ADVECTION OF AIR AND THE FORECASTING OF PRESSURE CHANGES

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Abstract

The advection method of preparing prognostic pressure charts was tested using a number of different procedures. First, it was assumed that the density change at a given level can be regarded as representative of the density change throughout a layer which includes this level. Accordingly, isopycnic lines were drawn for five levels between sea level and 14,000 meters, and forecasts of the density changes at these levels were prepared on the hypothesis that the air moves with the geostrophic wind velocity, preserving its density. The sea level pressure change is obtained by adding the density changes for all levels after making allowance for the thickness of each layer. The effects of advection above 14,000 meters were neglected. The examples selected for a test gave poor results, and a statistical check showed that, even with perfect forecasts of the density, satisfactory results cannot be expected because the density changes at a given level are not sufficiently representative of the density changes in the whole layer.

In the major part of the study the pressure forecasts were based on weight charts for four layers, from sea level to 13,000 meters. Since the weight, represented by the pressure difference between the bottom and the top of the layer, is proportional to the mean density of the layer, this method appears more satisfactory than the one studied first. To obtain prognostic weight changes, the motion of the weight lines has to be known. Three different methods for the determination of the motion of the weight lines were tested:

  1. The motion was assumed to be geostrophic.
  2. The trajectory of the air was found by successive approximations taking into account the variation of the pressure distribution during the forecast interval.
  3. The trajectory of the air was found under the assumption that the absolute vorticity is conserved (Rossby method).

In all three cases the weight was considered as a conservative property of the air. The sea-level pressure changes were obtained by adding the prognosticated weight changes in all layers and the pressure change at the top level. The prognostic pressure changes thus prepared were quite unsatisfactory, and statistical checks gave correlation coefficients between the observed and prognostic pressure changes which were too small to be of forecasting significance.

It is concluded that the advection method used by itself cannot be recommended as a forecasting tool. In addition to advection, other factors such as divergence and vertical motions make themselves felt too. Their effects have also to be taken into account if satisfactory prognostic charts are to be constructed in a purely objective fashion.

Abstract

The advection method of preparing prognostic pressure charts was tested using a number of different procedures. First, it was assumed that the density change at a given level can be regarded as representative of the density change throughout a layer which includes this level. Accordingly, isopycnic lines were drawn for five levels between sea level and 14,000 meters, and forecasts of the density changes at these levels were prepared on the hypothesis that the air moves with the geostrophic wind velocity, preserving its density. The sea level pressure change is obtained by adding the density changes for all levels after making allowance for the thickness of each layer. The effects of advection above 14,000 meters were neglected. The examples selected for a test gave poor results, and a statistical check showed that, even with perfect forecasts of the density, satisfactory results cannot be expected because the density changes at a given level are not sufficiently representative of the density changes in the whole layer.

In the major part of the study the pressure forecasts were based on weight charts for four layers, from sea level to 13,000 meters. Since the weight, represented by the pressure difference between the bottom and the top of the layer, is proportional to the mean density of the layer, this method appears more satisfactory than the one studied first. To obtain prognostic weight changes, the motion of the weight lines has to be known. Three different methods for the determination of the motion of the weight lines were tested:

  1. The motion was assumed to be geostrophic.
  2. The trajectory of the air was found by successive approximations taking into account the variation of the pressure distribution during the forecast interval.
  3. The trajectory of the air was found under the assumption that the absolute vorticity is conserved (Rossby method).

In all three cases the weight was considered as a conservative property of the air. The sea-level pressure changes were obtained by adding the prognosticated weight changes in all layers and the pressure change at the top level. The prognostic pressure changes thus prepared were quite unsatisfactory, and statistical checks gave correlation coefficients between the observed and prognostic pressure changes which were too small to be of forecasting significance.

It is concluded that the advection method used by itself cannot be recommended as a forecasting tool. In addition to advection, other factors such as divergence and vertical motions make themselves felt too. Their effects have also to be taken into account if satisfactory prognostic charts are to be constructed in a purely objective fashion.

