components of $\mathbf{L^{s}}$.

Figure 22 shows the $\mathbf{G}$ correlation, which is the equivalent to
$\mathbf{L}$ for the vorticity field. The volumetric (panel a) and planar
(panel b) distributions of $G$ are found to closely resemble the enstrophy
field (as shown in figure 17), where the $G$ field at high magnitudes also
forms worm-like structures. Like $L$, the $G$ field also yields high values in
regions which have both high vorticity, and a high degree of alignment of the
vorticity streamlines. The strong correspondence with high enstrophy regions
shows that these regions form small scale vorticity-jets, the size of which is
smaller than the high $E_{k}$ velocity-jets. The PDFs of the $G_{i}$
components show that the correlation yields a long-tailed, positively-skewed
distribution, similar to the positive half of the vorticity PDF (figure 18c).
This again reflects that the vorticity streamlines are well aligned in the
core of high enstrophy regions, since the product of
$\boldsymbol{\omega}(\mathbf{x})$ and the integral of the vorticity in the
neighbourhood $-\Lambda<x_{i}<\Lambda$, yields positive values, reflecting
that the two quantities have the same sign. The CDFs, in panel (d), show that
approximately $84\%$ and $94\%$ of the $G_{i}$ fields are below
$G_{i}^{\prime}$ and $2G_{i}^{\prime}$, respectively. In comparison, $70\%$
and $95\%$ of the $\omega_{i}$ fields are belowe $\omega_{i}^{\prime}$ and
$2\omega_{i}^{\prime}$, respectively. This is consistent with the visual
impression of figures 17 and 22 and it indicates that the vorticity field in
the range $\omega_{i}^{\prime}<\omega_{i}<2\omega_{i}^{\prime}$ is not as
closely associated with regions of well-aligned vorticity streamlines as for
higher values of $\omega_{i}$.

(a)

(b)

(c)

(d)

Figure 22: The $\mathbf{G}$ correlation is shown for a single field snapshot,
at $500\tau_{k}$. Panel (a) shows the three-dimensional field as a volume
rendering, while panel (b) shows the cross-sectional view of
$G=\sqrt{\mathbf{G}\cdot\mathbf{G}}$. Panels (c) and (d) show the time-
averaged PDFs and CDFs of the three components of $\mathbf{G}$.

Figure 23 shows the $\mathbf{G^{s}}$ correlation, which is the vorticity field
equivalent of $\mathbf{L^{s}}$. The volumetric distribution and planar profile
of the correlation field are shown in panels (a) and (b), respectively. The
$G^{s}$ field appears ‘patchy’ in comparison to $\mathbf{G}$, and its kernels
are found to be more sparse and fragmented. This is because the
$\mathbf{G^{s}}$ correlation is sensitive to the symmetries in the vorticity
field (in the $\Lambda-$neighbourhood) along each direction. Visually, we find
that there are no high amplitude $G^{s}$ regions that do not coincide with
high amplitude $G$, showing that anti-parallel vorticity streamlines are
uncommon. The PDFs in panel (c) shows that the components $G^{s}_{i}$ have a
positively-skewed distribution, although negative values of $G^{s}_{i}$ are
slightly more prevalent than negative values of $G_{i}$. The positive-skew of
the high magnitude $G^{s}_{i}$ shows that the stronger vorticity regions have
vorticity streamlines that are well aligned. The CDFs in panel (d) show that
approximately $80\%$ and $96\%$ of $G^{s}_{i}$ is under ${G^{s}_{i}}^{\prime}$
and $2{G^{s}_{i}}^{\prime}$, respectively. Interestingly, for the same rms
thresholds, the CDFs of the $\omega_{i}$ and $G_{i}$ fields yield $70\%-95\%$
and $84\%-94\%$, respectively. This suggests that, while high vorticity
regions are closely associated with well-aligned vorticity jets, “intermediate
high” vorticity (roughly, in the range
$\omega_{i}^{\prime}<\omega_{i}<2\omega_{i}^{\prime}$) has a less organized,
weaker structure but containing more complex local symmetries and anti-
symmetries, leading to a more “patchy” $G_{i}^{s}$ field. This is consistent
with the work of She et al. (1990, 1991) and the vorticity “ranges” they
proposed, however, in this work we will not explore this further.

