October 1970735UDC 651.6!57.31:651.642.1~DIURNAL VARIATIONS IN BOUNDARY LAYER WINDSOVER THE SOUTH-CENTRAL UNITED STATES IN SUMMER WILLIAM D. BONNER 1 and JAN PAEGLE2Department of Meteorology, University of California, Los Angeles, Calif.ABSTRACTAnalysis of 1 week's dat,a in August 1960 shows significant diurnal variations in surface geostrophic wind over thesouth-central United States. The oscillation in the southerly component (V,) is driven by the response of the thermalwind to the diurnal temperature cycle over sloping krrain. A smaller oscillation in U, derives from spatial variationsin the arnplitudc of the diurnal pressure wave. The amplitudc of the oscillation in V, is about 3 to 5 m sec-1 at thesurface, decaying exponentially with height to near 0 at 2 km.Examination of 11 yr of summertimc rawinsonde data at Fort Worth, Tex., shows a very regular diurnal variationin boundary layer wind with maximum amplitude of about 3 m sec-1 at 600 m above the ground. This oscillation isforced by periodic variations in both eddy viscosity and geostrophic wind. Using a simplified model of the boundarylayer, we obtain solutions for the diurnally periodic wind resulting from "reasonable" variations in eddy viscosity and"observed" variations in geost,rophic wind.1. INTRODUCTIONBoundary layer winds oscillate diurnally reaching amaximum speed at night and a minimum during the day atelevations between about 30 and 2000 m above the ground.The amplitude of this oscillation is 2 to 3 m sec-l at levelsbetween 0.5 and 1.0 km nbore the ground (Hering andBorden 1962). Maximum speed occurs between 00 and 03local time (LT) (Bonner 1968).The wind variation is especially pronounced with south-erly flow along the central plains to the east of the RockyMountains. In this region, superposition of a strongdiurnal oscillation upon the large-scnle geostrophic flowmay lead to nocturnal jet streams with speeds of more than25 m sec" within the first kilomet.er nbove the ground(Bonner et nl. 1968).Boundary layer wind oscillations may arise from periodicvariations in the horizontal pressure force as in mountainvalley wind circulations, or they may be driven by day-to-night variations in the frictional stress. Numericalexperiments with constcant geostrophic wind and diurnallyvarying eddy viscosity duplicate reasonably well theobserved oscillations over the central plains (Buajitti andBlackadar 1957, Est,oque 1963, and Krishna 1968). Whilethis mechanism alone may explain the oscillation over levelterrain, it has become increasingly apparent that t,here is astrong diurnal variation in the surface geostrophic windjust east of the Rocky Mountains (Hoecker 1965 andSangster 1967).Using the altimeter correction system of Bellamy (1945),Sangster (1967) has shown a day-to-night change ingeostrophic wind as large ns 9 m sec" in northern Texas.Minimum speed occurs near 06 CST, maximum speed near16 CST. An oscillation this large and nearly 180' out ofI Now at the Techniques Development Laboratory, Weather Bureau, ESSA, Silver2 Now at the University of Utah at Salt Lake CitySpring, Md.phase with the oscillation of the real wind certainly can-not be ignored in explanations of the low-level jet.As a corroboration and extension of Sangst.er's work,we have at,tempted to describe the diurnal variation ofgeostrophic wind over the Great Plains, its variation withaltitude, and its interaction with the viscous forcingfunction to produce the observed variation in boundarylayer wind. Sections 2 and 3 describe variations in low-level geostrophic winds in Texas and Oklahoma during a1-week period in August 1960. Section 4 describes observedT-ariat,ions in boundary layer wind. as det.ermined from twosets of data that, are combined to give the equivalent of3-hr observations of t.he wind. In section 5, we examinethe effects of variable viscosity and variable pressure-gradient force upon the boundary layer wind, comparingderived results with observations in section 4.2. VARIATIONS IN SURFACE GEOSTROPHIC WINDUsing the altimeter correction system of Bellamy(1945), one may write t.he eastward and northward com-ponents of tjhe surface geostrophic wind aswhere z is the height of the terrain, S* is