The instruments used to collect the measurements that lead to the variables in this section are described on the EOL web site, in the “State Parameters” section at this URL. The data acquisition and processing for these variables and the calibration coefficients used where applicable are described.
| Character | location |
|---|---|
| B | bottom (or bottom-most) |
| B | (obsolete) boom |
| F | fuselage |
| G | (obsolete) gust probe |
| R | radome |
| T | top (or top-most) |
| W | wing |
In addition, a true letter ’X’ (not replaced by the above letters) may be appended to a measurement to indicate that it is the preferred choice among similar measurements and is therefore used to calculate derived variables that depend on the measured quantity. Other suffixes sometimes used to distinguish among measurements are these: ’D’ for a digital sensor; ’H’ for a heated (usually, anti-iced) sensor, ’L’ for port-side sensors, and ’R’ for starboard-side sensors.
The atmospheric pressure at the flight level of the aircraft, measured by a calibrated absolute (barometric) transducer at location x. PSx is the measured static or ambient pressure before correction, and it may be affected by local flow-field distortion. PS_A is the pressure measurement taken from the avionics system on the aircraft, processed via unknown algorithms in the avionics system that may smooth, correct, and perhaps delay the result. PSxC is PSx corrected for local flow-field distortion (see RAF Bulletin #21 and the discussion in this memo and this supplement), and PSXC is the preferred corrected measurement used for derived calculations. These measurements have been made using various sensors, so it is best to consult the project documentation for the transducer used. Recent measurements from both the C-130 and the GV have been made using a Paroscientific Model 1000 Digiquartz Transducer.
Corrections to the pressures have been determined by reference to some standard, including a “trailing cone” sensor, the pressure PS_A from the cockpit avionics system, or (since 2012) the Laser Air Motion Sensing System (LAMS). The latter correction is discussed in the memo referenced above, where corrections used prior to 2011 are also discussed. Beginning in 2012, the deduced corrections Δp to the measured pressures as functions of dynamic pressure q, angle of attack α,12 and the Mach number M are described by the following equations and coefficients:
For the C-130,13 \[\begin{equation} \frac{\Delta p}{p_{m}}=d_{0}+\frac{q_{m}}{p_{m}}(d_{1}+d_{4}\frac{\alpha^{2}}{a_{r}^{2}})+d_{2}\frac{\alpha}{a_{r}}+d_{3}\thinspace M \tag{4.1} \end{equation}\]
where, for pm = PSFD , qm = dynamic pressure (QCF), α = ATTACK and ar = 1∘ (included to keep the equation and coefficients dimensionless), and {d0, d1, d2, d3, d4} = {-4.389e-03, -2.966e-02, -6.831e-05, 2.672e-02, 2.4466e-03}. For PSFRD, the corresponding coefficients are {0.007372, 0.12774, −6.8776e-4, −0.02994, 0.001630}. The latter coefficients are significantly different from the coefficients for PSFD, but the static ports where PSFRD is measured are at a different location on the fuselage so different flow-distortion effects are expected.
For the GV,14 \[\begin{equation} \frac{\Delta p}{p}=a_{0}+a_{1}\frac{q}{p}+a_{2}M^{3}+a_{3}\frac{\alpha}{a_{r}} \tag{4.2} \end{equation}\]
where, for p = PSF, q = QCF, α = ATTACK, and ar = 1∘ (included to keep the equation and coefficients dimensionless) {a0, a1, a2, a3} = { − 0.012255, 0.075372, − 0.087508, 0.002148}}.
In equations (4.1) and (4.2) the Mach number is calculated from the uncorrected measurements of p and q, using dry-air values for R, cv and cp, via
\[\begin{equation}
M=\left\{ \left(\frac{2c_{v}}{R}\right)\left[\left(\frac{p+q}{p}\right)^{R/c_{p}}-1\right]\right\}^{1/2}\,\,\,.
\tag{4.3}
\end{equation}\]
The pressure PSTF is measured at the static-pressure port of a pitot-static tube mounted on the top of the fuselage. The correction to this pressure (leading to PSTFC) differs from the pressure corrections listed above in that it is based on an empirical fit to other measurements of static pressure. The airflow at that sensor is distorted by the fuselage, so that measurement is usually less reliable than other measures of static pressure. See this note for additional discussion of this measurement.
For additional information on these correction coefficients, see this note and Cooper et al. (2014).15
The pressure excess caused by bringing the airflow to rest relative to the aircraft. These quantities represent the difference between the total pressure pt as measured at the inlet of a pitot tube or other forward-pointing port and the ambient pressure that would be present in the absence of motion through the air. The variable QC_A is provided by the avionic data system and is subject to an unknown delay and smoothing. The variables ending in “C” have been corrected for flow-distortion effects, mostly arising from errors in the measurement of static pressure. Since 2012, the corrections are based on measurements from the LAMS system as described for PSxC, and they have the same functional form as in (4.1) and (4.2) except that the correction applied to q is − Δp with reversed sign because q = pt − pa and the error arises primarily from the error in pa. The same correction is applied to QCR because it is also measured relative to the static pressure ports so errors in the pressure sensed at those ports affect QCR in the same way that QCF is affected. See the notes referenced in the preceding section, and also RAF Bulletin 21 for the corrections applied to earlier data files.16
A Rosemount Model 1221 differential pressure transducer is used for current measurements of dynamic pressure on the C-130, and a Honeywell PPT transducer is used on the GV. This measurement enters the calculation of true airspeed and Mach number and so is needed to calculate many derived variables.
In the case of QCRC from the GV, one additional correction is applied (beginning 2017). The uncorrected measurement QCR is affected by flow angles, while QCF is not (for modest angle of attack or sideslip), so an additional adjustment is needed. The needed correction can be found by using an empirical relationship matching QCR to QCF, which leads to the following equation:
\[\begin{equation}
\mathrm{\{QCR\}}=b_0+b_1\mathrm{\{QCR\}}+b_2\mathrm{\{AKRD\}}^2+b_3\mathrm{\{SSRD\}}^2-\Delta p
\tag{4.4}
\end{equation}\]
where \(\Delta p\) is given by (4.2) and the coefficients are {b0 − 3}={ − 0.5635, 0.9982, 0.0273, 0.0562}. Some justification for this correction is contained in this note. A similar correction is not made for measurements from the C-130 radome because they do not appear to be necessary, as discussed in that note.
The measurement QCTF is made at the dynamic-pressure port of a pitot-static sensor mounted on the top of the fuselage. Because this is located in a region of airflow distortion around the fuselage of the GV, special processing is required. The method proposed in this note is to calculate a corrected dynamic pressure from QCTFC=QCTF+PSTF-PSXC. For high-rate files, PSXC is low-pass-filtered with a cut-off frequency of 0.5 Hz to eliminate high-frequency noise in this measurement.
