Sure, I’ll try, and I’ll stay away from definite integrals if you could grace me with a tiny bit of religious faith.
Let’s look at the phrase “Watts per square meter”. That is the rate of energy transfer per unit area. Yes, that word Watts (W) has the exact same meaning as what’s printed on your light bulbs (the old kind). So if your 100W light bulb has, say, 10 sq inches of surface area, then it is correct to say the flux of the light bulb is 10 W/in^2. There’s a mess of sq inches in a sq meter, so that would shoot up the value if we were to use metric units. Yes … if the Earth shined like a light bulb, we wouldn’t call it a planet, we’d call it a star.
Flux is measuring how much energy flows through an area over so much time. We use Joules (J) for measuring the amount of energy, a square meter (m^2) as a standard unit area, and the second (s) as the basic unit of time. Thus J per s per m^2 or J/s-m^2 and since W=J/s therefore W/m^2.
From the 1st Law of Thermodynamics, the total energy the Earth absorbs from the sun must be equal to the energy the Earth radiates out into space for the total energy the Earth is holding to remain the same. Temperature is a direct measure of this energy content. The forcing in this case is zero … the positive value of energy input is added to the negative value of energy output gives exactly zero. Input flux + Output flux = 0.
Turns out this forcing is actually positive. The Earth is either receiving more sunlight, or something is interfering with the Earth’s ability to radiate. Whchever is happening, the total energy being held by the Earth is increasing, and it’s direct measure of temperature is also increasing. For example, if you’re pouring gas into a tank at one gallon per hour, and the engine the tank is feeding consumes one gallon per hour, the level in the tank stays the same. Now, if a wolf comes and bites you in the ass, you’re going to slosh more gas in, the engine consumes the same, the level in the tank will go up.
I came up with +0.000031 W/m^2 for average forcing over the 10ºC temperature rise we’ve seen these past 10,000 years. The calculations were giving to a doper for double checking, currently with no response … and it involved multiplication so we know the yo-yos can’t understand it.