Integrated Calendar

Many processes in biology, chemistry, physics, medicine, and engineering are modeled by systems of differential equations. These systems describe the interrelationships between the variables involved, and depend in a complicated way on unknown quantities (e.g., initial values, constants or time dependent parameters). Most often, the researcher would like to execute important tasks such as testing the validity of a model, analyzing its qualitative behavior or predicting future states of the system. In order to execute these tasks, one usually needs to estimate the unknown quantities of the system from real data. However, in the case of differential equations, the inverse problem of parameter estimation is considered as the bottleneck in modeling dynamical systems and estimating parameters based on observed noisy state variables has a relatively sparse statistical literature.

In this talk we focus on the fairly general and often applied class of systems of ordinary differential equations linear in (functions of) the parameters. For such systems we first characterize a necessary and sufficient condition for identifiability of parameters. Then we present a novel estimation approach and support it by a general statistical theory that enables the development of estimators tailored for a variety of experimental scenarios. In particular, we present estimators corresponding to some common experimental setups and discuss their statistical properties. Simulation studies and application of the method to real data will be discussed.

The general circulation of the ocean is forced by surface fluxes of momentum, heat, and freshwater at basin scales. The kinetic and available potential energy sources associated with these external forces drive a circulation which exhibits flow features that vary on a wide range of spatial and temporal scales. Understanding how the different forcing mechanisms lead to the observed large-scale ocean circulation patterns and to what degree do the various smaller scale processes modify them have been long standing problems for oceanographers.
A large fraction of the kinetic energy in the ocean is stored in the mesoscale eddy field. This `balanced' eddy field is expected, according to geostrophic turbulence theory, to transfer energy to larger scales. In order for the general circulation to remain approximately steady, sub-mesoscale instabilities leading to `loss of balance' (LOB) have been hypothesized to take place so that the eddy kinetic energy (EKE) may be transferred to small scales where it can be dissipated.
We examine the kinetic energy pathways in fully resolved direct numerical simulations of flow in a flat-bottomed re-entrant channel, a configuration that resembles the Antarctic Circumpolar Current. The flow is allowed to reach a statistical steady state at which point it exhibits both a forward and an inverse energy cascade. We show that EKE is dissipated preferentially at small scales near the surface via sub-mesoscale instabilities associated with LOB and a forward energy cascade rather than by bottom drag after an inverse energy cascade. These results highlight the importance of sub-mesoscales dynamics to the general circulation of the oceans.