PSXC = ambient pressure [hPa]
HGME = (radar) altitude above the surface [m]
TVIR = virtual temperature [\(^{\circ}\mathrm{C}\)]
PSURF = estimated surface pressure [hPa]
\(g\) = acceleration of gravity\(^{\dagger}\)
\(R_{d}\) = gas constant for dry air\(^{\dagger}\)
\(c_{pd}\) = specific heat of dry air at constant pressure\(^{\dagger}\)
\[\begin{equation} T_{m}=(\mathrm{\{TVIR\}}+T_{0})+0.5\mathrm{\{HGM\}}\frac{g}{c_{pd}} \tag{4.5} \end{equation}\] \[\begin{equation} \mathrm{PSURF}=\mathrm{\{PSXC\}}\,\exp\left\{ \frac{g\,\{\mathrm{HGME}\}}{R_{d}T_{m}}\right\} \tag{4.6} \end{equation}\]
Moist-Air Considerations
Primes on the symbols denote that these values should be moist-air
values, appropriately weighted averages of the dry-air and water-vapor
contributions. The practice prior to 2014 was to use the dry-air values
for specific heats and the gas constant, except as described in connection
with TASHC below. Since 2014, calculations use the appropriate values
for moist air, except that to avoid errors introduced by unrealistically
high measurements of humidity the humidity correction was limited
to be less than or equal to the equilibrium value at the measured
temperature. The formulas used for the specific heats and gas constant
of moist air in terms of the water vapor pressure \(e\), the specific
heats for dry air (\(c_{pd}=\frac{7}{2}R_{0},\,c_{vd}=\frac{5}{2}R_{0}\))
and water vapor (\(c_{pw}=4R_{0},\,c_{vw}=3R_{0}\)), and the ratio
of molecular weights (\(\epsilon=M_{W}/M_{d}\)) are those of Khelif
et al. 1999:
\[\begin{equation}
R^{\prime}=R_{d}/[1+(\epsilon-1)\frac{e}{p}]
\end{equation}\]
\[\begin{equation}
c_{v}^{\prime} = \frac{(p-e)R^{\prime}}{pR_{d}}\frac{5R_{0}}{2M_{d}}+\frac{eR^{\prime}}{pR_{w}}\frac{3R_{0}}{M_{w}}=c_{vd}\frac{R^{\prime}}{R_{d}}\left(1+\frac{1}{5}\frac{e}{p}\right)
\tag{4.9}
\end{equation}\]
\[\begin{equation}
c_{p}^{\prime} = c_{pd}\frac{R^{\prime}}{R_{d}}\left(1+\frac{1}{7}\frac{e}{p}\right)
\tag{4.10}
\end{equation}\]
\[\begin{equation}
\gamma\,^{\prime} = \gamma_{d}\frac{1+\frac{1}{7}\frac{e}{p}}{1+\frac{1}{5}\frac{e}{p}}
\tag{4.11}
\end{equation}\]
See also the discussion of TASHC below
and the reference there for Khelif et al. 1999.
In (4.8) {Ua, Ur, Ut} are respectively the aircraft true airspeed, the airspeed relative to the aircraft of the air in thermal contact with the sensor, and the airspeed of air relative to the aircraft when fully brought to the motion of the sensor (i.e., zero). Then, from (4.8) \[\begin{equation}
T_{a}=T_{r}-\alpha_{r}\frac{U_{a}^{2}}{2c_{p}^{\prime}}
\tag{4.12}
\end{equation}\]
The temperature sensors used on RAF aircraft are designed to decelerate the air adiabatically to near zero velocity. Recovery factors determined from wind tunnel testing for the Rosemount sensors are approximately 0.97 (unheated model) and 0.98 (heated models).20
These values have also been confirmed from flight maneuvers, often from “speed runs” where the aircraft is flown level through its speed range and the variation of recovery temperature with airspeed is used with (4.12), with the assumption that Ta remains constant, to determine the recovery factor. Data files and project reports normally document what recovery factor was used for calculating the true airspeed and ambient temperature for a particular project.
Because the values used in processing have varied, the project reports should be consulted to find what was used for particular projects. The Goodrich Technical Report 5755 documents wind-tunnel testing of the probes formerly made by Rosemount. Their plot showed that, for heated sensors, there is a significant variation with Mach number (M); cf their Eq. (38). The dependence in their plot is represented well by the following equations, where \(\alpha_r^{[h]}\) refers to heated probes and \(\alpha_r^{[u]}\) to unheated probes:
\[\begin{equation}
\alpha_r^{[h]} = 0.988+0.053(\log_{10}M)+0.090(\log_{10}M)^2+0.091(\log_{10}M)^3
\tag{4.13}
\end{equation}\]
\[\begin{equation}
\alpha_r^{[u]}=0.9959+0.0283(\log_{10}M)+0.0374(\log_{10}M)^2+0.0762(\log_{10}M)^3
\tag{4.14}
\end{equation}\]
Some studies of the recovery factor are discussed further in this memo and this technical report.
The true airspeed \(U_a\) is used in (4.12) to calculate the ambient temperature \(T_a\). However, the ambient temperature is also needed to calculate the true airspeed. Therefore the constraints imposed on ambient temperature and true airspeed by the measurements of recovery temperature, total pressure (the pressure measured by a pitot tube pointed into the airstream and assumed to be that obtained when the incoming air is brought to rest relative to the aircraft), and ambient pressure must be used to solve simultaneously for the two unknowns, temperature and airspeed.
The relationship is conveniently derived by first calculating the dimensionless Mach number \(M\), which is the ratio of the airspeed to the speed of sound \(U_{s}=\sqrt{\gamma^{\prime}R^{\prime}T_{a}}\), where \(\gamma^\prime\) is the ratio of specific heats of (moist) air, \(c_p^\prime /c_v^\prime\) and \(R^\prime\) is the gas constant for moist air. The Mach number is a function of air temperature only and can be determined as follows:
a). Express energy conservation, as in (4.12), in the form
\[\begin{equation}
d\left(\frac{U^{2}}{2}\right)+c_{p}^{\prime}dT=0\,\,\,\,.
\tag{4.15}
\end{equation}\]
where the total derivatives apply along a streamline as \(U\) changes from \(U_a\) to \(U_t=0\) and \(T\) changes from \(T_a\) to \(T_t\).
b). Use the perfect gas law to replace \(dT\) with \(\frac{pV}{nR}(\frac{dV}{V}+\frac{dp}{p})\) where \(V\) and \(p\) are the volume and pressure of a parcel of air. Then use the expression for adiabatic compression in the form \(pV^\gamma = \mathrm{constant}\) to replace the derivative \(\frac{dV}{V}\) with \(-\frac{1}{\gamma}\frac{dp}{p}\), leading to \(dT=\frac{R^{\prime}T}{c_{p}^{\prime}}\frac{dp}{p}\) or, after integration, \(T(p)=T_{a}\left(\frac{p}{p_{a}}\right)^{R^{\prime}/c_{p}^{\prime}}\). Using this expression for \(T\) in the formula for \(dT\) and then integrating both total derivatives in (4.15) along the streamline leads to
\[\begin{equation}
\frac{U_{a}^{2}}{2}+c_{p}^{\prime}T_{a}=c_{p}^{\prime}T_{a}\left(\frac{p_{t}}{p_{a}}\right)^{\frac{R^{\prime}}{c_{p}^{\prime}}}
\tag{4.16}
\end{equation}\]
where \(p_t\) is the total pressure (i.e., PSXC+QCXC) and
\(p_a\) is the ambient pressure (PSXC).
c). Use the above definition of the Mach number \(M\) (\(M=U_a/U_s\)) in the form
\(U_a^2=\gamma^\prime M^2 R^\prime T_a\) to obtain:
\[\begin{equation}
M^{2}=\left(\frac{2c_{v}^{\prime}}{R^{\prime}}\right)\left[\left(\frac{p_{t}}{p_{a}}\right)^{\frac{R^{\prime}}{c_{p}^{\prime}}}-1\right]
\tag{4.17}
\end{equation}\]
which is the same as (4.3). This equation shows that \(M\) can be found from measurements of \(p_t\) and \(p_a\) alone, except for the moist-air corrections.