J 0 IJ RNA I,0 I;M ET EO I< 0 I, 0 G YVOI,I!ME 2, NT:MBRR 2JIINE 194.5ADVECTION OF AIR AND THE FORECASTING OF PRESSURE CHANGESBy Bernhard Haurwitz and Collaborators" Massachusetts Institute of Technology (Manuscript Received March 31, 1945)ABSTRACTThe advection method of preparing prognostic pressure charts was tested using a number of differentprocedures. First, it was assumed that the density change at a given level can be regarded as representativeof the density change throughout a layer which includes this level. Accordingly, isopycnic lines were drawnfor five levels between sea level and 14,000 meters, and forecasts of the density changes at these levels wereprepared on the hypothesis that the air moves with the geostrophic wind velocity, preserving its density.The sea level pressure change is obtained by adding the density changes for all levels after making allowancefor the thickness of each layer. The effects of advection above 14,000 meters were neglected. The examplesselected for a test gave poor results, and a statistical check showed that, even with perfect forecasts of thedensity, satisfactory results cannot be expected because the density changes at a given level are not sufficiently representative of the density changes in the whole layer.In the major part of the study the pressure forecasts were based on weight charts for four layers, fromsea level to 13,000 meters. Since the weight, represented by the pressure difference between the bottomand the top of the layer, is proportional to the mean density of the layer, this method appears more satisfactory than the one studied first. To obtain prognostic weight changes, the motion of the weight lines hasto be known. Three different methods for the determination of the motion of the weight lines were tested:u. The motion was assumed to be geostrophic.6. The trajectory of the air was found by successive approximations taking into account the variation ofc. The trajectory of the air was found under the assumption that the absolute vorticity is conservedIn all three cases the weight was considered as a conservative property of the air. The sea-level pressurechanges were obtained by adding the prognosticated weight changes in all layers and the pressure changeat the top level. The prognostic pressure changes thus prepared were quite unsatisfactory, and statisticalchecks gave correlation coefficients between the observed and prognostic pressure changes which were toosmall to be of forecasting significance.It is concluded that the advection method used by itself cannot be recommended as a forecasting tool.In addition to advection, other factors such as divergence and vertical motions make themselves felt too.Their effects have also to be taken into account if satisfactory prognostic charts nre to be constructed in apurely objective fashion.the pressure distribution during the forecast interval.(Rossby method).1. Introduction. It has been suggested that theforecast of the future pressure field may be improvedby determining for the air column above each stationthe changes in weight which would be caused by thehorizontal advection of air. It is recognized by theproponents of this "advection method" that otherphenomena also play a part in producing the dayto-day pressure changes, but it is argued thatthese effects are not strong enough to invalidate theprognostic chart obtained by the advection method.It is the purpose of the present paper to report on acheck of the practical value of the advection method.In applying the advection method it is necessary,Dstrictly speaking, to know the density and wind distribution at every level in the atmosphere and tocompute the weight changes due to advection forevery level. In practice, only a limited number oflevels or layers can be used. Under these conditions,there are two possible methods of approach. First, onemay construct lines of equal density for selectedheights. These isopycnic lines are then displaced withthe speed indicated by the wind and pressure field atthis level. In doing this it is assumed that the advectivechanges of mass obtained in this manner are representative not only of the changes of mass at the levelunder consideration, but also through a certaindistance above and below this level (Section 2). Theother method consists in constructing charts of thepressure difference between fixed levels (weight charts)and determining the displacement of the lines of equalpressure difference or weight through the forecast* The investigation reported in this paper has been carried outunder the direction of B. Haurwitz and T. F. Malone. The mainpart of the synoptic and statistical work was done by Lt. M. A.Eaton, U.S.N.R., Mr. R. C. StaleY, and Lt. A. E. Tancreto,U.S.N.R., with the liberal assistance of other staff members ofthe Institute's Meteorology Department.8384 JOURNAL OF 'METEOROLOGYperiod (Section 3). In this case, it is necessary toassume a mean motion for the whole layer under consideration but:, despite this shortcoming, the lattermethod appears more satisfactory from a practicalviewpoint than the first one. Hence, the first methodand its check are discussed rather briefly here.Even if advection is the major factor in determiningthe future pressure field the path of advection wouldstill be doubtful. Three possible procedures of determining the horizontal advection have been employedin connection with the work on the weight charts. Thesimplest metbod is based on the assumption that theisobars are streamlines and that the flow patternchanges so little with time that the isobars can beregarded as tlne lines along which the pressure-difference lines are displaced. The second procedure usestrajectories determined by successive approximationfrom the isobars and from the displacement of theisobars during the forecast period (3). As a thirdmeans of determining the path of advection, themethod of finding trajectories on the basis of conservation of vorticity suggested by Rossby is utilized (4).By the selection of these three different methods ofadvection for a check, it is not implied that