A Method of Forecasting Development of Tropical Cyclones WILLIAM KRASNER and JULIUS LONDON * ABSTRACT A method is devised (based on the theory of Bjerknes and Holmboe on the deepening of cyclones), for the forecasting of the deepening of tropical cyclones. It is shown that the distribution of divergence around an easterlies trough ("easterly wave"), as revealed through consideration of the vorticity equation agrees with this application to the deepening of tropical cyclones. The conclusion is reached that an estward tilt of the pressure, or streamline, axis of a trough or developing tropical disturbance is most favorable for deepening; whereas, in deep easterlies, a westward tilt of the axis would result in filling and dissipation. A technique is also suggested for a quick qualita- tive evaluation of the direction of slope of the axis of such a pressure or streamline system. HE FOLLOWING is an attempt to arrive at ing cyclonic curvature."* The second, the a qualitative method by which the factor of latitude, is a result of the change of T development of tropical storms may be velocity with latitude, so that . "diver- forecasted using data available at any first- gence [occurs] in currents with a component order station in the tropics. There has long towards the equator and convergence in cur- been a need for such a method, especially in rents with a component towards the pole."f the central and western North Pacific Ocean In westerlies troughs, these factors are op- areas, where there are great open stretches posite in sign and effect. That is, west of the with few or no reporting stations—stretches trough line increasing cyclonic curvature (con- which, many of them criss-crossed by heavily vergence) is associated with equatorward dis- traveled air and water routes, are the areas of placement (divergence); and east of the trough incubation of many of our most destructive line the reverse occurs. storms. The net effect is chiefly dependent on the This paper originated as a project of the velocity of the wind with changing curvature Research Section, Tropical Weather School, effects dominating above certain critical wind AAF Weather Service, Pacific Ocean Areas. speeds, changing latitude effects below them. Army Weather Central at Guam has since re- The vertical picture then, with increasing examined its typhoon data in the light of these westerlies aloft, often shows the low-level ideas and its comments are appended at the pattern of divergence becoming reversed end of this report. above a "level of non-divergence." If we assume balanced gradient flow as a Since surface tendencies are the result of first approximation to stream-line flow in the mass divergence* within a vertical column of tropics, this technique may be considered as air from the surface to infinity, that is being based essentially on the ideas described dp foo by Bjerknes and Holmboe in their article, — = — g I div (pv)dz, "On the Theory of Cyclones", in the Journal dt of Meteorology, September 1944 (Vol. I, Nos. the total effect, as it affects these surface 1 and 2). Their conclusions are reviewed tendencies, depends for the most part on the briefly in the following. height of the level of non-divergence. The distribution of convergence and diver- The pressure tendency at the center of a gence around a sinusoidal streamline wave— wave and therefore the rate of deepening or disregarding acceleration effects—is the result filling becomes a function of the degree and of primarily two factors. In the first, the direction of vertical tilt of the associated factor of changing curvature, . "diver- trough, which determines whether the diver- gence prevails where particles go along their gence pattern in the eastern or western half trajectory toward decreasing cyclonic or in- of the trough will dominate the center. "... creasing anticyclonic curvature. Convergence A baroclinic westerly current is dynamically prevails where particles along their trajectory * Bjerknes, J., East Asian Circulation (Winter), go toward decreasing anticyclonic or increas- AAF Wea. Serv. Bull., July-Aug. '45, p. 6, "Tool of Reasoning No. 2". * Department of Weather, Chanute Field, 111. t Ibid., "Tool of Reasoning No. 3". Unauthenticated | Downloaded 10/01/21 03:39 AM UTC unstable for waves whose level of non-diver- state, common in the tropics, there is little or gence is sufficiently low and tends to make no temperature gradient, and therefore no tilt these waves grow strong, however weak they to the pressure axis. In the baroclinic state may be at the start."