by Academician Vasily SHULEIKIN
FROM DESCRIPTION TO ANALYSIS
Every year newspapers and periodicals carry dramatic and tragic accounts of natural disasters caused by tropical cyclones and hurricanes in the Atlantic, the Indian and the Pacific oceans. Tropical cyclones and hurricanes are calling for closer studies in view of the growing fishing fleets operating in the tropical latitudes of all oceans, especially in the Atlantic. Soviet fishing fleets are also operating in this plentiful belt which means that studies of tropical hurricanes, apart from their fundamental scientific significance, have obvious economic importance.
When they spot the signs of an approaching hurricane on the horizon, there is still time to move away from its path if you know the laws of its origin and progression. Although wind velocity in a hurricane system can approach 100 m/sec, and be even higher in some cases, it takes the hurricane several days before it achieves full force. And it travels along its trajectory no faster than 6 - 8 m/sec.
But what about a hurricane's progression? What laws guide it on its paths which can be very winding? To get the right answers to these questions it is not enough conducting systematic observations of tropical storms like those carried out by meteorologists and oceanographers in various countries. One has to understand the physics of such phenomena, to trace the mechanism of "energy supply" of a tropical cyclone and find the source of that energy. One has to establish the top levels of a hurricane force depending on the conditions of its development and study the conditions in which it dies down. And finally we have to study the forces which control the movement of tropical hurricanes across the oceans and, sometimes, push them on shore.
It was not too long ago when they tried to link the development of hurricanes with the instability of the atmosphere over the oceans at some seasons of the year. But, firstly, no one has been able to trace an analytical dependence between the degree of instability, expressed in arbitrary units, and the mounting strength of hurricanes. Second, at the present time we have objective and reliable data to the effect that the strongest hurricanes in regions like the Caribbean or the Mexican Gulf occur when instability is either very low, or is of a negative kind-that is the atmosphere is stable and its maximum instability is associated with seasons with very rare and weak hurricanes.
Today one can still find in scientific publications claims that tropical hurricanes originate in some intermediate
region, between two or even three regions dominated by anti-cyclones. It is assumed that under the effect of clockwise air currents in the northern hemisphere there should develop in the intermediate region anti-clockwise air currents, typical of hurricanes. We shall not dwell upon this hypothesis because it makes impossible building some physico-mathematical model for a qualitative description of the development of a tropical hurricane and contradicts the observed movement of hurricanes along their trajectories. At the same time it was noticed for quite some time that the real "guides" for the movements of hurricanes are warm surface currents like the Gulf Stream, the North-Atlantic current in the Atlantic, the Kuroshio in the Pacific and warm currents in the Indian Ocean.
TROPICAL HURRICANE - THERMAL MACHINE
Tropical hurricanes have never been observed in the Atlantic south of the equator where there are no warm currents. And the obvious question is: is it not from such warm currents that hurricanes receive their staggering power? Can they be regarded as thermal, or "heat machines" of a kind? And according to the laws of thermodynamics mechanical work produced by machines depends on the amount of heat supplied by some heater and transferred in part to some kind of a cooler. If in our particular case the "heater" are the surface layers of warm currents, then we can regard as a "cooler" the whole semi-space over the ocean covering the hurricane system from all sides. Is that not so?
This approach has made it possible for the author of this article to suggest a physico-mathematical model of a tropical hurricane, to trace a connection between its power and the temperature of the underlying oceanic water surface, to formulate the laws of development and termination of a tropical hurricane in different thermal conditions, to outline the basics of calculating the trajectories of tropical hurricanes on the basis of sufficiently reliable data of weather maps.
And it was necessary to give up attempts of solving all of these problems by standard methods: even the best modern computers cannot solve completely the most complicated thermo-hydrodynamic problems in the integration of complete equations. Simple omission of some of the members of these equations has led, for example, Japanese scientists to results which contradict direct observations in nature. Therefore the work published by the author of this article in 1970 was based in part on the sufficiently reliable, but purely empirical data obtained by the Finnish geophysicist Prof. E. Palmen and others as a result of large number of measurements in natural conditions.
