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Suppose we spot an Atlantic hurricane very early. How much energy should we expend to alter its track:

  • with nuclear detonations (let's forget about the side effects for the moment)?
  • with a large orbital geostationary mirror illuminating say, the subtropical ridge?
  • by any other means (microwave heating in the troposphere?)

References on these and other methods from the NOAA Hurricane FAQ:

Why the above-mentioned FAQ isn't enough

  • It's incomplete (doesn't include heating/illumination from satellites)
  • Doesn't include a fully-developed nuclear scenario
  • Has no formula linking time since hurricane formation, energy applied and cross-track difference at landfall on the East Coast
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    $\begingroup$ I added some information into my answer about a recent study that suggests that large offshore windfarms can reduce the severity of a hurricane. $\endgroup$ – user889 Sep 27 '14 at 10:19
  • $\begingroup$ Related question: earthscience.stackexchange.com/questions/2300/… $\endgroup$ – milancurcic Sep 27 '14 at 15:22
  • $\begingroup$ Sorry, this question is way too broad. It is also phrased badly: How much energy? and then you list all kinds of scenarios for which the amount of energy would be the same. Maybe you meant 'effort' but that isn't easily quantifiable. $\endgroup$ – Jan Doggen Sep 12 '17 at 12:57
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This is not a complete answer, but offers insights to research of some perspectives of the question you raised.

There has been a document recently written about this, Controlling Hurricanes Can hurricanes and other severe tropical storms be moderated or deflected? (Hoffman, 2004). In it, there are several main considerations, one major consideration in relation to your questions is that studies show that changes in the energy available can potentially alter the path of a hurricane. The studies reported in the article

confirms that these massive, chaotic systems are susceptible to minor changes in their initial conditions—for instance, the air temperature and humidity near the center of the storm and in the surrounding regions

Using computer simulations of hurricanes, they confirmed that

changes in precipitation, evaporation and air temperature could alter a storm’s path or weaken its winds.

Even though the article suggests one of your suggestions, about microwave heating of the air or the water, this power would be harnessed by satellite based solar panels, the main simulated methods appear not to be a case of adding the energy to the hurricane itself - but rather robbing the hurricane of its energy sources, examples include

Cloud seeding with silver iodide or similar material is suggested, these would promote precipitation, which would rob the eye wall of

the water it needs to grow and intensify

Another method, and rather unique is the use of biodegradable oil on the water's surface, specifically

Biodegradable oil could be distributed across the sea surface in the path of a hurricane to limit evaporation—the source of a storm’s energy

A study by Stanford University, reported in the article Offshore wind farms could tame hurricanes before they reach land, Stanford-led study says (Carey, 2014), suggest by computer modelling that

"We found that when wind turbines are present, they slow down the outer rotation winds of a hurricane," Jacobson said. "This feeds back to decrease wave height, which reduces movement of air toward the center of the hurricane, increasing the central pressure, which in turn slows the winds of the entire hurricane and dissipates it faster."

The data indicated that

In the computer model, by the time Hurricane Katrina reached land, its simulated wind speeds had decreased by 36-44 meters per second (between 80 and 98 mph) and the storm surge had decreased by up to 79 percent.

For Hurricane Sandy, the model projected a wind speed reduction by 35-39 meters per second (between 78 and 87 mph) and as much as 34 percent decrease in storm surge.

In terms of using a nuclear detonation to stop or divert a hurricane, the author of the article Can You Stop a Hurricane by Nuking It? (2012) describes the potential effectiveness of this method

might be about as effective as trying to stop a speeding Buick with a feather.

mainly due to as you said in your question:

the same amount of energy released by exploding a 10-megaton nuclear bomb every 20 minutes.

A nuke on water nearby may make the hurricane worse, by heating the water, thus 'feeding' the hurricane.

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