{"Bibliographic":{"Title":"Studies of nocturnal stable layers at BAO","Authors":"","Publication date":"1983","Publisher":""},"Administrative":{"Date created":"08-17-2023","Language":"English","Rights":"CC 0","Size":"0000168193"},"Pages":["P P\nno.4\nSTUDIES OF NOCTURNAL\nSTABLE LAYERS AT BAO\nA","QC\n851\nB6\nno.4\nSTUDIES OF NOCTURNAL\nSTABLE LAYERS AT BAO\nJ.C. Kaimal, Editor\nLIBRARY\nMAR 3 0 1990\nUS\nReport Number Four\nJanuary 1983\nNOAA\nBoulder Atmospheric Observatory\nATMOSPHERIC\nAND\nU.S. Department of Commerce\nNOAA\nNational Oceanic and Atmospheric Administration\nEnvironmental Research Laboratories\nA NOAA publication available from NOAA/ERL, Boulder, CO 80303.","NOTICE\nMention of a commercial company or product does not constitute\nan endorsement by NOAA Environmental Research Laboratories.\nUse for publicity or advertising purposes of information from\nthis publication concerning proprietary products or the tests\nof such products is not authorized.\nii","CONTENTS\nPage\nV\nFOREWORD\nOBSERVATIONS OF TURBULENCE STRUCTURE IN STABLE LAYERS\n1.\nAT THE BOULDER ATMOSPHERIC OBSERVATORY (J.C.R. Hunt,\n1\nJ.C. Kaimal, J.E. Gaynor, and A. Korrell)\n1\nABSTRACT\n2\n1.1\nINTRODUCTION\nTOPOGRAPHIC AND SYNOPTIC CONDITIONS OF THE EXPERIMENTS\n3\n1.2\n6\nMEAN WIND SPEED AND STRATIFICATION DATA\n1.3\nANALYSIS OF VELOCITY AND TEMPERATURE FLUCTUATIONS\n1.4\n10\nAND HEAT FLUX IN TERMS OF WAVES AND/OR TURBULENCE\n23\n1.5 TENTATIVE CONCLUSIONS\n26\n1.6 ACKNOWLEDGMENTS\n26\n1.7 REFERENCES\n30\nAPPENDIX A: BACKGROUND DATA FOR OBSERVATION PERIODS\nAPPENDIX B: VARIANCES AND SCALES OF TEMPERATURE AND\n41\nVELOCITY FLUCTUATIONS IN THE STABLE BOUNDARY LAYER\nAPPENDIX C: SCALES OF TEMPERATURE AND VELOCITY FLUCTUATIONS\n44\nAPPENDIX D: HYPOTHESES CONCERNING INTEGRAL SCALES AND\n47\nDISSIPATION SCALES IN SLOWLY VARYING TURBULENT FLOWS\nWAVE AND TURBULENCE STRUCTURE IN A DISTURBED NOCTURNAL\n2.\nINVERSION (Lu Nai-Ping, W.D. Neff, and J.C. Kaimal)\n53\n53\nABSTRACT\n53\n2.1 INTRODUCTION\n54\n2.2 SITE CHARACTERISTICS AND INSTRUMENTATION\n55\n2.3 GENERAL DESCRIPTION OF CASE STUDY\n55\n2.3.1 Meteorological Situation\n56\n2.3.2 Acoustic Sounder and Mean Flow Data\n60\nInternal Wave Behavior\n2.3.3\n61\n2.4 SPECTRAL ANALYSIS\n61\n2.4.1 Spectra Under Steady Conditions\n62\n2.4.2\nDisturbed Cases\n66\n2.5 INTERPRETATION\n2.5.1 Effects at the Frontal Interface, 0100-0140 MST\n66\n68\nTrends/Steps in the Data, 0420-0540 MST\n2.5.2\nStrong Disturbances in the Inversion,\n2.5.3\n68\n0320-0440 MST\n71\n2.6\nCONCLUSIONS\n71\n2.7\nACKNOWLEDGMENT\n72\n2.8\nREFERENCES\niii","3.\nA STUDY OF MULTIPLE STABLE LAYERS IN THE NOCTURNAL LOWER\nATMOSPHERE (Li Xing-sheng, J.E. Gaynor, and J.C. Kaimal)\n75\nABSTRACT\n75\n3.1\nINTRODUCTION\n75\n3.2\nFORMATION OF MULTIPLE LAYERS\n76\n3.3\nA\nCASE STUDY\n77\n3.4\nEDDY KINETIC ENERGY AND THE DISTRIBUTION OF\nTURBULENT FLUXES OF HEAT AND MOMENTUM\n82\n3.5\nTEMPERATURE BUDGET PROFILE\n85\n3.6\nCONCLUSIONS\n89\n3.7\nACKNOWLEDGMENTS\n89\n3.8\nREFERENCES\n90\n4.\nA METHOD FOR MEASURING THE PHASE SPEED AND AZIMUTH OF GRAVITY\nWAVES IN THE BOUNDARY LAYER USING AN OPTICAL TRIANGLE\n(Li Xing-sheng, Lu Nai-ping, J.E. Gaynor, and J.C. Kaimal)\n93\nABSTRACT\n93\n4.1\nINTRODUCTION\n93\n4.2\nMEASUREMENTS AND COMPUTATIONS\n94\n4.3\nRESULTS\n99\n4.4\nCONCLUSIONS\n105\n4.5\nACKNOWLEDGMENTS\n106\n4.6\nREFERENCES\n106\n5.\nRICHARDSON NUMBER COMPUTATIONS IN THE PLANETARY BOUNDARY\nLAYER (R.J. Zamora)\n109\nABSTRACT\n109\n5.1\nINTRODUCTION\n109\n5.2\nGOVERNING EQUATIONS\n111\n5.2.1 Dry Atmosphere\n111\n5.2.2 Moist Atmosphere\n112\n5.3\nCOMPUTATIONAL SCHEME\n114\n5.3.1 Instrumentation and Calibration\n114\n5.3.2 The Algorithm\n116\n5.4\nERROR ANALYSIS\n117\n5.5\nANALYSIS\n117\n5.6\nCONCLUSIONS\n127\n5.7\nACKNOWLEDGMENTS\n128\n5.8\nREFERENCES\n128\niv","FOREWORD\nThis report, the fourth in the Boulder Atmospheric Observatory\n(BAO) series, brings together in one volume five studies of the noc-\nturnal stable layer. The work was conducted in the Wave Propagation\nLaboratory during 1981 and 1982. Impetus for the first four of these\nstudies came from visiting scientists who found in data collected at\nBAO a unique source of information on mean and turbulent character-\nistics of stable layers. The structure of turbulence in nocturnal\nstable layers, unlike that of daytime convective boundary layers,\ncannot be described in simple universal terms. Stable layers are\nalmost always in a state of evolution. At BAO, the problem is even\nmore challenging because of the rolling terrain, the drainage flow\nfrom the Rocky Mountains, and the frequent presence of gravity waves\nand elevated inversion layers. To the scientist eager to unravel\nthese complexities, the BAO records offer conditions of unending\nvariety. Each paper in this report explores a different aspect of\nthe nocturnal stable layer. It is our hope that this collected\nwork will stimulate comments from other workers in the field and\nlead to new insights into the structure of the stable atmosphere.\nJ.C. Kaimal, Editor\nAtmospheric Studies Program Area\nWave Propagation Laboratory\nV","OBSERVATIONS OF TURBULENCE STRUCTURE IN STABLE LAYERS\n1.\nAT THE BOULDER ATMOSPHERIC OBSERVATORY\nJ.C.R. Hunt\nDepartment of Applied Mathematics\nand Theoretical Physics\nUniversity of Cambridge\nCambridge CB39EW, United Kingdom\nJ.C. Kaimal, J.E. Gaynor, and A. Korrell\nNOAA/ERL/Wave Propagation Laboratory\nBoulder, Colorado 80303\nABSTRACT. Three days of measurements in stable conditions (Monin-\nObukhov length L between 15 and 300 m) at the Boulder Atmospheric\nObservatory (BAO) tower are presented. Winds came off the Rocky\nMountains on 2 days and off the plains on the other day. Vertical\nprofiles of the mean horizontal velocity U, and the Brunt-Väisala\nfrequency N, and of the variances, fluxes, spectra, cospectra,\nquadspectra, and correlations are examined in this study. New\ntheoretical arguments are developed for relating variances to\nscales of temperature 0 and vertical velocity W fluctuations and to\nscales of heat and momentum fluxes in the stable boundary layer.\nSome tentative conclusions drawn from this study are the following:\n(1) In terrain, even gently rolling as it is at BAO\n(slopes <7%), the profiles of dU/dz and d/dz are significantly\ndifferent from those over level ground. For example, in stable\nconditions, Z (dU/dz) and Z (d/dz) often do not increase with\nheight z.\n(2) The presence of coherent wave motions cannot be detected\nfrom U and O profiles. The most sensitive tests are w0 cospectra\nand quadspectra.\n(3) Wave motions may be apparent only in the 0 spectra (mod-\nerate wave conditions) or may be apparent in both W and 0 spectra\n(strong wave conditions). Typical inertial subrange spectra are\nfound in all cases.\n(4) Whether or not waves are present, a theoretical prediction\nfor local parameterization of temperature fluctuation of and heat\nflux wt based on ow' N, and d/dz appears to order the data fairly\nwell where strong uniform density gradients exist. In particular\nwe find the dimensionless temperature fluctuation parameter 50 =\n0.8 + 0.25 and thermal diffusivity parameter F = 0.2 + 0.1.","(5) In the boundary layer, where dU/dz > N, the integral\nscale of the W component L (w) is found to agree (within about 50%)\nwith the theoretical formula,\n[L(w)1-1\n(6) The large values of the cospectra of W and 0 at low\nfrequencies show that wavelike motions can transfer significant\nquantities of heat (or pollutant). This is probably due to\nsmall-scale mixing on a time scale of about 5 N-Superscript(1) accompanying\nthe wave-induced, large-scale motions across the temperature\ngradient.\n1.1\nINTRODUCTION\nThe general aim of this paper is to present some interesting measure-\nments, taken on the Boulder Atmospheric Observatory (BAO) 300-m tower, of the\nmean and fluctuating velocity and temperature in stably stratified atmospheric\nair flows within 300 m of the ground. This depth may or may not exceed the\ndepth of the stable boundary layer. From the analysis of these measurements,\na number of general features of these flows can be discerned; we particularly\nemphasize those features that are of practical importance for estimating flow\nand diffusion in stable conditions, both over level ground and in hilly ter-\nrain, and those features of the flow that are affected by the terrain at BAO.\nThe BAO tower is situated in rolling terrain 30 km from the foothills of the\nRocky Mountains. It is of interest to know what features of the flow are\nsimilar to those measured over the flat terrain at Kansas, Minnesota, and St.\nLouis, Mo. The results presented here are not chosen with the view to study\nthe development of the boundary layer; rather we concentrate on the structure\nof these flows averaged over periods on the order of 20 or 60 min.\nThe measurements bring out some interesting features of the large-scale\nturbulence in and above the stable boundary layer (SBL). In the upper part\nof the SBL, in all kinds of terrain, there appears to be a significant amount\nof energy in internal wave motion mixed in with the turbulence. By use of\nvarious statistical analyses we attempt to identify these various kinds of\nfluctuations, to estimate the relative energies in these fluctuations, and\nto find out their relative contributions to the heat fluxes and to the de-\nstruction of turbulent energy by buoyancy flux and by viscous dissipation.\n2","More than in convectively unstable or neutral flows, the dynamics of a\nstably stratified turbulent flow depend on the rate at which fluid elements\nmix with each other and change their density. If there is no such mixing,\nthe vertical displacements of fluid elements are constrained by the amount of\nenergy in 'the turbulence to lie within a distance on the order of O W /N of\ntheir equilibrium static height, where o is the standard deviation of ver-\nW\ntical velocity and N is the Brunt-Vaisala frequency (Pearson et al., 1983;\nPearson and Britter, 1980). By measuring the vertical velocity distribution\nand spectra, the temperature fluctuations, and the heat flux, and by making\nuse of the Lagrangian model of Csanady (1964) and Pearson et al., , we can\nlearn a good deal about the particle displacements and mixing. We are inter-\nested in knowing to what extent internal wave motion affects these relations.\nFrom these measurements, predictions (by means of the Lagrangian model) for\nvertical diffusion of plumes can be made, which we hope to test later in the\nforthcoming BAO diffusion experiments.\nAnother aspect of the turbulence structure we measure and analyze here\nis the effect of stable stratification on the length scales of velocity and\ntemperature. Using an analysis (unpublished but summarized in Appendix B)\nbased on the turbulence kinetic equation, we examine the main effects of\nstable stratification on the turbulence in the lower part of the boundary.\nSince the length scales affect the dissipation of turbulent energy and the\ndestruction of temperature variance, there are some important dynamical and\nmodeling consequences from our findings. Inter alia, we show that, in the\nlower part of the stable boundary layer, the main effect of an increase in\nstable stratification on the turbulent energy is not so much the loss of\nkinetic energy by the buoyancy flux (the usual explanation) as the larger (by\na factor of 4) increase of viscous dissipation caused by the reduction in\nlength scale associated with the increased mean velocity gradient.\n1.2 TOPOGRAPHIC AND SYNOPTIC CONDITIONS OF THE EXPERIMENTS\nThe measurements were made on the BAO tower about 30 km east of the\nfoothills, and about 60 km from the Continental Divide of the Rocky Mountains,\n3\nwhere the mountains are at an effective height H (=1.5 X 10 m) above the\nground level at BAO. In stable conditions, unlike unstable conditions, the\n3","5\n4\n(a)\n3\nBAO Tower\n2\n1\n0\n-80\n-70\n-60\n-50\n-40\n-30\n-20\n-10\n0\n+10\n+20\nDistance from BAO Tower (km E-W)\n80\n-70\n-60\n-50\n-40\n-30\n-20\n10\n0\n+ 10\n+20\n20\n(b)\n1\nN\n10\nBoulder\nBAO\n0\nErie\n-10\n-20\nFigure 1.1. -- Topography of the BAO site and surroundings: (a) E-W sec-\ntion through the BAO site; (b) map showing the location of BAO with\nrespect to the Continental Divide.\neffects of large hills can affect the flow many kilometers downwind or upwind;\nalso, the flow over the local topography (where slopes are at most of order\n7%) has to be considered more carefully (Kaimal et al., 1982). Details of\nthe topography are given in Figs. 1.1 and 1.2.\nSurface weather maps and rawinsonde plots bracketing the observation\nperiods on 18 April 1978, 22 April 1978, and 15 April 1980, and upper-air\nsoundings at Denver (30 km from BAO) are given in Appendix A.\nOn the second and third days analyzed, 22 April 1978 and 15 April 1980,\nthe general air flow over 300 km around the site, and the local air flow,\nwere from the west to northwest and were therefore coming from the mountains.\nThe tower is at a distance of about 30 to 40 times H from the mountains.\n4","105°00'\nBoulder Creek\n4950\nRte. 52\n5000\n5050\n5100\n105° 00'12\" W\nBAO\n40° 02'54\" N\nTower\nElev. 5174'\nBldg.\nErie\n40°03'\nCounty Rd. 8\n5250\n5150\n5300\n5150\nScale\n(mile)\n0\n0.5\n1.0\nFigure 1.2. -- A conventional contour map of the immediate BAO terrain\n(elevation in feet).\nLaboratory experiments (e.g., Hunt and Snyder, 1980), calculations (e.g.,\nKlemp and Lilly, 1975), , and aircraft measurements show that the effects of\nthe Rockies on the wind extend at least as far as this in stable conditions.\nSo on these days we expect the air flow at BAO to have characteristics quite\ndifferent from that over a plain, whereas in the unstable conditions measured\nby Kaimal et al. (1982) such winds behaved much as they did over the plains\nof Kansas or Minnesota. [It would be interesting to analyze the upper-air\nsoundings given in Appendix A to see if and what kinds of internal gravity\nwaves are expected; perhaps calculations such as those performed by Einaudi\nand Finnigan (1981) might be possible. ]\n5","1.3 MEAN WIND SPEED AND STRATIFICATION DATA\nFigure 1.3 shows the vertical profiles of the mean horizontal velocity U\nand direction A, and the mean potential temperature gradient d/dz calculated\nin the form of the Brunt-Vaisala frequency N = [g(d0/dz)/T_] 1/2 , where\nT\nis\nthe ground-level temperature and g is the acceleration due to gravity. On\nthe first day of measurements, 18 April 1978, the wind came from between\nnorth and northeast both locally and generally. The only reason in this case\nto expect results different from those over a plain is because of the local\nterrain around the tower, which, as we shall show, is sufficiently sloping to\nhave a measurable effect in stable conditions. On 22 April 1978 and 15 April\n1980 the wind came off the mountains. (Numerical values of U and other mean\nand fluctuating velocity and temperature data are presented in Appendix A.)\nA striking feature of the profiles of U is how only in the third case,\nwhere there was a collapsing daytime mixed layer, is the mean velocity shear\ndU/dz effectively confined to about 200 m. In the second case, in the middle\nof the night with a strong wind, there is appreciable shear all the way up\nto 300 m. (The value of L, the Monin-Obukhov length, is about the same at\n1700 MST on the first day as between 0006 and 0436 MST on the second day). In\nthe first case considered here, 18 April 1978, the shear is significant at the\ntop of the tower. Only the profiles on 15 April 1980 are similar to the\nMinnesota profiles measured by Caughey et al. (1979), where dU/dz was usually\nnegligible above about 100 m in strongly stable conditions.\nThe upper-air soundings at 1700 on 18 April and at 0500 on 19 April\n1978 were too different to infer the conditions at 2100 to 2300. On 22 April\n1978 there was also little change in the wind's direction or strength, but the\nvalue of N varied with height z: N = 0.007 rad/s for Z < 0.8 km; 0.001 rad/s\nfor 0.8 km < Z < 1.5 km; and 0.037 rad/s for 1.5 km < Z < 3 km. On 15 April\n1980 the soundings show that the direction and strength of the wind and the\nvalue of N did not change significantly up to about 4 km above the ground.\nTo what extent are the tower profiles affected by the local terrain when\nthe winds are from the west, given a local upslope of about 2% to the west of\nthe tower? The effect of a slope on U(z) depends on the shear in the approach\n6","0.03\n(b)\n(d)\nFigure 1.3. --Horizontal wind speed (U), , wind direction (A), , and buoyancy frequency (N) profiles based on\n0.05\no\n0.02\nN (rad/s)\nN (rad/s)\n0.01\nDD\nDOA\n0\n0\n276 h, h being the thickness of the stable surface layer (e.g.,\nSmith, 1980).\nThe calculations of (N/U) 2 and are plotted in Fig. 1.4b\nand show that, for the 3 days considered here, the mean velocity curvature\nlargely determines l for Z V 2 300 m. Consequently in rolling or hilly ter-\nrain, when calculations are to be made of air flow over a specific hill or\nover a limited region of hills, the effects of the incident shear profile may\nwell be as important as the dynamical effects of the stable stratification.\nThe curvature may extend farther up into the boundary layer in stable con- -\nditions than in neutral conditions.\n8","1.0\n(a)\nMST\nDate\nL\n0.8\n2120\n18 Apr 78\n148\n2220 18 Apr 78 166\n0226 22 Apr 78 189\n0.6\n1.15 + 3.2 z/L\n0.4\n8\n0.2\n= 16,36,75 m\n0\n0\n1.0\n2.0\n3.0\n4.0\nkz (dU/dz)/u.\n1000\n(b)\nA\n100\n10\ni2\nd2U\nA\n15 Apr 80\n18 Apr 78\n15 Apr 80\n18 Apr 78\n1\n10-7\n10-6\n10-5\n10-4\nd2U\n/u\n(N/U)²\n(m-2)\nor\nFigure 1.4. -- Analysis of mean velocity and density profiles: (a) com-\nparison of the dimensionless velocity gradient with the Monin-Obukhov\nsimilarity law for 18 April 1978 (k = 0.4 is assumed) ; (b) comparison\nof the profile curvature term (-d2U/dz²) 2 /U term with (N/U) 2\n9","1.4 ANALYSIS OF VELOCITY AND TEMPERATURE FLUCTUATIONS AND HEAT FLUX\nIN TERMS OF WAVES AND/OR TURBULENCE\nA cursory look at Figs. 1.5-1.8 shows that the fluctuations in vertical\nvelocity and temperature exhibit waves as well as turbulence, waves being\ndefined in the sense of regular oscillations. (Since these were tower mea-\nsurements, we could not make a more decisive test of waves by distinguishing\nwhether these oscillations were or were not propagating relative to the mean\nflow.) On the days chosen here for analysis, varying amounts of wave motion\nrelative to the turbulence are observed; the wave conditions are weak, moder-\nate, or strong on these occasions, and are classified accordingly.\nIn Fig. 1.5a, vertical velocity W and temperature 0 fluctuations at 10\nand 150 m are plotted as functions of time for 22 April 1978. Note the strong\nstrong wavelike behavior in W at 150 m. By contrast, in Fig. 1.5b the traces\nfor 15 April 1980 show weak wavelike behavior. The waves are certainly or-\nganized, but their amplitude is weak.\nIn Figs. 1.6a,b the spectra of W and 0--ns (n) and ns 0 (n)--and the co-\nW\nspectra and quadspectra of w0--nC we (n) and nQ we (n) --are plotted for the\nturbulence on 18 April 1978. Note that there appear to be concentrations\nof energy at discrete frequencies in all these spectra, a fact that can be\ntested by looking for discrete frequencies in autocorrelation functions.\nThere is a much greater contribution by these wavelike fluctuations to the\ntemperature than to the velocity spectra. The object of plotting both the\ncospectra and the quadspectra is to show how much heat flux is actually\ntransferred by the low-frequency, wavelike motions. For a nearly pure sine\nwave with no mixing in a temperature gradient, the cospectra would be very\nsmall compared with the quadspectra. In Figs. 1.6a,b we observe that, com-\npared with the velocity spectra, a significant portion of the heat flux is\ntransferred by the low-frequency, wavelike motions, because Cwo (n) is of the\nsame order as Owo(n). It is also observed that, at the frequencies where\nthe 02 \"energy\" is large* and S 0 (n) is near its maximum, two (n) has the form\nof a double \"spike\" as it rapidly swings from one sign to the other over a\nThe drift in O produces about 10% of the variance.\n*\n10","Figure 1.5. -- Typical time series for strong and weak wave conditions: (a) W and 0 fluctuations at 10 and\n20\n20\n150 m for 22 April 1978; (b) W and 0 fluctuations at 10 and 150 m for 15 April 1980.