Three things for flat earth fanatics to ponder.
1) The surface area of the flat earth model is approximately ten times the surface area of the spherical earth model. The distance from the North to South pole (or "edge") is 40,075km, giving a surface area of 5045415473.12km^2. The radius of the earth is approximately 6,400km, giving a surface area of 510,072,000 kmē.
A similar ratio can therefore be applied to the troposphere (First ~10km of the atmosphere, 10km << radius of the earth, so the surface area of the top of the troposphere is negligibly larger than the surface area at ground level).
The magnitude of greenhouse gas emission causes an increase of GHG concentration in the atmosphere. The majority (>99%) of these emissions remain in the troposphere. Observed increases in concentration accurately reflect measured GHG emissions for the spherical earth. (See IPCC AR5, chapter 8
http://www.ipcc.ch/report/ar5/wg1/)
This doesn't work for a flat earth model, unless the troposphere is ten times thinner. This clearly isnt true. (Temperature profiles show an inversion taking place at 10km (The tropopause), and in addition, almost all weather processes require heights in excess of 1km, which can only occur in the troposphere).
2) Measurements of the ACC (Antarctic Circumpolar Current) are incorrect on a flat earth model. This is most notable through looking at the observed current speeds (
http://www.nature.com/ngeo/journal/v7/n2/full/ngeo2037.html). The time taken for a tracer to complete one full "lap" of this current would be magnitudes higher than observed on the flat earth model.
3) The coriolis force. More specifically, the concept of geostrophic balance. there is no equivalent mechanism on a flat earth model. The coriolis force is an essential component of processes such as vorticity conservation, longwave rossby waves at mid-latitude, and in limiting equatorial weather signals (e.g. the MJO) to keeping at low latitude.
These three concepts are unequivocally agreed upon in the scientific community, because of the wealth of observation and measurement in favour of them for a spherical earth, and lack of alternative explanation. More importantly, they can only function correctly on a spherical earth.
These points provide a thorough explanation as to why the earth must be spherical.