(a)

(b)

(c)

(d)

Figure 23: The $\mathbf{G^{s}}$ correlation is shown for a single field
snapshot, at $500\tau_{k}$. Panel (a) shows the three-dimensional field as a
volume rendering, while panel (b) shows the cross-sectional view of
$G^{s}=\sqrt{\mathbf{G^{s}}\cdot\mathbf{G^{s}}}$. Panels (c) and (d) show the
time-averaged PDFs and CDFs of the three components of $\mathbf{G}$.

Figure 24 shows the $\mathbf{H}$ correlation, which relates the local
vorticity $\boldsymbol{\omega}(\mathbf{x})$ to the velocity field along
directions $r_{i}$. Similarly to the $G$ and $G^{s}$ fields, the volumetric
and planar $H$ fields, in panels (a) and (b), respectively, also closely
resemble the enstrophy field in figure 18. The PDFs of the $H_{i}$ components
are highly positively-skewed, and have a long-tailed distribution. Together,
these results show that the velocity field in the vicinity of strong vorticity
regions has an angular velocity closely positively-aligned with the jet-like
vorticity streamlines. This is because (i) there exists a strong spatial
correspondence between the occurrence of strong enstrophy regions and high
magnitude $\mathbf{H}$, and (ii) the distributions of $H_{i}$ dominantly show
positive values, which means that the local flow, in the
$\Lambda-$neighbourhood, has an angular velocity well correlated with the
vorticity, which tends to form small vorticity jets in the strong vorticity
regions. The CDFs of $H_{i}$ in panel (d) show that approximately $82\%$,
$94\%$ and $99\%$ of the $H_{i}$ fields are below $H_{i}^{\prime}$,
$2H_{i}^{\prime}$ and $3H_{i}^{\prime}$, respectively.

(a)

(b)

(c)

(d)

Figure 24: The $\mathbf{H}$ correlation is shown for a single field snapshot,
at $500\tau_{k}$. Panel (a) shows the three-dimensional field as a volume
rendering, while panel (b) shows the cross-sectional view of
$H=\sqrt{\mathbf{H}\cdot\mathbf{H}}$. Panels (c) and (d) show the time-
averaged PDFs and CDFs of the three components of $\mathbf{H}$.

Lastly, figure 25 shows the $\mathbf{H^{p}}$ correlation, which is very
similar to the $\mathbf{H}$ correlation. Similarly to the $G$, $G^{s}$ and $H$
fields, the volumetric and planar $H^{p}$ fields, in panels (a) and (b),
respectively, also closely resemble the enstrophy field in figure 17.
Similarly to $H_{i}$, the PDFs of the $H^{p}_{i}$ components are highly
positively skewed, and have a long-tailed distribution, with slightly more
extreme values than $H_{i}$. The similarity between $H^{p}$ and $H$, together
with the other results, shows that the high enstrophy regions form small scale
vorticity jets surrounded by a small region of swirling-flow. Note that the
different correlation measures address different aspects of the structure of
the velocity and vorticity fields and it is the combination of the different
results and the similarities involved that leads to this conclusion (see
figure 15, where the correlations are applied to Burgers vortices). For
example, a small-scale high vorticity jet is not necessarily surrounded by a
swirling-flow region, nor a small region of strong alignment between the
angular velocity and the vorticity necessarily implies a swirling-flow.
However, comparing the high enstrophy structures
($\omega^{2}>5\left\langle{\omega^{2}}\right\rangle$) in figure 17 with
regions of high $G$, $G^{s}$, $H$ and $H^{p}$, in figures 22, 23, 24 and 25,
respectively, we can conclude that strong enstrophy regions are almost
invariably associated with a surrounding region of swirling-flow. The CDFs of
${H^{p}_{i}}$ show that roughly $86\%$, $94\%$ and $97\%$ of the fields are
within $1{H^{p}_{i}}^{\prime}$, $2{H^{p}_{i}}^{\prime}$ and
$3{H^{p}_{i}}^{\prime}$, respectively, which are values similar to the
relative levels of the $H$ field.