As is known, located in the central part of a hurricane system is a narrow area of practically complete calm. Close to zero there are the radial component of wind velocity (u) and tangential component (v), which is perpendicular to the radius and is directed to the right from it. So let us assume that on the outer border of that area, called "hurricane's eye", that tangential component reaches its maximum v = Ao. And at even greater distance from the center of the system it decreases according to the law of the logarithm. On the outside border of the hurricane system we assume v = 0. We have chosen a particular case of a typical strong hurricane with the maximum value of Ao = 75 m/sec at the altitude of 500 m above sea level (theoretical calculations are conducted relative to this level; from them one can proceed to the figures of velocities at the ocean level, using a transitional multiplier which we shall mention later). Analyses of current photos of cloud formations in hurricane systems obtained from space-probes show that air particles in hurricane systems move along curves which (in projection) can be regarded as very close to logarithmic spirals. And the conclusion is that particles' trajectories everywhere are at a permanent angle to radii-vectors drawn from the center of the system (a feature of the spiral). In this particular case this angle is close to 18° and therefore the ratio of the radial component of velocity (u) to the tangential component (v) remains constant and is equal to the tangent of this angle u/v = 0.325. On the basis of that a curve was built describing the reduction of the radial component u* and wind velocity with growing distance from the border of the "hurricane's eye".
Having formed an idea about angles of the radial component, one can use the continuity factor for the determination of the vertical component of the air currents from the radius of the external border of the "eye" to the radius of the external border of the whole area of the hurricane. Some simple calculations made in 1970 led the author of the article to formulating the law of variation of the vertical component w. For our typical hurricane we obtained the velocity values w (cm/sec).
It is very important that at a certain point-at a distance of 220 km from the center - the latter curve crosses the abscissa axis. That means that from the area of a circle, "curved" on the surface of the ocean with an inner circumference with the radius of 15 km and the outer one of 220 km, air streams upwards from the surface of the ocean. And it carries away water vapor which almost saturates the near-water layer of the atmosphere. When this vapor gets into much cooler atmospheric layers, located hundreds and thousands of meters above the sea level, the vapor is condensed forming the well familiar dense clouds in the hurricane area with showers and thunderstorms. And every gram of condensed vapor releases 539 cal. of heat.
It is this factor which determined the vertically averaged density of the air "overheated" in the "eye of the hurricane"-an area limited on the ocean surface by circumferences with the aforesaid radii. It was assumed that the other components of the heat balance
* E. Palmen and other researchers point not only to this empirical curve describing the reduction of the tangential component "v", but a similar curve describing the reduction of the radial component "u". But this component is small as compared with full wind velocity "V". Therefore its measurements in natural conditions are unreliable. The author of the article therefore favors a different method of measurement "u". - Auth.
of the atmosphere are mutually compensated: the additional supply of heat through the contact and radiant heat exchange with water surface is "balanced off' by additional heat emission by heated air into the outer space. But is this assumption legitimate? The answer is "yes" which is proved by calculations of the vertically averaged air temperature at different distances from the "eye" within the confines of the "hurricane core" by two independent methods. On the one hand, hydrodynamic equations make it possible to determine how atmospheric pressure "behaves" in the hurricane system.
Calculations performed by the author of the article in 1970 showed that atmospheric pressure should drop from the outer border of the whole hurricane system towards the outer border of its "eye". In this case of our typical hurricane the pressure dropped down along the whole distance by 85 mb. And one should note with satisfaction that it is this complete drop of pressure which is observed in nature during the formation of hurricanes. And the formula obtained from our theory in 1970 is not only identical by its composition with the empirical formula obtained by practical observations of 14 hurricanes of different strength, but also by the numerical coefficient which appears in the formula.
If one knows the law of pressure reduction, one can calculate by how many degrees rises the air temperature, averaged along the vertical line at this or that distance from the "eye". Then one of the curves represents the results of calculations of overheating by that method and the other expresses the regularity of distribution of air overheating found on the basis of our simplified notion of the thermal balance of air - the supposition that the release of latent heat during water vapors condensation determines the overheating of air within the confines of the "hurricane eye". Both these curves are located very close to one another. And that means that the assumption made by the author is legitimate: at the cost of some inaccuracies it has been possible to get important information about the sources of energy of a hurricane are located on the surface of the ocean with its higher temperature.
