Balanced atmospheric models are systems of equations based on an assumption of approximate balance between the wind and mass fields. Existing balanced models have played a key role in advancing our understanding of atmospheric dynamics. Their usefulness, however, is limited by the accuracy of the underlying balance assumption, and none is capable of representing asymmetric flows with large curvature, such as are encountered in hurricanes. The goal of this project is to remedy this situation, by extending the theory of balanced atmospheric models, developing efficient and robust methods for their numerical solution, and using the resulting models to study aspects of hurricane dynamics and midlatitude weather phenomena.
The principal theoretical task is to develop two new balanced models in continuous form: a three-dimensional f-plane model for limited-area midlatitude dynamics and tropical cyclones and its generalization to spherical coordinates for global flows. Both will be based on the nonlinear balance assumption. The resulting models will be applied to explore basic dynamical principles, including available potential energy and angular momentum principles, and necessary conditions for combined instability, and wave-mean flow interactions; to study aspects of hurricane dynamics, including potential vorticity rings, secondary eyewalls, the role of asymmetries in the evolution of the mean vortex, responses to lower- and upper-level potential vorticity anomalies, and the role of small-scale fluctuations of temperature and moisture not controlled by the invertibility principle; to investigate weather-scale phenomena, including effects of horizontal shears on frontogenesis in three dimensions; and to high Rossby number turbulence, focusing on the extent to which features of quasi-geostrophic turbulence carry over to the new spherical model.
Broader impacts of the project are in developing user-friendly software for the new balanced models, which will be made available to other researchers to facilitate additional studies of atmospheric dynamics. The graduate students involved in this work will receive training which will prepare them for careers in research. Increased knowledge of hurricane dynamics may someday contribute to an improved ability to predict hurricane intensity changes. Finally, this collaborative project will foster increased interaction between the computational mathematics and atmospheric dynamics communities.
Balanced models have played a key role in advancing our understanding of atmospheric dynamics. Such a model consists of a system of equations which incorporates an assumption of approximate balance between the wind and mass fields. Our primary effort in this project was to extend the theory of balanced models, with the goal of providing better theoretical and computational tools which may to a deeper understanding of atmospheric dynamics. This work focused on four areas. First, a new computer code for the balanced vortex model was developed. Based on multigrid methods, this approach is approximately one thousand times faster than previous methods for this problem. This model can be used to study the dynamics of strong hurricanes, under the assumption that the flow is axially symmetric. Second, we showed that formulating a more general balanced model without the assumption of symmetry requires using a horizontal coordinate transformation based on vorticity (a measure of the local rotation of the fluid). We formulated an improved characterization of such coordinates and developed an efficient computer code for computing this transformation, again based on multigrid methods. Computations using the observed flow near a hurricane show that the transformation properly handles strong asymmetric flows with large curvature. Third, we studied two related types of vertical coordinates for such balanced models, namely, potential temperature coordinates and specific entropy coordinates. This work showed that with appropriate nondimensionalization, the resulting equations are similar in both formulations. Using specific entropy leads to a somewhat more natural set of equations; however, using potential temperature leads to a better treatment of the lower boundary of the model. Fourth, we explored different balance assumptions for such models. The standard nonlinear balance which we had planned to use does not in fact lead to a practical model formulation; however, we developed a more general balance assumption which may do so. Model development based on this assumption is being explored in our subsequent research project. Numerical methods and computer codes developed in this work have been applied by our collaborators to study atmospheric flows associated with the earth's topography, including flows around large mountain ranges such as the Andes. The generalized balanced model being developed will facilitate further studies of atmospheric dynamics, and the computer codes and underlying numerical methods developed will be useful tools in other areas of research.
