Research Areas

To limit the scope of the school, we focus on three specific methodological Research Areas as outlined below. At the same time, we identify three topical research fields in Earth Sciences - Solid Earth Dynamics, Atmosphere-Hydrosphere Dynamics, and Hydrological Flow and Transport Processes - each of which will be addressed under these three Research Areas.

Observational and experimentally measured data form the basis of any modeling study. For the creation of a (predictive) mathematical model representing a physical process of interest, scientists need to be able to view and analyze - explore - their data in very flexible, creative ways. This will require advanced skills in e.g. data mining, time series analysis and numerical mathematics of deterministic and stochastic processes. In many cases a mismatch exists between the available observational data and the model state variables. This calls for data assimilation strategies that take into account the physical and mathematical structure of both the observation process and the computational predictive model. The most challenging situation for projects within this research area arises when there is no agreement yet with respect to a mathematical description of the physical system. In this case, the testing andselection of a simulation model requires a process of iterative interaction between the model developers and the scientists providing constraints from observations or experiments.

Application to the topical research fields:

Solid Earth Dynamics: The understanding of earthquake patterns in space and time is of prime interest for risk management. However, it is restricted due to the lack of long enough instrumental time series and incomplete understanding of the rupture process itself. The key questions are whether earthquakes follow a predictable pattern and how the related hazard in terms of probability theory should be treated. These questions may be addressed by, for example, dynamically scaled laboratory models of the earth's upper crust producing extended series of earthquakes which allow the development of statistical earthquake occurrence models, by probabilistic modeling of aftershocks or by multivariate analysis of observational data taking into account the knowledge about the physical properties of the system.

Atmosphere-Hydrosphere Dynamics: Observational data of Earth system dynamics from satellites, airborne systems, ground-based stations or (paleo)proxies are generally inhomogeneous in time and space as well as of unequal quality, and they exhibit rich, highly non-trivial dynamics for which most standard assumptions of classical statistical methods are not applicable. Advanced data exploration and time series analysis tools are thus needed which reliably and efficiently operate on high-dimensional, statistically non-stationary, non-Markovian data, and allow us to query the data in a number of very different ways, e.g., to check their mutual consistency, to extract the input for specific runs of a computational model, or to test hypotheses regarding underlying causal chains.

For example, in weather forecasting and climate change assessments, assimilation of observational data (required to obtain continually updated initial conditions for forward simulation runs) may be achieved by methods based on Kalman filtering and Bayesian learning. These techniques could be used to integrate proxy data from ice cores, tree rings, or lake sediments into coupled atmosphere-ocean models for paleo climate. Simulations will then generate new insight into past abrupt climate changes and into the general dynamics of the climate system. Hydrological Flow and Transport Processes: The dynamics of the near-surface hydrosphere, which is governed by the behavior of the soil and plant systems, is as yet insufficiently understood. Improved prediction of flow and transport processes can be achieved through laboratory studies on artificial (disturbed and undisturbed) soil samples, by imaging patterns of water uptake and biochemical interaction of roots and soil, the dynamics of infiltration events and resulting ground water table fluctuations, as well as the heterogeneity of plant water uptake with high spatial and temporal resolution.

In Earth sciences, the presence of multiple spatio-temporal scales and the coupling of processes across scales represent a ubiquitous challenge. While a number of processes and features appear to be scale-invariant over a large range of spatial scales, others exhibit a trend to scale dependence. In studying multiscale processes, measurements and observations as well as computational models tend to be "underresolved", i.e., the measured and computed data are not distributed sufficiently densely in space and time to capture accurately all the variability of the actual physical process. In this case, the data - both measured and computed - can only be meaningfully interpreted if a reliable model exists that relates coarse-grained data distributions to the non-resolved fine-scale structure. Moreover, in many situations the fine scale structure of the system exerts a nontrivial influence on the (evolution of) the large scales.

To meet the challenges posed by multiscale processes and phenomena, we will combine expertise in stochastic and multiscale asymptotic analysis with space-time adaptive computational techniques, and we will encourage close cooperation with projects from Area I.

Application to the topical research fields:

Solid Earth Dynamics: Processes in solid earth dynamics occur on very different scales both in space and time. Deformation processes occurring in the crystal lattice and along grain boundaries are ultimately underlying earthquake rupture or plate motion at the scale of hundreds to thousands of kilometers. Linking the earthquake timescale with the geological time-scale (106-108 years) of global deformation processes and distribution of stresses in the Earth`s lithosphere is a key challenge of Earth science research.