the specific virtualtemperature anomaly T*- T,s*="---1TPand D is the difference z-zp at terrain height. VariablesT, and z, are temperature and pressure altitude in thestandard atmosphere. Thus, each component of the surfacegeostrophic wind is a sum of two terms, one depending7 36MONTHLY WEATHER REVIEW Vol. 98, No. 10primarily upon the gradient of D values along the slopingterrain, the other upon the product of specific temperatureanomaly S* and the terrain slope.Suppose, for simplicity, that the gradient of terrain is inthe minus x direction and that the average daily value ofS* is zero. In the morning, a negative S* gives a negativecontribution to V,. If there is no counterbalancing oscil-lation in aD/ax, there mill be a diurnal oscillation of thegeostrophic wind with an a.mplitude that depends upon theterrain slope and the amplitude of the diurnal temperaturewave. In the case of southerly flow east of the RockyMountains, the geostrophic wind mill oscillate in phasewith the temperature cycle-reaching maximum speed atthe time of maximum temperature. With northerly windsor with a reversal in terrain slope, the sense of the oscilla-tion is reversed with maximum speed at the time of mini-mum temperature.METHODGeostrophic surface winds were computed from equa-tions (1) at 28 grid point,s in Texas and Oklahoma (fig. 1).Grid points are at intervals of 1' latitude and 1.25'longitude. Centered differences were used so that the basiclength unit is approximately 220 km.Use of equations (1) requires a smoot,hed representationof the terrain. We used a determination by McClain (1960)of average terrain heights within areas bounded by 1'latitude and longitude lines (fig. 2). Terrain heights wereint.erpolated at grid points; values of az/az and jay werecomputed from centered finite differences and thensmoothed to represent only the large-scale features ofterrain.Altimeter settings, temperatures, and dew points weretabulated at 3-hr intervals at approximat.ely 50 stationsin the area of interest. Altimeter settings were convertedto D values (see, for example, Haltiner and Martin 1957),temperatures and dew points to virtual temperature T*.Virtual temperatures and D values were averaged overthe period from Aug. 2 to 8, 1960, and maps were con-structed of mean D and T* fields at 00, 03, 06, . . .,21 CST. Virtual temperatures were converted to specifictemperature anomalies (equation 2), and values of S*,D, /ax, lay at grid points were used to compute Uand V components of the surface geostrophic wind.RESULTSFigure 3 shows isotachs of the mean geostrophic windat 00, 06, 12, and 18 CST for the period from Aug. 2 t,o 8,1960. Geostrophic minds during the period were constantlyfrom the south or southwest. Front's remained to the northof the area, and skies were mainly clear except for after-noon showers and thunderstorms in Texas and scatterednocturnal thunderstorms in Oklahoma.Each map shows a zone of strong geostrophic windbetween 96" and 98' W. There is a weak maximum inOklahoma and a second stronger maximum in centralTexas. Strongest computed winds are 13.7 m sec" at06 CST and 23.7 m sec" at 18 CST.FIGURE 1.-Grid used in determination of geostrophic winds. Gridpoint separation is 1' latitude by 1.25'' longitude. The map showst>he position of radiosonde stations used in thermal wind calcula-tions (section 3).Sangster (1967) det,ermined the average geostrophicsouth wind component between Amarillo and OklahomaCity for the month of June 1966 to be 6 m sec -* at 06 CSTand 15 m sec -l at 18 CST. Our values in this area for theshorter period in August 1960 are 7 m sec and 15 msec", respectively.Figure 4 shows the time variation in the northward Vcomponent of the geostrophic wind at two locations inOklahoma. Near Oklahoma City (grid point 35.2 infig. l), the oscillation is quite regular with minimum V,near 0430 CST and a maximum near 17 CST. At grid point36.4 in northeastern Oklahoma, the t'errain slopes upwardto the east (fig. 2). Here, the sense of the oscillation ischanged with minimum geostrophic speed during theafternoon and maximum speed near 06 CST.Daily variations in U, and V, are summarized in figures5, 6, and 7 that represent averages of