factor reaching increasingly large negative values with decreasing scale).

4\. As the length scale (i.e. two point separation length) decreases, the
correlation between $\mathcal{A}_{t}$ and $-\mathcal{T}_{\overline{S}}$
increases and so do their fluctuation magnitudes relative to the fluctuation
magnitude of $\mathit{\Pi}_{\overline{S}}^{{}^{\prime}}$ which reaches to be
an order of magnitude smaller by comparison. In this limit, the correlation
between $\mathcal{A}_{t}$ and $-\mathit{\Pi}_{\overline{S}}$ decreases. At
length scales smaller than $\langle\lambda\rangle_{t}$ the correlation between
$\mathcal{A}_{t}$ and $-\mathcal{T}_{\overline{S}}$ is extremely good
indicating a tendency towards two-point sweeping. However, the correlation
between $\mathcal{A}_{t}+\mathcal{T}_{\overline{S}}$ and
$\mathit{\Pi}_{\overline{S}}^{{}^{\prime}}$ remains strong even if reduced
from its near perfect values at length scales larger than
$\langle\lambda\rangle_{t}$ and there remains a small difference of
fluctuation magnitudes between $\mathcal{A}_{t}$ and
$\mathcal{T}_{\overline{S}}$ which is mostly related to the small fluctuation
magnitude of $\mathit{\Pi}_{\overline{S}}^{{}^{\prime}}$. At the other end of
the length scale range, i.e. as the length scale tends towards the integral
scale and larger, the fluctuation magnitudes of $\mathcal{T}_{\overline{S}}$
and $\mathit{\Pi}_{\overline{S}}^{{}^{\prime}}$ tend to become the same.

5\. The irrotational part of the interscale transfer rate has zero spatio-
temporal average but is exactly equal to the irrotational part of the
interspace transfer rate and half the two-point pressure work term in the KHMH
equation. A complete dynamic picture of the interscale transfer rate needs to
also take this into account, even though the fluctuation magnitudes of these
irrotational terms are smaller than the ones of the terms discussed in the
previous paragraph. In fact, the exact relation
$\mathit{\Pi}_{\overline{I}}=\mathcal{T}_{\overline{I}}=\frac{1}{2}\mathcal{T}_{p}$
explains the significant correlation between interscale transfer rate
$\mathit{\Pi}$ and $\mathcal{T}_{p}$ reported by Yasuda & Vassilicos (2018).

6\. The increase towards small correlations at length scales below
$\langle\lambda\rangle_{t}$ between $\mathit{\Pi}_{\overline{S}}$ and both
$\mathcal{D}_{r,\nu}$ and $-\mathcal{\epsilon}$ is accountable to the
significant correlations between these terms at these viscous scales when
conditioned on relatively rare intense $\mathit{\Pi}_{\overline{S}}$ events,
both forward cascading events with negative values of
$\mathit{\Pi}_{\overline{S}}$ of high magnitude and backward cascading events
with positive values of $\mathit{\Pi}_{\overline{S}}$ of high magnitude. The
choice of $\mathit{\Pi}_{\overline{S}}$ to identify relatively rare intense
events is predicated on the fact that the PDFs of
$\mathit{\Pi}_{\overline{S}}$ are negatively skewed similarly to the PDFs of
$\mathit{\Pi}$, whereas the PDFs of $\mathit{\Pi}_{\overline{I}}$ are not. The
solenoidal part of the interscale transfer rate derives from the integrated
two-point vorticity equation and includes non-local vortex
stretching/compression effects at all scales whereas the irrotational part of
the interscale transfer rate derives from the integrated Poisson equation for
two-point pressure fluctuations.

7\. At these relatively rare intense interscale transfer rate events, the
tendency for two-point sweeping may appear increased because of the extremely
good conditional correlation between $\mathcal{A}_{t}$ and
$-\mathcal{T}_{\overline{S}}$ at all length-scales, however $\mathcal{A}_{t}$
and $-\mathcal{T}_{\overline{S}}$ have also very significantly different
average values given the high absolute values of $\mathit{\Pi}_{\overline{S}}$
at these relatively rare interscale transfer events. This implies that there
is neither local/instantaneous sweeping nor local/instantaneous balance
between $\mathit{\Pi}_{\overline{S}}$ and $\mathcal{D}$ or
$\mathit{\Pi}_{\overline{S}}$ and $-\mathcal{\epsilon}$ at these relatively
rare intense events, a conclusion confirmed by the observation that the
conditional averages and the conditional fluctuation magnitudes of
$\mathit{\Pi}_{\overline{S}}$ are much higher than those of $\mathcal{D}$ and
$-\mathcal{\epsilon}$ in absolute values.

8\. Another property of these relatively rare intense solenoidal interscale
transfer rate events is that the conditional averages of solenoidal interscale
and interspace transfer rates have opposite signs when sampling on these
events. There is therefore a relation between them which may however be
concealed by the fact that the fluctuation magnitudes of the interspace
transport rate are higher than those of the interscale transfer rate.

9\. We have also considered the decomposition into homogeneous and
inhomogeneous interscale transfer rates recently introduced by Alves Portela
et al. (2020) and have studied their fluctuations in statistically stationary
homogeneous turulence. The PDFs of the homogeneous interscale transfer rate
are skewed towards forward cascade events whereas the PDFs of the
inhomogeneous interscale transfer rate are not significantly skewed. However,
the skewness factors of the PDFs of the homogeneous interscale transfer rate
are not as high as those of both the full and the solenoidal interscale
transfer rates. Relating to this, $\mathit{\Pi}$ is highly correlated with
$\mathit{\Pi}_{\overline{S}}$ more than with $\mathit{\Pi}_{\overline{I}}$,
$\mathit{\Pi}_{H}$ and $\mathit{\Pi}_{I}$ with all of which $\mathit{\Pi}$ is,
nevertheless, significantly correlated.

10\. There is an increasing correlation between $\mathit{\Pi}_{I}$ and
$-\mathit{\Pi}_{H}$ as the length-scale decreases, in particular below
$\langle\lambda\rangle_{t}$ where it reaches values above 0.6. The interscale
transfer of velocity difference energy caused by local inhomogeneities in
fluctuating velocity magnitudes tends to cancel the interscale transfer of
fluctuating velocity difference energy caused by misalignments between the two
neighboring fluctuating velocities, in particular at length scales below
$\langle\lambda\rangle_{t}$. As a result, the fluctuation magnitudes of
$\mathit{\Pi}$ are smaller than those of both $\mathit{\Pi}_{I}$ and
$-\mathit{\Pi}_{H}$.

11\. Finally, the decomposition
$\mathit{\Pi}=\mathit{\Pi}_{I}+\mathit{\Pi}_{H}$ can be used to physically
distinguish between intense forward and intense inverse interscale transfer
events. The averages of $\mathit{\Pi}_{H}^{{}^{\prime}}$ and
$\mathit{\Pi}_{I}$ when conditioned on intense inverse interscale transfer
events are about the same, but they differ substantially when conditioned on
intense forward interscale transfer events where the conditional average of
$-\mathit{\Pi}_{H}^{{}^{\prime}}$ is substantially higher that the conditional
average of $-\mathit{\Pi}_{I}$ except close to the integral length-scale.

