2. Relevant Equations

The relevant equations for performing measurements on a fast-moving research aircraft in a dry atmosphere are well documented. The applicability of these equations to a moist atmosphere is examined and confirmed in this section. Finally, the specific equations for downlinking water vapor mixing ratio information from a relative humidity sensor are provided. The total temperature (TT) measured on an aircraft is equal to the static temperature (Ts) or ambient temperature plus the dynamic effects of the moving aircraft. Therefore, taking into account the internal energy and kinetic energy (of the moving aircraft) we have:
          (1)


where Cp = specific heat at constant pressure = 1004 m2 s-2 K-1 and where T is always in degrees Kelvin.

Eq. (1) can be reduced as follows:

but V = MC where M is the Mach number and C = velocity of sound = (gRTs)1/2
where R = gas constant for dry air = 287 m2 s-2 K-1
one can divide the above expression for TT by Ts and obtain:
           (2)
One can deduce the ratio of dynamic pressure to static pressure by the above and from the Poisson equation (where PT is total pressure, static plus dynamic, and Ps is static pressure) and obtain
          (3)
In the real world, the actual measured or recovered measurement of a quantity on a moving aircraft is related to a recovery factor defined as
where re = recovery factor, Tm = temperature measured or recovered, TT = total temperature, and TS= static temperature.

It can be shown that Eq. (2) then becomes

         (4)
BFG, manufacturer of most of the world's aircraft temperature probes, determined the recovery factor to be greater than 0.9987 for speeds up to Mach number =0.6 and about 0.998 for Mach number = 0.8 (close to the maximum speed of most existing commercial aircraft). Therefore, re is assumed to be 1 for the remainder of this paper.

The above discussion was for a dry atmosphere. Are these equations valid for a moist atmosphere? This is always assumed to be true, but not having found a direct
reference and wishing to keep this report short, we guide the interested reader to his own proof as follows. The equation of state for moist unsaturated air is

pawa = RwaT
where awa and Rwa are the specific volume and the gas constant of moist, unsaturated air respectively. [Note that here and in the following, we have modified Fleagle and Businger's (1963) subscript notation to match our previous definitions for R, Cp and Cv.]

It can easily be shown (e.g., Fleagle and Businger, 1963) that

Rwa = (1 + 0.61q)R»(1 + 0.61r)R
Cpwa = (1 + 0.90q)Cp»(1 + 0.90r)Cp
Cvwa = (1 + 1.02q)Cv»(1 + 1.02r)Cv
where q is specific humidity (mass of water per unit mass of air) and r is the mixing ratio (mass of water per unit mass of dry air.) In order to have the above equation of state contain only three variables rather than four, one can combine the variability of temperature and water vapor with the virtual temperature defined as
T* = (1 + 0.61q)T » (1 + 0.61r)T
Working back through Eqs. (1) through (3) with these definitions, and assuming a value of r of 10 g/kg (far wetter than expected at flight level), one finds changes in TT and PT to be less than 0.1%. One can conclude that the dry atmosphere assumption holds for the dynamic heating effects on temperature and pressure due to the speed of the aircraft. Thus, in the following sections when referring to measurements within the probe of the moving aircraft, we will be using Tprobe (TP) and Pprobe (PP) as defined in Eqs. (2) and (3) with TP and PP replacing TT and PT respectively.

The most common form of relating the amount of moisture in the air is via the relative humidity (RH). This is defined (with respect to water, per the World Meteorological Organization) as:

RH = (e/es) 100           (5)
where RH is a percent, e is the atmospheric vapor pressure (Pascals), es is the saturation vapor pressure with respect to water (Pascals); es is defined with respect to water by Fan and Whiting (1987) as
es = 10[(10.286T - 2148.909) / (T - 35.85)]           (6)
A form of measurement of the water vapor content in the atmosphere (used by most meteorological prediction models) is the mixing ratio (mass of water to mass of dry air).
r = 0.62197e / (P - e)           (7)
Another measure of the amount of water vapor in the atmosphere is the dewpoint (used by meteorologists in plotting temperature and water as a function of height). Since the definition of dewpoint is the temperature at which the atmosphere is saturated with respect to water, one can invert Eq. (6) for T and replace es with e. Thus,
Tdew = (-2148.909 + 35.85 log10 es)/(log10 es - 10.286)           (8)
The WVSS provided by LMC measures RH via a Vaisala thin-film capacitor and downlinks mixing ratio (Hills and Fleming, 1994 described why this should be more accurate than downlinking the converted static RH).

One can convert RH to mixing ratio from its definition giving

          (9)
Now since r is conserved, it can be calculated from static values (RH, T, P)static or from dynamic values in the aircraft probe (RH, T, P)probe. For the WVSS the dynamic values are used, thus we have
          (10)
In the WVSS probe, the RH and T (thus es) are measured next to each other. However, the dynamic pressure in the probe at the choke point (PP) is not measured in the probe but deduced from the total pressure measured elsewhere on the aircraft.

However, LMC and BFG find that the temperature (pressure) is not quite adiabatic in the choke area of the probe (that is, TP is slightly less than the total temperature and PP slightly less than the total pressure PT). Therefore, they have found in wind tunnel tests that

PP = PT (0.9992 + 0.0027M - 0.0144M2 )           (11)
This final minor correction is used in determining the final downlinked mixing ratio (r) in Eq. (10). There is one final very important equation derived in Hills and Fleming (1994) that must be recognized in assessing the calibration of the WVSS.

Since

it can be shown that this plus the definition of RH leads to
          (12)
The impact of Eq. (12) is primarily felt at "flight level" where Mach number is high and it has relatively little impact on "ascent" and "descent" where Mach numbers are much lower. Figure 1 is a plot of Eq. (12) where the ratio of RHstatic to RHprobe is shown as a function of Mach number and temperature. One can see from the figure that for high Mach numbers and very cold temperatures, this ratio becomes substantial. This "Mach effect" is due to the highly nonlinear nature of Eq. (6) and the effects of dynamic heating through Eqs. (2) and (3). A numerical example of this ratio is shown below. Consider an aircraft traveling at Mach=0.8 with an outside air temperature of Ts = -60°C (213.15 K). Eq. (2) gives the temperature in the probe as 240.4 K and Eq. (3) gives Pp/Ps as 1/1.524. Eq. (6) gives es,s(213.15) = 1.76 and es,p (240.4) = 38.41. From Eq. (12) we have:
Rhstatic/RHprobe = (38.41/1.76) / (1.524)
                          = 14.32