measurements of fine-scale structure at the top of marine

MEASUREMENTS OF FINE-SCALE STRUCTURE AT THE TOP OFMARINE STRATOCUMULUS

DONALD H. LENSCHOW?

National Center for Atmospheric Research,?? P.O. Box 3000, Boulder, CO 80307, U.S.A.

MINGYU ZHOUNational Research Center for Marine Environmental Forecasts, Beijing, People’s Republic of China

XUBIN ZENGDepartment of Atmospheric Sciences, University of Arizona, AZ, U.S.A.

LIANSHOU CHEN and XIANGDE XUChinese Academy of Meteorological Sciences, Beijing, People’s Republic of China

Abstract. During the Dynamics and Chemistry of the Marine Stratocumulus (DYCOMS) experi-ment in July–August 1985, the NCAR Electra aircraft flew a series of flight legs just at the top ofthe marine stratocumulus cloud decks that cap the mixed layer off the coast of southern California.Because of the corrugated structure of the cloud-top, the aircraft, which was flown at a nearly constantlevel and adjusted only to maintain its altitude at the average cloud-top height, was alternately withinand above the clouds – roughly half the time in each domain. These legs were used to examinethe structure of the cloud-top by compositing the segments on either side of the cloud/clear-airinterface, which was identified by the transitions of liquid water measured by the Forward ScatteringSpectrometer Probe (either increasing or decreasing) through a threshold of 0.04× 10−3 kg m−3.An equivalent vertical distance (EVD) from the cloud-top was obtained from the horizontal flightlegs by estimating the average slope of the cloud-top from the cloud-top radiation temperature. Theresults show that a near discontinuity occurs in variables across cloud top over an EVD of 0.3 m, butthat above this, the air has already been modified by boundary-layer air. Thus, cloud-top is not thelimit of mixing of boundary-layer air. This mixing may extend to tens of metres or more. The bulkRichardson number in the vicinity of cloud-top increases from near zero within the cloud to about1.2 at an EVD of 3–6 m above cloud. Fluctuations of the three velocity components within cloudare nearly equal; above cloud the vertical component structure function is about half the horizontalcomponents. The scalar structure functions are about an order of magnitude higher above cloud thanin cloud. The structure parameters of temperature and humidity measured just below cloud-top agreereasonably well with predicted values based on a previously-developed model for the clear convectiveboundary layer. Above cloud, the scalar structure parameters are much larger, but their interpretationis questionable, since this region does not contain isotropic turbulence.

Keywords: Aircraft measurements, Entrainment, Stratocumulus, Structure functions, Turbulence.

? E-mail: [email protected]?? The National Center for Atmospheric Research is sponsored by the National Science

Foundation.

Boundary-Layer Meteorology97: 331–357, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

332 DONALD H. LENSCHOW ET AL.

1. Introduction

Marine stratiform clouds play an important role in the earth’s radiation budget,and therefore its climate (Randall et al., 1984). In addition, they modulate thesurface heat and moisture fluxes, and the exchange of air between the planetaryboundary layer (PBL) and the overlying free atmosphere. Therefore, understandingtheir structure and evolution is important in global climate studies, as well as forlocal weather forecasting. One of the important problems in studying their structureand evolution is determining the details of the entrainment process through thetop; that is, the rate at which the turbulent PBL incorporates nonturbulent fluidthrough engulfment of the nonturbulent fluid by turbulent eddies. This has receivedconsiderable attention in the literature, beginning with the pioneering study of Lilly(1968). Subsequent work on this topic includes laboratory experiments (e.g., Saylerand Breidenthal, 1998; Siems et al., 1990), numerical simulations (e.g., Moeng etal., 1996) and observations (e.g., Nicholls and Turton, 1986; Paluch and Lenschow,1991).

Entrainment is inherently a small-scale process that occurs sporadically overa limited domain of cloud-top, within a relatively thin layer. Therefore, directobservations of entrainment events are difficult to obtain with presently availableinstrumentation. Observing the mean and turbulence structure of the atmosphere inthe vicinity of cloud-top is considerably simpler, and yet can give some insight intothe entrainment process, since the structure is, to a large extent, shaped by entrain-ment. Here we use observations of wind components, temperature, humidity, liquidwater and ozone obtained from the NCAR Electra aircraft during the Dynamicsand Chemistry of Marine Stratocumulus (DYCOMS) Experiment to investigatethe structure in the vicinity of the capping inversion at cloud-top. This experimentwas carried out a few hundred kilometres west of San Diego, California in thesummer of 1985. Further details are reported by Lenschow et al. (1988) and Kawaand Pearson (1989).

