Lombardo Gregg1989

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 94, NO. C5, PAGES 6273-6284, MAY 15, 1989

Similarity Scaling of Viscous and Thermal Dissipation in a Convecting Surface Boundary Layer

C. P. LOMBARDO 1 AND M. C. GaEGG

Applied Physics Laboratory and School of Oceanography, College o] Ocean and Fishery Sciences University o] Washington, Seattle

By continuously deploying a turbulence profiler in the upper ocean, we observed the rate of viscous dissipation of turbulent kinetic energy, e, and the rate of diffusive smoothing of turbulent temperature fluctuations, X, during eleven diurnal cycles of the surface boundary layer (SBL). Even though we restricted our analysis to times when the ocean lost buoyancy at a nearly constant rate, we observed a wide range of conditions, including dominance of the turbulent production by the wind stress. Throughout, e was normalized very well by the sum of the similarity scalings for turbulence produced by wind stress and by convection. Scaling of X was less successful and applied only when turbulent production was dominated either by wind stress or by convection, and then only within part of the SBL.

1. INTRODUCTION

As an unexpected bonus while looking for turbulent patches in the upper thermocline, we observed eleven cycles in the diurnal growth and decay of the surface boundary layer (SBL) (Figure 1). By analogy with the planetary boundary layer (PBL) of the atmosphere, we take the SBL as the near-surface zone directly affected by the wind stress and buoyancy flux applied at the surface. By profiling continuously, we sampled a wide range of turbulent regimes in the SBL. Because turbulence is a major factor in the dynamics of the boundary layer, our immediate goal is to determine how well similarity scaling describes profiles of e, the rate of viscous dissipation of turbulent kinetic energy, and X, the rate of diffusive smoothing of turbulent temperature fluctuations. As the first step, we restricted ourselves to those times during the night when the ocean lost buoyancy to the atmosphere at a steady rate. Our ultimate goal, however, is to use these unique observations to increase our understand- ing of the SBL, and thereby improve our abihty to model it.

Oceanographers have learned many things about the SBL, but they still lack even an approximate energy budget and are as uncertain about the disposition of the energy as they are about its sources. For example, Richman and Garrett [1977] estimated the rate of energy entering the ocean from the wind as (0.02- 0.1)Et0, where Et0 is the wind work 10m above the surface, in units of W m -2. Relatively lit- tle of this energy goes into deepening the SBL. Typically, increases in the potential energy as the layer thickens ac- count for only (0.001- 0.002)El0 [Denman and Miyake, 1973]. By contrast, viscosity dissipates 10 times as much of the wind work. For instance, in analyzing their observations of 20 m thick SBL, Oakey an Elliott [1982] estimated

1Now at Naval Postgraduate School, Monterey, California.

Copyright 1989 by the American Geophysical Union.

Paper number 88JC04308. 0148-0227/89/88JC-04308505.00

f52ø pe dz • 0.01 El0, where p is the density in kg m -3 and e has units of W kg- 1. In view of the disparity between the en- ergies going into deepening and into dissipation, oceanographers will not close the energy budget or realistically model the SBL until they accurately parameterize the turbulence.

Although for 20 years atmospheric scientists have param- eterized turbulence in the PBL with similarity scahng, the difficulty of making the measurements has impeded a corre- sponding development for the SBL. Seawater is much harder on sensors than air. Also, in contrast to thicknesses of 1 km in the atmosphere, the oceanic boundary layer usually extends only tens of meters, and it has the added comphcation of breaking waves. Therefore, only recently did Oakey and Elliott [1982] make the first direct comparisons of e measurements in the SBL with Et0, and as yet only a handful of other studies have been done. Dillon et al. [1981] in- ferred the dissipation rate from the spectral shape of temperature microstructure measured close to the surface of a lake and found that e(z) ,,• z -t, the structure predicted by wind stress scaling. Shay and Gregg [1984, 1986] reported the first evidence for convective scaling, using e profiles from two ocean sites. Brubaker [1987] then demonstrated convective scaling of X at shallow depths in a lake. Taken together these studies prove that similarity scahng applies in the SBL at some times and places. However, no attempt at X scaling in the ocean has been reported, nor do we know the limitations on when and where different parts of similarity scaling apply. Our purpose here is to examine these ques- tions systematically for the nighttime phase of the diurnal cycle.

As necessary background, in section 2 we summarize the similarity scaling of e and X in the PBL. In section 3 we de- scribe the what and where of our observations in the SBL, followed in section 4 by comparisons of e and X with similarity scaling. We summarize and briefly discuss the impli- cations of our results in section 5.

2. SIMILARITY SCALING APPLIED TO THE PBL

In most cases, the PBL develops vertically, being set in motion and controlled by the wind stress and heat flux at

6273

6274 LOMBARDO AND GREGG: SIMILARITY SCALING DURING NIGHTTIME CONVECTION

0.1

0.2

•. 0.3

Q- 0.4.

0.5

0.6

0.7

-8

• • •,,, •,, , • ?, !

13 14 15 16 17 18 19 20 21 22 23

October 1986

Fig. 1. Diurnal cycles during the Patches Experiment (PATCHEX). Each day the ocean lost heat and buoyancy, starting several hours before sunset and continuing until a few hours after sunrise. These losses are shown by the shaded portions of the surface heat and buoyancy fluxes in the top panel. In response, the SBL slowly deepened (lower panel). The solid line marks D, the middle of the entrainment zone, and the lightest shading

7 1 shows 10 -s W kg -1 < e < 10- W kg- . The shading increases by decades, so that the darkest shade is e > 10-• Wkg -1 .

the surface. Because of this control from the bottom, much of the average and turbulent structure within the boundary layer is determined by only four variables: the heat flux at the surface, jq0, in units of W m-2; the wind stress at the surface, r, in pascals; the distance above the surface, z, in meters; and the innate buoyancy of heated air, g/T, in units of m s-2K -1. (Here g is the gravitational accel- eration.) In the atmospheric literature, the wind stress is usually stated in terms of the friction velocity, u. ---- •/•, and the heat flux is expressed as Q = Jqø/pcp, where Cp is the specific heat at constant pressure, in units of J kg- 1 K- 1 ß

Then J• = (g/T)Q is the buoyancy flux, in units of W kg- 1 The use of these four variables to normalize boundary layer measurements is known as similarity scaling [Monin and Obukhov, 1954]. Similarity scaling works well when the PBL is controlled vertically by horizontally uniform fluxes at the surface.