\n15\n15\nTime (min)\nTime (min)\n10\n10\n(b) Weak waves, 1800-1820 MST, 15 April 1980\n(a) Strong waves, 0206-0226 MST, 22 April 1978\n5\n5\n0\n0\n810\n150\n1°C\n010\n150\n1°C\nT\nT\n1\n1\n20\n20\n15\n15\nTime (min)\nTime (min)\n10\n10\n5\n5\n0\n0\nW150\n1 m/s\n10\n1 m/s\nW10\n150\nT\nT\nW","wave conditions. All spectra, cospectra, and quadspectra are from 150 m except for the 10-m W spectrum in\n1980)\n10-1\n-2\n10-superscript(3)\n10-4\n10\n10-1\n10-2\n10-3\n10-4\n101\n101\nFigure 1.6 6. -- Typical spectra for (a) and (b) moderate (18 April 1978) and (c) and (d) weak (15 April\n(d)\n(c)\nW150\n10°\n150\n10°\nnQ(n)\n0\nnC(n)\n1700-1800 MST\n1700-1800 MST\n15 Apr 1980\n15 Apr 1980\n10-1\n10-1\nn (Hz)\nn (Hz)\n10-2\n10-2\n10-3\n10-3\n10-4\n10-4\n10°\n10-1\n10-2\n10-3\n10-1\n10-2\n3\n10-4\n10\n(a) . The modulus of the quadspectrum Q(n) is plotted.\n10-1\n2\n10-4\n10\n10-3\n1\n10-2\n10-3\n10-4\n10\n101\n101\nW10\n(a)\n(b)\n10°\n150\nnQ(n)\n10°\n150\nnC(n)\nW\n2120-2220 MST\n2120-2220 MST\n18 Apr 1978\n18 Apr 1978\n10-1\n10-1\nn (Hz)\nn (Hz)\n10-2\n10-2\n10-3\n10-3\n10-4\n10-4\n10°\n2\n10-3\n101\n10\n10-1\n-2\n-3\n10\n10\n10-4","Figure 1.7. -- Typical spectra, cospectra, and autocorrelations for the strong wave conditions of 22 April\n1978. Spectra of W for four heights are shown in (a), , but only the 150-m cospectra and quadspectra are\n(d)\n(c)\n160\n150 m\n0206-0306 MST\n100 m\n150 m\n140\n10 m\nshown in (b). . Autocorrelations of W on small and large time scales are shown in (c) and (d).\n22 Apr 1978\n120\n880\n100\nT (s)\n704\n80\nT (s)\n628\n60\nLarge Scale\nSmall Scale\n252\n40\n176\n20\n0\n1.0\n0.5\n0\n-0.5\n-1.0\n0.2\n0\n1.0\n0.8\n0.6\n0.4\n101\n10\n(b)\n(a)\npos. nC(n)\n0206-0306 MST\n150 m\nZ = 150 m\nnSg(n)\n10 m\n22 m\n50 m\n10°\n22 Apr 1978\n10°\nnC(n)\nnQ(n)\n10-1\n10-1\nn (Hz)\nn (Hz)\n10-2\n10-2\n10-3\n10-3\n10-4\n10-4\n10-2\n10-3\n10-4\n10-2\n10-3\n10-1\n-1\n10°\n10","relation spectra of w for the most energetic (n1) waves and the next energetic (n,) waves compared with cross\n150-m tower levels respectively. (c) Profiles of energy in most energetic (n1) wave motions for two periods,\nmotions\n(d) Cor-\nFigure 1.8. -- -Cross spectra and correlations between different heights in strong wave conditions. (a) W4W5\ncospectra and (b) W 4 5 quadspectra for 0006-0106 MST, 22 April 1978. Subscripts 4 and 5 refer to 100- and\n0006-0106 MST Hz) and 0206-0306 MST (n 1 Hz), and the next energetic (n2) wave\n202\nW\n(c)\n(d)\ncorrelation coefficients of observed W and 0 at different heights with observations at 100 m.\ndenotedly\n300\nA\nRWzW4\nR8z04\n(o) for n = n1\n(.) for n n2\n300\n0\nW\nfor n = n1\n0006-0106 () for n = n2\nheights,\nW\n200\n0006-0106 (o)\n0206-0306 (A)\n200\nw\nzm(\nZ (m)\ndifferent\n8\n100\n100\n22 Apr 1978\nat\n50\n02/2018\n0\nnormalized\n0\n1.6\n1.2\n0.8\n0.4\n0\n1.0\n0.8\n0.6\n0.4\n0.2\n0\nwith\n101\n10\n(a)\nn1: most energetic mode\nn2: next energetic mode\n(b)\ncompared\n10°\n10°\n10-1\n10-1\nn2\n0-2\nn (Hz)\nn (Hz)\n10\n10-2\n10\n0006-0106 MST\nn1\n22 April 1978\n10-3\n10-3\n(n2\n10-4\n10\n0.10\n0.05\n0\n-0.05\n-0.10\n0.10\n0.05\n-0.10\n0\n-0.05\nMST\n0006-0106\nfor","narrow frequency range. This \"spiky\" form is indicative of large phase shifts\nbetween W and 0 and is a nonlinear phenomenon.\nThe W spectrum in Fig. 1.6c, for 15 April 1980, indicates that on this\noccasion, when L, N, and U have about the same values as on 18 April 1978,\nthere is no discernible wave motion. But the 0 spectrum in Fig. 1.6c and\nthe wO cospectrum and quadspectrum in Fig. 1.6d indicate some weak concen-\ntration of energy into bands of frequencies. Note that Owo (n) does not have\na spiky form at frequencies where Cwo (n) is a maximum, another indication\nof the weakness of wave motion. For frequencies where the wave motions are\nweak, it is possible to define a length scale L X = U/2TT even for the tem-\nperature fluctuations. Here n is the value of frequency n where S(n) or\nm\nc (n) is a maximum. The turbulent flow with very weak waves is comparable with\nthe Kansas air flow analyzed by Kaimal (1973). The graphs are not shown\nhere. However, it was found that the ratio (6)/IL(w) is in the range 6 to 8\nat 22 and 50 m, but this ratio is only about 3 at 150 m (as may be inferred\nfrom Fig. 1.6c); L(w0)/I((w) is about 2 at 150 m.\nFigure 1.7 shows typical spectra, cospectra, and autocorrelations\nRW(T) for strong wave conditions. Figure 1. .7a shows how the strength of\nW\nthe wave motion centered at a few frequencies increases with height. (These\nfrequencies can be picked out from the autocorrelations in Figs. 1.7c,d.)\nAt 10 m the W spectrum has much the same form as the moderate and weak wave\nspectra of Figs. 1.6a,c. At greater heights, note the concentration of the\nheat flux at a few frequencies (Fig. 1.7b) and the spiky shape of the Owg(n)\nquadspectrum. The contribution of these wavelike motions to we is even\ngreater than in the moderate wave conditions shown in Fig. 1.6b.\nFigures 1.8a,b show how the energy in the cospectrum and quadspectrum of\nthe vertical velocity measured at 100 and 150 m is also concentrated around a\nfew frequencies, the first and second most energetic waves (denoted by no1 and\n*\nn2) being at about 7 10-3 and 5 10- Hz respectively. In Fig. 1.8c,\nwe\nexplore the variation of the energy in these two modes with height. We ob-\nserve a noticeable maximum in (n=n1) at Z = 200 m, but not in ns (n=n2).\n* The first of these frequencies is close to the value of N/2TT in the deep\nstratified layer between 1.5 and 3 km.\n15","The latter, monotonic increase is somewhat similar to the energy in the wave\nstudied by Einaudi and Finnigan (1981) The former's distribution is distinctly\ndissimilar. The correlation of the vertical turbulence and temperature, and\nalso the correlation spectra of vertical velocity at the first two most\nenergetic frequencies, are plotted in Fig. 1.8d. Note how the correlation\ncoefficient, Rw2W4 (subscript 4 denotes 100-m level), falls off with z-100 -\n,\nthe height above or below 100 m, i.e., at a rate similar to the falloff of\nCw2w4(n=n2). But the correlation in Cw2w4 (n=n1) is much higher and is com-\nparable with that of R O 0 , showing again how the temperature fluctuations\n4\nZ\nare driven by wave motions to a greater extent than the vertical velocity\nfluctuations are.\nEstimating the relative amounts of energy in turbulence and waves cannot\nbe unambiguous since waves may well have random phases and be rotational, two\nfeatures that are also important in low-frequency turbulence. If waves are\nmotions with energy centered over a narrow frequency range, they can be identi-\nfied if the cospectra and quadspectra are spiky. To estimate the energy of\nturbulence in the presence of waves, we assume that the contribution of tur-\nbulence to the spectrum is of the same form as that in the absence of waves\n(Figs. 1.9a,b) . This was the assumption of Caughey et al. (1979) in their\nanalysis of the Minnesota data. Thence we can say that on the basis of\nFigs. 1.6a,c on 18 April 1978 and 15 April 1980 for Z < 300 m, and of Fig. 1.7a\non 22 April 1978 for Z < 10 m, the contribution of waves to the variance 22 2\nwas weak. But waves provided about 50% of 02/2018 at 150 m on the strong-wave\noccasion of 22 April 1978.\nA further distinction between different kinds of wave-turbulence com-\nbinations appears on examining the temperature and heat flux spectra. As\nis shown schematically in Figs. 1.9a-d on different stable days, W spectra\nmay be largely similar to those in turbulent shear flows, but 0 spectra and we\ncospectra may be quite different: on one occasion dominated by waves and on\nthe other not. In the first of these two situations, we shall refer to the\nwaves as \"strong\" (22 April 1978) or \"moderate\" (18 April 1978); on the\nsecond, we shall refer to them as \"weak\" (15 April 1980).\n16","(c)\n(a)\nN/ow\n3\n2E/O\n-\nStrong\n~ N/ow (z>h)\nWeak\nWaves\nn-2/3\nModerate\nn+1\nWeak\nTurbulence\nWaves\nk1 (=2nn/U)\nk1 (=2nn/U)\n(d)\n&\n(b)\nN/ow\n1/UT sample\nN/ow\nStrong\nN/U\nModerate\nWeak\nTurbulence\nStrong Waves\nTurbulence\nWaves\nk1 (=2nn/U)\nk1 (= 2wn/U)\nFigure 1.9. --Schematic form of spectra in stable conditions: (a) vertical\nvelocity in the presence of weak waves (15 April 1980) ; (b) vertical\nvelocity with strong waves (22 April 1978); (c) heat flux spectra; (d)\ntemperature spectra.\nTo understand the dynamical processes in a stratified turbulent flow, one\nmust estimate the rate of energy dissipation E per unit mass, and relate it\nto the rate of buoyant production per unit mass, B = g(w0)/T and to the\nratio of the macroscopic velocity and integral scales, 0.4 3/L((w) (see\nAppendix C) . It is also of interest to estimate which part of the energy\nspectrum is contributing most to B. Only if the major contribution to B\noccurs at frequencies less than n can E be estimated from the inertial sub-\nrange of the spectrum. (Otherwise the turbulent energy in the subrange would\nbe transforming itself into potential energy.)\nOur analysis of the measurements of C we (n) shows that, even in the\nweak-wave case where L (w0) = 2L (w) , the turbulence with scales larger than\nL(w)\nproduces almost all the buoyancy flux. Consequently E should be given\nby the value of S (n) in the inertial subrange according to the result\n(n) 12 (4/3) 0.5 W223 (Kaimal et al., ,\n1972).\nS\nW\n17","Using our measurements of 18 April 1978 at 2120 MST (moderate waves),\n,\nwe estimate E = 0.0046 m²/s3 at 50 m and 0.4 For\nthis\ncase we find B/E = 1/5. On the strong-wave occasion of 22 April 1978 at\n150 m, = 0.0037 and B/E = 1/2. It is interesting to compare\n(E/0.4 03)-1 with two length scales: the scale of the peak of the nsw(n)\nspectrum, 2n m /U, and the scale /. In this case, inspection of the\nspectra in Fig. 1.7 shows that the turbulent component of the W variance is\nabout half the total variance. Thence the scale [e/ (0.4 w(t) 11 59 m,\nwhereas (2Tn_/U)-1 = 40 m and (N/OW)-1 = 100 m. They are all of the same\norder (as they are in non-wavy conditions).\nSince in strongly stable conditions the integral scale of the vertical\ncomponent of turbulence is small compared with the distance above the surface\n(for Z 2 25 m), , it is plausible that the turbulence scale is largely deter-\nmined by local conditions, namely the values of dU/dz, ow' and N, which\ndetermine the two natural inverse length scales (dU/dz)/o. and (see\nFig. 1.9). (Theoretical dependence dU/dz,\narguments for the of\nand N are summarized in Appendix D.) An integral scale is most easily de-\nrived by using the spectrum of W to infer that = U/(2Tn_), where is\nthe value of n where nsw(n) W is a maximum. (This definition is equal to the\ntrue integral scale if the autocorrelation is an exponential function.) Evi-\ndently this definition is not appropriate for turbulence with strong waves,\nwhere the spectrum (as in Fig. 1.7a) does not have a maximum.\nThe values of n 'm' even in the spectra chosen here, are uncertain to\nwithin a factor of about +25%. First, Figs. 1.10a-c show how L(w) increases\nto a large but finite value where the shear vanishes (dU/dz 0). This finite\nvalue approximates to /N. Second, the results show that in the lower part\nof the shear layer the vertical profile of [L(w),1-1 more closely approximates\nthe profile of (dU/dz)/o W than the profile of N/o. To within 50%, the BAO\ndata agrees with the formula for level ground [(D.14) of Appendix D] that\n12 0.7 (dU/dz)/o. + 1/z (when dU/dz > N).\nNote in Fig. 1.10c how the values of L(w) measured in the Kansas ex-\nperiment at the same Monin-Obukhov length L are about half the values of\n18","and 50 m respectively for the 1-h periods in (a) and (b)\nvertical velocity [L(w) 1-1 (= 6/X( w), , where (w) = U/n m\n(c). The Kansas curve is based on a 1-h average (2120-\n(b) on 15 April 1980 and moderate waves (c) on 18 April\nscales, (dU/dz)/o and N/O (imposed by shear and buoy-\nas shown in Appendix C. Monin-Obukhov length L is 100\nand 148 and 166 m for two successive 20-min periods in\nFigure 1. .10. Comparison of the reciprocals of length\nin conditions with weak waves for two periods (a) and\n1978. Comparison is also made in (c) with the stable\nz/L)\nancy), with the reciprocal of the integral scale of\ndata from Kansas where = 2.9 Z -1 (1 + 4\n(b)\n0.3\n1800-1900 MST\n15 Apr 1980\n[Lx and (m-1)\n0.2\nW\nX\n2220 CST): ; L = 142 m.\n0.1\nW\n0\n200\n100\n0\n300\n(c)\n(a)\n0.3\n0.3\n2120-2140 MST\n2220-2240 MST\n2120-2220 MST\n2120-2220 MST\nOw\n1700-1800 MST\n/\nN/ow\nKansas\n(w)\ndU\n18 Apr 1978\n15 Apr 1980\ndz\nan] (m-1)\nas] (m-1)\n0.2\n0.2\n000\n0.1\n0.1\no\n0\n0\n0\n200\n100\n300\n100\n0\n300\n200","L(w)\nin these experiments, where wave motions (even though weak) may increase\nL(w)\nDespite the scatter in our results, this difference is probably real.\nThe spectra of 0 and cospectra of we in Figs. 1.6 and 1.7 show how the\ntemperature fluctuations and the heat flux are controlled by motions with\nlarger scales than that of the vertical velocity. The results of Fig. 1.10\nindicate that such scales are likely to be affected by the buoyancy length\nscale o W /N. In the moderate- and strong-wave cases, these large motions are\nmostly waves that are also likely to be controlled by the local value of N if\nthe vertical variation of N is small. If N(z) varies sharply, then other\nmodes can be generated as in the case studied by Einaudi and Finnigan (1981).\nThe upper-air soundings in Appendix A are rather different to their situation.\nThese are the immediate reasons for suspecting that ow' N, and de/dz are the\nmain parameters determining o O and the heat flux. For the idealized situation\nof homogeneous turbulence in a strongly stable stratification, the Lagrangian\nmodel of Pearson et al. (1982) suggests that\n(1.1)\n,\nwhere 50 is a temperature fluctuation parameter of the order of unity. In fact\nfor a wide class of turbulence, 50 = 1/22. The model also predicts that the\ncoefficient of thermal diffusivity is given by\n(1.2)\n,\nwhere F is a thermal diffusivity parameter much smaller than unity and roughly\nindependent of The relationship in (1.2) can also be written to\nshow that the effective length scale of turbulence governing heat flux (i.e.,\nkg/ow) is equal to F ow/N. It is important to underline that this theory\nmakes no distinction between wavy and turbulent motions at low frequency, and\ntherefore suggests that in measuring 50 and F we should not expect any great\nsensitivity to waviness.\nConsequently in Fig. 1.11 we have plotted (d0/dz) as a function of\now/N for all three of the days and conditions considered here from 16 m (where\nd0/dz is evaluated) to 275 m (00 is interpolated). We have only included points\nwhere d0/dz can be calculated. In some cases, do/dz is indeterminate because\ninstruments were not working or because its value is too small (0.0005°C/m).\n20","300\n0620 MST 18 Sept 78 (Same period analyzed by\nFinnigan and Einaudi, 1981)\no 0226 MST 22 Apr 80 (L = 189 m)\n1700 MST 15 Apr 80 (L = 123 m)\n1820 MST 15 Apr 80 (L = 36 m)\n4 2120 MST 18 Apr 80 (L = 148 m)\n200\nD 2300 MST 18 Apr 80 (L=33m)\n100\nAs\nA\nD\nD\n0\n100\n110\n120\n130\n10\n20\n30\n40\n50\n60\n70\n80\n90\n0\now/N (m)\nFigure 1. 11. -- Plot of the relationship in eq. (1.1) showing 50 12 0.8, ir-\nrespective of variations in Z or L or the existence of waves.\nIn the latter case, temperature fluctuations persist, but there can no longer\nbe a local connection between of and do/dz. (There are only about three ex-\ncluded points, and all are at 225 m.)\nThe agreement with (1.1) is reasonably good, the observed constant 50\nbeing about 0.8 (see Fig. 1.11), compared with 0.7 in the theoretical model.\nNote that the correlation of the Kansas surface layer data for strong stabil-\nity (z < 25 m) gives 50 between 0.5 and 1.0 (see Appendix B). . There is no\napparent trend in the value of 50 with Z or with the existence or absence of\nwave motions.\nFigure 1.12a shows how the effective length scale of the heat flux (Kg/ow)\nvaries with height in the stable boundary layer. Note the marked variation of\nthe reciprocal heat flux W/KO even over an hour. Figure 1.12b shows that ow/kg\nis an approximately linear function of N/OW; it does not correlate well with\nthe velocity gradient length scale (dU/dz)/o_.. Finally in Fig. 1.13, F (i.e.,\nKg/Ow = OW/N) is plotted as a function of the effectively largest scale of\nmotion in these stratified flows (o //N) and as a function of the inverse\nsquare root of the Richardson number Ri-1/2 = (dU/dz)/N. There is no obvious\ntrend in this data. [According to surface boundary similarity arguments F\n21","300\n(a)\n0226 MST, 22 Apr 1978, L= 189 m (Strong Waves)\n1700 MST, 15 Apr 1980, L= 123 m\n1820 MST, 15 Apr 1980, L=36 m (Weak Waves)\n200\n100\n0\n(b)\n0.4\n0.16\n0.3\nA\n0.12\n0.2\n0.08\nOAD N/ow\n0.1\n0.04\ndU\n0\n0\n0.2\n0.4\n0.6\n0.8\n1.0\n1.2\now/Kg (m-1)\nFigure 1. 12.\n-Reciprocal heat flux scale w/kg W plotted as a function of\n(a) height and (b) reciprocal length scales imposed by buoyancy N/O and\nby shear (dU/dz) ow.\nW\nshould be a function of z/L or z/(o./N), when z/L >> 1/5.] The fact that the\nW\naverage value of F is about 0.17, which is half the value in the Kansas ex-\nperiment, might be explained by the relatively larger values of o associated\nW\nwith the wave motions present here. However there is no direct correlation\nbetween wave motion and any of the results in Fig. 1.13.\nRecent measurements by Nieuwstadt (1982) at the Cabauw tower in the\nNetherlands also showed that in stable conditions above about 100 m the tem-\nperature fluctuations and heat flux could be scaled on the local turbulence\nand temperature gradient. His results yield so = 1.3 and F = 0.25. Unlike us,\nhe found that F systematically decreased as Z decreased, reaching a value of\nabout 0.1 at z/L = 0.1.\n22","0226 MST 22 Apr 1978 (L=189m) =\n(Strong-Wave Case)\n1700 MST 15 Apr 1978 (L=123m)\n1820 MST 15 Apr 1978 (L=36m)\n2120 MST 18 Apr 1978 (L=148 m)\n2300 MST 18 Apr 1978 (L=33 m)\n(a)\nKansas Surface Layer Value\n0.30\n0.25\nAD\n0.20\nD\nF 0.17\n0.15\n0.10\n0.05\n150\n200\n50\n100\n0\now / N (m-1)\n0.3\n(b)\n0.2\n4\no°\n0.1\nR=1/4\n0\n5\n4\n2\n3\n0\n1\n(dU/dz) / N\nFigure 1. 13. -- Thermal diffusivity parameter F = Kg/(02/N) as a function\nof (a) ow/N and (b) (dU/dz)/N R1-1/2. ).\nTENTATIVE CONCLUSIONS\n1.5\n(1) The presence of coherent wave motions in stable atmospheric flows\ncannot be detected from the mean temperature and velocity profiles, nor even\non some occasions from spectra of the vertical velocity.\n(2) Vertical wave motions are most likely to be detected by temperature\nspectra and/or temperature-vertical velocity cospectra. Cospectra, quad-\nspectra, and autocorrelations help identify the frequencies where wave energy\n23","is particularly concentrated. Waves of higher frequency appear to be cor-\nrelated over smaller vertical distances. Temperature fluctuations are better\ncorrelated over larger vertical distances than are the velocity fluctuations.\n(3) It may be useful to consider turbulence in stably stratified condi- -\ntions as belonging to one of three categories: weak, moderate, or strong wave\nconditions, depending on whether significant wave effects are present in\nneither velocity nor temperature spectra, only temperature (or we cospectra),\nor both W and 0 spectra, respectively.\n(4) Wave motions and large-scale turbulence are both agents for the\ntransport of heat and the production of temperature fluctuations. In the\npresence of a stable density gradient, where the buoyancy frequency N is\napproximately uniform with height, it appears that both processes are asso-\nciated with vertical motions on a scale approximately equal to /N, over a\ntime scale of order N-Superscript(1). . Consequently dimensionless parameters can be defined\nto estimate (or correlate) temperature fluctuations °0 or thermal diffusivity\nK0:\n,\nwhere typically\nso = 0.8 + 0.25\nF = 0.2 + 0.1 .\n[Note that these general ideas are not appropriate when vertical oscillations\nare driven by elevated inversions, i.e., strong variations in N. Indeed, as\nFinnigan (CSIRO, Canberra, Australia; personal communication) has pointed out,\nwave-induced heat flux may then be in the opposite direction to the high fre-\nquency wave and turbulent flux. ]\n(5) The integral scale for the vertical turbulence velocity is shown to\nbe approximately given by\n24","in the lower part of the boundary layer, at Z n 25 m. Below this height, the\ndirect effect of the ground on the eddies has to be considered; other data and\ntheoretical arguments given in Appendix D suggest, in stable neutral and\nunstable conditions,\nIn the upper part of the boundary layer and above it (to within a factor of\nabout 2)\n(when N >> dU/dz) .