The primary analysis tool for carrying out this study is a compositing techniquethat is used to obtain variables as a function of distance above or below cloud-top.The analysis is carried out on a series of horizontal aircraft flight legs at the topof the cloud layer. The flight level was specified to be as nearly at cloud-top asthe pilot could fly. The pilot was instructed to fly at a level that would place theairplane inside and outside the cloud roughly half the time, with gradual adjust-ments in the flight level to compensate for long-term trends in cloud height. Sincethe cloud-top height also varies on a scale characteristic of individual eddies ofseveral hundred metres to several kilometres length, the airplane passed repeatedlyin and out of cloud-top. In this way, we were able to collect data from multiplecloud penetrations, which can then be composited using the cloud-top interface asa fiducial point.

FINE-SCALE STRUCTURE AT CLOUD TOP 333

Figure 1. Schematic diagram showing the categories of segments referenced to cloud top that areused in the composited profiles.

2. Technique

We define cloud as that part of the flight track where the liquid water, measured byintegrating over all 15 size channels (measuring particle sizes from about 3µm to40 µm) of the Particle Measurements System Forward Scattering SpectrometerProbe (FSSP), exceeds 0.04 × 10−3 kg m−3. This threshold was chosen to besignificantly above the noise level of the FSSP, and yet about an order of magnitudeless than the average expected jump across cloud-top reported by Lenschow et al.(1988). The number of particles in each size channel was recorded every 0.1 s(which corresponds to about 10 m at an airplane speed of 100 m s−1). Therefore,our reference for compositing in-cloud and out-of-cloud segments is the middle ofthe 0.1 s interval over which the integrated liquid water contentq` passes through(either increasing or decreasing) the 0.04× 10−3 kg m−3 threshold. We then com-posited the variables used in this analysis – potential temperatureθ , humidity q,liquid water q` measured by both the FSSP as well as the CSIRO liquid waterprobe (King et al., 1978), ozone O3, and the three wind components,u, v andw –for two-second intervals on both sides of the threshold value. Since the sample ratefor all these variables except for the FSSP liquid water was 20 s−1 (or about 5 m),a total of 40 data points (20 for the FSSPq`) from each segment were used in thecomposited results. We also specified that the interval over whichq` must remainabove (or below) the threshold value must be at least 4 s in length [actually 4.1 s toallow computation of the structure functions (described below) out to a lag of 2 s];otherwise it is rejected. This criterion is necessary to ensure that when the airplanepasses into (or out of) a cloud, we include the corresponding segment when theairplane passes out of the cloud (or into the next cloud).

We see that this compositing technique leads to four categories of 2-s segments,which we denote by the superscript index(l): (1) flying in clear air toward thecloud-clear air interface, (2) flying in cloud away from the interface, (3) flying incloud towards the interface, and (4) flying in clear air away from the interface.Figure 1 is a schematic sketch showing the segments defined by the compositingtechnique.


We then calculate composite mean profiles for each of the four categories of 2-ssegments listed above. The composite profiles for a variables are defined as

s̃(l)(j) = 1

N

N∑i=1

s(l)i (j), (1)

where i denotes a particular 2-s segment satisfying criterion(l), N is the totalnumber of(l) segments, andj is the number of points away from the interface.Typically, the total number of segments in each category amounts to about 20 to40 for a 900-s (about 90 km) leg. In the subsequent discussion, the different casesthat are considered are each a 90-km leg at cloud-top. Some flights contain morethan one case, but the flight legs are always several hours apart and in a completelydifferent location.

In order to investigate the turbulence structure in this interface region, we definedepartures from the composited profiles (1),

s(l) ′i (j) = s(l)i (j)− s̃(l)(j). (2)

We use these departures to compute generalized structure functions of twovariables, defined as

D(l)s (j, k) =

1

N

N∑i=1

[s(l) ′i (j + k)− s(l) ′i (j)

]2, (3)

whereD(l)s (j, k) are matrices with indicesj and k, andk is the number of lags

(i.e., the separation interval over which the structure function is calculated). Themaximum range of the indexj that we use isM = ±40 for the 20 s−1 data, andM = ±20 for q` from the FSSP. Thus,−M ≤ j ≤ 0 for categories (1) and (3),and 0≤ j ≤ M for categories (2) and (4); andk ≤ M.