d). Use the expression for ambient temperature in terms of recovery temperature and airspeed, (4.12), to obtain the temperature in terms of the Mach number and the recovery temperature:
\[\begin{align}
T_{a} & = T_{r}-\alpha_{r}\frac{U_{a}^{2}}{2c_{p}^{\prime}}=T_{r}-\alpha_{r}\frac{M^{2}\gamma^{\prime}R^{\prime}T_{a}}{2c_{p}^{\prime}}\notag\\
& = \frac{T_{r}}{1+\dfrac{\alpha_{r}M^{2}R^{\prime}}{2c_{v}^{\prime}}}
\tag{4.18}
\end{align}\]
e). Express the true airspeed \(U_a\) as
\[\begin{equation}
U_{a}=M\sqrt{\gamma\,^{\prime}R^{\prime}T_{a}}
\tag{4.19}
\end{equation}\]
RTX = recovery temperature (\(T_{r})\)
QCxC = dynamic pressure, corrected (\(q_{a}\))
PSXC = ambient pressure, after airflow/location correction
(\(p_{a}\))
MACHx = Mach number
based on QCxC and PSXC; cf. (4.17)
MACHX = best Mach number, based on QCXC and PSXC
\(\alpha_{r}\) = recovery factor for the particular temperature sensor
\(R^{\prime}\), \(c_{v}^{\prime}\) and \(c_{p}^{\prime}\) as defined
above and in the list of symbols
From (4.17),
\[\begin{equation}
\mathrm{MACHx}=\left\{ \left(\frac{2c_{v}^{\prime}}{R^{\prime}}\right)\left[\left(\frac{\mathrm{\{PSXC\}+\{QCxC\}}}{\mathrm{\{PSXC\}}}\right)^{\frac{R^{\prime}}{c_{p}^{\prime}}}-1\right]\right\} ^{1/2}
\tag{4.20}
\end{equation}\]
From (4.18),
\[\begin{equation}
\mathrm{ATx}=\frac{\mathrm{\left(\{RTx\}+T_{0}\right)}}{\left(1+\dfrac{\alpha_{r}\mathrm{(\{MACHX\})}^{2}R^{\prime}}{2c_{v}^{\prime}}\right)}-T_{0}
\tag{4.21}
\end{equation}\]
The dew point obtained from the original measurement after correction for the housing pressure, the enhancement of the equilibrium vapor pressure arising from the total pressure (discussed below), and conversion from frost point if appropriate, The result is the temperature at which the equilibrium vapor pressure over a plane water surface in the absence of other gases would match the actual water-vapor pressure. Dew/frost-point hygrometers measure the equilibrium point in the presence of air, and the presence of air affects the measurement in a minor way that is represented by a small correction here named the “enhancement factor.” In the case where the dew-point or frost-point sensor is exposed to ambient air directly, the enhancement factor is defined so that the ambient water vapor pressure \(e_a\) is related to \(T_p\), the measured dew or frost point in the presence of air having total pressure \(p\), by \(e_a=f(p,T_P)e_s(T_p)\) where \(e_s(T_p)\) is the vapor pressure in equilibrium with ice or water at the dew or frost point \(T_p\) in the absence of air. Calculation of DPxC removes this dependence, so the vapor pressure obtained from \(e_s(\{DPxC\})\) will be that vapor pressure corresponding to equilibrium in the absence of air. In addition, if the measurement is below \(0^\circ\mathrm{C}\), it is assumed to be a measurement of frost point and a corresponding dew point is calculated from the measurement (also with correction for the influence of the total pressure on the measurement). Some changes were made to these calculations in 2011; for more information, see this memo.
An additional correction is needed in those cases where the pressure in the housing of the instrument (measured as PSDPx or CAVP_x) differs from the ambient pressure, because the changed pressure affects the partial pressure of water vapor in proportion to the change in total pressure and so changes the measured dew point from the desired quantity (that in the ambient air) to that in the housing. This is especially important in the case of the GV because the potential effect increases with airspeed. If the pressure in the housing is measured or otherwise known (e.g., from correlations with other measurements), then this correction can be introduced into the processing algorithm at the same time that the correction for the presence of dry air is introduced, and the enhancement factor should be evaluated at the pressure in the housing.
The relationship between water-vapor pressure and dew- or frost-point temperature is based on the Murphy and Koop23
(2005) equations.24
They express the equilibrium vapor pressure as a function of frost point or dew point and at a total air pressure \(p\) via equations that are equivalent to the following:
\[\begin{equation}
e_{s,i}(T_{FP})= b_{0}^{\prime}\exp(b_{1}\frac{(T_{0}-T_{FP})}{T_{0}T_{FP}}+b_{2}\ln(\frac{T_{FP}}{T_{0}})+b_{3}(T_{FP}-T_{0}))
\tag{4.22}
\end{equation}\]
\[\begin{equation}
e_{s,w}(T_{DP})=c_{0}\exp\left((\alpha-1)c_{6}+d_{2}(\frac{T_{0}-T_{DP}}{T_{DP}T_{0}})\right)+d_{3}\ln(\frac{T_{DP}}{T_{0}})+d_{4}(T_{DP}-T_{0})
\tag{4.23}
\end{equation}\]
\[\begin{equation}
f(p, T_P)= 1 + p(f_1 + f_2T_P + f_3T_P^2)
\tag{4.24}
\end{equation}\]
where \(e\) is the water vapor pressure, \(T_{FP}\) or \(T_{DP}\) is the frost or dew point, respectively, expressed in kelvin, \(T_0\) = 273.15 K, \(e_{s,i}(T_{FP})\) is the equilibrium vapor pressure over a plane ice surface at the temperature \(T_{FP}\), \(e_{s,w}(T_{DP})\) is the equilibrium vapor pressure over a plane water surface at the temperature \(T_{DP}\) (above or below \(T_0\)), and \(f(p,T_P)\) is the enhancement factor at total air pressure \(p\) and temperature \(T_P\), with \(T_P\) equal to \(T_{DP}−T_0\) when above \(T_0\) and \(T_{FP}−T_0\) when below \(0^\circ\)C.
The coefficients used in the above formulas are given in the following tables, with the additional definitions that \(\alpha_T=\tanh(c_5(T-T_x))\), \(T_X=218.8\) K, and \(d_i=c_i+\alpha_Tc_{i+5}\) for i = {2,3,4}:
|
|
Tests of these interpolation formulas against high-accuracy numerical inversion of formulas (4.23) and (4.22) showed that the maximum error introduced by the interpolation formula was about \(0.004^\circ\mathrm{C}\) and the standard error about \(0.001^\circ\mathrm{C}\). This inversion then provides a corrected dew point that incorporates the effects of the enhancement factor as well as differences between the ambient pressure and that in the housing. The algorithm is documented in the box below.
