It is important to assess the conditions under which it is valid to set . Now the tracer terms are, of course, fictitious and the act of setting them to zero is only valid if the terms in which they appear are small relative to those that appear in the exact equations. As we have seen, setting and equal to zero results in a solution for that has exactly the same vertical structure to that of the
exact, linear solution. Clearly, errors resulting from the omission of the terms associated with and (i.e. in the continuity equation and in the vertical momentum equation) will be found elsewhere. The influence of the term traced by in eq.(27) is only small provided that:
If represents a characteristic depth scale for the motion then the above inequality can be expressed as and only then will the pressure perturbation be accurately modelled by this approximated set. We also require that the error in be small for the accurate representation of the dependent variables (other than ). This will be achieved provided that the term in the dispersion equation (37) is small compared to the last term . Using the a posteriori estimate this condition may be expressed as:
which becomes
or,
where is the phase speed of inertia-gravity wave ( ). Therefore, obtained from the approximated set (with ) with be accurate provided that the phase speed of the wave mode is much smaller than the speed of sound. In fact, the term above has already been proven to be small since it is identical (apart from a factor of 4) to expression (35).
For all but the deepest scales of atmospheric motions (e.g. vertically-propagating planetary Rossby waves) the condition is well-satisfied. Similarly, the phase speed of meteorologically-relevant waves is much less than the speed of sound and so setting is justified. Setting suppresses the time-derivative term in eq.(13) but leaves the advective term }.