they arethe only possibilities, but rather that they are themost promising ones.It must be added here too, that neither the densityof an air particle nor the mean density of an air layeras represented on the weight charts is a conservativeproperty of the air, although either of these two quantities may occasionally remain unchanged for a givenair volume. For this reason, it cannot be assumed thatthe lines of elqual density or of equal weight movealong with the velocity indicated by the trajectory ofthe air. Nevertheless, some such assumption seemsnecessary if advection is to be used in practice. Analternate method suggested by Braun and Douglis (2)would be to displace the density, or the weight lines,by the same'distance and in the same direction asthey moved during the time before the forecast period.However, this method appears more unsatisfactorythan those employed here because its physical basis issomewhat obscure. Moreover, the study carried outat the Massachusetts Institute of Technology showedthat advection is rather erratic and is, if anything,better indicated by the flow pattern than by the pastmotion.2. Density charts. As explained in the introduction,the first attempt at a utilization of the advectionmethod for the preparation of prognostic pressurecharts was based on charts of the density distributionat different atmospheric levels. The density at theradiosonde sta.tions (about 20-30 for each map) wasobtained for certain levels (see below) by means oftables and was plotted on a map. It was then assumedthat the density distributian at each level was representative of the density distribution through a certainlayer. The following levels were chosen and regardedas representative for the corresponding layers :Level Layer5,000 feet 1-2 km10,000 feet 2-4 km. 20,000 feet 4-8 km , 8-12 km10 km13 km 12-14 kmAbove 13 km, sufficient data for constructing <adensity chart usually were not available; therefore theeffects of advection at higher levels were neglected.This does not mean that they were considered unimportant. But the only way to take changes at higherlevels into account would have been to prepare prognostic charts of the pressure at 14 km, and it was feltthat the preparation of a prognostic chart for the toplevel, by the customary methods, based on the forecaster's experience and empirical rules, would renderthe check of the advection method considerably lessobjective and too much dependent Ion the skill of theforecaster. The effect of this omission on the accuracyof the forecasts is discussed below.After the charts of the density distribution weredrawn, the changes of the density during the forecastinterval were determined. It was assumed that the airmoves without changing its density and according tothe geostrophic wind relation. This assumption impliesthat the temperature is conserved while the air flowis along the isobars. Thus, the density changes at eachof the selected levels over each station were determined. From these density changes at representativelevels in each layer the change of, the pressure difference between the bottom and the top of the layer, andthus the contribution to the surface pressure change,could easily be computed. Even thLough the densityitself varies appreciably with the a1 titude throughoutthe layer, the vertical variation of the density changewith time will be smaller. Hence, it may be permissibleto consider the density change at an intermediate levelas representative for the whole layer. Actually, thissupposition is not quite correct as will be shown at theend of this section.The pressure changes at.a given level were found byadding all the pressure-difference clnanges above thislevel. In particular, for the sea-level pressure variation, the changes in all layers above this level hadto be added, except that the layer from sea level to1000 meters was disregarded. This omission seemedadvisable for two reasons. In the first place, the elevation of the ground cuts off a substantial part of thislowest layer over large parts of the United States.Secondly, the motion of the atmosphere in the lowestlayer is so much affected by friction that any advection on the basis of the geostrophic wind relationwould be illusory. The omission of the lowest layer inB E R N HA R DH A U R W I T Zthe computation of advection may falsify the resultssomewhat and thus give too unfavorable an impressionof the results obtained with this method. However, aswill be shown below, the assumption of the densitychange at a level being representative of the densitychange in the layer is so unsatisfactory that themethod has to be discarded or this reason alone.The computed pressure changes were finally addedto the pressure at the start of the forecast period. Inthis manner a prognostic pressure chart was obtained.This method of preparing prognostic sea-levelpressure charts was used in connection with theregular map discussions held by the staff of theMeteorology Department during the period fromAugust 31 to September 18, 1944. As the starting mapthe 06302 map was used and a prognostic map wasconstructed, at first for 24 hours ahead, later for only12 hours ahead because it seemed that the failurc ofthe method to give satisfactory prognostic chartsmight have been due, in part at least, to the variationof the isobaric pattern during the forecast period. Aforecast for only 12 hours has the disadvantage thatthe diurnal variation of the pressure is not eliminated,but it was felt that the shorter forecast interval shouldnevertheless lead to an improvement in the forecast,especially since the normal 12-hour pressure changesare generally less than one millibar in Scptembcr.During the daily map discussion the prognostic sealevel pressure chart was not only compared with theactual pressure distribution, but in addition theforecast density values were compared with thosewhich actually occurred. In all cases the result wasdisappointing. In particular, the prognostic chartprepared in the customary manner by judging thedisplacement and deepening and filling of pressuresystems by experience lead