Some of these include:

• neglect advection by $\mathbf v_a$
• neglect advection by vertical velocity
• neglect the time tendency of $\mathbf v_a$
• neglect advection of $\mathbf v_a$ by $\mathbf v_g$
• replace $f$ with the beta plane approximation ($f = f_0 + \beta y$)
• neglect friction
• horizontally constant static stability ($\sigma$)

Starting with the frictionless horizontal momentum equations you end up with a number of "Q-G" equations: a thermodynamic energy equation, a vorticity equation and the omega equation. The Q-G omega equation:

$$\left( \nabla^2_p + \dfrac{f^2_0}{\sigma}\dfrac{\partial^2}{\partial p^2} \right)\omega = - \dfrac{f_0}{\sigma} \dfrac{\partial}{\partial p} \left[ - \mathbf v_g \cdot \mathbf \nabla_p(\zeta_g + f) \right] + \dfrac{R}{\sigma p}\left[ -\nabla^2_p(- \mathbf v_g \cdot \mathbf \nabla_p T) \right]$$

The first term on the RHS vertical change in advection of geostrophic absolute vorticity by the geostrophic wind. Positive vorticity advection increasing with height results in upward vertical motion. Negative vorticity advection increasing with height results in downward vertical motion. The second term on the RHS is related to temperature advection by the geostrophic wind. Cool air advection (CAA) correlates with upward vertical motion.

This is the traditional form of the equation and other forms exist to aid in diagnosing vertical motion with specific variables (e.g. the Sutcliffe-Trenberth recasts the equation using the thermal wind and Hoskins et al. (1978) defines the equation in terms of $\vec Q$, "Q vectors").

There isn't much NWP utility to the Q-G equations with todays computers, but they are good for diagnosing vertical motion in hand map analysis.

(will add the QG chi equation here)

Further reading:

• Real-time anlyses and forecasts of the QG diagnostic equations
• The QG equations (Thanks @DrewP84)
• An abbreviated derivation