12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

A SCALE-SPACE ANALYSIS OF SIMULTANEOUS ENERGY, HELICITY,
AND ENSTROPHY CASCADES

Douglas W. Carter
Department of Aeronautical andAstronautical Engineering
University of Southampton
Burgess Road, Southampton, UK SO17 1BJ
D.W.Carter@soton.ac.uk

Felipe Alves Portela
Laboratoire de Mécaniquedes Fluides de Lille
University of Lille, CNRS
Kampé de Fériet, F-59000 Lille, France
felipe.alvesportela@univ-lille.fr

Paweł Baj
Institute of Aeronautics and Applied Mechanics
Warsaw University of Technology
Nowowiejska 24, 00-665 Warsaw, Poland
pawel.baj@pw.edu.pl

ABSTRACT
We characterise the incompressible turbulence cascade interms of the concurrent inter-scale and inter-space exchangesof the scale-by-scale energy, helicity and enstrophy. The governing equations for the scale-by-scale helicity and enstrophy
are derived in a similar fashion to that of the second order
velocity structure function obtained by Hill (2002). We apply these equations to both forced periodic turbulence and a
von Kármán flow, focusing on scales smaller than r = λ , Taylor length scales. The well-known random sweeping effectin the energy cascade between unsteady and non-linear transport is found to extend across all quantities considered. Additional mechanisms within the individual cascades are identified. Across cascades, the helicity cascade was found to be
decorrelated from the others at all scales.

tensor. This quantity lies at the center of the RichardsonKolmogorov cascade phenomenology (Richardson, 1920; Kolmogorov, 1941a,b). Under the assumptions of local homogeneity, isotropy, and appropriate separation of scales, it balances the non-linear inter-scale energy transfer to yield theinfamous Kolmogorov’s 4/5-ths law. The dissipation ε is intimately linked with the enstrophy ω 2 = ωi ωi (where ωi =εi jk ∂ /∂ x j uk is the vorticity vector) as well as the second invariant of the NS equations known as the helicity h = ui ωi . It isclear that a full characterisation of the turbulence cascade must
involve not only the kinetic energy but also quantities such as
enstrophy and helicity, as they bear relation to dissipation andcapture the presence of coherence, intermittency, parity, andother such phenomena known to break with the classical picture of turbulence.

INTRODUCTIONSCALE-SPACE FRAMEWORKThe instantaneous interactions between eddies that govern turbulence are highly dynamic, taking place across a broad
range of scales. The distinction between the instantaneousinteractions in the turbulence and the statistically stationary
cascade that they give rise to are a well known problem inturbulence modelling. For example, in Large Eddy Simulations (LES) the directionality of the cascade (from large scales
to small scales, or vice-versa) at any given instant may differ from that predicted by statistically stationary models (Germano et al., 1991) in which energy is transferred on average, from large, energy-containing eddies, to small, energydissipating ones. Recent work by Goto & Vassilicos (2016)
and Yasuda & Vassilicos (2018) highlights how the statistically stationary viewpoint of the turbulence cascade effectively
overlooks important dynamics that underlay (and set) the stationary state itself. The present work seeks to expand uponthis viewpoint using a correlation-based analysis of cascadesof energy, helicity and enstrophy.
The core element at the heart of the turbulence cascade is the rate of kinetic energy dissipation ε = 2νSi j Si j , νbeing the fluid’s kinematic viscosity and Si j the strain-rate

Hill (2002) derived a budget for the scale-space energy
increment δ q2 = δ ui δ ui , where δ ui = ui − u′i is the difference of the velocity at two independent points xk and xk′ . Thisbudget can be obtained directly from the Navier-Stokes equations without requiring any information regarding the structureof the flow, reducing to the Kármán-Howarth equation under
the assumptions of homogeneity, isotropy and stationarity. An
identical procedure is carried out to derive the evolution equations for the scale-space helicity δ h ≡ δ ui δ ωi and scale-spaceenstrophy δ ω 2 ≡ δ ωi δ ωi . This results in three generalised
equations capturing the instantaneous dynamics of the energy,
helicity, and enstrophy (in the absence of explicit forcing) for
incompressible turbulent flow:
A

T

z
}|
{
z }|t {
ΠTpz }| {z }| {∂ δ q2Σuuδ q22+ ∇r · δ uδ q = − ∇X ·−2 ∇ X · δ u δ p
∂t
2


