One class of surface water models is the shallow water models obtained by depth-averaging the microscale mass and momentum balances. Application of shallow water models to large scale problems (on the order of 1000's of km) requires the use of spherical coordinates. Traditionally, balance laws in spherical coordinates are derived by simply expanding the spatial operators in the standard depth-averaged equations. However, the equations themselves are based on an assumed planar surface so that an inconsistency exists between the derivation and the interpretation. In this article, a method is presented that properly accounts for the curvature of the Earth during the depth-averaging procedure. The derivation gives rise to new terms in both the continuity and momentum balances, terms that we refer to as curvature terms. A scaling analysis evaluates the magnitude of the terms. It is shown that the curvature term in the continuity balance is insignificant when the vertical velocity is small, i.e., at least four orders of magnitude less than lateral components. Similarly, when the vertical velocity is small, the curvature term in the momentum balance that comes from the convective terms is insignificant. The second curvature term in the momentum balance is associated with the macroscopic stress tensor; it is on the same order of magnitude as other stress terms and should be retained when momentum dissipation is modeled. Finite element discretization of the resulting balance laws utilizes the generalized wave continuity equation which, because of its monotonie dispersion relation, eliminates noise in the solution without resorting to artificial damping. A cartographic mapping simplifies finite element implementation in spherical coordinates so that Cartesian master elements can be used. Finally, an application to the Western North Atlantic and Gulf of Mexico illustrates the importance of using equations in spherical coordinates for large scale applications.
RESUMEUne categorie de modèles d'écoulement a surface libre est constitué par les modèles de Saint-Venant, obtenus par moyenne sur la verticale du débit élémentaire et des quantités de mouvement. L'application de 3 Downloaded by [Rutgers University] at 18:00 04 April 2015 tels modèles a des problèmes de grande emprise (de l'ordre du millier de Km) implique l'utilisation des coordonnées sphériques. De facon classique, les relations de bilan en coordonnées sphériques sont dérivées par un simple développement des operateurs spatiaux dans les equations moycnnées de facon classique sur la verticale. Cependant, les equations elles-mêmes sont établies dans l'hypothèse d'une surface plane, de sorte qu'il y a une contradiction entre cette derivation et ('interpretation. Cet article présente une methode qui prend en compte correctement la courbure du globe pendant la procédure d'établissement de la moyenne sur la verticale. La derivation engendre de nouveaux termes, a la fois dans les relations de continuité et de quantité de mouvement, termes qui se rapportent a la courbure. Un...
Abstract. The future reliance on water supply and flood control reservoirs across the globe will continue to expand, especially under a variable climate. Construction of additional reservoirs is less likely compared to simultaneous flow and sediment management in existing reservoirs. One aspect of this sediment management is related to the control of upstream sediment sources.However, key research questions remain regarding upstream sediment loading rates. Highlighted in this article are research needs relative to measuring sediment transport rates and loading due to streambank and gully erosion within a watershed. For example, additional in-stream sediment transport and reservoir sedimentation rate measurements are needed across a range of watershed conditions, reservoir sizes, and geographical locations. More research is needed to understand the intricate linkage between upland practices and in-stream response. A need still exists to clarify the benefit of restoration or stabilization of a small-reach within a channel system or maturing gully on total watershed sediment load. We need a better understanding of the intricate interactions between hydrological and erosion processes to improve prediction, location, and timing of streambank erosion and failure and gully formation. Also, improved process-based measurement and prediction techniques are needed that balance data requirements regarding cohesive soil erodibility and stability as compared to simpler topographic indices for gullies or stream classification systems. Such techniques will allow the research community to address the benefit of various conservation and/or stabilization practices at targeted locations within watersheds.
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