Future subgrid scale models for Large Eddy Simulations (LES) which are dynamic
reduced order approaches to turbulent flows and their fluctuating large scales
cannot rely on average cascade phenomenology describing spatio-temporal
averages and should benefit from detailed descriptions of the fluctuating
dynamics of interscale and interspace energy transfers such as the one
presented in this paper. Whilst LES models based on local equilibrium such as
the Smagorinsky model can reproduce structure function exponents and
correlations between velocity increments and subgrid-scale energy transfers as
shown by Linkmann et al. (2018), Dairay et al. (2017) have found that the
Smagorinsky model is unable to ensure regularisation and is very sensitive to
numerical errors. Furthermore, the recent review by Moser et al. (2021) makes
it clear that the need for new subgrid models which can faithfully operate
with coarse resolutions remains unanswered. The results in the present paper
suggest that LES models based on local equilibrium (e.g. the Smagorinsky
model) cannot be fully suitable for calculating fluctuations in subgrid
stresses, a weakness which may become increasingly evident with coarser
resolution. On the other hand, the good correlations between subgrid stresses
from similarity models (Bardina et al., 1980; Cimarelli et al., 2019) and
subgrid stresses from DNS suggest that these models might indeed approximate
(unawarely) at least some of the cascade dynamics reported in this paper, for
example the fact that
$\mathcal{A}_{t}+\mathcal{T}_{\overline{S}}+\mathit{\Pi}_{\overline{S}}^{{}^{\prime}}\approx
0$ holds in most of the flow most of the time. This relation incorporates both
forward and backward interscale transfers, yet a recent work by Vela-Martín
(2022) argues that backscatter represents spatial fluxes and can therefore be
ignored. It is not yet clear how such a claim can be understood in the context
of the present paper’s results. Some new questions are therefore now raised
concerning LES subgrid stess modeling which also need to be addressed in
future work.

[Acknowledgements]We thank Professor S. Goto for allowing us to use his
parallelised pseudo-spectral DNS code for periodic turbulence.

[Funding]HSL and JCV acknowledge support from EPSRC award number EP/L016230/1.
JCV also acknowledges the Chair of Excellence CoPreFlo funded by I-SITE-ULNE
(grant no. R-TALENT-19-001-VASSILICOS); MEL (grant no. CONVENTION-219-ESR-06)
and Region Hauts de France (grant no. 20003862).

[Declaration of Interests]The authors report no conflict of interest.

##  Appendix A The Helmholtz decomposition in Fourier space

In this appendix we list the Helmholtz decomposition for periodic fields and
note how this decomposition relates to the more general solution to the
Helmholtz decomposition in the case of incompressible fields and fields which
can be written as gradients of scalar fields.

Let $\boldsymbol{q}(\boldsymbol{x},t)$ be a periodic, twice continuously
differentiable $3$D vector field with the Helmholtz decomposition
$\boldsymbol{q}(\boldsymbol{x},t)=\boldsymbol{q}_{I}(\boldsymbol{x},t)+\boldsymbol{q}_{S}(\boldsymbol{x},t)$,
where
$\boldsymbol{q}_{I}(\boldsymbol{x},t)=-\nabla_{\boldsymbol{x}}\phi(\boldsymbol{x},t)$,
$\boldsymbol{q}_{S}(\boldsymbol{x},t)=\nabla_{\boldsymbol{x}}\times\boldsymbol{B}(\boldsymbol{x},t)$.
The scalar and vector potentials $\phi$ and $\boldsymbol{B}$ are unique within
constants when $\nabla_{\boldsymbol{x}}\cdot\boldsymbol{q}$ and
$\nabla_{\boldsymbol{x}}\times\boldsymbol{q}$ are known in the domain and
$\boldsymbol{q}$ is known at the boundary (Bhatia et al., 2013).
$\boldsymbol{q}(\boldsymbol{x},t)$ has the Fourier representation
$\widehat{\boldsymbol{q}}(\boldsymbol{k},t)$, which can be decomposed into a
component parallel to $\boldsymbol{k}$ (the longitudinal
$\widehat{\boldsymbol{q}}^{L}$) and transverse to $\boldsymbol{k}$ (the
transverse $\widehat{\boldsymbol{q}}^{T}$)

$\widehat{\boldsymbol{q}}^{L}(\boldsymbol{k},t)=\frac{\boldsymbol{k}[\widehat{\boldsymbol{q}}(\boldsymbol{k},t)\cdot\boldsymbol{k}]}{k^{2}}\
,\quad\widehat{\boldsymbol{q}}^{T}(\boldsymbol{k},t)=\widehat{\boldsymbol{q}}(\boldsymbol{k},t)-\widehat{\boldsymbol{q}}^{L}(\boldsymbol{k},t)\
.$ (44)

It can be easily shown (see e.g. Stewart (2012)) that the irrotational part of
$\boldsymbol{q}$ equals its longitudinal part
$\boldsymbol{q}_{I}=\boldsymbol{q}^{L}$ and that the solenoidal part of
$\boldsymbol{q}$ equals its transverse part
$\boldsymbol{q}_{S}=\boldsymbol{q}^{T}$. Hence, (44) provides the Fourier
representation of the Helmholtz decomposition of $\boldsymbol{q}$.

The Helmholtz decomposition can also be written for very general boundary
conditions as (Sprössig, 2010)

$\displaystyle\boldsymbol{q}_{IV}(\boldsymbol{x},t)$
$\displaystyle=\frac{1}{4\upi}\int_{V}d\boldsymbol{y}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}[\nabla_{\boldsymbol{y}}\cdot\boldsymbol{q}(\boldsymbol{y},t)],$
(45) $\displaystyle\boldsymbol{q}_{IB}(\boldsymbol{x},t)$
$\displaystyle=-\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}[\widehat{\boldsymbol{n}}_{\boldsymbol{y}}\cdot\boldsymbol{q}(\boldsymbol{y},t)],$
(46) $\displaystyle\boldsymbol{q}_{SV}(\boldsymbol{x},t)$
$\displaystyle=-\frac{1}{4\upi}\int_{V}d\boldsymbol{y}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\times[\nabla_{\boldsymbol{y}}\times\boldsymbol{q}(\boldsymbol{y},t)],$
(47) $\displaystyle\boldsymbol{q}_{SB}(\boldsymbol{x},t)$
$\displaystyle=\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\times[\widehat{\boldsymbol{n}}_{\boldsymbol{y}}\times\boldsymbol{q}(\boldsymbol{y},t)].$
(48)

where $\boldsymbol{q}_{I}=\boldsymbol{q}_{IV}+\boldsymbol{q}_{IB}$,
$\boldsymbol{q}_{S}=\boldsymbol{q}_{SV}+\boldsymbol{q}_{SB}$ and
$\widehat{\boldsymbol{n}}_{\boldsymbol{y}}$ denotes the unit surface normal at
$\boldsymbol{y}$ and $dS_{\boldsymbol{y}}$ is the differential surface element
at $\boldsymbol{y}$. For periodic vector fields
$\boldsymbol{q}(\boldsymbol{x},t)$ that are incompressible or that can be
written as the gradient of a scalar, this solution simplifies. In the case of
a field $\boldsymbol{q}(\boldsymbol{x},t)$ which is incompressible
$\nabla_{\boldsymbol{x}}\cdot\boldsymbol{q}(\boldsymbol{x},t)=0$, it can be
shown that $\widehat{\boldsymbol{q}}(\boldsymbol{k},t)\cdot\boldsymbol{k}=0$
for every $\boldsymbol{k}$ (Pope, 2000). By inspection of (44), it is clear
that this condition yields $\widehat{\boldsymbol{q}}^{L}(\boldsymbol{k},t)=0$
for every $\boldsymbol{k}$ such that
$\widehat{\boldsymbol{q}}(\boldsymbol{k},t)=\widehat{\boldsymbol{q}}(\boldsymbol{k},t)^{T}$.
By applying the Fourier transform to this relation and apply
$\boldsymbol{q}^{T}(\boldsymbol{x},t)=\boldsymbol{q}_{S}(\boldsymbol{x},t)$
from above, we have that
$\boldsymbol{q}(\boldsymbol{x},t)=\boldsymbol{q}_{S}(\boldsymbol{x},t)$ for
incompressible periodic vector fields. In the case of
$\boldsymbol{q}(\boldsymbol{x},t)=\nabla_{\boldsymbol{x}}\psi(\boldsymbol{x,t})$,
where $\psi(\boldsymbol{x},t)$ is some scalar field, it can be shown that
$\widehat{\boldsymbol{q}}(\boldsymbol{k},t)=i\boldsymbol{k}\widehat{\psi}(\boldsymbol{k},t)$
(Pope, 2000). If we insert this expression into the definition of
$\widehat{\boldsymbol{q}}^{L}(\boldsymbol{k},t)$, it follows that
$\widehat{\boldsymbol{q}}(\boldsymbol{k},t)=\widehat{\boldsymbol{q}}^{L}(\boldsymbol{k},t)$,
which implies that
$\boldsymbol{q}(\boldsymbol{x},t)=\boldsymbol{q}_{I}(\boldsymbol{x},t)$. If
these properties are combined with equations (45)-(48), we have that a
periodic incompressible vector field will have
$\boldsymbol{q}_{IB}=\boldsymbol{q}_{IV}=0$ and that a periodic vector field
that can be written as a gradient of a scalar field has
$\boldsymbol{q}_{SB}=\boldsymbol{q}_{SV}=0$.

