If I'm wrong about the heat above the troposphere being no concern, then shouldn't the heat be distributed throughout the entire volume of the atmosphere and flood waters per day?
The
accumulation of heat would reasonably be distributed throughout the entire volume of the atmosphere and flood waters per day, but as Bill Murray says, "It
just doesn't matter!!". At the end of the 40 days you get all that heat built up. Heat can be thought of as a quantity of stuff; it's not like temperature, which is a
measurement of a quantity of stuff. You are adding a whole bunch of heat by your scenario, and subtracting a very little heat by radiation to space, and whether you add it over an hour or a day or 40 days or a year doesn't matter much. What people have been calculating is the total heat added, not the rate at which heat is added, and the total heat added doesn't depend on the time period over which it's added. Now, if you start calculating the radiation to space, you would typically calculate that as a rate of heat loss per day or whatever; a little more is going to be lost over 40 days than is lost over an hour. But people have done these calculations before … the result ain't good for your scenario.