Height & Pressure Coordinates

Introduction

Scientifically forecasting our weather involves an estimate of the behavior of an air and water vapor atmosphere above the surface of a planet (Earth). Though surface conditions primarily concern us, forecasting success requires accurate prediction of atmospheric flow aloft. Similar with surface charts, upper air charts are used for this.

Pressure as a Vertical Coordinate

As illustrated below, upper air analysis is much more convenient using pressure rather than height as the reference vertical coordinate. Other than in violent atmospheric circulations with locally rapid accelerations such as tornadoes, pressure always decreases with increasing altitude (else an upward acceleration greater than gravity's downward acceleration is necessary). Thus for all practical purposes, pressure is a continuously and smoothly decreasing function with respect to elevation (height above mean sea level). Just as the horizontal direction defines a surface of constant elevation or height, a surface of constant pressure can also be defined. Just as our previous surface pressure analysis was for a constant height of mean sea level, upper air air analyses are done for constant pressures (surfaces) aloft.

Pressure and Height Gradients

For mathematical analysis, transformations among height and pressure coordinates are necessary. Particularly, pressure gradient is 0 on a constant pressure surface (no change of pressure along a constant pressure surface), but height gradients exist (height changes along a constant pressure surface). We shall see that a height gradient on a constant pressure surface is analogous with a pressure gradient on a constant height surface. The required transformation is :

[¶P/¶x]_z = - [¶P/¶z]_x [¶z/¶x]_P
      A                       B                  C

P : pressure

for which [---]_a implies the quantity --- in brackets is valid for constant values of the variable a. Thus, the above equation states that the partial derivative of pressure with respect to horizontal on a constant height surface (A) equals the negative of the partial derivative of pressure with respect to height at some place along the horizontal (B) times the partial derivative of height with respect to horizontal on a constant pressure surface (C). You may recall in a previous discussion (not yet here) that hydrostatic balance implies :

¶P/¶z = - r g

r : density
g : Earth's gravitational acceleration

Thus,

[¶P/¶x]_z = r g [¶z/¶x]_p

I.e., pressure gradient on a constant height surface is proportional with height gradient on a constant pressure surface, as illustrated above & right.

You may notice that because constant height & pressure surfaces do not exactly correspond (as illustrated earlier), neither do the contours on corresponding maps, though the 700 mb & 3150 m contours would be exactly same for these maps for those height & pressure.

Geopotential Height

F = _0,z g dz

Because gravity is nearly constant in our atmosphere,

F @ g_o z

g_o : standard gravitational acceleration at mean sea level = 9.80665 m/sec²

Energy = Mass × Acceleration × Distance, and specifically, Potential Energy = m g z (assuming constant g). So as its name implies, Mass × geopotential is the gravitational potential energy a mass has if suspended at height z. More subtle dynamical meteorological consequences of this exist which I hope I can discuss later, but my purpose for mentioning it now is for describing geopotential height (Z) :

Z = F / g_o

which is used instead of height for most forms of meteorological upper air data. The main reason why is that mathematical analysis of dynamical equations is much easier after doing this. Perhaps you can see that for most locations (particularly aloft) geopotential height underestimates height; but this difference is generally small where weather occurs in our lower atmosphere (generally a few meters or less) - enough so that the difference is often ignored. The mention of "height" on a meteorological upper air sounding or chart more likely means geopotential height than actual height. This assumption is not very good above the tropopause though.

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