Abstract
The shape complexity of irregular surfaces is quantified by a dimensionless area-volume measure. A joint distribution of shape complexity and size is found for level-set islands and lakes in two-dimensional slices of the scalar field of liquid-phase turbulent jets, with complexity values increasing with size. A well-defined power law, over three decades in size (six decades in area), is found for the shape complexity distribution. Such properties are important in various phenomena that rely on large area-volume ratios of surfaces or interfaces, such as, turbulent mixing and combustion.