At the small scale, particle interactions may be represented by analytical functions. Efficient, accurate, and transferable interaction potentials that can be used reliably to predict properties of geomaterials over a wide range of pressures, temperatures, and compositions still need to be developed. These 'continuum' properties are needed as input for larger scale thermo-mechanical models, in which we aim to include both short-term and long-term deformation processes. In addition, scaled analogue models of the coupled asthenosphere-lithosphere may be used to explore the multiscale pattern resulting from the coupling of brittle-plastic and viscoelastic deformation over multiple seismic cycles. Finally, in the context of probabilistic seismic hazard analysis, multi-dimensional scaling techniques techniques (e.g. Sammon´s maps and Kohonen networks) may be used to select and compare models.

Atmosphere-Hydrosphere Dynamics: Understanding and modeling the multi-scale nature of atmosphere-hydrosphere dynamics requires a rich toolkit of techniques such as multiple scales asymptotics for small Froude, Rossby, or Mach numbers; dynamical systems theory (chaotic as well as deterministic) and stochastic modelling when there is a continuous spectrum of scales of which only a selected range is to be represented; multiscale adaptive numerical methods when localized processes, e.g., in boundary layers or pores and gaps of a porous medium, exert a net influence on the overall system dynamics on much larger scales.

When localized processes involve inherently different physics than the bulk of the considered flows, e.g., in land and sea ice covers or glaciers, the physically and mathematically sound construction and internal coupling of multi-physics models will be pursued.

Hydrological Flow and Transport Processes: The subsurface part of the hydrological cycle is coupled to surface water flows mainly by water exchange via streambeds in the so-called hyporheic zone. At the same time, this interface is a preferential location for biogeochemical mass transformations, fed by the solute fluxes on top of the water fluxes. However, the underlying water fluxes are heterogeneously distributed, e.g., due to heterogeneities of permeability, from the scale of a single stream cross-section via the scale of stream stretches up to the scale of entire streams. Also, the fluxes are transient due to changes in hydraulic boundary conditions, and this again occurs at different time scales of days via months to years.

Temperature measurements combined with heat flux simulations offer a promising possibility to characterize the variations of water fluxes at all of these scales. The modelling challenge will be to upscale the local water and heat flux patterns to the scale of an entire stream, and to thus provide the basis for further investigations on the fate of nutrients and pollutants in the stream catchment.

Here we will address phenomena about which there may be substantial empirical evidence and observational data, but for which no complete physical descriptions and/or abstract mathematical models are available yet. The latter may be achieved through verification of existing hypotheses by systematic exploitation of modern high-performance computing capacities. The iterative process of hypothesis creation and testing may be accelerated by flexible data exploration (see also RA I), rapid provision of numerical solutions to hypothesized model equations, and efficient implementation of validation procedures. Such explorative simulation will benefit from tools and techniques acquired and newly developed in Research Areas I and II in the form of software libraries with unified user interfaces or in common software frameworks.

Application to the topical research fields:

Solid Earth Dynamics: Because of the inaccessibility of the active earthquake source, our understanding of the physical process controlling earthquakes remains poor and the variety of sometimes complex models untested. A first step to improve our understanding would be to formulate mathematical descriptions for the behaviour of sequences of earthquakes produced in scaled analogue models which have relatively simple setups.

Changes in fluid pressure (either due to natural phenomena such as volcanic activity or caused by technological processes like borehole fluid injections) play a key role in seismicity. The understanding of fluid-induced seismicity requires numerical modelling of non-linear pore pressure in stochastic heterogeneous and anisotropic systems, including poro-elastic coupling, applying non-stationary boundary conditions, and comparison of these results with natural seismicity dynamics.

Atmosphere-Hydrosphere Dynamics: There is a range of phenomena in atmosphere-hydrosphere dynamics which are well-identified observationally but often only poorly accounted for in computational sub-models. Examples are cloud formation and precipitation, radiative transfer, chemical reactions, and molecular and turbulent transport. Yet, these rudimentary sub-models are incorporated in more complex aggregated atmosphere and ocean, ice sheet, and land surface and vegetation models, which are then again coupled to form highly complex atmosphere-ocean global circulation models. The interplay of the various components of such a complex system is very hard to understand in theoretical terms. We expect important progress in this field from the collaboration between theoreticians developing reduced but easier understood model versions (see also RA II) and computational modelers who will use their complex model systems as computational laboratories.

Hydrological Flow and Transport Processes: Hillslopes represent a spatial scale where processes of surface and subsurface water flow and the linked transport of physical and chemical matter (e.g. soil, nutrients, pesticides) are generated and may be monitored as well. In addition, the interactions of hillslopes with other landscape compartments (such as valley bottoms, flood plains and river reaches) are important to assess such fluxes at the larger scales. The challenge here is to develop more accurate and also computationally efficient process-based extensions of the current standard class of hillslope models. This may be done by combining detailed computational simulations (hydro-physical-chemical models), observations, and experiments as a basis for the construction of abstract mathematical descriptions of the relevant processes. The developed technology can be applied to assess physical-chemical fluxes, such as nutrients and pesticides, and to derive recommendations to reduce emissions of fertilizers and pesticides from hilly agricultural landscapes to river systems.