2
1
+ ν 2∇2r + ∇2X δ q2 −2Σ |{z}
εq ,
2
E
|
{z
}
Dν

1

(1)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022δ ω δ q2
Σuuδ q2∂δh∇X ·+ ∇r · δ uδ h −
= −∇
∂t
2
2

2
ωδq
Σω
1
−
− ∇X · δ ω δ p + ν(2∇2r + ∇2X )δ h − 2Σε h ,4
2

RESULTS
Characterisation of the data sets
In order to characterise the three scale-space quantities
of interest, we first investigate the ensemble-averaged scalespace energy δ q2 , helicity δ h, and enstrophy δ ω 2 structure
functions for the data sets. These are shown for the VK andJHU data in figure 1. Upon appropriate normalisation (as in
the figure), these functions are expected to plateau in the inertial range (beyond the Taylor microscale). The limited size ofthe data cubes in the experimental data set prevents us from detecting such a plateau, however there is close agreement withthe larger DNS cubes that do appear to plateau at r ≳ λ . Finite
Reynolds number effects must also be taken into account.
Despite the lack of explicit helical forcing in the DNS,
the scale-space helicity is non-zero for both data sets. Thisis consistent with the helicity spectrum computed over a wide
range of simulations (Chen et al., 2003; Mininni et al., 2006)
with and without helical forcing (Alexakis, 2017). This canbe seen directly through an expansion of δ h, i.e. for homogeneous turbulence (ui − u′i )(ωi − ωi′ ) = 2h − 2ui ωi′ . This expansion reveals that even when small-scale mirror symmetry
holds (h = 0) the non-zero scale space helicity (and spectrumof helicity) arises from the coherence in the velocity-vorticity
correlation (Levich & Shtilman, 1988).
We next consider the non-linear flux (NLF) structure
functions for each of the cascades. A small departure to justify this chosen terminology will be taken here. These structure functions are commonly referred to as “third-order structure functions”, but as the origin of “third order” is in thestatistical moment of the increment of a single quantity (i.e.
δ u3 ) a different terminology is adopted for improved generality. As noted by Hill (2002), making use of Gauss’ theorem
when integrating the non-linear energy transfer term of eq. (1)˝ q2
over the spherical scale space volume Vr gives
Π dVr =‚
2
∂ Vr δ uk δ q nk dS with nk the outward normal vector anddS the surface of the spherical shell. The orientation average over the spherical shell is identically zero, leaving only aflux in the radial direction. This motivates referring to thesequantities as non-linear flux (NLF) structure functions in thepresent context. In addition to the central role of NLF structure functions in classical turbulence theory, i.e. the 4/5ths(K41) and 2/15ths laws (Chkhetiani, 1996; L’vov et al., 1997),
their physical significance is well documented in the context ofa spherical scale-space coordinate system (Valente & Vassilicos, 2015; Gomes-Fernandes et al., 2015; Alves Portela et al.,2017).
The orientation-averaged NLF energy structure function
is shown in fig. 2(a) normalised using the classical 4/3rds law(a form of the 4/5ths law that avoids breaking down δ q2 into
components, see Hill, 2002). The subscript “r” is used to denote the radial flux in the spherical coordinate system. The
plateau seen in the JHU data, for scales beyond the Taylor microscale, suggests classical behavior of δ ur δ q2 . Consistent
with fig. 1, the VK data does not show a clear plateau develop2
ing for δ ur δ q2 / 43 ε q r, but does reach a maximum value close
to one at r ≈ λ .
The orientation-averaged NLF helicity structure functions
are shown in figures 2(b) and 2(c) normalised using the product of r and dissipation rate of helicity ε h . This normalisation is chosen to compare the two distinct mechanisms associated with Πh in equation 2. The first, δ uk δ h, originates fromthe advective term and the second, δ ωk δ q2 , from the vortex
stretching term of the vorticity form of Navier-Stokes equations (Yan et al., 2019). It is again seen that, due to its nonpositive definiteness and inherently high variation, the non-(2)