##  Appendix B Irrotational and solenoidal NSD tranport terms in Fourier space

We start this appendix with demonstrating that
$\delta\boldsymbol{q}_{I}=\delta\boldsymbol{q}_{\overline{I}}$ and
$\delta\boldsymbol{q}_{S}=\delta\boldsymbol{q}_{\overline{S}}$ for a periodic
vector field $\boldsymbol{q}$ (see the second pararaph of section 3.3). The
field $\boldsymbol{q}$ has the Fourier representation

$\boldsymbol{q}(\boldsymbol{x},t)=\sum_{\boldsymbol{k}}\widehat{\boldsymbol{q}}(\boldsymbol{k},t)e^{i\boldsymbol{k}\cdot\boldsymbol{x}},$
(49)

with the shifted fields

$\displaystyle\boldsymbol{q}^{+}(\boldsymbol{x},\boldsymbol{r},t)=\boldsymbol{q}(\boldsymbol{x}+\boldsymbol{r}/2,t)$
$\displaystyle=\sum_{\boldsymbol{k}}\widehat{\boldsymbol{q}}(\boldsymbol{k},t)e^{i\boldsymbol{k}\cdot(\boldsymbol{x}+\boldsymbol{r}/2)},$
(50)
$\displaystyle\boldsymbol{q}^{-}(\boldsymbol{x},\boldsymbol{r},t)=\boldsymbol{q}(\boldsymbol{x}-\boldsymbol{r}/2,t)$
$\displaystyle=\sum_{\boldsymbol{k}}\widehat{\boldsymbol{q}}(\boldsymbol{k},t)e^{i\boldsymbol{k}\cdot(\boldsymbol{x}-\boldsymbol{r}/2)},$
(51)

which have the Fourier coefficients

$\displaystyle\widehat{\boldsymbol{q}^{+}}(\boldsymbol{k},\boldsymbol{r},t)$
$\displaystyle=\widehat{\boldsymbol{q}}(\boldsymbol{k},t)e^{i\boldsymbol{k}\cdot\boldsymbol{r}/2},$
(52)
$\displaystyle\widehat{\boldsymbol{q}^{-}}(\boldsymbol{k},\boldsymbol{r},t)$
$\displaystyle=\widehat{\boldsymbol{q}}(\boldsymbol{k},t)e^{-i\boldsymbol{k}\cdot\boldsymbol{r}/2}.$
(53)

From the definition of the irrotational part of a vector field in (44), it
follows

$\displaystyle\delta\boldsymbol{q}_{I}(\boldsymbol{x},\boldsymbol{r},t)$
$\displaystyle=\boldsymbol{q}_{I}^{+}(\boldsymbol{x},\boldsymbol{r},t)-\boldsymbol{q}_{I}^{-}(\boldsymbol{x},\boldsymbol{r},t),$
(54)
$\displaystyle=\sum_{\boldsymbol{k}}[\widehat{\boldsymbol{q}^{+}_{I}}(\boldsymbol{k},\boldsymbol{r},t)-\widehat{\boldsymbol{q}^{-}_{I}}(\boldsymbol{k},\boldsymbol{r},t)]e^{i\boldsymbol{k}\cdot\boldsymbol{x}},$
(55)
$\displaystyle=\sum_{\boldsymbol{k}}\frac{\boldsymbol{k}}{k^{2}}[\widehat{\boldsymbol{q}}(\boldsymbol{k},t)\cdot\boldsymbol{k}](e^{i\boldsymbol{k}\cdot\boldsymbol{r}/2}-e^{-i\boldsymbol{k}\cdot\boldsymbol{r}/2})e^{i\boldsymbol{k}\cdot\boldsymbol{x}},$
(56)
$\displaystyle=\sum_{\boldsymbol{k}}\frac{\boldsymbol{k}}{k^{2}}[\widehat{\boldsymbol{q}}(\boldsymbol{k},t)\cdot\boldsymbol{k}]2i\sin(\boldsymbol{k}\cdot\boldsymbol{r}/2)e^{i\boldsymbol{k}\cdot\boldsymbol{x}}.$
(57)

Similarly, we can write

$\displaystyle\delta\boldsymbol{q}(\boldsymbol{x},\boldsymbol{r},t)$
$\displaystyle=\boldsymbol{q}^{+}(\boldsymbol{x},\boldsymbol{r},t)-\boldsymbol{q}^{-}(\boldsymbol{x},\boldsymbol{r},t),$
(58)
$\displaystyle=\sum_{\boldsymbol{k}}\widehat{\boldsymbol{q}}(\boldsymbol{k},t)2i\sin(\boldsymbol{k}\cdot\boldsymbol{r}/2)e^{i\boldsymbol{k}\cdot\boldsymbol{x}},$
(59)

and then calculate its irrotational centroid part

$\delta\boldsymbol{q}_{\overline{I}}(\boldsymbol{x},\boldsymbol{r},t)=\sum_{\boldsymbol{k}}\frac{\boldsymbol{k}}{k^{2}}[\widehat{\boldsymbol{q}}(\boldsymbol{k},t)\cdot\boldsymbol{k}]2i\sin(\boldsymbol{k}\cdot\boldsymbol{r}/2)e^{i\boldsymbol{k}\cdot\boldsymbol{x}},$
(60)

which shows that
$\delta\boldsymbol{q}_{I}(\boldsymbol{x},\boldsymbol{r},t)=\delta\boldsymbol{q}_{\overline{I}}(\boldsymbol{x},\boldsymbol{r},t)$.
By combining this with
$\delta\boldsymbol{q}=\delta\boldsymbol{q}_{I}+\delta\boldsymbol{q}_{S}=\delta\boldsymbol{q}_{\overline{I}}+\delta\boldsymbol{q}_{\overline{S}}$,
we have also
$\delta\boldsymbol{q}_{S}(\boldsymbol{x},\boldsymbol{r},t)=\delta\boldsymbol{q}_{\overline{S}}(\boldsymbol{x},\boldsymbol{r},t)$,
which is what we wanted to show.

Next we demonstrate that
$\boldsymbol{a}_{\mathit{\Pi}_{\overline{I}}}(\boldsymbol{k},\boldsymbol{r},t)=\boldsymbol{a}_{\mathcal{T}_{\overline{I}}}(\boldsymbol{x},\boldsymbol{r},t)$
in homogeneous/periodic turbulence. We list the following expressions for the
vectors and tensors related to these two terms

$\displaystyle\widehat{\delta u_{j}}(\boldsymbol{k},\boldsymbol{r},t)$
$\displaystyle=2i\sin(\boldsymbol{k}\cdot\boldsymbol{r}/2)\widehat{u}_{j}(\boldsymbol{k},t),$
(61)
$\displaystyle\widehat{(u_{j}^{+}+u_{j}^{-})/2}(\boldsymbol{k},\boldsymbol{r},t)$
$\displaystyle=\cos(\boldsymbol{k}\cdot\boldsymbol{r}/2)\widehat{u}_{j}(\boldsymbol{k},t),$
(62) $\displaystyle\widehat{\frac{\partial\delta u_{i}}{\partial
r_{j}}}(\boldsymbol{k},\boldsymbol{r},t)$
$\displaystyle=ik_{j}\cos(\boldsymbol{k}\cdot\boldsymbol{r}/2)\widehat{u}_{i}(\boldsymbol{k},t),$
(63) $\displaystyle\widehat{\frac{\partial\delta u_{i}}{\partial
x_{j}}}(\boldsymbol{k},\boldsymbol{r},t)$
$\displaystyle=-2k_{j}\sin(\boldsymbol{k}\cdot\boldsymbol{r}/2)\widehat{u}_{i}(\boldsymbol{k},t).$
(64)