The application of similarity scaling has changed a• atmospheric scientists have learned more about the internal structure of the PBL. Like the boundary layer in the upper ocean, the PBL waxes and wanes in a diurnal cycle. Over land, convection usually begins soon after sunrise, and by early afternoon it pushes the PBL up to about l km. The thickness, D , of the PBL is defined by the first inversion, where temperature reverts to its normal increase with height. Once D grows beyond a few hundred meters, atmo-

spheric scientists distinguish within the P BL a sequence of three sublayers: the surface layer, the mixed layer, and the entrainment zone [Panofsky and Dutton, 1984]. Similarity scaling is applied differently in each sublayer.

The Surface Layer

When the convecting PBL is strongly established, only its bottom 10% is directly affected by the surface. Within this surface layer, the height, z, establishes the energy-containing scale of the turbulence. Because only three of the four similarity variables are independent, a second length scale can be formed to give the distance above the surface where the wind stress and buoyancy are equally effective at producing turbulence. Known as the Monin-Obukhov length, this scale is defined as

-u. 3 L --- •J2 (1)

where von K•rm•.n's constant, • = 0.4, is included by tradi- tion, as is the minus sign, which makes L negative when J2 is positive. L is the fundamental length scale used to separate the two asymptotic regimes- where z/- L • 1, wind stress dominates the production of turbulence; where z/- L • 1, buoyancy controls production. During strong convection, -L _• 10m. Hence, in midafternoon the wind-stress regime

LOMBARDO AND GREGG: SIMILARITY SCALING DURING NIGHTTIME CONVECTION 6275

TABLE 1. Characteristic Similarity Scales for the Three Sublayers of the Planeta w Bounda• Layer

Surface Layer Characteristic Wind Stress Free Convection Mixed Layer

Range 0 <-z <<-L -L <<-z << D -L <<-z < D

Length z z z,D

f 31/2

u. j w. ß - _

%pu. %pu• %pw.

•3 g Es = -- Ef =j•0 g• =j•0

u.T. 2 ufTf 2 w. 0. 2 z • D

Terminology differs slightly from atmospheric usage. The range specified is meant to be general. reflecting the uncertainty in atmospheric literature.

is usually confined to the lower 10% of the surface layer, accounting for only the bottom !% of the PBL.

The scaling procedure is to nondimensionalize boundary layer parameters using the similarity variablqs, e.g., dimensionless temperature is T/T., where Table I shows T. to be the ratio of Q and u.. More complex parameters are con- structed from simpler ones. As an example, in Table I we see that Xs is formed from u., T., and z. When properly nondimensionalized, parameters are then functions of z/L that can be determined only by observation. For instance, after observing e between 0 < z/- L < 2.5 Wyngaard and Cotd, 1971] fitted e/es with

( z 2/3)3/2 --•- 1+ 0.sl•l (2) In the free convection regime, i.e., where z/- L >> 1,

u. is dropped because it is no longer important, leading to the similarity parameters shown in the "Free Convection" column of Table 1. With one less variable, z/L is no longer a dimensionless length scale. As a result, dimensionless parameters are constant with height [Wyngaard et al., 1971], e.g., clef -- el Ji5 ø = const and xlx! -- const. The uniformity of e with height is one of the distinguishing features of convection and carries up into the mixed layer.

The Mixed Layer

Above the surface layer, potential temperature and velocity are nearly constant, demonstrating that most of the PBL is well mixed. Thus, it is not surprising that the energy- containing scales in the mixed layer are limited only by D. Consequently, similarity scaling in the mixed layer is like that in the free convection regime of the surface layer, but with D in place of z, as shown in the rightmost column in Table 1. (Notwithstanding, some parameters still vary with z.) Since the vertical velocities, w., dominate the motions, the time scale of the mixed layer is

D =- [4 (s)

W,

Typically, w, • 2ms -x giving Trot • 8min for D = 1000m.

As in the free convection sublayer of the surface layer, e is approximately constant with height. However, X falls off with distance from the surface, a result obtained indirectly by matching the scalings in the indistinct transition between the free-convection sublayer and the mixed layer,

X___ _ X Xrnl _ X (Z) 4/3 XJ' Xml Xl Xml •' (4)

(Here, Xml/X.i = (z/D) 4/:3 is obtained by inserting the similarity variables.) In the free-convection sublayer, X/X.t' is constant. Therefore,

X a(3) -4/3 = Xrn/

where a is a constant determined from observations. This scaling often apphes from the free-convection sublayer to deep within the mixed layer, and sometimes even to the top of the layer: temperature fluctuations carried down from the entrainment zone can be larger than those coming up from the surface.

The Entrainment Zone

When the PBL is growing upward, the inversion rises by entrMning air from aloft. The thickness and character of the entrainment zone vary greatly, depending on subsidence of air above the PBL, as well as on the surface fluxes, but active entrainment typically extends over 0.8 D _< z _< 1.2 D. Al- though some scaling has been successful in the entrainment

6276 LOMBARDO A• GItEGG- SIMILARITY SCALING DURING NIGHTTIME CONVECTION

4

13 14 15 16 17 18 19 20 21 22 23

October 1986

Fig. 2. Except for brief interruptions to relocate the ship, we operated AMP continuously, first alongside FLIP, then following the RINO float, and finally by FLIP again. The rate of profiling was somewhat less in the second half because the MSP was operating more frequently.

zone [Wyngaard and LeMone, 1980], it is not well coupled to mixed layer scaling, and we do not apply it in this paper.