the geostrophiccomponents at 12 grid points in figure 1.Total U, and V, components are shown in figure 5.On the average, minimum V, occurs between 03 and 06CST and maximum V, near 17 CST. The total variation insoutherly geostrophic wind from morning to evening isabout 7.5 m sec-l. Changes are roughly in phase with thediurnal temperature wave; however, largest V, occursabout 2 hr after the time of maximum temperature. TheU, component is strongest near noon, weakest near 21CST, and undergoes a total variation of about 2.5 m sec".. . ,-October 1970William D. Bonner and Jan Paegle7379 '9 9 9 IO I1 II III MISSOURI9 (9 IO I1 I3 13 12 12431223111 I LOUISIANA433\2211'13 3 71,' 2 I I 2222;221 I2I I I,'I I I I II00,000000x0FIGURE 2.-Terrain heights in hundreds of feet within areas boundedby lo latitude and longitude lines, taken from data used byMcClain (1960) in construction of a smoothed topographic map.Diurnal variations in the contribution to U, and V,from the terrain slope term (fig. 6) are roughly as es-pected with minimum and masimum values at' 06 and 15CST. Total variations in U, and V, from the second termsin equations (1) are 0.8 and 5.4 m sec-', respect'ively.The upper half of figure 7 shows t,hat the observedvariations in U, result primarily from a diurnal rarintionin the y derivative of D values at the level of the t,errain.This effect dominates the t,errain slope term at mostlocations enhancing the westerly geostrophic winds in thelate morning nnd ea,rly afternoon. The 17, component. inthe lower half of figure 7 shows a relatively small, pri-marily semidiurnal oscillation with a maximum cont,ribu-tion near 18 CST, a relative maximum near 06 CST, andminima new 03 and 10 CST.OOCST'1I8CST IFIGURE 3.-Isotachs (met,crs per second) of average geostrophic wind from Aug. 2 to 8, 1960.terrain slope.METHOD AND ILLUSTRATION3. VARIATIONS IN THERMAL WINDBy subt,rncting the component equat,ions nt. two levelsa constant height above the ground and usillg the hydrostatic,relationship aD/az=S*/1 +S*, the follo\I-ing espressiousare derived for the thermal wind:Ulh=-g - - (S*/(l+S*))+9 (SY-S,*) - AZ a aZ f a?/ f a?J AZ aVlh=g - - (S*/(l+S*))".- (ST-S,*) -* Q aZ .f ax *f axand (3)The bnr indicat,es an average vnlue in tjhe layer from 0to 1 of thickness Az. Equat,ions (3) neglect small terms , .As an example, figure 8 shows the T* and S* curvesfrom mean dnta for Fort, Wort2h, Tes., at 06 and 18 CST.During the aft,ernoon, S* decreases with height.. Sinceazlax is negative, the second term in the espression for V,n(equations 3) is negative nnd acts to decrease the strongsoutherly geost,rophic, wind at. the surface. At 06 CST, s*is constant or increasing wit,h height and t,he second termis small or acting in the opposite sense.Similar curves were const,ructed from t,he averagetemperature data from ench of 11 radiosonde stations infigure 1. Menn values of S*/( 1 + S*) in 400-m la~yers weredetermined for each station at each time. These valueswere plotted and andyzed, and first terms were computedfrom dues interpolated at grid point,s. Sample analyses ofS*/(l+S*) are shown in figure 9. At both 06 and 18 CST,738 MONTHLY WEATHER REVIEW Vol. 98, No. 10EY0,>GRID PT. 36.4IO500 3 6 9 I2 15 18 21 0LOCAL TIMEFIGURE 4.-Southerly component of geostrophic wind at grid points35.2 and 36.4 (see also fig. 1). The values are averages from Aug. 2to 8, 1960. Smooth curves connect points representing totalgeostrophic wind. Unconnected points are contributions from theD value term alone. Note the phase reversal with the reversal ofterrain slope.LOCAL TIME64CONTRIBUTION TO Ug FROM TERRAIN SLOPEI I-I 0a3AVERAGE OF 12 GRID POINTS1 ;2m C-k-f"3T01 1 I I ICONTRIBUTION TO Vg FROM TERRAIN SLOPE~~~IO-"""_l/ I I IAVERAGE OF 12 GRID POINTS40 3 6 9 12 15 18 21 0 LOCAL TIME'IGURE 6.-Time variation of terrain contribution to geostrophicwind, averaged over the same grid points as in figure 5.D6 I I I I I I I IB. CONTRIBUTION TO Va FROM GRADIENT OFD0; 4 k b 1; Ib I; 211 ALOCAL TI MEFIQURE 7.-Time variation of the contribution to geostrophic windfrom the gradient of D values, averaged over the same grid pointsas in figure 5.warmest air lies over the northwestern section of theregion. Thermal winds from this term parallel the isoplethsof S*/(l+S*) and shift from northeasterly during themorning to northerly during the afternoon-acting at the-FIGURE 5.