Experimental study of two-dimensional enstrophy cascade

– We study the direct enstrophy cascade in a two-dimensional flow generated in an electromagnetically driven thin layer of fluid. Due to the presence of bottom friction, the energy spectrum deviates from the classical Kraichnan prediction k. We find that the correction to the spectral slope depends on the thickness on the layer, in agreement with a theoretical prediction based on the analogy with passive scalar statistics. Introduction. – Laboratory studies of two-dimensional turbulence are affected by the interaction with the three-dimensional environment in which they are embedded. This interaction is often frictional and generates an additional linear damping term to the equation of motion. A well known example, which is the object of the present letter, is the flow in a shallow layer of fluid, which is subject to the friction with the bottom of the tank where it is contained [1]. Another example is the motion of a soap film dumped from the interaction with the surrounding air [2]. Linear friction dumping is not restricted to laboratory experiments: an important example is the Ekman friction in geophysical fluid dynamics [3]. As predicted in a remarkable paper by Kraichnan in 1967 [4], when a two-dimensional layer of fluids is forced at intermediate scales, smaller than the domain-size and larger than the viscous dissipative length-scales, two different inertial ranges are observed. This is a consequence of the coupled conservation of energy (mean square velocity) and enstrophy (mean square vorticity) in the inviscid limit, a basic difference with respect to three-dimensional turbulence phenomenology. The energy flows toward the large scales giving rise to an inverse energy cascade characterized by the k Kolmogorov spectrum. Small scale statistics is governed by a direct enstrophy cascade which is expected to develop a smooth flow with k energy spectrum, with a possible logarithmic correction [5]. The presence of a linear friction term affects the direct cascade in a dramatic way: the enstrophy flux across scales is no longer constant and the scaling exponent of the spectrum differs from the Kraichnan prediction k by a correction proportional to the friction intensity [6, 7]. The origin of this correction can be understood exploiting the analogy between twodimensional vorticity field and a scalar field passively advected by a smooth velocity field [8].