We note some overall differences in the correlation fields in comparison to
the canonical flows. Unlike both the one-dimensional Oseen vortex examples and
the three-dimensional Burgers vortices, in (homogeneous isotropic) turbulence,
we do not find high $L$ regions being associated with high $H^{p}$ at their
cores. This shows that there are no large or intermediate scale swirling flow
regions, which instead are found only at the smaller scales. Additionally,
unlike the Taylor-Green example, the jet-like flow regions in turbulence,
which are also associated with high $E_{k}$, are not induced by the
interaction of swirling structures. These regions, however, are still
externally induced, as our analysis will reveal. This shows that the
organization of coherent structures in turbulence is more complex than the
simplified canonical flows shown here where there is a clear association
between the coherence of high vorticity regions and high kinetic energy
regions.

(a)

(b)

(c)

(d)

Figure 25: The $\mathbf{H^{p}}$ correlation is shown for a single field
snapshot, at $500\tau_{k}$. Panel (a) shows the three-dimensional field as a
volume rendering, while panel (b) shows the cross-sectional view of
$H^{p}=\sqrt{\mathbf{H^{p}}\cdot\mathbf{H^{p}}}$. Panels (c) and (d) show the
time-averaged PDFs and CDFs of the three components of $\mathbf{H^{p}}$.

We also applied these correlations to a reference dataset of homogeneous
isotropic turbulence obtained from the Johns Hopkins Turbulence Databases
(JHTD) (Perlman et al., 2007; Li et al., 2008), which was generated using a
pseudo-spectral method on a grid of $1024^{3}$ at $Re_{\lambda}=433$. The
results are shown in Appendix B; they are found to be essentially similar to
those presented here, from our in-house code. We use our own numerical
datasets for the remainder of this study, for our ease of access and control
over the data.

###  5.3 Influence of the choice of $\Lambda$

Before proceeding with further analysis of the correlation fields, it is
important to consider the influence of $\Lambda$ on the results. The obvious
values of $\Lambda$ that can be disregarded are those extremely small or
large. A value of $\Lambda$ that is too small is somewhat meaningless, since
we intend to capture non-local structures which have a finite physical size.
On the other hand, very large values of $\Lambda$ ($\sim N_{x}/2$) will
introduce periodicity induced artifacts in the correlation fields, which
should be avoided. However, there is a wide range of values of $\Lambda$ in
$0<\Lambda<N_{x}/2$ which are viable. Here, we do not intend to study in
detail the influence of $\Lambda$ on the correlation measures of the field
structures, hence, we would like to choose a value of $\Lambda$ large enough
to cover most structure sizes, such that the results do not depend strongly on
$\Lambda$.

The correlations $\mathbf{G}$ and $\mathbf{G^{s}}$ are not expected to vary
significantly for different choices of $\Lambda$ (once $\Lambda$ is large
enough to cover the size of these structures), since the structures in the
vorticity field are small-scaled, and more or less randomly distributed
throughout the volume. Similarly, the correlations $\mathbf{H}$ and
$\mathbf{H^{p}}$ are expected to be even more insensitive to the choice of
$\Lambda$, since the Biot-Savart influence of the vorticity decays with the
square of the distance. We consider the $\mathbf{L}$ correlation, since it
identifies intermediate sized structures in the velocity field, and can
potentially be influenced by the choice of $\Lambda$. $\mathbf{L}$ has the
form

$L_{i}(\mathbf{x},\Lambda)=\int_{-\Lambda}^{\Lambda}\mathbf{u}(\mathbf{x})\cdot\mathbf{u}(\mathbf{x}+\mathbf{r}_{i})\mathrm{d}r_{i}$
(39)

where $\mathbf{u}(\mathbf{x})$ can be placed outside the integral as

$\displaystyle L_{i}(\mathbf{x},\Lambda)$
$\displaystyle=\mathbf{u}(\mathbf{x})\cdot\int_{-\Lambda}^{\Lambda}\mathbf{u}(\mathbf{x}+\mathbf{r}_{i})\mathrm{d}r_{i}$
$\displaystyle L_{i}(\mathbf{x},\Lambda)$
$\displaystyle=2\Lambda\left(\mathbf{u}(\mathbf{x})\cdot\mathbf{\widetilde{u}}_{i}\right)$
(40)

where

$\mathbf{\widetilde{u}}_{i}=\frac{1}{2\Lambda}\int_{-\Lambda}^{\Lambda}\mathbf{u}(\mathbf{x}+\mathbf{r}_{i})\mathrm{d}r_{i}$
(41)