3. OBSERVATIONS AND BACKGROUND

The Observations

In October 1986, as our contribution to PATCHEX, we took the R/V Thompson to 34 o N, 127 o W, located in the outer reaches of the California Current and the site of the

Mixed-Layer Dynamics Experiment (MILDEX) [Paduan et al., 1988]. We started profiling next to the Floating Instru- ment Platform (FLIP), which was anchored, then followed a midwater float (called RINO for Richardson number), and finished by FLIP again (Figure 2). By the time we returned, the RINO float had drifted 19 km from FLIP.

For our primary turbulence observations, we took M- most 700 profiles with the Advanced Microstructure Profiler (AMP), each going to 3 MPa (300m). We averaged 2.4 profiles per hour (Figure 2). Although this rate is adequate for studying the diurnal cycle, it is three or four times less than we could have achieved by observing only the SBL.

Sensors on the AMP resolved the dissipation spectra of centimeter-scale velocity and temperature fluctuations, allowing us to compute e and X by integrating spectra [Shay and Gregg, 1986]. ¾Ve used standard definitions, i.e., ½ ---- 7.5•'(Ou•/Ox3) 2 and X ---- 6nT(OTt/Ox3) 2, where x3 is vertical distance, • is the kinematic viscosity, and n T is the thermal diffusivity. Both definitions assume isotropy, accounting for the multiplier of 7.5 for e and a factor of 3 in the X definition. (The additional factor of 2 for X is standard for oceanographic turbulence measurements.) To remove any contamination from the initial AMP wobble or from the Thompson's wake, we avoided all data above 0.05 MPa and some above 0.1 MPa.

In addition to the AMP, we launched and recovered the Multi-Scale Profiler (MSP) and operated the Thompson's 150 kHz Doppler acoustic profiler, made by RD Instruments. The MSP data confirmed the turbulence levels measured

with AMP, Mthough they were less densely sampled. As background for this study, the Doppler measurements pro- vide nearly continuous shear profiles through the SBL. They were digitized every 4m, and we calculated shear by first-

differencing adjacent points. The vertical resolution, however, is somewhat larger than 10 m.

Atmospheric Forcing

The winds were weak, with speeds usually less than 10ms -x and Ex0 rising only once to 1 W m -2. Rob Pinkel had sampled a full suite of meteorological sensors on FLIP at four times per second, and graciously gave us a copy of the data for processing. From the 16 min summaries in Figure 3, we see that the calms of October 13-16 and October 20-22

were separated on October 17-19 by mild winds, which produced El0 = (0.2- 0.6)W m -2. On October 23, a passing front briefly raised E•0 to lWm -2. Even then, the time scale of wind fluctuations,

El0

Ts = 10El0 /Otl [s] (6)

was at least several hours, allowing wind-produced turbulence in the SBL to remain in equilibrium with the surface.

In addition to having a diurnal cycle, the surface heat flux, j•0, also varied over several days, modulated by changes in the Mr-sea temperature contrast and in the winds. The sea- surface temperature stayed near 18.5øC, fluctuating daily by several tenths of a degree. The air was usually several degrees cooler, but warmed both times the winds increased, particularly while the front went by. Falling air temperatures on the 17th and 18th, combined with the moderate winds, produced the largest nighttime heat losses, j•0 = 200- 300 W m -2. By comparison, on the 23rd the heat loss was only 100 W m -2 because the Mr was nearly the same temperature as the sea. As a result of these fluctuations, the cumulative heat loss, H(t) = f• pjqO dr, varied between 4-10 MJ m -2. Spread uniformly over the upper 50m, the maximum H(t) would change the temperature by

i i

1.o

o.t3

0.2

19 :•t• Sea Surface

D o •,

• -12 • -o -24 I

13 14 15 1• 17 18 19 20 21 22 23 October 1986

Fig. 3. Two calm intervals were interrupted by mild wincls and followed by a sharp front (top panel). The sea surface temperature was steady, except for small diurnal fluctuations, until it increased about 0.5øC after the front passed (middle panel). Throughout, the air remained cooler than the sea, contributing to the positive heat flux each night. However, the average nighttime heat loss varied by about a factor of 3 (bottom panel), causing the cumulative integral of the heat flux, H, to fluctuate every few days between net waxming and cooling.

LOMBAP•O AND GREGG: SIMILARITY SCALING DURING NIGHTTIME CONVECTION 6277

Sunset Sunrise

- O•IDX X (•00Q•

I--• 15 hours I i• 18 hours

, I , I , I , I • I , I , I , I , I , I , I , ..,

-2 0 2 4 6 8 10 12 14 16 18 20 22

GMT / hours

Fig. 4. The times when J• changed sign are plotted •s circles and the times of sunset and sunrise as crosses. Convection began when the buoyancy flux went positive, several hours before sunset, and continued until J• went negative, I to 3 hours after sunrise.

+0.06øC. Since we observed a net increase of 0.08øC in

the temperature of the nighttime SBL, weak advection also affected the heat balance.