-Time variation of total geostrophic wind, average of later time to diminish with height the strong southerly12 grid points in western Oklahoma and north-central Texas. surface geostrophic winds.October 1970 William D. Bonner and Jan Paegle739IFORT WORTH, TEXAS (2-8 AUG 60)06 CST 18 CSTI S*.04 .05 .06 .06 .O? .08 .09 IFIGURE %-Height and time variation of T* (solid lines) and S*(dashed lines) at Fort Worth, Tex., averaged over the week fromAug. 2 to 8, 1960.RESULTSFigure 10 shows t.herma1 winds computed from equa-tions (3) at 06 and 18 CST and the vect.or change in thermalwind between t,he t,wo observat,ion times. At 06 CST,thermal winds within the first 2 km are directed mainlyfrom the east-northeast with speeds of roughly 4 to 7 msec-l. At 18 CST, thermal winds are from the north-northeast with speeds of S t,o 12 m sec" t'hrough most ofthe region. Chsnges in thermal wind are parallel to thet'errain contours and are northerly where the terrainslopes down t,o the east, and southerly where the terrainslope reverses (grid points 37.4 and 36.4). Vector changesin figure 10 show a rariat,ion in t,hermal wind withinthe first 2 km that is nearly equal and opposite to thevariation in surfme geostrophic, wind. Addition of theV components of the thermal winds in figure 10 to t,he 06and 18 CST surface winds in figure 3 yields geostrophic,southerly winds at 2 km t,hat are essentially the same at,06 and 1s CST. Since 06 a,nd 18 CST are nea.r the times ofminimum a,nd masimum surface geostrophic wind, theamplitude of the implied surface osci1lat)ion is roughlyone-half the magnitude of the t,hermal wind changesin figure 10.The oscillation in thermal wind is shown schematicallyin figure 11. Physically, t,he process is quite simply androughly as Sangster describes. At 2 km, t,he geostrophicwind does not vary from day to night,. A,t night, nir abovethe higher termin cools much more than the air at thesame pressure level farther to the east.. This introducesa southerly component to the thermal wind that impliesthat V, at the surface is less than Vg at 2 km. During the06 CSTi18 CSTFIGURE 9.-Fields of S*/(l+S*) from the surface to 400 m abovethe ground. 1-alues have been multiplied by 103. Not,e the warmtongue to the west and the change in orientation and spacing ofthe lines between 06 and 18 CST.the day, the air over the mountains becomes warmert,han the air to the east.; V,, is from t'he north, and T, at'the surface becomes larger t.han V, at 2 km. Thus, theoscillation in the surface geostrophic wind is driven byan alt,ernating thermal wind that results from the dailyheating cycle over sloping terrain. The pict.ure is asimplification of the thermal wind patterns in figure 10where a mean temperatmure gradient from east, to westkeeps a northerly component, to the t'hermal winds at.both 06 and 18 CST. However, t,he import,ant feature isthe pronounced daytime increase in the northerly c,om-ponent of the thermal wind (fig. 10) that produces anafternoon masimum in the southerly component of t'hesurface geostrophic wind.Figure 12 shows the rate of decay of this oscillationwith height as determined from thermal wind calculationin 400-m layers. The graph is based upon changes in tbeI' component, of the geostrophic wind at 15 grid pointsand assumes that t,here is no significant shift in the phaseof t,he oscillation with height. At each level, t,he changein ITK was espressed 11s a percenta,ge of the'surface change.Percentages at. individual grid points were then averagedto obtain the plotted points in figure 12. The decay withheight is logarithmic and follows very closely the relat'ion-ship r=e-z/fl.8 where z is in kilometers above the groundand r is the ratio between the amplitude at z and thesurface amplit.ude.4. VARIATIONS IN OBSERVED WINDThe period Bug. 2 to 8, 1960, was chosen initially be-cause of a high frequency of morning lowlevel jet, obser-vations (Bonner 1968), and winds during this period doshorn strong and regular diurnal variations. Our aim,however, is not to describe the part.icular events fromAugust 2 t,o 8, but to provide a much more general de-scription of the diurnal oscillation in boundary layer wind.403-235 0-70--3740MONTHLY WEATHER REVIEWVol. 