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.

In this work, we obtain the hydrostatic approximation by taking the small aspect ratio limit to the Navier-Stokes equations. The aspect ratio (the ratio of the depth to horizontal width) is a geometrical constraint in general large scale geophysical motions that the vertical scale is significantly smaller than horizontal. We use the versatile relative entropy inequality to prove rigorously the limit from the compressible Navier-Stokes equations to the compressible Primitive Equations. This is the first work to use relative entropy inequality for proving hydrostatic approximation and derive the compressible Primitive Equations.

On the hydrostatic approximation of compressible anisotropic Navier-Stokes equations -- rigorous justification

A widely used approach to mathematically describe the atmosphere is to consider it as a geophysical fluid in a shallow domain -- and thus to model it using classical fluid dynamical equations combined with the explicit inclusion of an {\epsilon} parameter representing the small aspect ratio of the physical domain. In our previous paper [15] we proved a weak convergence theorem for the polluted atmosphere described by the Navier-Stokes equations extended by an advection-diffusion equation. We obtained a justification of the generalised hydrostatic limit model including the pollution effect described for the case of classical, east-north-upwards oriented local Cartesian coordinates. Here we give a two-fold improvement of this statement. Firstly, we consider a meteorologically more meaningful coordinate system, incorporate the analytical consequences of this coordinate change into the governing equations, and verify that the weak convergence still holds for this altered system. Secondly, still considering this new, so-called downwind-matching coordinate system, we prove an analogous strong convergence result, which we make complete by providing a closely related existence theorem as well.

The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates.

&lt;p style='text-indent:20px;'&gt;A widely used approach to mathematically describe the atmosphere is to consider it as a geophysical fluid in a shallow domain and thus to model it using classical fluid dynamical equations combined with the explicit inclusion of an &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \epsilon $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; parameter representing the small aspect ratio of the physical domain. In our previous paper [&lt;xref ref-type="bibr" rid="b14"&gt;14&lt;/xref&gt;] we proved a weak convergence theorem for the polluted atmosphere described by the Navier-Stokes equations extended by an advection-diffusion equation. We obtained a justification of the generalised hydrostatic limit model including the pollution effect described for the case of classical, east-north-upwards oriented local Cartesian coordinates. Here we give a two-fold improvement of this statement. Firstly, we consider a meteorologically more meaningful coordinate system, incorporate the analytical consequences of this coordinate change into the governing equations, and verify that the weak convergence still holds for this altered system. Secondly, still considering this new, so-called downwind-matching coordinate system, we prove an analogous strong convergence result, which we make complete by providing a closely related existence theorem as well.&lt;/p&gt; 

The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates

Derivation of the inviscid compressible Primitive Equations

Considered as a geophysical fluid, the polluted atmosphere shares the shallow domain characteristics with other natural large-scale fluids such as seas and oceans. This means that its domain is excessively greater horizontally than in the vertical dimension, leading to the classic hydrostatic approximation of the Navier-Stokes equations. The authors of the \cite{azerad2001mathematical} article have proved a convergence theorem for this model with respect to the ocean, without considering pollution effects. The novelty of this present work is to provide a generalisation of their result translated to the atmosphere, extending the fluid velocity equations with an additional convection-diffusion equation representing pollutants in the atmosphere.

Nora Juhasz

Papers

The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates.

Weak solution of the merged mathematical equations of the polluted atmosphere

Weak solution of the merged mathematical equations of the polluted atmosphere