The $\mathbf{L}$ correlation is essentially the inner product of the velocity
field $\mathbf{u}$ with the $\mathbf{\widetilde{u}}$ field (which is a
function of $\Lambda$). Hence, if the $\mathbf{\widetilde{u}}$ field varies
significantly with $\Lambda$, so will $\mathbf{L}$. In figure 26, snapshots of
the $\widetilde{u}_{z}=|\mathbf{\widetilde{u}}_{z}|$ field are shown for a
wide range of $\Lambda/\lambda$ values, at a planar crossection of the volume
(at the same time instance shown in figure 17). At very small values,
$\Lambda/\lambda<1$, the $\mathbf{L}$ field looks very similar to the $E_{k}$
field, which is expected since the limit $\Lambda\to 0$ reduces
$\mathbf{u}\cdot\mathbf{\widetilde{u}}$ to $\mathbf{u}\cdot\mathbf{u}$. The
$\widetilde{u}_{z}$ field does not appear to change significantly for
$1\leq\Lambda/\lambda<7$, which is also true for $\Lambda/\lambda\geq 7$,
although those values of $\Lambda$ begin to approach the size of the
simulation domain and should be disregarded.

Figure 26: Crossectional slices of the $\tilde{u}_{z}$ field for varying
$\Lambda/\lambda$.

It is interesting that the $\widetilde{u}_{z}$ field appears to vary slowly
for $\Lambda/\lambda>1$. The variation of $\mathbf{\widetilde{u}}_{i}$ with
$\Lambda$ can be quantified by calculating

$\left\lvert\frac{\mathrm{d}\mathbf{\widetilde{u}}_{i}}{\mathrm{d}\Lambda}\right\rvert=\frac{|\mathbf{\widetilde{u}}_{i}^{\Lambda+\Delta\Lambda}-\mathbf{\widetilde{u}}_{i}^{\Lambda}|}{\Delta\Lambda}$
(42)

where $|\cdot|$ is the amplitude of the difference between the two fields.
This is shown in figure 27 for
$\mathbf{\widetilde{u}}_{x},\mathbf{\widetilde{u}}_{y}$ and
$\mathbf{\widetilde{u}}_{z}$, where $\left\langle{\cdot}\right\rangle$ denotes
spatial averaging over the entire volumetric domain, and over two independent
realizations of $\mathbf{\widetilde{u}}_{i}$ at $t\approx 500\tau_{k}$ and
$1000\tau_{k}$. The change in
$\left\langle{|\mathrm{d}\mathbf{\widetilde{u}}_{i}/\mathrm{d}\Lambda|}\right\rangle$
is large for $\Lambda/\lambda<1$. This reflects the fact that most of the
velocity structures in the flow are smaller than the Taylor microscale
$\lambda$, and they get averaged over in the $\mathbf{\widetilde{u}}$ fields
for increasing $\Lambda$. Next,
$\left\langle{|\mathrm{d}\mathbf{\widetilde{u}}_{i}/\mathrm{d}\Lambda|}\right\rangle$
seems to decay exponentially for $\Lambda/\lambda>1$, with a slope of
approximately $-1/5$. This change of behaviour occurs via a sharp transition
around $\Lambda/\lambda\approx 1$.

The behaviour of
$\left\langle{|\mathrm{d}\mathbf{\widetilde{u}}_{i}/\mathrm{d}\Lambda|}\right\rangle$
shown in figure 27 indicates that for large $\Lambda/\lambda$, i.e. in the
limit of $\Lambda\to\infty$: (i)
$\mathrm{d}\mathbf{\widetilde{u}}_{i}/\mathrm{d}\Lambda\to 0$, (ii)
$\mathbf{\widetilde{u}}_{i}\to 0$, (iii)
$\mathrm{d}\mathbf{L}/\mathrm{d}\Lambda\to 0$ and (iv) $\mathbf{L}$ goes to a
constant. This is consistent with equation 20 and the requirement that there
should not exist “an organized motion over an infinite distance”. Also, figure
27 shows that $\mathbf{\widetilde{u}}_{i}$ varies slowly for
$\Lambda/\lambda>1$ and suggests that $\Lambda|\mathbf{\widetilde{u}}_{i}|$
attains a maximum, roughly, in the range $1\leq\Lambda/\lambda\leq 4$, hence,
the results for $\mathbf{L}$ and $\mathbf{L^{s}}$ are not expected to vary
significantly in this range of $\Lambda$. This, together with the sharp
transition around $\Lambda/\lambda\approx 1$, suggests that the Taylor
microscale $\lambda$ is a good measure for the integration length of the
correlations. For the remainder of this study, we use correlation fields
calculated for $\Lambda=\lambda$. This value is also large enough to account
for vorticity field structures and vorticity-velocity (i.e. Biot-Savart
related) field structures.

