It takes energy to melt ice.
So if the world is losing over a trillion tonnes of ice a year, how much energy does that require?
It takes 334 kilojoules to melt a kilogram of ice -- or about 7,500 joules to melt an ice cube (1.5" x 1" x 1").
If it took three minutes to melt this ice cube, it'd require an average power of 42 Watts, which seems believable (as a check).
So melting 1,070 gigatons of ice a year requires 3.6e17 kJ/yr, or an average of 11 terawatts.
By contrast, human civilization now runs on about 16 TW.
Over the surface of the Earth, the energy to melt all this ice comes out to 0.02 Watts per square-meter -- or just a few percent of the Earth's current energy imbalance.
But it's a few percent, which isn't nothing.
With an ice density of about 900 kg/m3, this 1,070 Gt/yr of ice loss represents a decrease of about 1.2e12 cubic meters of water a year -- or 2.3 mm/yr over the entire Earth's surface.
There's your sea-level rise from ice melting -- about 2/3rds of the 3.2 mm/yr that is observed. The other third is due to the thermal expansion of water.
I like it when my numbers work out.
A pair of errors in the above needs to be corrected; they just about cancel each other out.
As someone pointed out on Twitter, I mistakenly included sea ice in my calculation of sea-level rise, and also divided by the surface area of the Earth instead of the surface area of the ocean.
Subtracting out the sea ice gives a volume change of about 8.7e11 m3/yr. Over the Earth's ocean, that comes to a sea-level rise of 2.4 mm/yr.
I know, of course, that floating ice that melts doesn't change sea-level. Just barfed on this one.