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658 MONTHLY WEATHER REVIEW Vol. 96,No. 9 THE EFFECT OF POLAR STEREOGRAPHIC PROJECTION ON THE CALCULATION OF THE CURVATURE OF HORIZONTAL CURVES* K. KRISHNA Poona, India ABSTRACT It is shown that, in any secant polar stereographic projection, a small circle on a sphere projects into a circle. This property provides a simple relationship between KH,the horizontal component of curvature of a horizonal curve and Kb, the curvature of its projection on a secant polar stereographic map. KHcan be computed by sub- tracting from the map factor times Kh the earth’s curvature multiplied by a correction factor that depends only on the latitude of the place and inclination of the curve to the latitude circle. This factor vanishes if tho curve is along a meridian but takes an extreme value if it is along a latitude. For a given orientation of the curve, the value of this factor incrcases gradually as the location of the curve moves from the Pole to the Equator and more rapidly after it crosscs the Equator. It is less than 1 in the Northern Hemisphere but can exceed unity in the Southern Hemisphere. 1. INTRODUCTION The polar stereographic projection is used for several purposes, especially for weather charts in the middle and higher latitudes. Sometimes it is of interest to compute the curvature of certain contours as well as other types of curves like isobars from the curvature of their pro- jections on this chart. The following derivation establishes a relationship between the curvatures of a horizontal curve and its projection on a polar stereographic chart. 2. DERIVATION rn figure 1, Point p (colatitude J. and longitude lies on a circle with its center below point C (colatitude $o). The longitude of C can be taken as 0’ without loss of generality. The angular radius, b, of the circle is given by jects into a circle on the curvature Of the latter projection is in question. T h e cos b=cos $o cos $+sin $o sin $ cos A. (1) In circumpolar stereographic projection with scale true at colatitude $1, the radius of projection of any latitude circle is r=c tan $12 (2) where c=a(l+cos $J, a being the radius of the earth. Eliminating $I between (1) and (2) nates, a circle of radius, rL=c sin b (costj0+ cos b) with center at distance from the Pole, d=c sin $o (cos$o+cos b)’ Equation (3) proves that a small circle on a sphere pro- *Theseresults were presented at a meeting of the Poona Branch of the Indian Meteore logical Society in 1960 in the course of a talk on “Conformal Map Projections Used in Meteorology.” S COS $O-COS b- cos$o+cos b-’ 2-2rd COS A-C’ which is a quadratic in r representing, in polar coordi- (4) (5) (3) FIGUR1E.-The coordinates of a point on a small circle on a sphere. K&= COS $OSCOS b G sin b Taking the spherical triangle NCP (fig. I), it can be shown that ~~(K=L-m ) (6) where KHis the horizontal curvature of the original circle, m is the scale factor, and /3 is the angle between the curve and the latitude N I a(l+cos sin cos$1/)3 September 1968 K. Krishna T h e v a l u e s o f KH a n d m a r e Equation (6) provides a relationship between KH and K‘H in terms of m, 0, +, &. I n the case of a tangent polar stereographic projection (+1=0), this reduces to as given by Haltiner and Martin [l]. Equation (6) provides a value for the error E, the difference between the curvature of the curve and curva- ture of its projection adjusted for scale. 659 E=KH-mKH=-- tan - cos /3. (8) a2 E is, however, independent of the location of the standard parallel. It vanishes at points V and V’ (see fig. 1) where the curve touches the meridian. Its magni- tude has maximum and minimum values at points X and M respectively, i.e., when the curve touches a latitude circle. The magnitude of the extreme value of the error is tan +/2 times the curvature of the earth. Thus, in the Northern Hemisphere (for projection from the South Pole) where +is less than go”, the error is always less than the curvature of the earth and it increases as the latitude decreases. As the location moves into the South- ern Hemisphere, it can exceed unity. 3. DISPLACEMENT OF THE CENTER OF CURVATURE Whereas a circle projects into a circle, its center does not project into the center of its projection. The center of the projection of the circle lies farther away from the Pole than the projection of the center of the circle. Figure 2 represents a cross section of the sphere shown in figure 1 along the meridional plane passing through C. The diameter MX of the small circle projects into M’X’ and the center C into C’. P’ is the midpoint of M’X’. It is evident that P’ and C’ are not one and the same, since MC and CX will not project into equal lengths. If the angular shift of the center of the circle (viz CP) is a degrees from the Pole, from (2) and (5) t1* tan*-0-+a - Expanding the above equation in terms of tangents of half angles of a , b, and +o and solving for tan a12 (9) If a and b are expressed in degrees we have, for small values of b, the approximate relationship, If the center of the circle lies on the Equator the appar- ent shift of the center of the projection of a circle is approximately 15’ and 1’ for circles of radius 5’ and 10’ respectively. The corresponding values for circles with centers at 30’N. lat. are 9’ and 35’ respectively. Since the center of the projected circle is not the same as the projection of the center of the small circle, con- centric small circles other