The buoyancy flux, J•, reached its nighttime value within about an hour after becoming positive and had only minor excursions until about a half hour before going negative in the morning. The positive sign resulted from the net heat flux at night, plus a small contribution from the effective salinity flux due to evaporation, i.e.,

J• = (g/p) [(a/cp)Jqø + (/5s/(1 - s) ) q ] is œe je , where t• the coefficient ot thermal expansion, /3 is the coefficient of line contraction, s is the salinity in concentration units,

is the latent heat, and J• is the latent heat flux. Each night, convection began I- 2 i hours before sunset and continued until I -3« hours after sunrise (Figure 4), when the sun finally overcame the evaporative and longwave fluxes. Dur- ing these 15-18 hours, Y• was nearly steady from (1 - 3) x 10-?W kg -1 (Figure 1) for at least 13 hours. For example, during the night of the 17th, J• = (1.3+0.3) x 10 -? Wkg -z, and the fluctuations took at least an hour (Figure 5). Con- sequently, the buoyancy time scale [ Wgmgaard, 1973]

s• N (•) To =- ios•o/ot I exceeded 5 hours, providing ample opportunity for turbulence in the SBL to equilibrate.

Oceanic Background

Shear was low across the pressure range of the SBL (Fig- ure 6). But, how low must it be to not compete with surface forcing in producing turbulence? Our only recourse is to compare the PATCHEX data with observations above the equatorial undercurrent. During Tropic Heat at the start

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I •-Sunset Sunrise-• I

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.::::::: ::::: ::: :::: :::: :::::::::::::::::::::::::::: :;::::::::::: :;: :;: :;:;:;½•:: ::::::::::::::::::::::::::::: :•:;:•:•:•:•:•:•:;:•:• •:•:•:•:•:•:•:;:•:•:•:•:•:•:•::.. • • -:::: ::::::::: :: .:: :::::::::::::::::::::::: :::: :::: :::: :::::: :::::: ::::: ::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::: .:::::::: :::::::::::::::::::::::: :::. :::::::•:::•:::::•. •.•:•::•::•:•:•:•:•:•:•:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::•:•:•:•:•:•::•:•:•:•::•::•::::::::::::::::::::::::::::•::: •1. 0.4

0.7 ,I ,I,.,,,, 0 4 8 12 16 2o

GMT / hours

Fig. 5. The six stages of the diurnal cycle during a typical day (refer also to Table 2). We analyzed profiles from stage II, entrainment, and stage III, equilibrium. During both stages, d• was relatively steady and -L (shown by solid squares) was less than 10m. The shading of e is the same as in Figure 1.


0.1

0.2

0.3

0.4

0.5

0.6

0.7

October 1986

i i

19 2o 21 22 23

Fig. 6. Two-hour averages of shear 2, contoured by decades, with D superimposed. No diurnal shear cycle is evident, but shear • in the pressure range of the diurnal SBL is lower than in the thermocllne.

of the nightly deepening, the near-surface zone was weakly stratified, with shear e • 3 x 10 -4 s -e and e about 10 times the similarity scaling. As the deepening proceeded, density was homogenized, shear e dropped to • 5 x 10 -5, and e decreased to the similarity level (H. Peters et al., Merid- ional variability of turbulence through the equatorial undercurrent, submitted to Journal of Geophysical Research, 1989). In contrast, during PATCHEX shear e was alter- nately greater than and less than I x 10 -5 s -e, averaging about a decade less than in the undercurrent. Furthermore, the absence of a diurnal shear cycle leads us to conclude that turbulent production by shear was not important.

Typical of the location, an irregular salinity minimum containing thermohaline intrusions lay near 0.6 MPa (60 m), the maximum depth of the SBL (Figure 7). Some days, we found the intrusions in the deeper part of the SBL, contributing to the restratification. Because Shay and Gregg [1986] observed similar features in the Bahamas, we believe

0.0

Nig

• 0.3

0.4

0.5 :Da) 0'61•.• ' i•.• ' •.432.94 33.;423.73 23.760 2 40.0 0.6 -1.2

O / øC 10 3 salinity (70 / kg m '3 102N / rad s '• 102S / s '•

Fig. 7. Typical day and night profiles. During the night (heavy lines), the SBL usually descended to about 0.SMPa, where it encountered a large increase in stratification. During the day (thin lines), restratification produced N • 0.003s -1 over the pressure range of the nighttime SBL. We found the shear to be the same day and night. These profiles were taken October 18.

that intrusions usually accompany the daytime restratification of diurnal boundary layers and are not simply a pecu- liarity of the PATCHEX site. Their characteristics, though, depend on the local horizontal temperature and sahnity gra- dients. Like the intrusions found in the Bahamas, those during PATCHEX were weak and apparently did not affect the convection.

Below the seasonal thermochne, internal waves and dissipation were unremarkable and had the magnitudes that we believe typify the background. Using the MSP data, shear e calculated over Az = 10 m was close to the levels predicted by Garrett and Munk [1975] [Gregg and Sanford,1988]. The shear appeared completely random, and we found no trace of shear from persistent near-inertial motions. Dissipation rates were low and decreased with depth as e ~ N e, as predicted by interaction calculations for background internal waves [Gregg and Sanford, 1988].

4. SBL EVOLUTION AND SIMILARITY SCALING

Evolution o• the SBL

Although the changing sign of J2 forces the diurnal cycle, the state of the boundary layer is not simply convecting or nonconvecting. It also depends on whether J2 is steady or changing and whether the thickness of the layer, D, is growing, shrinking, or constant. In Figure 5 and Table 2, we distinguish six stages: I-IV were convecting, and V-V! were restratifying under a negative buoyancy flux. We restricted this study to stages II, entrainment, and III, equilibrium, because only then was J2 positive and steady, making Tb >> Tml. Stage II, however, may differ from stage III; during entrainment some of the energy released at the surface by convection goes into the growing potential energy of the deepening layer, and some goes into internal waves radiating from the oscillating entrainment zone.

