Inclusion of the buoyancy forces in the Incompressible Schrödinger Flow algorithm to simulate inviscid fluids

In 1927, Erwin Madelung developed an analogy between hydrodynamics and the quantum mechanics world by linking wave functions and the velocity of inviscid fluids by a suitable coordinate transformation; in particular, a wave function satisfying a Schrödinger equation results in an associated velocity field compliant with the Euler equations for inviscid flows. Thus, Euler fluids can be numerically simulated with this method, lowering the computational complexity associated with the direct solution of the Euler equations through the Madelung transform under the name of Incompressible Schrödinger Flow (ISF). This work extends the framework of ISF implemented by the authors within a Finite Element framework to adopt the velocity to the advection-diffusion of passive scalars (such as temperature or concentration) and to include, at the 'quantum' level, buoyancy forces in the Schrödinger equation, assuming a Boussinesq-like approximation. This work sets up a simple case study to investigate the relation between wave function and velocity, showing how wave properties reflect in the fluid dynamics world and how temperature gradients can influence the wave functions through linear potential terms, which correspond to buoyancy forces.