Welcome to AMP

 

In data assimilation work funded through the NSF Collaboration between Mathematics and Geophysics (CMG) program, Greg Duane (CDP visitor), Jeff Weiss (CU) and Tribbia have been examining the relationship between synchronization and assimilation. In past work they showed that the synchronization approach is equivalent to standard approaches based on least-squares optimization, including Kalman filtering, except in highly non-linear regions of state space where observational noise links regimes with qualitatively different dynamics. In such narrow regions, the synchronization approach is expected to give an improvement to Kalman filtering that will apply in any situation where a computational model is intended to track a physical process. The synchronization approach is used to calculate covariance inflation factors from parameters describing the bimodality of a one-dimensional system. See (DTW) for more details. In the recent extension of this research, the use of synchronization ideas in parameter estimation has been explored and the promising results for this paradigm are detailed in (DT).

In addition to advances in the numerics, the next generation of atmospheric model dynamical cores will in all likelihood span a range of scales which will include those for which the hydrostatic approximation is questionable. This will require new understanding of global non-hydrostatic effects, including the role of the horizontal component of the Coriolis force.

In search of benefits that a more general formulation of the dynamical models for global weather prediction and climate projection will provide, Akira Kasahara continued his research to understand the role of the horizontal component of the Coriolis force, which is neglected on the basis of the "traditional approximation (TA)" in most of the current weather prediction and climate projection models. It has been known that the usual justification of the TA for the atmospheric and oceanic dynamics is too simplistic. Because of a mathematical complication involved in the analysis of the dynamica system without the TA, Kasahara has been focusing the analysis on a very simple yet nontrivial dynamical system, namely linear Boussinesq equations in Cartesian coordinates. The most detrimental effect of using the TA is on the physics of inertio-gravity motions. An accurate description of inertio-gravity motions requires the horizontal component of the Coriolis force as well as its vertical component. Yet, relatively little work has been done to understand the role of this often-neglected partner of the earth's rotation effect in the atmosphere and ocean.

This year Kasahara formulated a numerical model to solve initial-value problems with the linear Boussinesq equations without the TA. Motions are assumed to be horizontally periodic, but bounded vertically at the top and bottom. The time-evolution of the vertical structure of wave motions is calculated from given initial conditions with or without forcing/dissipation under variable thermal buoyancy stratification. This program is intended to serve as a simple numerical laboratory to study the time-evolution of inertio-gravity waves. As one example, the formation of near-inertial currents in the oceans generated by atmospheric storms is investigated in detail. It is shown that under a realistic vertical buoyancy stratification, the non-traditional wave mode is likely to be excited if the forcing is applied near the bottom of the ocean, resulting from, say, an up and down movement of barotropic tide over corrugated topographic features. This phenomenon may provide a new mechanism to energy dissipation of tidal motions unique only by taking into account of a full rotation effect, not a partial effect as done traditionally. More details are included in (K).

Tribbia continued investigating the limitations of the hydrostatic balance approximation in a different context, that of limited area modeling. With Roger Temam (Indiana University) and Antoine Rousseau (Universit'e Paris-Sud), he has been pursuing the examination of approximate equations which break the strong constraint of hydrostatic balance. The reason for their interest is the well-known deficiency of the hydrostatic primitive equations, ill-posedness as an initial-boundary value problem. The ill-posedness of the system imposes severe restrictions on the applicability of the system for limited area regional climate modeling and the use of adaptive mesh methods. In the recent work they have studied a linear differential system consisting of two coupled scalar evolution equations in one space dimension which was derived from a modal analysis of the Primitive Equations of the ocean. They have shown numerically that, by adjunction of a small viscosity, the system converges to an unusual, unexpected limit system thus producing boundary layers and reflections of waves at the boundary. They have proposed an alternate set of boundary conditions of transparent type for the viscous systems and, in this case, the viscous system does not produce boundary layers or reflections of waves at the boundary. This work is described fully in (RTT). Over the past year, this work has successfully been extended to three spatial dimensions and the manuscript delineating the results is currently under review.

In studies for which the RTT research noted above should have immediate application, Tribbia is also involved in a project that examines the efficiency of numerical modeling on parallel machines. The collaborative effort with Aime' Fournier (IMAGe), Mark Taylor (Sandia) and Ferd Baer and Houjun Wang (UMd), has developed a spectral element based, locally refined resolution version of CAM. The work is described in (BWTF and WTBFT).