98, No. 10(\5.9 5.3 5.8I\VTH (0-2KM) 06CIT YIi L3 L2 L9\V TH (0-2 KM 18 CST T+FIGURE 10.-Thermal winds from the surface to 2 km and the change in thermal wind from 06 to 18 CST. Values at grid points are inmeters per second. Winds are plotted in standard synopt,ic form with barbs representing speeds in knots. Dashed lines are mean terrain contours from figure 2 labeled in meters above sea level.06 CST06 CSTEs /18 CST18 CSTwjy EFIGURE 11.-Schematic represent,ation of oscillation in thermal wind.For this purpose, we examined daily rawinsonde obser-vations at Fort Worth, Tex., for July and August 1952 to1955 and 1958 to 1964. During the earlier period, obser-vations were regularly scheduled at 03, 09, 15, and 21 GMT;for the later period, observation times mere 00, 06, 12, and18 GMT. By combining the two series of observations, itis possible to obtain the equivalent of 3-hr observationsof the wind. The same technique has been used by Harriset al. (1966), Harris (1959), and Johnson (1955) to obtainfirst and second harmonics of daily variations in wind,PERCENT OF SURFACE VARIATIONFIGURE 12.-Decay of the oscillation with height. The values followclosely a curve with equation r=e-r'0.8 where z is in kilometersabove the ground and r is the percentage of the surface oscillationat level z.pressure, and temperature. Levels examined that arecommon to both series include the surface, 0.5, 1.0, 2.0,2.5, and 3.0 km above sea level.At each observation time, we computed the deviationof the wind from its average value for the particular day.Deviations were averaged over all days in each series, andthe two series were combined to give the average variationshown in figure 13.October 1970 William D. Bonner and Jan Paegle 7413.02.5nz30az 2.00W>0m I .5zaYI .o0.50.0U' COMPONENTV' COMPONENTr-0.5 -0.3 -0.1 -0.7 -0.2 0.6 0.8 0.4 -0.50 6 12 18 0TIMEBoth component's of the n-intl shou- ti (tail_\- vnriation of5 to 6 m sec-l between about 400 and 800 nl above theground. Maximum westerly wind occurs between 06 ant1 07CST; maximum southerly wind near 00 CST. The level ofmaximum amplitude in both components is approximatel5-600 m above the ground, and the oscillation disappearsbetween 2.0 and 2.5 km above t'he ground.Figure 14 shows hodographs of the wind rarintions at.selected levels above t'he ground. At. 0.15 km, observationsexist only for the later period. At' the surface, observationscould not be combined, bat both sets of data indicatecounterclockwise rot,at~ion of t>he deviation rector \\-itahtime. At, other levels, rot'at'ion is clockwise at, a rate thntnppears to be great,est at, night,, slowest during t,he after-noon. Hodogrnphs are nearly circular, wit.h slightly re-duced amplitude during the afternoon.The size of the data circles in figure 14 indicates t.heprobable error in tlhe estimate of each mean deviationrector RS determined from tlhe relationship r=0.939 ZIT';(Chapman 1951) where r is the radius of t'he probnbleerror circle, ;i is t,he mean distance between the vect,or endpoint for each individual year and the n-yr mean. Hodo-gl.aphs are, in general, well determined, dh r approxi-mately an order of magnitude smaller than the deviationrector itself. The tendency for larger probable errors at03, 09, etc. is a reflection of the smaller nnmber of years inthe earlier data sample. 5. THEORETICAL SOLUTIONSFOR THE BOUNDARY LAYER WINDIn t,his sect,ion, we use n model developed by Paegle(1970) to examine the effects of diurnally periodic eddyviscosity and pressure gradient force on the boundarylayer wind. SpecificnlI_vy we consider (1) time dependent.eddy riscosit.5-, constant geostrophic wind ; (2) constanteddy viscosity, variable geostrophic wind ; and (3) variable\*iscositsy, variable geostrophic wind.The first problem has been treated by Bunjitti andBlackadar (1957), Ooyama (1957), Estoque (1963), andKrishnn (1968); the second by Lettau (1964) and Holton(1967) ; and t.he third by Sangster (1970) and Paegle (1970).METHODHere, we will briefly outline the model and the method742MONTHLY WEATHER REVIEW Vol. 98, No. 10"II11958 - 1964I_m 0.15 km1958 -1964 1952- 1955FIGURE 14.