We can directly relate the composited profiles, which are calculated as a func-tion of the time interval away from the interface, to horizontal distance away fromthe interface, since the airplane was flown approximately at constant level. On theother hand, convection (as well as possibly some contribution from shear across theinterface) in the underlying mixed layer distorts the interface so that the airplanepasses in and out of the cloud layer. At each of these intersections, the layer of airjust above cloud-top is likely to also be tilted so that it is parallel to the underlyingcorrugated surface of the cloud-top. Assuming that this is the case, and that thelayer is horizontally homogeneous with respect to its height above cloud-top, therelationship between the horizontal displacementδx of a particular layer of airfrom the interface along the trajectory of the airplane and the vertical displacementof that same layer away from the interfaceδz is given by

δz = δx sinα, (4)


Figure 2.Schematic diagram showing the slope of the cloud at the airplane penetration pointα andthe assumed slopes of the isotherms above cloud-top (parallel to cloud-top) and inside the cloud(horizontal).

whereα is the slope of the interface at the penetration point. Later we will showthat the slope is small, and thus with little error we can assume that sinα ' α. Wealso note that for aircraft measurements,

δx = U1t, (5)

whereU is the airplane speed (hereU ' 100 m s−1) and1t is the sampling period.The geometry is illustrated schematically in Figure 2.

We estimate the interface slope by the following technique. In the cases studiedhere we flew, in addition to the cloud-top leg, another leg a few hundred metresimmediately above the cloud-top leg. Two identical downward-looking infraredradiation thermometers (Model PRT-5, manufactured by Barnes Engineering Com-pany) were mounted close to each other on the Electra. They detect radiation in theatmospheric window to reduce absorption by the intervening atmosphere, and thusmeasure the radiation temperature of the surface, or of intervening clouds. Theiroutputs were available to us at a sample rate of 1 s−1. We assume that within thecloud the air is well mixed by turbulence, and that therefore variations in cloud-top height are proportional to variations in the measured radiation temperatureδTR. The constant of proportionality is given by the wet adiabatic lapse rateγw;therefore,

|δz| = γ −1w |δTR|. (6)

Dividing (6) by δx and going to the limit ofδx → 0, we have∣∣∣∣ ∂z∂x∣∣∣∣ = γ −1

w

∣∣∣∣∂TR∂x∣∣∣∣ ' α. (7)

As a first step, we calculated the coherence between the two radiometers todetermine to what maximum frequency they both measure the same temperature.


We believe that this limit is determined by the ability of the radiometers to resolvesmall temperature differences. Typical spectra (multiplied by wavenumber) of theradiation temperature flying over stratiform cloud, and over a clear boundary layerwhere the radiometer is measuring the ocean temperature, are shown in Figure3. From this we see that the cloud-top radiation temperature spectrum is aboutan order of magnitude larger than the ocean temperature spectrum, and that thespectral slope of the cloud-top radiation temperature for wavelengths shorter thanabout 7 km is about−1. Furthermore, the coherence between the two downward-looking radiometers (not shown here) is close to one at long wavelengths, andremains significant down to a wavelength of about 500 m. From this we concludethat the temperature measured by the two radiometers is representative of the actualcloud radiation temperature for wavelengths> 500 m.

On this basis, we low-pass filtered the temperature with a 0.2 Hz (500 m) fil-ter to eliminate the noise. An example of both filtered and unfiltered time seriesof radiation temperature from both radiometers on a leg above cloud is shownin Figure 4. We next calculated the cumulative probability distribution function(CPDF) of the temperature during the cloud-top leg. From this, we estimated thetemperature at which the CPDF is equal to the ratio of time in cloud to the total timeof the cloud-top leg. We assume that this temperature is at the height of the cloud-top leg. We next calculated the time derivative of the low-pass filtered radiationtemperature at this reference value. Before we use this derivative to compute theslope of the cloud, we note thatTR is low-pass filtered to eliminate contributions> 0.2 Hz (or< 500 m wavelength) while the minimum time interval requiredfor computing the composited variables either inside or outside cloud is 4 s. Thismeans that the minimum period of an entire cycle of an inside-cloud leg plus anoutside-cloud leg is 8 s, or 800 m. We low-pass filteredTR at several frequencies toestablish the functional form of∂TR/∂t as a function of cutoff frequency and thenused the value at 0.125 Hz (i.e., 800-m wavelength) to correspond to the shortestwavelength of cloud-top variation that is passed by our compositing procedure.We can then evaluate the slopeα from (7) at the temperature, and consequentlythe height, at which the airplane is flying on the cloud-top leg, and the equivalentvertical distance (EVD)δz from (4). We then average all of the estimates ofα (i.e.,20 to 40) over the leg to obtainα.