Figure 27: Rate of change in the $\mathbf{\widetilde{u}}_{i}$ fields with
increasing $\Lambda$. Here $\left\langle{\cdot}\right\rangle$ denotes spatial
averaging.

###  5.4 Spectral characteristics of the correlation fields

The spectra of the three-dimensional correlation fields are shown in figure
28, in comparison to the kinetic energy ($E_{k}$) and enstrophy ($\omega^{2}$)
spectra. The correlation spectra are calculated in the same way as the $E_{k}$
spectra, where the three-dimensional Fourier transforms of the correlation
fields are squared and spherically averaged over wavenumber shells. The
spectra have also been time-averaged over $20$ realizations, each separated by
$50\tau_{k}$. Panels (a) and (b) show the $\mathbf{L}$ and $\mathbf{L^{s}}$
spectrum respectively, that are found to have very similar shapes. This
reflects that the spatial distribution of the $\mathbf{L^{s}}$ field is very
similar to that of the $\mathbf{L}$ field, reaffirming that there are no large
symmetries or anti-symmetries in the velocity field. The spectra, in
comparison to the $E_{k}$ spectrum, have a shift in the peak to higher
wavenumbers. This is a consequence of the fact that the $\mathbf{L}$ and
$\mathbf{L^{s}}$ spectra reflect the structure of the correlation fields,
which comprise of relatively smaller correlation kernels associated with
larger flow structures.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 28: Time-averaged premultiplied spectra of correlation fields (a)
$\mathbf{L}$, (b) $\mathbf{L^{s}}$, (c) $\mathbf{G}$, (d) $\mathbf{G^{s}}$,
(e) $\mathbf{H}$ and (f) $\mathbf{H^{p}}$ shown together with the kinetic
energy ($E_{k}$) and enstrophy ($\omega^{2}$) premultiplied spectra.

The spectra of $\mathbf{G}$ and $\mathbf{G^{s}}$ in panels (c) and (d),
respectively, are found to closely resemble the enstrophy spectrum, with a
slight shift in the spectral peak to higher wavenumbers for both cases, which
is again due to the correlation kernels being relatively smaller than the
vorticity field structures. This shift is more pronounced in the
$\mathbf{G^{s}}$ spectrum, which is consistent with the fact that the
$\mathbf{G^{s}}$ field has sharper features, as can be seen in both figure
23(a) (volume rendered $\mathbf{G^{s}}$ field snapshot) and figure 10 (one-
dimensional Oseen vortex-pair example).

The $\mathbf{H}$ and $\mathbf{H^{p}}$ spectra in panels (e) and (f),
respectively, are found to have a striking similarity to the enstrophy
spectrum. This reflects that the spatial variation of enstrophy consistently
yields a similar spatial variation in the $\mathbf{H}$ and $\mathbf{H^{p}}$
fields. This also indicates that the “energy containing” wavelengths of the
velocity field do not have any significant swirl, showing that the swirling
flow structures are coupled with the vorticity structures, and decoupled from
the strong velocity structures. Changing the correlation integration length
$\Lambda$ to different values can possibly influence the spectral
characteristics of the correlation fields, which we do not investigate in this
study.

Overall, the similarity between $\mathbf{L}$ and $\mathbf{L^{s}}$ spectra
shows that there are no larger symmetries or anti-symmetries in the velocity
field. Further, the similarities between $\mathbf{G}$, $\mathbf{G^{s}}$,
$\mathbf{H}$ and $\mathbf{H^{p}}$, together with the enstrophy spectrum, shows
that high enstrophy regions invariably form concentrated vorticity-jets,
surrounded by swirling velocity regions. These results also support the
statistics of the correlation fields shown so far. For the remainder of this
study, we shall use the $\mathbf{L}$ and $\mathbf{H^{p}}$ correlations to
identify structures, since the nature of (homogeneous isotropic) turbulence
fields leads to a strong correspondence between (i) $\mathbf{L}$ and
$\mathbf{L^{s}}$ correlations and (ii) $\mathbf{G}$, $\mathbf{G^{s}}$,
$\mathbf{H}$ and $\mathbf{H^{p}}$ correlations, while in general, this need
not be the case.