STRENGTH OF A HURRICANE
The formula of calculating the power supplied by the ocean to the whole hurricane system has been drawn up on the basis of the known volume of steam in a unit of volume of the near-water layer of air over a warm current. But the content of steam at a known relative humidity fully depends on the temperature of the water surface.
Fortunately, only an insignificant share of this power (about 3 percent) can turn into mechanical energy producing fantastic waves in the ocean and grave devastations on shore. But even that share is enough to produce - at the maximum wind velocity near the edge of the "hurricane eye" of 60 m/sec - waves 12 and more meters high in about 12 hours.
Supply of a hurricane system with thermal energy according to the aforesaid scheme can take place only when there appears in the atmosphere some initial movement in the form of a whirlwind with a vertical axis. Only then water vapor (steam) can be carried upwards from the surface of the ocean, become condensed with the resease of latent heat. In the tropical conditions the most likely cause of the appearance of such "initial whirlwinds" must be the overflow with orderly air currents from the ununiformities of the underlying surface. These ununiformities include pointed promontories, peninsulas cutting into the ocean or even such geographical objects as the Lake Chad over which the air is heated less than over the surrounding African sands. In reality, from Lake Chad and from Cape Verde they begin their journey along the parallel initial vortexes which turn into tropical hurricanes already west of the Cape Verde islands. So, how are they progressing? This question was examined by the author of this article in 1972. The solution of this very complicated - unstationary - problem had to be divided into two stages. To begin with the author made an indispensable simplifying assumption: it was assumed that from the moment of a hurricane's inception until its complete development only its dynamic and thermal parameters are changing while the geometrical parameters of its area, or field, remain the same as those which will be achieved in its full development. It was established that a tropical hurricane is not a simple "heat machine", but a self-excited, "self-driven" system: wind velocities in the hurricane system increase with time and, accordingly, the vertical components of these velocities. Also increased are the volumes of water steam rising upwards from the ocean surface into air layers where steam is condensed into water. This is accompanied by increased overheating of masses of air with the limits of "hurricane eye", which means that atmospheric pressure continues to drop in this most important part of the field. That means growing pressure gradient which serves to accelerate wind velocity. Thus a closed cycle of phenomena is produced and there begins the self-exitation of the heat machine-the tropical hurricane.
How long can continue this "pumping up" of power into this self-exiting machine? It will continue until heat capacity, turned into mechanical energy (3 percent of the total supplied by the ocean!) is absorbed by the friction of air currents against the surface of the ocean without anything left over.
If one marks on the abscissa axis the time, expressed in 24 hours, since the birth of the initial vortex in the atmosphere, and on the ordinates axis - the values of the tangential component of wind velocity at the altitude of 500 m above ocean level, on the border of the "hurricane eye", then in order to determine full wind velocity over the surface of the ocean itself it is necessary to reduce all these figures by 5 percent. Growth of the hurricane power and wind velocities in it is limited by a period of one week. And it was assumed that thereafter the hurricane crosses on the mainland. And what changes then? It is quite obvious that this ends the supply of energy from the ocean together with vapors rising upwards (the volume of vapors rising from dry land is negligible). At the same time there increase the energy losses of air
Tornado over the ocean.
currents from friction over the underlying surface (energy losses from friction over land surface are greater than those over the surface of water). As a result the hurricane begins to die down.
After that, primary analysis of the phenomena, the problem was solved in a secondary approximation: bearing in mind the insignificant original dimensions of the space covered by the hurricane and the gradual expansion of that space. To obtain more authentic data on the development of hurricanes it turned out to be necessary to resort to that more complicated solution, applicable to certain particular values of the characteristic parameters in the investigated movement equations. In connection with that, for each of the four values of temperature of the underlying water surface we obtained three versions of curves for the initial and intermediate stages of hurricane development. The curves, calculated with the second approximation, differ substantially from their predecessors only at the initial and partially at intermediate stages of development. The closer we are to the complete development, the smaller are the differences in the behavior of the hurricane.
