San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

Power Dissipation in Cyclones Such
as Hurricanes, Typhoons and Tornadoes

It is alleged, notably by Kerry Emmanuel a meteorologist at the Massachusetts Institute of Technology, that the rate of power dissipation in hurricanes, and therefore in other wind storms, is given by the formula

P = CDρv³

In this equation P is the rate of energy dissipation per unit time per unit horizontal surface area, v is the wind speed, ρ is the air mass density and CD is a coefficient depending upon the surface irregularities.

Kerry Emmanuel developed a theory about thirty years ago that tropical cyclones are essentially heat engines powered by the difference in the temperature of the sea surface and the upper atmosphere. (Hurricanes, typhoons and cyclones are the same thing but just given different names in different oceans.)

Any equation or theory should be viewed as a hypothesis or conjecture in search of empirical verification. There are even theories which have verification in the laboratory which are not valid for in nature because they leave out phenomena which are important in nature but are eliminated in the controlled setting of the laboratory. This point is driven home in the book Useless Arithmetic by Orrin H. Pilkey and Linda Pilkey-Jarvis. (The subtitle of this book is Why Environmental Scientists Cannot Predict the Future.)

Proposed theories concerning natural phenomena must always be supported empirically. This is always true but is more crucial in this day and age when purported scientific results are presented to support an ideological agenda. Kerry Emmanuel's work got tied to the issue of global warming in that if global warming increased sea surface temperature there would be more energy to dissipate in terms of more hurricane and typhoons and/or greater severity of these phenomena. Kerry Emmanuel has spent about the last twenty five years trying to empirically verify his model. However he was not able to establish a trend in number of hurricanes or in the severity of hurricanes to correspond with the upward trend in sea surface temperature. About five years ago Emmanuel took a different approach. He asserted that the power dissipation in a hurricane is proportional to the cube of its wind velocity. Thus the severity of a hurricane would be proportional to the cumulative sum of the cube of its wind velocity over time. This material examines the assertion that the rate of power dissipation of a cyclone satisfies the equation given previously.

Theoretical justification of an equation is significant in that it lends support for seeking the empirical verification. Here is a theoretical justification for the above power dissipation equation. Let m be the mass of a gas molecule which is traveling at a group velocity v. The kinetic energy of that molecule is ½mv². Suppose the collision of the molecule with an upright surface is inelastic; i.e., the molecule loses all of its forward motion. The energy loss in that collision would be ½mv². Now the problem is determining the number of such molecules hitting a unit area of vertical surface per unit time. That number would be the number of molecules contained in a prism having a unit area base and a length equal to v. The number would be the number density of the molecules times the volume, which in this case is v. The product of the number density and the mass per molecule is the mass density of the air, ρ. The energy dissipation per unit area per unit time is thus ρv³.

The coefficient CD then accounts for the amount of vertical surface area per unit of horizontal surface area.

Thus the velocity cubed law for power dissipation for hurricanes and typhoons has a simple and solid theoretical justification and justifies an effort at empirical verification.

Consider some of the implications of the equation. A category 5 hurricane has wind speeds in excess of 155 miles per hour (mph). A category 4 hurricane has wind speeds in the range of 131 to 155 mph. A category 5 hurricane with winds of 170 mph would have a rate of destructive power almost 80 percent greater than that of a category 4 hurricane with wind speeds of 140 mph. A tornado with a wind speed of 300 (F5 on the Fujita scale) would have a rate of destructive power per unit area nearly ten times that of a 140 mph hurricane and about 5.5 times that of a 170 mph hurricane.

Thus Kerry Emmanuel's formula for the power dissipation of a wind has a solid justification. The question of whether his heat engine model of tropical cyclones is verified or not is a different question. For that see Global Warming and Cyclones and Moving Averaging.

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