In carrying out this procedure, we found that legs over solid cloud cover gavereasonably consistent estimates of the average slopeα, but that legs with manybreaks in the cloud cover gave considerably larger estimates of the slope due toincorporation of the ocean radiation temperature, which is several◦C warmer thancloud temperature, into the filtered time series. Figure 4 shows an example of thisat, for example, 52.1 and 52.4 minutes.

A more direct approach to measuring the cloud-top slope would be to use thebackscatter from a lidar to measure range to cloud. This was done in a later programflown in the same area in 1987 as part of the First International Satellite CloudClimatology Project Regional Experiment (FIRE). In addition to the measurements


Figure 3.Wavenumber times spectra of cloud-top temperature and ocean surface temperature meas-ured with a downward-looking Barnes PRT-5 radiation thermometer plotted versus both wavenumberand wavelengthλ. The dash-dotted line has a+1 slope characteristic of a white noise spectrum.

discussed above, a downward-pointing pulsed CO2 lidar provided direct distancemeasurements from the aircraft to cloud top (Schwiesow et al., 1988). The resultsshowed that variations in the radiation-temperature derived cloud-top height weresimilar to the lidar cloud-top height.


Figure 4.Unfiltered and filtered (low-pass cutoff of 500-m wavelength) of both surface temperatureradiometers for a portion of the flight leg above cloud on the first DYCOMS flight used to measurecloud-top temperature.

3. Results

3.1. EXAMPLES

Examples of the results obtained by our compositing technique are shown in thefollowing figures. Figure 5 shows the variation of liquid waterq` in the vicinityof the cloud-top. Sinceq` is our tracer of cloud edge, we expect that the drop to‘zero’ q` should be rapid, but we also note thatq` increases to> 0.40 g m3 withina horizontal distance of< 20 m (i.e., 2× the resolution of theq` measurement);thus, the cloud edge is very sharp. The ordinate scale on the right side has beenconverted to an equivalent vertical distance (EVD) scale (which we assume appliesto the profile above cloud-top, denoted by positive numbers) using (4); our estimateof the slope of the interface for the cloud-top flight legs from (7) isα ' 0.06±0.01.Therefore,δz ' 0.06× U1t ' 0.6 m forq` and 0.3 m for the other variables.


Figure 5. Composited profiles of liquid water content versus horizontal distance and equivalentvertical distance (defined in the text) from cloud-top from flight 1, 30 July 1985. Negative valuesare in cloud; positive out of cloud.

Figure 6 shows composited profiles of temperature and ozone from a cloud-topflight leg on the first DYCOMS flight. These profiles are obtained from an averageof segments (1) and (4) for the above-cloud profile and segments (2) and (3) for thein-cloud profile. The results are typical of subsequent flights as well. We see that,to the resolution of the measurements (about 0.3 m EVD), a jump occurs in bothvariables across cloud-top. In the first three metres EVD or so, a large gradientexists; above this level, the gradient is much smaller. The entire profile containsonly a small fraction of the total jump across cloud-top obtained from verticalaircraft soundings. The soundings give jumps in temperature and ozone mixingratio in the first 30 m above cloud-top of about 8 K and 35 ppbv, respectively.That is, when the aircraft is actually in a descent, as in Figure 11, the measuredjumps are considerably larger than what is obtained from the horizontal flight legs


Figure 6.Same as Figure 5, for temperature and ozone.

at cloud-top. This also illustrates a limitation in the constant-level flight legs – theydo not extend far enough above the clouds to reach the unmixed free atmosphere.

Another way to look at this interface region is to plot one scalar versus anotherin this interfacial region. Figure 7 shows a plot ofθ versus the total water (vapourplus liquid) qt for two minutes of the first cloud-top leg of the first DYCOMSflight. The figure includes data from segments: (1) in clear air towards the interfaceand (2) in cloudy air away from the interface. [Segments (3) and (4) show nearlyidentical behaviour.] The results show a nearly discontinuous jump in bothθ andqtacross the interface, as well as a linear relationship between them in clear air. Fromthese two figures, we can conclude that some boundary-layer air leaks across thecloud-top inversion (i.e., is detrained) and is mixed into air in the free atmosphere.(Or alternatively, the top of the cloud is not the top of the PBL, and a layer ofair containing a mixture of PBL and free atmospheric air lies above the cloud.)Furthermore, a nearly discontinuous jump occurs in these scalars across cloud-top.

