I created a program to calculate the temperature, pressure, density, gravity, mols/m³, and root-mean squared velocity profile for the troposphere. It is based on the US/International Standard. The height used is the real geometric height, and not the geopotential height.
#!/usr/bin/bash
# Zoe Phin, 2019/12/03
seq 0 1000 11000 | awk 'BEGIN {
Ts=288.16; Ps=101.325; Ds=1.2249356; Gs=9.80665
Re=6371009; L=0.0065; R=8.3144626; M=0.0289644
} {
H = (Re*$1)/(Re+$1)
T = Ts-L*H
G = Gs*Re/(Re+$1)
P = Ps*(T/Ts)^(G/L*M/R)
D = Ds*(T/Ts)^(G/L*M/R-1)
Mols = P*1000/(R*T)
Vrms = sqrt(3*R*T/M)
printf "%+5s %.3f %7.3f %.3f %.3f %.3f %.3f\n", $1, T, P, D, G, Mols, Vrms
# printf "%+5s %.3f %7.3f %.3f %.3f %.3f %.3f, %.3f = %.3f\n", $1, T, P, D, G, Mols, Vrms, P*M*1000, D*R*T
}'
Copy and paste above code to a new file called stdatmo.sh. Make it executable and run it:
> chmod +x stdatmo.sh
> ./stdatmo.shThe result is:
0 288.160 101.325 1.225 9.807 42.291 498.152
1000 281.661 89.879 1.112 9.805 38.379 492.503
2000 275.164 79.508 1.007 9.804 34.753 486.790
3000 268.669 70.135 0.909 9.802 31.396 481.010
4000 262.176 61.681 0.820 9.800 28.296 475.163
5000 255.685 54.077 0.737 9.799 25.437 469.244
6000 249.197 47.253 0.661 9.797 22.806 463.251
7000 242.710 41.148 0.591 9.796 20.390 457.182
8000 236.225 35.700 0.526 9.794 18.177 451.033
9000 229.743 30.854 0.468 9.793 16.153 444.801
10000 223.262 26.558 0.414 9.791 14.307 438.483
11000 216.783 22.760 0.366 9.790 12.628 432.074| Column 1 | Height (m) |
| Column 2 | Temperature (K) |
| Column 3 | Pressure (kPa) |
| Column 4 | Density (kg/m³) |
| Column 5 | Gravity (m/s²) |
| Column 6 | Mols/m³ |
| Column 7 | Root Mean Square Velocity (m/s) |
To verify that the troposphere does indeed follow the ideal gas law, I swap the commented line in the code to:
printf "%+5s %.3f %7.3f %.3f %.3f %.3f %.3f, %.3f = %.3f\n", $1, T, P, D, G, Mols, Vrms, P*M*1000, D*R*TThis gave me the result:
0 288.160 101.325 1.225 9.807 42.291 498.152, 2934.818 = 2934.818
1000 281.661 89.879 1.112 9.805 38.379 492.503, 2603.278 = 2603.278
2000 275.164 79.508 1.007 9.804 34.753 486.790, 2302.915 = 2302.915
3000 268.669 70.135 0.909 9.802 31.396 481.010, 2031.408 = 2031.408
4000 262.176 61.681 0.820 9.800 28.296 475.163, 1786.561 = 1786.561
5000 255.685 54.077 0.737 9.799 25.437 469.244, 1566.299 = 1566.299
6000 249.197 47.253 0.661 9.797 22.806 463.251, 1368.668 = 1368.668
7000 242.710 41.148 0.591 9.796 20.390 457.182, 1191.825 = 1191.825
8000 236.225 35.700 0.526 9.794 18.177 451.033, 1034.038 = 1034.038
9000 229.743 30.854 0.468 9.793 16.153 444.801, 893.679 = 893.679
10000 223.262 26.558 0.414 9.791 14.307 438.483, 769.223 = 769.223
11000 216.783 22.760 0.366 9.790 12.628 432.074, 659.240 = 659.240Verified! To change the height increment, change the second parameter of the ‘seq’ command on line 4. I prefer 100. Full result:
0 288.160 101.325 1.225 9.807 42.291 498.152
100 287.510 100.130 1.213 9.806 41.887 497.590
200 286.860 98.946 1.202 9.806 41.485 497.028
300 286.210 97.773 1.190 9.806 41.087 496.464
400 285.560 96.612 1.179 9.806 40.691 495.900
500 284.910 95.462 1.167 9.806 40.298 495.336
600 284.260 94.323 1.156 9.806 39.909 494.770
700 283.610 93.196 1.145 9.806 39.522 494.204
800 282.961 92.079 1.134 9.805 39.138 493.638
900 282.311 90.973 1.123 9.805 38.757 493.071
1000 281.661 89.879 1.112 9.805 38.379 492.503
1100 281.011 88.794 1.101 9.805 38.004 491.935
1200 280.361 87.721 1.090 9.805 37.631 491.365
