The spectrum of fractal dimensions of passively convected scalar gradients in chaotic fluid flows

The passive convection of scalar fields by an incompressible fluid flow in two dimensions is investigated numerically. The prescribed flow is chaotic meaning that nearby fluid elements diverge exponentially with time. The gradient of the convected scalar field is of primary interest, and a measure is defined, reflecting the spatial distribution of the regions having large gradient. The dimension spectrum for this measure is computed by the standard box counting technique, and it is found to be fractal. A recent theory proposes that the fractal structure of the scalar gradient can be related to the nonuniform stretching properties of the flow. Using this theory, the fractal dimension spectrum is computed from the distribution of finite time Lyapunov exponents of the flow, and it is found to be in reasonable agreement with the dimension spectrum computed directly by means of box counting.