Turbulent stretching of line and surface elements

Material line and surface elements transported in a turbulent velocity field increase in length or area at an exponential rate. In this paper we investigate how the stretching rates are related to the statistical properties of the velocity field both analytically and numerically in simple models of turbulence. In a Gaussian model the statistics exhibit time-reversal invariance. We demonstrate that, as pointed out by Kraichnan (1974), this leads to equality of line and area stretching rates. We also construct a model which violates the time-reversal property and splits the values of the rates for lines and surfaces. The sign of the splitting depends on the sign of the time-reversal breakdown.