1300 279.712 86.658 1.079 9.805 37.262 490.796
1400 279.062 85.606 1.069 9.804 36.895 490.225
1500 278.412 84.564 1.058 9.804 36.531 489.654
1600 277.763 83.533 1.048 9.804 36.170 489.083
1700 277.113 82.511 1.037 9.804 35.812 488.510
1800 276.463 81.500 1.027 9.804 35.456 487.938
1900 275.814 80.499 1.017 9.804 35.103 487.364
2000 275.164 79.508 1.007 9.804 34.753 486.790
2100 274.514 78.528 0.997 9.803 34.405 486.215
2200 273.865 77.557 0.987 9.803 34.060 485.639
2300 273.215 76.595 0.977 9.803 33.718 485.063
2400 272.566 75.644 0.967 9.803 33.379 484.486
2500 271.916 74.702 0.957 9.803 33.042 483.908
2600 271.267 73.770 0.947 9.803 32.707 483.330
2700 270.617 72.847 0.938 9.802 32.376 482.751
2800 269.968 71.933 0.928 9.802 32.047 482.172
2900 269.319 71.029 0.919 9.802 31.720 481.591
3000 268.669 70.135 0.909 9.802 31.396 481.010
3100 268.020 69.249 0.900 9.802 31.075 480.429
3200 267.370 68.373 0.891 9.802 30.756 479.846
3300 266.721 67.505 0.882 9.802 30.440 479.263
3400 266.072 66.647 0.873 9.801 30.126 478.680
3500 265.422 65.797 0.864 9.801 29.815 478.095
3600 264.773 64.957 0.855 9.801 29.506 477.510
3700 264.124 64.125 0.846 9.801 29.200 476.924
3800 263.475 63.302 0.837 9.801 28.896 476.338
3900 262.826 62.487 0.828 9.801 28.595 475.750
4000 262.176 61.681 0.820 9.800 28.296 475.163
4100 261.527 60.884 0.811 9.800 28.000 474.574
4200 260.878 60.095 0.802 9.800 27.705 473.985
4300 260.229 59.314 0.794 9.800 27.414 473.394
4400 259.580 58.541 0.786 9.800 27.124 472.804
4500 258.931 57.777 0.777 9.800 26.837 472.212
4600 258.282 57.021 0.769 9.800 26.553 471.620
4700 257.633 56.273 0.761 9.799 26.270 471.027
4800 256.983 55.533 0.753 9.799 25.990 470.433
4900 256.334 54.801 0.745 9.799 25.713 469.839
5000 255.685 54.077 0.737 9.799 25.437 469.244
5100 255.037 53.360 0.729 9.799 25.164 468.648
5200 254.388 52.652 0.721 9.799 24.893 468.051
5300 253.739 51.951 0.713 9.798 24.625 467.454
5400 253.090 51.257 0.706 9.798 24.358 466.856
5500 252.441 50.571 0.698 9.798 24.094 466.257
5600 251.792 49.893 0.690 9.798 23.832 465.657
5700 251.143 49.222 0.683 9.798 23.573 465.057
5800 250.494 48.559 0.675 9.798 23.315 464.456
5900 249.845 47.902 0.668 9.798 23.060 463.854
6000 249.197 47.253 0.661 9.797 22.806 463.251
6100 248.548 46.612 0.653 9.797 22.555 462.648
6200 247.899 45.977 0.646 9.797 22.306 462.044
6300 247.250 45.349 0.639 9.797 22.060 461.439
6400 246.602 44.729 0.632 9.797 21.815 460.833
6500 245.953 44.115 0.625 9.797 21.572 460.226
6600 245.304 43.508 0.618 9.797 21.332 459.619
6700 244.656 42.908 0.611 9.796 21.093 459.011
6800 244.007 42.315 0.604 9.796 20.857 458.402
6900 243.359 41.728 0.597 9.796 20.623 457.793
7000 242.710 41.148 0.591 9.796 20.390 457.182
7100 242.061 40.575 0.584 9.796 20.160 456.571
7200 241.413 40.008 0.577 9.796 19.932 455.959
7300 240.764 39.447 0.571 9.795 19.706 455.346
7400 240.116 38.893 0.564 9.795 19.481 454.732
7500 239.467 38.345 0.558 9.795 19.259 454.118
7600 238.819 37.804 0.551 9.795 19.039 453.503
7700 238.170 37.269 0.545 9.795 18.820 452.886
7800 237.522 36.740 0.539 9.795 18.604 452.270
7900 236.874 36.217 0.533 9.795 18.389 451.652
8000 236.225 35.700 0.526 9.794 18.177 451.033
8100 235.577 35.190 0.520 9.794 17.966 450.414
8200 234.929 34.685 0.514 9.794 17.757 449.794
8300 234.280 34.186 0.508 9.794 17.550 449.173
8400 233.632 33.693 0.502 9.794 17.345 448.551
8500 232.984 33.206 0.496 9.794 17.142 447.928
8600 232.335 32.724 0.491 9.793 16.940 447.304
