  Modeling mass flows is classically based on hydrostatic balance equations.
However, if momentum transfers scale similarly in slope parallel and flow depth
directions, then the gravity and acceleration can have the same order of
magnitude effects, urging for a non-hydrostatic model formulation. Here, we
extend existing single-phase Boussinesq-type models by developing a new
non-hydrostatic model for multi-phase mass flows consisting of solid and
fine-solid particles and viscous fluid (Pudasaini & Mergili, 2019). The new
model includes enhanced gravity and dispersion considering various interfacial
momentum transfers. We outline new contributions in the non-hydrostatic
Boussinesq-type multi-phase gravity waves emerging from phase-interactions. We
present a general, well-structured framework of multi-phase mass flows with
enhanced gravity and dispersion, setting a foundation for comprehensive
simulations. We discuss situations where non-hydrostatic, dispersive effects
are more pronounced for such flows. Reduced models demonstrate the importance
of non-hydrostatic contributions. Simplified analytical solutions reveal how
the new dispersive model can be reduced to non-dispersive ones, yet largely
generalizing existing models. We postulate a novel, spatially varying
dissipative force, called the prime-force, which physically controls the
dynamics and run-out of the mass flow in a precise way. Practitioners and
engineers may find this force very useful in technical applications. This
illuminates the need of formally including the prime-force in momentum
equations. A simple dispersion equation is derived highlighting the essence of
dispersion on mass flow dynamics. Dispersion produces a wavy velocity field
about a reference state without dispersion, the first of this kind for
avalanching debris mass. Dispersion intensity increases energetically as the
solid volume fraction or friction decreases.