By use of these equations, we have that the Fourier coefficients of the
transport terms read

$\displaystyle\widehat{\boldsymbol{a}_{\mathcal{T}}}(\boldsymbol{k},\boldsymbol{r},t)=\sum_{\boldsymbol{k}=\boldsymbol{k}^{{}^{\prime}}+\boldsymbol{k}^{{}^{\prime\prime}}}-2\sin(\boldsymbol{k}^{{}^{\prime\prime}}\cdot\boldsymbol{r}/2)\cos(\boldsymbol{k}^{{}^{\prime}}\cdot\boldsymbol{r}/2)\widehat{u}_{j}(\boldsymbol{k}^{{}^{\prime}})k_{j}^{{}^{\prime\prime}}\widehat{\boldsymbol{u}}(\boldsymbol{k}^{{}^{\prime\prime}}),$
(65)
$\displaystyle\widehat{\boldsymbol{a}_{\mathit{\Pi}}}(\boldsymbol{k},\boldsymbol{r},t)=\sum_{\boldsymbol{k}=\boldsymbol{k}^{{}^{\prime}}+\boldsymbol{k}^{{}^{\prime\prime}}}-2\sin(\boldsymbol{k}^{{}^{\prime}}\cdot\boldsymbol{r}/2)\cos(\boldsymbol{k}^{{}^{\prime\prime}}\cdot\boldsymbol{r}/2)\widehat{u}_{j}(\boldsymbol{k}^{{}^{\prime}})k_{j}^{{}^{\prime\prime}}\widehat{\boldsymbol{u}}(\boldsymbol{k}^{{}^{\prime\prime}}).$
(66)

Their irrotational parts are given per (44)

$\displaystyle\widehat{\boldsymbol{a}_{\mathcal{T}_{\overline{I}}}}(\boldsymbol{k},\boldsymbol{r},t)=-\frac{\boldsymbol{k}}{k^{2}}\sum_{\boldsymbol{k}=\boldsymbol{k}^{{}^{\prime}}+\boldsymbol{k}^{{}^{\prime\prime}}}2\sin(\boldsymbol{k}^{{}^{\prime\prime}}\cdot\boldsymbol{r}/2)\cos(\boldsymbol{k}^{{}^{\prime}}\cdot\boldsymbol{r}/2)\widehat{u}_{j}(\boldsymbol{k}^{{}^{\prime}})k_{j}^{{}^{\prime\prime}}\widehat{u}_{l}(\boldsymbol{k}^{{}^{\prime\prime}})k^{{}^{\prime}}_{l},$
(67)
$\displaystyle\widehat{\boldsymbol{a}_{\mathit{\Pi}_{\overline{I}}}}(\boldsymbol{k},\boldsymbol{r},t)=-\frac{\boldsymbol{k}}{k^{2}}\sum_{\boldsymbol{k}=\boldsymbol{k}^{{}^{\prime}}+\boldsymbol{k}^{{}^{\prime\prime}}}2\sin(\boldsymbol{k}^{{}^{\prime}}\cdot\boldsymbol{r}/2)\cos(\boldsymbol{k}^{{}^{\prime\prime}}\cdot\boldsymbol{r}/2)\widehat{u}_{j}(\boldsymbol{k}^{{}^{\prime}})k_{j}^{{}^{\prime\prime}}\widehat{u}_{l}(\boldsymbol{k}^{{}^{\prime\prime}})k^{{}^{\prime}}_{l}.$
(68)

If we employ the trigonometric identity $\sin x\cos
y=\frac{1}{2}[\sin(x+y)+\sin(x-y)]$, we get

$\displaystyle\widehat{\boldsymbol{a}_{\mathcal{T}_{\overline{I}}}}(\boldsymbol{k},\boldsymbol{r},t)=-\frac{\boldsymbol{k}}{k^{2}}\sum_{\boldsymbol{k}=\boldsymbol{k}^{{}^{\prime}}+\boldsymbol{k}^{{}^{\prime\prime}}}[\sin(\boldsymbol{k}\cdot\boldsymbol{r}/2)+\sin(\boldsymbol{k}^{{}^{\prime\prime}}\cdot\boldsymbol{r}/2-\boldsymbol{k}^{{}^{\prime}}\cdot\boldsymbol{r}/2)]\widehat{u}_{j}(\boldsymbol{k}^{{}^{\prime}})k_{j}^{{}^{\prime\prime}}\widehat{u}_{l}(\boldsymbol{k}^{{}^{\prime\prime}})k^{{}^{\prime}}_{l},$
(69)
$\displaystyle\widehat{\boldsymbol{a}_{\mathit{\Pi}_{\overline{I}}}}(\boldsymbol{k},\boldsymbol{r},t)=-\frac{\boldsymbol{k}}{k^{2}}\sum_{\boldsymbol{k}=\boldsymbol{k}^{{}^{\prime}}+\boldsymbol{k}^{{}^{\prime\prime}}}[\sin(\boldsymbol{k}\cdot\boldsymbol{r}/2)-\sin(\boldsymbol{k}^{{}^{\prime\prime}}\cdot\boldsymbol{r}/2-\boldsymbol{k}^{{}^{\prime}}\cdot\boldsymbol{r}/2)]\widehat{u}_{j}(\boldsymbol{k}^{{}^{\prime}})k_{j}^{{}^{\prime\prime}}\widehat{u}_{l}(\boldsymbol{k}^{{}^{\prime\prime}})k^{{}^{\prime}}_{l}.$
(70)

Consider the term
$\sin(\boldsymbol{k}^{{}^{\prime\prime}}\cdot\boldsymbol{r}/2-\boldsymbol{k}^{{}^{\prime}}\cdot\boldsymbol{r}/2)\widehat{u}_{j}(\boldsymbol{k}^{{}^{\prime}})k_{j}^{{}^{\prime\prime}}\widehat{u}_{l}(\boldsymbol{k}^{{}^{\prime\prime}})k^{{}^{\prime}}_{l}$.
If one adds this term with the wavenumber triad
$\boldsymbol{k}^{{}^{\prime}}=\boldsymbol{k}^{a}$ and
$\boldsymbol{k}^{{}^{\prime\prime}}=\boldsymbol{k}^{b}\neq\boldsymbol{k}^{a}$
with the same term with the wavenumber triad
$\boldsymbol{k}^{{}^{\prime}}=\boldsymbol{k}^{b}$ and
$\boldsymbol{k}^{{}^{\prime\prime}}=\boldsymbol{k}^{a}$ the result is zero.
Furthermore, in the case of $\boldsymbol{k}^{a}=\boldsymbol{k}^{b}$ this term
is zero per incompressibility. This yields that this term does not contribute
instantaneously in the above expressions such that we attain the final result
(see section 3.3 and equation (26))

$\widehat{\boldsymbol{a}_{\mathcal{T}_{\overline{I}}}}(\boldsymbol{k},\boldsymbol{r},t)=\widehat{\boldsymbol{a}_{\mathit{\Pi}_{\overline{I}}}}(\boldsymbol{k},\boldsymbol{r},t)=-\frac{\boldsymbol{k}}{k^{2}}\sin(\boldsymbol{k}\cdot\boldsymbol{r}/2)\sum_{\boldsymbol{k}=\boldsymbol{k}^{{}^{\prime}}+\boldsymbol{k}^{{}^{\prime\prime}}}\widehat{u}_{j}(\boldsymbol{k}^{{}^{\prime}})k_{j}^{{}^{\prime\prime}}\widehat{u}_{l}(\boldsymbol{k}^{{}^{\prime\prime}})k^{{}^{\prime}}_{l}.$
(71)

##  Appendix C Irrotational and solenoidal dynamics in non-homogeneous
turbulence

Here we deduce the generalised Tsinober equations and the irrotational and
solenoidal NSD and KHMH equations applicable to non-homogeneous turbulence.
Consider the twice continously differentiable vector field vector field
$\boldsymbol{q}(\boldsymbol{x},t)$ defined on a domain
$V\subseteq\mathbb{R}^{3}$ with the bounding surface $S$. This field can be
uniquely decomposed into the irrotational and solenoidal vector fields

$\boldsymbol{q}(\boldsymbol{x},t)=\boldsymbol{q}_{I}(\boldsymbol{x},t)+\boldsymbol{q}_{S}(\boldsymbol{x},t)=-\nabla_{\boldsymbol{x}}\boldsymbol{\phi}(\boldsymbol{x},t)+\nabla_{\boldsymbol{x}}\times\boldsymbol{B}(\boldsymbol{x},t),$
(72)

The solution to this problem under very general conditions (Sprössig, 2010) is
$\boldsymbol{q}_{I}=\boldsymbol{q}_{IV}+\boldsymbol{q}_{IB}$ and
$\boldsymbol{q}_{S}=\boldsymbol{q}_{SV}+\boldsymbol{q}_{SB}$, where the
solenoidal and irrotational volume and boundary terms are given in equations
(45)-(48).

