FacebookTwitterGoogle+RedditEmail

The Energy of a Hurricane

In Fidel Castro’s account of Hurricane Gustav’s impact on Cuba, he speculates about the energy such a storm can hold:

In all honesty, I daresay that the photos and film footage shown on national television on Sunday reminded me of the desolation I saw when I visited Hiroshima, victim of the first nuclear strike in August 1945. With good reason, it is said that hurricanes release an enormous amount of energy, equal, perhaps, to thousands of nuclear weapons like the ones used on the cities of Hiroshima and Nagasaki. It would be worthwhile for a Cuban physicist or mathematician to do the relevant calculations and make a comprehensible presentation.

Being Cuban (from New York) , and a physicist (when I was employed), I felt challenged to provide details. What follows are the results of a simple model.

I consider a hurricane to be a vortex with a local rotational speed that increases as the point of observation moves to smaller radius. The center of the hurricane is a zone of low pressure, while the exterior of the hurricane is a region of high pressure; air is drawn radially inward by this pressure gradient. Air drawn into the outer radius of the hurricane, Ro,  rotates with angular (or circular) speed vo. Let us say this condition applies to a ring (inward along the radial direction) of thickness dr. This ring has a height L, from sea level up to the top of the storm (what satellite cameras photograph). The rotation of this large cylindrical shell carries a total angular momentum (the product of total air-mass in the shell times average rotational speed), which we will label Mr.

The air in the outermost shell moves inward to the next nested shell of thickness dr (these shells are arbitrary mathematical conceptions), which must also carry momentum Mr since there is no significant force to impede the flow (this is the law of conservation of angular momentum). However, since there is less volume in this next inner ring, and since the air is not significantly compressed as it moves, the circular velocity of the flow must increase (the radial velocity also increases, but is of lower magnitude). This effect continues so that the circular wind speed, vi, at the inner radius, Ri, is very high. The total number of rings in the storm is given by (Ro-Ri)/dr, or simply Ro/dr when Ro is much larger than Ri. Since each ring has the same momentum Mr, the total momentum of the storm is equal to the product M = Mr x (Ro-Ri)/dr, or to a reasonable approximation M = Mr x (Ro/dr). [x has been used as the multiplication sign.]

The circular speed v at any radius r can be expressed as the product v = w x r, where w is the rotational or circular frequency. This frequency is low at the outer edge of the hurricane and increases as the point of observation moves inward. From the model here, one can find that the circular frequency varies with radius as follows,

w = M/(2 pi rho Ro L r r).

[I have dropped the x for multiplication.] This quantity is the total momentum of the storm, M, divided by the product of 2, pi = 3.14159…, rho = density of air, Ro, L, and the square of the radius in question.

Hang on, we’re almost there.

Because M, 2, pi, rho, Ro and L are constants, the angular frequency can be written as w = A/(r r), where A is a constant equal to M/(2 pi rho Ro L). Notice that for any pair (r1, w1), where perhaps these specific values were measured at radius r1, that

A = r1 r1 w1,

the product of w1 times the square of r1. So, once we measure the wind speed at any given radius, we can infer the rotational frequency there (w1 = v1/r1), and then we can define constant A and use this in the formulas shown to get w and v at any radius.

Now, we find the kinetic energy of every layer (shell), which is essentially the product of its mass times the square of its average circular velocity, divided by 2. Then, we add up all the layers for the total energy of the storm. I do this with differential layers and form the double integral, radially and vertically, but we will skip over that discussion. The result for the total energy of the storm is

E = pi rho L A A LOGe(Ro/Ri)

in units of energy called joules (if you fell asleep, this is the answer).

E is the product of five factors, the four leading ones being: pi, rho, L and the square of A; and the product of these four is multiplied by the mathematical function called the natural logarithm (LOGe) of the ratio Ro/Ri. The natural logarithm of 1 is equal to 0; LOGe(2.7182818…) = 1; LOGe(10) = 2.3026; LOGe(100) = 4.6052.

Let’s list parameters, choose values and find some numerical answers:

pi = 3.14159…

rho = 1.2 kilogram/(cubic meter)
[air at sea level]

L = 5000 meters (5 km)
[height]

A = 4,000,000 (meters squared)/seconds
[value chosen for rotational acceleration parameter]

Ro = 400,000 meters (400 km)
[outer radius]

Ri = 40,000 meters (40 km)
[inner radius].

Our choice for the value of A is consistent with:

v1 = 10 m/s,
r1 = 400,000 m (400 km) and
w1 = 0.000025 radians/s; and/or

v1 = 20 m/s,
r1 = 200,000 m (200 km) and
w1 = 0.0001 radians/s; and/or

v1 = 100 m/s,
r1 = 40,000 m (40 km) and
w1 = 0.0025 radians/s.

