Standard Troposphere

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.sh

The 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 1Height (m)
Column 2Temperature (K)
Column 3Pressure (kPa)
Column 4Density (kg/m³)
Column 5Gravity (m/s²)
Column 6Mols/m³
Column 7Root 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*T

This 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 = 659.240

Verified! 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 12.628 432.074

Enjoy 🙂 -Zoe

4 thoughts on “Standard Troposphere

  1. 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?

    Liked by 1 person

    1. 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.

      Like

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