Abstract
Two-layer shallow water equations describe flows that consist of two layers of inviscid fluid of different (constant) densities flowing over bottom topography. Unlike the single-layer shallow water system, the two-layer one is only conditionally hyperbolic: the system loses its hyperbolicity because of the momentum exchange terms between the layers and as a result its solutions may develop instabilities. We study a three-layer approximation of the two-layer shallow water equations by introducing an intermediate layer of a small depth. We examine the hyperbolicity range of the three-layer model and demonstrate that while it still may lose hyperbolicity, the three-layer approximation may improve stability properties of the two-layer shallow water system.
Highlights
Shallow water models are widely used as a mathematical framework to study water flows in rivers and coastal areas as well as to investigate a variety of phenomena in atmospheric sciences and oceanography
Speaking, the hyperbolic region of the system (1.1) mainly depends on the difference between the velocities of the two layers: when |u1 − u2| is large, the system is not hyperbolic and one may expect appearance of Kelvin–Helmholtz-type instabilities. Another numerical challenge is related to the presence of nonconservative momentum exchange terms on the right-hand side (RHS) of (1.1)
We study a three-layer approximation of the twolayer system (1.1), which is obtained by introducing an intermediate layer of a depth hm
Summary
Shallow water models are widely used as a mathematical framework to study water flows in rivers and coastal areas as well as to investigate a variety of phenomena in atmospheric sciences and oceanography. Speaking, the hyperbolic region of the system (1.1) mainly depends on the difference between the velocities of the two layers: when |u1 − u2| is large, the system is not hyperbolic and one may expect appearance of Kelvin–Helmholtz-type instabilities Another numerical challenge is related to the presence of nonconservative momentum exchange terms on the right-hand side (RHS) of (1.1). As we demonstrate in our numerical experiments reported, even if having a “buffer” layer does not completely remove instabilities, the oscillations become much smaller This suggests that the three-layer shallow water system may be viewed as an improved version of its two-layer counterpart.
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