8700 231.687 32.248 0.485 9.793 16.741 446.680
8800 231.039 31.778 0.479 9.793 16.543 446.055
8900 230.391 31.313 0.473 9.793 16.347 445.428
9000 229.743 30.854 0.468 9.793 16.153 444.801
9100 229.094 30.401 0.462 9.793 15.960 444.174
9200 228.446 29.953 0.457 9.793 15.770 443.545
9300 227.798 29.510 0.451 9.792 15.581 442.915
9400 227.150 29.073 0.446 9.792 15.393 442.285
9500 226.502 28.640 0.440 9.792 15.208 441.653
9600 225.854 28.214 0.435 9.792 15.024 441.021
9700 225.206 27.792 0.430 9.792 14.842 440.388
9800 224.558 27.375 0.425 9.792 14.662 439.754
9900 223.910 26.964 0.420 9.791 14.484 439.119
10000 223.262 26.558 0.414 9.791 14.307 438.483
10100 222.614 26.156 0.409 9.791 14.131 437.846
10200 221.966 25.760 0.404 9.791 13.958 437.209
10300 221.318 25.368 0.399 9.791 13.786 436.570
10400 220.670 24.981 0.394 9.791 13.616 435.931
10500 220.022 24.599 0.389 9.791 13.447 435.290
10600 219.374 24.222 0.385 9.790 13.280 434.649
10700 218.727 23.850 0.380 9.790 13.114 434.007
10800 218.079 23.482 0.375 9.790 12.951 433.363
10900 217.431 23.119 0.370 9.790 12.788 432.719
11000 216.783 22.760 0.366 9.790 12.628 432.074Enjoy 🙂 -Zoe
Zoe's Insights 
HI Zoe,
Nice website.
I am wondering where you get your data for Vrms = sqrt(3*R*T/M). Is the source measured wind speed?
Thanks,
James McGinn
Solving Tornadoes: Woke Meteorology
https://anchor.fm/james-mcginn
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Vrms comes directly from Kinetic Theory:
http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/kintem.html
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Thank you for doing this. It would be very interesting to know the mean average temperature of the entire mass of the Earth’s atmosphere below the tropopause – especially if it turned out to be somewhere near 255K. Just imagine if your result found that the mean temperature of the entire atmosphere actually coincided with Earth’s effective S-B temperature. It would show that the Sun’s insolation was responsible for holding up the Earth’s atmosphere and the atmosphere held in Earth’s gravity field was delivering the adiabatic lapse rate that ensured that the near-surface temperature was 288K. Such a finding would entirely debunk the radiative Greenhouse effect theory. Of course, doubling the concentration of CO2 from, say, 410ppm to 820ppm would result in a change in the specific heat capacity per mole and a change to the molar mass, both of which affect the lapse rate. It would be ironic if a doubling of the CO2 concentration changed the lapse rate such that near surface temperature was raised by 3K to 291K. Any chance you could do these two calculations?
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Thank you for the comment, Steve.
The lapse rate is from the surface – upward. L = – g/Cp. It is not causative downward. I could do the calculations, but it would probably be meaningless. There is no single radiating surface. At the bottom of the atmosphere there is 42 mols/m^3, and 12 mols/m^3 at the top. The bottom shines through the top due to the density difference. Instruments in space detect an average of this:
. . . . .
…………….
(note the extra dots shining from bottom to space)
Even if the average emission was found for the average mass (somewhere in the middle of the troposphere), it would not work on Venus – and hence not a be a general theory.
I hope that helps.
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