Σuuδ ω 2∂ δ ω2∇X ·+ ∇ r · δ u δ ω 2 = −∇+ ⟨δ ω δ ω T , ΣSS ⟩
∂t
2


1
ω T , ΣSS ⟩ + ν 2∇2r + ∇2X δ ω 2 − 2Σε ω . (3)+ ⟨δ ω Σω2
In eqs. (1) to (3) the symbol Σ indicates the two-pointsum (akin to the two-point difference δ ), angled brackets denote the inner product, S is the strain-rate tensor, p is the ratio
of pressure to (constant) density, ν is the kinematic viscosityand ε is the viscous dissipation of the quantity in the superscript. The gradient operator corresponds to gradients withrespect to the scale variable ri and the centroid position Xi(Mollicone et al., 2018). The shorthand notation for the various transfers in equation 1 apply analogously to equations 2
and 3, with the exception that the generation terms of equation
3 appear in lieu of the pressure transport with the shorthandω T , ΣSS ⟩.
GS = ⟨δ ω δ ω T , ΣSS ⟩ + ⟨δ ω Σω
Notice that these three budgets are similarly structured.
At is an unsteady term and represents the temporal increaseor decrease of energy/helicity/enstrophy at each instant and ateach scale. Π represents the nonlinear transfer in scale space.
It describes the transfer of energy/helicity/enstrophy from aspherical shell at scale rk centred on Xk to either an adjacentshell or to an adjacent location within the same shell rk + drk .
T is the nonlinear turbulent transport in physical space. It captures the transport of energy/helicity/enstrophy from a spherical shell of radius rk centred at Xk to an adjacent sphericalshell at the same rk centred at Xk + dXk . T p results from theinteraction of the pressure and velocity/vorticity fields to produce a pressure transport (similar to T ) that acts to transportenergy/helicity at a particular scale to a neighboring shell. GS
is a generation term in scale space resulting from the coupling
between the rate-of-strain and the enstrophy. Dν represents theviscous diffusion of energy/helicity/enstrophy in scale spaceand physical space. E represents the two-point average dissipation rate. This can be seen, for example for the scale to scaleenergy transfers, dividing equation 1 by 4 on both sides suchthat E = 12 (ε + ε ′ ) where ε = ν(∂ ui /∂ xk )2 .
We refer to Valente & Vassilicos (2015); GomesFernandes et al. (2015); Alves Portela et al. (2017); Mollicone
et al. (2018) who give a more detailed interpretation of transport, production and destruction terms, as such. Henceforth,superscript notation is adopted to distinguish the various transfers of energy (q2 ), helicity (h), and enstrophy (ω 2 ).

Data Set Descriptions
The instantaneous cascades are evaluated using two data
sets of homogeneous turbulence. The first is a direct numerical simulation (DNS) of homogeneous isotropic turbulence
at Taylor-microscale Reynolds number Reλ = 433 from theJohns Hopkins Turbulence Database (JHU, Li et al., 2008).
We analyse 2560 full resolution sub-cubes from the full simulation of side length 5λ . A separate analysis on time-resolved
3D-3C scanning particle image velocimetry data from the center of a Von-Kármán mixing tank (VK) from the work of Knutsen et al. (2020) was also performed for comparison, for whichReλ = 199 and 2×105 snapshots were used. The experimental
cubes of data are slightly smaller, with side lengths of approximately 1.5λ .

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

(a)

(b)

(c)

Figure 1: Normalized mean energy structure function δ q2 (a), helicity structure function δ h (b), and enstrophy structure
function δ ω 2 (c).

(a)

(b)

(c)

(d)

Figure 2: Normalised orientation-averaged NLF structure functions corresponding to the non-linear scale-to-scale flux ofenergy δ q2 (a), helicity δ h via advection (b) and vortex stretching (c), and enstrophy δ ω 2 (d).

linear flux of helicity is very slow to converge, leading to large
uncertainty bars. Despite this, they are found to be non-zeroand the JHU data exhibits a plateau (within large uncertainty)
for both non-linear helicity flux mechanisms.

scale-space coordinate system. This necessitates compensating −δ ur δ ω 2 by r for a one-to-one comparison).

Correlations of Instantaneous Cascades

Finally, the NLF enstrophy structure functions can beseen in fig. 2(d) normalised using the dissipation rate of energy,
the Kolmogorov length scale η, and r. Similarly to the enstrophy structure function, a collapse of the data sets is observed in
the near-dissipation range between r/η = 10 and r/λ = 1. The
flux is seen to decrease as r−3 (r−2 in the figure due to compensating −δ ur δ ω 2 by r) into the inertial range. This powerlaw decrease is consistent with the results of Davidson et al.
(2008) (in their case, an alternative scale-by-scale enstrophy
flux is defined that is not motivated by the present sphericalIn the following we consider the correlations within the
cascades (i.e. terms of equations 1-3 individually) and between cascades (i.e. terms across equations 1-3). We focus
on the separation r = λ , the Taylor microscale, as this is theupper bound for scales conditioned by viscosity (Yasuda &Vassilicos, 2018; Valente & Vassilicos, 2015).