The entraining layer advanced slowly. For instance, on October 17 it took 10 hours to grow 40m. As the layer


TABLE 2. Stages of the Diumal Cycle

Stage Characteristic Duration, hours

I. Initiation

II. Entrainment

1TI. Equilibrium

IV. Decay

V. Suppression

VI. Stratification

j•o positive and increasing

Jff steady and D increasing

j•o and D steady

j•o decreasing and D steady

j•o negative and D decreasing

j•o negative and SBL stratifying

Refer to Figure 5 for an illustration.

1%-2 •, T=2

4-16, T=9

0-12,T=7

T=2

T=3

deepened: D increased from 13 to about 45 m, w. increased from 11 to 20 mm s-!, and -L shrank from 13 to 8 m. Con- sequently, D/-L increased from less than 2 to over 15, and Tml lengthened from 15 to 45 min. As a result, the production of turbulence changed so slowly from wind stress to convection that Tml was always much less than the 5 hours estimated for Tb.

While the front passed over us on October 23, strong winds kept the SBL in the turbulent regime throughout the night (Figure 8). Although stage II began with D/-L < 1, after a few hours the winds rose and the buoyancy flux de-

creased with the dropping air temperature, plunging the Monin-Obukhov length well below the boundary layer. This one night let us observe the vertical structure of a deep, stress-driven layer and accounted for all but one of the hours having D/-L < 1 in the summary of turbulent regimes in Figure 9.

Scaling

Averaged throughout the boundary layer, dissipation rates tend toward cs and ½ml for the asymptotic forcing regimes. To compare all the data on the same criteria, we

j I

I-•- Sunset Sunrise -• I I I, , I . I , , I,

I I I I [ I

23 oct 1986

-4 , I , , , , I , , , I

0 ' I ' I' ' I ' ' ' I ' I

• 0.3 .... • ..... ?.;iiii;i,½1i•,.;i{i'ii,liliiiii,•iiii ................................ . ........................... ?,:';!;!'{i',•!:•ii!•,•,!?•?::: i:•'i }! 'ii!•iiii!111!11ii:;'i!!11iii!i:ii':! i!!i';• {! !!i{ if?'

[ II ':-::•i}•i•{{{•i{!•i•?:•:{i•½/•::::::.:.:.:..... ..... .:--:•:•:!:-:::--:-. ..............

=== • ---• 0.6 '• i••'"""" "• I 8=10 -8 W kg -1 I •- ' I

I ß . I • I I-

0.7 , i , •, . I , , , ! ,•,' , • , , •, , , , , , , 0 4 8 12 16 20

GMT / hours

•ig. 8. D•ing the •ght of October 23, a reduced buoyancy fl•, •d rising winds, &ove --• below the b•e of the SBL.

I I I , J I

6280 LOMBARDO AND GREGG: SIMILAR/TY SCALING DURING NIGHTTIME CONVECTION

-L/m 0 10 20 30 40 50 60 70 80 90 100

10 D/-L < 1 -

20 -

• 30

o o o o o

60 ................. ' ............... ' ......... ' ......... ' .........

Fig. 9. Hourly averages of D/-L reveal a wide range of conditions. The llne to the right is D/-L = 1, and that to the left is D/-L = 10. Most ratios duster near 10, indicative of strong convection in the bulk of the SBL. A few, however, have D/-L < 1; all of these, except one, occurred on October 23.

plotted 11(D - 5) f/½i•, dz and 11(D - 5) f/½iJ2 dz versus D/-L (Figure 10). (We began the integration at 5 m to avoid contamination by the questionable data close to the surface.) Ratios normalized by es cluster near 1 when D/-L < 1 but are between 10 and 100 when D/-L > 10. Conversely, those normalized by J2 approach 0.6 for large D/-L and rise to 1-10 when D/- L < 1. Therefore, wind-

10 3 ..... •

10 2

Cb '----,• 10 1

I

v 10 0

10-1

N

Q ,---,t.r) 10 0

I

v

101 ..... ,

x x o

x x

x

o

x

x

x

___

x x

x x • x o x

'xX x (ko

o

x

I I I I i I , [

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o .... Entraining x .... Equilibrium

o o x x x

o x o

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o x ' .,% x x 0.6 oø ø• ß

7o Xø øo(• ;'. .... x ø,, xX x f o % xX,.o2. 'b---o ..... x ............ x x o x .,_.x

o x o oo x- ..• .... x o x ø o x x o o Xo o o xo x XOoO o o o o 0% 0

x o o o o ø o

10 -1 .... ,I • I , 10 o 1 1 10 2

D/-L

Fig. 10. Vertical averages of •/•s and •/J•. Each point is an average of all drops taken in an hour. Wind-stress scaling (top panel) is a good normalization when D/-L < 1, but greatly underestimates the dissipation when convection dominates, D/-L > 10. Convective scaling (bottom panel) behaves in the opposite manner. Dashed and dotted lines show averages during entrainment (circles) and during equihbrium (crosses). Al- though data from the entrainment and equilibrium stages show no differences for wind-stress scaling, the data during equilibrium are consistently higher that those during entrainment for convective scaling (bottom).

0.0

0.5

1.0

1.5

œ/œs 0 -1 100 10

' ' ' ' .... i I' ....... i i

126 • i

D/-L < 1

œ/Jb ø 10 -1 10 0

i i

0.58

D/-L > 10

Fig. 11. When D/-L < 1, normalized dissipation profiles follow wind-stress scaling throughout the boundary layer (left panel). When D/-L > 10, profiles obey convective scaling (right panel). Shading marks the 95% confidence estimates calculated with the bootstrap procedure [Elton and Gong, 1983].

stress scaling works well when D/- L < 1, and convection scaling works well when D/- L > 10.

