On the Decadal Modes of Oscillation of an Idealized Ocean-atmosphere System

Axially-symmetric, linear, free modes of global, primitive equation, ocean-atmosphere models are examined to see if they contain decadal (10 to 30 years) oscillation time scale modes. A two-layer ocean model and a two-level atmospheric model are linearized around axially-symmetric basic states containing mean meridional circulations in the ocean and the atmosphere. Uncoupled and coupled, axially-symmetric modes of oscillation of the ocean-atmosphere system are calculated. The main conclusion is that linearized, uncoupled and coupled, ocean-atmosphere systems can contain axially-symmetric, free modes of variability on decadal time scales. These results have important implications for externally-forced decadal climate variability

Harder-Narasimhan Filtrations which are not split by the Frobenius maps

Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural question to ask is whether this still holds in higher dimension. Indranil Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that there is always a counterexample to this over any algebraically closed field of positive characteristic which is uncountable. However, we will produce a smooth projective variety over $\mathbb Z$ and a rank 2 vector bundle on it, which, restricted to each prime $p$ in a nonempty open subset of \spec\mathbb Z, constitutes a counterexample over $p$. Indeed, given any split semisimple simply connected algebraic group $G$ of semisimple rank $>1$ over $\mathbb Z$, we will show that there exists a smooth projective homogeneous space $X_Z$ over $\mathbb Z$ and a vector bundle $V$ on $X_Z$ of rank 2 such that for each prime $p$ in a nonempty open subset of \spec\mathbb Z, the restriction $V\otimes\mathbb F_p$ as a vector bundle over $X_Z\otimes\mathbb F_p$ is a counterexample. We only use the Borel-Weil-Bott theorem in characteristic 0 and Frobenius Splitting of $G/B$ in characteristic $p$.Comment: 3 page