-Hodographs of the wind variations at selected levels(same data as in fig. 13). Size of the data circle indicates probableerrors in determination of the mean deviation vector. Heights arein meters above the ground; timcs are CST.of solution. A more complete description is given by Paegle(1970).In the complex plane, the horizontal momentum equa-tion may be writtenwhere (U,, V,) are eastward and northward componentsof the geostrophic wind, t' is time, and K is t,he eddy vis-cosity. The geostrophic mind can have an arbit,rary timeand height dependence; the eddy viscosity ma.y be t,imedependent, but is assumed to be constant in height. L Y 011-linear terms are ignored, although Bonner et al. (1968)and Paegle (1969) have shown that they may contributesignificantly to ageostrophic winds in the vicinity of awell-developed jet.Solut,ions of equation (4) are more compact when theindependent variables are nondimensional:Equation (4) has the exact solutionwhere W,, W,, and W, are defined by the followingintegrals :where x()=(U+iV)(, ~=o)-(U,+iv,>(, T=O),If the eddy viscosity is constant in t,ime, then r=t. If theeddy viscosity varies with time, then r is a more compli-cated function of time. We will consider an eddy viscositywith time dependence :K=A(l-v COS fit'), Ovl (8)in which caseand the scaling in (5) is acc,omplished with A replacing K.The first integral, W,, accounts for initial conditions, TIT2for the surface boundary condition, Wr3 exists for nonzeroG that occurs only if the geostrophic wind is time tlepend-ent or has curvature with height. Ching and Businger(1968) give similar int,egral solut,ions for the nonsteadyboundary layer.Integrals in equation (6) were evaluated by Gaussianquadrature for prescribed values of K, U,, and T7,. Solu-tions are for the initial value problem, and transientcomponents are present together with uny steady andperiodic modes. With the winds initially in geostrophicbalance, diurnally periodic modes dominate the solutionsaft.er several days. Results to be shown were taken fromthe fourth day of the integrations.In problem 1, geostrophic components U, md V, areassumed t,o be functions of height, alone. Eddy viscosit'pis given by equation (8) which is the same formulation RSused by Ooyama (1957) except that Ooyama allowed Ato vary with height. In the calculations t,o be shown, A isassumed to be 8 m2 sec-' implying, for the steady Eckmanproblem, a geostrophic wind level of 1.4 km. We set v=O.8.The maximum K value of 14.4 mz set" correspondsclosely to numerical results by Krishna (1968). MinimumK is then 1.6 m2 sec-1 (equation 8), and the oscillation isphased so that the minimum eddy viscosity occurs atIn problem 2, eddy viscosity is given a mean due of8 m2 sec-I, independent of height or time. The geostrophicwind is allowed to vary in the following way:01 LT.u,+iv,=(~,+i~~)-~Voe-'~~ cos (Qt') (9)October 1970William D. Bonner and Jan Paegle743A. VARIABLE KCONSTANT Vg:min OlLOCALTlME-4 -2 0 i3. CONSTANT KVARIABLE Vg'gmin 05 LOCALTIMEO= 32.54-202Y0537.5-4 -2 0 2u rn set'C. VARIABLE KVARIABLE Vg6-4-2 0 2 4 60=37*5 ti-4 -2 0 2 4u mse2FIGURE 15.-Hodographs of wind variations from (A) variableviscosity and constant geostrophic wind, (B) constant viscosit,yand variable geostrophic wind, and (C) variable viscosity andvariable geostrophic wind. Upper diagrams are for 32.5" latitude;lower, for 37.5" latitude. Altitude is 500 m; times arc CST. Oscilla-tions in geostrophic wind and eddy viscosit,y arc phased to giveminima at 05 and 01 CST, respectively.where U,+iV, is the time-averaged geostrophic wind at,some height z above the terrain. The geostrophic variationin (9) is entirely in the y component of the wind. The nu"-ation is diurnally periodic with amplitude AVO at thesurface, decreasing exponentially with height. H is setequal to 0.8 km, AVO is 3 m sec". The oscillation is phasedto give minimum and maximum V, at 05 and 17 LT,respectively. (See preceding sections.)In problem 3, eddy viscosity varies according to (8),geostrophic wind according t,o (9). We use the same raluesfor A, Y, AVO, and H as in the