To examine the vertical structure during the wind-stress regime, we averaged •/½• for all times when D/-L < 1 (Fig- ure 11, left panel). The uniformity with depth clearly shows that similarity scaling normalizes the vertical decay, with no distinction between entrainment and equihbrium stages, although the scatter is large. (The shading shows the 95% confidence limits from the bootstrap estimator [E/ron and Gong, 1983]) The normalized dissipation rate begins dropping off near 0.SD, which is often the top of the entrainment zone. Plotting the wind-stress comparison versus z/-L (not shown) reveals the same uniformity with depth over the range 0.15 < z/-L < 0.8. They average 1.76, comparable to atmospheric observations. For example, evaluating (2) for z/-L = 0.5 gives e/es = 1.5. Therefore, when the wind stress dominates the turbulence, the PATCHEX dissipation rates match both the vertical decay and the magnitude predicted by similarity scaling.

Turning now to the convection regime and averaging ½/J2 for all times when D/-L > 10, the normalized dissipation rates fall off gradually with depth and scatter much less than during the wind-stress regime (Figure 11, right). The trend is very close to that previously found in both atmosphere and ocean (Figure 12), although the vertical average of 0.58 is slightly smaller. Returning to Figure 10 (bottom), we find that e/J2 is significantly lower during entrainment than during equihbrium. Separately averaging e/J2 for stage II and stage III reveals that the bias is statistically significant in the upper and lower parts of the SBL (Figure 13). Across the entire layer, the net differences are large: vertical averages of ½/J2 are 0.66 q- 0.13 and 0.44 q- 0.06 during equilibrium and entrainment, respectively. (The confidence hmits are 95%.) The size of the contrast suggests that the variability of vertical averages of •/J2 among observations at different times and places may result from varying ratios of equilib-

LOMBARDO AND GP•EGG: SIMILARITY SCALING DURING NIGHTTIME CONVECTION 5281

œ / Jb 0 œ / Jb 0 œ / Jb 0 iO-a lO-t lO o iO-a lO-t lO o lO-a iO-• lO o

: .. :: :;

0.4 --•-6 i • L-- • • • + • - • / Jb = 0.58• , •2• • / Jb = 0.72 :: • . • / Jb ø = 0.64 • • o. 6 - :: '•?'?•:: - • • - • -

•. o .................... 7"•:•"•=•? .... r .......................... • ...... '•"r ............................... • ................

..=.• HEX ' • •ing • ' + Atmosphere '

Fig. 12. The •ver•ge of • PATCHEX pro• when D/-L > 10 (left) is si• •o obseNa[lom d•g • cold- •k ou•bre• over • w•-core ring [•a• a•d •rcgg, 1986] (•d•e) •d [o •[mospheric me•emen[s [•en in Minneso• [I{aima• • a•., 1976] •d near Asch•ch in •he U•[ed K•g•om [Oa•gh•y and Pa•m•r, 1979] (right). However, [he ver•ic• •ver•ge fo•d d•ing PATCHEX was s•lx sm•er. ShrUg, on ghe left, •d horizon• b•s, •d•e •d •gh[, in, cage gS• co•dence

rium and entrainment stages in the data ensembles in the averages. For instance, both stages were included by Shay and Gregg [1986]. Having relatively lower dissipation rates during entrainment is consistent with some of the energy rele•ed by convection going into potential energy and some going into internal waves.

So far, we have matched data during the two asymptotic regimes with similarity scaling, but have not dealt with those

in between, i.e., when 1 < D/-L < 10. As noted in section 2, when working in part of this range, atmospheric scientists make empirical fits to e/e•. We reasoned that both wind stress and convection are significant sources in this midrange. Consequently, their sum, 1.76 e•-F0.58 J•, should be a better normalization, as indeed it is (Figure 14, left). Since it fits the intermediate data so well, we applied it to all our observations, with results that are almost as good

10 -1 10 0

Entraining I

0.5

1.0

½.-"•i!i ........ D /-L > 10

101 , i , , , ,11

Equilibrium _

Fig. 13. When D/-L > 10, •/J• dnring equilibrium is larger than it is during entrainment. The vertical average during equi- librim is 0.65, compared to 0.44 during entrainment.

.0 , i , i

0.5

1.5

s_, / (0.58Jlb • 1 76Ss) 10 -1 ø+ ' 10 ''''l

I

,-4- 0.84

I

s_,/(0.58Jlb • + 1.76Ss) 10-1 o 10

r' J•- 0.87

i - '""•ii :] -

_

D/-L>O

Fig. 14. Scaling • with 1.76•s +0.58•/J• works very well when 1 < D/-L < 10. Using this snm for all profLies (D/-L > O) works as wen as using 0.58•/J• alone works when D/-L > 10, as seen by comparing with Figure 11.


10 5

10 4 N

• 10 3

• '----,u') 10 2

I • 101

10 0

10-•

10

8

6

4

• 2 o

o• 0

o x o _

_

o

o

x x x x x - x o x -

o x o o x o o x o o o o o x

o o oX•oo x• o Xx •o - - x o Oo% xi• x o % •xx o o x x x o x o ø o x x x

x •o• xxo x •ffo • x x • x x -- Xo o o x- o ø øoøXx x o

o (• x ..... I I I

' ' ' ''1 I I

- 00 0 -

- o .... Entrainlug - - x .... Equilibrium - _ _

o o

- o 8 o - _ _

x o _ _

- o . o (1• - - x -4/3 • x %oo to o• 2 -

- •__x x oø •-•o '% x• .... x x o o - -2 -- •-•--- --F--•?•.½-•?•-• <•-•¾.•--•. ...... ½---o----- o o x (• 0'o-'•-- ' .... x .... x- o o ß x x •)"'"' x .... •<.x x

-4- o •'x x -•.... x •-xX - o ox o o x x x-

_ _

-6 o - o x - i i i I

'1'•)o 10 • 1• a D/-L

Fig. 15. Vertical averages of X/X• scatter much more than similar comparisons for dissipation (compare with Figure 10, top). In addition, they fail to converge at low D/-L during both entrainment and equilibrium stages. Slopes of log(x/x.•) converge to -4/3 when D/-L > 10, but not at smaller D/-L. Profiles taken during equilibriuxn approach -4/3 more closely than do those taken during entrainment.