to better agreement withreality than the chart prepared with the aid of thedensity charts.The failure of the method to give satisfactory prognostic charts is not necessarily attributable ,to thedeficiency of the basic assumption of the advectionmethod, viz., that advection is predominant in determining pressure variations as compared to otherfactors. The following three causes may contribute tofaulty prognostic charts even though the basic principle of the advection method is sound:i)ii)iii)The representative density change determinedfor each layer by finding the density change ata.fixed level in this layer may not be truly representative.The omission of the layers above 14,000 metersmay produce a considerable error.The paths selected for advection, that is theisobars, deviate too much from the actual pathsto give satisfactory prognostic pressure patterns.AN DC 0 L L A B 0 RAT 0 RS85iv) The omission of the lowest one-kilometer layergives rise to some error.To check the first two points the radiosonde observations at Omaha, Nebraska, during February,1940, 1941, and 1942 were used. These three monthsgave 67 sets of data. To test the first point (i) theactual surface pressure changes were correlated withthe pressure changes as computed from the observeddensity changes plus the actual pressure changes at14,000 meters elevation. The correlation coefficientbetween these two quantities was found to be(9 0.90The standard error of estimate is 3.24 nib when theappropriate linear regression equation is used.If the pressure changes at 14,000 meters are disregarded in computing the surface pressure change,the correlation coefficient becomes(ii)0.87The standard error of estimate is 3.8 mb. The materialon which these statistical characteristics are based isnot large and refers only to winter. Nevertheless, itshows clearly that even with perfect forecasts of therepresentative density changes the forecasts of thesea-level pressure by no means would be perfect. Also,if the pressure changes at 14,000 meters could be forecast with complete accuracy, the forecast of the sealevel pressure would still be substantially less thanperfect. Since the entire air column is considered inthis test, the error arising from (iv) is eliminated. Themain cause of this imperfect correlation is evidentlythat the so-called "representative" density changes atfixed levels are not sufficiently representative for themean density change throughout the layer.It follows that the application of weight charts inthe advection method is much more promising thanthe use of density charts, because a weight chartrepresents the field of mean density of a layer. For thisreason the third point raised above, viz. the questionof the proper paths of advection, was studied only inconnection with the weight charts.3. Weight charts. The pressure difference betweentwo fixed levels in the atmosphere gives directly theweight of an air column of unit cross section betweenthese two levels. Hence the pressure difference betweenthe two levels is directly proportional to the meandensity of this air column. Thus, in applying theadvection method it seems preferable to use lines ofequal pressure difference between two layers ratherthan isopycnic lines at a level intermediate betweenthe upper and the lower boundary of the layer. In thiscase, the question of the representativeness of thedensity does not arise. Most of the work done atM.I.T. in connection with the advection method86 JOURNAL OF METEOROLOGYFIG. 1. Pressure difference between 10,000 feet and 20,000 feet elevation (broken curves). Mean isobar:, (full curves). September 20, 1944, 04002.centered around the weight charts, and the densitycharts were considered mainly in order not to leaveunexplored any possible method of applying advection.Since pressure maps were drawn by the AAFWeather Station at M.I.T. for sea level, 10,000 ft,20,000 ft, 10,000 m and 13,000 m, the pressure-difference or weight charts were constructed for the following layers :Sea level to 10,000 ft (with isobars at 5,000 ft),10,000 ft to 20,000 ft (with isobars at 15,000 ft),20,000 ft to 10,000 in (with isobars at 20,000 ft),10,000 m to 13,000 m (with isobars at 10,000 m).The upper-level pressure charts are based on observations taken at 04002 while the sea-level reportsare for 06302. To make the sea-level reports simultaneous with the upper-air pressure data the threehour pressure tendency was used to correct the sealevel pressures. In order to make forecasts three different methods of advection were tried as mentionedin t,he introduction.(a) Advective motion determined by the geostrophic zehdAs in the case of the density charts, it was assumedfirst that the air in each layer retains its weight andmoves with a speed and direction given by the geostrophic wind relation. To obtain the flow pattern theisobar patterns as indicated above were used for eachlayer. The forecasting period was 12 hours, as, in themajority of cases when the density charts were usedand for the same reasons. The change of weight in eachlayer during the forecast period was computed bymoving the weight lines with the speed and in thedirection indicated by the isobars. To take intoaccount changes above the 13,000-meter level, it wasnecessary to forecast the pressure changes at 13,000meters. This forecast is based largely on rather limitedexperience and is admittedly not much better than aguess. The summation of all these changes for a givenspot gave the pressure change at sea. level which uponaddition to the present pressure resulted in the prognostic pressure. This procedure was applied to therelation.BERNHARD HAURWITZAND COLLABORATORS 87radiosonde stations, and the resulting pressures wereused to draw a prognostic chart.As an example the weight chart for the level from10,000 feet to 20,000 feet on September 20, 1944,04002 is reproduced in Figure 1. Figure 2 shows thesea-level map for the same time. Figure 3 representsthe prognostic sea-level pressure chart for September20, 1944, 16002, obtained from the four weight chartsby the advection method. Figure 4 shows