  In this paper an SPH version of the alpha turbulence model devised by Holm
and his colleagues is formulated for compressible flow with a resolution that
varies in space and time. The alpha model involves two velocity fields. One
velocity field is obtained from the momentum equation, the other by averaging
this velocity field as in the version of SPH called XSPH. The particles (fluid
elements) are moved with the averaged velocity. In analogy to the continuum
alpha model we obtain a particle Lagrangian from which the SPH alpha equations
can be derived. The system satisfies a discrete Kelvin circulation theorem
identical to that obtained with no velocity averaging. In addition, the energy,
linear and angular momentum are conserved. We show that the continuum
equivalent of the SPH equations are identical to the continuum alpha model, and
we conjecture that they will have the same desirable features of the continuum
modelincluding the reduction of energy in the high wave number modes even when
the dissipation is zero. Regardless of issues concerning turbulence modelling,
the SPH alpha model is a powerful extension of the XSPH algorithm which reduces
disorder at short length scales and retains the constants of the motion. The
SPH alpha model is simple to implement.
