Abstract
The rotation of the earth influences both atmospheric and oceanic flows. In fact, the fast rotation and small aspect ratio are two main characteristics of large-scale atmospheric and oceanic flows. The small aspect ratio characteristic leads to primitive equations, and the fast rotation leads to quasi-geostrophic equations. A main objective in climate dynamics and in geophysical fluid dynamics is to understand and predict the periodic, quasi-periodic, aperiodic, and fully turbulent characteristics of large-scale atmospheric and oceanic flows. The general model of atmospheric and oceanic flows is very complicated. In this chapter, we study a simplified model of dynamic convection in a sea due to Ovsiannikov. First we perform a symmetry analysis on the equations. Then we use a new moving-line variable to solve the equations. An approach that uses the product of the cylindrical invariant function with the vertical variable is introduced. Finally, we reduce the three-dimensional (spatial) equations into a two-dimensional problem and then solve it using three different ansatzes (assumptions).
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Xu, X. (2013). Dynamic Convection in a Sea. In: Algebraic Approaches to Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36874-5_7
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DOI: https://doi.org/10.1007/978-3-642-36874-5_7
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