** the pressure axis tilts toward colder air, which If we apply these principles to the easter- lies, the picture is even more simple: the two factors (effects of changing curvature, and changing latitude) act in the same direction, reinforcing each other, and all waves have the distribution of divergence and convergence shown belown (FIG. 1). FIG. 3 with east-west isotherms, would be to north or south. Isotherms in the tropics, however, like streamlines, are seldom straight east- west; perturbations form and move along FIG. 1. Easterlies Trough ("Easterly Wave") them, accompanying northerly or southerly transport of air. Where, for example, the In a closed center, the effect of changing easterly component of the wind in an easterlies curvature is no longer operative, but the same trough is greater than the trough speed, a net distribution of divergence exists (FIG. 2). wind moving toward the south will make itself felt on the west side of the trough, and a northbound wind on the east,* and the iso- therms will be bent correspondingly, causing temperature ridges and troughs.** There will be, therefore, a certain amount of east or west tilt in each easterlies trough depending on the thermal distribution, and it is these tilts which are the most important. FIG. 2. Distribution of Convergence in a Closed Low. Thus due to latitudinal displacement alone, divergence is present west of the closed center and convergence to the east. It follows therefore, that the rate of deepen- ing or filling in a trough in the easterlies de- pends on the tilt of the streamline axis even more obviously than in the case of trough in FIG. 4. Baroclinic State, Case I. the westerlies. A tilt forward (to the west) In the baroclinic state, there are two major must result in filling; a tilt backward (to the cases: CASE I—warm air originally to the east) must result in deepening (FIG. 3). south; east wind decreasing with height, often The tilt of the axis of a pressure system is reversing to westerlies (FIG. 4). Only shallow the result of horizontal thermal asymmetry stable waves will result from such a distribu- across the system, and . "the magnitude tion, with forward tilt and therefore a tend- of the slope varies inversely as the horizontal ency to fill almost as soon as formed. This temperature gradient."J In the barotropic case is more common in winter. ** Bjerknes and Holmboe, op. ext., p. 18. * Riehl, H., "Waves in the Easterlies and the Polar t Panofsky, H. A., The Slope of Axes of Pressure Front in the Tropics", Univ. of Chicago Press, Misc. Systems, Bull. Amer. Met. Soc., April '45 (Vol. 26, Report No. 17, p. 14. No. 4) pp. 101-2. ** Bjerknes and Holmboe, op. cit., p. 18. Unauthenticated | Downloaded 10/01/21 03:39 AM UTC CASE II—warmer air originally to the after the storm has recurved and resembles north, east wind increasing with height the pattern discussed by Bjerknes and Holm- (FIG. 5). With this distribution—more com- boe.* The storm will continue to deepen only mon during the summer when the heat equator if the axis of its center tilts to the west or northwest. Special reference is made to just such a case in the comments appended to this report. The simplified picture of balanced gradient flow, on which the foregoing development was based, is not, however, completely applicable to the tropics. Consideration of changes in vorticity gives a fuller analysis of tropical FIG. 5. Baroclinic State, Case II. stream flow, since these changes are concerned with the actual wind patterns. In the tropics is in the northern hemisphere—the tilt of the the density gradient is normally so small that axis is toward the east, or southeast; and mass divergence as used in the tendency equa- troughs tend to deepen and move slowly west- tion can correctly be considered from the ward. High-level divergence is present, in- viewpoint of divergence of stream flow neglect- tensified by the high velocities aloft—an effect ing lateral changes in density. which will continue even after a closed low The broad easterly stream is held in its has formed in the lower levels. This, then, is sinusoidal path by the effect of the constantly the ideal case for deepening of tropical cy- changing vorticity relative to the rotation of clones. the earth. This is accompanied by shrinking or stretching of a significant depth of the column of air and hence is an application of the vorticity theorem developed by Rossby, which states that /+ $ P = constant, where / is the angular rotation of the earth, £ is the relative vorticity of the air particles relative to the earth's rotation (measured positive for cyclonic rotation), and D is the depth of the significant portion of the column FIG. 6. Closed LowlwithSOpenJWave Aloft of air considered. Getting the change of vorticity along the An interesting development is the case trajectory of the air parcels we have in an where the closed low moves into higher lati- easterlies wave (FIG. 8), tudes and is surmounted by an open westerlies d trough (FIG. 7). This will be the development ( l\ \ dt JT \dtST \dt/r FIG.