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I am trying to read following article: http://arxiv.org/pdf/1410.1262v1.pdf

According to the equation (2.10) and (2.11), the random force is defined as

$ \langle f_i(x) \ f_j(x) \rangle = \delta(t-t') \int_{k\geq m} \frac{d^dk}{(2\pi)^d} D_{0} k^{4-d-y} \{ P_{ij}^{\perp} + \alpha P_{ij}^{\parallel} \} \exp\{ i\mathrm{kx} \} $

where $ P_{ij}^{\perp} = \delta_{ij} + k_i k_j / k^2,\ P_{ij}^{\parallel} = k_i k_j / k^2 $ are transverse and longitudinal projection operators.

My question is: How can I imagine purely transverse or purely longitudinal random force? Can you give me a real physical example of such a random forces?

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I received an answer from a professor, in short:

This is a random force introduced for the energy pumping in the compressible Navier-Stokes turbulence in order to maintain the turbulent behaviour. The transversal part of the random force generates the classical "incompressible" velocity modes while the longitudinal is responsible for the generation of sound waves. In the case of incompressible turbulence, there are no sound-waves and this is why in general we can omit the longitudinal part. In the case of compressible turbulence this is no longer true and in principle we are free to choose any combination of these two components we want. An important point also is that limit $ \alpha \rightarrow 0 $ is not generally an incompressible limit since in the compressible turbulence the sound waves are also generated by the non-linearities in the Navier-Stokes equation.

I also found some recent numerical simulations and visualizations of highly compressible (Mach 17) turbulence: http://www.ita.uni-heidelberg.de/~chfeder/pubs/supersonic/supersonic.shtml?lang=en.

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