\n(6) The integral scales of temperature L(6) and we fluctuations L (w0) are\ngreater than those of L(w) (They can only be defined in weak wave conditions.)\nTypically when N 100 in such conditions n\nNear the surface, Z L(w) m decreases where so at 22 m or 50\nm,\n(7) Whether waves are present or not, inertial subrange spectra are\nobserved for W and O. Measurements of E from this range are consistent to\nwithin a factor of 2 with the usual estimate based on the turbulent component\nof the total variance (02) and with local integral scales estimated on the\nbasis of o /N and the wavenumber at the peak in the spectra.\nW\n(8) From a practical point of view, the data show unmistakably how heat\nis transferred vertically by wave or wavelike motions as well as by low-\nfrequency turbulent motions. Consequently contaminants can also be trans-\nferred by such processes. Therefore it is important that, in the presentation\nand recording of measurements of atmospheric turbulence, the wavelike motions\nshould be included. [By excluding spectral contributions from frequencies\nbelow 0.001 Hz (Caughey et al., , 1979) in the analysis of the Minnesota data,\nthe data appeared to be very consistent, presumably because it was just a\nturbulence record. But the analysis may be misleading if it is to be used to\nestimate heat or contaminant fluxes. ]\n(9) The second practical point about the data is that they provide\nevidence for the theoretical suggestion of Pearson et al. (1982) that in a\n25","stably stratified flow, diffusion heat or pollutant transfer takes place more\nby slow mixing between fluid elements, which are constrained in their vertical\nmotions to a distance of about o /N, rather than by large-scale advection or\nw\ntransport by fluid elements. If F = 0.2, it means that the time scale for a\nfluid element to change its temperature is about 5 N-Superscript(1). Therefore vertical\ndiffusion of any pollution released from an elevated source into the atmo-\nsphere is limited to o \"/N W for some distance downwind of a source until mixing\ncan occur. This prediction for the vertical extent of a plume in strongly\nstable flows can also be estimated from the variance of temperature fluctua-\n2\ntions of 0 using eq. (1.1).\n1.6 ACKNOWLEDGMENTS\nJCRH is grateful for stimulating conversations with J.C. Wyngaard and\nvery helpful comments by S.P.S. Arya, H.J. Pearson, W. Rodi, J.J. Finnigan,\nand P.J. Mason.\n1.7 REFERENCES\nBritter, R.E., , J.C.R. Hunt, L. Marsh, and W.H. Snyder, 1983: Turbulent\ndiffusion in a stratified fluid. J. Fluid Mech. (in press).\nBrost, R.A., and J.C. Wyngaard, 1978: A model study of the stably stratified\nplanetary boundary layer. J. Atmos. Sci., 35, 1427-1440.\nBusch, N.E., , 1973: The surface boundary layer. Bound.-Layer Meteorol.,\n4, 213-240.\nCaughey, S.J., J.C Wyngaard, and J.C. Kaimal, 1979: Turbulence in the\nevolving stable boundary layer. J. Atmos. Sci., 36, 1041-1052.\nCsanady, G.T., 1964: Turbulent diffusion in a stratified flow. J. Atmos.\nSci., 7, 439-447.\n26","Einaudi, F., , and J.J. Finnigan, 1981: The interaction between an internal\ngravity wave and the planetary boundary layer. Part I: The linear\nanalysis. Quart. J. Roy. Meteorol. Soc., 107, 793-806.\nFinnigan, J.J., and F. Einaudi, 1981: The interaction between an internal\ngravity wave and the planetary boundary layer. Part II: Effect of the\nwave on the turbulence structure. Quart. J. Roy. Meteorol. Soc., 107,\n807-832.\nHunt, J.C.R., 1981: Turbulent stratified flow over hills. Proceedings,\nColloquium on \"Construire avec le Vent,\" Centre Sci. Tech. du Bat,\nNantes, France, paper I-1.\nHunt, J.C.R., 1982: Turbulent diffusion in the stably stratified boundary\nlayer. Proceedings, Short Course on Boundary Layer Meteorology in Air\nPollution Modeling, Royal Netherlands Meteorological Institute, De Bilt,\nF.T.M. Nieuwstadt and H. Van Dop, eds., Reidel, Dordrecht, Netherlands,\n231-274.\nHunt, J.C.R., 1983: Turbulence structure in convective and shear-free\nboundary layers. J. Fluid Mech. (submitted).\nHunt, J.C.R., and J.M.R. Graham, 1978: Free-stream turbulence near plane\nboundaries. J. Fluid Mech., 83, 209-235.\nHunt, J.C.R., and W.H. Snyder, 1980: Experiments on stably and neutrally\nstratified flow over a model three-dimensional hill. J. Fluid Mech.,\n96, 671-704.\nHunt, J.C.R., and A.H. Weber, 1979: A Lagrangian statistical analysis of\ndiffusion from a ground level source in a turbulent boundary layer.\nQuart. J. Roy. Meteorol. Soc., 105, 423-443.\nKaimal, J.C., 1973: Turbulence spectra, length scales and structure param-\neters in the stable surface layer. Bound.-Layer Meteorol., 4, 289-309.\n27","Kaimal, J.C., J.C. Wyngaard, y. Izumi, and O.R. Coté, 1972: Spectral charac-\nteristics of surface layer turbulence. Quart. J. Roy. Meteorol. Soc.,\n98, 563-589.\nKaimal, J.C., R.A. Eversole, D.H. Lenschow, B.B. Stankov, P.H. Kahn, and\nJ.A. Businger, 1982: Spectral characteristics of the convective\nboundary layer over uneven terrain. J. Atmos. Sci., 39, 1098-1114.\nKlemp, J.B., and D.K. Lilly, 1975: The dynamics of wave-induced downslope\nwinds. J. Atmos. Sci., 32, 320-329.\nMonin, A.S., and A.M. Yaglom, 1971: Statistical Fluid Mechanics, Vol. 1.\nMIT Press, Cambridge, Mass., 769 pp.\nNieuwstadt, F.T.M., 1982: Observations on the turbulent structure of the\nstable boundary layer. Quart. J. Roy. Meteorol. Soc. (submitted).\nOkamoto, M. , and E.K. Webb, 1970: The temperature fluctuations in stable\nstratification. Quart. J. Roy. Meteorol. Soc., 96, 591-600.\nPearson, H.J., and R.E. Britter, 1980: A statistical model for vertical\nturbulent diffusion in stably stratified flows. Proceedings, 2nd\nInternational Symposium on Stratified Flows, Trondheim, 24-27 June,\nTapir Publishers, Trondheim, Norway, 269-279.\nPearson, H.J., J.S. Puttock, and J.C.R. Hunt, 1983: A statistical model\nof fluid element motions and vertical diffusion in homogeneous stratified\nfluid. J. Fluid Mech. (in press).\nPrandtl, L., 1952: The Essentials of Fluid Dynamics. Hofner, New York,\n452 pp.\nScorer, R.S., 1949: Theory of waves in the lee of mountains. Quart. J.\nRoy. Meteorol. Soc., 75, 41-56.\nSherman, F.S., , J. Imberger, and G.M. Corcos, 1978: Turbulence and mixing in\nstably stratified waters. Ann. Rev. Fluid Mech., 10, 267-288.\n28","Smith, R.B., 1980: Linear theory of stratified hydrostatic flow past an\nisolated mountain. Tellus, 32, 348-364.\nTownsend, A.A., 1958: Turbulent flow in a stably stratified atmosphere.\nJ. Fluid Mech., 3, 361-372.\nTownsend, A.A., 1976: Structure of Turbulent Shear Flow. Cambridge\nUniversity Press, Cambridge, England, 429 pp.\nTurner, J.S., , 1973: Buoyancy Effects in Fluids. Cambridge University Press,\nCambridge, England, 367 pp.\nWyngaard, J.C., , O.R. Coté, and Y. Izumi, 1971: Local free convection,\nsimilarity, and the budgets of shear stress and heat flux. J.\nAtmos.\nSci., , 28, 1171-1182.\n29","APPENDIX A: BACKGROUND DATA FOR OBSERVATION PERIODS\nPresented here are background data on synoptic and boundary layer scales\nfor the three nights discussed in this paper. The daily weather maps for\n12 Z (0500 MST), published by NOAA's Environmental Data and Information Ser-\nvice, are reproduced in Figs. A.1-A.3. Two maps bracketing the selected\nnights are shown. The rawinsonde plots for Denver (30 km from the BAO) cover-\ning the same observation periods are presented in Figs. A.4-A.6. The upper-\nair soundings at 1700 MST the evening before and 0500 MST the morning after\nare shown to highlight changes occurring during the night. On a smaller\nscale, the profiles of the mean and turbulent properties of the stable bound-\nary layer to a height of 300 m over 20-min averaging periods can be found in\nTables A.1-A.4.\n30","TUESDAY, APRIL 18, 1978\n1016\n1020\n1012\n1016\n1020\n1024\n32\"\n1028\n2542\nL\no\nI\n1016\nSURFACE WEATHER MAP\nAND STATION WEATHER\nAT 7:00 A.M. E.S.T.\n115\nWEDNESDAY, APRIL 19, 1978\n1016\n1020\n1016\n1012\n1012\n32*1020 1024\n1032 1028\n1024\n1020\n1028\nL032\n0a\nHIGH\n0°1036\n28/201\n431985\n30\n1008\n555/199\n35\n508+24\n9\n1008\n1016\n1004\nSURFACE WEATHER MAP\n1004\nAND STATION WEATHER\n1012\n1008\nAT 7:00 A.M E.S.T\n112\nFigure A.1. -- National Weather Service maps for 18 and 19 April 1978.\n31","FRIDAY, APRIL 21, 1978\n1012\n1012\n1012\n1012 1008, 1004 3000\nrood 1004\n32*\nLOW\nRx\nHIGH\nLOW\nLOW\n1016\n1008\n1004\nSURFACE WEATHER MAP\nAND STATION\nWEATHER\n1004\nAT 7:00 A.M. E.S.T\n1008\n35\nSATURDAY, APRIL 22, 1978\n1012\n1016\n1012\n101 1016\n1016 1012_1008 10041000 32*1000 1004\nLOW\n1004\n1016\n1008\nSURFACE WEATHER MAP\nAND STATION\nWEATHER\n1004\nAT 7:00 A.M E.S.T\n1008\nFigure A.2. --National Weather Service maps for 21 and 22 April 1978.\n32","TUESDAY, APRIL 15. 1980\n1012 1008 1004 1000 32*1900 1004 1008 1012 1016 1020\n1000 1004 1008-1012 1016\n0*\n1016\n1000 1008\n1008 1004 1000\n1004\nHTS\n1020\nHIGH\na\nx\ncy\n1020\n1012\n1016\n1012\nHIGH\nWEATHER MAP\n1016\nH\n1016\nATETIONHEATHER\n1020\n1020\nFD0MEST\nWEDNESDAY, APRIL 16, 1980\n1008 1684 1000. 382 1000 -1004 1008 10120 1016\n1016101210081004 1000\n996\n1000 0890121012\nD\nBUO\nLOW\n1012\n0\n1012\n37.7\nBUOY\nSURFACE WEATHER MAP\n1012\nAND STATION WEATHER\nY020\n1016\n1020\nAT 7:00 A.M. E.S.T.\nos\nor\nFigure A.3. -- -National Weather Service maps for 15 and 16 April 1980.\n33","","","","33\n67\n107\n81\n132\n167\n256\n195\n148\n250\n216\n645\n(m)\nL\n-0.008\n-0.018\n-0.014\n-0.020\n-0.156\n-0.006\n-0.007\n-0.010\n-0.006\n-0.014\n-0.017\n-0.015\n-0.037\n-0.040\n-0.026\n-0.011\n-0.006\n-0.005\n-0.009\n-0.026\n-0.026\n(°C m/s)\n*\n*\n*\nwe\n-0.052\n-0.091\n-0.063\n-0.124\n-0.078\n-0.201\n-0.123\n-0.117\n-0.190\n-0.094\n-0.028\n-0.038\n-0.173\n-0.239\n-0.170\n-0.300\n-0.239\n-0.052\n+0.007\n-0.123\n-0.143\n2,2\n*\n*\n*\nTable A.1.--Data summary for 18 April 1978\nuw\n0.006\n0.004\n0.005\n0.005\n0.018\n0.026\n0.051\n0.022\n0.008\n0.006\n0.022\n0.017\n0.015\n0.015\n0.018\n0.017\n0.017\n0.049\n0.034\n0.057\n0.041\n0.026\n0.022\n0.020\n(°C)\n2\n2\n0\no\n)\n0.160\n0.507\n0.564\n0.185\n0.094\n0.245\n0.403\n0.480\n0.190\n0.280\n0.341\n0.354\n0.317\n0.275\n0.277\n0.451\n0.376\n0.282\n2\n0.343\n0.402\n0.443\n2\n*\n*\n*\nO\n2\nindicates missing or suspect data\n(m\n-2.24\n-3.88\n-2.87\n-2.02\n-2.63\n-1.99\n-2.80\n-2.86\n-2.51\n-4.32\n-3.07\n-2.52\n-3.15\n-2.48\n-3.31\n-3.32\n-2.73\n-4.50\n-2.80\n-3.44\n-2.92\n-3.51\n-3.46\n-3.25\n(°C)\nd\nT\nparameter not computed\n1.14\n0.67\n0.28\n2.44\n2.26\n1.93\n1.47\n3.88\n2.30\n1.98\n2.37\n3.52\n3.08\n2.66\n2.41\n3.92\n3.96\n4.46\n4.48\n4.38\n4.00\n3.55\n3.18\n2.75\n(°C)\nT\n(deg)\n29\n24\n349\n14\n15\n23\n48\n50\n45\n41\n43\n35\n342\n43\n44\n36\n49\n53\n52\n53\n47\n*\n*\n*\nA\n2.30\n4.25\n5.46\n6.69\n9.00\n12.06\n12.13\n14.22\n16.12\n1.66\n2.11\n12.83\n14.86\n16.21\n7.50\n8.70\n10.27\n8.06\n9.25\n10.71\n12.41\n(m/s)\n*\n*\n*\nU\n50\n100\n150\n200\n250\n300\n100\n150\n200\n250\n300\n10\n22\n150\n200\n250\n300\n10\n22\n50\n10\n22\n50\n100\n(m)\nZ\n*\n2300 to\n2220 to\n2120 to\nNotes:\nPeriod\n2320\n2240\n2140\n(MST)","702\n624\n488\n(m)\n272\n212\n299\n347\n438\n16\n78\n31\n135\nL\n-0.024\n-0.022\n-0.011\n-0.011\n+0.003\n+0.018\n+0.041\n-0.028\n-0.022\n-0.026\n-0.046\n-0.065\n-0.016\n-0.021\n-0.025\n-0.037\n-0.026\n-0.034\n-0.055\n-0.043\n-0.032\n(°C m/s)\n*\n*\n*\nwO\n-0.355\n-0.308\n-0.165\n-0.108\n-0.125\n+0.308\n-0.175\n-0.190\n-0.229\n-0.394\n-0.749\n-0.186\n-0.029\n-0.109\n-0.047\n-0.148\n-0.699\n+0.252\n(m /s )\nTable A.2.- -- Data summary for 22 April 1978\n2,2\n*\n*\n*\n*\n*\n*\nuw\n0.027\n0.026\n0.026\n0.026\n0.030\n0.025\n0.028\n0.029\n0.068\n0.048\n0.045\n0.044\n0.036\n0.019\n0.017\n0.024\n0.148\n0.108\n0.085\n0.076\n0.065\n0.043\n0.037\n0.031\n)\n(°C4)\n2\n2\n0\no\n(m2)'s2,\n0.527\n0.478\n0.395\n0.323\n0.364\n0.416\n0.601\n0.265\n0.233\n0.226\n0.297\n0.381\n0.346\n0.588\n0.132\n0.225\n0.359\n0.724\n0.691\n0.852\n1.012\n2\n*\n*\n*\nindicates missing or suspect data\n(°C)\n-13.07\n-15.05\n-14.97\n-13.42\n-14.28\n-13.60\n-13.65\n-15.10\n-12.59\n-14.84\n-17.36\n-12.91\n-13.92\n-13.38\n-13.42\n-15.00\n-13.09\n-15.13\n-19.95\n-13.75\n-14.72\n-14.12\n-14.07\n-15.72\nd\nT\no\nparameter not computed\n(°C)\n4.85\n4.84\n4.72\n4.37\n3.97\n3.58\n3.15\n2.73\n4.34\n4.41\n4.33\n4.04\n3.72\n3.38\n2.96\n2.56\n4.01\n4.18\nT\n4.15\n3.92\n3.56\n3.23\n2.80\n2.40\n(deg)\n253\n254\n256\n259\n261\n264\n265\n265\n265\n269\n273\n279\n276\n276\n277\nA\n284\n287\n284\n*\n*\n*\n*\n*\n*\n(m/s)\n9.12\n10.83\n12.36\n13.86\n14.34\n14.13\n5.92\n7.45\n8.65\n10.03\n10.49\n11.28\n3.99\n5.01\n5.94\n7.32\n8.37\n10.50\nU\n*\n*\n*\n*\n*\n*\n10\n(m)\n22\n50\n100\n150\n200\n250\n300\n10\n22\n50\n100\n150\n200\n250\n300\n10\n22\n50\n100\n150\n200\n250\n300\nZ\n*\n0006 to\n0026 to\n0046 to\nPeriod\n(MST)\n0026\nNotes:\n0046\n0106","133\n147\n89\n86\n105\n195\n850\n222\n89\n98\n189\n276\n(m)\nL\n-0.040\n-0.020\n-0.024\n-0.029\n-0.011\n-0.003\n-0.026\n-0.027\n-0.045\n-0.011\n-0.011\n-0.036\n-0.005\n-0.035\n-0.040\n-0.032\n-0.048\n-0.033\n-0.065\n-0.045\n-0.051\n(°C m/s)\n*\n*\n*\nwe\nTable A. 3.--Data summary continued for 22 April 1978\n-0.174\n-0.078\n-0.087\n-0.249\n+0.115\n-0.106\n-0.161\n-0.399\n+0.244\n-0.152\n-0.117\n-0.603\n-0.054\n-0.223\n-0.310\n-0.136\n-0.203\n-0.124\n(m2/s2)\n*\n*\n*\n*\n*\n*\n2\nuw\n0.014\n0.049\n0.035\n0.030\n0.025\n0.021\n0.019\n0.016\n0.036\n0.033\n0.022\n0.016\n0.012\n0.069\n0.222\n0.130\n0.070\n0.043\n0.333\n0.275\n0.039\n0.046\n0.068\n0.051\n)\n2\n2\n0\n(°C\no\n1s2\n0.381\n0.396\n0.582\n0.403\n0.539\n0.306\n0.316\n0.268\n0.317\n0.394\n0.407\n0.405\n0.299\n0.394\n0.587\n0.928\n0.900\n1.310\n1.576\n0.409\n0.415\n2\n*\n*\n*\nW\no\n2\nindicates missing or suspect data\n(m\n-15.02\n-14.78\n-15.40\n-16.19\n-15.48\n-15.20\n-15.64\n-16.87\n-14.75\n-14.84\n-14.61\n-15.29\n-16.16\n-15.55\n-15.30\n-16.09\n-14.97\n-15.13\n-15.20\n-15.02\n-15.55\n-16.24\n-15.49\n-15.26\n(°C)\nd\nT\no\n2.01\n1.57\n2.99\n3.20\n3.30\n3.04\n2.68\n2.27\n1.85\n1.41\nparameter not computed\n3.36\n3.15\n2.81\n2.44\n3.44\n3.74\n3.80\n3.49\n3.06\n2.63\n2.19\n1.76\n3.10\n3.28\n(°C)\nT\n(deg)\n281\n283\n286\n28\n285\n277\n280\n281\n281\n280\n286\n286\n286\n287\n286\n281\n276\n276\n*\n*\n*\n*\nA\n*\n*\n15.77\n7.27\n8.93\n10.41\n14.13\n12.14\n12.96\n13.24\n8.12\n9.88\n11.59\n13.53\n14.45\n6.05\n7.64\n8.83\n10.39\n11.32\n(m/s)\n*\n*\n*\n*\n*\n*\nU\n50\n100\n150\n200\n250\n300\n100\n150\n200\n250\n300\n10\n22\n10\n22\n50\n100\n150\n200\n250\n300\n10\n22\n50\n(m)\nZ\n*\n0246 to\n0226 to\n0206 to\nNotes:\nPeriod\n0306\n0246\n0226\n(MST)","123\n(m)\n165\n263\n208\n36\n16\n39\n18\n76\n89\nL\n-0.025\n-0.021\n-0.010\n-0.014\n-0.007\n-0.004\n-0.004\n-0.005\n-0.019\n-0.015\n-0.007\n-0.004\n-0.003\n-0.003\n+0.003\n-0.003\n-0.015\n+0.029\n-0.017\n-0.036\n-0.039\n+0.013\n+0.025\n+0.027\n(°C m/s)\nwe\n-0.109\n-0.118\n-0.102\n-0.106\n-0.118\n-0.060\n-0.056\n-0.014\n-0.040\n-0.020\n-0.021\n+0.020\n+0.026\n+0.089\n+0.133\n+0.203\n-0.022\n+0.036\n-0.060\n-0.114\n-0.216\n-0.214\n-0.219\n-0.226\nTable A. 4. -- Data summary for 15 April 1980\n22\nuw\n0.084\n0.042\n0.019\n0.058\n0.010\n0.006\n0.006\n0.003\n0.084\n0.060\n0.108\n0.017\n0.007\n0.005\n0.004\n0.005\n0.162\n0.472\n0.272\n0.170\n0.147\n0.065\n0.062\n0.065\n(°C2)\n2\n2\n0\no\n\"/s\")\n2.\n0.164\n0.153\n0.172\n0.173\n0.152\n0.135\n0.263\n0.133\n0.108\n0.093\n0.097\n0.066\n0.096\n0.120\n0.228\n0.237\n0.067\n0.079\n0.072\n0.130\n0.173\n0.153\n0.305\n0.203\n2\nW\no\n2\n(m\n-10.29\n-11.34\n-12.55\n-13.40\n-14.05\n-15.04\n-15.29\n-15.43\n8.89\n9.68\n-10.46\n-12.08\n-12.29\n-13.26\n-13.10\n-13.13\n7.56\n8.46\n9.18\n-10.19\n-10.08\n-11.11\n-10.89\n-10.67\n(°C)\nd\nT\n-\n-\n-\n-\n-\n17.67\n17.75\n17.67\n17.33\n16.92\n16.49\n16.04\n15.63\n15.57\n16.18\n16.71\n16.96\n16.63\n16.26\n15.78\n15.32\n14.32\n15.53\n16.17\n16.12\n15.94\n15.74\n15.32\n14.36\nparameter not computed\n(deg) (°C)\nT\n276\n276\n276\n277\n278\n275\n274\n273\n303\n297\n290\n290\n289\n285\n286\n283\n258\n263\n265\n269\n275\n274\n276\nA\n272\n(m/s)\n6.22\n7.40\n8.50\n9.24\n10.33\n10.77\n10.54\n11.09\n3.16\n4.73\n7.17\n9.31\n9.64\n9.23\n9.15\n8.83\n4.93\n6.86\n8.39\n8.70\n9.52\n9.87\n10.12\n10.26\nU\n10\n(m)\n22\n50\n100\n150\n200\n250\n300\n10\n22\n50\n100\n150\n200\n250\n300\n10\n22\n50\n100\n150\n200\n250\n300\nZ\n1700 to\n1820 to\n1900 to\nPeriod\n1720\n(MST)\n1840\n1920\nNote:","APPENDIX B: VARIANCES AND SCALES OF TEMPERATURE AND VELOCITY\nFLUCTUATIONS IN THE STABLE BOUNDARY LAYER*\nMonin-Obukhov similarity reasoning and measurements over flat terrain\nshow that the mean temperature gradient in stable conditions (z/L > 0) is\ngiven by\n(B.1)\nwhereTand - = k and B are constants (Monin and\nYaglom, 1971). Since fluxes are taken as constant in the surface layer,\now (=1.3 ux) is approximately constant with height, as is we. To is the mean\nsurface temperature. Consequently for similarity,\n(B.2)\n= Og/I* = constant, when z/L>0.\nThe Kansas data give a range for I e = 1.5 to 2.0 (Wyngaard et al., 1971),\nwhereas Okamoto and Webb (1970) suggest 2.5 to 3.3, whence Two = -w0/060==0.35\n(Kansas and Minnesota data, Caughey et al., 1979). The ranges of values of\nre and Two measured at BAO (z < 22 m) are 2.9 + 0.6 and 0.23 + 0.1.\nGiven (B.1), we can write (B.2) as a relation between the rms temperature\nfluctuation of' the mean temperature gradient de/dz, and a length scale ()\ndefined by\n(B.3a)\nwhere\n(B.3b)\n* Summary of a yet-to-be-published note by J.C.R. Hunt. It is on the basis\nof the results presented here that some of the measurements are discussed in\nthe paper. [Some aspects were published by Hunt (1982a).]\n41","So\nfor weak stratification, = I and for strong stratification, where\nB z/L >> 1,\n+\n0.5)\nL\n(B.3c)\n,\nif B = 5.4, as in the Kansas experiments (Busch, 1973). The Monin-Obukhov\nlength scale L can be expressed as\n(-w0)k\n(B.4a)\nThus for strong stratification,\nL = 5 o / N for the Kansas value of B.\n(B.4b)\nAt BAO, L/ (OW/N) varies over the range 3 to 10 for Z < 50 on the strongly\nstratified days considered here when z/L 1/1 Thus (B.3a) becomes for the\nKansas data\n(B.5)\n,\nwhere So varies from 0.5 to 1.0.\nThe thermal diffusivity parameter F = Kg/(02/N) can also be derived\nfor surface-layer stable turbulence. From (B.1), the coefficient of thermal\ndiffusivity is\nz/L)]\nThen from (B.4a)\n42","B 1/2\nF =\n(B.6a)\n1/2\n(ow/ux) 2 [1 + 1/(B (z/L)]\nThence\n0.3\n(B.6b)\nF =\n[1 + (ow/N)/z] 1/2\nwhen B(z/L) >> 1, using Kansas data.\n43","APPENDIX C: SCALES OF TEMPERATURE AND VELOCITY FLUCTUATIONS\nIf in stable conditions in the surface layer there is a local balance of\nturbulent kinetic energy (TKE) between production, dissipation, and the local\nflux of buoyancy, then the steady-state TKE equation becomes\n(C.1)\nwhich provides a definition for the energy dissipation scale\nSince it\n€,U*\nis observed that in stable conditions\ndU/dz\n(C.2)\nwhere a = 5 (Monin and Yaglom, 1971), then from (C.1) and (C.2)\n(a-1)z/L]\n(C.3a)\n(Busch, 1973), and\n(C.3b)\n.\n2\nThe implication of (C.3a) is that for given shear stress u4, or for given\nW variance 22 the increase of dissipation with stable stratification is\n(a - 1), or about 4 times greater than the damping of turbulence by the\nbuoyancy flux. Only if one assumes that the mean horizontal velocity U or\nmean velocity gradient dU/dz is held constant as z/L increases do the dissipa-\ntion and turbulence decrease. Most physical discussions of the damping\naction of stratification are rather vague as to what aspect of the flow is\nkept constant or determined by boundary conditions when z/L or Ri increases.\nAn explanation for this result is given in Appendix D. [Usually Prandtl's\n(1952, p. 381) discussion of \"balls\" of fluid moving vertically influences\nmost qualitative descriptions, though Turner (1973, p. 136) emphasizes that\ndissipative viscous losses greatly exceed those caused by buoyancy. However,\nhe fails to consider how this dissipation changes with stratification or why\nit does. ] Clearly such an assumption has to be made (implicitly or expli-\ncitly) and this then determines which is the dominant physical process.\n44","Observations and physical arguments based on the local nature of small-\nscale turbulent motion indicate that\neven in stably stratified boundary layers where C E is a coefficient that\nvaries little between different shear flows (see Appendix D) . It follows\nfrom (c.1) that\n(C.4a)\nwhence from (c.3b)\n(C.4b)\n+ (a-1)z/L]\n.\n= and k 0.4,\n(c.5)\n+\n.\nThe results in (C.3a) and (C.5), which derive from the observed form of\n= zz(du/dz)/ux and the TKE equation, agree well over the range of\n0 < z/L < 1.0 with Kaimal's (1973) Kansas data. (His points at z/L = 1.5 and\n3.0 do not agree!) This equivalence between LE,Ux and L(w) was an important\nhypothesis in Townsend's (1958) argument for the critical Richardson number;\nit certainly seems to be substantiated by Kaimal's data.