the actualpressure distribution on September 20, 1944, 18302.The maps have only been reproduced south of 50-55"N latitude because the upper-air data farther northare too scanty to permit a reliable forecast.At the start of the forecasting period (Fig. 2), ahigh is situated over the southwestern part of theUnited States; another one is found along the NEcoast of the continent with its center over the Atlantic.These highs are separated by a weak trough. Thecirculation over the northwestern part of the continent is dominated by a low centered over northernManitoba and Saskatchewan. According to the prognostic chart (Fig. 3), the high in the southwesternUnited States should move about 400 miles to ESEwithout changing its intensity appreciably. Actuallythe high moved about 200 miles in an almost easterlydirection and, while the central pressure did notincrease appreciably, the 1017-mb isobar spread considerably. The forecast position of the weak troughdid not agree well with the actual position and thelow which was forecast to occur as the northeasterncontinuation of the trough is not quite deep enoughand too far to the east on the prognostic chart. Theposition of the high over the western North Atlanticcannot be determined accurately from the map forSeptember 20, 1944, 18302 (Fig. 4) because of theabsence of ships' observations but the forecastedFIG. 2. Weather map for sea level. September 20, 1944, 06302.88 JOURNAL OF METEOROLOGYFIG. 3. Prognostic sea-level pressure chart. September 20, 1944, 16002.pressures along the Atlantic coast agree fairly wellwith the observed pressures. This motion of the lowover the Canadian prairie provinces is quite wellreproduced by the prognostic chart (Fig. 3) but, whilethe forecast called for a strengthening of the pressuregradient, the low actually became weaker and thegradient less steep. The prognostic chart (Fig. 3) calledfor the formation of a high over Lake Superior extending toward James Bay, a development which wasnot shown by the actual map (Fig. 4). However, inthis last case it must be pointed out that the failureof the advection method may be due to the inadequacyof the network of radiosonde stations in the northernpart of the continent, which makes the construction ofweight lines and isobars rather subjective.On the whole, the example presented here is one ofthe more succe:jsful prognostic charts plotted by meansof the advectioa method.Such prognostic charts were drawn for about threeweeks during ,4ugust and September, 1944, and discussed at the regular daily map discussions. It wasfound that the prognostic charts constructed in thismanner were certainly not superior to the prognosticcharts produced by a skilled forecaster drawing on hisexperience.In order to have a more objective measure oftheusefulness of the advection method for the preparationof prognostic pressure charts than can be obtainedfrom a simple comparison of the prognostic and theactual map, it is desirable to have a numerical quantityexpressing the degree of similarity between the twomaps. As such a quantity Braun and Douglis (2) havechosen the mean of the absolute values of the deviations of the forecast from the actual pressures, takenfor a sufficiently large number of stations. However,this quantity would be smaller the smaller the variation of the pressure over the map, so that twodifferent cases could not be compared directly. Thisdifficulty can be overcome by dividing the mean absolute value of the deviations by the mean absolutechange over the forecast period, a quantity which iscalled the error ratio by Braun and Douglis. However,even the error ratio does not appear to be a verysuitable quantity for comparison. For instance. if the,B E R N HA R DHA I; KW I T Zforecast were everynhere too high by the same amountthe relative error as well as the mean absolute errormight be quite large, even though the pressure patternand the pressure gradient are forecast perfectly. Onthe other hand, both quantities might be rather small,bht the pressure pattern might still be forecast ratherpoorly, for instance if large changes are forecastwhere only small changes or none occur and vice versa.As another test of the accuracy of the forecasts,Braun and Douglis compared the signs of the forecastchanges 'with those of the observed changes. But hereagain apparently good results might be indicated byhigh percentages of agreement while in reality theprognostic pressure pattern may diverge widely fromthe actual one.Because of the absence of a satisfactory objectivemethod of comparing prognostic and actual charts,correlation coefficients were computed between theactual and forecast pressure changes. Correlation coefAX D COLLA B 0 RA TO RS 89ficients between pressure changes rather than betweenthe pressures themselves were computed because itwas found that for the selected forecast interval highcorrelations, from .80 to .90, were obtained even bycorrelating the pressures at the beginning with thoseat the end of the forecast interval. Hence a high correlation between the forecast and the actual pressurewould imply that the advection method gives resultswhich are as good as the assumption that the pressuredoes not change at all.All the'correlation coefficients are given in Table 1.The first part of this table shows the correlationbetween the forecast changes of the weights of thedifferent layers and of the sea-level pressure and thechanges which actually occurred. The correlation coefficients are only between .41 and .48-too small to beof significant forecast value. The highest correlationwas obtained for the forecasts of the weight of thelayer between 10 and 13 km, the lowest for the layerFIG. 