Spectrum of anisotropic exponents in hydrodynamic systems with pressure

We discuss the scaling exponents characterizing the power-law behavior of the anisotropic components of correlation functions in turbulent systems with pressure. The anisotropic components are conveniently labeled by the angular momentum index l of the irreducible representation of the SO(3) symmetry group. Such exponents govern the rate of decay of anisotropy with decreasing scales. It is a fundamental question whether they ever increase as l increases, or they are bounded from above. The equations of motion in systems with pressure contain nonlocal integrals over all space. One could argue that the requirement of convergence of these integrals bounds the exponents from above. It is shown here on the basis of a solvable model (the "linear pressure model") that this is not necessarily the case. The model introduced here is of a passive vector advection by a rapidly varying velocity field. The advected vector field is divergent free and the equation contains a pressure term that maintains this condition. The zero modes of the second-order correlation function are found in all the sectors of the symmetry group. We show that the spectrum of scaling exponents can increase with l without bounds while preserving finite integrals. The conclusion is that contributions from higher and higher anisotropic sectors can disappear faster and faster upon decreasing the scales also in systems with pressure.