1.3. Basdevant reduction

Writing the nonlinear term as $\boldsymbol{\nabla}\cdot(\omega\mathbf{u})$, we need to apply the inverse fft to three functions, namely on $\hat\omega$, $\hat u$, $\hat v$, and then two direct fft, namely on $\omega u$ and $\omega v$, before taking the gradient in spectral space.

Alternatively, we can keep it as $\boldsymbol{\nabla}\omega \cdot \mathbf{u}$ and compute four ifft and one fft.

The best, however, is to perform the Basdevant reduction to reduce to four ffts (two inverse and two direct). This is obtained by writing

This way, we need to compute the inverse fft of $u$ and $v$, compute the product $uv$ and the term $v^2 - u^2$ in physical space, and finally compute the FFT of these two functions.