4. Weather map for sea level. September 20, 1944, 18302.90JOURNAL OF METEOROLOGY TABLE 1Statistical Evaluation of Studies in Advection n = number of casesIsobars as trajectories- Standard trajectory12 hours method-12 hours Sept. 18-22, 1944Sept. 18-22, 1944Layers and levels for which correlations Correl. Standard Correl. Standardof pressure changes are computed n Coeff. Error n Coeff. ErrorA. Layer: Sea level to 13 km.B. (A) above + observed 13-km changes 136 +.45 f.06 100 +.38 f.09C. Layer: 10,000 ft to 13 kmD. (C) above + observed 13-km changesE. Layer: Sea level to 10,000 ft. 116 +.41 f.08 100 +.62 f.06F. Layer: 10,000 It to 20,000 ft 130 +.43 1.07 100 f.45 f.08G. Layer: 20,000 It. to 10 km. 120 +.43 f.07 100 +.34 f.09H. Layer: 10 km to 13 km. 110 +.48 f.07 100 +.25 f.10between sea level and 10,000 feet, but the differencecan hardly be considered significant in view of themagnitude of the standard errors of the correlationcoefficients. The correlation between the forecast andthe observed changes of the sea-level-pressure was .45(line B). In considering this value it should be takeninto account that here the actual pressure changes at13 km rather than forecast figures have been usedso that this correlation coefficient presupposes perfectpressure forecasts at 13 km, a highly unlikely accomplishment. Assuming the existence of a linear regression equation between the actual and the forecastpressure change, the standard error of estimate of theforecast pressure was found to be 3.7 mb. This valuedoes not compare favorably with the standard errorwhich a reasonably skilled forecaster could attain.An attempt was made to estimate the percentagefrequency of different causes for failures of the advection method. These frequencies were obtained byjudging in every case which of the following causescontributed mainly to the difference between forecastand actual weight change:A. Speed of isopleths slower than speed of geoB. Speed of isopleths faster than speed of geoC. Lack of data (resulting in uncertainty aboutD. Change in pressure pattern in 12 hours.E. Slight mclvement or none.F. Calculation of advective change unsatisfactory.G. Reason nlot apparent at the present time.H. Good forecast.The percentages for the different causes are shownin Table 2. It is hardly necessary to state that theyare rather subjective since the forecaster cannot becertain about the actual cause of the failure of theforecast. G comprises the cases where the forecasterfelt definitely that other factors besides advectionwere active, although this may also be true in otherstrophic wind.strophic wind.pattern).Rossby trajectorymethod-24 hours.Correl. Standardn Coeff. Error79 +.34 42.1079 +.40 41.1078 +SO f.0978 +.61 f.0778 +.54 +.0880 +.64 f.0778 +.64 f.0773 +.34 f.10cases even though it is not quite so obvious. In thecase H, the forecast was considered successful by theforecaster.(b) Advective motion determined by successive approxiThe assumption that the air motion follows theisobars as given at the beginning of the forecastingperiod is doubtless incorrect. As shown in Table 2, ina number of instances the failure of the advectionmethod ' to give satisfactory results was directlyattributed to this cause by the forecaster. Even if thewind always follows the isobars closely, the path ofthe air during the next 12 hours cannot be given bythe isobars at the beginning of the period since thepressure field itself is changing. Thus, even if horizontal advection alone were determining the pressurechanges, a method of advection along the isobars atthe beginning of the forecast period could not giveperfect results. Better agreement between prognosticand actual pressure fields could be expected if thevariation of the air trajectories due to the variationin the pressure field were taken into account. This wasdone next in the manner outlined by Petterssen (3).In applying Petterssen's method, it is necessary tomake assumptions about the future positions of thepressure systems. Instead of forecasting a future position of the pressure systems, the trajectories have beendetermined from the actual positions of the pressurecenters at the beginning and at the end of the forecastmation to the trajectory. TABLE 2Sources of Error in Forecasting by the Advection MethodCause 0-10,000 ft 10,000 ft-20,000 ft 20,000 ft-10 km 10-13 kmA 18%B 8C 17D 5E 3F 7G 29H 13;%8100144324%418 184 19B E R N H A R DH A U R W I T Zperiod in order to avoid any possibility of errors whichare not directly due to the assumptions of the advection method. Furthermore, since the pressure at thetop level, 13 km, cannot be forecast by the advectionmethod, an error may be introduced in the prognosticpressures for which the advection method is notdirectly responsible. Hence, the actual 13-km pressurerather than a prognosticated 13-km pressure has beenadded to the forecast weight of the layer from sealevel to 13 km. These two procedures require aknowledge of the position of the pressure systems andof the 13-km pressures at the end of the forecastingperiod. Therefore they cannot be used in actual forecasting and the results of the present check must beexpected to be more favorable than they would beunder actual working conditions.The results are statistically appraised in the secondpart of Table 1 under the heading "standard trajectorymethod." Only the correlation for the layer betweensea level and 10,000 ft is now appreciably larger thanbefore. On the other hand, the correlation betweenactual and prognostic sea-level pressure is even smallerthan before although the difference is hardly significant. The result of this comparison can be summarized by saying that this method does not yield anybetter results than the other method of advection.This procedure is a rigorous test of the applicabilityof the advection method inasmuch as the changes inflow pattern and the pressure changes at the top levelwere considered as known.