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From the weather forecasting perspective: what is the main reason for showing just map of 500 hPa geopotential? Are there any practical or theoretical reasons? Why do not show 600 hPa geopotential or 400 hPa instead? I understand that forecast model usually operates at pressure levels, so this is the reason why choose pressure level instead of quantity at some altitude.

Let us take e.g.:

https://www.ecmwf.int/en/forecasts/charts/catalogue/medium-z500-t850-public?facets=Range,Medium%20(15%20days)&time=2018112900,0,2018112900&projection=classical_europe

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    $\begingroup$ BTW Did you notice that the text below the map offers some explanation? ;-) $\endgroup$ – Jan Doggen Nov 30 '18 at 10:58
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    $\begingroup$ Not too much. :-) " The contours effectively show the main tropospheric waves that "control" our weather". But there is no explanation why just this level. There is only statement and not reason. $\endgroup$ – weatherman Nov 30 '18 at 11:01
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The 500 hPa geopotential is usually called the Level of non divergence. It is understood that beneath that level there is a level of convergence and above that level there is a level of divergence so that this surface(500 hPa) is the surface to look for vertical motions. To a first approximation one can assume the time derivative of density of a particular surface to be a constant and then set the continuity equation to be zero.

$$\nabla. u = 0 $$

We can then expand on the terms on the LHS and separate the horizontal components from the vertical components.

This turns out to be(in isobaric coordinates)

$$ \nabla_H. u_H = -\frac{ \partial \omega}{\partial p}$$

One can then conclude from this equation the following

1) If the LHS is less than zero then there is an increase in vertical velocity with decreasing pressure

2) If the LHS is greater than zero then there is a decrease in vertical velocity with decreasing pressure

Since the atmosphere is bounded at two levels (the surface and tropopause) the vertical velocity is close to zero at these two levels. However in between these two boundaries the vertical velocity starts to increase and achieves it's maximum value before again tapering off at the tropopause. The level of maximum vertical motion is usually the 500 hPa and this is also called the level of non divergence.

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The link you provided states at the bottom of the page that 500hPa (500mb) is used because the weather systems as the Earth's surface move with the winds at this height (approximatly 5.5km above the surface). It makes sense to show just the 500hPa on these maps as this is the most applicable height for forecasting weather system movement. More pressures/heights would make a single map confusing and not really be worthwhile. I hope this helps. Feedback would be welcome.

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