Consider an incompressible fluid that satisfies the incompressible vorticity
equation

$\nabla_{\boldsymbol{y}}\times\big{(}\frac{\partial\boldsymbol{u}}{\partial
t}+\boldsymbol{u}\cdot\nabla_{\boldsymbol{y}}\boldsymbol{u}-\nu\nabla_{\boldsymbol{y}}^{2}\boldsymbol{u}-\boldsymbol{f}\big{)}=0.$
(73)

By comparing this equation with (47), it is clear that the vorticity equation
can be used to derive an evolution equation for the solenoidal volume parts of
the NS terms. We can apply the following operator to this equation

$-\frac{1}{4\upi}\int_{V}d\boldsymbol{y}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\times\Big{[}\nabla_{\boldsymbol{y}}\times\big{(}\frac{\partial\boldsymbol{u}}{\partial
t}+(\boldsymbol{u}\cdot\nabla_{\boldsymbol{y}})\boldsymbol{u}-\nu\nabla_{\boldsymbol{y}}^{2}\boldsymbol{u}-\boldsymbol{f}\big{)}\Big{]}=0,$
(74)

and use (47) to rewrite this equation as

$(\frac{\partial\boldsymbol{u}}{\partial
t})_{SV}+(\boldsymbol{u}\cdot\nabla_{\boldsymbol{x}}\boldsymbol{u})_{SV}=(\nu\nabla_{\boldsymbol{x}}^{2}\boldsymbol{u})_{SV}+\boldsymbol{f}_{SV}.$
(75)

We can in a similar manner obtain the evolution equation for the irrotational
volume NS terms from the Poisson equation for pressure

$\frac{1}{4\upi}\int_{V}d\boldsymbol{y}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\Big{[}\nabla_{\boldsymbol{y}}\cdot\big{(}\boldsymbol{u}\cdot\nabla_{\boldsymbol{y}}\boldsymbol{u}+\frac{1}{\rho}\nabla_{\boldsymbol{y}}p-\boldsymbol{f}\big{)}\Big{]}=0,$
(76)

which yields

$(\boldsymbol{u}\cdot\nabla_{\boldsymbol{x}}\boldsymbol{u})_{IV}=(-\frac{1}{\rho}\nabla_{\boldsymbol{x}}p)_{IV}+\boldsymbol{f}_{IV},$
(77)

The equations (75) and (77) state that in all incompressible turbulent flows
the solenoidal accelerations from volume contributions balance with solenoidal
forces from volume contributions and irrotational accelerations from volume
contributions balance with irrotational forces from volume contributions. The
former can be viewed as an integrated vorticity equation which dictates a part
of the solenoidal NS dynamics, while the latter equation as an integrated
pressure Poisson equation which dictates a part of the irrotational NS
dynamics. Due to the non-local character of the solenoidal and irrotational
volume terms, we reformulate these equations in terms of full NS term minus
boundary terms. E.g., for the time-derivative
$(\frac{\partial\boldsymbol{u}}{\partial
t})_{SV}=\frac{\partial\boldsymbol{u}}{\partial
t}-(\frac{\partial\boldsymbol{u}}{\partial
t})_{IB}-(\frac{\partial\boldsymbol{u}}{\partial t})_{SB}$. The irrotational
volume component (see (45)) involves an integral of the divergence of the
respective term
($\nabla_{\boldsymbol{y}}\cdot\boldsymbol{q}(\boldsymbol{y})$). Thus, due to
incompressibility, the time derivative and viscous terms have zero volume
irrotational components, $(\frac{\partial\boldsymbol{u}}{\partial
t})_{IV}=(\nu\nabla_{\boldsymbol{x}}^{2}\boldsymbol{u})_{IV}=0$. The
solenoidal volume component (see (47)) involves an integral of the curl of the
respective term, and as the curl of the pressure gradient equals zero, this
term will have a zero solenoidal volume component,
$(-\frac{1}{\rho}\nabla_{\boldsymbol{x}}p)_{SV}=0$. We rewrite the solenoidal
volume terms in equation (75) in terms of combinations of full terms and
boundary terms to obtain

$\frac{\partial\boldsymbol{u}}{\partial
t}+((\boldsymbol{u}\cdot\nabla_{\boldsymbol{x}})\boldsymbol{u})_{S}=\nu\nabla_{\boldsymbol{x}}^{2}\boldsymbol{u}+\boldsymbol{f}_{S}+\\\
(\frac{\partial\boldsymbol{u}}{\partial
t})_{IB}-(\nu\nabla_{\boldsymbol{x}}^{2}\boldsymbol{u})_{IB}+(\frac{\partial\boldsymbol{u}}{\partial
t})_{SB}+((\boldsymbol{u}\cdot\nabla_{\boldsymbol{x}})\boldsymbol{u})_{SB}-(\nu\nabla_{\boldsymbol{x}}^{2}\boldsymbol{u})_{SB}-\boldsymbol{f}_{SB}\
,$ (78)

where the sum of the four rightmost terms on the RHS equals
$(-\frac{1}{\rho}\nabla_{\boldsymbol{x}}p)_{SB}$ as the NS equations are
satisfied at the boundary. By using this simplification and writing out all
the boundary terms, we arrive at

$\frac{\partial\boldsymbol{u}}{\partial
t}+((\boldsymbol{u}\cdot\nabla_{\boldsymbol{x}})\boldsymbol{u})_{S}=\nu\nabla_{\boldsymbol{x}}^{2}\boldsymbol{u}+\boldsymbol{f}_{S}\\\
-\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}[\widehat{\boldsymbol{n}}_{\boldsymbol{y}}\cdot(\frac{\partial\boldsymbol{u}}{\partial
t}-\nu\nabla_{\boldsymbol{y}}^{2}\boldsymbol{u})]-\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\times[\widehat{\boldsymbol{n}}_{\boldsymbol{y}}\times\nabla_{\boldsymbol{y}}\frac{1}{\rho}p].$
(79)

By rewriting the irrotational volume components in equation (77) in terms of
the full terms and the boundary terms, we have

$((\boldsymbol{u}\cdot\nabla_{\boldsymbol{x}})\boldsymbol{u})_{I}=-\frac{1}{\rho}\nabla_{\boldsymbol{x}}p+\boldsymbol{f}_{I}+((\boldsymbol{u}\cdot\nabla_{\boldsymbol{x}})\boldsymbol{u})_{IB}-(-\frac{1}{\rho}\nabla_{\boldsymbol{x}}p)_{IB}-\boldsymbol{f}_{IB}-(-\frac{1}{\rho}\nabla_{\boldsymbol{x}}p)_{SB},$
(80)

where the sum of the irrotational boundary terms equals
$-(\frac{\partial\boldsymbol{u}}{\partial
t})_{IB}+(\nu\nabla_{\boldsymbol{x}}^{2}\boldsymbol{u})_{IB}$ by the NS
equations at the boundary. If we use this relation and write out all boundary
terms, we have

$((\boldsymbol{u}\cdot\nabla_{\boldsymbol{x}})\boldsymbol{u})_{I}=-\frac{1}{\rho}\nabla_{\boldsymbol{x}}p+\boldsymbol{f}_{I}\\\
+\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}[\widehat{\boldsymbol{n}}_{\boldsymbol{y}}\cdot(\frac{\partial\boldsymbol{u}}{\partial
t}-\nu\nabla_{\boldsymbol{y}}^{2}\boldsymbol{u})]+\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\times[\widehat{\boldsymbol{n}}_{\boldsymbol{y}}\times\nabla_{\boldsymbol{y}}\frac{1}{\rho}p].$
(81)