(Figure 14, right). Therefore, the sum of the two asymptotic regimes offers a good scaling for conditions between the two regimes and an adequate scaling for all PATCHEX observations deeper than 5 m.

Scaling

Wind-stress scaling is less successful with X. Vertical av- 5 D ß ) X/X &, t,l to convg low

(Figure 15, top). Nevertheless, averaging all profiles when D/-L < 1, we find that X/Xs is uniform near 1 in the upper half of the boundary layer, in agreement with similarity scaling (Figure 16). The vertical averages fail to converge, owing to the large increases near the bottom of the boundary layer. Although most of the profiles were take n during equilibrium, we presume that the deep rise in X/Xs was produced by temperature structure from the entrainment zone and will investigate this further as part of a study of that zone.

Scaling for the convection regime also works in only part of the boundary layer. Computing the average slope of log(x/X•) between z/D: 0.1 and z/D = 0.7, we find a rough convergence toward -4/3 when D/-L > 10 (Fig- ure 15, bottom). Averaging X/Xrnt for all profiles when D/- L > 10 shows that the expected -4/3 slope fits the data well for 0.3 < z/D < 0.7 (Figure 17), ending near the depth where the entrainment zone is first encountered. By comparison, Brubaker [1987] found the -4/3 slope between 0.04 < z/D < 0.7 in a 5 m thick SBL in a lake, and Guillemet et al. [1983] reported it between 0.03 < z/D < 0.4 in the atmosphere (both are included in Figure 17). The scaling factor is a • 4, compared to 2 in the lake and 1.1 in the atmosphere. (Because Brubaker [1987] defined Xml to be

twice as large as our definition, we multiplied his value for a by 2 to put it on the same basis as ours. Similarly, atmospheric scientists usually define X without the factor of 2 we used. So we also multiphed the reported atmospheric value for a by 2.) Overall, in terms of z/D, the convective scaling regime for X extended as deep into the SBL as found previously in the atmosphere and in a lake. We, however, did not approach as close to the surface as was done in the lake and atmosphere.

5. SUMMARY AND DISCUSSION

While profiling for eleven days thrOUgh the surface boundary layer (SBL) of the ocean, we encountered a wide range of regimes producing turbulence. In this paper, we focused on those times when the ocean lost buoyancy at a steady rate to the atmosphere and asked how well • and X in the SBL were described by similarity •caling. The short answer is: very well for • and somewhat for X, provided that in both cases we consider turbulent production by wind stress as well as by convection. Because the range of D/-L is much more restricted in the ocean than in the atmosphere, in most situations both mechanisms of producing turbulence must be considered to obtain a complete description of the full boundary layer. (D is the depth of the SBL, and L is the Monin-Obukhov length.) Previous comparisons with similarity scaling in the SBL dealt with only one of the asymptotic regimes.

More specifically, the following cases are c•nsidered: 1. When D/-L < 1, wind-stress scaling removed the

z -• decay of stress-produced dissipation, at least for depths greater than 5 m, the shallowest we considered. In this comparison, we find no difference between entrainment and equi-

0.0

0.5

1.0

1.5

Iog[z / Zs] -1 0 1 2 3 4

I I I I

D/-L< 1

Fig. 16. The average of X/Xs for all profiles with D/-L < 1 demonstrates that wind-stress scaling works only in the upper half of the SBL. The shading marks the 95% confidence limits determined with the bootstrap procedure Elton and Gong, 1983].


10 -2

r• 10 '1 N

10 -2 10 4

10-1

I

Lake

I ' I

PATCHE (Z / 'D)-4/3

"•" 2 (z/D) '4/3 _

o

10 0 10 2 10 4 10 -2 17,, / Y,,ml

, I J I •

10 -2 10 ø 10 2 10 4 17,, / Y,,ml

10-1

Fig. 17. During PATCHEX when D/-L > 10, X/Xrat followed the expected -4/3 slope only for 0.3 < z/D < 0.7. Observations from a lake [Brubaker, 1987] and the atmosphere [Guillgrnet et al., 1983] had more extensive -4/3 ranges. As explained in the text, we multiplied the reported amplitude coefficients, a, for the lake and atmosphere by a factor of 2 to put them on the same basis as ours.

librium stages. The vertical average, 1/(D-5) • e/es dz = 1.76, is consistent with observations in the atmosphere [Wyngaard and Cord, 1971]. We believe that the 1.76 multiplier represents the additional contribution from the production of turbulence by convection.

2. When D/- L > 10, mixed layer scaling normalizes •, leaving only the gradual decrease with depth found before. The vertical average is 1/(D- 5)ff 6/J•)dz = 0.58, compared to 0.72 [ShagI and Gre##, 1986] and 0.(34 in the atmosphere. However, during entrainment the average is 0.44 q- 0.06 compared to 0.65 q- 0.13 during equilibrium, sug- gesting to us that some of the Variability in •/J• results from differing ratios of entraining and equilibrium stages in the data.

3. When 1 < D/-L < 10, both wind stress and convection are important and neither of the asymptotic scalings for • fits the observations. However, normalizing by 1.76 ½s + 0.58 J•, the sum of the two asymptotic scalings, fits the data very well when 1 < D/-L < 10 and fairly well for D/-L > 1.

4. When D/-L < 1, wind stre'Ss scaling normalized X only in the upper half of the SBL. Departures from the scaling were equally frequent during entrainment and equilibrium stages.

