Positive periodic convolution filter and subfilter stress

Assume the Axiom of Countable Choice for the normalized periodic torus convention used here. Let $d\in \{1,2,3\}$ and let $L>0$ be a period length. Put ${\mathbb{T}}_L^d=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^d$, with normalized Haar measure $d{\mu }_L=L^{-d}dx$; writing $y=x\mathrm{/}L$ and $\xi =2\pi y$ identifies this normalized measure with normalized Haar measure on ${\mathbb{T}}^d=(\mathbb{R}\mathrm{/}2\pi \mathbb{Z}{)}^d$, the periodic Fourier convention fixed in the referenced Fourier-Sobolev definition. Let $K:{\mathbb{R}}^d\to [0,\mathrm{\infty })$ be smooth, vanish outside a compact subset of $B_1(0)$, and satisfy ${\int }_{{\mathbb{R}}^d}K(s)ds=1$. For $0<\mathrm{\ell }<L\mathrm{/}2$, define the periodized kernel

$G_{\mathrm{\ell }}(r)={\sum }_{n\in {\mathbb{Z}}^d}{\mathrm{\ell }}^{-d}K\text{\hspace{0.17em}⁣}\left(\frac{r+Ln}{\mathrm{\ell }}\right),r\in {\mathbb{T}}_L^d\mathrm{.}$

For a field $q$ with finitely many real components, each smooth and $L$-periodic, define its filtered field by

${\bar{q}}_{\mathrm{\ell }}(x)={\int }_{{\mathbb{T}}_L^d}{G}_{\mathrm{\ell }}(r)q(x-r)dx=L^d{\int }_{{\mathbb{T}}_L^d}{G}_{\mathrm{\ell }}(r)q(x-r)d{\mu }_L(r)\mathrm{.}$

Here $dx$ is the volume measure on one period cell. For any smooth periodic vector field $u:{\mathbb{T}}_L^d\to {\mathbb{R}}^d$, define the subfilter stress and transfer by

${\tau }_{\mathrm{\ell }}={\overline{u\otimes u}}_{\mathrm{\ell }}-{\bar{u}}_{\mathrm{\ell }}\otimes {\bar{u}}_{\mathrm{\ell }},{\mathrm{\Pi }}_{\mathrm{\ell }}=-{\tau }_{\mathrm{\ell }}:\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }},$

where $(u\otimes v{)}_{ij}=u_i{v}_j$, $(\mathrm{\nabla }v{)}_{ij}={\mathrm{\partial }}_j{v}_i$, and $A:B={\sum }_{i,j}{A}_{ij}{B}_{ij}$. We call ${\mathrm{\Pi }}_{\mathrm{\ell }}$ the forward-transfer scalar by sign convention: ${\tau }_{\mathrm{\ell }}:\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}=-{\mathrm{\Pi }}_{\mathrm{\ell }}$, so ${\mathrm{\Pi }}_{\mathrm{\ell }}>0$ exactly when this stress-gradient contraction is negative. This convention does not assert an evolution equation, an energy balance, or a sign for ${\mathrm{\Pi }}_{\mathrm{\ell }}$. With length in metres and time in seconds, $x,L,\mathrm{\ell }$ have units m, $u$ has units m s${}^{-1}$, $G_{\mathrm{\ell }}$ has units m${}^{-d}$, ${\tau }_{\mathrm{\ell }}$ has units m${}^2$ s${}^{-2}$, and ${\mathrm{\Pi }}_{\mathrm{\ell }}$ has units m${}^2$ s${}^{-3}$. The nonnegative-kernel hypothesis is part of this definition; no positivity or transfer-sign property is assigned to signed kernels or sharp spectral cutoffs.

normalized Fourier-Sobolev torus convention

Countable Choice axiom

University of Washington, ME 544, Large-Eddy Simulation lecture notes, §2, printed pp. 5–6, filter definition and commutation conditions

Gregory L. Eyink, The ‘Multifractal Model’ of Turbulence and A Priori Estimates in Large-Eddy Simulation, §1, printed pp. 2–3, filter and exact subscale stress