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I read the paper, but found nowhere a breakdown of the temperature increase. Did I miss something? How much is due to the river discharge, how much due atmospheric forcing and due to the insolation. Their two dates 14th June to 5th July is exactly during peak insolation and in 20 days it can warm the water just by this.
Respectfully, you have missed something. In Section 4 of the paper they provide evidence of a 6degC average change in surface water temperature from 14 June to 5 July 2012. They also give spatial evidence that the greatest warming is closest to the river mouths of the Mackenzie, and warmth is distributed through the Beaufort sea in water eddies. They also compare to the much smaller increases in T in the Canada Basin well away from river mouths (but with the implication of similar insolation).
While I agree with you that they do not give quantitative estimates of the balance of heating from a) insolation, and b) river water influx, the qualitative balance giving significance to river water influx is there.
Thus, they do give estimates of the river discharge over that 3 week period
"The Mackenzie discharge typically peaks in June and remains high in July [Woo and Thorne, 2003]. For a peak flow of 33,300 m3/s [Environment Canada, 2013], the volume of the total discharge over the 3 week period is equivalent to a layer thickness of 0.19 m of warm waters across the entire open water area of 316,000 km2 on 5 July 2012."
So perhaps someone would like to estimate the observed warming to surface water expected over 20 days from insolation alone as a comparison with the 6degC change observed? That would help us evaluate their interpretation of the observational data.
There is also an interesting discussion of the likely effects of continental rivers on the Siberian side.
20 days of insolation upon _water_ should produce many times higher warming, even based on entirely napkin-kind of consideration: we know insolation 24/7 to "mostly melt ponds" area is able to melt dozens centimeters of ice thickness, while on the other hand, 0.19 meters of "warmer by 6 degrees celcius" water is roughly an equivalent to 6K * 0.19m * 4.187 / 334 = 0.014m (where 4.187 kJ/kg*K - water specific heat; 334 kJ/kg is latent heat of melting 1 kg of ice).
I.e., this means that all that "19 centimeters thick layer of 6°C warmer water" is only good enough to melt 1.4 centimeters of ice if all "extra heat" of that "warmer water" would be spent to melt extra ice. Which obviously not the case - some extra will be lost to evaporation, some more will directly radiate as IR, some more will be mixed into deeper waters. So it's more like 1 centimeter or even less.
Peak insolation in Arctic June is nearly 500 W/m2, in other words 0.5kJ/s*m2. Being generous, let's say only half of that is absorbed by water, and only 5 days out of 20 are sunny. Then 0.25*60*60*24*5 = 108000 kJ/m2 during those "20 days" from insolation. Very roughly, 1 m2 of 1-meter-thick ice is 1000kg (for simplicity), so this is 108kJ/kg if all that heat would be melting ice. Pretty much enough to melt 108/334=0.32 meters of ice thickness, that insolation heat alone.
That all said, to add to the discussion, - personally i see direct effect of Arctic rivers as indeed small, but in the same time i can fathom how it hits the ice exactly before maximum insolation period and thus produces lots of positive feedback in doing so; by itself, those "warmer river waters" melt very little (in terms of whole Arctic, i mean), but that "very little" then absorbs much sunlight while without river waters it would not (absorb much) still being much higher albedo (ice), and then whole thing cascades outwards.
It's sort of a primer on a gun's ammo: by itself, it can't propel the bullet, but it ignites the powder which can. Same with river waters (not always, but quite often) - rather large places which could remain frozen for much longer start increasingly fast melting if large river creates initial area with low albedo (obviously, the closer to the river's delta, the higher influence river waters will have locally).
