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In ERA5, and often in meteorology, heights (the height of the land and sea surface, or specific heights in the atmosphere) are not represented as geometric height, or altitude (in metres above the spheroid), but as geopotential as geopotential height (in geopotential metres above the geoid, which is represented by the mean sea level in ERA5). Note, that ECMWF usually archive the geopotential (in m2/s2), not the geopotential height.
To obtain the geopotential height (h) in metres (of the land and sea surface or at particular heights in the atmosphere), simply divide the geopotential by the Earth's gravitational acceleration, which has a fixed value of 9.80665 m/s2 in the IFS. This geopotential height is relative to the geoid (over land and the ocean, mean sea level over ocean is assumed to be coincident with the geoid) - for more information see ERA5: data documentation - spatial reference systems.
Geometric height
The Geometric height is not represented in the IFS. However, the geometric height or altitude (alt) is given bycan be approximated by the following formula (neglecting horizontal variations in the Earth's gravitational acceleration):
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where Re is the radius of the Earth. This geometric height is relative to the geoid (over land and the ocean, mean sea level over ocean is assumed to be coincident with the geoid) and it is assumed that the Earth is a perfect sphere - for more information see ERA5: data documentation - spatial reference systems.
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