Figure 8 shows the corresponding wind component profiles. The lateral com-ponent of the wind with respect to the airplane is obtained from a vane, while thelongitudinal component is obtained from a Pitot tube pressure difference measure-


Figure 7.Potential temperature versus total water for segments: (1) in clear air towards the interface(‘o’) and (2) in cloud away from the interface (‘x’) for the first 260 s of the first cloud-top flight legon the first DYCOMS flight (30 July 1985).

ment. We believe that the discontinuous jump in the longitudinal component acrosscloud-top may be due to an offset introduced in either the static or the dynamicpressure measurement by wetting in cloud. Paluch and Lenschow (1991) noted thatpressure difference and flow angle measurements on the Electra were affected uponentering and exiting clouds. However, we can avoid this problem by subsequentlyconsidering air velocity measurements only on segments (1) and (3); i.e., segmentspreceding the cloud-edge transition.

We can obtain an estimate of the significance of the mean profiles by calculatingthe standard error (i.e., the standard deviation divided by the square root of thenumber of samples) for each of the composited variables. The results for the firstcloud-top leg of the first DYCOMS flight are shown in Table I. The standard errorsin the velocity components are calculated only for segments (1) and (3), while thescalars include all four sets of segments. The results indicate that the general shapesof the mean profiles and the jumps across the interface are statistically significant.


Figure 8.Same as Figure 4 for lateral and longitudinal (with respect to the aircraft) wind components.

We can compute a bulk Richardson number for the cloudy/clear-air interfaceregion using the relation

Ri = (g/T )δθvδz

(δU)2+ (δV )2 , (8)

whereθv is the virtual potential temperature. This was done using (8) with EVD=δz = 1.5 m (i.e.,1t = 0.25 s) for eight cases [1B, 1C, 2C, 3B, 3C, 7B1, 7B2and 10B2; the initial number denotes the flight number, the following letter (andnumber) denote the cloud-top leg within the particular flight], then averaged. Somecases (4B, 5B, 5C, and 10B1) with poor quality wind data were not used. Figure 9shows the distribution of average Ri with EVD, where EVD= 0 denotes cloud-top. We can see from Figure 9 that the maximum of Ri is not at cloud-top; rather


TABLE I

Standard error of variables measured on the first cloud-top flight leg of DYCOMS, 30 July 1985. Thenumbers of samples for the scalar variables are 37 for cloud segments and 44 for clear segments; forthe velocity components the numbers of samples are 19 for cloud segments and 22 for clear segments.

Variable In-cloud Clear

u – m s−1 0.10 0.13

v – m s−1 0.14 0.13

w – m s−1 0.07 0.07

θ – K 0.05 0.18

O3 – ppbt 0.2 0.8

q – g m−3 0.01 0.18

q` – g m−3 0.005 0

Ri increases from near zero within the cloud to a maximum of about 1.2 at an EVDof 3–6 m above cloud-top.

Figure 10 shows the structure functions of the three wind componentsu, v,andw averaged over six different cases for the region within and above cloud.Here we sum the generalized structure function (3) over the indexj , so that thevariables plotted in Figure 10 and Figure 12 are functions only of the separationindexk. The structure functions ofu, v, andw in cloud are shown in Figure 10a,and above cloud in Figure 10b. The structure functions were not used if the datafor one of the wind components were obviously in error. Figure 10 shows that thefluctuation levels ofu, v, andw in cloud are basically the same, while above cloudthe fluctuation level ofw is less than those ofu andv by about a factor of two. Thestructure functions ofu, v, andw in cloud are close to the 2/3 power relation forinertial subrange turbulence, although for small lags, the slope is somewhat steeperthan 2/3. Thus, in-cloud turbulence at scales of a few tens of metres is consistentwith what is expected for inertial subrange turbulence.

Above cloud the strong inversion (an example of a strong inversion from anaircraft sounding is shown in Figure 11) suppresses vertical fluctuations, but thefluctuations of horizontal velocities maintain about the same intensity as those incloud. Again, for small lags, the slopes are somewhat steeper than 2/3. However,for lags greater than about 10 m the slope of thew structure function above cloud isless than 2/3 and flattens out with increasing lag. We must remember, however, thatthis structure function calculation is dependent on the distance from the interface,so it cannot be interpreted as representative of what would be measured in a homo-geneous field of turbulence. For example, if the fluctuation level of the turbulencedecreases with distance away from the interface, this would also decrease the slopeof the structure function with increasing lag.


Figure 9.Variation of Richardson number with equivalent vertical distance (EVD). Negative valuesof EVD are in cloud, positive above cloud.