\(T_{p}\) = DPx from instrument x [\(^{\circ}\)C], or alternately
RHO = water vapor density measurement
[\(\mathrm{g\ }\mathrm{m}^{-3}\)]; only one is used in any calculation
ATX = reference ambient temperature [\(^{\circ}C\)]
\(T_{K}\)=ATX+\(T_{0}\) \(^{\dagger}\) = ambient temperature [K]
\(p\) = PSXC = reference ambient pressure [hPa]
\(p_{h}\) = CAVP_x = pressure in instrument “x” housing [hPa]
\(e_{t}\) = intermediate vapor pressure used for calculation only
\(e\) = EWx = water vapor pressure from source x [hPa]
\(M_{w}\) = molecular weight of water\(^{\dagger}\)
\(R_{0}\) = universal gas constant\(^{\dagger}\)
\(f(p_{h},T_{p})\) = enhancement factor (cf. (4.24))
\(F_{d}(e)\) = interpolation formula giving dew point temperature from water vapor pressure
For dew/frost point hygrometers, producing the measurement DPx:
if DPx < 0\(^\circ\)C:
obtain \(e_{t}\) from (4.22) using \(T_{FP}\)=DPx + \(T_{0}\)
else (i.e., DPx \(\geq 0^\circ\)C):
obtain \(e_{t}\) from (4.23)
using \(T_{DP}=\mathrm{DPx}+T_{0}\)
correct \(e_{t}\) for enhancement factor and internal pressure
to get ambient vapor pressure \(e\):
\[\begin{equation}
e=f(p_{h},T_{P})\,(\frac{p}{p_{h}})\,e_t
\tag{4.26}
\end{equation}\]
obtain DPxC by finding the dew point corresponding to the
vapor pressure \(e\):
\[\begin{equation}
\mathrm{\{DPxC\}} = F_{d}(e)
\tag{4.27}
\end{equation}\]
- - - - - - - - - - - - - - - - - - - -
For other instruments producing measurements of vapor density (RHO
[g m\(^{-3}\)]:(a)
find the water vapor pressure in units of hPa:
\[\begin{equation}
e = (\mathrm{\{RHO\}}\,R_{0}\,T_{K}\,/\,M_{w})\times 10^{-5}
\tag{4.28}
\end{equation}\]
find the equivalent dew point:
\[\begin{equation}
\mathrm{\{DPxC\}} = F_{d}(e)
\tag{4.29}
\end{equation}\]
__________
(a) prior to 2011 the following formula was used:
\[Z=\frac{\ln((\mathrm{\{ATX\}}+273.15)\,\mathrm{\{RHO\}}}{1322.3}\]
\[\mathrm{\{DPxC\}}=\frac{273.0\,Z}{(22.51-Z)}\]
For other instruments that measure vapor density, such as a Lyman-alpha or tunable diode laser hygrometers (including the Vertical Cavity Surface Emitting Laser (VCSEL) hygrometer), a similar conversion is made from vapor density to dew point, as described in the next paragraph.
The ambient vapor pressure of water, also used in the calculation of several derived variables. It is often obtained from an instrument measuring dew point or water vapor density. In the case where it is derived from a measurement of dew point (DPx), a correction is applied for the enhancement factor that influences dew point or frost point measurements.28
The formula for obtaining the ambient water vapor pressure as a function of dew point is given in the discussion of DPxC above, Eqs. (4.23) and (4.24), where the calculation of the variables EWx and EWX are also discussed. EWX (or previously EDPC) is the preferred variable that is selected from among the possibilities {EWx} for subsequent calculation of derived variables.
For the case where water vapor pressure is determined by the VCSEL hygrometer, EW_VXL is determined from CONCV_VXL: EW_VXL=Ck{CONCV_VXL}{ATX+273.15) where k is the Boltzmann constant and C = 10 − 4(cm/m)3(hPa/Pa) converts units to hPa.
In the case where the water vapor pressure is determined from the UV Hygrometer data, this variable is calculated using one of two methods:
The ratio of the water vapor pressure to the water vapor pressure in equilibrium over a plane liquid-water surface, scaled to express the result in units of per cent or Pa/hPa:
EWX = atmospheric water vapor pressure (hPa)
ATX = ambient air temperature [\(^{\circ}\mathrm{C}\)]
\(T_{0}=273.15\) K
\(e_{s.w}(\mathrm{ATX+T_{0}})\) = equilibrium water vapor pressure
at dewpoint ATX (hPa)
(see Eq. (4.23) for the formula
used.)
\[\begin{equation} \mathrm{\{RHUM\}}=100\%\,\times\,\frac{\mathrm{\{EWX\}}}{e_{s,w}(\mathrm{\{ATX\}+T_{0}})} \tag{4.32} \end{equation}\]
To follow normal conventions, the change in equilibrium vapor pressure that arises from the enhancement factor is not included in the calculated relative humidity, even though the true relative humidity should include the enhancement factor as specified in (4.24) in the denominator of (4.32).
The ratio of the water vapor pressure to the water vapor pressure in equilibrium over a plane ice surface, scaled to express the result in units of per cent or Pa/hPa:
EWX = atmospheric water vapor pressure (hPa)
ATX = ambient air temperature [\(^{\circ}\mathrm{C}\)]
\(T_{0}=273.15\) K
\(e_{s.i}(\mathrm{ATX+T_{0}})\) = equilibrium water vapor pressure
at frostpoint ATX (hPa)
(see Eq. (4.22) for the formula
used.)
\[\begin{equation} \mathrm{\{RHUMI\}}=100\%\,\times\,\frac{\mathrm{\{EWX\}}}{e_{s,i}(\mathrm{\{ATX\}+T_{0}})} \tag{4.33} \end{equation}\]
To follow normal conventions, the change in equilibrium vapor pressure that arises from the enhancement factor is not included in the calculated relative humidity, even though the true relative humidity should include the enhancement factor as specified in (4.24) in the denominator of (4.33).
The water vapor density computed from various measurements of humidity as indicated by the ’x’ suffix, and conventionally expressed in units of g kg − 1 or per mille. The calculation proceeds in different ways for different sensors. For sensors that measure a chilled-mirror temperature, the calculation is based on the equation of state for a perfect gas and uses the water vapor pressure determined by the instrument, as in the following box:
ATX = ambient temperature (\(^{\circ}\mathrm{C}\))
EWX = water vapor pressure, hPa
\(C_{mb2Pa}\)= conversion factor, hPa to Pa} = 100 Pa hPa\(^{-1}\) (conversion factor to MKS units)
\(C_{kg2g}=10^{3}\,\mathrm{g\,kg}^{-1}\) = (conversion
factor to give final units of g\(\,\)m\(^{-3}\))
\(T_{0}\) = 273.15,K
\[\begin{equation} \mathrm{\{RHOx\}} = C_{kg2g}\frac{C_{mb2Pa}\mathrm{\{EWX\}}}{R_{w}\mathrm{(\{ATX\}+T_{0})}} \tag{4.34} \end{equation}\]
For instruments measuring the vapor pressure density (including the Lyman-alpha probes and the newer version called the UV hygrometer), the basic measurement from the instrument is the water vapor density, RHOUV or **** RHOLA, determined by applying calibration coefficients to the measured signals (XUVI or VLA). In addition, a slow update to a dew-point measurement is used to compensate for drift in the calibration. The processing used for early projects with the Lyman-alpha instruments is similar but more involved and won’t be documented here because the instruments are obsolete. See RAF Bulletin 9 for the processing previously used for archived measurements from the Lyman-alpha hygrometers.
The mass of water vapor per unit mass of (moist) air, conventionally measured in units of g/kg or per mille.