previous problems."RESULTSResults-of R series of integrations for latitudes 32.5'and 37.5' are shown in figure 15. The altitude in eachcase is 0.5 km, which is near the level of maximum oscil-lation in the model and in the observed wind.Results from problem 1 are shown in figure 15A. Atboth latitudes, the oscillation from eddy viscosity raria-t.ion alone is elliptical with the major axis directed slightlyto the right of the geostrophic wind. The mean amplitudeof the oscillation is about 2.5 m sec-' at 37.5' and 3 msec" at 32.5'. Phase, orientation, and amplitude of theoscillation agree closely with results by Ooyama (1957).Maximum speed occurs between 00 and 03 LT, minimumspeed near noon. The phase of the oscillation advnnceswith latitude as would be expected from the change inthe inertial period (see also Rrishna 1968).Hodographs for problem 2 (fig. 15B) show an amplitudesmaller than that for problem 1. Maximum speed occursbetween 1s and 21 LT-well ahead of the observed max-imum in figure 14. The solution at 32.5' agrees closelywith results by Holton (1967) for an isothermal atmos-phere at. 30' latit,ude.Hodographs for problem 3 are shown in figure 15C. Theamplitude of t,he oscillation is 4 m sec" at 37.5' androughly 5 nl set" at 32.5'. Oscillations are nearly cir-cular. Maximum wind speed occurs near midnight, mini-mum speed between 09 and 12 LT.Giren the very simplified treatment of t.he boundarylayer, particularly the assumption that eddy viscosityis independent of height, we cannot expect. exact. agree-ment with result's in figure 14. The amplitude of the os-cillation is fairly sensitive to t,he mean geostrophic wind,the selection of V, and the phase difference between eddyviscosity and pressure gradient oscillations (Paegle 1970).Solutions to problem 3, however, duplicat,e t.he majorfeatures of the observed oscillation in figure 14. most^important, comparison of results from problems 1 and3 shows that the prescribed variation in geostrophic windincreases the amplitude of the oscilhtion that wises fromeddy viscosit;v alone. With, cliurna,lly periodic ~~scosity,a, thermal wind osci.llation gibing maxim.um geostrophicwind in late a:fternoon yi,elds a large circular osci,llation inthe real wind with maximurn speed near midnight.7. SUMMARY AND CONCLUSIONSummert~ime geostrophic winds at t,errain level in Texmand Oklahoma show day-to-night speed variations of theorder of 5 t.0 9 m set". Maximum speed occurs near 17CST, minimum speed near 05 CST. The oscillation is drivenby 811 dternat$ing thermal wind within the first 2 kmt,hat arises from t.he daily heating cycle over sloping ter-rain. The surface varintion is explained fnirly well byconsidering only a term S* azfax in the equation for V,.However, t,here are significant variations in both U, and17, t,hat derive from diurnal variations in the gradient ofD ralues over the sloping terrain (Bonner and Paegle1969).Observed winds at Fort Worth, Tex., show a veryregular diurnal oscillation with an amplitude of nearly3 m set" at 600 m above the ground. This oscillation isdescribed rather poorly by n model with constant viscosityand variable geostrophic wind, fairly well by assumingconstant geostrophic wind and variable viscosity. Simula-tion of the real situation in this area with variable viscosity144 MONTHLY WEATHER REVIEW Vol. 98, No. 10and variable geostrophic wind yields solutions with roughlythe same shape and phase as tho oscillation in observedwind. The time of the geostrophic minimum is known.If we prescribe a phase lag of at least 2 hr between thegeostrophic wind and eddy viscosity oscillations, thegeostrophic oscillation acts to increase the amplitudethat would arise from variable viscosity alone (Paegle1970). This provides at least a partial explanation for thepronounced diurnal oscillations in boundary layer mindobserved with southerly flow over the south-centralUnited States.ACKNOWLEDGMENTSWe wish to thank Prof. Morton Wnrtele for suggesting theapproach used in the theoietical portions of the study. We thankalso Miss Ylm-Mei Chang and Mr. Ted Tsui who did much of thecomputational work required in determining the geostrophic winds,and Dr. Julia Noglws de Paeglc who