Comparisons of theoretical calculations with the results of direct studies of hurricanes in nature produced good results. That applies to typical hurricanes, moving along the parallel near 15° of north latitude westwards from the Cape Verde islands, where maximum wind velocities approach 75 m/sec, and also to the powerful hurricane IN ESS which developed along the same route in later September 1966 and reached its climax over the Gulf of Mexico with surface water temperature of >0° and maximum wind velocity of 80 m/sec. Finally, one should notice the fine coincidence between the theoretical velocity of wind calculated for the CAMILLA hurricane of record strength, which raged in the second half of August 1969 over the Caribbean and the Mexican Gulf at surface water temperature of 32° and the results of wind velocity measurements. Maximum wind velocity reached 90 m/sec, which was clearly due to the overheating of water surface by 3° over the mean climatological norm. The power of that hurricane was one and a half times greater than that of the ordinary and very strong hurricanes observed in that region, and was twice greater than that of the ordinary hurricanes moving from Cape Verde to the Antilles.
So, why hurricanes, originating near Cape Verde, not always reach the Antilles moving along the parallel of 15o of north latitude? Why have they chosen this "beaten track" at all and why, having traveled over some distance, they deviate-first to the northwest, then to the north and sometimes to the north-east? One can find answers to these questions only if one knows what forces impact the hurricane system from the surrounding masses of air.
FORCES PROPAGATING TROPICAL STORM
One very common opinion shared by researchers to this day is that potential air currents drag after them giant whirlwinds - tropical hurricanes - which, like floats, passively drag after them in the same direction as the surrounding masses. But one can easily see that this view really "holds no water" if one remembers at least that near the Cape Verde islands the poten-
For calculations of the forces of friction of air against the surface of the ocean.
tial passat current is directed at southwest as different from local tropical hurricanes which move straight to the west. The hypothesis about "passive floats" cannot hold water also because on the border of the "hurricane eye" horizontal components of wind velocities soar up from zero to tremendous figures. Here, like on the border of a hurricane, there occur inevitable breaks of all functions used in hydrodynamics for describing the movements of liquids and gas. And as for a hurricane, the results of such "breaks" can be seen with a naked eye. The hurricane axis, stretching from a storm-cloud to the ocean surface, is "S" shaped. And this axis should be a straight line if the wind did not impact the hurricane system, as some foreign body present in the air. The "band" of this axis is caused by changing wind direction under the storm-cloud; and if the wind direction was constant, the hurricane axis would have been in the form of a chain line.
Like a whirlwind, a tropical hurricane should be seen as a foreign body located in the surrounding air current, like a passat stream. As different from a whirlwind, the surface that should be impacted by the surrounding potential flux. How? The answer lies in the Zhukovsky* theorem, formulated by him in a most general form, which found important application in the aircraft. It provided a quantitative explanation of lifting force of the aircraft wing and made it possible to calculate this force. Like the wing which supports a flying aircraft, the "hurricane eye" leads forward the whole hurricane system, directs its movements relative to the surrounding masses of air (like in passat). The Zhukovsky force, applied to a unit of body length, proportional to the velocity of incident flux, air density and, what is most important-proportional to what we call circulation velocity around the investigated body. In this particular case velocity circulation is equal to product of the tangential component of wind velocity on the border of the "hurricane eye" multiplied by the length of the circumference with which the "eye" can be surrounded.
Apart from the Zhukovsky factor, the hurricane system is also impacted by the head pressure of the external flux. But, as demonstrated by the author of the article in 1973 - this force, as applied to tropical hurricanes, is negligibly small as compared to the Zhukovsky force. That means that hurricanes should be moving in the direction of the Zhukovsky force which acts only in the direction perpendicular to the relative velocity of the external current and to the left of it.
One can say that in the passat region - the starting point of hurricanes - the relative velocity of a passat (in relations to a moving hurricane) is
* N. Ye. Zhukovsky (1847 - 192 l)-founder of modern aerodynamics, Corresponding Member of the St. Petersburg Academy of Sciences. - Ed.
