  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.