wrote the progran~ for dcter-mining time variations in thc observed winds. Rcscarch v-as s11p-ported by the National Scicnce Foundat,ion Grant GA-698.REFERENCESBellamy, John C., "The Use of Pressure Altitude and AltimeterCorrections in Meteorology," Journal of Meteorology, Vol. 2,No. 1, Mar. 1945, pp. 1-79.Bonner, William I)., "Climatology of the Low Level Jet," MonthlyWeather Review, Vol. 96, No. 12, Dec. 1968, pp. 833-850.Bonner, William I)., Eshensen, S., and Greenberg, It., "Kinematicsof the Low-Level Jet," Journal of Applied Meteorology, Tol. 7,No. 3, June 1968, pp. 339-347.Bonner, William I)., and Paegle, Jan, "Diurnal Variations in Windand Geostrophic Wind Over the South-Central United States inSummer," Technical Report, Dept. of Meteorology, Universityof California, Los Angeles, 1969, 30 pp.Buajitti, Kajit, and Blackadar, A. K., "Theoretical Studies ofDiurnal Wind-Structure T-ariations in the Planetary BoundaryLayer,'' Quarterly Journal of the Royal Meteorological Society,Vol. 83, No. 3.58, Oct. 1957, pp. 486-500.Chapman, Sydney, "Atmospheric Tides and Oscillations," Com-pendium of Meteorology, 1951, pp. 510-530.Ching, Jason K. S., and Businger, J. A., "The Itesponsc of thePlanetary Boundary Layer to Time Varying Pressure GradientForce," Journal of Atmospheric Sciences, Vol. 25, No. 6, Nov. 1968,Estoque, Mariano A., "A Numerical Model of the AtmosphericBoundary Layer," Journal of Geophysical Research, lvol. 68, No. 4,Feb. 15, 1963, pp. 1103-1113.pp. 1021-1023.Haltiner, George J., and Martin, Frank L., Dynamical and PhysicalMeteorology, McGraw-Hill Book Co., Inc., New York, 1957,470 pp.Harris, Miles F., "Diurnal and Semidiurnal Variations of Wind,Pressure, and Temperature in the Troposphere at WashingtonD.C.," Journal of Geophysical Research, VoI. 64, No. 8, Aug. 1959,Harris, Miles F., Finger, Frederick G., and Teweles, Sidney,"Frictional and Thermal Influences in the Solar SemidiurnalTide," Monthly Weather Review, 1'01. 94, No. 7, July 1966, pp.427-447.Hering, Wayne S., and Borden, Thomas R., Jr., "Diurnal Vari-ations in the Summer Wind Field Over the Central UnitedStates," Journal of the Atmospheric Sciences, Vol. 19, No. 1, Jan.Hoccker, Walter H., "Comparative Physical Behavior of SoutherlyBoundary-Layer Wind Jets," Monthly Weather Review, Vol. 93,No. 3, Mar. 1965, pp. 133-144.Holton, James R., "The Diurnal Boundary Layer Wind OscillationAbove Sloping Terrain," Tellus, Vol. 19, No. 2, 1967, pp. 199-203.Johnson, D. I%., "Tidal Oscillations of the Lower Stratosphere,"Quarterly Journal of the Royal Mdaorological Society, Vol. 81,No. 347, Jan. 1955, pp. 14.Krishna, K., (`A Numerical Study of the Diurnal Variation ofMeteorological Parameters in thc Planetary Boundary Layer: I.Diurnal Variation of Winds," Monthly Weather Review, Vol. 96,No. 5, May 1968, pp. 269-276.Lettau, Heinz H., "Studies of the Effects of Variations in BoundaryConditions on the Atmospheric Boundary Layer," AnnualReport 1964, Department of ?c.leteorology, The University ofWisconsin, Madison, 1964, pp. 99-11.5.McClain, E. Paul, "Some Effects of the Western Cordillera ofNorth America on Cyclonic Activity," Journal of Meteorology,Vol. 17, No. 2, Apr. 1960, pp. 104-115.Ooyama, Katsuyuki, "A Study of Diurnal Variations of WindCaused by Periodic \`ariation of Eddy Viscosity," Final Report,Contract No. AF19(604)-1368, College of Engineering, NewYork University, May 1957, pp. 80-13.5.Paegle, Jan, "Studies of Diurnally Periodic Boundary Layer Winds,"Ph. I). thesis, Department of Meteorology, University of tal-ifornia, Los Angeles, 1969, 131 pp.Paegle, Jan, "Studies of Diurnally Periodic Boundary Layer Winds,"Technical Report, NSF Grant GA-698, Dept,. of Meteorology,University of California, Los Angeles, 1970, 69 pp.Sangster, Wayne E., "Dillrnal Surface Geostrophic Wind VariationsOver the Great Plains," Proceedings of the Fifih Conference onSevere Local Storms, St. Louis, Missouri, October 19-90, 1967,American Meteorological Society, Greater St. Louis Chapter,hlo., 1967, pp. 146-153.Sangster, Wayne E., Weather Bureau Central Region, ESSA,Kansas City, Mo., 1970 (personal communication).pp. 983-995.1962, pp. 81-86.[Received November 18, 1969; revised June 5, 19701