PSXC = ambient total air pressure. hPa
EWX = ambient water vapor pressure, hPa
\(C_{kg2g}=10^{3}\,\)g\(\,\)kg\(^{-1}\) (conversion factor to give
final units of g\(\,\)kg\(^{-1}\))
\(M_{w}=\)molecular weight of water\(^{\dagger}\)
\(M_{d}=\)molecular weight of dry air\(^{\dagger}\)
\[\begin{equation} \mathrm{\{SPHUM\}} = C_{kg2g}\frac{M_{w}}{M_{d}}(\mathrm{\frac{\{EWX\}}{\mathrm{\{PSXC\}-(1-\frac{M_{w}}{M_{d}})\{\mathrm{EWX}\}}}}) \tag{4.35} \end{equation}\]
PSXC = ambient total air pressure. hPa
EWX = ambient water vapor pressure, hPa
\(C_{kg2g}=10^{3}\,\)g\(\,\)kg\(^{-1}\) (conversion factor to give
final units of g\(\,\)kg\(^{-1}\))
\(M_{w}=\)molecular weight of water\(^{\dagger}\)
\(M_{d}=\)molecular weight of dry air\(^{\dagger}\)
\[\begin{equation} \mathrm{\{MR\}}=C_{kg2g}\frac{M_{w}}{M_{d}}\frac{\mathrm{\{EWX\}}}{(\mathrm{\{PSXC\}-\{EWX\})}} \tag{4.36} \end{equation}\]
The absolute temperature reached if a dry parcel at the measured pressure and temperature were to be compressed or expanded adiabatically to a pressure of 1000 hPa. It does not take into account the difference in specific heats caused by the presence of water vapor, and water vapor can change the exponent in the formula below enough to produce errors of 1 K or more.
ATX = ambient temperature, \(^{\circ}\)C
PSXC = ambient pressure (hPa)
\(p_{0}\) = reference pressure = 1000 hPa
\(R_{d}\) = gas constant for dry air\(^{\dagger}\)
\(c_{pd}\) = specific heat at constant pressure for dry air\(^{\dagger}\)
\(T_0=273.15\,\mathrm{K}\)
\[\begin{equation} \mathrm{\{THETA\}}=\left(\mathrm{\{ATX\}}+T_{0}\right)\left(\frac{p_{0}}{\mathrm{\{PSXC\}}}\right)^{R_{d}/c_{pd}} \tag{4.37} \end{equation}\]
The absolute temperature reached if a parcel of air were to be expanded pseudo-adiabatically (i.e., with immediate removal of all condensate) to a level where no water vapor remains, after which the dry parcel would be compressed to 1000 hPa. Beginning in 2011, pseudo-adiabatic equivalent potential temperature is calculated using the method developed by Davies-Jones (2009).30
This is discussed in the memo available at this link. The following summarizes that study. The Davies-Jones formula is:
\[\begin{equation}
\Theta_{P}=\Theta_{DL}\exp\{\frac{r(L_{0}^{*}-L_{1}^{*}(T_{L}-T_{0})+K_{2}r)}{c_{pd}T_{L}}\}
\tag{4.38}
\end{equation}\]
\[\begin{equation}
\Theta_{DL}=T_{K}(\frac{p_{0}}{p_{d}})^{0.2854}\,(\frac{T_{k}}{T_{L}})^{0.28\times10^{-3}r}
\tag{4.39}
\end{equation}\]
where \(T_K\) is the absolute temperature (in kelvin) at the measurement level, \(p_d\) is the partial pressure of dry air at that level, \(p_0\) is the reference pressure (conventionally 1000 hPa), \(r\) is the (dimensionless) water vapor mixing ratio, \(c_p\) the specific heat of dry air, \(T_L\) the temperature at the lifted condensation level (in kelvin), and \(T_0=273.15\,\mathrm{K}\). The coefficients in this formula are \(L_0^* = 2.56313\times 10^6\mathrm{J\,kg^{-1}}\),
\(L_1^* = 1754\,\mathrm{J\,kg^{-1}K^{-1}}\), and \(K_2 = 1.137\times 10^6\mathrm{J\,kg^{-1}}\). The asterisks on \(L_0^*\) and \(L_1^*\) indicate that these coefficients depart from the best estimate of the coefficients that give the latent heat of vaporization of water, but they have been adjusted to optimize the fit to values obtained by exact integration. Note that, unlike the formula discussed below that was used prior to 2011, the mixing ratio must be used in dimensionless form (i.e., kg/kg), not with units of g/kg. The following empirical formula, developed by Bolton (1980),31
is used to calculate \(T_L\):
\[\begin{equation}
T_{L}=\frac{\beta_{1}}{3.5\ln(T_{K}/\beta_{3})-\ln(\mathrm{e/\beta_{4}})+\beta_{5}}+\beta_{2}
\tag{4.40}
\end{equation}\]
where \(e\) is the water vapor pressure, \(\beta_1 = 2840\,\mathrm{K}\), \(\beta_2 = 55\,\mathrm{K}\), \(\beta_3 = 1\,\mathrm{K}\), \(\beta_4 = 1\,\mathrm{hPa}\), and \(\beta_5 = -4.805\). (\(\beta_3\) and \(\beta_4\) have been introduced into (4.40) only to ensure that arguments to logarithms are dimensionless and to specify the units that must be used to achieve that.)
\(T_K\) = ATX + \(T_0\) = ambient temperature [K]
\(e\) = EWX = water vapor pressure
\(p_d\) = PSXC - EWX = partial pressure of dry air [hPa]
\(p_{0}\) = reference pressure = 1000 hPa
\(r\) = MR = water vapor mixing ratio
\(R_{d}\) = gas constant for dry air\(^{\dagger}\)
\(c_{pd}\) = specific heat at constant pressure for dry air\(^{\dagger}\)
\(T_L\) = temperature at the lifted condensation level (LCL) [K]
\(L_0^*+L_1^*(T_L-T_0)\) = latent heat of vaporization at the LCL
\(L_0=2.56313 × 10^6\) J\(\,\)kg\(^{-1}\), \(L_1=1754\) J\(\,\)kg\(^{-1}\)K\(^{-1}\)
\(K_2=1.137 × 10^6\) J\(\,\)kg\(^{-1}\)
\(\beta_{1-5}\) = {2840 K, 55 K, 1 K, 1 hPa, −4.805}
\[\begin{equation} T_{L}=\frac{\beta_{1}}{3.5\ln(T_{K}/\beta_{3})-\ln(\mathrm{e/\beta_{4}})+\beta_{5}}+\beta_{2} \tag{4.41} \end{equation}\] \[\begin{equation} \Theta_{DL}=T_{K}(\frac{p_{0}}{p_{d}})^{0.2854}\,(\frac{T_{k}}{T_{L}})^{0.28\times10^{-3}r} \tag{4.42} \end{equation}\] \[\begin{equation} \Theta_{P}=\Theta_{DL}\exp\{\frac{r(L_{0}^{*}-L_{1}^{*}(T_{L}-T_{0})+K_{2}r)}{c_{pd}T_{L}}\} \tag{4.43} \end{equation}\]
Prior to 2011, the variable called the equivalent potential temperature32
and named THETAE in the output data files was that obtained using the method of Bolton (1980), which used the same formula to obtain the temperature at the lifted condensation level (\(T_L\)) and then used that temperature to find the value of potential temperature of dry air that would result if the parcel were lifted from that point until all water vapor condensed and was removed from the air parcel. The formulas used were as follows:
\(T_{L}\)= temperature at the lifted condensation level, K
\(T_0=273.15\,\mathrm{K}\)
ATX = ambient temperature [\(^{\circ}\mathrm{C}\)]
EDPC = water vapor pressure [hPa] – now superceded by EWX
MR = mixing ratio [g/kg]
THETA = potential temperature [K]
\[\begin{equation} T_{L}=\frac{2840.}{3.5\ln(\mathrm{\{ATX\}+T_{0}})-\ln(\mathrm{\{EDPC\}})-4.805}+55 \tag{4.44} \end{equation}\] \[\begin{align} \mathrm{\{THETAE\}} = & \mathrm{\{THETA\}}\left(\frac{3.376}{T_{L}}-0.00254\right)\notag \\ & \times (\mathrm{\{MR\}})(1+0.00081(\{MR\})) \tag{4.45} \end{align}\]
Differences vs the new formula are usually minor but can be as much as 0.5 K.****
The temperature of dry air having the same pressure and density as the air being sampled. The virtual temperature thus adjusts for the buoyancy added by water vapor.