###  5.5 Spatial distribution of correlation fields

The different correlation measures identify distinct forms of coherence in the
flow. The $\mathbf{L}$ correlation identifies intermediate sized regions where
the velocity field has a high magnitude and parallel streamlines. The
$\mathbf{H^{p}}$ correlation identifies smaller scale high vorticity regions
associated with a swirling velocity. The relative spatial distribution of the
correlation fields, hence, sheds light on the distribution of the different
coherent structures in physical space. In this section, we discuss the spatial
distribution statistics of $\mathbf{L}$ and $\mathbf{H^{p}}$, in relation to
$E_{k}$ and $\omega^{2}$.

First, figure 29(a) shows iso-surfaces of
$E_{k}=2\left\langle{E_{k}}\right\rangle$ (in blue), together with iso-
surfaces of $L=2L^{\prime}$ (in red). At the chosen threshold levels, the $L$
regions are mostly contiguous, and consistently contained within the $E_{k}$
regions, reflecting the fact that increasingly higher values of the $L$ field
occupy successively smaller regions of space. This also shows that high
kinetic energy regions, $E_{k}\geq 2\left\langle{E_{k}}\right\rangle$, yield
high $L$ values, in the range $L\geq 2L^{\prime}$. This also suggests that as
$E_{k}$ gets higher, the level or flow organization increases.

Figure 29(b) shows iso-surfaces of vorticity at $\omega=3\omega^{\prime}$ (in
blue) together with iso-surfaces of $H^{p}$ at $8{H^{p}}^{\prime}$ (in red).
The $H^{p}$ iso-surfaces are shown at a high value to demonstrate that only a
fraction of the intermediate vorticity ‘worms’ yield very high $H^{p}$ values
in their core regions. For instance, in panel (d), the
$H^{p}=5{H^{p}}^{\prime}$ iso-surfaces shown have a comparable size to the
$3\omega^{\prime}$ iso-contours in panel (b). It is found that $H^{p}$
occupies equivalent or more volume than $\omega$ at low thresholds, while at
increasingly higher thresholds, the $H^{p}$ field occupies successively
smaller fractions of the volume, as shall be quantified with the spatial
statistics.

(a) Contours of the turbulent kinetic energy
$E_{k}=2\left\langle{E_{k}}\right\rangle$ (blue) shown together with contours
of $L=2L^{\prime}$ (red).

(b) Contours of vorticity at $\omega=3\omega^{\prime}$ (blue) shown together
with contours of $H^{p}=8{H^{p}}^{\prime}$ (red).

(c) Contours of $E_{k}=2\left\langle{E_{k}}\right\rangle$ (blue) along with
$\omega=3\omega^{\prime}$ (red) shows that these fields are spatially
exclusive.

(d) Contours of $L=2L^{\prime}$ (blue) along with $H^{p}=5{H^{p}}^{\prime}$
(red) shows that these correlation fields are spatially exclusive.

Figure 29: Spatial distribution of correlations $L$ and $H^{p}$ in comparison
to turbulent kinetic energy $E_{k}$ and vorticity $\omega$.

Figure 29(c) shows contours of $E_{k}$ at $2\left\langle{E_{k}}\right\rangle$
(in blue) shown together with contours of vorticity at
$\omega=3\omega^{\prime}$. It appears that high kinetic energy regions and
high vorticity regions are spatially exclusive to a large extent. This spatial
distribution of the two quantities might be a consequence of the dynamical
separation between them, as high $E_{k}$ and high $\omega^{2}$ are influenced
by different aspects of the Navier-Stokes dynamics, as also observed by
Tsinober (2014). The two fields, as expected, begin to also overlap when the
thresholds are lowered, and become more exclusive and distanced at higher
thresholds. Lastly, panel (d) shows the distribution of correlations $L$ and
$H^{p}$, where contours of $L=2L^{\prime}$ (in blue) are shown together with
contours of $H^{p}=5{H^{p}}^{\prime}$. Since these correlations closely
resemble $E_{k}$ and $\omega$, respectively, they also remain spatially