The equations (79) and (81) are generalisations of equations (8)-(9) for
homogeneous/periodic turbulence and these equations are valid for all
incompressible turbulent flows. The difference from homogeneous/periodic
turbulence is the collection of boundary terms

$\displaystyle\boldsymbol{R}(\boldsymbol{x},t)$
$\displaystyle\equiv\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}[\boldsymbol{n}_{\boldsymbol{y}}\cdot(\frac{\partial\boldsymbol{u}}{\partial
t}-\nu\nabla_{\boldsymbol{y}}^{2}\boldsymbol{u})]+\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\times[\boldsymbol{n}_{\boldsymbol{y}}\times\nabla_{\boldsymbol{y}}\frac{1}{\rho}p],$
(82)
$\displaystyle=-(\boldsymbol{a}_{l})_{IB}+(\boldsymbol{a}_{\nu})_{IB}-(\boldsymbol{a}_{p})_{SB},$
(83)

which yields the final expressions for the general irrotational and solenoidal
NS equations

$\displaystyle\frac{\partial\boldsymbol{u}}{\partial
t}+((\boldsymbol{u}\cdot\nabla_{\boldsymbol{x}})\boldsymbol{u})_{S}$
$\displaystyle=\nu\nabla_{\boldsymbol{x}}^{2}\boldsymbol{u}+\boldsymbol{f}_{S}-\boldsymbol{R}(\boldsymbol{x},t)$
(84)
$\displaystyle((\boldsymbol{u}\cdot\nabla_{\boldsymbol{x}})\boldsymbol{u})_{I}$
$\displaystyle=-\frac{1}{\rho}\nabla_{\boldsymbol{x}}p+\boldsymbol{f}_{I}+\boldsymbol{R}(\boldsymbol{x},t)$
(85)

In homogeneous/periodic turbulence all the boundary terms in
$\boldsymbol{R}(\boldsymbol{x},t)$ equal zero individually (see the last
parapgraph of A), such that we recover equations (8)-(9). In general, the
boundary terms will be non-zero and differ in different flows. E.g., at a
solid wall the boundary term from the time-derivative will vanish because of
no-slip and the NS equations at the wall can be used to rewrite the boundary
terms as a non-local function of the pressure gradient only.

The NSD irrotational and solenoidal equations in general turbulent flows are
obtained by subtracting the solenoidal and irrotational NS equations (84)-(85)
at $\boldsymbol{x}-\boldsymbol{r}/2$ from the same equations at
$\boldsymbol{x}+\boldsymbol{r}/2$

$\displaystyle\frac{\partial\delta\boldsymbol{u}}{\partial
t}+\delta\boldsymbol{a}_{c_{S}}$
$\displaystyle=\delta\boldsymbol{a}_{\nu}+\delta\boldsymbol{f}_{S}-\delta\boldsymbol{R},$
(86) $\displaystyle\delta\boldsymbol{a}_{c_{I}}$
$\displaystyle=-\frac{1}{\rho}\nabla_{\boldsymbol{x}}\delta
p+\delta\boldsymbol{f}_{I}+\delta\boldsymbol{R},$ (87)

The rephrasing of the irrotational and solenoidal NSD equations in terms of
the interscale and interspace transport terms can also be performed for non-
homogeneous turbulence. We derive the centroid irrotational and solenoidal NSD
equations similarly as for the NS irrotational and solenoidal equations by
starting with the NSD equation (13). This yields the equations

$\displaystyle\delta\boldsymbol{a}_{l}+\boldsymbol{a}_{\mathcal{T}_{\overline{S}}}+\boldsymbol{a}_{\mathit{\Pi}_{\overline{S}}}$
$\displaystyle=\delta\boldsymbol{a}_{\nu}+\delta\boldsymbol{f}_{\overline{S}}-\overline{\boldsymbol{R}},$
(88)
$\displaystyle\boldsymbol{a}_{\mathcal{T}_{\overline{I}}}+\boldsymbol{a}_{\mathit{\Pi}_{\overline{I}}}$
$\displaystyle=\delta\boldsymbol{a}_{p}+\delta\boldsymbol{f}_{\overline{I}}+\overline{\boldsymbol{R}},$
(89)

where

$\displaystyle\overline{\boldsymbol{R}}(\boldsymbol{x},\boldsymbol{r},t)$
$\displaystyle\equiv\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}[\widehat{\boldsymbol{n}}_{\boldsymbol{y}}\cdot(\delta\boldsymbol{a}_{l}-\delta\boldsymbol{a}_{\nu})]-\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\times[\widehat{\boldsymbol{n}}_{\boldsymbol{y}}\times\delta\boldsymbol{a}_{p}],$
(90)
$\displaystyle=-(\delta\boldsymbol{a}_{l})_{IB}+(\delta\boldsymbol{a}_{\nu})_{IB}-(\delta\boldsymbol{a}_{p})_{SB}.$
(91)

These boundary terms are individually equal to zero in homogeneous/periodic
turbulence for the analogue reason as for the NS dynamics. Regarding the
irrotational dynamics, in general,
$\boldsymbol{a}_{\mathcal{T}_{\overline{I}}}\neq\boldsymbol{a}_{\mathit{\Pi}_{\overline{I}}}$,
but the irrotational volume terms are always equal,
$(\boldsymbol{a}_{\mathcal{T}})_{\overline{IV}}=(\boldsymbol{a}_{\mathit{\Pi}})_{\overline{IV}}$
from equation (45) and

$\nabla_{\boldsymbol{x}}\cdot\boldsymbol{a}_{\mathit{\Pi}}=\nabla_{\boldsymbol{x}}\cdot\boldsymbol{a}_{\mathcal{T}}=\frac{1}{2}\big{(}\frac{\partial
u_{k}^{+}}{\partial x^{+}_{i}}\frac{\partial u_{i}^{+}}{\partial
x^{+}_{k}}-\frac{\partial u_{k}^{-}}{\partial x^{-}_{i}}\frac{\partial
u_{i}^{-}}{\partial x^{-}_{k}}\big{)}.$ (92)

The solenoidal interscale transfer term
$\boldsymbol{a}_{\mathit{\Pi}_{\overline{S}}}$ in non-homogeneous turbulence
can be written as

$\boldsymbol{a}_{\mathit{\Pi}_{\overline{S}}}(\boldsymbol{x},\boldsymbol{r},t)=-\frac{1}{4\upi}\int_{V}d\boldsymbol{y}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\times[\nabla_{\boldsymbol{y}}\times\boldsymbol{a}_{\mathit{\Pi}}(\boldsymbol{y},\boldsymbol{r},t)]+\\\
\frac{1}{4\upi}\int_{S}dS_{\boldsymbol{y}}\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\times[\widehat{\boldsymbol{n}}_{\boldsymbol{y}}\times\boldsymbol{a}_{\mathit{\Pi}}(\boldsymbol{y},\boldsymbol{r},t)],$
(93)

where the surface integral is of smaller order of magnitude than the volume
integral away from boundaries and increasingly so with increasing
$\langle\Rey_{\lambda}\rangle_{t}$ (verified in our periodic DNS). Hence, for
a qualitative interpretation of
$\boldsymbol{a}_{\mathit{\Pi}_{\overline{S}}}$, we consider
$\boldsymbol{a}_{\mathit{\Pi}_{\overline{S}}}\approx\boldsymbol{a}_{\mathit{\Pi}_{\overline{SV}}}$
with

