Permanent hurricane, one order of magnitude more powerful than your run of the mill ones.
And you can forget "the eye of the hurricane" - that happens when the hurricane gathers energy over large expanses of warm ocean water.
For this one, it is the center of the thing where the energy is pumped into the air column - you will get a permanent warm (say at 293K) whirlwind, ascending in the center, fed at the base by a cold air (at 253K) rushing in from all sides and subject to Coriolis "force"
(if you remember, the strength of the Coriolis effect is the strongest at the poles, it goes with the $sin(latitude)$).
A fire-tornado of sorts, not that hot as one over a forest fire, but quite strong with the amount of incoming power (over a small surface) it has to dissipate.
Hawaii has a surface area of 28,311 km² = 28.311e9m². Is situated on 19.8968° N latitude - in average per year it receives (Lambert's cosine law) 94% of the total energy flux of a surface with the solar radiation at normal incidence.
The solar constant at Earth surface after passing through atmosphere on a sunny day is 1025W/m².
The total power your island receives is $28.311e9 m^2 \cdot 1025W/m^2 \cdot 0.94 = 2.73e13W = 27.2TW$
First perspective point - the total power the island receives is:
- about a Hiroshima bomb exploring every 2.4 seconds (and having the energy dissipated over the entire area of the island)
- about 12 times the average electric power generated on Earth in 2008 (which is 2311.4GW = 2.3114TW)
Let's say that the island is all green and lush at a moment (won't be for long, I promise). Some energy will go into photosynthesis - say about 5% of it (yes, plants have terrible efficiencies in using the energy - lucky us, we can keep warmer)
Part of the energy will go in making the rain. Oh, hang on, the same energy that goes into evaporating water is released when the water condenses and falls as rain. So no, while there may be some fluctuations, on average there's no actual consumption, just forget I considered it.
Part of the energy will escape to the vacuum of space as radiation. That's again a bit 6/117 = 5.1% (look at "The Radiation Balance at Earth’s Surface" and note the "Only 6 of these 117 units are emitted into space beyond Earth’s atmosphere").
actually, that's a nice diagram
Let's say part of it is used by the inhabitants - hang on, unless they use the energy to create fuel (eerrr.... energy rich substances, that they'll export as such, as an energy sink), that part of the energy they are "using" is actually transformed back into heat. No dice.
So, bottom line. 10% of the energy lost on any other ways except heating the air. Which means the 90% (= 24.48TW) rest of the 27.7TW is going to heat the air above the island. That's gonna be quite an impressive thermal, the paragliders there should be delighted, isn't it? Well, isn't it?
Except... that the temperature differential between the island and the rest of the frozen sea around and that lotsa cold air that will want to get warm itself above your beautiful island! And that spells a huge trouble
Second perspective point
The power developed by a hurricane winds is a puny 1.5 terrawatts. And you have 16 times more to dissipate in an area thousands time smaller than the one a hurricane spans!
So while wind is only a small part of the overall energy output of a hurricane, it still generates vast amounts of power: around 1.5 terawatts, or just over a quarter of the world’s current total electrical generating capacity of 5.25 terawatts.
What happens when an underwater volcano dumps 1-2TW in the ocean - megaplumes dispersing ashes over 10–150km3 areas.
Bottom line, just forget about the magic, the nature is way more than your magic can imagine.