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## Data assimilation in paleoclimatology

Reliable reconstructions of the past climate states are essential for a comprehensive understanding of the climate system, more accurate climate predictions and projections. A new but highly appealing approach to reconstruct the past climate states is data assimilation. The main purpose of data assimilation is to estimate the state of the climate system as accurately as possible incorporating all the available information: numerical modelling of the behavior of the system, observations, and uncertainties of the model and of the observations. Though data assimilation is well developed in some domains, its application to the past climate state reconstruction faces new challenges. First, the observations in paleoclimatology have more sparse spatial and temporal resolution than in meteorology. Second, the observations based on instrumental records start, in general, no earlier than year 1850 at best. For earlier times the only available observations are indirect reconstructions based on proxy records (e.g. tree rings, ice cores, sediments). The proxy-based observations have more uncertainties than the instrumental-based observations such as time uncertainties, uncertainties associated with the methods used to relate the climate signal recorded in the proxy records to the climate variables; moreover, they are subjected to a greater noise. Third, the variables which can be assimilated in a paleoclimatological application are limited, since proxy-based observations of only a few variables, e.g. surface temperature, precipitation, appear to be the most reliable over long periods of time. My recent research is devoted to development of a data-assimilation approach in paleoclimatology and to the investigation of its performance.

- S. Dubinkina and H. Goosse, ”An assessment of particle filtering methods and nudging for climate state reconstructions”, Clim. Past, 9 (2013), pp. 1141–1152.
- P. Mathiot, H. Goosse, X. Crosta, B. Stenni, M. Braida, H. Renssen, C.J. Van Meerbeeck, V. Masson-Delmotte, A. Mairesse and S. Dubinkina, ”Using data assimilation to investigate the causes of Southern Hemisphere high latitude cooling from 10 to 8 ka BP”, Clim. Past, 9 (2013), pp. 887–901.
- H. Goosse, E. Crespin, S. Dubinkina, M.F. Loutre, M.E. Mann, H. Renssen, Y. Sallaz-Damaz and D. Shindell, ”The role of forcing and internal dynamics in explaining the medieval climate anomaly”, Climate Dynamics, 39 (2012), pp. 2847–2866.
- H. Goosse, J. Guiot, M.E. Mann, S. Dubinkina and Y. Sallaz-Damaz, ”The medieval climate anomaly in Europe: comparison of the summer and annual mean signals in two reconstructions and in simulations with data assimilation”, Global and Planetary Change 84–85 (2012), pp. 35–47.
- S. Dubinkina, H. Goosse, Y. Sallaz-Damaz, E. Crespin and M. Crucifix, ”Testing a particle filter to reconstruct climate changes over the past centuries”, International Journal of Bifurcation and Chaos 21(12) (2011), pp. 3611–3618.

## Thermostat-based parameterisation of small-scale eddies in geophysical flows

Inviscid fluid models are natural in a number of application areas, such as atmosphere and ocean science, where flows are characterised by conservation of total energy, the cascade of vorticity to ever finer scales, and sensitive dependence on initial conditions. For the numerical simulation of such flows, the lack of a viscous diffusion length scale presents the challenge that–due to the vorticity cascade–any direct discretisation of the equations of motion must eventually become underresolved, as vorticity is transferred to scales below the grid resolution. It therefore becomes necessary to close the numerical model by some means. In the course of my PhD research, I have proposed a new perspective for looking at unresolved dynamics, via the reservoir concept by considering the thermostatting–a tool used in molecular dynamics to model a system in thermal equilibrium with a reservoir, whereby the system is augmented by a few degrees of freedom that model the exchange with the reservoir. The goal of thermostatting is to force the system to sample the canonical equilibrium distribution by continually perturbing it. I have shown for a two-scale point vortex flow that subgrid scale effects in fluid dynamics can be parameterised using suitably adapted thermostats.

- S. Dubinkina, J. Frank and B. Leimkuhler, ”Simplified Modelling of a Thermal Bath, with Application to a Fluid Vortex System”, SIAM Multiscale Model. Simul. 8 (2010), pp. 1882–1901.

## The statistical mechanics of numerical discretizations

In studies of climate variability, long numerical simulations are run for dynamical systems that are known to be chaotic, and for which it is consequently impossible to simulate a particular solution with any accuracy in the usual sense of numerical analysis. Instead, the goal of such simulations is to obtain a data set suitable for computing statistical averages i.e., to sample an associated probability distribution. A numerical method defines a discrete dynamical system with its own statistical mechanics, not necessarily in agreement with the continuum problem it models. It is important to understand the statistical bias induced by the method and to compare its results with theoretical expectations for the continuum. In the course of my PhD research, I have examined the effects of a numerical method on statistical results of long-time integrations for simple geophysical fluid models, establishing that a particular choice of method does strongly bias the statistical results obtained from simulations. I have demonstrated this on two classes of numerical methods for quasigeostrophic flow with topography: the classical Arakawa finite difference schemes and a modern particle method – the Hamiltonian Particle-Mesh method. I have shown that these numerical methods give very different statistics based on their conservation properties.

- S. Dubinkina and J. Frank, ”Statistical relevance of vorticity conservation with the Hamiltonian particle-mesh method”, J. Comput. Phys. 229 (2010), pp. 2634–2648.
- S. Dubinkina and J. Frank, ”Statistical mechanics of Arakawa’s discretizations”, J. Comput. Phys. 227 (2007), pp. 1286–1305.

## Motion of fluids with free interface under weightlessness

The laws of motion of free liquid films are important for understanding the processes occurring in the foams. In the general case, the hydrodynamic model is two-velocity, but in the limiting case when the velocity profile is almost uniform over the film thickness it is one-velocity. Due to the Marangoni effect, the viscous forces create shear stresses, but in the long-wave limit the energy dissipation is vanishingly small. If the surface surfactant concentration gradient is fairly large, the shear stresses may result in the rupture of the film. In the course of my M.Sc. research, I have studied a model of deformation of a weightless liquid film with rims fixed at a plane contour under thermocapillary forces. I have obtained an estimation of lifetime of the film as a function of the temperature profile on the free boundary as well as the critical values of the temperature, when the film breaks.

- V. Pukhnachev and S. Dubinkina, ”A model of film deformation and rupture under the action of thermo-capillary forces”, Fluid Dynamics 41(5) (2006), pp. 755–771.