\nThe relative scale of temperature fluctuations can also be inferred from\nobservations and the equation for temperature variance 02 (see Townsend,\n1958); viz,\n(c.6)\nThen let where LEO is the temperature dissipation\nscale. As\nTKE (c.1), the diffusion term is negligible in the\nin the equation\nstable surface layer. Since de/dz is observed to have the form of (B.1) in\n45","Appendix B, it follows that\nkz\n(C.7)\nz/L)\n.\nFrom (c.3b) and (C.7) the ratio of temperature to vertical velocity dis-\nsipation scales can be estimated as\nz/L\n(C.8)\nIn the Kansas data T we = 0.35, and Assuming that the tem-\nperature integral and dissipation scales are related to each other as are the\nvertical scales, then when z/L = 0,\n=\n(c.9a)\nand when B z/L >> 1, since B n 6.4 (Busch, 1973),\n(c.9b)\nIn the Kansas measurements (8)/IL(w) n 8 when z/L = 0 and 3 when z/L = 1.0,\nwhereas in the BAO measurements (Kaimal et al., 1982), (8)/I(w) 5 when\nz/L = 0. These results give some support to the arguments leading to (c.9a)\nand (c.9b). We conclude that there may be at most a reduction by a factor of 2\nin the relative scales of (e) and L((w) as the stability increases or as Z\nincreases in stable conditions.\n46","APPENDIX D: HYPOTHESES CONCERNING INTEGRAL SCALES AND\nDISSIPATION SCALES IN SLOWLY VARYING TURBULENT FLOWS\nThe ideas in this Appendix are derived from studies of grid turbulence\n(near and far from the wall), and of convective and neutral atmospheric bound-\nary layers. Some of the new results are applicable in these other flows as\nwell as in stable boundary layers.\nHypothesis 1\nIn many turbulent flows it is observed that\nE = CE 03/LCW\n(D.1)\nIn turbulent boundary layers, whether stable, neutral, or convective, C E\nvaries from 0.4 to 0.6, whereas in grid turbulence C E 1 1.5. [For some justi-\nfication see the subsequent analysis as well as Appendix C, and Hunt (1982b),\nKaimal (1973), and Townsend (1958).]\nHypothesis 2\nIn neutral and stable shear flows\n(D.2)\nwhere ag-1.3\n(Townsend, 1976).\nThe integral scale of L(w) is defined in an unconfined stably stratified\nX\nshear flow by dU/dz, N, and ow A linearized analysis similar to that of\nTownsend (1976, pp. 46-49) for the rate of growth T-1 of a disturbance with\nwavenumber k in such a flow shows that, if N/(dU/dz) << 1,\n(D.3)\ndU/dz +\nwhere the constants X1' X2 are functions of k and hereafter taken to be equal\nto 1.0.\n47","Weak stable stratification affects turbulent energy only when combined\nwith shear; without shear, laboratory experiments of decaying turbulence show\n(Britter et al., 1983) that there is little effect until the turbulence is\nvery weak.\nIf the shear is so small that dU/dz << N, and the turbulence intensity\nis weak enough that N 3, then the density gradient may directly\naffect the straining of small eddies by the large eddies, and thence determine\nE and L(w). . Such a situation is likely to occur only above the boundary layer\nin decaying turbulence or in turbulence maintained by large-scale wave motion.\nFor these reasons, in the boundary layer where dU/dz = N and NL(W) low V 1 there\nappears to be no physical reason why the turbulence scale should be controlled\nby the local value of N/OW. However, when the flux and gradient Richardson\nnumbers are constant, as assumed by Brost and Wyngaard (1978) in their model\nfor the stable boundary layer, dU/dz ~ N and then the local value of ow/N is\nan appropriate scaling for L(w)\nIf the local Richardson number Ri = N2/(du/dz)2 V 1/4, then it follows\nfrom (D.3) that length L(w) in an unconfined shear\nthe natural local scale\nflow is given by\n(D.4)\nwhere A1 is an unknown constant.\nIf the turbulence is close to a rigid boundary at Z = 0, the turbulence\nscale is reduced in proportion to Z (Hunt and Graham, 1978). If the turbu-\nlence is far from a boundary and controlled by the overall scale of the flow\n(e.g. , a grid mesh scale L or a wake depth L or a convective boundary layer\no\nscale L or the scale of some large-scale wave motion), then L is the rele-\nvant scale.\nThus combining these suggestions with (D.4) leads to the next hypothesis\nfor boundary layer flows where Ri VR 1/4.\n48","Hypothesis 3\n(D.5)\nwhere AS and AB are constants for the shear and \"blocking\" effects.\nFrom the kinematic theory of shear-free turbulence near a rigid interface\n(Hunt, 1982; Hunt and Graham, 1978), it can be shown that close to the sur-\nface when\nz/Lo - o or [2/L(w)(2 + 00)] 0, then L(w) = 1.7 z ,\n(D.6)\nthe factor 1.7 being determined by the experimental constant in Kolmogorov's\ninertial subrange law. Since in the limit the expression in (D.5)\nfor (L (w) 1-1 equals AB/Z, it follows that\n(D.7)\nAg = 1/1.7 = 0.6 .\n.\nThe putative constant AS in (D.5) can be estimated by either of two\nalternative observations.\n(1) The turbulent kinetic equation in a boundary layer and (D.1), (D.2),\nand (D.5) show that, where\n(D.8)\nSince it is observed that\n(D.9)\nwhere a = 5 (Monin and Yaglom, 1971), then combining (D.2), (D.8), and (D.9),\nwhen z/L >> 1, gives\n(D.10a)\n=\n(D.10b)\n49","(2) Alternatively, the value of AS can be fixed by comparing the expres-\nsion for L(w) in (D.5) when 1 with the observation that in a neutral\nboundary layer = (0.4 0.1)z (Hunt and Weber, 1979). In that case,\nAs1.30.4(2.50.5)-0.6) =\n(D.11)\n= 1.0 + 0.25 ,\n.\nThus either method gives roughly equivalent values of As. On the other hand,\nif C FC E = 0.6, and AS is fixed by the first alternative, then (D.5) provides a\nprediction that in a neutral SBL\nL(w)/z = 0.5 .\nThe physical implication is that in a neutral boundary layer L(w) is\nreally determined by two comparable effects--the local shear and the wall\nproximity effect.\nA second important prediction can be made by substituting the value of\nthe \"constant\" As into the turbulent kinetic energy equation (C.1) for con-\nditions near the ground when there is significant stable stratification, i.e.,\nz/L > 0. Then from (c.1), (D.1), (D.5), and (D.9)\nso\n(1 + /L)\n(D.12)\nkz\nwhere\nand\n(D.13)\nNote that when /L >> 1, z/L >> N2//du/dz)2, so dU/dz /L). Since\nAS is chosen to agree with the measurements of the neutral case, k k = 0.4,\n50","but a is not fixed. However by these arguments a is found to be 3.2. Experi-\nments over flat ground give a value of a equal to about 5, which is certainly\nof the same order of magnitude.\nWe see in Appendix B that when a stable stratification is applied to a\nturbulent boundary layer, the increase in viscous dissipation is a factor\n(a-1) times as great as the loss of energy caused by the buoyancy flux. We\ncan now see that this must be so. For given or ow' the generation of a\nsmall buoyancy flux OB must require an increased energy input by an increased\nshear S (dU/dz). If this were the only balance to be made, then s (dU/dz) =\nSB/u. 2 But an increase of S (dU/dz) reduces the integral scale L(w) and in-\ncreases the dissipation by about 0.7 ux 2 S (dU/dz). Thus only 30% (by our\ncalculation) of the increased shear can be used to balance the buoyancy flux.\nOnly by realizing the relative sensitivity of L(w) to both dU/dz and Z 2-1 can\npoint be appreciated. If L(w) is assumed to be entirely controlled by\nthis\nthe distance from the surface, then an alternative explanation would have to be\nsought for the observed sensitivity of E to z/L. Since the turbulent struc-\nture at scales less than L(w) is not measurably affected by stable stratifi-\ncation, there is no evidence for any other explanation. [The same effects are\nfound in free shear layers where about 70% to 80% of the energy of Kelvin-\nHelmholtz billows is dissipated, the rest providing a buoyancy flux (Sherman\net al., 1978); perhaps the same explanation is appropriate. ]\nAs a third prediction, the hypothesis (D.5) and the observation (D.9)\nimply that when az/L >> 1, dU/dz = au+/(0.4L), and that\nTHE\n(D.14)\nor\n1\nL\nor\n(D.15)\nL 1 0.55 ,\nin approximate agreement with the observations (c.5) and (B.4b).\nThus, given the more general turbulence hypotheses (D.1), (D.2), and (D.5),\nwe have found that in strongly stable conditions L(w) must be proportional to\n51","L and o W /N near the surface, a result otherwise only deducible from the kind\nof Monin-Obukhov similarity and dimensional arguments used in Appendix B.\nIn boundary layers with strong stable stratification, Brost and Wyngaard\n(1978) argued that on physical grounds WW and Z are the two scales that\nW\ndetermine L\nE,Ux , so by interpolating from the outer edge of the boundary layer\nto the layer nearest the surface, they postulated that\n1-1 = = N/OW+\n(D.16)\n.\nThis result is consistent with the arguments and results presented here when\nthe stratification is strong and at the outer edge of the boundary layer (even\nwhen the Richardson number is not constant). When the stratification is weak,\n(i.e., L(w) N/O W VI 1) we are suggesting on the basis of theoretical arguments,\nand surface-layer and BAO observations, that (D.16) is not the appropriate\nscaling, even in boundary layers. The more general scaling suggested here is\nappropriate.\n52","2. WAVE AND TURBULENCE STRUCTURE IN A DISTURBED NOCTURNAL INVERSION\nLu Nai-ping\nInstitute of Atmospheric Physics\nAcademia Sinica\nBeijing, China\nW.D. Neff and J.C. Kaimal\nNOAA/ERL/Wave Propagation Laboratory\nBoulder, Colorado 80303\nABSTRACT. Acoustic sounder and tower data obtained at the Boulder\nAtmospheric Observatory (BAO) are used to examine several features\nof the wave and turbulence structure associated with a disturbed\nnocturnal inversion. General features, including mean fields and\nRichardson number, for the case selected for this study are pre- -\nsented. Spectral analysis of the tower data reveals a separation\nof energy into wavelike and turbulent fluctuations. Analysis of\nthe heat flux, however, shows upward counter-gradient fluxes in the\nvicinity of a low-level jet and near the top of the inversion. Co-\nspectral analysis shows that the major contribution to the upward\nheat flux occurs at frequencies that would normally be considered\ncharacteristic of waves. In some cases, the upward flux is asso-\nciated with a phase shift between vertical velocity W and fluctuat-\ning temperature 0 different from the quadrature relation that would\nbe expected of internal waves. Time series analysis reveals that\nthese unexpected positive fluxes occur in relatively short bursts.\nAnalysis of time series of 0 and W in other cases, as well as in-\nspection of acoustic sounder records, shows that sometimes such\nupward fluxes can result from a combination of wave motion and\nhorizontal temperature advection. In this case the advection is\nassociated with a shallow cold front.\n2.1 INTRODUCTION\nRemote-sensing devices using microwaves or acoustic waves have often been\nused to provide visual documentation for special case studies of waves and\nturbulence within the nocturnal inversion layer (e.g., Gossard et al., , 1970;\nEmmanuel et al., 1972; Merrill, 1977; Einaudi and Finnigan, 1981). Whereas\nmany of the details of such events are well understood, the data on which such\nanalyses are often based arise from idealized circumstances (such as situations\nwith nearly monochromatic internal waves). More commonly, the nocturnal\n53","inversion as seen by remote-sensing devices reveals a complex mixture of waves,\ninstabilities, and turbulence. Often, the complexity of the inversion seems to\nbe related to particular synoptic situations as noted by Zhou Ming-yu et al.\n(1980) and Neff (1980). The acoustic sounder and the sensors on the 300-m\ntower at the Boulder Atmospheric Observatory (BAO), operating almost con-\ntinuously over the last 4 years, provide the data base needed to relate typical\npatterns in nocturnal inversion development to synoptic and mesoscale events.\nOne event that occurs frequently in winter at BAO is the passage of a\nshallow cold front. Using data from BAO, we present a case study of one such\nevent. We will first describe the meteorological setting, the general features\nof the low-level front, and the complexity of the waves, turbulence, and ad-\nvective processes that are present. We will then outline the spectral charac-\nteristics of the turbulence and waves followed by detailed interpretation of\nthe features contributing to an anomalous counter-gradient heat flux that\nappears in a conventional 20-min covariance of W (vertical velocity) and 0\n(fluctuating temperature).\n2.2 SITE CHARACTERISTICS AND INSTRUMENTATION\nBAO is located about 30 km north of Denver and 30 km east of the foothills\nof the Rocky Mountains. This site, on the high plains of Colorado, is charac-\nterized by gentle slopes (Kaimal et al., 1982); at the time of the present\nstudy, the ground cover consisted of areas of wheat stubble mixed with bare\nareas planted with winter wheat.\nThe tower instrumentation at BAO has been described in some detail by\nKaimal and Gaynor (1983). Briefly, the tower is instrumented at eight levels\n(10, 22, 50, 100, 150, 200, 250, and 300 m) with three-axis sonic anemometers,\npropeller-vane anemometers, fast-response platinum wire and slow-response\nquartz thermometers, and a cooled-mirror dewpoint hygrometer. A network of\nfive sensitive microbarographs, measuring pressure fluctuations P, is centered\nabout the tower to detect the phase speed and direction of internal waves pro-\npagating across the site. Two acoustic sounders were used: one close to the\n54","tower with a 340-m vertical range, and a second about 600 m away with an\n850-m vertical range.\nData from the tower and the microbarographs are transmitted via phone\nline to the central data archiving computer system in Boulder. Standard\nsystem programs are used to compute fluxes, variances, and 20-min spectra in\nreal time. Post-processing programs are available to compute spectra as well\nas cospectra and quadrature spectra over 80- or 160-min periods. A special\n\"beamsteering\" program (Kaimal and Gaynor, 1983) is used to calculate wave\nphase speeds and directions from the microbarograph array. This program has\nprovisions for filtering time series of tower data in any specified spectral\ninterval with varying passbands and cutoff frequencies.\n2.3 GENERAL DESCRIPTION OF CASE STUDY\n2.3.1 Meteorological Situation\nThe events under consideration here occurred on the morning of 27 March\n1981. Earlier in the evening, the development of the nocturnal inversion had\nfollowed an uneventful course in response to surface cooling. Winds aloft\nwere from south-southwest reflecting the presence of a cutoff low centered\nover southwestern Nevada. The combined circulation of the associated surface\nlow- and a surface high-pressure system located over the Great Lakes main-\ntained southeasterly surface flow over the eastern plains of Colorado. A\nsurface low over central Wyoming meanwhile formed and moved in a southerly\ndirection along the Continental Divide and by 0500 MST on 27 March was located\nsouth of Denver. The 0500 MST rawinsonde ascent made by the National Weather\nService at Denver showed a surface-based inversion capped by a weak isothermal\nlayer that extended from 800 to 700 mb. The winds were light and variable\nfrom the surface to 700 mb, above which the winds abruptly increased to 18 m/s\nfrom the southwest. While the relatively coarse rawinsonde network revealed\nlittle complexity in the flow, acoustic sounder and tower data at BAO showed\nan abrupt change in the nocturnal inversion shortly after midnight. As shown\nin the next section, winds at the top of the tower became light from the west\n55","while in the lower half of the tower a 10-m/s jet developed, blowing from\nnorth-northeast. Associated with this wind shift were a drop of about 3°C in\ntemperature and a sudden decrease in surface pressure of about 0.5 mb. Mea-\nsurement of the wave and turbulence properties associated with this unsteady\njet will be of primary interest here.\n2.3.2 Acoustic Sounder and Mean Flow Data\nFigure 2.1 shows the nocturnal inversion, as seen by the acoustic sounder,\nduring the period from midnight to 0800 MST. Three distinct periods can be\nnoted. The first is the initial transition period between midnight and\n0100 MST when a north-northeast jet first appears. The second is between 0330\nand 0400 MST when the inversion is strongly disturbed. The third is after\n0600 MST, just before the onset of convection, when the inversion height is\nrelatively stationary.\nThe intensity of the echoes depicted in Fig. 2.1 is a function of the\nstrength of the temperature fluctuations. Explicitly, the scattering cross\nsection is proportional to (Neff, 1980)\n((\n1/3\n5/3\nRi\n2/3\nao\n(2.1)\nKM\ndz\nRi\nThe first term in brackets measures the strength of the turbulence, where K\nH\nis the eddy diffusion coefficient for heat, KM is the coefficient for momen-\ntum, and Ri is the gradient Richardson number. With a given mean potential\ntemperature (0) profile, the distribution of turbulence is a significant\nfactor in the echo patterns seen in the acoustic backscatter. Results from\nNeff (1980) also suggest that the distribution of wind shear, in turn, is the\nmost significant factor determining the distribution of turbulence. These\nconclusions are illustrated fairly clearly in Figs. 2.2 and 2.3. Figure 2.2\nshows the wind and temperature fields as well as isopleths of Ri calculated\nusing data from fixed levels of the tower. The most notable feature of these\nfigures is the presence of two scattering layers: one near the ground, the\n56","860 m\n800\n300\n700\n600\n500\n400\n200\n100\n0\n2300\na site some 600 Acoustic m south sounder of the record tower. for Maximum 2300 MST range on 26 is March 850 to 0900 MST on 27 March 1981 from\n0000\n0100\n0200\nm.\n0300\n27 March 1981\n0400\nTime\n0500\n0600\n0700\n0800\nFigure 2.1.\n0900","3\n3 3\n3\n3\n5\n300\nV\n5\n3\n13\nV\n5\n3\ni\n3\n35\n7\n9\n9\n11\nV\n250\n3\n5\n9\n200\n5\n5\n150\n7\n100\n5\nO7\n5\n3\n50\n3\n5\n(1)\n5\nWind speed,\n3\n17\n0\n6 7 8 8 8\n8\n8\n88\n8\n9 9 11\n8\n9\n9\n9\n13\n300\n5\n6\n13\nC\n250\n5\n12\n4\n3\n200\n12\n6\n150\n100\n4\n11\n50\nTemperature\n2\n10\n0\n13 31\n11\n331\n300\nU\n1\n1\n250\n1\n1\n200\n3\n(s\n150\n3\n100\n1\n3\n50\nRichardson number\n0\n0500\n0400\n0300\n0200\n0100\n0000\nTime (MST)\nFigure 2.2. . -- Isopleths of wind speed, temperature, and Ri developed from\ntower fixed-level - data for 0000-0540 MST, 27 March 1981. (Wind speeds\nare in meters per second, and temperatures are in degrees Celsius.)\n58","T(°C)\n12\n14\n4\n6\n8\n10\n300\n236°\n250\n284°\n354°\n200\n150\n359°\nV\nT\n100\n4°\n50\n351°\n0115\n0100\n10\n0130\n0\n2\n4\n6\n8\nTime (MST)\nV (m/s)\nFigure 2. 3. Tower wind and temperature profiles averaged over 5 min frc\n0105 to 0110 MST compared with the acoustic sounder record from 0100 to\n0130 MST obtained at a site just 160 m north of the tower. The dark,\nhorizontal bands correspond to reflections from the tower.\nsecond elevated above a clear echo region on the record. Comparison of\nFigs. 2.1 and 2.2 indicates that the clear region is characterized by large\nvalues of Ri. This can also be seen in a detailed comparison between the\nsounder record and the tower wind and temperature profiles as shown in\nFig. 2.3. Within the resolution permitted by the 50-m spacing of fixed tower\nlevels, the non-echo region is closely associated with the maximum of the low-\nlevel jet where the shear is the least. The elevated scattering layer appears\nto be connected with the region of increased shear and stability above the\njet. In accord with Eq. (2.1), the strongest scattering occurs in layers of\nlargest potential temperature gradient.\n59","2.3.3 Internal Wave Behavior\nOf interest throughout the event shown in Fig. 2.1 are the oscillations\nwith periods of several minutes at the height of the inversion base capping\nthe low-level jet. Analysis of the microbarograph data from the array sur-\nrounding the tower shows waves of several frequencies propagating from a\ngenerally southwesterly direction at speeds from 6 to 22 m/s.\nTo determine the characteristics of these waves, we used the BAO beam-\nsteering program to calculate wave phase speed, direction, and coherence as a\nfunction of frequency. This program requires an 80-min time series. Because\nwaves of a particular frequency may occur only over a short period of time, we\nincremented the starting time of the calculation by 20 min. We found that\nprior to 0300 MST, only one wave component dominated for each 80-min calcula-\ntion; that at the time of frontal passage, the wave phase speed increased\nmarkedly; and that after 0300 MST, many waves of different speeds and direc-\ntions were present. These results are shown graphically in Fig. 2.4, where\ndata are plotted at the center time of each period and the vertical bars\nrepresent the range of significant wave speeds and directions found. This\nbroadening of the wave spectrum coincides with the large disturbance in the\ninversion shortly after 0300 MST as seen in Fig. 2.1.