spectralEnstrophy
plain-language theorem explainer
The spectral enstrophy for a 3D Galerkin truncation of order N is the weighted sum of squared Fourier coefficients, with weights given by the squared wave numbers. Researchers studying finite-mode approximations to the Navier-Stokes equations cite this quantity when deriving a priori bounds on enstrophy growth under viscosity. The definition is realized as an explicit finite sum over the product of the mode set and the three velocity components.
Claim. Let $M_N$ denote the finite set of integer triples $(k_1,k_2,k_3)$ with each $|k_i|leq N$. For a coefficient vector $u$ in the Euclidean space indexed by $M_N times {1,2,3}$, the spectral enstrophy is $sum_{(k,j) in M_N times {1,2,3}} |k|^2 u_{k,j}^2$, where $|k|^2 = k_1^2 + k_2^2 + k_3^2$.
background
In the 3D spectral Galerkin truncation the velocity field is represented by its Fourier coefficients on the torus $T^3$ restricted to modes with wave numbers in $[-N,N]^3$. The state space is the Euclidean space of real coefficients for these modes across the three velocity components. The auxiliary function assigning to each mode $k$ the squared Euclidean norm $|k|^2$ enters the viscous dissipation term. This construction extends the 2D Galerkin bridge to three dimensions, as described in the module documentation for the Navier-Stokes spectral approximation. The mode set is the product of three intervals from $-N$ to $N$, and the mode type is the integer triple.
proof idea
The definition is a direct summation: for each pair consisting of a mode and a component index, multiply the squared wave number by the square of the corresponding coefficient and sum the results.
why it matters
This definition supplies the spectral enstrophy that appears in the enstrophy bound for the Galerkin system, which is controlled by the viscosity and the truncation level $N$ squared. It is used by the nonnegativity lemma that follows immediately. Within the Recognition Science framework the quantity supports the discrete sub-Kolmogorov analysis for the Navier-Stokes equations in the paper RS_NavierStokes_BKM.tex §4.
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