18 October 1995
A. Given T and T_{d}:
Suppose we are given values of the temperature, T, and the dewpoint, T_{d}, at some set of specified height levels. We have the following relationships. The Hydrostatic Equation is:
where p denotes pressure, r denotes density, z denotes height, and g is the acceleration due to gravity. The Equation of State for moist air is:
where R_{d} is the gas constant for dry air and T_{v} is the virtual temperature, defined by
where e is a constant (= 0.6217 for w in grams per gram) and w is the mixing ratio found from
where e is the vapor pressure, given by
where T is given in deg C, and this last relationship is more or less empirical.
For the record, if we knew e_{s} somehow and we wished to know T_{d} (in deg C), we would have to invert this empirical relationship, to find that
Also, if we have w and wish to know e, then we invert the equation for w to find
Returning to our problem, which is to find the pressure at the given height levels, one needs the density. To know the density, one must have the virtual temperature. To know the virtual temperature, one must have the mixing ratio. To know the mixing ratio, one must have the vapor pressure and the pressure. The relationship between pressure and the input variables (T,T_{d}) appears to be implicit, thereby requiring iteration. However, in fact, it can be solved approximately as a differential equation, in what will turn out to be a familiarlooking way.
We begin with the Hydrostatic Equation combined with the Equation of State:
Substitution from the definition of virtual temperature gives
Next, we substitute for the mixing ratio
Now, consider the term in brackets:
Substitution into the preceding result gives
This simplifies to
Therefore, the differential equation can be written in two terms:
The first term is a simple computation of the height change associated with the temperature without a virtual correction, and the second term is a correction term. Integration of the first term (replacing "d" with "d" where d denotes a finite difference) gives a familiar expression (recall the Hypsometric Equation):
or, put another way:
where the overbar denotes an average of T over the interval dz. In doing this "integration," the value of p used is that at the bottom of the zlayer. In principle, one could refine this via an iteration process (in effect confirming the implicit nature of the relationship), using the new top value to find an average value for p in the layer and recalculating the correction using the average value, and thereafter iterating with the average. I have tested this using an "artificial sounding" with dz increments of 50 m ... such an exercise may be fun for the reader to try ... and the correction made to dp converges quite rapidly to zero, as can be seen in the following table.
































The difference in dp after the first iteration is of order 0.01mb, and the difference between the inital value and the final value doesn't change much with more iterations. The answer to which the iteration converges differs only by slightly less than 0.3% from the original value (0.0153415 divided by 5.54756), so the correction should be virtually negligible for many practical purposes. If really accurate solutions are desired, then this correction might be worthwhile. Note that the accuracy should increase as one goes upward, because p will be decreasing and, hence, its relative contribution to (dp)_{1} is decreasing.
The second term accounting for the contribution of water vapor looks a bit different:
This second order correction does not require a pvalue and so, presumably, does not require iteration. The values turn out to be quite small ... of order 0.10.01 mb, even when using a relatively large moisture content. Simple orderofmagnitude arguments should convince the reader that this is a small correction.
Thus, one can find the pressure from the input values numerically, assuming that the simple average gives a good representation to the integrals of T and e_{s} over the height interval; if these functions are linear in z over the height interval, then the average is exact (recall the Mean Value Theorem). The pressure is found by starting at the bottom and working up, assuming that the pressure at the lowest level is given. At the i^{th} pressure level, then
Note that the averaging is between levels (i) and (i1).
B. Given T and w:
Now suppose that rather than the dewpoint, one is given the mixing ratio, w. In this case, we return to
This can be used to find T_{v} directly, and the problem them becomes the much easier one of
or put in the same way as (1) above:
with no correction term needed. As with (1), since a pvalue is needed, the use of the value at the bottom of the layer is not quite appropriate, and iteration could refine this somewhat. Thus, at the i^{th} pressure level,
C. Given q and w:
As a final kicker, suppose we are given q and w, where we have the definition of potential temperature:
with p_{o} usually selected to be 1000 mb. In this case, we have to back out the pressure in a quite different way. First of all, we use the obvious relationship
in the Equation of State, so the Hydrostatic Equation now reads
Note that we have defined q_{v} analogously to T_{v}:
Now, doing a little rearranging of terms, our hydrostatic equation reads
Hence, our solution looks like
At the risk of repetition, a pvalue is needed here, so using the value at the bottom of the layer is only an approximation. The quality of that approximation is considered in Table 2, below.
































In comparison with the results using formula (1), Table 2 shows that the convergence is even more rapid than shown in Table 1. The convergence is virtually complete after the second iteration. Obviously, this confirms the notion that iteration is not a very useful improvement, especially for this type of initial data specification (q and w); the change is only about 0.1% (0.0053333 divided by 5.46827066) in this case. The reason that this calculation is slightly more accurate even with the bottom pressure than that using (1) is because it depends on p^{1k} instead of p. At the i^{th} pressure level, one has what should be by now a familiar expression: