JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL.
89, NO. C4, PAGES 6403-6414, JULY
20, 1984
Universal Similarity and the Thicknessof the Viscous Sublayer
at the Ocean
Floor
T. M. ellRISS! AND D. R. CALDWELL
Schoolof Oceanography,OregonState University
Experimentsconductedon the Oregon continentalshelfin June 1979 indicatethat the boundarylayer
flow at the seafloorwas hydrodynamically smooth. Fine-resolution velocity profiles are used to test the
assumptionthat the flow behavedlike a universallysimilar, neutrally buoyant flow over a smoothwall.
The non-dimensionalthicknessof the viscoussublayeris far more variable than has been observedin
laboratory studiesover perfectlysmoothwalls. This variability may conceivablybe related to upstream
changesof surfaceroughnessor to the presenceof distributedroughnesselements.Laboratory results
indicate that the near-bed flow adjusts extremely slowly to upstream roughnesschangesand suggests
that flow near the sublayerin the oceanmay be influencedby roughnesschangesas far as tensof meters
upstream.
Data gatheredduring that experimentwill be usedto test the
applicability
of the assumptionsthat lead to the conventional
The simplestmodel that might describethe flow near the
seabedis provided by laboratory studiesof neutrally stratified equationsfor suchflows. This questionis far from academic.
turbulent boundary layers [Monin and Yaglorn,1971; Town- Papers dealing with sedimenttransport and boundary layer
send,1976; Yaglorn,1979]. This model is especiallyappealing processes[e.g., Bowden, 1978; Wirnbushand Munk, 1971;
becauseuniversal similarity applies to these flows; experi- Komar, 1976] often cite equationswhich imply that, if a vismental results,when expressedin the appropriate nondimen- coussublayerexists,the flow behaveslike a universallysimisional forms, are independent of flow conditions and fluid lar, neutrally buoyant flow over a smoothwall.
properties.In particular, profiles of mean velocity become
THEORETICAL BACKGROUND
"universal"when velocitiesand distancesfrom the boundary
The developmentof Monin and Yaglorn[1971, chapter 3] is
are appropriatelynondimensionalized.
recastslightly here to emphasizethe significanceof the viscous
In an earlier paper [Caldwell and Chriss,1979], we demonsublayer.Given a sublayerin which the molecular viscosity,v,
strated the existenceof a viscoussublayerin the near-bed flow
dominates the vertical transport of momentum, the stressat
on the Oregon shelf. A more detailed examination of adthe wall, Zo,is givenby
ditional data from that experiment revealed a "kink" in the
% = •v (•U/•z)
(1)
velocity profiles 11-15 cm above the bed. We concludedthat
the flow above the kink was significantlyinfluencedby form wherep is the densityof the fluid and • is the meanvelocity
drag due to topographic irregularities [Chriss and Caldwell, at a distancez from the wall. Defining the friction velocity by
1982]. Although the flow was "hydrodynamicallysmooth" in
u, = Co/p)'/2
(2)
the sensethat a viscoussublayerwas always present,the true
bed stress(computed from the velocity profile in the viscous integrating(1) yields,for the profile in the sublayer,
sublayer)was severaltimes smaller than the stresscomputed
U(z)= u,2z/v
(3)
from the portion of the velocity profile influenced by form
drag. A subsequentexperiment(June 1979) again confirmed In the overlying turbulent logarithmic layer, the shearis given
the existenceof a viscous sublayer. Although the profiling by
sensorssampled only the lowermost few centimetersof the
boundary layer, independently measured currents 59 cm
- u,/k
(4)
above the bed agree very well with those predicted from the
c3(lnz)
sublayer data and a constant stress assumption, suggesting
that form drag effectswere insignificantduring this later ex- which defines von Karman's constant, k. Integrating (4), we
obtain
periment.We might, then, picturethe near-bedflow as closely
resemblinglaboratory flow over a smooth wall, but perturbed
U(z) = (u,/k)In (z) + C
(5)
in somelocations by the effectsof small-scaletopography.
where C is a constant of integration. Defining the sublayer
Because the data from June 1979 show no measurable influenceof topographicirregularities,we use them here to exam- thickness,rS,by the intersectionof the extrapolated sublayer
ine the hypothesisthat, in the absenceof such influence,the profile(3) and the logarithmicvelocityprofile (5), we obtain
INTRODUCTION
flow can be appropriately describedas a universallysimilar,
neutrally buoyant boundary layer flow on a smooth wall.
U,2(•/¾--(u,/k)In (rS)
+C
(6)
Therefore C is explicitly a function of rS'
C -- u,2(•/¾
- (u,/k) In r5
• Now at Departmentof Oceanography,
DalhousieUniversity.
Copyright 1984 by the AmericanGeophysicalUnion.
(7)
Thus for the logarithmic portion of the turbulent boundary
layer,
Paper number 4C0327.
0148-0227/84/004C-0327505.00
U(z)= (u,/k)In (z/•5)
+ u,2•5/v
6403
(8)
6404
CHRISSAND CALDWELL' UNIVERSAL SIMILARITY AND VISCOUSSUBLAYER
cludesnot only the "true" viscoussublayer (the linear velocity
profile region) but also a portion of the "buffer" region, in
which molecular viscosityand turbulence both contribute to
the vertical momentum transport. Although some authors
define the sublayer thicknessby the linear region alone, our
definition has the advantage that it can be directly related to
the constantB in the logarithmic layer equation. Furthermore,
becausedeterminationof the highestlinear point is strongly
influenced
bythevertical
sampling
intervalandmeasurement
error, estimatednondimensionalthicknessesin laboratory experiments show more variability when defined by the linear
profiles.For universalsimilarity to hold, the nondimensional
thicknessof the viscoussublayer,definedin either way, must
be constant [Monin and Yaglom, 1971]. Data presentedlater
in this paper demonstrate that our conclusionsregarding
/
variabilityof • + are independent
of our choiceof definition.
Not all flows over smooth walls obey universalsimilarity.
For example,the "constant"B in (10) may be larger than 25 in
Fig. 1. Theoreticalmean velocity profile for a logarithmiclayer some smooth-walled boundary layer flows of polymer soluoverlyinga viscoussublayer.A logarithmicscalehas been usedfor
tions [Huang, 1974]. In wind tunnel studiesby Antonia and
the distancefrom the boundary.
Luxton [1972], Mulhearn [1978], and Andreopoulosand Wood
[1982], boundary layer flows adjusted very slowly to changes
from rough to smooth surfaces.The "constant"B in equation
This form of the logarithmicprofile equationdemonstrates (10) may be as large as 15 near the change of roughness,and
that the mean velocity in the logarithmic layer above a tens of boundary layer thicknessesdownstreamit may still be
smoothwall depends
not onlyon u, but alsoon 6 (Figure1).
as large as 7.
U(z)
Definingthe nondimensional
distancefrom the wall,z +, the
nondimensional
sublayerthickness,
6 +, and a nondimensional
EXPERIMENT
meanvelocity,U +, by z/(v/u,),6/(v/u,),andU/u,, respectively, The experiment was carried out June 5-13, 1979, on the
equation(8) becomes
U + = (l/k) In (z+/g+) + g+
(9)
If k and • + are "universalconstants,"
thisequationdefinesthe
logarithmic portion of all neutrally buoyant turbulent
boundary-layerflowsover a smoothwall.
It is important to clarify the relationship between (9) and
the commonly cited nondimensional equation [Monin and
Yaglom,1971,pp. 276--277],
U + =Alnz
+ +B
(10)
Comparison of (9) and (10) demonstratesthat B is related to
/i+ by
90-m and 185-m isobathsalong 45ø20'N on the Oregon shelf.
The surface sediment at the 90 m station is a fine sand,
whereasat 185 m it is silty sand [Runge, 1966]. A roughness
Reynoldsnumber,u,d/v, basedon the maximumparticlediameter(d = 0.007cm),the largestu, (0.48cm/s),and the kinematicviscosity
(0.015cme/s),is 0.22,wellbelow5, the upper
limit for hydrodynamicallysmoothflow [Schlichting,1968]. A
time-lapsemotion picture camera,mounted on the instrumented tripod to monitor the condition of the sensors,showedno
bedformslikely to influence the near-bed flow. However, becauseof the relativelypoor quality and restrictedfield of view
(•1 me) of the photographs,
the absenceof suchfeaturesin
the photos is inconclusive.The velocity data were obtained
from profiling heated-thermistorsensorsmounted on the 2-mB = -A In (rS+) + r5+
(11)
high tripod. Most data (24 mean velocity profiles)come from
In laboratory studies of flows over smooth walls without two deploymentsat the 185-m station. Four additional proroughnesschanges,A and B do not vary significantly with
fileswere obtainedduring one deploymentat the 90-m station.
fluid propertiesor flow conditions,thus lending support to A data acquisitionsystemon the tripod sampledeachthermisuniversal similarity [Monin and Yaglom, 1971, p. 277]. The tor at an interval set between2 s and 4 s, dependingon the
most reliable data [Monin and Yaglom, 1971' Yaglom, 1979] deployment duration, 1-3 days. Additional instrumentation
indicate that A is close to 2.5 (corresponding to avon
included profiling and stationary temperature sensors,a SaKarman's constant of 0.4) and that B is between 5 and 5.5. vonius rotor, a 25-cm path-length beam transmissometer,and
(Although various investigatorscite slightly different values a high-resolutionpressuretransducer.
Each thermistor (ThermometricsInc. seriesFP14) consisted
for A and B, the differencesare small and are usually attributed to measurement errors or to differences in the obof a 0.02-cm-diameterthermistor bead sealedin the tip of a
servedz + ranges.)It shouldbe clear from (11) that universal 0.2-cm-diameter,1.2-cm-longglassrod. This rod was epoxied
similarityrequiresthat the nondimensional
sublayerthickness to the end of a 0.4-cm-diameter,9-cm-longpiece of hypoder•+ is constant and that its value lie• between 11 and 11.6
mic tubing attachedto the profiling mechanism.Current was
(correspondingto B = 5-5.5). Becausethe differencebetween suppliedto each heated thermistor to heat it approximately
•5+ = 11•1and •5+ = 11.6 corresponds
to a changeof at most 20øCabove the water temperature.The temperaturedifference
2% in U(z), the differencebetween these two values is not betweenthe thermistor and the water dependson the power
significant,so we will refer to the laboratory•+ as 11.1.We dissipatedand on the rate of removal of heat by the water
emphasizethat this value,like equation(11), is basedon oper- flow. Because the calibration is a function of both the water
ationally defining the sublayerthicknessby the intersectionof temperatureand the orientation of the flow with respectto the
the extrapolatedlinear and logarithmicprofiles.Thus •+ in- thermistor,each thermistorwas postcalibratedat the temper-
CHRISS AND CALDWELL:
UNIVERSAL SIMILARITY AND VISCOUS SUBLAYER
6405
atures and flow directionsobservedduring the experiment.
Calibrations were performedby towing in a 1-m radius annular channel. The power dissipatedin the thermistor per unit
changein temperaturewasrelatedto the flow velocityby
P/AT = a + bUN
(12)
where P is the power dissipatedin the thermistor, AT is its
temperaturerise, U is the speed,and a, b, and N are experimentally determined. Inversion of this formula allows the calculation of speedfrom measurementsof P/A T computedfrom
the output of a bridge circuit [Caldwell and Dillon, 1981]. By
using(12) with empiricallydetermineda, b, and N, speedcan
be determinedwithin 0.1 cm/s in the laboratory. No changes
in calibration
were observed for sensors calibrated
0.5-
both before
and after field deployments.The heated thermistorsmeasure
speedonly, flow directionbeingdetermined(to within 5ø)by a
0.20
small vane.
The heated thermistors were carried up and down by a
crank-and-pistonmechanismdriven by an underwatermotor.
The profiler mechanism was mounted 0.5 m outside one
tripod leg, assuringunobstructedflow through an arc of 300ø.
Only casesof unobstructedflow were chosenfor analysis.The
profiling period was set at either 57 or 566 s. The vertical
travel was 6 cm. To ensurethat the thermistorspenetratedthe
viscoussublayer,we allowed the thermistorsto penetrate the
sediment(in somecasesas much as 4 cm). Becausethe lighting
and resolution of the Super-8 camera systemwere inadequate
to define details in the area where the thermistorspenetrated
the sediment, there is no direct information on the extend to
which the penetration disturbed the interface. However, becausethe sensordiameter was very small, we believethat the
resulting disturbance did not affect the measurements.The
agreement of the heated thermistor profile with Savonius
rotor measurementssupports this assumptionand suggests
that the effects of sensorfouling were insignificant(see Discussionsection).
The vertical position of the thermistorswas determined(to
within 0.03 cm) from the output of a potentiometerconnected
to the profiler motor. The position of the sediment-water
interface(for each mean velocity profile) was taken to be the
I
2
3
4
5
6
0 (cm/s)
Fig. 3. Samedata as in Figure 2 but with a logarithmicscalefor the
distance above the sediment.
zero-velocityinterceptof the linear velocityprofile in the viscous sublayer. The calculated current velocity does not decrease exactly to zero within the sediment, but instead becomes constant at 0.1-0.4 cm/s, dependingon the particular
sensor. This happens because the velocity calibrations for
water are not valid in the sediment, which has different ther-
mal properties.The varianceof the signaldropsnearly to zero
at the intercept,confirmingit as the positionof the sedimentwater interface.
Becausethe profiling mechanismhad its own referencing
"foot," its vertical positionwas relativelyconstantin spite of
potential tripod settling.Nevertheless,the calculatedposition
of the zero-velocity intercept sometimesincreasesslightly
(typically by less than 0.1 cm) from one data interval to the
next, suggestingthat someminor settlingof the profiler "foot"
may have occurred.Becausethe mean velocity data for each
of the studied intervals is referencedto the position of the
zero-velocityintercept for the same interval, the effect of settling on the sublayerdata is negligible.
The Savoniusrotor was calibratedin the tow facility at the
Bonneville Dam laboratory of the U.S. Army Corps of Engineers.The standarderror of the calibrationequationis 0.3
cm/s,and the rotor thresholdis approximately2.0 cm/s.
ANALYSIS
•D
-
W
W
Mean velocity profileswere constructedby averagingover
periods9-132 min long, eachperiod containingmultiple traverses.With the exceptionof a few short (10-21 min) intervals
in which the flow was accelerating or decelerating
(0.002< IOU/Otl< 0.012 cm s-e), the studiedintervalswere
chosenfor steadinessof speedand direction.The lower por-
W
Z
0
I
;)
3
4
5
6
U (cm/s)
Fig. 2. Typical mean velocity profile for the June 1979 experiment.The lowermoststraightline represents
a linear fit to the points
in the viscoussublayer,whereasthe uppermostline representsa fit to
thepointsin the lowerportionof the logarithmiclayer.
tion of each traversewas divided into segments0.1 cm thick
for averaging,and the mean for eachsegmentwas determined
by averagingall measurementswithin it during the repeated
traverses.In a typical averaged profile (Figure 2), the linear
profile just above the sedimentdemonstratesthe existenceof a
viscoussublayer.A semilogarithmicplot (Figure 3) showsthat
the upper sectionof the profile looks like a logarithmiclayer.
(In the logarithmic layer, velocitieswere averaged over 0.3
cm.) The shear in the sublayerwas determinedby linear regression,and the bed stress and the friction velocity were
computedby using(1) and (2).
6406
CHRISS AND CALDWELL:
-
/
-
E
UNIVERSAL SIMILARITY AND VISCOUS SUBLAYER
/
$=14.0
u/u. / • /
$: 12.9•/u.'"""¾
/ •
ence our estimate of k. Becauseof potential tripod settling
and/or irregular microtopographysurroundingthe tripod, the
effectivelocation of the rotor may not have been exactly at 59
cm above the mean position of the sediment.Having no evidenceof either effect, we simply note that, for thesedata, k
changesonly by 1.5% for each 3-cm error in the effectiverotor
position.Basedon the physicalnature of the sediment,we feel
that tripod settling in excessof 2-3 cm was unlikely and thus
that a "corrected"
estimate
of k would
be no less than 0.42.
(Even a very large tripod "settling"of 10 cm would reduceour
estimateonlyto 0.41).For typicalconditions
(u, • 0.25cm/s),
0
0045
0 090
0135
•/u.(cm)
Fig. 4. The thicknessof the viscous sublayer plotted versus the
ratio of the kinematic viscosityto the friction velocity.The solid line
representsthe expected relationship for a nondimensionalsublayer
thicknessequal to 11.1. Broken lines representnondimensionalsublayer thicknesses
equal to 8.3, 9.3, 12.9,and 14.0,respectively.
Sensorssometimesmalfunctioned(due to seawaterleakage)
or were broken during recovery.We allowed for thesecontingenciesby usingseveralsensorsand alwaysrecoveredat least
one functioning sensor for postcalibration.Comparison of
profilesfrom two thermistorsseparatedhorizontally by 11 cm
during one of the 185-m deploymentsshowscloseagreement
in sixof tenintervals,
theaveragedifference
between
u, values
an error of 0.4 cm/s in the measuredrotor velocity resultsin
an error of 0.07 in k. Considering the standard error of the
rotor calibration (0.3 cm/s), the error bounds(+0.05) on our
estimateare not surprising.Somesuggestions
in the discussion
section,if correct, make our estimate of k somewhat uncertain.
The Thicknessof the ViscousSublayer
One of the implicationsof universal similarity is that the
nondimensional
sublayerthickness,
3 +, is constant.We have
determinedthe thicknessof the viscoussublayerfrom 28 mean
velocityprofiles.When the thermistordata did not encompass
enough of the logarithmic layer to determinethe log layer
shearaccurately,(4) was usedwith k -0.4 to extrapolatethe
velocityat the top of the heatedthermistorprofile downward
in order to determinethe log-linear intercept.We have used
the acceptedvalue of k = 0.4 for subsequentcomputations
because this value is within
the error bounds for our estimate.
being only 3%. The four remainingintervalsshow an average Use of the measuredvalue, k = 0.43, in calculationsof • + and
U (59 cm) would change the calculated quantities by only
differenceof 25%. Although thesefour profilesdo not agreein
the sublayer,they agree very well above, suggestingthat the + 1% and -3%, respectively,changesinsignificantwith redifferencesmay reflect real differencesin the near-bed flow spectto other potentialerrors.By this procedure,3 + values
rather than sensor malfunction.
If the differences were caused
by sensormalfunction,the malfunctionsmust have been episodic, for theseintervals are bounded by others in which the
two sensorsagree closely. The differencesin the flow could
have been the result of tiny topographic irregularities,their
influence being limited to flow from certain directions. Becauseour photographyis inadequate to show such features
and becausethe only argumentsfavoring sensormalfunction
are that the two sensorsdo not agree, the data from both
sensorswere used when they differed. Inclusion of data from
both sensorsdoes not significantlyaffect the conclusionsof
thispaper.
Estimationof yon Karman's Constant
varies from 8 to 20.
DISCUSSION
The variabilityin 3 + suggests
somequestions:
1. Does the scatter (Figure 4) reflect true variation or experimental error .9
2. If the variationsare real, are they causedby small-scale,
local irregularitiesof the seabed?
3.
Given u, (from the sublayerprofile)and the shearin the
logarithmic layer, (4) can be solvedfor k. Becausethe traverse
was limited to 6 cm to increase resolution
are determined from heated thermistor data alone, and so are
independent
of the rotordata.A plot of $ againstv/u, (Figure
4) showsthat $ doesscalewith v/u,, but not perfectly.
Rather
than lying between11 and 12 as in laboratory studies,3 +
in the viscous sub-
layer, and becausethe thermistorspenetratedthe sedimentat
the bottom of the profile, enoughof the logarithmiclayer was
not always covered for accurate determination of the shear
(and hence k) by using heated-thermistordata alone. Therefore we determined the log layer shear from the difference
betweenthe mean velocity at the top of the traverse and the
mean velocity from the Savonius rotor, the center of which
was approximately 59 cm above the sediment. For the 23
profiles in which the rotor was above threshold, k is
0.43 + 0.05 (95% confidenceinterval for the mean). This compareswell with other estimates,including0.419 + 0.013 from a
turbulent Ekman layer in the laboratory [Caldwell et al.,
1972], 0.41 + 0.025 from a summaryof atmosphericestimates
[Garratt, 1977] and 0.40 + 0.01 from recent marine estimates
[Soulsbyand Dyer, 1981]. Several types of errors may influ-
What is the effect of a variable 3 + on the flow above?
The most significanterrors are probably causedby the sampling scheme.The close agreement of predeploymentand
postdeployment
calibrations,
as well as the agreement
of u,
values calculated independently from two sensors,makes
seriouscalibrationdrift unlikely,and the useof singleprofiling
sensorsto determinethe profiles(and hence3+) eliminates
intercalibrationerrors.This advantageof usinga singleprofiling sensoris somewhatoffset,however,by the samplingerror
which resultsfrom the lack of continuoussamplingat each
elevation.Although each sensorwas sampledevery few seconds, the samples at any given elevation occurred far less
frequently,the exact timing being determinedby the profiling
period and the position within the profile. (Becauseof the
crank-and-piston nature of the profiling mechanism, the
number
ofsamples
andthetimingof thesesamples
varie
s with
the position in the profile.) If the flow has significantvariability on time scalesshorter than the interval between successivesamplesat a given height,the calculatedmean velocity
elIRISS AND CALDWELL'
UNIVERSAL SIMILARITY AND VISCOUS SUBLAYER
may differ from the mean which would have been measured
by samplingcontinuouslyat that elevation.
A sampling error analysis (appendix) was performed by
taking data from another experimentduring which the sensors
remained almost stationary within the viscoussublayer and
buffer layer (and for which continuous time seriesare available) and then subsamplingby usingthe profiler periodsand
sampling scheme used to collect the data reported in this
6407
16
14
paper.This analysissuggests
that,if •5+ for the "true"velocity
profile were 11.1, sampling error would result in approxi-
matelyone •+ measurement
in twentybeingsmallerthan 8.3
or larger than 14.0. The bounds of 8.3 and 14.0 are conservative becausethey are calculated assumingthe simultaneous
occurrenceof two types of errors which may be uncorrelated.
0
More realistic estimates are 9.3 and 12.9. The number of •+
4
6
8
I0
ROTOR U (59crn)
measurementsoutside even the conservativelimits (Figure 4)
is far greaterthan would be expectedif • + were always11.1
2
12
14
16
(crn/s)
from the laboratory value cannot be attributed to sampling
Fig. 6. The velocity at 59 cm determined by the Savonius rotor
versusthat calculated by using the measuredvalues of the friction
velocity and the sublayerthicknessin equation (8). The straight line
indicatesa 1:1 relationship between the measuredand calculated ve-
error alone.
locities.
and if all deviations
from this value were the result of sam-
pling error. Thereforedeviationof the measured•5+ values
The conclusion that the measured variations in • + are real
is supported by a comparison of the mean velocities determined by the Savoniusrotor 59 cm above the sedimentwith
the velocitiesat 59 cm expectedbased on sublayermeasurements alone. We first evaluate the hypothesisthat the varia-
average difference between the rotor-derived velocities and
thosecalculatedfrom the heated thermistorsis only 0.5 cm/s,
not much larger than the standard error of the rotor calibration (0.3 cm/s). Since the Savoniusrotor was separatedhori-
bility in •5+ is due to error in determining•5,the true valueof
zontallyfrom the thermistors
by 1.3 m, the measuredu, and
•+ being11.1.By usingu, for eachtimeintervaland the •5+ valuesmust representthe averageflow conditionsover
assumption
that •5+ - 11.1,we have used(8) to calculatethe
expectedmean velocity at 59 cm for each interval, taking
k -0.4. The calculatedand measuredvelocitiesdo not agree
very well (Figure 5). The agreementis markedly improvedif
several square meters (or more) of the seabed, rather than
merely reflectinglocal influences.
As a furthertest,we usethe u, valuesfrom the sublayerto
calculatethe • + requiredaccordingto (8) to predictthe rotor
the measured•5+ valuesare used(Figure6). The observation velocity.If the deviationsof the measured• + valuesfrom a
that a variable•+ yieldsbetteragreement
than a constant•+
constant value were caused by experimental error or very
againsuggests
that • + is truly not constantand that the scat- small scale irregularities,the "required"and measured•5+
ter in Figure 4 is not primarily due to experimentalerror.
values should be uncorrelated. On the other hand, if the mea-
The fact that the u, and •5valuesusedin (8)came entirely sured thicknessesrepresent near-bed flow conditions over a
from heated thermistorsand yet closelypredict the rotor velocities is evidence for the quality of the heated thermistor
data. This agreementalso suggeststhat form drag does not
measurably influence the flow in the lowest 59 cm. The
16
14
horizontal scaleof severalmeters,then there should be a sig-
nificantcorrelation.A plot of the "required"and measured•5+
values shows a definite relationship (Figure 7). Testing the
hypothesisthat thesevariablesare uncorrelatedis equivalent
to testing the hypothesisthat a regressionline through the
points in Figure 7 would have a slope of zero. Results of a
statistical analysis using the t test [Draper and Smith, 1966]
indicate that this hypothesiscan be rejected at the 99.9%
confidencelevel.This supportsour conclusions
that • + is not
constantat the ocean floor and that most of the variability in
6+ is not the result of local small-scaleirregularities,but
rather reflectsflow variability extendingover horizontal scales
of several meters.
Are deviationsfrom the "universalvelocity profile" (10) significantly greater than the deviations of typical laboratory
data? Figure 8 is a standard nondimensionalplot of our mean
velocityprofiles.The line U + - z+ (for thelinearprofilein the
viscoussublayer)and the line U + -2.5 In z + + 5.1 (for the
0
2
4
6
8
I0
12
14
16
ROTOR 0 (59crn) (crn/s)
Fig. 5. The velocity at 59 cm determinedby the Savoniusrotor
versusthat calculatedfrom the friction velocity determinedin the
viscoussublayerwith the assumptionof a nondimensionalsublayer
thicknessof 11.1. The straight line indicatesa 1:1 relationshipbetween measured
and calculated
velocities.