Correlations within cascades The correlationsof the transfer terms for the energy, helicity, and enstrophy312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Correlation between various transfers of scale-space energy δ q2 (a) & (d), helicity δ h (b) & (e), and enstrophyδ ω 2 (c) & (f) for the experimental VK data (a)-(c) and the DNS of JHU (d)-(f) at the scale r = λ .

cascades are presented in figure 3. Starting with the energy,
the correlations of terms are in good agreement between both
the JHU and VK data. The most2 robust correlation lies beq
tween the unsteady transport At and the turbulent transport
2q
T . Recalling that T is associated with transport in physical2q2space Xk , the high correlation between At and T q is characteristic of the advective nature of turbulence. This is known
from single-point analysis as the random sweeping effect andextends to scale space (Yasuda & Vassilicos, 2018). As explained by Tsinober (2001), it is caused by the tendency of the∂ui to anti-alignunsteady acceleration of the NS equations ∂t∂with the convective acceleration uk ∂ x ui , with increasing antik
alignment for increasing Reynolds number.

downscale), the overall tendency is for the pressure transport
to facilitate downscale energy transfer. The non-local nature of
the pressure field (Tsinober, 2001) and its resulting influenceon transport motivates the interpretation of this correlation asa signature of non-locality on the non-linear energy transfer.
Despite the difference in domain and boundary conditions, forq2
the energy cascade the agreement in the correlation of T p
2and Πq between the two data sets is high.
The correlations of the transfer terms of the scale toscale helicity reveal that the same mechanisms within the energy cascade are present in the helicity cascade. The random
sweeping effect is evident from the correlation between At hand T h , being only slightly lesser in magnitude when compared to the corresponding terms for the energy. In contrast to
the energy, the pressure transport T ph appears to have only aweak correlation with the non-linear transfer Πh . If the same
picture holds here as for the energy cascade, this indicates the
non-local role played by the pressure in facilitating the nonlinear cascade of helicity is not as significant at r = λ .
For the scale to scale enstrophy transfers, due to the “curling out” of the pressure transport, instead a generation term
2
Gsω features in equation 3. Caution must be exercised in interpreting the results for the enstrophy transfers. Noise prop2
2
agation in the VK data renders correlations of At ω and E ωparticularly erroneous. As a result in the following we focus
on the results of the JHU data in figure 3(f).
A striking similarity with the energy and helicity cascades
is immediately apparent through the anti-correlation of the un-

It should be noted that the random sweeping effect is,q2
more accurately, represented by the correlation between At22
and the difference T q − Πq (Yasuda & Vassilicos, 2018).
Nevertheless, the present analysis is restricted to correlations
between individual terms of equation 1. Despite this differ2
ence, a strong individual correlation with T q is observed,
consistent with the results of Yasuda & Vassilicos (2018).
2

Turning attention to the non-linear transfer Πq , a signif2
icant anti-correlation between Πq and the pressure transport2q
T p is clear. Yasuda & Vassilicos (2018) used geometric analysis to show that this is due to the tendency of δ ui and δ ∂∂ xpi
to align in such a way that compressing motions move energy
downscale and stretching motions upscale. Although the opposite happens too (e.g. compressing upscale and stretching4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
between At and T . There is a positive correlation between
q2
the transient energy At and the turbulent transport of enstro2
phy T ω (and vice-versa). Consider the turbulent transport
2of enstrophy expressed as T ω = 12 Σuk ∂∂X δ ω 2 . When Σuk
kaligns with the physical space gradient of the scale enstrophy∂2δ ω , the turbulent transport is positive. The sweeping mo∂ Xktions (embodied by Σuk ) assist the scale enstrophy gradient (inother words, they sweep with the gradient). When Σuk antialigns with ∂∂X δ ω 2 , the sweeping motions resist the scale enk
strophy gradient. Consider also the close relationship between
the dissipation of energy and the scale enstrophy. Together, the2q2
positive correlation of T ω and At relates to energy decay
for sweeping motions assisting the physical space gradient of
the dissipation of energy. Similarly, it corresponds to energy
growth for sweeping motions resisting the physical space gradient of the dissipation of energy. This effect is mirrored by22
the positive correlation of T q and At ω . This hints at the significance of the alignment of sweeping motions with the scale
velocity gradients of the flow.