\nSince the direction of propagation of these waves is opposed or perpen-\ndicular to the low-level flow, it may be that the waves originated from a\ncritical level aloft. The most likely region, determined from the 0500 MST\nrawinsonde launched from Denver, is at 2 km where, in a strongly sheared re-\ngion, the winds increase from less than 1 m/s to nearly 18 m/s. This seems a\nreasonable conclusion based on the work of Keliher (1975) who studied the\ncorrelation of surface-measured waves with jet-stream velocities and direc-\ntions. Atkinson (1981) suggests, though, that although a low-level inversion\nmay amplify surface pressure fluctuations, wavelengths less than 5 km at\ntropopause level are unlikely to be associated with surface pressure fluctua-\ntions. Prior to 0300 MST, the dominant wave period was about 15 min; after-\nwards wave periods ranged from 4 to 15 min. On the basis of the wave phase\nspeeds in Fig. 2.4 after 0300 MST, only the long-period oscillations in the\ninversion, from 10 to 20 min in length, are likely to be associated with an\n60","30\n20\n\"\n10\n0\n360\nWind at\n300m\n270\nFigure 2.4. -- Wave phase speed and\ndirection calculated at 20-min inter-\nI\nvals from 1900 MST on 26 March to\n180\n0800 MST on 27 March 1981. Calcula-\ntion is in overlapping 80-min seg-\nments. Vertical bars represent the\nrange of significant wave speeds and\n90\ndirections.\n2000 2200 0000 0200 0400 0600 0800\nTime (MST)\nupper-level - source. The origin of the shorter period waves is not determined\nalthough one might speculate that they could be an artifact of the frontal\ndisturbance and/or its interaction with the foothills of the Rocky Mountains\nsome 30 km to the west.\n2.4 SPECTRAL ANALYSIS\n2.4.1 Spectra Under Steady Conditions\nThe evidence of waves in both the microbarograph and acoustic sounder\ndata prompted a closer examination of the tower wind and temperature data by\nspectral techniques. The first period chosen for this analysis was one with a\nfairly constant inversion height, just before the onset of convection.\n61","A number of authors have discussed spectra under stable conditions.\nSteward (1969) pointed out some of the difficulties of distinguishing rigor-\nously between waves and turbulence. However, a number of subsequent papers\n(Axford, 1971; Kaimal, 1973; Caughey and Reading, 1975; Caughey, 1977; Caughey\net al., 1979) have shown that in the analysis of power spectra there is often\na clear separation into wavelike and turbulent regions. Figure 2.5 presents\nsuch results from data taken between 0620 and 0720 MST, 27 March. These data\nshow a clear spectral gap at about 10-2 Hz. In the W spectra, the gap is much\nless noticeable near the ground, primarily because W must approach zero at the\nsurface. The high-frequency portion of the spectra follow the characteristic\n-2/3 power law. Cross-spectra at 250 m for the same time period show an\nabrupt change in spectral slope at 10 2 Hz. The slope at lower frequencies is\nalmost -5; the slope at higher frequencies tends to be a bit less than -4/3\npredicted from surface layer similarity theory (Wyngaard and Coté, 1972).\nIntegrating the wavelike and turbulent portions of the spectra sepa- -\nrately, we get the results in Table 2.1. Here it can be seen that the low-\nfrequency portion of the spectra and cospectra contributes more strongly to\nthe variances and covariances than the high frequency portion does.\nTable 2.1 -- Ratios of low-frequency (L) to high-frequency turbulent (H)\ncontributions to standard deviations and covariances during the\nperiod (0620-0720 MST, 27 March 1981)\n(OB'L\n(OV)L\n(w0)\n(wu)\n(wv)\nZ\nu'L\nw L L\nV\nL\nL\nL\n(m)\n(OB)\n(o.)\n(of)\n(o.)\nH\nu H\nv H\n(w0)\nW H\n(wu)\n(wv)\nH\nH\nH\n10-100\n8.9\n9.0\n13.0\n0.4\n3.1\n2.6\n6.2\n150-300\n13.0\n22.7\n21.7\n3.6\n92.4\n63.1\n30.0\n2.4.2 Disturbed Cases\nFigure 2.6 shows time series of temperature from 2100 to 0540 MST, and\nFig. 2.7 shows time series of temperature flux we from 0000 to 0540 MST.\n(Prior to 0000 MST, the wind was through the tower rendering the sonic mea-\nsurements useless.) Data for these time series were obtained as 20-min means.\n62","10\nFigure 2.5. -- Variance spectra for 0, u, W, and V together with cross-spectra of we and uw during 0620-\n-4/3\nn 4/3\n10°\nn\nn\n10\nn (Hz)\n-2\n10\n10-3\n10-4\n-2\n-3\n103\n-4\n104\n-5\n0\n10°\n-1\n0\n-2\n-1\n-3\n-4\n10\n10\n10\n10\n10\n10\n10\n10\n10\n-2/3\n-2/3\n10°\nn\nn\n100m\n250m\n22m\n-1\n10\nn (Hz)\n-2\n10\n-3\n10\n10-4\n-2\n-3\n-4\n-2\n10\n-3\n-4\n10°\n-1\n10\n1\n0\n-1\n10\n10\n10\n10\n10\n10\n10\n10\n101\n-2/3\n0720 MST, 27 March 1981.\nn\n-2/3\n10°\nn\n10-1\nn (Hz)\n-2\n10-2\n-3\n10\n10-4\n-2\n-3\n-4\n1\n-1\n10\n-2\n-3\n-4\n0\n1\n0\n10\n10\n10\n10\n10\n10\n10\n10\n10\n10","20\n15\n10\n300m\n5\n200m\n100m\n10m\n0\n2100\n2300\n0100\n0300\n0500\n0700\nTime (MST)\nFigure 2.6. -- Time series of temperature averaged over 20 min at four\ntower levels from 2100 MST on 26 March through 0540 MST on 27 March.\nThe covariance of temperature and vertical velocity was computed in the usual\nmanner after removal of the means. The temperature and temperature flux data\nused are the standard archived values in the BAO files. As we will show\nbelow, use of these standard flux values requires some caution. In principle,\nsimple removal of mean values should be satisfactory if no more than one time\nseries has a trend. In general it is assumed that the time series of W is\nfree of trends. An example will be given later where this is not the case.\nNotable in the temperature traces is the interruption of the general\ncooling trend by a warming at midlevels after 0300 MST. Unusual in the flux\ntime series are the positive fluxes observed several times during the night.\nSince these are counter-gradient, they are most puzzling. Applying the tech-\nnique used in the previous section, of integrating the cospectrum of W and 0\nseparately over wave and turbulence intervals, we find that the positive con-\ntributions to the temperature flux come from low-frequency motions. Inspec-\ntion of Fig. 2.7 shows positive fluxes mostly in the midlevels of the tower\n64","300m\n250m\n200m\n150m\n100m\nScale\n0.2 °C m/s\n50m\n0500\n0400\n0300\n0200\n0100\n0000\nTime (MST)\nFigure 2.7. -- Time series of temperature flux we calculated over 20 min\nfrom 0000 to 0540 MST, 27 March 1981.\n(100-200 m) with negative fluxes at 250 and 300 m. Comparison with the\ntemperature traces in Fig. 2.6 shows that this midlevel flux convergence\ncorresponds to a period of net warming at 200 m. From Fig. 2.7, , taking an\naverage flux difference of 0.2°C m/s between 150 m and 250 m respectively,\nover the period from 0320 to 0440 MST, produces a net heating of 9.6 . O C in the\nabsence of advective or radiative temperature changes. This temperature\nchange compares with the observed heating of about 4°C at 200 m over the same\nperiod. In the next section we examine these unusual counter-gradient fluxes\nin more detail.\n65","2.5\nINTERPRETATION\nOur analysis of the temperature flux measured at each level showed\nperiods during which there was a net upward flux that was counter to the mean\ngradient. The first of these occurred shortly after 0100 MST, the second\nafter 0500, and the third after 0330 MST.\nIn the analysis of the w0 cospectrum, upward temperature flux was as-\nsociated with frequencies characteristic of internal waves. The spectral\ntechnique, however, has the limitation that it does not allow an identifica-\ntion of contributions to the positive flux from any isolated events. In\naddition, it is difficult to separate idiosyncracies in the time series (such\nas trends or steps) from waves after they have been Fourier-transformed.\nInspection of the time series of the product we often showed singular\nflux events where an excursion in one direction was followed by one of the\nopposite sign. Since the net contribution appeared to come from the imbalance\nbetween the two, which was not always evident from the actual time series of\nw0, a time series of the cumulative sum of w0 is more useful for identifying\nthe contribution of such individual events. Such time series, plus those of W\nand 0, are shown for each of three cases in Fig. 2.8. Case 1 shows the effect\nof a long-period wave combined with a step decrease in temperature associated\nwith the passage of the frontal interface. Case 2 shows a similar case with\nthe added complication of a slow drift in the mean vertical velocity. Case 3\nrepresents a strong disturbance in the inversion.\n2.5.1 Effects at the Frontal Interface, 0100-0140 MST\nA key feature in Case 1 is the sharp step in the contributions to the\nflux associated with the time when the inversion interface moves through the\n150-m tower level. As indicated by the vertical velocity trace, the passage\noccurs at the particular phase of the wave so that a negative velocity is\nassociated with a decrease in temperature. Were the interface to remain\nstationary and only perturbed by a wave, there would be no net flux. However,\nwith advection of the tilted frontal interface and its associated waves, this\nbecomes a singular event.\n66","Figure 2.8. -- Time series of the cumulative heat flux, W, and 0, for the time periods 0100-0140 MST,\n0440\nz=150m\n0420\nTime (MST)\n0420-0540 MST, and 0320-0440 MST, which correspond to the record displayed in Fig. 2.1.\n0400\n0340\nCase 3\n0320\n0\n-2\n-4\n-2\n-4\n8\n6\n4\n2\n0\n-2\n-4\n6\n4\n2\n0\n10\n8\n6\n4\n2\nz=150m\n0420 0440 0500 0520 0540\nTime (MST)\nCase 2\n6\n0\n-2\n-4\n4\n2\n1\n0\n-1\n-2\n8\n-2\n2\n-4\n6\n4\n2\n0\nz=100m\n0100 0110 0120 0130 0140\nMy\nN\nMy\nTime (MST)\nCase 1\n-10\n10\n0\n-5\n20\n15\n5\n-4\n-6\n4\n2\n0\n-2\n0\n-2\n-4\n6\n6\n4\n2\n67","2.5.2 Trends/Steps in the Data, 0420-0540 MST\nCase 2 shows a similar set of traces for the period after 0500 MST. In\nthis case there is a gradual increase in the net positive flux after an ini-\ntial step. An inspection of the time series of W, however, shows a downward\ntrend in W, of about 0.25 m/s over 40 min, after the abrupt drop in tempera-\nture at 0450 MST. This trend, together with an offset in temperature, con-\ntributes the net upward flux. Separate calculations, based on means taken\nbefore and after the step, show no positive flux.\n2.5.3 Strong Disturbances in the Inversion, 0320-0440 MST\nAnother anomalous case (Case 3) that we noted earlier occurred between\n0320 and 0400 MST. Again, calculating the cumulative contributions to the\ntemperature flux, we see that the contributions occur in discrete steps\n(Fig. 2.8). However, inspection of the time series shows no steps or trends\nin either the temperature or vertical velocity data at, for example, 150 m as\nshown in Fig. 2.8. Returning to Fig. 2.7, it should also be noted that the\nfluxes at 250 and 300 m due to the same disturbance are downward.\nSpectral analysis of W and 0 from 150 m reveals several peaks in the\nrange from 0.001 to 0.004 Hz (Fig. 2.9). Use of a linear frequency scale here\nallows identification of three distinct frequency components common to both W\nand 0. Of note is that the central peaks match well while the adjacent peaks\nare shifted in frequency. The cospectrum of W and 0, using standard BAO\nprograms, shows a broad peak in the same region. However, logarithmic smooth-\ning of the spectra by these programs does not allow detailed examination of\nthe cospectrum. Averaging of the cospectra and quadrature spectra over the\nentire band gives an average phase shift of -62°.\nNot shown in Fig. 2.9 is the spectrum of pressure fluctuations at the\nsurface. earlier In this presented spectrum. in\ncase the pressure spectrum closely followed the W\nFrom our analysis of microbarograph data, such as that\nFig. 2.4, we also found a spectrum of waves propagating opposite in direction\nto the inversion jet. To correlate surface pressure fluctuations with events\n68","4\nW\n3\n10\n2\n5\n1\n0\n0\n0\nn1\n-5\n0\n-4\n-3\n-2\n-1\n10\n0.004\n0.003\n10\n10\n10\n10\n0.002\n0.001\nn (Hz)\nn (Hz)\nFigure 2. 10. -Cospectrum of W and p\nFigure 2. .9. -- Spectra of W and 0 from\nfor the time period in Fig. 2.9.\n150 m in the frequency range 1-4 X\n-3\n10-\nHz on linear scales, for 0320-\n0440 MST, 27 March 1981.\nwithin the inversion, we computed the cospectrum of W at 150 m with surface\npressure at the base of the tower. This is shown in Fig. 2.10 where smoothing\nof the spectrum has again produced one central peak in the cospectrum. The\nquadrature spectrum in this case is small compared with the cospectrum. This\nin-phase relationship may reflect, in part, a phase shift introduced by the\nslower time constant of the pressure sensor compared with that of the sonic-\nderived W measurement. The finding that variations in W within the inversion\nare well correlated with surface pressure fluctuations remains valid. Similar\ncomparisons with temperature are not as clear except in those cases where the\ntemperature sensor happened to be near the top of the elevated inversion (see\nFig. 2.3) where temperature fluctuations are more pronounced.\nGiven the presence of these cospectral peaks, we used our filtering\nprogram to produce the time series of w and 0 in the frequency range shown in\nFig. 2.9. These are shown in Fig. 2.11 together with identifiable phase\nshifts at each cycle of the wave at heights of 300 and 200 m. The fact that\nthe wave packet centered on the time of the primary disturbance is fairly\n69","300m\n=\n124°\n105°\n97°\n95°\n138°\n141°\n150°\n10°\nAvg.=120°\nW\n200m\n0\n90°\n80°\n64°\n59°\n56°\n54°\n56°\n38°\n48°\nAvg.= 61°\n0320\n0330\n0340\n0350\n0400\n0410\n0420\n0430\n0440\nTime (MST)\nFigure 2. 11. -- Narrow bandpasses time series of W and 0 at two fixed\nlevels for 0320-0440 MST on 27 March 1981 for the frequency range\nshown in Fig. 2.9. 0 indicates individual phase shifts.\nbroad is a consequence of the narrow bandwidth used to filter the original\ntime series.\nThe breadth of the packet, At in time, obeys the relation\nAf.At = 1, where Af is the bandwidth. For an ordinary internal wave, density\nvariations are 90° out of phase with vertical velocity (e.g., Turner, 1973).\nAs can be seen in comparison with Fig. 2.7, a shift of less than 90° leads to\na positive flux, whereas a shift greater than 90° produces a negative flux.\nAlthough these results suggest vertical transfer of heat at wavelike\ntime scales, the actual transfer of heat by a wave seems physically implausi-\nble. However, one can imagine a combination of circumstances involving large-\namplitude solitary waves acting in conjunction with horizontal temperature\nadvection to produce such isolated fluxes. Profiles at a single location are\nnot, however, sufficient to resolve such questions.\n70","2.6 CONCLUSIONS\nThe analysis of a nocturnal inversion disturbed by the passage of a shal- -\nlow cold front shows the dominance of low-frequency motions over turbulence in\nthe calculation of variances and covariances. In addition, the unsteadiness\nof the flow, with both wave and advective processes present, was shown to\ncreate difficulties in standard flux calculations. Such effects are critical\nin the interpretation of long-term flux data archived from towers such as BAO.\nThree cases were presented. The first one showed that anomalous fluxes\ncould occur with the fortuitous superposition of wave motion and the advection\nof the frontal interface through a fixed tower level. The second revealed a\npositive flux arising from a combination of a trend in W together with a sharp\ndecrease in temperature. The third, involving a large-amplitude disturbance\nin the inversion, proved more intriguing. Analysis of this third case moti-\nvated the introduction of two additional analysis techniques. The first\ninvolved forming a cumulative sum of the product of W and O. This showed that\nthe apparent wavelike contributions to the upward vertical temperature flux\nactually occurred in short bursts. Cospectral analysis of W and 0 in this\ncase showed equal and opposite phase shifts, away from the 90° expected of\ninternal wave motion, above and below the elevated inversion base. A second\ntechnique was introduced to relate events within the inversion to wave motion\nsensed by the surface microbarograph array. To accomplish this, we formed\ncospectra of tower variables such as W with the pressure time series at the\nbase of the tower. The cospectrum of W and p showed strong peaks at the same\nfrequencies as those in the spectra of W and 0, suggesting that the large-\namplitude disturbances in the inversion layer flow were wave related. A1-\nthough some of the tower data show that advection may be an important element\nin the interpretation of such low-frequency effects, sufficient data to re-\nsolve the problem do not exist.\n2.7 ACKNOWLEDGMENT\nWe are grateful for the additional map analysis and discussion provided\nby R. Zamora.\n71","2.8 REFERENCES\nAtkinson, B.W., 1981: Meso-scale Atmospheric Circulations. Academic Press,\nNew York, 495 pp.\nAxford, D.N., 1971: Spectral analysis of an aircraft observation of gravity\nwaves. Quart. J. Roy. Meteorol. Soc. 97, 313-321.\nCaughey, S.J., 1977: Boundary-layer turbulence spectra in stable conditions.\nBound.-Layer Meteorol., 11, 3-14.\nCaughey, S.J., and C.J. Readings, 1975: An observation of waves and turbu-\nlence in the Earth's boundary layer. Bound.-Layer Meteorol., 9, 279-286.\nCaughey, S.J., J.C. Wyngaard, and J.C. Kaimal, 1979: Turbulence in the evolv-\ning stable boundary layer. J. Atmos. Sci., 36, 1041-1052.\nEinaudi, F., and J.J. Finnigan, 1981: The interaction between an internal\ngravity wave and the planetary boundary layer. Part I: The linear\nanalysis. Quart. J. Roy. Meteorol. Soc., 107, 793-806.\nEmmanuel, C.B., B.R. Bean, L.G. McAllister, and J.R. Pollard, 1972: Obser-\nvations of Helmholtz waves in the lower atmosphere with an acoustic\nsounder. J. Atmos. Sci., 28, 886-892.\nGossard, E.E., J.H. Richter, and D. Atlas, 1970: Internal waves in atmosphere\nfrom high resolution radar measurements. J. Geophys. Res., , 75, 903-913.\nKaimal, J.C., 1973: Turbulence spectra, length scales, and structure param-\neters in the stable surface layer. Bound.-Layer Meteorol., 4, 289-309.\nKaimal, J.C., and J.E. Gaynor, 1983: The Boulder Atmospheric Observatory.\nJ. Clim. Appl. Meteorol. (accepted).\n72","Kaimal, J.C., R.A. Eversole, D.H. Lenshow, B.B. Stankov, P.H. Kahn, and J.A.\nBusinger, 1982: Spectral characteristics of the convective boundary\nlayer over uneven terrain. J. Atmos. Sci., 39, 1098-1114.\nKeliher, T.E., 1975: The occurrence of microbarograph-detected gravity waves\ncompared with the existence of dynamically unstable wind shear layers.\nJ. Geophys. Res., 80, 2967-2976.\nMerrill, J.T., 1977: Observational and theoretical study of shear instability\nin the airflow near the ground. J. Atmos. Sci., , 34, 911-921.\nNeff, W.D., 1980: An observational and numerical study of the atmospheric\nboundary layer overlying the east Antarctic ice sheet. Ph.D. Thesis,\nUniversity of Colorado, Boulder, Colo., 272 pp.\nSteward, R.W., 1969: Turbulence and waves in a stratified atmosphere. Radio\nSci., 4, 1269-1278.\nTurner, J.S., 1973: Buoyancy Effects in Fluids. Cambridge Univ. Press,\nCambridge, England, 367 pp.\nWyngaard, J.C., and O.R. Coté, 1972: Cospectra similarity in the atmospheric\nsurface layer. Quart. J. Roy. Meteorol. Soc., 98, 590-603.\nZhou Ming-yu, Lu Nai-ping, and Qu Shaohou, 1980: Applications of sodar data\nin weather analysis and local weather forecasting. Kexue Tongbao, 25,\n328-331.\n73","","3. A STUDY OF MULTIPLE STABLE LAYERS IN THE\nNOCTURNAL LOWER ATMOSPHERE\nLi Xing-sheng\nInstitute of Atmospheric Physics\nAcademia Sinica\nBeijing, China\nJ.E. Gaynor and J.C. Kaimal\nNOAA/ERL/Wave Propagation Laboratory\nBoulder, Colorado 80303\nABSTRACT. The structure of nocturnal inversions in the first 300 m\nof the atmosphere is analyzed using the observational data from the\nBoulder Atmospheric Observatory from March through June 1981. The\ntemperature profiles show more than one inversion layer 41% of the\ntime during the observational period. The vertical distributions of\nwind speed and moisture also show evidence of stratification during\nthese multiple-layer events. The relation between the radiative\ncooling rate in time and height, including moisture, and the vertical\nstructure of the multiple layers is calculated. The vertical dis-\ntribution of eddy kinetic energy and the turbulent vertical fluxes\nof heat and momentum are also calculated. Turbulent structure in\nthe elevated inversion layers is more complicated than that in the\nsingle-layer, stable nocturnal boundary layer. The total heat bud-\nget for a multiple-layer case is calculated, and turbulent cooling\nis found to be negligible relative to radiative cooling and to hori-\nzontal advection and/or horizontal divergence of heat flux.