(c) Advective motion determined by Rossby's trajectorymethod.A third method of determining the horizontal flowwhich determines advection is found in Rossby'sprocedure (4) of constructing trajectories under theassumption that the vorticity of individual air particles is conserved. It is not the purpose of this paperto evaluate this method. The reason for applying ithere is merely to include every procedure which mightgive a reasonable basis for the application of the advection method, and thus to make sure that the failureof the advection method to give better results is notcaused by the use of an incorrect method of advection.As before, maps of the pressure at sea level, 10,000feet, 20,000 feet, 10,000 meters and 13,000 meterswere chosen because these charts were constructed inconnection with the daily forecasting routine. Graphical subtraction (1) of the appropriate pressure chartsprovided weight charts for the same layers as mentioned above, viz.Sea level to 10,000 feet,10,000 feet to 20,000 feet,20,000 feet to 10,000 meters,10,000 meters to 13,000 meters.LA N D C 0 L L A B 0 R -4 T 0 R S91Graphical addition of the pressures at the top andbottom of each layer and division by two provide amean isobar and flow pattern for the layer. The appearance of these charts is similar to the sample shownin Figure 1. From the pressure patterns the geostrophic winds were determined. The latter wereregarded as the current velocities for the layers.In the application of the Kossby trajectory methodthe drawing of trajectories is greatly facilitated if onlycertain points in the flow patterns are selected, viz.i) Inflection points,ii) Points of maximum curvature where the flow isFor the purpose of this part of the investigation theassumption was made that an atmospheric columnbetween two levels moves in its layer along a pathdetermined by the trajectory of the current in themean flow pattern of the layer. Since the period ofthis motion can be determined by Rossby's method,the 24-, 48-, and 72-hour displacements of this columnalong its trajectory could be forecast. When a numberof air columns have been treated in this manner, anarea could be selected for which a prognostic chartcould be drawn showing the forecast positions of thecolumns. Since a weight for each column was determined, the prognostic chart is actually a prognosticweight chart. When all the prognostic weight chartsare added, a sea-level prognostic chart is obtained.As in the previous section, it was assumed that aperfect forecast of the pressure at 13,000 meterscould be made; and therefore, the actual pressure at13,000 meters was simply added to the sum of theweights, a procedure which should be kept in mindwhen judging the merits of the advection method.Because of the nature of the Rossby trajectorymethod it is impossible to secure a uniform networkof points for the prognostic weight charts. The pointsat which the weights of different layers are determinedare not in the same vertical. Hence the prognosticpressure charts are best prepared by graphical additionof the different weight charts. Furthermore, it isdifficult to obtain a large enough number of startingpoints that satisfy the limitations set up by theassumptions, so that the distribution of the final pointson the prognostic chart is as a rule rather uneven.Moreover, in many cases no values of the weights ofthe layers are available in the regions from which theair is coming. For example, no data could be obtainedfor the portion of the Pacific bordering upon the westcoast of North America. The result is a restriction inthe area in which verification may be made.The series of charts selected for this part of theinvestigation was that of September 1-5, 1944, 04002.This set had an adequate network of points at therequired levels so that verification charts could befrom the west.92 JOURNAL OFFIG. 5. Sea-level pressure chart. September 5, 1944, 04002.constructed. The forecast interval was always 24hours. Since the sea-level charts were based on the06302 observations while .the upper-air data wereobtained at 041002, the three-hour pressure tendencywas applied as a correction to the sea-level charts.The prognostic charts fall into two categories,weight charts for layers and constant-level pressurecharts. An example of sea-level prognostic pressurecharts is presented in Figure 6. Figure 5 showstheactual pressure distribution at this time. The comparison between these two charts is summarized inTable 3.The prognosticated intensities of the systems foundin Table 3 are either too low or too high and the displacement is too far eastward. In addition, thegradientof pressure is much greater on the prognostic chartthan on the observed chart. A system which conTABLE 3Comparison between Observed and Forecast Sea-Level Pressure Chart September 5, 1944, 04002- . Observed ForecastObserved Forecast In tensity IntensityTypeofSystem Position Position . (mb) (mb)Low New York Ottawaand 1004 994Trough Atlantic Coast Atlantic Coast 1004-1014 994-1010 of US. fromState and Montreal' Lake Ontarioof U.S. fromHarrisburg, BinghampPa. and ton, N. Y.southward southwardTrouph St. Lawrence St. Lawrence 1004-1014 994-996YValley ValleyWedge Kansas High center 1014 1012-1022over Milpvl E T E 0 R 0 L 0 G Ytributes to the intensity of the gradient on the prognostic chart is the high over the Atlantic at 40' Nand 66" W. No evidence for this high can be found onthe actual sea-level pressure chart because reports arelacking in that immediate area.In examining all the prognostic charts it was observed that the isobars form very definite patterniswith extremely steep gradients. The magnitude of thepressure as forecast differs widely from the magnitudeas observed. The difference in the magnitude of theactual and forecast pressures is more noticeable forthe sea-level prognostic charts than for the 10,000-ftcharts. This may be explained in part by the fact thatthe layer froin sea level to 10,000 ft is fictitious overthe portion of the continent where elevations of thesurface run' to great altitudes-especially over theCordilleras. Yet this layer is displaced along the trajectory to form a prognostic weight chart which isadded to the other weight charts and the 13-kmobserved pressure chart in order to produce the sealevel prognostic pressure map.As brought out by the maps the geographical location of the pressure systems is not forecast satisfactorily. While the general pressure patterns haveoften some similarity, the errors of several hundredmiles in the displacement restrict still further theforecasting value of the method. In computing thetrajectory, if the speed of the current were determinedby some fraction (80 to 90%) of the value of thegeostrophic wind, it might be possible to secure betterqualitative and quantitative results. However, it isvery doubtful that a uniform reduction of the speedwould lead to an over-all improvement of the results.A further study of the results was undertaken bystatistical methods, correlating forecast and observedwaukee andwedge overTennesseeFIG. 