Conservation properties of smoothed particle hydrodynamics applied to the shallow water equations

Conservation of potential vorticity (PV) and Kelvin's circulation theorem express two of the fundamental concepts in ideal fluid dynamics. In this note, we discuss these two concepts in the context of the Smoothed Particle Hydrodynamics (SPH) method. We show that, if interpreted in an appropriate way, one can make a statement about conservation of circulation and PV in a particle fluid model such as SPH. We also indicate some limitations where the analogy with ideal fluid dynamics breaks down.