ATX = ambient temperature, \(^{\circ}\mathrm{C}\)
\(r\) = mixing ratio, dimensionless {[}kg/kg{]} = {MR}/(1000 g/kg)
\(T_{0}=273.15\),K
\[\begin{equation} \mathrm{TVIR}=(\mathrm{\{ATX\}}+T_{0})\left(\frac{1+\frac{M_{d}}{M_{w}}r}{1+r}\right)-T_{0} \tag{4.46} \end{equation}\]
A potential temperature analogous to the conventional potential temperature except that it is based on virtual temperature instead of ambient temperature. Dry-adiabatic expansion or compression to the reference level (1000 hPa) is assumed. As for THETA, use of dry-air values for the gas constant and specific heat at constant pressure can lead to significant errors in humid conditions. For further information, see this note.
TVIR = virtual temperature [\(^{\circ}\mathrm{C}\)]
PSXC = ambient pressure [hPa]
\(R_{d}=\)gas constant for dry air\(^{\dagger}\)
\(c_{pd}=\)specific heat at constant pressure for dry air\(^{\dagger}\)
\(T_{0}=273.15\,\)K
\(p_{0}\) = reference pressure, conventionally 1000 hPa
\[\begin{equation} \mathrm{THETAV}=\left(\mathrm{\{TVIR\}}+T_{0}\right)\left(\frac{p_{o}}{\mathrm{\{PSXC\}}}\right)^{R_{d}/c_{pd}} \tag{4.47} \end{equation}\]
\(e=\){EWX}\(*100\) = water vapor pressure [Pa]
ATX = ambient temperature (\(^{\circ}\mathrm{C}\))
\(r=\){MR}/1000. = mixing ratio (dimensionless)
\(p_{d}=\)({PSXC}\(-\){EWX})\(*100\) = ambient dry-air pressure [Pa]
\(p_{0}=\)reference pressure for potential temperature, 10\(^{5}\)Pa
\(\chi=\){PLWCC}/1000.=cloud liquid water content [kg\(\,\)m\(^{-3}\)]
\(R_{d}=\)gas constant for dry air\(^{\dagger}\)
\(\rho_{d}=\)density of dry air = \(\frac{p_{d}}{R_{d}(\{ATX\}+T_{0})}\)
\(c_{pd}=\)specific heat of dry air\(^{\dagger}\)
\(c_{w}=\)specific heat of liquid water\(^{\dagger}\)
\(L_{v}=L_{0}+L_{1}\mathrm{\{ATX\}}\) where \(L_{0}=2.501\times10^{6}\mathrm{J}\,\mathrm{kg^{-1}}\)
and \(L_{1}=-2370\,\mathrm{J\,\mathrm{kg^{-1}\,\mathrm{K^{-1}}}}\)
\[\begin{equation}
r_{t}=r+(\chi/\rho_{d})
\tag{4.49}
\end{equation}\]
\[\begin{equation}
c_{pt}=c_{pd}+r_{t}c_{w}
\tag{4.50}
\end{equation}\]
If outside cloud or below 100% relative humidity, define
\[\begin{equation}
F_{1}=\left(\frac{e}{e_{s,w}(T)}\right)^{-\frac{rR_{w}}{c_{pt}}}
\tag{4.51}
\end{equation}\]
otherwise set \(F_{1}=1\). Then
\[\begin{equation}
T_{1}=\mathrm{(\{ATX\}}+T_{0})\left\{ \frac{p_{0}}{(\mathrm{\{PSXC\}}-\mathrm{\{EDPC\})}}\right\} ^{\frac{R_{d}}{c_{pt}}}
\tag{4.52}
\end{equation}\]
\[\begin{equation}
\mathrm{\{THETAQ\}}=T_{1}F_{1}\exp\left\{ \frac{L_{v}r}{c_{pt}(\{\mathrm{ATX\}}+T_{0})}\right\}
\tag{4.53}
\end{equation}\]
RAF Bulletin 23 documents the calculation of wind components, both with respect to the earth (UI, VI, WI, WS and WD) and with respect to the aircraft (UX and VY). In data processing, a separate function (GUSTO in GENPRO, gust.c in NIMBUS) is used to derive these wind components. That function uses the measurements from an Inertial Navigation System (INS) as well as aircraft true airspeed, aircraft angle of attack, and aircraft sideslip angle. The wind components calculated in GUSTO/gust.c are used to derive the wind direction (WD) and wind speed (WS). Additional variables UIC, VIC, WSC, WDC, UXC, and VYC are also calculated based on the variables VNSC, VEWC discussed in Section 3.4, which combine INS and GPS information to obtain improved measurements of the aircraft motion. Those are usually the highest-quality measurements of wind because the merged INS/GPS variables combine the high-frequency response of the INS with the long-term accuracy of the GPS.
There is an extensive discussion of the wind-sensing system and the uncertainties associated with measurements of wind in this Technical Note. The details contained therein and in Bulletin 23 will not be repeated here, so those documents should be consulted for additional information. There are two exceptions that are discussed in more detail here:
The calculation of vertical wind is described in more detail below for the variables WI and WIC.
Because measurements obtained by a GPS receiver are often used, the motion of the GPS receiving antenna relative to the IRU must be considered. Standard processing corrects for the motion of the gust system relative to the IRU arising from aircraft rotation, but a similar correction is needed because the GPS antenna is displaced from the IRU. The displacement is almost entirely along the longitudinal axis of the aircraft, so GPS-measured velocities like GGVNS, GGVEW, and GGVSPD (denoted here \(v_n\), \(v_e\), \(v_u\)) need correction as follows to give measurements that apply at the location of the IRU. Then these variables can be used in place of or to complement similar measurements from the IRU in the processing algorithms. The equations are:
\[\begin{align}\begin{split}
\delta v_{u} = & -L_{G}\dot{\theta}\notag \\
\delta v_{e} = & -L_{G}\dot{\psi}\,{\cos\psi}\notag \\
\delta v_{n} = & L_{G}\dot{\psi}\,{\sin\psi}
\end{split}
\tag{4.54}
\end{align}\]
where \(\theta\) and \(\psi\) respectively are the pitch and heading angles and \(L_G\) is the distance forward along the longitudinal axis from the IRU to the GPS antenna (\(−4.30\) m for the GV and \(-9.88\) m for the C-130 during and after 2015). The negative signs indicate that the GPS antennas are behind the IRUs. The dots over the attitude-angle symbols represent time derivatives, so for example \(\dot{\theta}\) is the rate of change of the pitch angle. All angles are expressed in radians. The correction terms should be added to the GPS-measured velocity components so that they represent the motion of the IRU relative to the Earth. This is done for the vertical wind, beginning in 2017, but for horizontal wind the complementary filter (discussed below) removes high-frequency fluctuations from the GPS-derived measurements so incorporation of these changes would have negligible effect. For more information, see this note.
The variables pertaining to the relative wind are described in the next subsection, and the variables characterizing the wind are then described briefly in the last subsection. Some additional detail is included in cases where procedures are not documented in that earlier bulletin.
Wind is measured by adding two vectors, the measured air motion relative to the aircraft (called the relative wind) and the motion of the aircraft relative to the Earth. The following are the measurements used to determine the relative wind. The motion of the aircraft relative to the ground was discussed in Section 3.1, and the combination of these two vectors to measure the wind is described in RAF Bulletin 23.
