= Helicity vector of a solenoidal Fourier mode
{title2=$\tfrac12\operatorname{Re}(i\hat{\mathbf u}^*\times\hat{\mathbf u})=H\mathbf k$}
For a transverse complex velocity amplitude $\hat{\mathbf u}$ and real nonzero <wavevector> $\mathbf k$, write $\hat{\mathbf u}=\mathbf a+i\mathbf c$. Solenoidality makes both real vectors perpendicular to $\mathbf k$, so $\tfrac12\operatorname{Re}(i\hat{\mathbf u}^*\times\hat{\mathbf u})=-\mathbf a\times\mathbf c=H\mathbf k$. The corresponding spatial average of <kinetic helicity density> is $-Hk^2$. Cross-product order determines this sign.
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