Examples of the structure functions of temperature and humidity (for case 1B)are shown in Figure 12 for the layers within and above cloud. The results showthat the fluctuations of both temperature and humidity are about two orders ofmagnitude higher above cloud than in cloud. This is typical of the other daysas well. On the basis of the nearly discontinuous jump across cloud-top and thelarge Richardson number, we conclude that the fluctuations above cloud are dueto the large gradients in the scalars that result from the intermittent transport ofboundary-layer air across the cloud-top interface into the region above cloud, andthe relatively inefficient rate of mixing of these detrained parcels of air. In addition,there may be contributions due to gravity waves in this very stable layer.

Thus, based on the above analysis, above cloud is a region of stable stratificationthat contains large fluctuations in scalars, and that is likely perturbed by gravitywaves generated by turbulence below the cloud-top inversion. The Richardson

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Figure 10.Structure functions of the three wind components,u, v, andw. Panel (a) is within cloud and (b) above cloud. The abcissa is separation distance.


Figure 11.An example of a strong inversion obtained from an aircraft sounding on flight 1, 30 July1985.

number is large enough that on the average, we would not expect turbulence, but notso large that locally it is quite plausible that shear-generated turbulence may occur,which can lead to transport of boundary-layer air across the cloud-top inversionand into the overlying free atmosphere.

3.2. CLOUD-TOP ENTRAINMENT INSTABILITY

To maintain a stable cloud layer, Lilly (1968) considered that the temperature in-version must be strong enough that the equivalent potential temperatureθe mustremain constant or increase across cloud-top. Thus, defining the above-cloud valueminus the in-cloud value as1θe (we use1( ) to refer to a generic jump acrosscloud-top, without regard to a specific process for selecting it, and1`( ) to thelarge-scale jump that is estimated by eye from profiles across the entire transition

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Figure 12.Structure functions of temperature (left panel) and humidity (right panel).


region above cloud-top, whileδ( ) refers to the small-scale jump just across thecloud/clear-air interface),

1θe ≥ 0. (9)

Although the mixed-layer equation set proposed by Lilly took into account the ef-fects of water vapour and liquid water on buoyancy, his cloud-top stability analysesdid not. Randall (1980) and Deardorff (1980) included these additional buoyancyeffects and proposed what they considered to be a more realistic stability criterion,

1θe ≥ k Lcp1qt , (10)

where1qt is the cloud-top jump in total water mixing ratio andk ' 0.23 is adimensionless parameter.

Points on the(1`θe,1`qt) plane obtained from the DYCOMS aircraft data areshown in Figure 13. The jumps are estimated by eye using conventional “large-scale” estimates of the jump across the inversion capping the cloud layer. That is,the jump occurring across the entire region that shows an enhanced gradient in thescalar. For example, in Figure 11, the jump inθ would be about 10 K. The solidline in Figure 13 represents1`θl = 0, where

1θl ≡ 1θe − L

cp1qt (11)

is the liquid water potential temperature. The interesting aspect of this figure is theapproximate arrangement of the points along a line parallel to the1`θl = 0 line;that is, along a1`θl = C line, whereC is a constant. It can be seen from Figure 13that1`θl ' 8 K, a little less than the1`θl ' 9 K that Kuo and Schubert (1988)report, based on data from many experiments and a variety of marine stratocumulusregimes.

The jump can also be estimated from the compositing technique discussedearlier for looking only at the region at the cloud/clear-air interface, as shown in theexample on Figure 6 for temperature and ozone. Points on the(δθe, δqt ) plane (seeFigure 14) based on these data also show a linear behaviour, but withδθl ' 2.4 K.This different behaviour has, as far as we know, not been previously reported.

Figure 15 shows the probability distribution of1`θ and1`q based on DY-COMS sounding data. The number of temperature values in the interval1`θ = 7–9K is very high (48%) compared to other intervals of1`θ , while the distributionof 1`q is broader. This figure also illustrates why the points in Figure 14 lieapproximately along a1`θl = 9 K line, since

1`θ = 1`θ` + L

cp1`q`. (12)


Figure 13.(1`θe, 1`qt ) plane based on DYCOMS aircraft data. The jumps are estimated by eye.The solid line is1`θ` = 1`θe − L

cp1`qt = 0.

In DYCOMS, 1`q` is typically in the range of−0.1 to −0.2 × 10−3, so that− Lcp1`q` < 0.5 K.