6. Prognostic sea-level pressure chart. September 5, 1944, 04002.BERNHARD HAURWITZ AND COLLABORATORS 93weight and pressure changes. Because of the difficultyin finding a sufficient number of well distributed pointswhen using Rossby's trajectory method, the correlations for each map between the prognostic and theactual values were based on only 20 points. Since therewere four verification periods (September 2, 3, 4, and5, 1944, 04002) there was a maximum of 80 cases forthe computation of each correlation coefficient. Thecorrelation coefficients are shown in Table 1. TheRossby trajectory method shows the lowest correlations for sea-level pressure (B), for the layer from sealevel to 13 km (A), and for the layer from 10 to 12 km(H). The sea-level pressure forecasts are somewhatbetter than those of the weight of the air column fromsea level to 13 km because in the former case theactual 13-km pressure has been added to the prognosticated weight of the air column from sea level to13 km. The highest correlations were obtained for theweights of the layer between 10,000 ft and 20,000 ft(F) and the layer between 20,000 ft and 10 km (G).As in the case of the sea-level pressure, it will be notedthat the forecasts of the 10,000-ft pressure are betterthan those of the weight of the air column between10,000 ft and 13 km since the actual 13-km pressurehas been added to these figures.In comparing the correlation coefficients for theforecasts obtained by means of the Rossby trajectorymethod with those representing the results of theother two methods, it will be seen that in some casesthe differences are quite substantial and larger thanthe limits set by the probable errors. A direct comparison is not possible since in the investigation onthe basis of Rossby's trajectory method differentmaterial and a different forecasting interval wereused. However, as far as the basic problem is concerned, namely that of constructing prognosticpressure charts, all three methods show correlationcoefficients of about the same magnitude. There areno correlation coefficients in Table 1 large enough tomake it appear that the application of the advectionmethod in any of the forms tried would yield satisfactory prognostic charts.4. Conclusions. From the foregoing discussion itfollows that the advection method in any of the investigated forms does not yield results which recommend its application as a forecasting tool, at least byitself. To some extent its short-coming may be due tothe fact that advection in each layer was based on theflow pattern at a fixed level in this layer. In general,the flow pattern changes somewhat through each layer,a fact which must of course give rise to errors. But inview of the slight variation of the flow patterns withelevation, particularly above 10,000 ft, it appears-Dunlikely that appreciably better results would beobtained if the change of the flow pattern with thealtitude were taken into account.Some of the errors which arose in the study of theadvection method at M.I.T. could have been avoidedif prognosticated changes in obvious disagreementwith all synoptic experience had been disregarded. Ifthe advection method were to be used in practice suchcritical judgment by the forecaster should doubtlessbe employed. However, in doing so one of the mainadvantages of the advection method would be lost,namely, its objectivity; the forecaster, as heretofore,would to a large extent have to rely on his experienceand on his judgment. It is not implied that experienceand judgment can be eliminated from weather forecasting at the present time so that forecasting technique would become as completely mechanical as thecomputation of a nautical almanac. But in view of thefact that the physical basis of the advection methodconsiders only advection and ignores other factorssuch as divergence and vertical motions, it appearsdesirable to bring a subjective element into themethod only after inclusion of these other factors. Ifsuccessful forecasts of the future position of the weightlines and the isobars at a top level could be made, theconsideration of these other factors would be unnecessary, because in that case all the other influenceswould be taken into account automatically. As thepresent investigation has shown, this is impossible.The main reason is that the changes in weight are notdue to advection in horizontal direction only, asimplied by the advection method, but also due todivergence, vertical motions, and other, probablyminor, causes. It would be wrong to discard advectionas one of the factors determining atmospheric developments. But, since there is not just one predominantcause determining these developments, it is necessaryto evaluate and combine all other important causes inorder to arrive at a promising rational forecastingprocedure. REFERENCES(1) Bjerknes, V. and collaborators1911 Dynamic Meteorology and Hydrography, Vol. 2, chap. 8. Carnegie Inst., Washington, D. C.Weight Charts. Weather Service Bulletin, Vol. 3,No. 1. AAF Headquarters Weather Wing, pp. 1924.(2) Braun, R. and Douglis, A.1945(3) Petterssen, S.1940 Weather Analysis and Forecasting. McGraw-HillBook Co., Inc., Sew York and London, p. 222.(4) Rossby, C.-G.1942 Appendix to V. P. Starr, Basic Principles of Weather Forecasting. Harper and Brothers, New York and London.

*The investigation reported in this paper has been carried out under the direction of B. Haurwitz and T.F. Malone. The main part of the synoptic and statistical work was done by Lt. M.A. Eaton, U.S.N.R., Mr. R.C. Staley, and Lt. A.E. Tancreto, U.S.N.R., with the liberal assistance of other staff members of the Institute's Meteorology Department.

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