According to one popular theory, the sun circles closest to the North Pole in June, then spends the next six months spiraling slowly outward toward the ice wall at the edge of the world.

The diameter of these sun-circles governs the seasons. The light of the sun is confinedto a limited area, and its light acts like a spotlight upon the Earth.' 'The sun moves in circles around the North Pole. Here's how members of the Flat Earth Society (one of the foremost flat-Earth activist groups in the world) describe the idea on their official wiki page:

Many modern flat-Earthers now believe that the sun sits about 3,000 miles (5,000 km) over the Earth, but Rowbotham's general idea remains popular in the community. How teeny and how close is it? According to the early flat-Earth thinker Samuel Birley Rowbotham, who published the influential treatise 'Zetetic Astronomy: Earth Not a Globe' in 1881, the sun is only about 32 miles (52 km) in diameter and hovers anywhere from 400 to 700 miles (640 to 1,130 km) above the Earth, depending on the month.

While you might envision the sun as an enormous ball of exploding gas located 93 million miles (150 million kilometers) away, a flat-Earther would see it as a teeny, tiny spotlight hovering just over the Earth. Many flat-Earthers agree that the sun perfectly circles the ring of the equator on the equinox however, to account for the equal hours of daytime and nighttime, the models make a few tweaks to how the sun itself looks and behaves.

High resolution gleason flat earth map