The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

An Introduction to Magnetohydrodynamics

The Kuramoto model in complex networks

An introduction to the theory of point processes

The Kuramoto model: A simple paradigm for synchronization phenomena

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data or for data with monotonicity properties. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solutions with non-degenerate critical points. Interestingly, the strong instability is due to vicosity, which is coherent with well-posedness results obtained for the inviscid version of the equation. A numerical study of this instability is also provided.

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On the ill-posedness of the Prandtl equation

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data, or for data with monotonicity properties. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solutions with non-degenerate critical points. Interestingly, the strong instability is due to vicosity, which is coherent with well-posedness results obtained for the inviscid version of the equation. A numerical study of this instability is also provided.

/pdf/on-the-ill-posedness-of-the-prandtl-equation-4stulfo512.pdf

It has been thought for a while that the Prandtl system is only well-posed under the Oleinik monotonicity assumption or under an analyticity assumption We show that the Prandtl system is actually locally well-posed for data that belong to the Gevrey class 7/4 in the horizontal variable x Our result improves the classical local well-posedness result for data that are analytic in x (that is Gevrey class 1) The proof uses new estimates, based on non-quadratic energy functionals

Well-posedness for the Prandtl system without analyticity or monotonicity

In the lines of the recent paper (J. Amer. Math. Soc. 23(2) (2010), 591-609), we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary non-monotonic shear flows, we show that for some C ∞ initial data, local in time H 1 solutions of the linearized Prandtl equation do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.

/pdf/remarks-on-the-ill-posedness-of-the-prandtl-equation-wf397290yt.pdf

David Gérard-Varet

Papers

On the ill-posedness of the Prandtl equation

On the ill-posedness of the Prandtl equation

Well-posedness for the Prandtl system without analyticity or monotonicity

Remarks on the ill-posedness of the Prandtl equation

On compressible Navier-Stokes equations with density dependent viscosities in bounded domains