"universalvelocity profile" in the logarithmiclayer) have been
includedfor reference.The scalingimplied by universalsimilarity fails to collapse our data on the "universal profile."
Comparisonwith Figure 9 (reprinted from Monin and Yaglom
[1971]) demonstratesthat deviationsfrom the "universalprofile" are much larger than the deviationsin the elevenlaboratory investigationsreported by Monin and Yaglom.
When a different symbol is used to identify each of the
profiles upon which Figure 8 is based, it becomes obvious
6408
ellRISS AND CALDWELL'
UNIVERSAL SIMILARITY AND VISCOUS SUBLAYER
23
eralizationof the Blackwelderand Haritonidis velocityprofile
data to other boundarylayer experimentsmay yield erroneous
results,and because
we haveno evidence
that U,,ogis larger
thanU,vi•• in our study,wefeelthat otherexplanations
for our
5+ variabilityshouldbe explored.
What are the consequences
of this variability of 6 +? Computations of shear stress,based on a single log-layer sensor
and the assumption
that 5+ = 11.1,will overestimate
the true
stressif 5 + is larger(equation(8)).This is alsotrue for calculations using (10) with B = 5.1. With our data, differencesbetween the true stress and that calculated
from the rotor veloci-
ty assumingB = 5.1 (5+ = 11.1) are as large as 60% of the
true
77 , , , ad
II , , , 1,5
I , , , 19I , , ,
MEASURED 8+
Fig. 7. The measurednondimensionalsublayer thicknessversus
the sublayerthickness"required" to predict the measuredrotor velocity (at 59 cm), given the friction velocityfrom the sublayermeasurements.The error bars indicate the changein the "required" nondimensionalsublayerthicknesswhich would be causedby an error of
0.3 cm/s in the measuredrotor velocity.
(Figure 10) that the deviationsfrom the "universalprofile" are
not due to measurementerror, but rather to variationsin 6 +
The slope of the profilesis consistent,however,with A • 2.5
(that is, with k • 0.4). Not only does 6 + (as we have operationally definedit) vary (Figure 10), but so doesa nondimensional sublayerthicknessdefinedby the linear velocity profile
alone. Thus it is clear that our conclusion(that 6 + is not
constant)does not depend upon any particular definition of
stress.
While our experimentsare, to our knowledge,the only studies in which velocity measurementshave been made within the
viscoussublayer of a geophysicalflow, several other experi-
mentsalsosuggest
that 5+ may vary.Csanady[1974] summarized the resultsof experimentsby Portman [1960] and $heppard et al. [1972] in which wind velocity profiles over lakes
indicated that Zo was significantlyless than expectedfor a
universallysimilar,neutrallybuoyantflow. Equating[Y(z)=
(u,/k) In (Z/Zo)with (8), it is clearthat for smoothflow, the
so-calledroughnesslength Zois simplya functionof the thicknessof the viscoussublayer:
Zo= 6 exp (-6u,k/v)
(13)
and that the commonlyusedexpression
zo = v/9u,
(14)
is appropriateonly for the specialcasein which 6 + = 11.6.
Becausethe Portman [1960] and Sheppardet al. [1972] data
indicatedthat Zowasoftensmallerthan v/9u,, Csanadyconcludedthat 5 + was often larger than 11.6. He further con-
6 +'
For completeness,
we note that velocity profile data from a
recentlaboratory study of a smooth-walledturbulent boundary layer [Blackwelderand Haritonidis, 1983, Figure 2] show a
degreeof scatterin the logarithmiclayer which is not tremendouslydifferentthan that of our Figure 10. Although individual mean velocityprofilesare not identifiedin their figure,the
scatterin the log layer data is not inconsistentwith that ex-
pectedfrom(9) if 6+ is allowedto vary fromapproximately
13
to 16 (assumingk = 0.4). Becausethe aboverangeof 6 + esti-
40
i
'
' ....
'l
'
'
......
I
'
'
' .....
