Differences among Height & Geopotential Height

You may recall from my previous discussions regarding gravity (not yet here), that gravitational acceleration at any location in the Earth-Atmosphere system is approximately :

gf,z = g45,0 (1 - 2.59 × 10-3 cos(2f))(1 - 3.14 × 10-7 z)

gf,z : gravitational acceleration at latitude f & elevation z
g45,0 : mean gravitational acceleration at 45° latitude & mean sea level = 9.80616 m/s2

MKS units are used in the above and subsequent equations. Using this equation, let's examine the difference among elevation (height above mean sea level) and geopotential height. Though a general equation for geopotential height can be written :

F = 0,z 9.80616 m/s2 (1 - 2.59 × 10-3 cos(2f))(1 - 3.14 × 10-7 z) dz

A more useful way may be separately considering affects of elevation and location on Earth.

Because of Elevation
For a location where gravitational acceleration equals its standard value, an accurate approximation for its vertical variation is :

g = g0 (1 - 3.14 × 10-7 z)

Thus

F = 0,z g dz

implies

F = 0,z g0 (1 - 3.14 × 10-7 z) dz
F = g0 (z - 1.57 × 10-7 z2)

The 2nd term of this equation is of concern, because it is that which makes Z differ from z :

Z = F / g0 = z - 1.57 × 10-7 z2

Thus, differences (Z - z) because of elevation are : 

  z (km)   - 1.57 × 10-7 z2(m)
   -1            .157
    0           0
    1           -.157
    2           -.628
    3          -1.413
    5          -3.925
    7          -7.693
   10         -15.7
   15         -35.325
   20         -62.8
   30        -141.3
   50        -392.5
  100       -1570
Because of Location on Earth
The gravity equation implies maximum gravitational force at the poles (where Earth radius is smallest and rotational centrifugal force is 0) and minimum at the Equator (where Earth radius is nearly largest and rotational centrifugal force is maximum). At mean sea level :

gf,0 = g45,0 (1 - 2.59 × 10-3 cos(2f))

Thus, the maximum at f = 90° latitude is 9.832 m/sec2 and the minimum at f = 0° latitude is 9.781 m/sec2. Thus,

gmax / g0 = 9.832 / 9.80665 = 1.00258
gmin / g0 = 9.781 / 9.80665 = .99738

You may notice that these values are essentially obtained from the multiplying factor in the equation above, with a slight difference because g45,0 is slightly less than g0. Neglecting gravitational height variations,

Z = F / g0 = gmax z / g0 = 1.00258 z   at the Poles
Z = F / g0 = gmin z / g0 = .99738 z   at the Equator

Thus, differences (Z - z) because of Earth location are : 

   z (km)   Poles (m)   Equator (m)
    -1        -2.58         2.62
     0         0            0
     1         2.58        -2.62
     2         5.16        -5.24
     3         7.74        -7.86
     5        12.9        -13.1
     7        18.06       -18.34
    10        25.8        -26.2
    15        38.7        -39.3
    20        51.6        -52.4
    30        77.4        -78.6
    50       129         -131
   100       258         -262

With much less variation approaching mid-latitudes.

Combined Effects
Adding both of the above factors provides an estimation of the differences among elevation and geopotential height, though integrating the top equation for a location and elevation for any specific location is not difficult. Doing this, the combined affects can be calculated. For example, 5645 m elevation at 37 °N :

F = 0,5645 9.80616 (1 - 2.59 × 10-3 cos(2f))(1 - 3.14 × 10-7 z) dz
F = 0,5645 9.79916 (1 - 3.14 × 10-7 z) dz
F = 9.79916 (z - 1.57 × 10-7 z2)
F = 9.79916 (5645 - (1.57 × 10-7)(56452)) = 55267.2 m2/sec2

Thus,

Z = F / g0 = 55267.2 / 9.80665 = 5635.7 m

Thus geopotential height is 9.3 m less than elevation for this situation.


Text is copyright of Joseph Bartlo, though may be used with proper crediting.

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