Prolonged extreme heat usually arrives under a stalled ridge of high pressure rather than from a hot air mass moving in. The dome is a circulation pattern, and its persistence is what produces the extremes.
Why sinking air warms
Under a ridge, air aloft descends toward the surface. As it descends it moves into higher pressure and is compressed.
Compression raises temperature without any heat being added, which is the same reason a bicycle pump warms in the hand.
Air arriving at the surface from above is therefore warmer than the air it displaces, and the process continues for as long as the ridge holds.
How the pattern suppresses its own relief
Sinking air is drying air, because warming raises its capacity to hold moisture while the moisture content stays fixed.
Clouds cannot form in descending, drying air, so skies stay clear and the surface receives the full daily dose of sunlight.
Showers that would normally interrupt an afternoon are suppressed by the same motion, removing the one process that reliably breaks heat.
Why the heat accumulates rather than resets
Each clear day loads heat into the ground, and each clear night releases only part of it before the sun returns.
The unreleased remainder becomes the starting point for the following day, so the peak climbs as the pattern persists.
This is why the fourth or fifth day of a heat event is typically hotter than the first, even with no change in the pattern itself.
What makes a ridge stall
Weather systems normally move because the jet stream steers them, and when the jet flows strongly from west to east nothing sits still for long.
A weak or heavily meandering jet leaves large waves that migrate slowly or stop, and a ridge caught in that configuration can hold for a week or more.
Blocking patterns downstream reinforce the stall by leaving the ridge nowhere to move, which is the difference between a hot spell and a heat event.
Why terrain sharpens the effect
Air descending the lee side of a mountain range warms by compression on the way down and arrives dry as well as hot.
Valleys and basins trap that air, since it has no easy path out and cool air cannot readily flow in beneath it.
The most extreme readings under a dome therefore appear in downslope and enclosed terrain rather than on exposed ground.