At Boulder, Colorado, elevation 1650 m, the raw (uncorrected) surface pressure measurement is 84 kPa while the surface temperature is 16 C. “Correct” the pressure to sea level using each of the following assumptions about the temperature in the “missing” atmosphere beneath the station.
a) The temperature is constant at its surface value.
b) The temperature increases with depth at the average lapse rate of 6.5 C/km.
The Correct Answer and Explanation is :
To correct the atmospheric pressure at Boulder, Colorado (elevation 1,650 meters) to sea level, we can use the barometric formula, which relates pressure to altitude and temperature. The formula is:
PSL=Pobs×(1+L×hTSL)g×MR×LP_{\text{SL}} = P_{\text{obs}} \times \left(1 + \frac{L \times h}{T_{\text{SL}}}\right)^{\frac{g \times M}{R \times L}}
Where:
- PSLP_{\text{SL}} = sea-level pressure (Pa)
- PobsP_{\text{obs}} = observed pressure at altitude (Pa)
- LL = temperature lapse rate (K/m)
- hh = altitude (m)
- TSLT_{\text{SL}} = standard sea-level temperature (K)
- gg = acceleration due to gravity (9.80665 m/s²)
- MM = molar mass of Earth’s air (0.0289644 kg/mol)
- RR = universal gas constant (8.3144598 J/(mol·K))
Given:
- Pobs=84 kPa=84,000 PaP_{\text{obs}} = 84 \, \text{kPa} = 84,000 \, \text{Pa}
- h=1,650 mh = 1,650 \, \text{m}
- TSL=16∘C=289.15 KT_{\text{SL}} = 16^\circ\text{C} = 289.15 \, \text{K}
a) Assuming constant temperature at the surface value:
If the temperature is constant at 16°C, the lapse rate L=0L = 0. The formula simplifies to:
PSL=Pobs×exp(g×M×hR×TSL)P_{\text{SL}} = P_{\text{obs}} \times \exp\left(\frac{g \times M \times h}{R \times T_{\text{SL}}}\right)
Calculating:
PSL=84,000×exp(9.80665×0.0289644×1,6508.3144598×289.15)P_{\text{SL}} = 84,000 \times \exp\left(\frac{9.80665 \times 0.0289644 \times 1,650}{8.3144598 \times 289.15}\right)
PSL≈84,000×exp(0.0565)P_{\text{SL}} \approx 84,000 \times \exp(0.0565)
PSL≈84,000×1.0582P_{\text{SL}} \approx 84,000 \times 1.0582
PSL≈88,900 PaP_{\text{SL}} \approx 88,900 \, \text{Pa}
PSL≈88.9 kPaP_{\text{SL}} \approx 88.9 \, \text{kPa}
b) Assuming a temperature lapse rate of 6.5°C/km:
For a lapse rate of 6.5°C/km (0.0065 K/m), the formula becomes:
PSL=Pobs×(1+0.0065×1,650289.15)5.2561P_{\text{SL}} = P_{\text{obs}} \times \left(1 + \frac{0.0065 \times 1,650}{289.15}\right)^{5.2561}
Calculating:
PSL=84,000×(1+0.0065×1,650289.15)5.2561P_{\text{SL}} = 84,000 \times \left(1 + \frac{0.0065 \times 1,650}{289.15}\right)^{5.2561}
PSL=84,000×(1+0.0370)5.2561P_{\text{SL}} = 84,000 \times \left(1 + 0.0370\right)^{5.2561}
PSL=84,000×(1.0370)5.2561P_{\text{SL}} = 84,000 \times (1.0370)^{5.2561}
PSL≈84,000×1.221P_{\text{SL}} \approx 84,000 \times 1.221
PSL≈102,300 PaP_{\text{SL}} \approx 102,300 \, \text{Pa}
PSL≈102.3 kPaP_{\text{SL}} \approx 102.3 \, \text{kPa}
Explanation:
The barometric formula models how atmospheric pressure changes with altitude, considering temperature variations. In part (a), assuming a constant temperature at the surface value, the pressure decreases exponentially with altitude. In part (b), with a lapse rate of 6.5°C/km, the temperature decreases linearly with altitude, leading to a different pressure correction. The lapse rate significantly influences the pressure correction, with a higher lapse rate resulting in a higher sea-level pressure.