The shallow water equations are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid, crucial for modeling weather events, ocean currents, and inundation due to tsunamis. These equations assume the horizontal length scale is much larger than the vertical scale, allowing for simplifications that make them essential in geophysical fluid dynamics applications.
The continuity equation is a fundamental principle in fluid dynamics that expresses the conservation of mass in a fluid flow system. It states that the mass flow rate of a fluid must remain constant from one cross-section of a pipe to another, assuming steady flow and incompressibility of the fluid.
Hydrodynamic models are mathematical frameworks used to simulate the movement of water and its interactions with natural and human-made environments. They play a critical role in understanding fluid dynamics, predicting weather patterns, and managing water resources efficiently.