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Around that time someone in the US government proposed to use nukes to widen Panama Channel.

Project Plowshare was the overall United States program for the development of techniques to use nuclear explosives for peaceful construction purposes.

Using nuclear power to thaw Greenland perfectly fits the enthusiasm of those years toward the use of nuclear power.

Sonic Cannon

From the Israelite army's trumpet-blaring priests at the battle of Jericho 3,500 years ago to today's modern LRAD (long-range acoustic device) cannons, sound has been used to harm and destroy.

Granted, it would take a lot of it.

Assuming you have 1 gram of snow at 0 C, the amount of energy needed to melt that is 334 Joules. The sound from an entire orchestra only amounts to 1 W of energy. If you could somehow focus all of the energy from the symphonies music onto that ice, it would take 334 seconds to melt it, a full 5 minutes. And that's an entire symphony focused directly on a little more than a tablespoon of freshly fallen snow. (Source)

However, orchestras are not amplified and the sound is highly distributed. That same orchestra, pumped through my meager 25W-per-channel high-school-era stereo amplifier would melt 50g of that same snow in 5 minutes, or 1g in 6 seconds.

Now let's back that up with the electrical power generating abilities of the Iowa-class U.S.S. Missouri battleship!

The four engine rooms each has a pair of 1,250 kW Ship's Service Turbine Generators (SSTGs), providing the ship with a total non-emergency electrical power of 10,000 kW at 450 volts alternating current. Additionally, the vessels have a pair of 250 kW emergency diesel generators. (Source)

Ignoring the details of what 450 VAC can do with a speaker (a lot...), that's 510KW of power! In that same 5 minute period we can now melt 510 Kg (half a metric ton) of snow!

To be fair, it's not efficient.¹ And I'm ignoring a lot of stuff that would get in the way (like how much power would be absorbed by liquid run-off (heating the water) rather than being used to melt the ice and snow.) But! It's a technology of the time that could be used to solve the problem with its own set of pros and cons. And you get to use an Iowa-class battleship! How cool is that?

_{¹ Certainly not as efficient as L.Dutch's nukes! Not by a long shot. But it does have the advantage of leaving the landscape radiation-free.}

One possible strategy would be to take many thousands of large black plastic sheets and place them on top of ice sheets during the summer, weighted down with rocks or clumps of ice. The plastic should heat up in the sunlight and melt some of the ice below it, possibly down to the ground.

Or lots and lots of black carbon particles could be strewn on top of the ice to melt their way down into it.

Possibly atomic bombs could be exploded over glaciers seeded with materials that would adsorb the various types of radiation from the bombs and turn that radiation into heat that would melt the glaciers.

Or large flat objects with mirror-like surfaces to reflect sun light could be laid on the ground right below the southern edges of glaciers. They would reflect sunlight toward the glaciers and melt them back.

A statite (a portmanteau of static and satellite) is a hypothetical type of artificial satellite that employs a solar sail to continuously modify its orbit in ways that gravity alone would not allow. Typically, a statite would use the solar sail to "hover" in a location that would not otherwise be available as a stable geosynchronous orbit. Statites have been proposed that would remain in fixed locations high over Earth's poles, using reflected sunlight to counteract the gravity pulling them down. Statites might also employ their sails to change the shape or velocity of more conventional orbits, depending upon the purpose of the particular statite.

https://en.wikipedia.org/wiki/Statite1

A vast fleet of statites could be placed over the north polar regions with their solar sails angled to reflect sunlight down onto selected Greenland glaciers to melt them, possibly in conjunction with other methods to melt the glaciers.

Ice sheets contain enormous quantities of frozen water. If the Greenland Ice Sheet melted, scientists estimate that sea level would rise about 6 meters (20 feet).

https://nsidc.org/cryosphere/quickfacts/icesheets.html2

So melting too much of the Greenland Ice Sheet could be considered a hostile act by many other governments ruling low lying coasts.

For example, Cape May, New Jersey, has been flooded by the sea during at least two or three storms since 1956, and has an elevation of 10 feet (3 meters), the highest point in the city, at the corner of Washington and Jackson streets is 14 feet (4.3 meters) above sea level. Residents of Cape May, and New Orleans, and many other coastal communities, would demand that the US government prevent any project that would melt enough ice to raise sea level by several feet.

I think other answers assumed you were trying to terraform Greenland. If this is just a mining operation, people do mine in Greenland and there is apparently more interest in this now as the ice melts and access is easier.

For purposes of mining, ice is treated as low strength rock, and removed with standard mining methods.

Open-Pit Glacier Ice Excavation: Brief Review. Copyright 2013

Open-ice-pit mining, in order to recover a subglacial mineral deposit, is dependent on safe and predictable large-scale ice excavation...Three distinct ice-excavation tech- niques are reviewed: blasting,
melting, and mechanical excavation, providing a case study of each.
The authors summarize the unique advantages and disadvan- tages of
each technique and conclude that an optimal open-ice-pit mining opera-
tion would most likely rely primarily on mechanical excavation and
secondarily on blasting.

