Grothendieck-Serre formula and bigraded Cohen-Macaulay Rees algebras

The Grothendieck-Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two m-primary ideals I and J in a local ring (A,m) in terms of local cohomology modules of Rees algebras of I and J. The cohomology of a variation of the Kirby-Mehran complex for bigraded Rees algebras is studied which is used to characterize the Cohen-Macaulay property of bigraded Rees algebra of I and J for two dimensional Cohen-Macaulay local rings.Comment: 23 page

Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results

The basic entity of nonlinear interaction in Navier-Stokes and the Magnetohydrodynamic (MHD) equations is a wavenumber triad ({\bf k,p,q}) satisfying ${\bf k+p+q=0}$. The expression for the combined energy transfer from two of these wavenumbers to the third wavenumber is known. In this paper we introduce the idea of an effective energy transfer between a pair of modes by the mediation of the third mode, and find an expression for it. Then we apply this formalism to compute the energy transfer in the quasi-steady-state of two-dimensional MHD turbulence with large-scale kinetic forcing. The computation of energy fluxes and the energy transfer between different wavenumber shells is done using the data generated by the pseudo-spectral direct numerical simulation. The picture of energy flux that emerges is quite complex---there is a forward cascade of magnetic energy, an inverse cascade of kinetic energy, a flux of energy from the kinetic to the magnetic field, and a reverse flux which transfers the energy back to the kinetic from the magnetic. The energy transfer between different wavenumber shells is also complex---local and nonlocal transfers often possess opposing features, i.e., energy transfer between some wavenumber shells occurs from kinetic to magnetic, and between other wavenumber shells this transfer is reversed. The net transfer of energy is from kinetic to magnetic. The results obtained from the studies of flux and shell-to-shell energy transfer are consistent with each other.Comment: 27 pages REVTEX; 14 ps figure