[There are 6.283 radians along a circle,
so 57.3 degrees/radian;
recall 360 degrees = 1 circle]

Note that 1 m/s = 2.24 mph (miles per hour);
1.61 km = 1 mile;
1 km = 0.62 mile.

For the values shown,

E = 6.944 x (10 to the 17th power) joules.

The energy released by the explosion of 1000 tons of TNT (a kiloton, abbreviated kt) is 4.182 x (10 to the 12th power) joules. So, E = 166,055 kt (or equivalently, 166.05 megatons). The atomic bomb exploded at Hiroshima on August 6, 1945 produced about 15 kt, so the model storm has the energy equivalent of 11,070 Hiroshima bombs. Most of the energy of a hurricane is dissipated as atmospheric turbulence and heating, and friction along the Earth’s surface, only a very tiny portion of it is absorbed by the structures built by humans.

Bear in mind that the energy of the hurricane is spread over a much larger volume than that of a nuclear explosion (so hurricane energy per unit volume is smaller), and it is released over a much longer period of time. But it is of awesome scale, and we are still as powerless before it as were our first ancestors four million years ago.

MANUEL GARCIA, Jr./strong>. is a retired physicist; e-mail = mango@idiom.com

 

Your Ad Here
 

 

 

 

More articles by:

Manuel Garcia, Jr, once a physicist, is now a lazy househusband who writes out his analyses of physical or societal problems or interactions. He can be reached at mangogarcia@att.net

April 25, 2018
Stanley L. Cohen
Selective Outrage
Dan Kovalik
The Empire Turns Its Sights on Nicaragua – Again!
Joseph Essertier
The Abductees of Japan and Korea
Ramzy Baroud
The Ghost of Herut: Einstein on Israel, 70 Years Ago
W. T. Whitney
Imprisoned FARC Leader Faces Extradition: Still No Peace in Colombia
Manuel E. Yepe
Washington’s Attack on Syria Was a Mockery of the World
John White
My Silent Pain for Toronto and the World
Dean Baker
Bad Projections: the Federal Reserve, the IMF and Unemployment
David Schultz
Why Donald Trump Should Not be Allowed to Pardon Michael Cohen, His Friends, or Family Members
Mel Gurtov
Will Abe Shinzo “Make Japan Great Again”?
Binoy Kampmark
Enoch Powell: Blood Speeches and Anniversaries
Frank Scott
Weapons and Walls
April 24, 2018
Carl Boggs
Russia and the War Party
William A. Cohn
Carnage Unleashed: the Pentagon and the AUMF
Nathan Kalman-Lamb
The Racist Culture of Canadian Hockey
María Julia Bertomeu
On Angers, Disgusts and Nauseas
Nick Pemberton
How To Buy A Seat In Congress 101
Ron Jacobs
Resisting the Military-Now More Than Ever
Paul Bentley
A Velvet Revolution Turns Bloody? Ten Dead in Toronto
Sonali Kolhatkar
The Left, Syria and Fake News
Manuel E. Yepe
The Confirmation of Democracy in Cuba
Peter Montgomery
Christian Nationalism: Good for Politicians, Bad for America and the World
Ted Rall
Bad Drones
Jill Richardson
The Latest Attack on Food Stamps
Andrew Stewart
What Kind of Unionism is This?
Ellen Brown
Fox in the Hen House: Why Interest Rates Are Rising
April 23, 2018
Patrick Cockburn
In Middle East Wars It Pays to be Skeptical
Thomas Knapp
Just When You Thought “Russiagate” Couldn’t Get Any Sillier …
Gregory Barrett
The Moral Mask
Robert Hunziker
Chemical Madness!
David Swanson
Senator Tim Kaine’s Brief Run-In With the Law
Dave Lindorff
Starbucks Has a Racism Problem
Uri Avnery
The Great Day
Nyla Ali Khan
Girls Reduced to Being Repositories of Communal and Religious Identities in Kashmir
Ted Rall
Stop Letting Trump Distract You From Your Wants and Needs
Steve Klinger
The Cautionary Tale of Donald J. Trump
Kevin Zeese - Margaret Flowers
Conflict Over the Future of the Planet
Cesar Chelala
Gideon Levy: A Voice of Sanity from Israel
Weekend Edition
April 20, 2018
Friday - Sunday
Paul Street
Ruling Class Operatives Say the Darndest Things: On Devils Known and Not
Conn Hallinan
The Great Game Comes to Syria
Jeffrey St. Clair
Roaming Charges: Mother of War
Andrew Levine
“How Come?” Questions
Doug Noble
A Tale of Two Atrocities: Douma and Gaza
Kenneth Surin
The Blight of Ukania
Howard Lisnoff
How James Comey Became the Strange New Hero of the Liberals
FacebookTwitterGoogle+RedditEmail