(a) Fill in the details of the dimensional-analysis derivation of the Kolmogorov spectrum (15.27) for a quantity such as n that is transported by the fluid and thus satisfies dn/dt = 0. In particular, convince yourself (or refute!) that the only quantities S_n can depend on are k, q, and qn, identify their dimensions and the dimension of S_n, and use those dimensions to deduce Eq. (15.27).

(b) Derive the spatial version (15.29) of the Kolmogorov spectrum for a transported quantity by two methods: 

(i) Fourier transforming the wave-number version(15.27), 

(ii) Dimensional analysis.

Transcribed Image Text:

Sn~ ang -1/3k-11/3. (15.27)