Figure 4:

Correlation coefficients between terms ofequations 1 and 3 (respectively transfers terms of δ q2
and δ ω 2 on the horizontal and vertical axis) at r = λ in
the JHU data.
2q2

An anti-correlation between the unsteady terms At and2At ω is also observed, with increasing anti-correlation for
decreasing scale. This implies that instantaneous enstrophy
growth corresponds to energy decay (and vice-versa). If weagain consider that the enstrophy positively correlates to thedissipation of energy, this anti-correlation is interpreted as simply the tendency for the energy to decay when the local dissipation is growing in time (and vice versa). This is unsurprising considering the role of the dissipation as a sink. Due tothe tendency of ∂ ui /∂t to anti-align with uk ∂∂x ui (the randomksweeping effect), the same phenomenon follows for the anti2
2
correlation of T q and T ω .

2steady transport At ω and turbulent transport T ω . This confirms that the random sweeping effect persists across cascades.
2
The generation term Gsω has a different instantaneous rolein the transfers of the enstrophy compared to the role of thepressure transport in energy and helicity. It exhibits a posi2
2
tive correlation with E ω and a negative correlation with Dνω .
This implies a concurrent enstrophy generation and dissipation mechanism (Davidson et al., 2008). This simultaneous
generation and dissipation is tied to a reduction (during highdissipative events) or increase (during low dissipative events)
in the diffusion of enstrophy.

Correlations between cascades Correlations
between transfers of energy and helicity and between transfersof enstrophy and helicity were computed and found to be universally near zero. This is due to the non positive definitenessof helicity (Moffatt & Tsinober, 1992). At first glance, this appears to contradict a wealth of research that has identified clear
causal relationships between the three quantities (Bershadskii
et al., 1994; Biferale et al., 2013; Alexakis, 2017; Bos, 2021).
As discussed by Tsinober (2001), near-zero correlations arenecessary, but not sufficient, to determine that quantities areunrelated. For example, the unsteady term ∂∂tui is almost eni
tirely decorrelated with the material acceleration Du
Dt simply∂ ui∂ ui
due to the underlying anti-correlation of ∂t with uk ∂ x .
kCorrelations between the transfers of energy and enstrophy are not affected by non positive definiteness and are shownfor the JHU data in figure 4. The most robust correlation appears in the dissipation, where the rate of energy dissipation
2
2
E q and the rate of enstrophy dissipation E ω are highly correlated. Following from this correlation between the dissipations, there is a negative correlation between energy dissipa2
2tion E q and enstrophy diffusion Dνω and a positive correla2
qtion between energy dissipation E and enstrophy generation
2
Gsω . Together, this supports the classical picture that the generation of small-scale structures leads to large gradients and inturn the dissipation of energy (as well as enstrophy generation,
Siggia, 1981).
Several other observations are of note between the energy
and enstrophy cascades. The signature of the random sweeping effect can be clearly seen amongst the weak correlations

CONCLUSIONS
We have presented an instantaneous analysis of the simultaneous energy, helicity, and enstrophy cascades of forced
homogeneous incompressible turbulence focusing on the separation at the Taylor microscale. The scale-space framework, already well established for generalisation of the energy
cascade via the KHMH equation, has been extended analogously to the scale-space helicity and enstrophy. It was foundthat large-scale sweeping is present across the three cascades:
this is identified through the significant anti-correlation of thetransient and physical-space transport terms of each cascade.
In addition, the correlation between the pressure and interscale transfer terms, already reported by Yasuda & Vassilicos (2018) in the context of the energy cascade, is observedin the helicity cascade. The energy and enstrophy cascades arefound to correlate with each other mostly through the generation/dissipation of enstrophy and dissipation of kinetic energy.
The signature of the random sweeping mechanism is also identified between them. The present work represents a rich framework for future analyses of turbulence cascades in scale spacecoordinates.

ACKNOWLEDGEMENTS
The authors are indebted to B. Ganapathisubramani forsupporting this work and enabling its dissemination at TSFP
12.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
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