5. When D/- L > 10, mixed layer scaling for X showed the expected (z/D) -4/3 slope for 0.3 < z/D <'0.7. The scaling constant, a •-. 4, is twice that found by Brubaker [1987] in a lake and 4 times that reported from the atmosphere.

Initially, the general success of similarity scaling in the SBL surprised us. During the two periods of elevated winds, we observed the elongated surface streaks marking the convergence lines of Lungmuir cells. Although we could not relate them to individual AMP profiles, the streaks were frequent. Viewing the Lungmuir cells as sources of turbulence additional to those considered by similarity staling, we expected that ½ would greatly exceed the similarity levels in much of the data. Although additional turbulence from the Lastgmuir cells may contribute to the scatter, evidently the Lungmuir circulation is part of the large-scale energy of the SBL, just as large-scale coherent structures are an integral component of the PBL. Although we still expect to find some differences in dissipation rates in Lungmuir cells versus else- where, the net effect does not seem to invalidate similarity scaling for ½.

We are intrigued by finding that •/J• is less during entrainment than during equilibrium. In a separate study, K. Purvis (personal communication, 1988) takes 1- ½/Jo ø as the energy available for entrainment at the base of the SBL. The PATCHEX result indicates that the energy balance shifts between entrainment and equilibrium stages, per- haps reflecting greater energy losses during entrainment to internal waves and the potential energy of the stratification. If verified, this will need to be incorporated into models of the SBL.

Acknowledgments. The Office of Naval Research supported this work under contract N00014-84-C-0111. The success of the

6284 LO•B•O AND GREGG: SIMILARITY SCALING DURING NIGHTTIME CONVECTION

observatiohs was the result of careful work at sea and aahore by the AMP crew: Jack Miller, Wayne Nodland, Steve Bayer, Dale Hirt, Donna Sorensen, Hartrout Peters, joel Wesson, Pat McKe- own, and Gordy Welsh. The officers and crew of the R/V Thomp- son assisted the observations by patiently enduring the tedium of a long sequence of AMP stations. Dave Winkel, Keith Brain- erd, Hartrout Peters, and Harvey Seim gave us useful comments. Contribution 1792 from the University of Washington School of Oceanography.

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Caughey, S. J., and S. G. Palmer, Some aspects of turbulence structure through the depth of the convective boundary layer, Q. J. R. MeteoroL Soc., 105, 811-827, 1979.

Denman, K. L., and M. Miyake, Upper layer modification at Ocean Station Papa: Observations and simulations, J. Phys. Oceanogr., 3, 185-196, 1973.

Dillon, T. M., J. G. Richman, C. G. Hansen, and M.D. Pearson, Near-surface turbulence measurements in a lake, Nature, 290, 390-392, 1981.

Efron, B., and (•. Gong, A leisurely look at the bootstrap, the jadmife, and cross-validation, Am. Statist., 37, 36-48, 1983.

Garrett, C. J. R., and W. H. Munk, Space-time scales of internal waves: A progress report, J. Geophys. Res., 80, 291-297, 1975.

Gregg, M. C., ahd T. B. Sanford, The dependence of turbulent dissipation on stratification in a diffusively stable thermocline, J. Geophys. Res., 93, 12,381-12,392, 1988.

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Kaimal, J. C., J. C. Wyngaard, D. A. Haugen, O. R. Cot•, Y. Izumi, S. J. Caughey, and C. J. Readings, Turbulence structure in the convective boundary layer, J. Atmos. $ci., 33, 2152-2169, 1976.

Monin, A. S., and A.M. Obukhov, Basic laws of turbulent mixing in the ground layer of the atmosphere, Tr. Geofiz. Inst. Akad. Nauk $SSR, 2J (151), 163-187, 1954.

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Paduan, J. D., R. A. de Szoeke, and J. A. Richman, Balances of heat and momentum at 33.5 ø N, 127 ø W in the upper ocean during the Mixed-Layer Dynamics Experiment, J. Geophys. Res., 93, 8147-8160, 1988.

Panofsky, H. A., and J. A. Dutton, Atmospheric Turbulence: Models and Methods for Engineering Applications, 400 pp. Wiley-Interscience, New York, 1984.

Richman, J., and C. Garrett, The transfer of energy and momentum by the wind to the surface mixed layer, J. Phys. Oceanogr., 7, 876-881, 1977.

Shay, T. J., and M. C. Gregg, Turbulence in an oceanic convecting mixed layer, Nature, 31 O, 282-285, 1984.

Shay, T. J., and M. C. Gregg, Convectively driven turbulent mixing in the upper ocean, J. Phys. Oceanogr., 16, 1777-1798, 1986.

Wyngaard, J. C., On surface layer turbulence, in Workshop on Micrometeorology, edited by D. A. Haugen, pp. 101-148, American Meteorological Society, Boston, Mass., 1973.

Wyngaard, J. C., and O. R. Cot•, The budgets of turbulent kinetic energy and temperature variance in the atmospheric surface layer, J. Atmos. $ci., 28, 190-201, 1971.

Wyngaard, J. C., and M. A. LeMone, Behavior of the refractive index structure parameter in the entraining convective boundary layer, J. Atmos. $ci., 37, 1573-1585, 1980.

Wyngaard, J. C., O. R. Cot•, and Y. lzumi, Local free convection, similarity, and the budgets of shear stress and heat flux, J. Atmos. Sci., 28, 1171-1182, 1971.

M. C. Gregg, Applied Physics Laboratory and School of Oceanography, College of Ocean and Fishery Sciences, Univer- sity of Washington, Seattle, WA 98105.

C. P. Lombardo, Department of Oceanography, Naval Post- graduate School, Monterey, CA 93943-5010.

(Received October 4, 1988; accepted November 7, 1988)

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