RAF uses the radome gust-sensing technique33 to measure incidence angles of the relative wind (i.e., angles of attack and sideslip). The pressure difference between sensing ports above and below the center line of the radome is used, along with the dynamic pressure measured at a pitot tube and referenced to the static pressure source, to determine the angle of attack. The sideslip angle is determined similarly using the pressure ports on the starboard and port sides of the radome. A Rosemount Model 858AJ gust probe has occasionally been used for specialized measurements. The radome measurements are made by differential pressure sensors located in the nose area of the aircraft and connected to the radome by semi-rigid tubing.
The flight speed of the aircraft relative to the atmosphere. This derived measurement of the flight speed of the aircraft relative to the atmosphere is based on the Mach number calculated from both the dynamic pressure at location x and the static pressure. See the derivation for ATx . The different variables for TASx (TASF, TASR, etc) use different measurements of QCxC in the calculation of Mach number. The variable TASxD is the result of calculations for which the Mach number, air temperature, and true airspeed are determined for dry instead of humid air. See the discussion of ATX for an explanation of how humidity is handled in the calculation of true airspeed.
(see (4.17) and (4.18) for MACHx and ATX)
Note dependence of MACHx on choices for QCXC and PSXC
TASx depends on QCXC, PSXC, ATX
where PSXC and ATX are the preferred choices
\(\gamma^{\prime}\), \(R^{\prime}\), and \(T_{0}\): See the List of Symbols
\[\begin{equation} \mathrm{TASx}=\mathrm{\{MACHx\}}\sqrt{\gamma^{\prime}R^{\prime}\mathrm{\,(\{ATX\}}+T_{0})} \tag{4.55} \end{equation}\]
This derived measurement of true airspeed accounted for deviations of specific heats of moist air from those of dry air. See List, 1971, pp 295, 331-339, and Khelif, et al., 1999. This variable is no longer used because the standard
calculation of TASX (documented in the preceding paragraph) now uses moist-air
values of the specific heats and gas constant. The equation previously used for this variable, given by Khelif et al. 1999,34
added a moisture correction to the true airspeed derived for dry air, as follows:
\(q\) = specific humidity (dimensionless) = SPHUM/1000.
for SPHUM expressed in g/kg
\(c=0.000304\,\mathrm{kg\,g^{-1}}=0.304\) (dimensionless)
\[\begin{equation} \mathrm{\{TASHC\}} = \mathrm{\{TASX\}} (1.0 + c\,q) \tag{4.56} \end{equation}\]
The angle of attack of the aircraft. This derived measurement represents the angle between the longitudinal axis of the aircraft and the component of the relative wind vector in the plane of port-starboard symmetry of the aircraft. The tangent of the angle of attack is the ratio of the vertical to longitudinal component of the relative wind. Positive values indicate flow moving upward (in the aircraft reference frame) relative to the longitudinal axis. The calculation is based on ADIFR and a measurement of dynamic pressure, and so is the measurement produced by a radome gust-sensing system. Empirical sensitivity coefficients for each aircraft, determined from special flight maneuvers, are used; see RAF Bulletin 23 and this Technical Note for more information. The sensitivity coefficients listed below have changed when the radomes were changed or refurbished, so the project documentation should be consulted for the values used in a particular project. For more information on the latest C-130 calibration, see this note.
Prior to 2017, the procedure was based on the following algorithm:
ADIFR = attack differential pressure, radome [hPa]
QCF = uncorrected dynamic pressure [hPa]
MACH = uncorrected Mach number based on QCF and PSF
without humidity correction
\(e_{0},\,e{}_{1},\,e_{2}\) = sensitivity
coefficients determined empirically; typically:
{4.7532, 9.7908, 6.0781} for the C-130(a)
{4.605\(\,[^{\circ}]\), \(18.44\,[^{\circ}]\), \(6.75\,[^{\circ}]\)}
for the GV
__________
(a) Prior to Jan 2012, when the GV radome was changed: {5.516, 19.07, 2.08}
\[\begin{equation} \mathrm{\{AKRD\}}=e_{0}+\frac{\{\mathrm{ADIFR}\}}{\{\mathrm{QCF}\}}\left(e_{1}+e{}_{2}\mathrm{\{MACH\}}\right) \tag{4.57} \end{equation}\]
See also this memo.
Beginning in 2017, a different strategy was used, as documented in more detail in this memo. Two variables were used to represent the angle of attack, \(A\)={ADIFR}/{QCF} and \(q\)={QCF}. However, each was filtered into complementary low-pass and high-pass components, with the cutoff frequency at (1/600) Hz, and the separate components were used to represent the separate components of angle of attack according to the following formula:
ADIFR = attack differential pressure, radome [hPa]
QCF = uncorrected dynamic pressure [hPa]
\(A\) = (ADIFR/QCF) = \(A_{f}+A_{s}\) where \(A_{f}\) is the high-pass
and \(A_{s}\) the low-pass component
\(e_{1},\,d_{0},\,d{}_{1},\,d_{2}\) = sensitivity
coefficients determined empirically; typically, for the GV,
\(e_{1}=21.481\,[^{\circ}]\)
\(d_{1-3}\) = {\(4.5253\,[^{\circ}]\), \(19.9332\,[^{\circ}]\), \(-0.00196099\,[^{\circ}\mathrm{hPa}^{-1}]\)}
\[\begin{equation} \mathrm{\{AKRD\}}=d_{0}+d_{1}A_{s}+d_{2}\mathrm{\{QCF\}_{s}+}e_{1}A_{f} \tag{4.58} \end{equation}\]
****
BDIFR = differential pressure between sideslip pressure ports, radome
[hPa]
QCXC = dynamic pressure [hPa]
\(s_{0},\,s{}_{1}\) = empirical
coefficients dependent on the aircraft and radome configuration
= {-0.000983, (1/0.08189) \(^\circ\)} for the
C-130
= {-0.0025, (1/0.04727) \(^\circ\)} for the GV(a)
\[\begin{equation}
\mathrm{\{SSRD\}} = s_{1}(\frac{\mathrm{\{BDIFR\}}}{\{\mathrm{QCXC}\}}+s_{0})
\tag{4.59}
\end{equation}\]
__________
(a) The technical note on wind uncertainty
recommended using SSRD=\(e_{0}+e_{1}\){BDIFR}/{QCF} with \(e_{0}=0.008\)
and \(e_{1}=22.302\). This has not yet been used in processing as of
May 2022.
The magnitude and direction of the horizontal wind. These variables are obtained in a straightforward manner from UI and VI. The resulting wind direction is relative to true north and represents the direction from which the wind blows. That is the reason that 180∘ appears in the following algorithm.
UI = easterly component of the horizontal wind
VI = northerly component of the horizontal wind
atan2 = 4-quadrant arc-tangent function producing output in radians
from -\(\pi\) to \(\pi\)
\(C_{rd}\) = conversion factor, radians to degrees, = 180/\(\pi\) [units: \(^{\circ}\),/,radian]
\[\begin{align} \mathrm{WS} = & \sqrt{\mathrm{\{UI\}}^{2}+\{\mathrm{VI\}}^{2}}\tag{4.60}\\ \mathrm{WD} = & C_{rd}\mathrm{\,atan2(\{UI\},\,\{VI\})}+180^{\circ} \tag{4.61} \end{align}\]
The horizontal wind vector relative to the frame of reference attached to the aircraft. UX is parallel to the longitudinal axis and positive toward the nose. VY is along the lateral axis and normal to the longitudinal axis; positive is toward the port (or left) wing.
