Wavelet analysis of the energy transfer caused by convective terms: Application to the Burgers shock

Orthonormal wavelet analysis, which can deal with the information about both space and scale simultaneously, is applied to analyze the energy transfer due to spatial structures. To utilize the concept of ''triad interaction'' in non-Fourier bases, a simple and appropriate definition of transfer functions is proposed. An essential problem in the use of orthogonal wavelets is a fast oscillation observed in the temporal variations of energy and transfer functions. This oscillation is intrinsic to a wavelet base function and corresponds to ''phase'' in spatial information. A way to remove the phase is also proposed. These prescriptions are applied to examine the energy transfer process of the Burgers shock as a preliminary work. It is shown that the energy transfer is well separated into ones caused by the mean how and the velocity field of the shock. Within a scale, those correspond to sweeping and compression, respectively. The mean how contributes even to the energy transfer across a scale, but it is not substantial.