Shot noise multifractal model for turbulent pseudo-dissipation

Multiplicative cascades have been used in turbulence to generate fields with multifractal statistics and long-range correlations. Examples of continuous and causal stochastic processes which generate such a random field have been carefully discussed in the literature. Here a causal lognormal stochastic process is built to represent the dynamics of pseudo-dissipation in a Lagrangian trajectory. It is introduced as the solution of a stochastic differential equation, driven by a source of noise which has sudden jumps at periodic intervals, its period being the dissipative time scale of the flow. This random field has scale invariance for a continuum of scales, and displays discontinuous jumps in time, with a smooth time evolution below the Kolmogorov scale. Its multifractal and correlation properties are demonstrated numerically.