Figure 16 shows the probability distribution ofδθ andδq based on the interfacedata used for Figure 14. We see that the highest probability forδθ is 2–3 K which,based on (12) is consistent with the value ofδθl = 2.4 K in Figure 14. Thus, itseems that the1θl = C line in the(1θe,1qt ) plane is dependent on how the jumpsin temperature and water content across the top of the PBL are determined; i.e.,whether they are determined from profile measurements extending across the entiretransition region or from measurements just across the cloud/clear-air interface.

3.3. STRUCTURE PARAMETERS AT CLOUD-TOP

Mixed-layer similarity theory predicts that under conditions of local free convec-tion in the convective PBL, the structure parameters, defined as

C2T (z/zi) ≡ DT (x, z/zi)x

−2/3, (13)

C2q(z/zi) ≡ Dq(x, z/zi)x

−2/3, (14)

whereDT (x) andDq(x) are the standard structure functions ofT and q, re-spectively (Tatarskii, 1971), andx is the separation distance, are predicted to befunctions of the normalized height,z/zi, wherezi is the depth of the convectivePBL. The predicted functional relationship in the lower part of the mixed layer is


Figure 14.(δθe, δqt ) plane based on the interface data. The solid line isδθ` = δθe − Lcpδqt = 0.

(z/zi)−4/3 (Obukhov, 1960; see also Wyngaard and Lemone, 1980). Observations

reported by Kaimal et al. (1976), and Wyngaard and Lemone (1980) indicate thatC2T andC2

q do vary as(z/zi)−4/3 in the lower part of the mixed layer. However,near the top of the PBL, large values of the structure parameters occur, which aregenerated by entrainment. Wyngaard and Lemone (1980) developed a model toestimate the structure parameters in the upper part of the PBL where entrainmentflux predominates. Their formulations for the average over the interfacial layer are

〈C2T 〉 = Ti

θ∗z

2/3i

, (15)

〈C2q〉 = 3.9

(1q)2θ∗1θvz

2/3i

, (16)

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Figure 15.Probability distribution of1`θ and1`q based on sounding data.

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Figure 16.Probability distribution ofδθ andδq based on cloudy/clear-air interface data.


where

Ti = 1θv[

0.5− 2.6T 1q

1θv+ 1.4

(T1q

1θv

)2], (17)

and1θv and1q are the jumps of virtual potential temperature and specific hu-midity at the top of the PBL. The temperature and convective velocity scalesare

θ∗ ≡ (wθv)0

w∗, (18)

w∗ ≡[ gT(wθv)0zi

]1/3, (19)

where(wθv)0 is the virtual potential temperature flux at the surface.The test of this model with observations by Wyngaard and Lemone (1980)

showed that

C2T (z/zi)

〈C2T 〉= C2

T (z/zi)z2/3i

Tiθ∗(20)

and

C2q(z/zi)

〈Cq2〉 =C2q(z/zi)z

2/3i 1θv

3.9(1q)2θ∗(21)

could somewhat exceed one at the height corresponding to the structure parameterpeak – the maximum a little less than 2.0, with considerable scatter. The scatterwas due largely to the difficulty of obtaining statistically reliable averages near thehighly intermittent interfacial layer and accurately determining the top of boundarylayer. In addition, the sounding data for determining jumps of temperature andhumidity, and the leg data for computing the structure parameters, were obtainedfrom different time periods – separated by as much as several hours.

We can test this model with the DYCOMS observations, but we need to keepin mind that this data set is for a cloud-capped mixed layer, while the model wasdeveloped for a clear mixed layer. However, instead of usingw∗ given by (19) weuse the expression suggested by Nicholls (1989) for a cloud-capped mixed layer,

w∗ =(g

T

∫ zi

02.5(wθv)(z)dz

)1/3

. (22)

With the DYCOMS data we can determine the top of the PBL, the jumps of temper-ature and humidity at the cloudy/clear-air interface, and the structure parameters


TABLE II

Temperature structure parameterC2T× 10−3 K2 m−2/3.

Flight No Date (C2T )mod (C2

T )meas (C2T )meas/(C

2T )mod

1B 7/30 0.75 0.68 0.91

1C 7/30 0.81 0.58 0.72

2C 8/2 0.32 0.31 0.97

3B 8/4 0.87 0.65 0.75

3C 8/4 0.49 0.30 0.61

5B 8/9 0.65 0.58 0.89

5C 8/9 0.78 0.67 0.86

7B1 8/16 0.40 0.41 1.03

7B2 8/16 0.25 0.34 1.36

10B1 8/21 0.66 0.48 0.73

10B2 8/21 0.73 0.64 0.88

Average 0.88

on the same flight segments. The top of the PBL,zi, is defined as the cloud-topheight, and is well-defined for each case (Figures 5, 6, and 8). We can estimatethe structure parametersC2

T andC2q from fluctuation data just beneath cloud top

on the same flight segments that are used for measuring jumps of temperature andhumidity.

