Spectrum of anisotropic exponents in hydrodynamic systems with pressure - art. no. 056302

We discuss the scaling exponents characterizing the power-law behavior of t he anisotropic components of correlation functions in turbulent systems wit h pressure. The anisotropic components are conveniently labeled by the angu lar momentum index / of the irreducible representation of the SO(3) symmetr y group. Such exponents govern the rare of decay of anisotropy with decreas ing scales. It is a fundamental question whether they ever increase as / in creases, or they are bounded from above, The equations of motion in systems with pressure contain nonlocal integrals over all space. One could argue t hat 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 introduc ed here is of a passive vector advection by a rapidly Varying velocity fiel d. The advected vector held is divergent free and the equation contains a p ressure term that maintains this condition. The zero modes of the second-or der correlation function are found in all the sectors of the symmetry group . We show that the spectrum of scaling exponents can increase with / withou t bounds while preserving finite integrals. The conclusion is that contribu tions from higher and higher anisotropic sectors can disappear faster and f aster upon decreasing the scales also in systems with pressure.