\n3.1 INTRODUCTION\nIn recent years, the study of the nocturnal stable boundary layer has\nreceived much attention. Many nocturnal boundary layer models have been\ndeveloped (Delage, 1974; Businger and Arya, 1974; Wyngaard, 1975; Blackadar,\n1976; Brost and Wyngaard, 1978; Andre et al., 1978; Zeman, 1979; Li et al. ,\n1982a), and many studies have been made of the parameterization of the noc-\nturnal boundary layer as well as of the basic vertical structure of the\nstably stratified atmospheric boundary layer (Rao and Snodgrass, 1979;\nSundararajan, 1979; Mahrt et al., 1979). However, observational studies of\nthe nocturnal lower atmosphere, of which the boundary layer is only a part,\nare rather incomplete. Its description has been restricted because of the\n75","lack of detailed turbulence and mean field measurements with height. The\nBoulder Atmospheric Observatory (BAO) tower provides such information.\nAn experiment directed toward the study of the nocturnal lower atmo-\nsphere was conducted at BAO between March and June 1981. Measurements taken\nduring this intense observational period included, in addition to standard\ndata from the 300-m BAO tower, profiles of wind, temperature, and humidity\nusing instruments attached to a movable carriage on the tower, and facsimile\nrecords of echo intensities from an acoustic sounder located near the tower.\nThe sensors on the tower and data acquisition procedures have been dis-\ncussed by Kaimal and Gaynor (1983). The carriage sensors were mounted on two\nhorizontal booms separated 10.3 m vertically. The lower boom supported a\nsonic anemometer measuring the mean and fluctuating components of the wind\nalong three orthogonal axes, a quartz thermometer measuring mean temperature,\na platinum-wire thermometer measuring temperature fluctuations, a Lyman-alpha\nhygrometer measuring fluctuations in specific humidity, and a propeller vane\nanemometer measuring mean wind speed and direction. For gradient measure-\nments across the 10.3-m vertical spacing, the upper boom supported another\nquartz thermometer, Lyman-alpha hygrometer, and propeller-vane anemometer.\nThe movable carriage ascends the full tower length in about 8.8 min and\ndescends in about 8.5 min.\n3.2 FORMATION OF MULTIPLE LAYERS\nMeasurements made with the acoustic sounder and with sensors mounted on\nthe tower indicate that the structure of the nocturnal lower atmosphere is\nvery complex. The inversion structure is often correlated with isolated echo\nlayers in the facsimile records. The statistical results in Fig. 3.1 show\nthat of the 87 nights examined (25 March-22 June 1981), 35 had at least one\noccurrence of multiple layers. The probability of occurrence of multiple\nlayers on any one night, therefore seems reasonably high. The starting time\nof such events is variable, but Fig. 3.1 shows a high concentration (86% of\nthe cases) between 2000 and 0200 mountain standard time (MST). The duration\nof a multiple-layer event varies from 3 to 12 h.\n76","12\n10\n8\n6\n4\nFigure 3. 1. . -- Frequency distribution\nof times when multiple inversion\n2\nlayers initially occurred between\n25 March and 22 June 1981.\n0\n06\n24\n02\n04\n18\n20\n22\nTime (MST)\nThe formation of multiple layers is almost always associated with cer- -\ntain atmospheric conditions and/or local topographical features. Following\nare some of the conditions that favor such formation:\n(1) Subsidence and advection associated with the passage of a cold\nfront in uniform and complex terrain (Neff, 1980).\n(2) Cold-air outflows from local thunderstorms (Goff, 1976).\n(3) Drainage winds in the vicinity of mountain ranges, as observed\noften at BAO (Hootman and Blumen, 1981).\nLi et al. (1982b) have discussed a multiple-layer event caused by advec-\ntion. This study will focus on a more typical event at BAO, one caused by\ndrainage flow from the Rocky Mountains 60 km to the west.\n3.3 A CASE STUDY\nThe event under consideration occurred between 0100 and 0530 MST,\n24 April 1981. One carriage profile (either ascending or descending) was\n77","0100\nFigure 3.2. -- The acoustic record of multiple inversion layers on 24 April 1981.\n0200\n0300\nTime (MST)\n0400\n0500\n350\n300\n250\n200\n150\n100\n50\n0","270°\n270°\n300\n1509\n270°\n210° 270°\n270°\n150.°\n180°\n270°\n270°\n180°\n250\n300°\n210°\n200\n300°\nO\n150\nOC\n270°\n270°\n300°\n240°\n240°\n100\n300°\n270°\n240°\n50\n270°\n120°\n240°\n210°\n180°\n180°\n60\n120°\n180°\n0\n120°\n0430\n0500\n0530\n0400\n0300\n0330\n0200\n0230\n0130\nTime (MST)\nFigure 3.3. The time-height cross section of the wind direction on\n24 April 1981. Isolines are at 30° O intervals.\nrecorded every 10 min during that period. The observed time series on all the\ndata channels were block-averaged over consecutive 10-s blocks. This averag-\ning corresponds to a 6-m spatial averaging in the vertical and is clearly more\nuseful for detection of elevated layers than the 50-m spacing between the\nfixed levels on the tower. These smoothed vertical profiles from the carriage\nsensors were used to construct time-height cross sections of the measured\nvariables.\nThe acoustic record (Fig. 3.2) indicates that multiple-layer structures\nstarted to form between 0100 and 0200 MST and between 0230 and 0330 MST. The\ntime-height cross section of wind direction between 0120 and 0530 MST on\n24 April 1981 is shown in Fig. 3.3. Before 0230 MST, the wind direction was\nessentially southerly above 150 m, northwesterly between 150 and 50 m, and\nquite variable below 50 m. After 0230 MST, a significant change in wind\ndirection occurred within the tower height. The wind direction became gen-\nerally northwesterly above 50 m and southeasterly below 50 m.\n79","12.4\n300\n12.2\n12.4\n12.2\n12.0\n12.0\n12.2\n12.4 12.2 12.4\n12.6\n12.0\n12.4 12.6\n12.0\n12.8\n250\n12.8\n11.4\n12.4\n11.6\n12.6\n13.0\n13.2\n11.2\n12.6\n12.6\n200\n12.2\n12.4\n13.2\n12.4\n11.8\n12.6\n11.8\n13.4\n150\n13.2\n11.2\n12.2\n13.6\n12.0\n13.8\n12.0\n100\n11.8\n11.8\n11.6\n11.6\n11.4\n11.2\n50\n10.5\n11.0\n8.5\n10.5\n6.5\n0\n9.5\n9.5\n9.5\n4.5\n5.5 6.5\n8.5\n6.5\n5.5\n7.5\n0130\n0200\n0230\n0300\n0330\n0400\n0430\n0500\n0530\nTime (MST)\nFigure 3.4. - The time-height cross section of the temperature on 24 April\n1981. Solid lines are isotherms at 0.2°C O intervals; thin dashed lines\nare at 0.5°C o C intervals; thick dashed lines indicate tops of the inversions.\nThe time-height cross section of temperature for the same case is shown\nin Fig. 3.4. The thick dashed lines indicate tops of the inversions Tz at >\n0\nbelow the thick dashed lines) The temperature field shows multiple inver-\nsion layers between 0120 and 0200 MST, in agreement with the multiple echo\nlayers on the acoustic record (Fig. 3.2). After 0230 MST, there was an\ninflow of cold air within the middle and lower tower levels. The inversion\nthen divided into two parts, the lower part continuing to exist as a radia-\ntion inversion at 50 m, the upper part lifting to 250 m by 0300 MST. The\ntemperature within the upper inversion increased about 1°C with height in a\n20- to 30-m thickness. These layers correspond to strong echo layers in the\nacoustic record of Fig. 3.2. The thermal stratification between the two\ninversions was nearly isothermal. After 0300 MST an elevated inversion seems\nto hover around 150 and 200 m as seen in Fig. 3.4. Although Fig. 3.4 shows\nonly two inversion layers, the acoustic sounder record in Fig. 3.2 displays\nat least four echo layers above 50 m. The multiple echo layers may represent\nthe top and bottom of each inversion layer, or they may represent a smaller\nscale structure in the temperature profile not easily resolvable in Fig. 3.4.\n80","3.0 2.5 1.5\n3.0\n4.0\n4.0\n2.5\n2.5.3.0\n3.5 2.0. 2.5\n(2.0\n2.0\n300\n2.0\n3.0\n3.0\n4.5\n2.5\n5.0\n1.5\n5.5\n2.0\n3.5\n250\n3.0\n3.0\n1.0\n3.5\n3.5\n4.0\n1.5\n6.0\n3.5\n200\n1.0\n5.5\n5.5\n6.5\n3.0\n0.5\n3.0\n6.0\n1.5\n3.5\n5.5\n150\n4.0\n1.0\n4.0\n5.0\n3.5\n1.0\n3.0\n100\n6.0\n2.5\n4.0\n2.0\n1.0\n3.5\n2.5\n50\n2.5\n3.0\n2.0\n3.0\n3.0\n1.5\n2.0\n1.0\n1.0\n1.0\n0\n1.0\n1.5\n2.0\n2.0\n1.5\n1.0 0.5\n0530\n0500\n0400\n0430\n0330\n0230\n0300\n0200\n0130\nTime (MST)\nFigure 3.5. -- The time-height cross section of the wind speed on 24 April\n1981. Solid lines are isotacks at 0.5-m/s intervals; thick dashed lines\nindicate maximum values of wind speed, i.e., the axes of the low-level\njets.\nThe time-height cross section of wind speed for this case is shown in\nFig. 3.5. The wind field shows multiple wind speed maxima from 0120 to 0200\nand 0230 to 0330 MST. Their height levels nearly coincide with the stratifi-\ncations in temperature (Fig. 3.4) . After 0330 MST, the low - -level jet cen-\ntered at mid-tower level expanded to a broad maximum as values of wind speed\nnear the surface diminished gradually. After 0500 MST, there were two wind\nspeed maximas at 100 and 200 m, at nearly the same levels as the inversions.\nThese observations are in agreement with the statistical results of Li et al.\n(1982b), which indicate a strong relationship between multiple wind speed\nmaxima and the existence of multiple inversion layers. The heights of the\nwind speed maxima roughly coincide with the tops of the inversions.\nFigure 3.6 shows the time-height cross section of specific humidity for\nthe same time as the temperature field in Fig. 3.5. It is worth noting that\nthe distribution of the isolines of specific humidity is similar to the\ndistribution of isotherms (Fig. 3.4) . After 0230 MST, the moisture field\nalso developed multiple layers. However, the height level of the maximum\n81","300\n3.7\n3.6\n3.6\n3.5\n3.7 3.8 3.7\n3.7\n3.8\n3.9 4.0 4.0 3.9\n3.8\n3.6\n3.8\n3.8\n250\n4.1\n3.9\n4.2\n4.3\n200\n4.4\n3.9\n4.1\n150\n4.0\n4.2\n4.3\n4.5\n100\n4.0\n4.5\n4.3\n4.1\n4.2\n50\n4.7\n4.4\n4.8\n4.5\n4.8\n4.7\n4.5\n4.4\n0\n4.6\n4.6\n4.6\n4.6\n4.6\n4.5\n4.7\n0130\n0200\n0230\n0300\n0330\n0400\n0430\n0500\n0530\nTime (MST)\nFigure 3.6. -- The time-height cross section of humidity on 24 April 1981.\nSolid lines are isolines of specific humidity at 0.1-g/kg intervals;\nthick dashed lines indicate maximum values of specific humidity.\nvalue of moisture is below the top of the inversion. The locations of the\nstrong moisture gradients, on the other hand, very nearly correspond to those\nof the temperature gradients. Radiative-transfer relations show that the\ngreater the moisture content in the inversion layer near the surface, the\nstronger the radiative cooling at the top of the inversion. Cooling rates in\nthe presence of moisture will be discussed in Sec. 3.5.\n3.4 EDDY KINETIC ENERGY AND THE DISTRIBUTION OF TURBULENT FLUXES OF\nHEAT AND MOMENTUM\nThe vertical distributions of Richardson number (Ri), horizontal eddy\nkinetic energy (e h , variances (w 2 and 0 2 ), , and fluxes\n(wu, WV, and w0) for\nthe 10-min period from 0250 to 0300 MST are calculated using the observa-\ntional data at eight fixed levels on the BAO tower. [Overbars indicate time\naverage (10 min); u, V, and W are the longitudinal, lateral, and vertical\nfluctuations of the wind; and 0 is the temperature fluctuation.]\n82","The turbulent fluxes and energy were calculated by integrating spectral\nand cospectral estimates in the frequency band 0.02-5 Hz. The results are\npresented in Fig. 3.7 and in Table 3.1. Ri and en were calculated using the\nfollowing relations:\nand\n2\nwhere g is the acceleration of gravity, 10 is the mean virtual temperature,\nU and V are the mean horizontal wind components along the E-W and N-S axes,\nand A denotes an increment over a finite height interval Dz.\nThe Richardson number achieves its maximum value at approximately the\njet level and has another maximum near the center of the inversion (Fig. 3.7).\nTherefore, the Richardson number also has a two-layer structure. Its struc-\nture is in agreement with the acoustic record of Fig. 3.2. The observations\nof Mahrt et al. (1979), based on aircraft soundings at Buckley Field, Colo.,\nin September 1975, also show that Richardson numbers reach a maximum and\nturbulence levels a minimum near the core of a jet. The profiles of and\nRi in Fig. 3.7 show the same behavior. Between 50 and 150 m the values of Ri\nwe\n82\n2\nWV\nen\nW\nwu\n300\nRi\n200\n100\n0\n8\n16\n24\n-8\n0\n-16\n8\n12\n0\n4\n0\n2\n4\nWA (10 -4 °Cm/s)\n02(10-3°C2)\nwu (10-4 -4 m²/s2\nen m²/s2)\nFigure 3. 7. -- The distributions of Ri, 0 w2, en' WV, we, and wu with\nheight at 0250 MST on 24 April 1981.\n83","(10-4 °C m/s)\n-6.8\n3.2\n-0.6\n-12.0\n-12.0\n-10.0\n-1.6\n-5.3\nwe\nTable 3.1. -- Profiles of the turbulent fluxes and energy in the frequency band 0.02-5 Hz\n(10-2 -4 m2/s2 )\n-1.2\n9.5\n1.6\n-1.4\n24.0\n6.0\n-14.2\n-17.0\n-\nWV\n(10-4 -4 m2/s2\n-2.7\n-1.6\n-1.3\n8.5\n-9.6\n23.0\n7.0\n1.0\n-\nwu\n(10-3 3 m2/s2)\n1.3\n2.2\n2.1\n8.2\n2.8\n1.5\n2.4\n3.9\nw2\nW\n(10-3 °C2)\n23.0\n4.6\n6.4\n2.5\n1.8\n1.6\n0.4\n0.6\n02\n(10-3 m2/s2,\n6.2\n11.7\n8.2\n15.1\n5.9\n4.4\n4.6\n7.4\nen\n10\n(m)\n22\n50\n100\n150\n200\n250\n300\nZ","are close to its subcritical value (Ri = 0.25); between 250 and 300 m,\nRi < 0.25. These regions have large en values. That the temperature field\nis uniform between the two inversion layers, combined with the low Ri and\nhigh en values between the layers, indicates that this region is well mixed\nin the vertical. The distribution of W w2 is similar to that of en Andre\net al. (1978) gave the following interpretation for the vertical structure of\nnocturnal turbulence: the horizontal components u and 2 V are generated by\n2\nwind shear, du/dz and dv/dz; their transformation into the vertical component\nw2 by pressure fluctuations would tend to give rise to a strong negative heat\nflux.\nThe results in Fig. 3.7 indicate that wu is maximum and positive near\nthe jet core. Therefore, the vertical flux of longitudinal momentum is\nupward. This profile is consistent with the upward movement of the jet core\nbetween 0250 and 0300 MST indicated in the wind speed cross section in\nFig. 3.5. However, WV is negative above the jet core and positive below, so\nthat the vertical transport of the lateral momentum is into the jet. Fig-\nure 3.7 shows that there is a general tendency for the temperature variance\n2\n0 to decrease slightly with height above about 50 m.\n3.5 TEMPERATURE BUDGET PROFILE\nTo help understand the maintenance or breakup of multiple layers in the\nstable nocturnal atmosphere, it is important to study the different factors\ncontributing to variations in the temperature profile. The factors include\nradiative flux divergence, turbulence heat flux divergence, advection, and\nsubsidence.\nThe profile of nocturnal radiative cooling with height is complex when\ntemperature and humidity vary with height. To evaluate this, the Brooks\n(1950) tabular method is used to calculate nocturnal radiative cooling rates\nwith the BAO profile data. The derived formula is the following:\n85","dw\ndw\n(3.1)\nwhere q is the specific humidity; K the pressure correction factor; CP the\nspecific heat of air at constant pressure; W the optical thickness of the\nradiating substance; O the Stefan's constant; IA the absolute temperature; E\nthe emissivity; W1 the upper boundary of the radiating substance; and the\narrows indicate the direction from which radiation reaches the reference\nlevel. Subscript E denotes the value at the top of the radiating layer.\nIf the integrals are simplified by considering (OI ) / aw constant within\neach integration interval and written approximately as A(OI4)/AW (3.1) can be\nwritten as\n#:\n(3.2)\nwhere is computed at 700 mb. The summation over n is\nfor layers above the reference level into which the atmosphere has been di-\nvided, and the summation over m is for layers below the reference level.\nA(OI 4) is zero at the ground surface.\nThe emissivity curve delaw was taken from Brooks (1950, Table c), which\nignores CO 2 absorption (assumed negligible). Brooks calculates the water\nvapor absorption from laboratory measurements using isothermal radiation and\nsmall path lengths. The vertical integration of (3.2) begins at the 2-m\nlevel.\nThe variation of radiative cooling rate with height has been calculated\nusing (3.2). . Figure 3.8 shows the results computed from the carriage profile\nthat began at 0250 MST. Because the surface inversion below 22 m is very\nstrong and moist, the radiative cooling rate is very strong (-0.47°C/10 min)\nat the top of this layer. This distribution of temperature and humidity also\n86","Total Cooling (°C/10 min)\nResidual Cooling (C/10 min)\nq(g/kg)\n0\n0.4\n0.8\n-0.8\n- 0.4\n1.6\n-1.2\n4.6\n5.0\n3.4\n3.8\n4.2\n-\n-\n300\nT\n200\n100\n0\n-0.2\n0\n0.2\n0.4\n14\n-0.4\n10\n12\n6\n8\n4\nRadiative Cooling (C/10 min)\nT(°))\nTurbulent Cooling (°C/10 min)\nFigure 3.8. -- Profiles of specific humidity (q), temperature (T), total\ncooling (---), , radiative cooling ( , residual cooling (....),\nand\n) for the case on 24 April 1981.\n(\nturbulent cooling\naccounts for the strong warming in the layer nearest the surface. In addi-\ntion, the cooling rate is larger at the levels of the temperature maxima.\nThe cooling rate maximum at the top of the surface inversion will cause the\ninversion strength below to weaken with time, and the temperature gradient\nabove to strengthen. The net effect of such cooling may be to maintain any\nelevated inversion present (Staley, 1965).\nWe can study the relative importance of radiative cooling in the tempera-\nture budget profile for this case using the relation\nAT\n= (Radiative cooling) Awo + (Residual cooling)\nA\n(3.3)\n.\nAt\nAI A/At is the total temperature change during the 10-min interval; the radia-\ntive cooling is calculated from (3.2) ; the turbulent cooling rates -Aw0/Dz\nare calculated from the results in Table 3.1; and the residual term contains\nthe horizontal advection of temperature and subsidence. The divergence of the\nhorizontal heat flux is presumed to be small.\nTable 3.2 presents the results for height intervals between successive\ntower levels, and Fig. 3.8 the plots of profiles of all the terms in the heat\n87","Table 3. 2. -Temperature budget (°C/10 min) for the\ncase of 24 April 1981\nTurbulent\ncooling\nAT\n-AwO/Az\nRi\nAt\nZ\nRadiative\nResidual\n10-1)\n(m)\n(interpolated)\ncooling\ncooling\n16\n0.98\n-1.10\n-0.50\n-0.17\n-0.88\n36\n2.24\n-0.28\n+0.08\n-0.35\n+0.06\n75\n0.38\n+0.70\n+0.14\n-0.08\n+0.77\n125\n0.49\n+0.02\n0\n-0.06\n+0.08\n175\n4.37\n0\n-0.02\n0\n0\n225\n0.36\n-1.43\n-0.10\n-0.09\n-1.33\n275\n0.15\n-0.60\n+0.04\n-0.05\n-0.55\nbudget. We see that the terms for radiative and turbulent cooling are much\nsmaller than the term for residual cooling at all levels. The horizontal\nadvection dominates the residual term, which is very likely a result of the\nrapid change in the height of the inversion seen in Fig. 3.4.\nThe relatively small turbulent cooling rates presented in Table 3.2 agree\nwith the model results of Andre et al. (1978) and Yamada and Mellor (1975).\nOther researchers have calculated layer-averaged temperature budgets in the\nnocturnal boundary layer for single-layer cases (Garratt and Brost, 1981;\nAndre and Mahrt, 1982). However, their calculations were performed in the\nnocturnal boundary layer, which is only a part of the nocturnal lower atmo-\nsphere. Within the boundary layer the temperature fluxes were more than an\norder of magnitude larger than the case presented here. Therefore, it is\ndifficult to compare their results with those in Table 3.2.\nThe importance of advection and/or horizontal divergence of heat flux is\nclearly shown by the large values of the residual cooling listed in Table 3.2.\nThe two-dimensional models cited above certainly do not apply to this fairly\ntypical case studied at the BAO, which is located in rolling terrain. Rea-\nlistic predictions for stable boundary layers must come from detailed three-\ndimensional models.