The component of the wind in the vertical direction. This is the standard calculation of vertical wind, obtained from the difference between the measured vertical component of the relative wind and the vertical motion of the aircraft (usually GGVSPD in recent projects).**** This should be used in preference to WI if the latter is present; see the discussion of WP3 in Section 3.1. Positive values are toward the zenith.
The magnitude and direction of the wind vector, obtained by combining measurements from GPS and IRU units. These variables are obtained in a straightforward manner from UIC and VIC, using equations analogous to (4.60) and (4.61) but with UIC and VIC as input measurements. They are expected to be the preferred measurements of wind because they combine the best features of the IRU and GPS measurements.
The above variables are normally included in the archived netCDF files from projects, but there are a few remote sensors that provide additional state-parameter measurements in some projects. These include:
The links provided connect to descriptions of these instruments on the EOL web site, and each provides a summary of how data are acquired and processed. These measurements are not normally part of the archived netCDF project files. Those interested in using these measurements should contact EOL data management (mailto:raf-dm@eol.ucar.edu) for access to the measurements and for information on how the measurements are processed.
A weakness is this form for the pressure correction is that occasionally the radome ports become plugged with ice and the measurement of angle of attack is not available. When the variable ATTACK representing angle of attack is invalid, the angle of attack is instead calculated from PITCH−VSPD/TASX, which approximates the angle of attack if the vertical wind is zero.↩︎
For C-130 measurements prior to 2012 but after September 2003, the correction applied to PSF was Δp = p + max ((3.29 + {QCX} * 0.0273),4.7915) using units of hPa. Prior to Sept 2003, the correction was Δp = max ((4.66 + 11.4405Δpα/Δqr), 1.113). For both PSFD and PSFRD, the correction prior to (2012?) was Δp = p + max ((3.29 + {QCX} 0.0273),4.7915). For GV measurements Aug 2006 to 2012, Δp= (-1.02 + 0.1565q) + q1(0.008 + q1(7.1979e-09q1 - 1.4072e-05). Before Aug 2006: Δp=(3.08 - 0.0894{PSF}) + {QCF}(-0.007474 + {QCF}4.0161e-06).↩︎
See this memo and this revisionfor details regarding implementation of this representation of Δp for the GV.↩︎
Atmos. Meas. Tech., 7, 3215-3231, 2014 doi:10.5194/amt-7-3215-2014.↩︎
C-130, prior to 2012: For QCFC: subtract max(4.66+11.4405 * \(\mathrm{\{ADIFR\}}\) / \(\mathrm{\{QCR\}}\), 1.113); For QCFRC prior to Sept 2003: same as for QCFC; after/including Sept 2003, subtract max(3.29+\(\mathrm{\{QCX\}}\) 0.0273, 4.7915); For QCRC: subtract max((3.29+\(\mathrm{\{QCX\}}\) 0.0273), 4.7915). GV Aug 2006 to 2012: For QCF, subtract (1.02+\(\mathrm{\{PSF\}}\)(0.215 - 0.04 * \(\mathrm{\{QCF\}}\)/1000.) + {QCF} * (\(-0.003266\) + \(\mathrm{\{QCF\}}\) * 1.613e-06))↩︎
The symbol † indicates that values are included in the table of constants in Sect. 1.3.↩︎
Prior to 2012, these variables were called “total temperature” and symbols starting with ’TT’ instead of ’RT’ were used. That name was misleading because these values are not true total-temperature measurements, for which the air would be at the same speed as the aircraft, but instead recovery-temperature measurements. The name has been changed to correct this mis-labeling, although this was a long-standing convention in past datasets.↩︎
See the related obsolete variables TTx, which are previously used names for these variables. The names were changed to clarify that the quantity represented is the recovery temperature, not the total temperature.↩︎
The recovery factor determined for the now-obsolete NCAR reverse-flow sensor was 0.6. The recovery factor for the now retired NCAR fast-response (K-probe) temperature sensor was 0.8.↩︎
A problem sometimes arises from use of the measured humidity, because that measurement might be obviously in error. For example, following descents the dew point determined from chilled-mirror hygrometers sometimes overshoots the correct value significantly, producing dew-point measurements well above the measured temperature. If such measurements are used, the result can produce a significant error in derived variables based on the humidity-corrected gas constant and specific heats. If the measurements are flagged as bad, there will be gaps in derived variables. To avoid these two errors, the corrections applied to the gas constant and specific heats are treated as follows: (i) The humidity correction is limited to not more than that given by the water-equilibrium humidity at the temperature ATXD, calculated using dry-air specific heats and gas constant. (ii) If the humidity from the primary sensor is flagged as a missing measurement (e.g., from a dew-point sensor), a secondary measurement is used (e.g., the VCSEL) in cases when the secondary sensor is almost always present in an experiment. (iii) As a backup, the variables TASxD and ATxD are always calculated omitting the humidity correction to the gas constant and the specific heats. These variables usually provide continuous measurements, although they will be offset from the humidity-corrected values. The offset indicates the magnitude of the correction when both are present, and one of the variables TASxD (ATxD) may be selected as TASX (ATX) in cases where missing values might cause a problem for derived variables.↩︎
See also DP_VXL and DP_CR2C below.↩︎
Q. J. R. Meteorol. Soc. (2005), 131, pp. 1539–1565↩︎
Prior to 2010, the vapor pressure relationship used was the Goff-Gratch formula as given in the Smithsonian Tables (List, 1980).↩︎
prior to 2011 the conversion was made using the formula DPxC = 0.009109 + DPx(1.134055 + 0.001038DPx). For instruments producing measurements of vapor density (RHO), the previous Bulletin 9 section incorrectly gave the conversion formula as DPxC = 273.0Z/(22.51 − Z), a conversion that would apply to frost point, not dew point. However, the code in use shows that the conversion was instead 237.3Z/(17.27 − Z), where Z in both cases is Z = ln ((ATX + 273.15)RHO/1322.3).↩︎
The conversion factor is given by this formula:
\[C^{\prime}=\frac{10^{6}\mathrm{cm}^{3}}{\mathrm{m}^{3}}\times\frac{M_{W}^{\dagger}}{N_{A}^{\dagger}}\]
where NA is the Avogadro constant, 6.022147 × 1026 molecules kmol − 1.↩︎
For details see Zondlo, M. A., M. E. Paige, S. M. Massick, and J. A. Silver, 2010: Vertical cavity laser hygrometer for the National Science Foundation Gulfstream-V aircraft. J. Geophys. Res., 115, D20309, doi:10.1029/2010JD014445.↩︎
prior to 2011, this variable was calculated using the Goff-Gratch formula. See the discussion of DPXC for more information on previous calculations.↩︎
For more details see Beaton, S. P. and M. Spowart, 2012: UV Absorption Hygrometer for Fast-Response Airborne Water Vapor Measurements. J. Atmos. Oceanic Technol., 29. DOI: 10.1175/JTECH-D-11-00141.1↩︎
Davies-Jones, R., 2009: On formulas for equivalent potential temperature. Mon. Wea. Review, 137, 3137–3148.↩︎
Bolton, D., 1980: The computation of equivalent potential temperature. Mon. Wea. Rev., 108, 1046–1053.↩︎
The AMS glossary defines equivalent potential temperature as applying to the adiabatic process, not the pseudo-adiabatic process; the name of this variable has therefore been changed.↩︎
Brown, E. N, C. A. Friehe, and D. H. Lenschow, 1983: Journal of Climate and Applied Meteorology, 22, 171–180↩︎
Khelif, D., S.P. Burns, and C.A. Friehe, 1999: Improved wind measurements on research aircraft. Journal of Atmospheric and Oceanic Technology, 16, 860–875.↩︎