The paper covers technology used in Greenland between the end of WW2 and the present. It is mining technology, adapted to the different density and mechanical properties of ice. They loosen it up and move it out with machines, as is done with open pit mines elsewhere. Not super sexy, and it doesn't really open up new areas for habitation because I gather the low lying mines tend to fill back up with water - a property also shared with mines elsewhere.

You are talking about permanently changing the climate of Greenland. Just melting the current ice is not quite enough.

Use the greenhouse effect

If you intentionally manufacture and release powerful greenhouse gasses, the global climate will warm enough that Greenland will defrost. This has some obvious flaws.

First, it is too slow for your purposes. Second, it would be expensive. Three, greenhouse effect was only fully understood in the late 60s and early 70s, too late for your purposes. Four, you'd more or less permanently mess up the rest of the planet and >99% of human population would have valid reason to want you dead.

The only real benefit this approach has is that it can happen accidentally. Maybe this gas is really useful and you manufacture lots of it. Maybe a nuclear explosion or volcanic eruption releases ridiculous amounts of a greenhouse gas.

Solar mirrors

By putting sufficient area of mirrors in space on polar orbits configured in away that reflects sunlight on Greenland you can in theory increase the temperature selectively.

The biggest downside of this is that Greenland is large, so you'd need a ridiculous amount of mirrors. Which you'd have to launch to orbit. The cost would literally be astronomical. There is nothing in Greenland AFAIK to justify it.

This would also still mess up the climate. And it would few decades ahead of its time for the 50s. This is clearly post Apollo Program (1960-1972) technology.

Just heat it up

Just directly apply heat to Greenland.

The simplest way to do this would probably be to take deep sea water off the coast which is always few degrees above freezing and pump it up. It will freeze and release heat to the environment. This would still be ridiculously expensive since you'd need to pump up ridiculous amounts of water but it is probably the most efficient way to apply heat.

Just have a nuclear reactor and transfer the heat it produces to deep ocean water. This will make the water to rise to the surface and melt the ocean ice. This might be done as a way to keep shipping lanes in Northern Greenland open all year for military purposes. Pretty sure it makes absolutely no sense from economic standpoint as the cost of building and maintaining the needed reactors would be far beyond any possible benefit.

Nuclear powered electric plants generate waste heat. Lots of it. Set up electric generating stations in Greenland to power all of North America. Use the waste heat to melt the ice.

The standard way to melt icebergs would have been entirely possible with 1940s technology.

It's as simple as it's effective - high pressure seawater. Very, very effective (high specific heat content, salt, almost trivial cost of deployment and inexhaustible). Low cost of deploying multiple of them, as well.

You do not want to be using flamethrowers or lasers and similar on sizeable icebergs or coastal ice buildups - they may be great for some things, but hopelessly outclassed in this job, for sheer ease, speed and efficiency, by high volume water canon.

For huge 'bergs, as the ice becomes cut up, the smaller bergs also become easier to separate, ending the cold microclimate that surrounds huge 'bergs, and making them more vulnerable to being tugged, pushed away (again with water jets), and exposing more surface to the sea/air/pressure hoses.

There's no reason this couldn't also be used with coastal and continental ice as well as floating ice, if it's either relatively close to the coast, or one can drill through it to seawater.

Are flamethrower brigades out of the question? According to Wikipedia, Germany started producing flamethrowers as early as 1911. I think it would not be far-fetched to be building fleets of flame tanks by the 40's.

Other options include: beaches and beaches of salt grit, large scale greenhouse construction, teams of people with tractors / dump trucks,

AND, my personal favourite, artificial explosive insemination to disrupt the active hotspot under all the ice, causing a massive volcanic eruption.

May I point you to the real-life U.S. military base of Camp Century, part of Project Iceworm (construction started in 1959):

Project Iceworm was the code name for a top secret United States Army
program of the Cold War, which aimed to build a network of mobile
nuclear missile launch sites under the Greenland ice sheet. The
ultimate objective of placing medium-range missiles under the ice —
close enough to strike targets within the Soviet Union — was kept
secret from the Government of Denmark. To study the feasibility of
working under the ice, a highly publicized "cover" project, known as
Camp Century, was launched in 1960. Unstable ice conditions within
the ice sheet caused the project to be canceled in 1966.

According to Science Leads the Way and other sources, the U.S. Army Corps of Engineers built an entire nuclear-powered Arctic research center into the glaciers:

Long ice trenches were created by Swiss made “Peter Plows”, which
were giant rotary snow milling machines. The machine's two operators
could move up to 1200 cubic yards of snow per hour. The longest of the
twenty-one trenches was known as “Main Street.” It was over 1100 feet
long and 26 feet wide and 28 feet high. The trenches were covered with
arched corrugated steel roofs which were then buried with snow.

The shifting glacier made the project unsustainable, so the project and its stash of irradiated waste were abandoned the ice in 1966 -- only to begin reemerging in recent years as Greenland's ice melts.