There is no missing heat in that scenario. The only heat that needs to be accounted for is the 0.5W/m2 imbalance at TOA because if that number is accurate then the only place it can be going is in the ocean.
A two part question on Energy imbalance .
Can we have a TOA imbalance of -0.5 W/m2? If not why not.
ANSWER we are neither a heat source or a heat sink
Energy in is energy out. In other words we cannot have a TOA imbalance because the TOA is where the energy in equals the energy out.
We can have a warmer atmosphere or ocean without having to violate that principal, but only if the input of energy [Sun] varies due to distance [Summer/Winter locally] North and south hemispheres depending on the elliptical orbit of the sun. Or due to intensity [solar Cycles]
In effect the temperature we have is balance of the energy in the ocean, land surface and air. In a mathematical model where the air and sea remained fixed the amount of heating up, the amount of clouds, would run like a clock and stay the same from 1 24 hour period to the next apart from the energy input.
In our world of currents and Coriolis forces and winds, erosion volcanoes etc where the heat is varies but if one area becomes hotter SOI, PDO, El Nino etc another becomes colder.
Adding CO2 to the air does not make the total energy in or out change one iota. It does modify where the heat is found and this should be more in the atmosphere [ Gates, Droedge, Mosher etc]. The air should be warmer Gates and when it isn’t for 16 years it is indeed a travesty for your argument and the IPCC.
What it implies is that the earth’s atmosphere is a lot more resistant to intemperate changes than most people here are prepared to realise.
The Sun’s rays are attenuated as they pass through the atmosphere, thus reducing the irradiance at the Earth’s surface to approximately 1000 W /m2 for a surface perpendicular to the Sun’s rays at sea level on a clear day.
, the sunbeam hitting the ground at a 30° angle spreads the same amount of light over twice as much area
Ignoring clouds, the daily average irradiance for the Earth is approximately 250 W/m2 (i.e., a daily irradiation of 6 kWh/m2)
The insolation of the sun can also be expressed in Suns, where one Sun equals 1000 W/m2 at the point of arrival, with kWh/m2/day expressed as hours/day