\n88","3.6 CONCLUSIONS\nThe above analysis indicates that the development and the structure of\nthe stable nocturnal lower atmosphere are fairly complicated. Multiple\nlayers are observed over both uniform and complex terrain and are most readily\napparent in acoustic sounder records. They are also apparent in plots of the\nwind field, the temperature field, and the moisture field. Atmospheric and\ntopographical factors contributing to such layering are many, but horizontal\nadvection is involved in some way in their formation.\nBecause the Richardson number achieves its maximum value near the center\nof a low-level jet, there appears to be a rough correspondence between the\nprofiles of Richardson number and mean wind speed. In the region between its\nmaxima, Ri tends to be subcritical and to be maximum. In the same region,\nthe temperature field is fairly uniform, suggesting the region between two\nelevated layers is well mixed. The turbulent vertical fluxes of heat within\nelevated layers and near the ground surface are negative, but the vertical\ndistribution of momentum flux is very complex.\nThe vertical distribution of radiative cooling rate in a nocturnal lower\natmosphere shows warming nearest the surface and cooling aloft. Cooling was\nmost pronounced at the top of the strong inversion layer near the surface\n(z 22 22 m) . The total heat budget for the multiple-layer case showed that the\nturbulent cooling rates were negligible compared with the radiative cooling,\nand both were much smaller than horizontal advection and/or subsidence.\n3.7 ACKNOWLEDGMENTS\nThe authors appreciate the helpful suggestions received from W.D. Neff,\nD. H. Lenschow, and R.A. Brost in the preparation of this paper.\n89","3.8 REFERENCES\nAndre, J.C., G. DeMoor, P. Lacarrere, G. Therry, and R. Du Vachat, 1978:\nModeling the 24-hour evolution of the mean and turbulent structures\nof the planetary boundary layer. J. Atmos. Sci., 35, 1861-1883.\nAndré, J.C., and L. Mahrt, 1982: The nocturnal surface inversion and influ-\nence of clear-air radiative cooling. J. Atmos. Sci., 39, 864-878.\nBlackadar, A.K., 1976: Modeling the nocturnal boundary layer. Preprints,\n3rd Symposium Atmospheric Turbulence, Diffusion and Air Quality, Raleigh,\nN.C., 19-22 October 1976, American Meteorological Society, Boston,\nMass., 46-49.\nBrooks, D.L., 1950: A tabular method for the computation of temperature\nchange by infrared radiation in the free atmospheres. J. Meteorol., 7,\n313-321.\nBrost, R.A., and J.C. Wyngaard, 1978: A model study of the stably stratified\nplanetary boundary layer. J. Atmos. Sci., 35, 1427-1440.\nBusinger, J., and S.P.S. Arya, 1974: Heights of the mixed layer in the\nstably stratified planetary boundary layer. Advances in Geophysics,\n18A, Academic Press, New York, 73-92.\nDelage, Y., 1974: A numerical study of the nocturnal atmospheric boundary\nlayer. Quart. J. Roy. Meteorol. Soc., 100, 351-364.\nGarratt, J.R., and R.A. Brost, 1981: Long-wave radiation fluxes and the\nnocturnal boundary layer. J. Atmos. Sci., 38, 2730-2746.\nGoff, R.C., 1976: Vertical structure of thunderstorm outflows. Mon. Wea.\nRev., 104, 1429-1440.\n90","Hootman, B., and W. Blumen, 1981: Observations of nighttime drainage flows\nin Boulder, Colorado during 1980. Preprints, 2nd Conference on Mountain\nMeteorology, Steamboat Springs, Colo., 9-12 November 1981, American\nMeteorological Society, Boston, Mass., 222-224.\nKaimal, J.C., and J.E. Gaynor, 1983: The Boulder Atmospheric Observatory.\nJ. Clim. Appl. Meteorol. (accepted).\nLi Xing-sheng, Liu Lin-gin, and Zheng Ai-ying, 1982a: Numerical studies of\nthe development of the nocturnal boundary layer. Sci. Atmos. Sin.\n(accepted).\nLi Xing-sheng, Zhu Cui-juan, Liu Lin-gin, Zheng Ai-ying, and Zhou Ming-yu,\n1982b: A study on multiple layers wind speed profile in the planetary\nboundary layer. Sci. Atmos. Sin. (accepted).\nMahrt, L., R.C. Heald, D.H. Lenschow, B.B. Stankov, 1979: An observational\nstudy of the structure of the nocturnal boundary layer. Bound.-Layer\nMeteorol., 17, 247-264.\nNeff, W.D., 1980: An observational and numerical study of the atmospheric\nboundary layer overlying the east Antarctic ice sheet. Ph.D. Thesis,\nUniversity of Colorado, Boulder, Colo., 272 pp.\nRao, K.S. and H.F. Snodgrass, 1979: Some parameterizations of the nocturnal\nboundary layer. Bound.-Layer Meteorol., 17, 15-28.\nStaley, D.O., 1965: Radiative cooling in the vicinity of inversions and the\ntropopause. Quart. J. Roy. Meteorol. Soc., 91, 282-301.\nSundararajan, A., , 1979: Some aspects of the structure of the stably strati-\nfied atmospheric boundary layer. Bound.-Layer Meteorol., 17, 133-139.\nWyngaard, J.C., 1975: Modeling the planetary boundary layer--extension to\nthe stable case. Bound.-Layer Meteorol., 9, 441-460.\n91","Yamada, T., and G. Mellor, 1975: A simulation of the Wangara atmospheric\nboundary layer data. J. Atmos. Sci., , 32, 2309-2329.\nZeman, 0., 1979: Parameterization of the dynamics of stable boundary layers\nand nocturnal jets. J. Atmos. Sci., 36, 792-804.\n92","4. A METHOD FOR MEASURING THE PHASE SPEED AND AZIMUTH OF GRAVITY\nWAVES IN THE BOUNDARY LAYER USING AN OPTICAL TRIANGLE\nLi Xing-sheng and Lu Nai-ping\nInstitute of Atmospheric Physics\nAcademia Sinica\nBeijing, China\nJ.E. Gaynor and J.C. Kaimal\nNOAA/ERL/Wave Propagation Laboratory\nBoulder, Colorado 80303\nABSTRACT. The phase speed and azimuth of gravity waves in the\nplanetary boundary layer are calculated using an optical tri-\nangle. The results are compared with those calculated using\na\nmicrobarograph array. The two methods are in essential agree-\nment.\n4.1 INTRODUCTION\nGravity waves, which are often synoptically generated, are frequently\nobserved in the nocturnal stable boundary layer. Because of the rapid develop-\nment of atmospheric remote sensing techniques in recent years, the temporal\nand spatial structure of gravity waves can be displayed and studied. Gossard\nand Richter (1970), , Emmanuel et al. (1972), and Hooke et al. (1973), , among\nothers, have used sodars and FM-CW radars to monitor the physical characteris-\ntics of gravity waves and to study the interaction between the waves and\nturbulence. These authors have also studied the relationship between internal\ngravity waves and synoptic and mesoscale phenomena. The studies have con-\nsidered the causes and the effects of the waves on mesoscale activity.\nThe phase speed and the azimuth of gravity waves are very important to help\nunderstand their source. In these studies, the phase speed and the azimuth of\ngravity waves generally were measured using the time series data of three or\nmore microbarographs. In addition, Gossard and Munk (1954) used single-point\ndata for pressure and wind speed to compute the phase speed. Recently, Eymard\nand Weill (1979) have measured the phase speed and the azimuth of gravity\nwaves (along with other characteristics) using a three-antenna Doppler sodar.\n93","The following describes the calculations of some basic characteristics of\nthe gravity waves utilizing an optical triangular array surrounding the 300-m\nmeteorological tower at the Boulder Atmospheric Observatory (BAO) in Colorado.\nThe wind speed measured by the optical systems is an average along the path\n(Lawrence et al., 1972). Therefore, high frequencies are automatically fil-\ntered. Because gravity waves are low-frequency phenomena by atmospheric\nboundary layer standards, the low pass filtering of the optical sensors may\nimprove phase speed and direction estimations when compared with an array of\nmicrobarograph sensors. Measuring the horizontal divergence of the wind speed\nwith an optical triangle (Kjelaas and Ochs, 1974; Tsay et al., , 1980) has\nproved effective. With this information in mind, measuring the phase speed\nand azimuth of gravity waves with optical methods appears to be a reasonable\napproach.\n4.2 MEASUREMENTS AND COMPUTATIONS\nThree optical systems have been operating at BAO. The optical paths form\nan equilateral triangle, 450 m on a side (Fig. 4.1). Because the optical\nsensors measure the mean wind perpendicular to each path and positive outward\nfrom the triangle center (Lawrence et al., 1972), the sum of the readings of\nall three paths represents the average horizontal wind speed out from the\ntriangle. The individual wind readings for each optical path, smoothed with a\n10-s running average along with the scalar sum of all three, were recorded on\ndigital tape. For the three optical systems, the horizontal area-average wind\nspeed and the wind direction can be expressed as\n(4.1)\nand\n180° + 4°,\n(4.2)\nwhere and VNV are the wind components oriented to the east and north respec-\ntively, vsw), and SE + vsw). VN' SE' and V SW\nare the wind components, transverse to the north, southeast, and southwest\ntriangle sides repectively. In Eq. (4.2), adding 180° makes the northerly winds\n94","X\nx\nTower\nv'i\naj\nX\nVi\nFigure 4. 1. . -- -Schematic of the optical triangle, the three microbarographs\nused in this analysis (x), , and the BAO tower (.). V1 = wind component\ntransverse to the triangle sides; = actual wind speed for each tri-\nangle side; ai = angle between V1 and V-1.\nrepresent a 0° direction, and the 4° is an adjustment for the difference\nbetween the north path of the optical triangle and the true east-west direc-\ntion. The wind directions 0 correspond to the standard meteorological defi-\nnition.\nThe theoretical wind-weighting function for a circular transmitter and\nreceiver with circular apertures, 3.0 and 1.95 Fresnel zones in diameter,\nrespectively, has its peak at the center of the path as shown in Fig. 4.2 (Ochs\net al., 1976). The curve of the wind-weighting function also shows the mini-\nmum value near the transmitter and receiver along the optical path. There-\nfore, the average wind speed measured by optical systems is strongly weighted\ntoward the center of the path, and it is assumed that a new triangle can be\nconstructed by joining the center points of each optical path.\nFor each optical path, the actual wind speed at the center point of the\npath (Fig. 4.1) can be written\n95","1\n0.5\nTheory\n0\n0\n0.5\n1\nT\nR\nWind weighting function\nFigure 4.2. -- Experimental wind-weighting function obtained by comparison\nof the optical measurement over a 500-m path with five pairs of anemom-\neters arranged to measure the horizontal cross-wind component (from\nOchs et al. , 1976). T = transmitter; R = receiver.\n(4.3)\n,\nwhere Vi is the wind component transverse to the triangle sides, is the\nactual wind speed for each triangle side, the subscript i (= 1,2,3) represents\nthe index of each side of the optical triangle, and a is the angle between\nthe average wind direction and the wind component transverse to each triangle\nside. Because the average wind speed and wind direction change with time, the\nangle a is also time variable. The angles made with each side of the optical\ntriangle may be instantaneously different. Therefore, with gravity wave\npropagation, the phase speed and direction appear as a variation in the time\nseries of the actual wind speed for the three sides. The propagation of a\nwave across the three separate optical systems can be distinguished by the\ntime delays between the three time series of wind speeds for pairs of optical\nstations. We can then obtain the phase difference between the pairs of\noptical stations for each frequency.\nIf the phase speed and the propagation direction of gravity waves are\nrepresented as C and Y respectively, then\n96","N\n1\n3\nY\nDirection of\nWave train\ngravity wave\n2\nFigure 4. 3. -- -Schematic of the geometry to determine the time delay with\nthe three separate optical systems. Y = propagation direction of gravity\nwaves; 0 = azimuth; l = length between the central points of the sides\nof the optical triangle.\n(4.4)\nCOSY = C 12/l\nand\n(4.5)\nwhere l is the length between the central points of the sides of the optical\ntriangle and t 12 and 13 are the time delays from vertex 1 to vertex 2 and to\nvertex 3 respectively (Fig. 4.3). . The corners of the triangle in Fig. 4.3 are\nat the center points of each optical path.\nFrom Eqs. (4.4) and (4.5), C and Y can then be written\n1 + tan 2 Y\n(4.6)\nc = (l/t12)\nV\nY r = tan -1 =\n(4.7)\n97","with\n0 = 210°-Y\n(4.8)\nwhere 0 is the azimuth.\nTo calculate C and 0, we must derive the time delays 12 and 11 If\nthe time series of the wind speed is vi(t) for vertex 1 and V2(t) for vertex\n2, the power spectrum is obtained by Fourier-transforming the auto-covariance\nfunction, and the cross spectrum is obtained by transforming the cross-covari-\nance function between the two time series (Gossard and Hooke, 1975; Brigham,\n1974). Therefore, the form of the cross-power spectrum for V1(t), V2(t) can\nbe written as\n+ dt dt\n(4.9)\n,\nwhere T is the time delay between time series, and W = 2Tf, with f the fre-\nquency.\nIft+t,(4.9)becomes\nH(w) dt\n(4.10)\n,\nwhere\ndo\n(4.11)\nFinally, from Eq. (4.10), the form of the cross-power spectrum can be\nwritten as\nE(w)=H(w)[R(w) +iI(w)],\n(4.12)\n,\nwhere\ndt\n(4.13)\nand\nsin(wt) dt\n(4.14)\n.\n98","The phase relationship between frequencies for two cross-spectral analyzed\ntime series is given by\n(4.15)\ntan WT = [(w)/R(w)\n,\nwhere T = 12' in this case. The time delay t 13 can be similarly calculated.\nThe phase speed and direction of the gravity wave can be obtained by Fourier-\ntransforming the auto-covariance function (from the time series of the wind\nspeed) using Eqs. (4.12), (4.13), and (4.14) and solving Eqs. (4.6), (4.8),\nand (4.15).\n4.3 RESULTS\nFour gravity wave cases are presented. The first, between 0000 and\n0700 MST on 13 November 1981 was embedded in strong stable stratification at\nthe lower levels. The gravity wave train is shown in the acoustic record pre-\nsented in Fig. 4.4. The phase speed and azimuth of the gravity waves have\nbeen calculated from 0210 to 0330 and from 0530 to 0650 MST. We first discuss\nthe gravity wave occurring between 0210 and 0330 MST. The optical wind sensor\nwas operating with a five-microbarograph array near the BAO tower. The three\nmicrobarographs closest to the tower were used for the analysis. Their loca-\ntions relative to the optical triangle are presented in Fig. 4.1. The time\nseries of the three microbarographs are shown in Fig. 4.5a. The coherence of\nthe traces is quite high. The time series of the three optical systems are\nshown in Fig. 4.5b. Similarly, their coherence is very good. The mean spec-\ntrum of the three optical wind measurements is presented in Fig. 4.6a. The\nmajor peak is at a 34.1-min period. Two minor peaks are at 19- and 14.2-min\nperiods. The phase speed and azimuth are 14.4 m/s and 200° respectively. The\nmean spectrum is also calculated for the same time period (Fig. 4.6b) using\nthe three microbarographs. The major peak appears at a 28.4-min period with\nminor peaks at 19- and 14.2-min periods. The phase speed and azimuth of the\nmajor peak are 17.7 m/s and 225° respectively.\nFor the 0530 to 0650 MST time period (Fig. 4.4), the two sets of three\ntime series (not shown) are also very coherent. Figures 4. 7a, b indicate a\nsimilarity between the optical wind and microbarograph mean spectra with\n99","0200\nFigure 4.4. Acoustic record between 0200 and 0700 MST on 13 November 1981.\n0300\n0400\nTime (MST)\n0500\n0600\n0700\n350\n300\n250\n200\n150\n100\n50\n0","mmm\n0\n0\n0\n60\n70\n80 (min)\n0\n10\n20\n30\n40\n50\nFigure 4.5a. -The time series of the three microbarographs between 0210\nand 0335 MST on 13 November 1981. The vertical scale between two lines\nis 80 ub/s, and the curves are relative to a zero mean.\nVN\n1.18\nMinh\nVSE\n1.22\nmmm\nVsw\n1.05\n80 (min)\n60\n70\n0\n10\n20\n30\n40\n50\nFigure 4.5b. The time series for three separate optical systems from\n0210 to 0335 MST on 13 November 1981. The vertical scale between two\nlines is 3 m/s, and the numbers on the far left are the individual means.\nand V are the wind components, transverse to the north, south-\nVN' 'SE'\nSW\neast, and southwest triangle sides, respectively.\n101","1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n0\n0\n2\n4\n6\n8\n10\nFrequency (10-3)\nFigure 4. 6a. -- The mean spectrum calculated from the optical triangle\ndata from 0210 to 0330 MST on 13 November 1981.\n4.0\n3.0\n2.0\n1.0\n0\n0\n2\n4\n6\n8\n10\nFrequency (10-3)\nFigure 4. 6b. -- The mean spectrum calculated from the three microbarographs\nfrom 0210 to 0330 MST on 13 November 1981.\n102","1.0\n0.8\n0.6\n0.4\n0.2\n0\n10\n6\n8\n4\n0\n2\nFrequency (10-3)\nFigure 4. 7a. The mean spectrum calculated from the optical triangle data\nfrom 0530 to 0650 MST on 13 November 1981.\n4.0\n3.0\n2.0\n1.0\n0\n10\n6\n8\n4\n0\n2\nFrequency (10-3)\nFigure 4.7b. The mean spectrum calculated from the three micorbarographs\nfrom 0530 to 0650 MST on 13 November 1981.\n103","Table 4. 1. -Periods of spectral peaks, phase speeds, and azimuths\ncalculated from the optical triangle (0) and microbarographs (M)\nfor the four cases discussed\nBandwidth\nPeriods of\nof wave\nPhase\nDate and time\nArray\npeaks\nanalysis\nspeed\nAzimuth\n(MST)\n(min)\n(min)\n(m/s)\n(deg)\n13 Nov. 1981\no\n34.1, 19, 14.2\n3.4-85.3\n14.4\n200\n0210-0330\nM\n28.4, 19, 14.2\n3.4-85.3\n17.7\n225\n13 Nov. 1981\no\n28.4, 21.3\n3.4-34.1\n10.0\n254\n0530-0650\nM\n24.4, 17.1\n3.4-34.1\n10.0\n270\n27 Oct. 1981\no\n34.1,\n24.4,\n17.1\n3.4-85.3\n12.0\n360\n1940-2100\nM\n34.1,\n24.4,\n13.1\n3.4-85.3\n10.0\n360\n28 Oct. 1981\no\n21.3, 14.2\n3.4-85.3\n11.6\n296\n0530-0650\nM\n21.3,\n14.2,\n12.2\n3.4-85.3\n10.0\n270\nrespect to the major peaks. The phase speed from both the optical array and\nmicrobarograph is 10 m/s, and the azimuths are 254° and 270° respectively.\nThe two other gravity wave events analyzed occurred on 27 and 28 October\n1981. A Summary of results for these periods is given in Table 4.1 along\nwith results for events of 13 November described above. The table indicates\nthat the agreement of the phase speeds and directions between the two methods\nis quite good.\nOne disadvantage in using the optical triangle for gravity wave studies\ncan be seen from Eq. (4.3) and Fig. 4.1. When the actual wind V! approaches a\ni\nparallel direction with one of the optical paths (a i 90°), the wind becomes\nindeterminate for that path. In this situation, a data spike will appear in\nthe V! i data stream for that path. The spikes can be removed by interpolation\nusing the acceptable points on either side.\nAs mentioned earlier, the spatial filtering of the optical array can be\nan advantage. One advantage to this built-in filtering is in the use of the\nimpedance relation (Gossard and Munk, 1954),\n- U ok (z)\n(4.16)\n,\n104","where both the background wind U and the wave-associated departure of the\nok\nwind u' can be measured with the optical triangle. In Eq. (4.16), p' is the\nwave-perturbed pressure, p the mean density, and k the wave number. Often,\nusing a time series of the wind from a single point sensor presents a diffi-\ncult filtering problem in ascertaining a proper u'. The optical triangle\nwinds alleviate many of the filtering problems (E.E. Gossard; NOAA/ERL/WPL;\nprivate communication).\nHowever, the spatial filtering of the optical winds can also present a\nlimitation not associated with the microbarograph technique. Waves with wave\nlengths equal to or smaller than the averaging distance (450 m) will be fil-\ntered out completely using the optical technique. Wave lengths slightly\nlonger than 450 m will be filtered considerably depending on the filter func-\ntion of the unit. For the major peaks analyzed in this study, all of the\nwavelengths are significantly longer than 450 m. A comparison of the optical\nwind and microbarograph spectra in Figs. 4.6 and 4.7 shows how the spatial\nfiltering with the optical triangle affects the higher frequencies.\n4.4 CONCLUSIONS\nThe phase speeds and directions of gravity waves are possible to calcu-\nlate, within the limits of the spatial averaging and nonparallel wind direc-\ntions, using an optical triangle wind-measuring system. The results agree\nquite well with those from a microbarograph array. Some of the differences\nbetween the optical wind spectra and microbarograph spectra can be explained\nby optical wind spatial filtering. The advantages of the optical triangle\nover a microbarograph array include the inherent spatial averaging of the\noptical triangle. The spatial filtering can be a disadvantage for wavelengths\nnear the pathlength.\nA discussion of the connection between wave-induced pressure fluctuations\nand wave-induced wind fluctuations near the ground under statically stable\nconditions is beyond the scope of this work. However, consideration of such a\nconnection is important to assess the ultimate applicability of optical wind\nsensing to atmospheric gravity wave measurement.