Exercise 10

The total kinetic energy of a fluid is,

$$E=\frac{1}{2}\int d^3x {\bf u}({\bf x})\cdot {\bf u}({\bf x}) =\frac{1}{2}\int d^3k u_{{\bf k}}u_{-{\bf k}}$$ (1)

Show the last identity (Parceval's theorem).

From this you can express the spectral energy density,

$$E=\int dk E_k.$$ (2)

The enstrophy is defined as,

$$Z=\frac{1}{2}\int d^3x {\bf\omega}({\bf x})\cdot {\bf\omega}({\bf x})$$ (3)

where the vorticity is the curl of the velocity, $\omega = \nabla \times {\bf u}$.

Express the Fourier transform of $\omega$ in terms of $u_{{\bf k}}$and show that the spectral enstrophy density is given as,

Z_k=k² E_k (4)