What would the effects be of a warm, sunny tropical island located in polar waters?

Question

So, suppose there exists an island (roughly the same size as Hawaii's big island) which is located well in the Arctic (or Antarctic) Circle, but magically has a tropical rainforest climate (warm temperatures, lots of rain, and bright sunlight even when the rest of the hemisphere is experiencing polar night.)

The climate is maintained by magical means, and has existed this way since before the evolution of modern humans, at the very least. It may have existed this way for much longer. At least part of the magic works by maintaining an artificial sun over the island (much smaller than the real one, located within the upper atmosphere. The artificial sun does not provide significant light or heat outside of the island's immediate vicinity.)

This artificial sun shines only on the island and the water within an arbitrary distance. Let's say that this distance is five miles, just for the sake of argument.

What climatic effects would this have on the island (outside of the artificial tropical climate), the local region, and the world as a whole? I am looking for weather, climate, and biological effects particularly, but any other notable effects I would like to hear.

Answer 1

Result

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.

Production

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)

Consumption

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.

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Bottom line, just forget about the magic, the nature is way more than your magic can imagine.

Answer 2

I assume you're asking, "what happens when the magic runs out?"

Your island is maintained by magical means. Therefore, the only weather patterns inside the limit of the magic is what is defined by the magic. Paradise. Wonderful!

Outside that magical limit we have standard polar behavior. Bears. Days that are six months long (give or take). Cold. Bears. Occasional seals and some left over equipment from various expeditions to the pole that got lost. I mentioned bears....

The only question that makes sense here is, what happens at the interface between those two worlds? For some insight, let's turn to the good folks at Britannica

Air masses are gigantic atmospheric volumes with very specific temperature and humidity characteristics. When two different air masses come into contact, they don't mix. They push against each other along a line called a front. When a warm air mass meets a cold air mass, the warm air rises since it is lighter. At high altitude it cools, and the water vapor it contains condenses. This type of front is called a warm front. It generates nimbostratus clouds, which can result in moderate rain. On the other hand, when a cold air mass catches up with a warm air mass, the cold air slides under the warm air and pushes it upward. As it rises, the warm air cools rapidly. This configuration, called a cold front, gives rise to cumulonimbus clouds, often associated with heavy precipitation and storms.

As air masses move, pushed by winds, they directly influence the weather in the regions over which they pass. In this way, they help to circulate heat and humidity in the atmosphere.

This is your classic problem of an immovable object (your magic) meeting an unstoppable force (the polar climate). What do you get? Storms. What are you probably going to get?

A permanent Arctic Cyclone.

So, the denizens of your island would have a lovely view straight up. But in all other directions, it would be a bit like being on the Truman Show from the inside.