\n105","4.5 ACKNOWLEDGMENTS\nThe authors wish to thank E.E. Gossard and W.D. Neff for their helpful\nsuggestions during preparation of this paper.\n4.6 REFERENCES\nBrigham, E.O., 1974: The Fast Fourier Transform. Prentice-Hall, Englewood\nCliffs, N.J., 252 pp.\nEmmanuel, C.B., B.R. Bean, L.G. McAllister, and J.R. Pollard, 1972: Observa-\ntions of Helmholtz waves in the lower atmosphere with an acoustic sounder.\nJ. Atmos. Sci., 29, 889-892.\nEymard, L., and A. Weill, 1979: A study of gravity waves in the planetary\nboundary layer by acoustic sounding. Bound.-Layer Meteorol., 17, 231-\n245.\nGossard, E.E., and W.H. Hooke, 1975: Waves in the Atmosphere. Elsevier, New\nYork, 456 pp.\nGossard, E.E., , and J.H. Richter, 1970: Internal waves in the atmosphere from\nhigh-resolution radar measurements. J. Geophys. Res., 75, 3523-3536.\nGossard, E.E., and W.H. Munk, 1954: On gravity waves in the atmosphere.\nJ. Meteorol., 11, 259-269.\nHooke, W.H., F.F. Hall, and E.E. Gossard, 1973: Observed generation of an\natmospheric gravity wave by shear instability in the mean flow of the\nplanetary boundary layer. Bound.-Layer Meteorol., 4, 511-523.\nKjelaas, A.G., and G.R. Ochs, 1974: Study of divergence in the boundary layer\nusing optical propagation techniques. J. Appl. Meteorol., 13, 242-248.\nLawrence, R.S., G.R. Ochs, and S.F. Clifford, 1972: The use of scintillations\nto measure average wind across a light beam. Appl. Opt., 11, 239-243.\n106","Ochs, G.R., , S.F. Clifford, and Ting-i Wang, 1976: Laser wind sensing: the\neffects of saturation of scintillation. Appl. Opt., 15, 403-408.\nTsay, M.K., Ting-i Wang, R.S. Lawrence, G.R. Ochs, and R.B. Fritz, 1980: Wind\nvelocity and convergence measurements at the Boulder Atmospheric Observa-\ntory using path-averaged optical wind sensors. J. Appl. Meteorol., 19,\n826-833.\n107","","5. RICHARDSON NUMBER COMPUTATIONS IN THE PLANETARY BOUNDARY LAYER\nR.J. Zamora*\nDepartment of Meteorology\nMetropolitan State College\nDenver, Colorado 80204\nABSTRACT. This paper focuses on a method of calculating the\ngradient Richardson number Ri using data gathered at the Boulder\nAtmospheric Observatory, a 300-m meteorological tower located\n30 km north of Denver, Colorado, and presents a case study of\nRichardson number (Ri) behavior at the 75-m level of the tower\nduring an active acoustic-gravity wave event. Ri is found to be\nmodulated by the gravity wave activity in addition to longer\nperiod oscillations that might be a result of inertial wave\nactivity.\n5.1\nINTRODUCTION\nRecent experiments conducted by NOAA/ERL/WPL have focused on the struc-\nture of the stable planetary boundary layer (PBL). One of the more important\nparameters used to quantitatively estimate the dynamic stability of such a\nfluid environment is the gradient Richardson number Ri. This has long been\nregarded as an important criterion for the onset of turbulence in a stably\nstratified fluid. Fluid parcels can be moved vertically against the force of\ngravity if sufficient kinetic energy is available in the velocity field.\nRi represents the ratio of work needed to bring about the exchange to the\nkinetic energy present to do the work. In general if Ri > 0.25 through the\nfluid, the fluid is assured to be stable everywhere. These results were\nobtained by Miles (1961) and Howard (1961) for incompressible fluids, and the\nproof was later extended to compressible fluids by Chimonas (1970).\nAlthough the Richardson number is of great theoretical importance, it is\na difficult parameter to calculate. In its usual form for atmospheric appli-\ncations (Gossard and Hooke, 1975),\nPresent affiliation: NOAA/ERL/Wave Propagation Laboratory, Boulder,\n*\nColorado 80303.\n109","Ri =\n(5.1)\ndu\n2\nwith\n(5.2)\n,\nwhere g is the gravitational acceleration, N the Brunt-Vaisala frequency, 0\nthe potential temperature, Z the vertical Cartesian coordinate, U the hori-\nzontal wind vector, and 10 the mean potential temperature over the gradient.\nTo fully appreciate the difficulty in calculating Ri, assume Ri = 0.25, a\ntypical value in a stable layer with shear, and (au/dz) = 0.004, a moderate\nvalue. Then\n01 = 1.5 x 10-3 ao °C/m.\nThese correspond to differences of approximately 0.05°C and 0.2 m/s over 50-m\nAz. Although it is possible to measure these differences with present BAO\ninstrumentation, it must be noted that absolute calibration errors can bias\nthe calculations, and aliasing due to the finite difference scheme can yield\npoor results. Thus, in cases where the wind shears and temperature gradients\nare weak, the limitations of instrumentation are most likely to be apparent in\nthe calculations.\nIn calculating Ri it is commonly assumed that the atmosphere is dry. If\nthe gradient of 0 in the layer over which the Richardson number is calculated\nis less than zero, the atmosphere is statically unstable and any perturbation\nin the atmosphere will grow. In this case the Richardson number is of little\nsignificance since the energy contained in the velocity field is not necessary\nfor vertical parcel exchange. If the gradient of potential temperature is\ngreater than zero, a restoring force exists and a displaced parcel will oscil-\nlate about its equilibrium position with a frequency equal to N in the absence\nof viscous damping. Since the potential temperature of a parcel depends only\non pressure and temperature, errors can be introduced into Ri computations if\nthe density of the parcel is assumed to depend only on the dry air pressure\nand temperature of the parcel.\n110","In an attempt to relax the assumption of a dry atmosphere, the virtual\ntemperature of a parcel can be substituted into the potential temperature\nequation, and in turn this virtual potential temperature can be used to cal-\nculate a density-adjusted static stability term and thus adjust Ri for the\neffect of moisture in the atmosphere.\nThe preceding stability analysis remains valid until the atmosphere\nreaches saturation, at which time the parcel cools at the moist adiabatic\nlapse rate. It has been demonstrated (Lalas and Einaudi, 1973) that at this\npoint a new Brunt-Väisala frequency must be used to describe parcel buoyancy\nadequately. This paper does not address the problem of dynamic stability in a\nsaturated atmosphere, or after condensation has occurred.\nTo examine the behavior of the Richardson number in the lower atmosphere\na data set was chosen from a period of active acoustic-gravity wave propaga- -\ntion beginning on the evening of 6 November 1981 at about 2000 MST and ending\non 7 November 1981 at 0400 MST at the Boulder Atmospheric Observatory (BAO).\nBAO is located 20 km east of Boulder on gently rolling terrain.\n5.2 GOVERNING EQUATIONS\n5.2.1 Dry Atmosphere\nThe gradient Richardson number is given by (5.1), where (5.2) represents\nthe static stability of the atmosphere, and\n(5.3)\nThis expression for the change in wind shear with height takes into account\nboth the directional and speed shears of the wind vector:\n13 = ui + vj\n(5.4)\n.\nThe parcel potential temperature 0 is given by\n0 = T K\n(5.5)\n,\n111","where K = 0.286, T is the Kelvin temperature of the parcel, and p is the\natmospheric pressure in millibars.\n5.2.2 Moist Atmosphere\nThe density-adjusted Richardson number Riv is\n(5.6)\nwhere\n(5.7)\n,\nand virtual potential temperature vv is given by Fleagle and Businger (1980)\nas\n(5.8)\nT* is given by Hess (1959) as\n(5.9)\nwhere q, the mixing ratio, is\n(5.10)\n,\nwith E = 0.622.\nThe vapor pressure e of the parcel is given by Murray (1967) as\ne = 6.1078 exp\n(5.11)\n112","20\nT =290 K\np=840 mb\n15\nT - Td (K)\nOv\n0\n10\n5\n0\n310\n308\n309\n304\n305\n306\n307\n0 or Ov (K)\n20\nT = 300 K\nP = 840 mb\n15\nT - Td (K)\n0\n10\n0\nV\n5\n0\n321\n320\n318\n319\n316\n317\n315\n0 or Ov (K)\nFigure 5.1. --0 and 0 v vs. T-Td for constant values of P and T.\n113","where\na = 21.8745584\nover ice,\nb = 7.66\na = 17.2693882\nover water,\nb = 35.86\nId is the parcel dewpoint.\nThe use of vv V in (5.7) allows the computation of a more realistic Richard-\nson number because the static stability of the atmosphere is no longer assumed\nto be a function of P and T only. Figure 5.1 illustrates the differences\nbetween 0 and 0 V for increasing I at constant values of P and T. This ad-\njustment is negligible at low temperatures because e is typically small.\n5.3 COMPUTATIONAL SCHEME\n5.3.1 Instrumentation and Calibration\nThe BAO facility includes a 300-m meteorological tower instrumented at\neight levels as shown in Fig. 5.2. Standard instrumentation at each level\nincludes fast-response wind measurements by sonic anemometers, slow-response\nwind measurements using R.M. Young propeller-vane (propvane) anemometers, mean\ntemperatures using Hewlett-Packard quartz thermometers in an aspirated shield,\nmean dewpoints using EG&G 110 Dewpoint Hygrometers, and fast-response tempera-\ntures using platinum wire thermometers. A full description of BAO is given by\nKaimal and Gaynor (1983).\nFor the purpose of Richardson number calculation the propvane anemometers\nwere chosen for velocity field measurements, and the mean temperature and\ndewpoint sensors were chosen to measure the thermodynamic variables. These\nchoices were made for operational reasons. Since Ri involves the gradients\n20/dz and di/dz, the absolute calibrations must be as accurate as possible.\nThe propvane anemometers, and mean-temperature and mean dewpoint sensors,\noffered the most accurate absolute calibration. The mean-temperature sensors\nare periodically placed in a precision temperature bath and intercompared.\nThe dewpoint hygrometers are calibrated with precision resistors and are\n114","300 m\nParameters Measured\n250 m\nStandard Levels and Carriage:\n1. Wind Components\n2. Mean Wind Speed and Direction\n3. Mean Temperature\n4. Fluctuation Temperature\n5. Mean Dewpoint Temperature\n200 m\nGround\n150 m\n1. Mean Atmospheric Pressure\n2. Fluctuating Pressure\n3. Solar Radiation\n4. Optical Triangle Crosswind and Cn2\n100 m\nInstrumented\nCarriage\n50 m\nComputer\nBuilding\n22 m\nSignal Processing\nand Calibration Vans\n10 m\n600 m\nFigure 5.2 2. -- --BAO tower, showing the location of fixed levels and the\nparameters measured.\n115","cleaned periodically to maintain precision. The propvane anemometers are\ncalibrated using a constant rpm motor for speed and are oriented with respect\nto the supporting boom for azimuth.\nThe accuracies for the respective measurements are\n+ 0.05°C for temperature,\n+ 0.5°C for dewpoint,\n+ 0.1 m/s for speed,\n+ 1° for azimuth.\n5.3.2 The Algorithm\nTo calculate the Richardson number, an interactive program, which resides\nin the permanent library of the BAO PDP 11/70 computer, was written by the\nauthor. The source code is written in Fortran IV. The program allows the\ninput quantities to be averaged over variable increments of time and allows\nthe option of calculating Ri or Riving\nCalculation of q, 0, and 0 V requires knowledge of p at each tower level,\nbut p is measured only at the surface. Thus, the hypsometric equation (Hess,\n1959) is solved in the form\n=\nexp\n(5.12)\nwhere and 21 are the tower levels chosen, and P1 =\nsurface pressure. Because T* also requires knowledge of P at each level,\n(5.12) cannot be solved directly.\nInstead, the mean temperature T is substituted for it in (5.12) and this\napproximate equation is solved for p at each tower level. This first guess is\nthen used to compute q and I* for the tower levels.\n*\nAfter this step is completed, (5.12) is then solved again obtaining a\ncorrected P for levels one through eight. This P is then used in successive\ncalculations.\n116","The partial derivatives 20/dz, and au/dz are approximated by\ncentered differences as\nthe\nV1\n(5.13)\nand\n2\n(5.14)\n5.4 ERROR ANALYSIS\nTo test the sensitivity of the algorithm to experimental and truncation\nerror, a separate version of the program was created in which the errors given\nin Sec. 5.3.1 were added to the measured quantities. The differences between\nthe two computer runs are as follows:\nARi_= 0.007\nARi = 0.008 .\nHence, the estimates of Ri and Ri are accurate to + 0.007 and + 0.008 re-\nV\nspectively. Much larger errors are possible if the differences in wind veloc- -\nity and temperature over the layer in question exceed the limits of the sensor.\n5.5 ANALYSIS\nThe echo patterns from an acoustic sounder located near the tower\n(Fig. 5.3) were used to determine the onset of wave activity, and the Richard-\nson number was computed between the 50- and 100-m levels of the tower starting\nat 2000 MST on 6 November 1981 and ending at 0420 on 7 November 1981, for an\naveraging period of 5 min using the algorithm of Sec. 5.3.2. A histogram of\nthe distribution of Rivy for the period is shown in Fig. 5.4.\n117","Tower\nSupport\nCables\nMicrobarograph\nBAO Tower\nAcoustic\nSounder\nN\nTemporary\nBuilding\nScale\n0\n200 m\nCounty Rd. 8\nFigure 5.3. -- BAO facility showing the location of the acoustic sounder\nand microbarograph.\n118","60\n50\nRiv = 2.456\nb = 10.126\n40\n30\n20\n10\n0\n50\n10\n20\n0.5\n1\n5\n0.3\n0.4\n0.1\n0.2\n0\nRiv at 75m\nFigure 5.4. Histogram of Ri V at 75 m.\nIt was hoped that since an acoustic-gravity wave represents the response\nof the atmosphere to accelerations caused by deviations from geostrophic\nbalance, one would observe a periodicity in Rivy that would represent the\nattempt of the atmosphere to gain geostrophic balance by transfering momentum\ninto the lower regions of the PBL through mechanical turbulence. Since the\ngradient Richardson number relates the mean field gradients of temperature and\nvelocity to the dynamic stability of the atmosphere and the onset of mechani-\ncal turbulence, any changes in the mean fields of velocity and temperature\nshould be reflected in the turbulence distribution (Neff, 1980).\nAs the wave propagates through the tower, parcels of differing 0 V will\nalternately move past the sensors, and thus ao V / dz should vary with the fre-\nquency of the wave. Since the vertical momentum distribution is changing,\n119","continuity requires that the horizontal momentum distribution must also\nchange. Thus, not only should ao /dz vary with time, but also (au/az)2\nmust vary. Also, dp/dt, at the ground, should reflect the wave propagation.\nThe evening of 6 November and early morning of 7 November are character-\nized by moderate static stability as shown by the National Weather Service\nsoundings taken at Stapleton Airport near Denver, Colo. (Fig. 5.5). In addi-\ntion the evening wind observation (Fig. 5.6) shows a weak shear zone in the\nlower kilometer of the atmosphere with the peak wind of 3 m/s at 200 m drop-\nping off to less than 0.5 m/s at 800 m. Although these conditions are not\nparticularly strong, the basic conditions for shear-generated instability are\nconcluded to be present. The distance from the upper-air observation and the\ntower might also reduce the representativeness of the rawinsonde observations.\nFigure 5.7 shows the time series of Ri dp/dt, (au/az)2, and N 2.\n2\nThe\ntime series of dp/dt documents the wave activity well, and the acoustic\nsounder records in Figs. 5.8a,b provide another look at the height distri-\nbution of the wave activity.\nYet the only clear evidence of wave propagation in the Richardson number\nbegins at 2220 and ends at 2330 MST. It is interesting to note that whereas\nthe shear term (du/dz) 2 appears to be modulated by the wave, the N 2 term does\nnot. It appears that the dominant factor in Richardson number reduction is\nthe wind shear. Further investigation by decomposing the shear into its\nazimuthal and speed parts (Fig. 5.9) and comparing with Ri, shows that early\nin the evening the azimuthal shear is large and seeks to drive the Richardson\nnumber lower (Fig. 5.10). Later, after the Richardson number falls below 0.25,\nthe speed shear increases.\nIf we assume that the onset of turbulence begins at Ri < 0.25, then the\nincrease in wind speed might be a result of the momentum transfer into the\nlower boundary layer by the turbulence, while the azimuthal shear reflects the\nageostrophic conditions that created the imbalance.\n120","400\n(a)\n500\nIn p\n600\n700\n800\n840\n30\n20\n10\n0\n-20\n-10\nT (°C)\n400\n(b)\n500\nIn p\n600\n700\n800\n840\n30\n20\n10\n0\n-10\n-20\nT (°C)\nFigure 5.5. -- -Rawinsonde observations taken by the National Weather\nService at Stapleton Airport, Denver, Colo., at (a) 1700 MST, 6 Novem-\nber 1981, and (b) 0500 MST, 7 November 1981.\n121","2000\nZ (m)\n1000\n0\n0\n1\n2\n3\n4\n5\n6\n7\n8\nSpeed (m/s)\nFigure 5.\n6. -- Wind speed VS. height for rawinsonde observations at\n1700 MST, 6 November 1981 (solid line), , and 0500 MST, 7 November 1981\n(dashed line).\n2\nThe lack of modulation in N might be an instrumentation problem.\nDuring calibrations the slow-response temperature probes were found to respond\nat differing rates to changes in ambient air temperature. This will result in\nsmoothing the potential temperature gradient. The poor correlation between\ndp/dt and Ri V could be due to this factor. But the overall behavior of Ri\nthrough the period shows evidence that it can be tied to the inertial oscil-\nV\nlations of the wind (Dutton, 1976) on a much longer period than that of acous-\ntic-gravity waves (Fig. 5.10).\n122","10\n8\n6\nRiv\n4\n2\n0.25 0\n5 X 10 -3\nN2\n(s-1)\n0\n-5 X 10-3\n1 X 10- 2\n5X 10-3\n2 0\n( dz\n(m/s)\n10\n5\ndp\nat\n0\n(ub/s)\n-5\n-10\n0440\n0235\n0354\n2357\n0119\n2238\n2000\n2119\nTime MST\nFigure 5. .7. . -- Time series of Riv, N V 2 , and (au/az) 2 at 75 m, and dp/dt\nmeasured at the ground.\n123","2000\n2200\nFigure 5.8a. -- Acoustic sounder record beginning at 2000 MST, 6 November 1981.\nTime MST\n2100\n2300\n2200\n0000\n340\n75\n0\n340\n75\n0","","10\n8\n6\nA Speed\n(m/s)\n4\n2\n0\n180\n135\nA Azimuth\n(deg)\n90\n45\n0\n2000\n2119\n2238\n2357\n0119\n0235\n0354\n0440\nTime MST\nFigure 5. .9.\nazimuth and A speed over the 50- to 100-m level of the\ntower.\n126","1.0\n0.8\n0.6\nRiv\n0.4\n0.25\n0.2\n0.0\n0354\n0440\n0119\n0235\n2238\n2357\n2119\n2000\nTime MST\nFigure 5.10. - Time series of Ri on an expanded scale showing long-\nV\nperiod oscillation about the critical Richardson number.\n5.6 CONCLUSIONS\nThe ability to calculate the gradient Richardson number using data from a\nboundary layer tower is demonstrated. It is concluded that the shear term\nplays an important role in the magnitude of the Richardson number and is in\nfact modulated by wave activity. It is also shown that the behavior of the\nshear term and the magnitude of Ri could be a result of inertial oscillations\nV\nin the wind field in addition to higher frequency wave activity.\n127","5.7 ACKNOWLEDGMENTS\nThe author wishes to thank J.E. Gaynor, E.E. Gossard, and W.D. Neff for\ntheir guidance, patience, and criticism during the preparation of this work\nand the creation and debugging of the software. I would also like to thank\nJ. Hart for her help in the software effort, the BAO tower crew, J. Newman\nand N. Szczepczynski, who provided me with their expert knowledge of the\ninstrumentation.\n5.8 REFERENCES\nChimonas, G., 1970: The extension of the Miles-Howard theorem to compressible\nfluids. J. Fluid Mech., , 43, 833-836.\nDutton, J., 1976: The Ceaseless Wind--An Introduction to the Theory of Atmo- -\nspheric Motion. McGraw-Hill, New York, 579 pp.\nFleagle, R.G., and J.A. Businger, 1980: An Introduction to Atmospheric\nPhysics, 2nd ed. Academic Press, New York, 432 pp.\nGossard, E.E., , and W.H. Hooke, 1975: Waves in the Atmosphere, Atmospheric\nInfrasound and Gravity Waves--Their Generation and Propagation. Else-\nvier, New York, 442 pp.\nHess, S.L., 1959: Introduction to Dynamic Meteorology. Holt, Rinehart and\nWinston, New York, 362 pp.\nHoward, L.N., 1961: Note on a paper of J.W. Miles. J. Fluid Mech., , 10, 509-\n512.\nKaimal, J.C., and J.E. Gaynor, 1983: The Boulder Atmospheric Observatory.\nJ. Clim. Appl. Meteorol. (accepted).\nLalas, D.P., , and F. Einaudi, 1973: On the stability of a moist atmosphere in\nthe presence of a background wind. J. Atmos. Sci. , 30, 795-800.\n128","Miles, J.W., 1961: On the stability of heterogeneous shear flows. J. Fluid\nMech., 10, 496-508.\nMurray, F.W., 1967: On the computation of saturation vapor pressure. J. Appl.\nMeteorol., 6, 203-204.\nNeff, W.D., , 1980: An observational and numerical study of the atmospheric\nboundary layer overlying the east Antarctic ice sheet. Ph.D. Thesis,\nUniversity of Colorado, Boulder, Colo., 272 pp.\n# U.S. Government Printing Office 1983 - 676-001/1216 Reg. 8\n129","3\n8398\nwanted\nSTATE\n0004\n-\nA\n44444"]}