The in-cloud turbulence behaves as inertial-subrange turbulence; therefore weestimateC2

T andC2q with the 2/3-slope inertial subrange hypothesis applied to

the average structure function curves for temperature and humidity. Table II andTable III show the structure parameters(C2

T )mod and(C2q)mod calculated with the

model Equations (20) and (21) using the temperature and humidity jump data, and(C2

T )measand(C2q)measobtained from the inertial subrange structure function estim-

ates at cloud top. We can see from Table II and Table III that(C2T )mod and(C2

q)mod

are somewhat larger than(C2T )measand(C2

q)meas. The ratios(C2T )meas/(C

2q)mod and

(C2q)meas/(C

2q)mod are less than one in most cases, but their averages are 0.92 and

0.88 respectively, close to one. The standard deviations of these ratios are 0.073 and0.078 respectively, which are reassuringly small. Thus the Wyngaard and Lemone(1980) model is a good predictor of the structure parameters in the interfacialregion just below cloud-top.

The fluctuations of temperature and humidity above cloud are much higher thanthose in cloud, likely due to the much larger vertical gradients and the effectsof gravity waves. The above-cloud estimates ofC2

T andC2q are about two orders

of magnitude higher than the in-cloud estimates. This has important implications


TABLE III

Humidity structure parameterC2q × 10−9 m−2/3.

Flight No. (C2q)mod (C2

q)meas (C2q)meas/(C

2q)mod

1B 1.47 0.96 0.65

1C 2.85 2.74 0.96

2C 0.79 0.73 0.92

5B 2.20 1.44 0.66

5C 3.44 2.91 0.85

10B2 1.32 1.37 1.04

Average 0.85

for the propagation of electromagnetic radiation in this transition region (e.g.,Tatarskii, 1971; Wyngaard and Lemone, 1980).

4. Conclusions

We have developed a technique to composite structure across the top of the cloud-capped boundary layer (summertime marine stratocumulus off the California coast)using level aircraft flight legs that repeatedly penetrate the corrugated cloud top.The cloud/clear-air transition is robustly defined by the presence of cloud droplets.We estimate the average slope of the interface using infrared measurements ofcloud-top temperature. The results show that there is a very sharp transition acrossthis interface that cannot be completely resolved with the instrumentation usedhere. However, we also found that air above this interface is a mixture of boundary-layer and free atmospheric air; that is, the interface leaks. This is consistent withestimates of the average gradient Richardson number across this interface, whichis between 0.5 and 1.0. This is small enough that episodic turbulence events arelikely to occur that would transport boundary-layer air across the cloud/clear-airinterface into the free atmosphere.

Estimates ofδθ`, the jump just across the cloud/clear-air interface, show aconsistent value of 2–3 K, as compared to the jump estimates across the entiretransition region of 7–9 K.

Measurements of the structure parametersC2T andC2

q just below cloud-top agreewell with the estimates obtained from the jumps across the top of the boundarylayer via the model of Wyngaard and Lemone (1980). The values of the structureparameters above cloud-top are about two orders of magnitude higher. Their largemagnitude suggest that this region is important for propagation of electromagneticwaves.


We were limited here to measurements on a horizontal scale of 10 m (20 mfor liquid water), which we converted to an equivalent vertical scale of 0.3 m byestimating the average slope of the cloud-edge penetration. It would be very usefulto apply the techniques presented here to even finer scale measurements (say by anorder of magnitude) than this. This should be feasible using hot-film measurementsof air velocity, smaller diameter cold wires for temperature and a faster liquid watersensor, e.g., Gerber et al. (1994). A much more satisfactory way of estimating theaverage slope of the penetrated cloud edge would be with lidar backscatter, whichis now available for airborne use.

5. Acknowledgements

We gratefully acknowledge the contributions of Vic Patel, Arthur Isbell and JohnDeSanto who carried out much of the computer calculations on which this work isbased, and Kyoko Ikeda who plotted most of the figures. We thank Andreas Musch-inski, Chin-Hoh Moeng, Bjorn Stevens and Douglas Lilly for useful discussionsand comments on the manuscript. This work was partially supported by NASAFIRE III under Grant No. L-9434, NOAA OGP under grant No. NA66GP0179, theTIPEX project, and by basic research grant No. G1998040902.

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measurements of fine-scale structure at the top of marine

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