than latitude circles will not project into concentric circles. The centers of the pro- jection of these concentric circles will be different and will lie on the same meridian; the center of a circle with larger radius will be displaced farther away from Pole. The complications introduced by the distortions due to the projection can be avoided if the diameter of the circle of curvature is taken as the difference in the lati- tudes of the points M and X where the meridian through the center intersects the circle of curvature and K H is calculated from its definition (see equation (6)). ACKNOWLEDGMENT I am grateful to Dr. Bh. V. Ramana Murty for his meticulous examination of the manuscript and for the suggestions which led to considerable improvement in the presentation of the paper. REFERENCE sin +o 2 (COS +~+COS b) [Received December 21, 1967; revised March I S , 19681 N S FIGUR2E.-Cross sectionofasphere,asmallcircleand its projection. 1. G. J. Haltiner and F. L. Martin, Dynamical and Physical Meteorology, McGraw-Hill Book Company, Inc., New York, 1957, 470 pp. (see p. 175). 660 MONTHLYWEATHER REVIEW PICTURE OF THE MONTH FRANCES C. PARMENTER National Environmental Satellite Center, ESSA, Washington, D.C. Vol.96,No.9 Mesoscale cloud patterns are strongly influenced by the terrain features of an area. A frequently observed example is the formation of wave clouds in the area of gravity waves to the lee of mountains. These clouds occur when: the wind direction .is perpendicular to the mountains through a deep layer, the mountain top wind is a minimum of 20 kt., and the atmosphere is stable for vertical dis- placements of air. Satellite photographs show that the unique parallel arrangement of small wave clouds is common to all major mountain chains throughout the world. In the Uhited States, lee waves are frequently observed, as in this case, along the northwestern ranges. On June 21, 1968, the 1200 GMT analysis showed a weak surface High centered in Wyoming with a low pressure area off the Washington coast. The 500- and 300-mb. analysis showed strong zonal flow across the northwestern United States. At this time, the 200-mb. jet stream was analyzed to cross the coast near Seattle and follow a path, due east, along 48'N. through Idaho and Montana and then northeastward into Canada. Upper air solindings for 1200 GMT at Lander, Wyo. (LND), and Great Falls, Mont. (GTF), accompany the 1435 GMT ESSA-2 photograph in figure 1. At this time, low clouds are present near LND and middle and high clouds at GTF. Little directional wind shear is indicated a t both these stations. The ESSA-5 picture (fig. 2), taken at 2309 GMT, shows that late morning and early afternoon convection in this area has resulted in a large area of wave clouds through Idaho, Montana, and Wyoming. The 0000 GMT soundings at both stations show the lapse rate to be dry adiabatic up to the base of the inversion. At GTF the winds aloft have increased due to a shift in the jet stream; now entering the coast at 49'N., it continues eastward to 115OW. and gradually turns southeastward passing through the extreme southwest corner of North Dakota. The brightest group of wave clouds (Q) is found near the 6,000- to 9,000-ft. Cabinet Mountains and the Bitter Root Range. The clouds become more widely spaced to the east in the vicinity of the Rocky Mountains of Montana. The upper air data at GTF indicates that the wind at the mountain top level is greater than 26 kt. from the west-southwest. The wave cloud pattern to the south (R) is in the vicinity of the higher Rocky Mountains in Wyoming and !i \ TEMP. GTF 8 i' SPEED 1200GMT June 21. 1960 ~ 8 DEWPDINT ('t) DEWPDINT 775 WIND (Knot.) T E M P LND 576 12DDGMT June 21. 1960 WIND (Knot.) SPEED FIGURE1.-ESSA 2, APT, Orbit 10705, 1436 GMT and 1200 GMT, umer air soundings for Lander, Wvo. (LND). and Great Falls, Mint. (GTF), JU& 21, 1968. I " . ., (.C.) September 1968 Frances C. Parmenter 661 southern Montana. These clouds lie to the north of Yellowstone Park, and along the north-south Absaroka and Wind River Ranges, and farther east along the Big Horn Mountains. The wind at the top of the 12,000- and 14,000-ft. mountains, indicated by LND, is 40 kt. from the west. Another area of wave clouds (S) can be seen along the eastern edge of the frontal cloudiness approaching Wash- ington and Oregon, in the vicinity of Mt. Adams and Mt. Hood. The presence of wave clouds in satellite photographs provides the aviation forecaster with visual information 2I0I'J0'I',L WIND SPEED (Knots1 OOOOGMT June 22,1968 FIGUR2E.-ESSA GMT, upper air soundings for Lander, Wyo. (LND), and Great Falls, Mont. (GTF), June 22, 1968. about the mesoscale wind patterns and general atmos- pheric structure in the vicinity of mountainous areas. Although the distribution of turbulence associated with lee waves is still under investigation, some preliminary results indicate that the turbuleiit layer is confined to the area within and below these clouds. Soaring and glideplane pilots were among the first to investigate wave clouds, and by flying these clouds, these pilots have established new height and distance records. Using the same tech- nique, light aircraft pilots have found that they can conserve fuel by "riding" the wave clouds. ~ ~ ~ TEMP. B DEWPOINT I'C) LND 576 TEMP B DEWPOINT (.C I GTF 7 7 1 OOOOGMT WIND SPEED (Knolrl June 22.1960 I11111I1I1IIAII121i -dli; ;I;1 1 1 -60 -40 -20 5, Orbit 5431, 2309 GMT, June 21, 1968, and 0000