How to calculate the climates.
Hi, recently I investigated how the winds affect the climates for a StackExchange Worldbuilding Section, and I believe it can help you a lot with how you calculate your climates.
Climates are indeed calculated from the monthly precipitation (rain) and temperature values in a year and the latitude.
For example, less than 3 mm of rain every month and high temperatures (more than 26 °C) mean that it's a hot desert, while the same precipitation values and a cold temperature (less than 3°C) and high latitudes mean that it's a cold desert, like Antartica.
It's all about the wind and the sun
The Hadley cell
In the equator, the sun heating the air near the surface generates an ascending current of air (because hot air is less dense than cold air, and less dense things float over more dense things). This ascending current drags all the humidity around. It ascends fast and it's condensed fast (because at higher altitudes the air is colder) and it precipitates most or all the humidity collected at the equator.
This wind reaches the higher layers of the atmosphere and expands and travels towards the poles. Because the Earth is rotating, and the equator is the fastest part of the Earth, the wind travelling towards the poles has more the speed of the equator, while the ground below doesn't, because it's not the equator, and this makes the wind to "get ahead" of the ground, in the rotation.
At middle latitudes, between 30° and 50°, the wind looses all its heat (radiated to the space) and becomes cold air, and it plummets towards the ground, creating a high pressure zone. If you go back a bit, you'll realize that this air lost all its humudity at the equator, so this descending cold air column is also dry air. This will originate tropical climates with extremely dry seasons.
Lastly, this cold dry air, will stay at low altitudes and will travel towards the equator, dragging with it any humidity it encounters. Again, as the higher latitudes are slower in the Earth's rotation than the lower latitudes, the wind will "lag behind" the ground, until friction with the ground gives the wind enough speed to stay still relatively to the ground. At the equator, the sun heats the air close to the ground and continues the cycle.
The Polar cell
There's another wind cell similar to the Hadley cell near the poles, and because the poles are perpendicular to the radiation of the sun, those are the descending column of the cold dry air, while the ascending column is between latitudes 49° and 65°, where the humidity is dragged and ascends due to the relative higher temperature and lower density. That ascending column of humid relatively hot air make the clouds condense and precipitate, bringing the cooling dry air to the poles.
The Ferrel cell
This cell works the same way as the others, but it goes inverted. At medium-high latitudes we have the column of humid hot air, while at medium-low latitudes we have the cdescending column of dry cold air. Also, the high altitude winds go towards the equator and lag behind the ground, while the low altitude winds go towards the poles "getting ahead" of the ground until they descend and friction with the ground make it stay still. From there it will go towards the pole to complete the cycle.
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How does this affect climates?
These cells define where the humid air precipitates, at low pressure places (ascending column of air), where the dry air robs the ground of its humidity and make sunny and dry places with high pressures (descending column of air), etc.
Temperature is mainly defined by latitude and closeness to the sea/ocean that can regulate it.
Precipitation is mainly defined by the atmospheric cells and influenced by the presence of hot or cold sea currents nearby.
Finally, disparity between seasons is defined by the tilt of the rotation axis, because the cells are delimited by the solar equator and the solar poles, not the geographical ones. So in summer in the north hemisphere, all the cells will be slided towards the north, originating the humid season in the north hemisphere tropical climates (perpendicular to te sun>ascending column of air>precipitations) and the dry season in the tropical climates of the south hemisphere. And vice-versa.
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