directed to the south. Consequently, the Zhukovsky force should propel the hurricane system westwards. And it is in this direction, as we know, the tropical hurricanes are moving here. We have suggested a method of calculating the "validity" of the hurricane system. That means that the speed of movement can be determined theoretically if we know what happens to power generated by the Zhukovsky force. The diagram helps us to understand how exactly it is spent. Thin arrows on the diagram indicate wind velocities V in 12 different points located at the same distance "r" from the hurricane center. Radii-sectors drawn from the center to 2 of these points ("3" and "9" make it possible to establish that velocities V are everywhere directed at the angle of 72° towards the radius-vector, that the tangential and normal components produce between themselves an angle of 90°-72° = 18°. These are wind velocities produced at the ocean level by the aforesaid thermohydrodynamic phenomena. But the actual wind velocities which can be measured at points marked on the diagram should differ from V: added to the V velocity (geometrically) should be another component (which we have just mentioned) produced by the movement of the hurricane system itself over the ocean. This is the sought for velocity "c" which on the previous scale is represented by vectors which are shown here by dots. On the example of vectors at points "3" and "9" one can see how the geometrical addition of vectors "V" and "c" produces the resulting velocities of V3 and V9 at the given points. Obtained in the same way are the resulting velocities V3and V9 at the given points. Obtained in the same way are the resulting velocities in all of remaining points out of the 12. We know that power consumed by the friction of air against the underlying surface (in this case-surface of the ocean) is proportional to cube of wind velocity. One can see on the diagram that the sum of the cubes of the actual wind velocities and taking into account the component "c" is greater than 12 V3 . One can calculate by simple methods to what extend the power consumed by the friction against the ocean surface exceeds in the whole hurricane system the one which would have been consumed in the absence of component "c", caused by the movement of the hurricane system. It is this additional power which is compensated by the action of the Zhukovsky force upon the hurricane, as on some "foreign body" in the atmosphere. Calculations conducted by the author of the article in 1973 showed that at passat velocities of 8 - 11 m/sec directed at an angle of about 45° to the meridian (to southwest) the hurricane with velocities of about 40 m/sec near the outer border of the "eye" should be moving westwards at 6 to 8 m/sec. The calculations were facilitated by the fact that the velocity of the passat itself in relation to the moving hurricane (directed to the south) by its absolute value (modules) was equal to "c". But a simple picture of this kind exists only at the initial stages of hurricanes' movement from the Cape Verde islands. Further on the conditions become more complicated: wind direction and velocities caused by the synoptic conditions over the central Atlantic, can change the trajectory of hurricanes movement in a most peculiar way.
We analyzed the complex movement of the three most typical hurricanes: Carrie, Debbie and Esther. All the three were observed in September of 1957 and 1961. Shown of the same diagram by thin lines the main warm currents in this part of the ocean. Taking a look at the diagram one may be prompted to "suspect" the warm currents influencing the paths of all the three hurricanes: the hurricanes kind of move around the closed system of cyclic currents shown on the map. An analysis of synoptic conditions by the maps of 1957 and 1961 over the periods of the movement of the investigated hurricanes confirmed these suspicions. Firstly, over this area the isobars had exactly the same configurations which were necessary for "tearing off' the hurricane trajectories from the parallel of 15° of North latitude. Second, everywhere along the routes of the hurricanes surface temperature of water was lower than the one which is enough for feeding the hurricane with thermal energy (according to our theoretical scheme). Stretches of the curves directed northwards along the meridian are all located upon a kind of "rift of increased atmospheric pressure" whose slopes decline to the north, south and west-over the streams of warm water. The only differences in the regimes of these currents in different years and at different times are caused by more earlier, or later changes in hurricanes' trajectories. And on these and other sections of hurricanes' routes one can observe one and the same regularity which confirms our theoretical considerations: it would have been natural to expect that the gradient of atmospheric pressure approximately coincides with the normals (standards) of warm currents; on the other hand, according to the examined scheme hurricane movement should usually be directed perpendicular to the gradient, to the right of it. This is what is observed in reality. And it is interesting that although the September hurricanes "Carrie" and "Debbie" were observed over an interval of 4 years, -over the distance with the coordinates of φ = 35° of north latitude, λ = 44° of west latitude and up to the southern border of Ireland-the dotted carve of "Carrie" and the solid curve of "Debbie" are strikingly close to one another.
For developing authentic methods of preliminary calculations of the progression of tropical storms - by the signals of their appearance - and for determining in advance their paths it is necessary to obtain authentic data on a non-stop basis about the surface water temperature and barometric relief in the especially dangerous regions of the world ocean.
Zemlya i Vselennaya (Earth and Universe), No. 1, 1975
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