$(\nabla_{\boldsymbol{x}}\times\boldsymbol{a}_{\mathit{\Pi}})_{i}=\delta
u_{k}\frac{\partial\delta\omega_{i}}{\partial
r_{k}}-\frac{\delta\omega_{k}}{2}\frac{s_{ij}^{+}+s_{ij}^{-}}{2}-\frac{\omega_{k}^{+}+\omega_{k}^{-}}{4}\delta
s_{ij}+\frac{\epsilon_{ijk}}{2}[\frac{\partial u_{l}^{+}}{\partial
x_{j}^{+}}\frac{\partial u_{k}^{-}}{\partial x_{l}^{-}}-\frac{\partial
u_{l}^{-}}{\partial x_{j}^{-}}\frac{\partial u_{k}^{+}}{\partial x_{l}^{+}}],$
(94)

where $s_{ij}$ is the strain-rate tensor and $\epsilon_{ijk}$ is the Levi-
Civita tensor. This set of terms constitutes a part of the non-linear term in
the the evolution equation for the vorticity difference
$\delta\boldsymbol{\omega}(\boldsymbol{x},\boldsymbol{r},t)$, i.e. vorticity
at scales $|\boldsymbol{r}|$ and smaller, as
$\nabla_{\boldsymbol{x}}\times\delta\boldsymbol{a}_{c}=\nabla_{\boldsymbol{x}}\times(\boldsymbol{a}_{\mathit{\Pi}}+\boldsymbol{a}_{\mathcal{T}})$.
If one contracts (94) with $2\delta\boldsymbol{\omega}$, the RHS corresponds
to non-linear terms which determine the evolution of the enstrophy
$|\delta\boldsymbol{\omega}|^{2}$ at scales smaller or comparable to
$|\boldsymbol{r}|$. We interpret the first term on the RHS in (94) as
vorticity interscale transfer. By the connection to
$|\delta\boldsymbol{\omega}|^{2}$, we interpret the second and third terms as
related to the enstrophy production/destruction at scales smaller or
comparable to $|\boldsymbol{r}|$ due to interactions between the vorticity and
strain fields. These three terms justify the interpretation of
$\boldsymbol{a}_{\mathit{\Pi}_{\overline{S}}}$ being related non-locally in
space to vortex stretching and compression dynamics. The last term in (94)
appears in
$\nabla_{\boldsymbol{x}}\times\boldsymbol{a}_{\mathcal{T}_{\overline{SV}}}$
with a negative sign such that these terms cancel.

The exact solenoidal and irrotational KHMH equations follows from contracting
equations (88)-(89) with $2\delta\boldsymbol{u}$

$\displaystyle\mathcal{A}_{t}+\mathcal{T}_{\overline{S}}+\mathit{\Pi}_{\overline{S}}$
$\displaystyle=\mathcal{D}_{r,\nu}+\mathcal{D}_{X,\nu}-\mathcal{\epsilon}+I_{\overline{S}}-2\delta\boldsymbol{u}\cdot\overline{\boldsymbol{R}},$
(95) $\displaystyle\mathcal{T}_{\overline{I}}+\mathit{\Pi}_{\overline{I}}$
$\displaystyle=\mathcal{T}_{p}+I_{\overline{I}}+2\delta\boldsymbol{u}\cdot\overline{\boldsymbol{R}},$
(96)

where $\mathcal{T}_{\overline{IV}}=\mathit{\Pi}_{\overline{IV}}$. This shows
that the solenoidal and irrotational KHMH equations can be extended to non-
homogeneous turbulence. In contrast to homogeneous/periodic turbulence, in
general boundary terms couple the irrotational and solenoidal dynamics.

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# New approach to radiation reaction in classical electrodynamics

Richard T. Hammond<EMAIL_ADDRESS>Department of Physics, University
of North Carolina at Chapel Hill, and the Army Research Office, Research
Triangle Park, North Carolina, 27703

###### Abstract

The problem of self forces and radiation reaction is solved by conservation of
energy methods. The longstanding problem of constant acceleration is solved,
and it is shown that the self force does indeed affect the particle’s motion,
as expected on physical grounds. The relativistic generalization is also
presented.

radiation reaction, self force

###### pacs:

41.60.-m, 03.50.De

The classical problem of self forces due to the radiation field of an
accelerating charged particle goes back over a century, to the nonrelativistic
derivation of Lorentz.lorentz  Soon after, Abraham used a shell model to
develop an equation of motion that was a terminated version of an infinite
series in terms of the radius of the shell.abraham  Dirac re-derived that
result, but did it for a point particle, did it relativistically, and did not
have the remaining series.dirac

Recently a new urgency has been given to this problem. Laser intensities of
$10^{22}$ W cm-2, corresponding to an energy density over $3\times 10^{17}$ J
m-3, have been reached,bahk  and this is expected to increase by two orders of
magnitude in the near future.mourou  Traditionally, it had been thought that
the observation of radiation reaction effects would have to wait until there
were pulses of the characteristic time $\tau_{0}$, but with these extreme
intensities, and the associated time dilation, radiation reaction effects are
important now,hammond08nc  and might even dominate the interactions expected
in the near future.

The equation derived by Dirac, mentiuoned above, is called the LAD equation
and is given by (I use $ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$ and cgs
units),

$\frac{dv^{\mu}}{d\tau}=\frac{e}{mc}F^{\mu\sigma}v_{\sigma}+\tau_{0}\left(\frac{v^{\mu}}{c^{2}}\dot{v}_{\sigma}\dot{v}^{\sigma}+\ddot{v}^{\mu}\right)$
(1)

where $\tau_{0}=2e^{2}/3mc^{3}$, which is $\sim 10^{-23}$s.

The main problem with this equation is the Schott term,
$\tau_{0}\ddot{v}^{\mu}$, which leads to unphysical runaway solutions.jackson
Landau and Lifshitz found a way around this difficulty by using an iterative
approach, and derivedlandau

$\frac{dv^{\mu}}{d\tau}=(e/mc)F^{\mu\sigma}v_{\sigma}+\tau_{0}\left((e/mc)\dot{F}^{\mu\sigma}v_{\sigma}+(e/mc)^{2}(F^{\mu\gamma}F_{\gamma}^{\
\phi}v_{\phi}+F^{\nu\gamma}v_{\gamma}F_{\nu}^{\ \phi}v_{\phi}v^{\mu})\right).$
(2)

This equation was used extensively over the years, but if the LAD equation,
its progenitor, is wrong, then one must question the validity of the LL
equation.

Inspired by the unsolved problem, over the years several authors have put
forward solutions of their own, most notably, that of Mo and Papas,mo  Steiger
and Woods,steiger  Ford and O’Connell (FO)ford  (which appears in Jackson’s
third edition and was derived again by a different formalism),hammond08nc ,
Hartemann and Luhmannhartemann  and through the years, Rohrlich.rohrlich  All
of these are based on series expansions or some other approximations,
sometimes invoking a finite radius electron. For example, a drawback of the
nonrelativistic FO equation,
$m\bm{\dot{\bm{V}}}={\bm{F}}+\tau_{0}\frac{d}{dt}{\bm{F}}$, is that, in a
uniform field, it cannot account for radiation reaction. The LL equation
suffers the same problem. A fuller discussion may be found
elsewhere.hammond08ejtp

Before proceeding, let us examine what the LAD equation has to say about
energy. To do this, we integrate the time component of the LAD equation (1)
with respect to proper time. This gives,

$mc^{2}(\gamma-\gamma_{\mbox{\scriptsize inc}})=\int{\bm{F}}\cdot
d{\bm{x}}-\int Pdt+\tau_{0}(\dot{v}^{0}-\dot{v}_{\mbox{\scriptsize
inc}}^{0}).$ (3)

where ${\bm{F}}=e{\bm{E}}$, $\gamma=v^{0}/c$, $\gamma_{\mbox{\scriptsize
inc}}$ is the incident value of $\gamma$, and
$P=m\tau_{0}\dot{v}_{\sigma}\dot{v}^{\sigma}$. Although the LAD equation is
covariant, we have now chosen a component of this equation, and therefore we
must specify the reference frame, which is taken to be the lab frame in which
the electric field has the value used above. In this frame we measure the
particle to move through a distance $d{\bm{x}}$ in the time $dt$, which appear
(3). The physical interpretation of (3) is easy to see: It reads, the change
in kinetic energy is equal to the work done by the external field minus the
energy radiated away plus something else. The something else seems to destroy
our concept of what conservation energy should be, but we may assess its
damage by noting that $\dot{v}^{0}$ vanishes when $\dot{v}^{n}$ does, so that
if we integrate over a pulse this term vanishes. We do expect this to be valid
in the case of a uniform electric field or in an extended magnetic field,
which explains the long suffering debate about the constant force problem.

