  Thin disk accretion is often modeled in highly dynamical settings using the
two-dimensional equations of viscous hydrodynamics, with viscosity representing
unresolved turbulence. These equations are supposed to arise after vertical
integration of the full three-dimensional equations of hydrodynamics, under the
assumption of a geometrically thin disk with mirror symmetry about the
midplane. But in the dynamical context, vertical dynamics are neglected by
incorrectly assuming instantaneous vertical hydrostatic equilibrium. The
resulting errors in the local disk height couple to the horizontal dynamics
through some viscosity prescriptions and gravitational softening models.
Furthermore, the viscous terms in the horizontal equations are only complete if
they are inserted after vertical integration, as if the system is actually
two-dimensional. Since turbulence breaks mirror symmetry, it is more physically
correct to insert a turbulence model at the three-dimensional level, and impose
mirror symmetry only on average. Thus, some viscous terms are usually missing.
We revisit the vertical integration procedure, restricting to the regime of a
Newtonian, non-self-gravitating disk. We obtain six evolution equations with
only horizontal dependence, which determine the local vertical position and
velocity of the disk surface, in addition to the usual fluid variables. This
"2.5-dimensional" formulation opens the door to efficiently study vertical
oscillations of thin disks in dynamical settings, and to improve the treatment
of unresolved turbulence. As a demonstration, by including viscous stress at
the three-dimensional level, we recover missing viscous terms which involve the
vertical variables. We also propose a resummation of the vertically integrated
gravitational force, which has a strikingly similar form to a gravitational
softening model advocated for in protoplanetary disk studies.
