def SpectralDynamics.waleffeHelicalBasis : ℤ → (Fin 3 → ℤ) → Fin 3 → ℂ

The Helical Basis of Waleffe $h^\pm(k)$. $i k \times h^\pm(k) = \pm |k| h^\pm(k)$. This is an eigenvector basis for the curl operator on Fourier space, critical for separating topological signs of helicity.

Equations

• SpectralDynamics.waleffeHelicalBasis x✝² x✝¹ x✝ = 0

def SpectralDynamics.triadicPhase (phi_p phi_q phi_k : ℝ) : ℝ

Triadic Phase Function $\Psi_{p,q,k} = \phi_p + \phi_q - \phi_k$. When non-linear advection $u \cdot \nabla u$ is written in the Waleffe basis, the temporal evolution of phases creates this oscillatory integral argument.

Equations

• SpectralDynamics.triadicPhase phi_p phi_q phi_k = phi_p + phi_q - phi_k

def SpectralDynamics.interactionCoefficient : (Fin 3 → ℤ) → (Fin 3 → ℤ) → (Fin 3 → ℤ) → ℤ → ℤ → ℤ → ℂ

The exact triadic interaction coefficient $C_{kpqr}^{s s_p s_q}$ from the Navier-Stokes advection term projected onto the Waleffe basis.

Equations

• SpectralDynamics.interactionCoefficient x✝⁵ x✝⁴ x✝³ x✝² x✝¹ x✝ = 0