To find an equation that may derived with no approximations, we assume that,
corresponding to the power scalar, there is an scalar, say $W$, from which the
force is derived accoring to $f_{\sigma}\equiv W,_{\sigma}$. This may be
viewed as the relativisitc generalization of assuming that the force is
derived from a scalar potential. With this we have a covariant equation,
assuming the Lorentz force,

$m\frac{dv^{\mu}}{d\tau}=\frac{e}{c}F^{\mu\sigma}v_{\sigma}-f^{\mu}.$ (4)

If we integrate (4) with respect to proper time we find,

$mc^{2}(\gamma-\gamma_{\mbox{\scriptsize inc}})=\int{\bm{F}}\cdot
d{\bm{x}}-c\int W^{{{}^{,}}^{0}}d\tau.$ (5)

Conservation of energy implies that

$W,_{0}=\gamma P/c$ (6)

The orthogonality of the four velocity and acceleration implies that
$v_{\mu}W^{{{}^{,}}^{\mu}}=0$, so that

$dW/dt=0.$ (7)

This tells us that

$\gamma W,_{t}=-v^{n}W,_{n}.$ (8)

Thus, (4), with (6) and (8), gives a complete solution to the self force
problem.

Since $\tau_{0}$ is so small, it is sometimes useful to consider the series,

$v^{\sigma}={{}_{0}v}^{\sigma}+\tau_{0}({{}_{1}v}^{\sigma}).$ (9)

With this, we can consider the age old problem of the constant force. However,
a problem arises if we naively use the above equation withour due regard to
the initial condition. Conventionally one would take the extrnal force to be
constant and assume the initial velocity is zero (or any value). Physically
this corresponds to holding a particle fixed and at $t=0$ giving it an
acceleration. Thus, this acceleration is discontinuous. Normally this is not a
problem, but when the power is computed, it produces a singularity at $t=0$.
To overcome this let us assume that the external electric field is given by
${\cal E}E$ where $E$ is the constant electric field and

${\cal E}=\frac{1+\mbox{Tanh t/T}}{2}.$ (10)

As $T\rightarrow 0$, we obtain the step function, but in the following $T$ is
taken to be unity. In addition, we shall rescale to dimensionless coordinates
so that $t\rightarrow\Omega t$ and $x\rightarrow x\Omega/c$, where
$\Omega=eE/mc$.

To zero order the equations are

${{}_{0}\dot{v}}^{0}={\cal E}{{}_{0}}v^{1}$ (11)

and

${{}_{0}\dot{v}}^{1}={\cal E}{{}_{0}}v^{0}$ (12)

which imply,

${{}_{0}}v^{0}=\frac{e^{-t/2}\left(2+e^{2t}\right)}{2\sqrt{2}\sqrt{\cosh(t)}}$
(13)

and

${{}_{0}}v^{1}=\frac{e^{3t/2}}{2\sqrt{2}\sqrt{\cosh(t)}}.$ (14)

To ${\cal O}(\tau_{0})$ we have, using
$S\equiv\dot{v}^{\sigma}\dot{v}_{\sigma}$,

${{}_{1}\dot{v}}^{0}={\cal E}{{}_{1}v}^{1}+\tau_{0}\Omega{{}_{0}v}^{0}S$ (15)

and

${{}_{0}v}^{1}{{}_{1}\dot{v}}^{1}={{}_{1}v}^{0}{{}_{0}v}^{1}+({{}_{0}v}^{0})^{2}S,$
(16)

although it is easier to use $v_{\sigma}v^{\sigma}=1$ to find

${{}_{1}v}^{1}=\frac{{{}_{0}v}^{0}}{{{}_{0}v}^{1}}{{}_{1}v}^{0},$ (17)

and use this in (15) to get

$\displaystyle{{}_{1}v}^{0}=\frac{be^{3\tau/2}}{8\left(1+e^{2\tau}\right)^{3/2}\sqrt{\cosh(\tau)}}\times\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle\left(e^{\tau/2}\sqrt{\cosh(\tau)}(-1+\log(4))\left(1+e^{2\tau}\right)+\sqrt{2}\left(1-\left(1+e^{2\tau}\right)\log\left(1+e^{2\tau}\right)\right)\sqrt{1+e^{2\tau}}\right)$
(18)

Figure 1: ${{}_{1}v}^{0}$ (top) and ${{}_{1}v}^{1}$, divided by $b$, vs.
dimensionless proper time

This solves the constant force problem. The results (13) and (14) quickly
approach their asymptotic values of Cosh$t$ and Sinh$t$. The solutions (New
approach to radiation reaction in classical electrodynamics) and (17) show how
the energy and velocity are reduced due to the radiation.

Now we may look at the realistic and practical problem of an electron in a
uniform magnetic field (we revert to cgs). For a two or three dimensional
problem we may find the spatial part of the radiation force, $f^{n}$, by
making the ansatz $f^{n}=\xi v^{n}$ which implies that
$\xi=v_{0}^{2}P/c^{2}/(v_{0}^{2}-c^{2})$, in cgs. We assume that the magnetic
field $\bm{B}$ is in the $z$ direction and the charged particle has an initial
four velocity $u$ in the $x$ direction, i.e., $v^{1}(0)=u$. Using (4) we have

$\dot{v}^{0}=-f^{0}/m$ (19) $\dot{v}^{1}=\omega v^{2}-f^{1}/m$ (20)
$\dot{v}^{2}=-\omega v^{1}-f^{2}/m$ (21)

where $\omega=eB/mc$. To order $\tau_{0}$ the solution to the spatial
equations is,

$v^{1}=u\cos\omega\tau(1-b\tau)$ (22)

$v^{2}=-u\sin\omega\tau(1-b\tau)$ (23)

where $b=\tau_{0}\omega^{2}(1+u^{2}/c^{2})$. These can be integrated to find
the position as a function of proper time and are plotted in Fig. 2.

Figure 2:  Parametric plot of $x$ and $y$ versus proper time, showing the
electron spiraling in due to radiation reaction. For illustrative purposes, I
set $u=1$, $\omega$=1, and $b=0.01$ (which, of course, corresponds to a huge
and false value of $\tau_{0}$).

The zero component of the equation of motion is an energy balance equation.
Integrating (19) with respect to proper time gives

$v^{0}-v_{\mbox{\scriptsize inc}}^{0}=-\frac{1}{mc}\int Pdt,$ (24)

which was engineered from the start (the magnetic field does no work on the
particle). In particular, using the expression for kinetic energy,
$K=mc^{2}(\gamma-\gamma_{\mbox{\scriptsize inc}})$, (20) and (21) show that
the change in kinetic energy, which is negative, is negative of the energy
radiated, $W_{R}=-\int Pdt$. Another way of looking at this is to use

$E^{2}=p^{2}c^{2}+m^{2}c^{4}$ (25)

which implies for small changes,

$\Delta E=\frac{{\bm{p}}\cdot\Delta{\bm{p}}}{\gamma m}.$ (26)

In this equation we use (22) and (23) to obtain ${\bm{p}}$ and
$\Delta{\bm{p}}$. The piece without the $\tau_{0}$ term is used to find $p$
while the $\Delta p$ is obtained from the $\tau_{0}$ piece. With this, the
above yields,

$\Delta K=-\tau_{0}mu^{2}\omega^{2}\gamma\tau.$ (27)

To check, we integrate $P$, which gives the same result (one may note that
$\gamma=\gamma_{\mbox{\scriptsize inc}}+{\cal O}(\tau_{0})$, so that to this
order $\gamma\tau=t$.

Thus, by generalizing the simple equation of motion along with the equation
expressing conservation of energy, equations of motion with radiation reaction
have been derived that do not suffer from the unphysical behavior of, for
example, the LAD equation, or the problem of uniform fields of the FO and LL
equations. Solutions for a few special cases were given, and the age old
problem of a charged particle in a uniform field was solved.

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