The Decomposition of Global Conformal Invariants: Some Technical Proofs. I

This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of "global conformal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand

Nonlocal Phenomenology for anisotropic MHD turbulence

A non-local cascade model for anisotropic MHD turbulence in the presence of a guiding magnetic field is proposed. The model takes into account that (a) energy cascades in an anisotropic manner and as a result a different estimate for the cascade rate in the direction parallel and perpendicular to the guiding field is made. (b) the interactions that result in the cascade are between different scales. Eddies with wave numbers $k_\|$ and $k_\perp$ interact with eddies with wave numbers $q_\|,q_\perp$ such that a resonance condition between the wave numbers $q_\|,q_\perp$ and $k_\|,k_\perp$ holds. As a consequence energy from the eddy with wave numbers $k_\|$ and $k_\perp$ cascades due to interactions with eddies located in the resonant manifold whose wavenumbers are determined by: $q_\|\simeq \epsilon^{{1}/{3}}k_\perp^{2/3}/B$, $q_\perp=k_\perp$ and energy will cascade along the lines $k_\|\sim C+k_\perp^{2/3} \epsilon^{1/3}/B_0$. For a uniform energy injection rate in the parallel direction the resulting energy spectrum is $E(k_\|,k_\perp)\simeq \epsilon^{2/3}k_\|^{-1}k_\perp^{-5/3}$. For a general forcing however the model suggests a non-universal behavior. The connections with previous models, numerical simulations and weak turbulence theory are discussed.Comment: Submited to Astophys. Let