I
30
mates (togetherwith potential samplingerrors) appearsade-
quate to explainmost of the 6+ variabilityobservedin our
study, we feel that these data deservefurther comment. Although the emphasisof their study was on the "bursting"
phenomenonand not on the form of the mean velocityprofile,
ß
20
the authorsdraw attentionto the atypicallylarge U + values
in the logarithmicregion as well as to the large scatterin these
values.They suggestthat the reasonfor theseobservationsis
relatedto the fact that the u, valuesusedto nondimensionalize the velocityprofileswere derivedfrom the velocitygradient in the viscoussublayer,rather than from the slopeof the
logarithmic
profile.Theyfurthernotethat U,,og(derivedfrom
the logarithmic profile, apparently assuming k = 0.41) was
typically15% larger than U,visc
(derivedfrom the sublayer).
Although generalization of these results might suggestthat
io
other experiments (see, for example, Bakewell and Lumley
[1967, Figure 1]; Wallace et al. [1972, Figure 1]) do not show
this effect. Furthermore, these data seem to be consistent with
theassumption
that U,,o•= U,vi•c.
Because
it appearsthatgen-
IOO
IOOO
z
logarithmicvelocityprofilesnondimensionalized
by U,visc
will
deviate significantlyfrom U+= 2.5 In z + + 5.1, data from
Z+=
Fig. 8. Nondimensionalplot of all the velocity data from this
study.The straightline is for the so-called"universalvelocityprofile"
in the logarithmiclayer. The gap in the data representsthe distance
betweenthe top of the heatedthermistorprofile and the rotor located
at 59 cm above the sediment.
CHRISS AND CALDWELL: UNIVERSAL SIMILARITY AND VISCOUS SUBLAYER
6409
2O
eludedthat•i+ variedinversely
withu, at lowwindvelocities,
with •i+ rangingfromlessthan 11 to 40.
What Causesthe Variabilityin • + 9
Energy balanceconsiderationsled Csanady[-1974,1978] to
concludethat sublayerthickeningis likely to result from any
processwhich extractsenergyfrom near-wallregion.Csanady
[1974] concludedthat sublayer thickening over lake surfaces
during light winds may have been causedby work against
surfacetension.Arya [1975] found an increasein •i+ with
U+: 2.5Inz++5.1
_
/
-
U+
=---•-n
I0-
increasingstabilityin thermallystratifiedflow over a flat plate
and attributed it to work against buoyancy forces.For very
unstablestratification,the sublayerwas thinner than for a
neutrallybouyantflow.
Gust [1976] reports laboratory experimentsin which the
thicknessof the viscoussublayerfor dilute seawater-claysuspensionswas two to five times larger than for comparable
suspension-free
flows. He suggestedthat this thickening was
related to elastic deformation of clay mineral aggregatesby
the turbulentflow.The valueof u, determinedfrom the slope
of the logarithmic profile was significantlylarger than that
from the sublayershear,an effectnot observedin our data.
0
show that
the bottom
few meters were within
a few milli-
degreesof isothermal. Caldwell [1978] found that when the
bottom layer was nearly isothermal, salinity variations within
it were lessthan 0.002 parts per thousand(below the resolution of the best instrumentation).Thus we cannot infer any
i
i
i i i i i I
I0
i
i
40
Z+= Z
We haveattemptedto correlateour •i+ measurements
with
someother quantities.Temperatureprofilesfrom a freely falling microstructureprofiler [Newberger and Caldwell, 1981]
i
i
Fig. 10. Nondimensionalplot of the velocitydata for three different time periods. The dashed line representsthe velocity profile expectedfor a nondimensionalsublayerthicknessof 11.1. The slope of
both straight lines is that expectedfor von Karman's constantequal
to 0.4 (cross,u, = 0.22 cm/s,6+= 11.2,intervallength= 22 min;
opencircles,u, = 0.24 cm/s,6+-- 14.7,intervallength= 21 min;
solidcircles,
u, = 0.25cm/s,6+ = 16.3,intervallength= 132min).
influenceof densitygradientson •i+
Data from the 25 cm path-length beam transmissometer
Neither the pressuretransducernor the heated thermistors
(optical axis located 21 cm above the sediment),and from a
similar unit mounted on the microstructureprofiler, indicate show any influence of surface gravity waves. The maximum
much lower suspended-sediment
concentrations(0.4-3 mg/li- measured friction velocity was significantly below the estiter) than the smallest(150 mg/liter) in Gust's experiments.The mated critical value (u,=0.88 cm/s) for sediment retransmissivity
showsno correlationwith •i+. Significantcon- suspension[Miller et al., 1977], so it is unlikely that either
centrationgradients,however,could have existedbetweenthe sedimentresuspensionor near-bottom oscillatory flow is responsible
for the observedvariationin •i+.
bed and the lowestpositionsof the optical sensors.
Gordon [1975] suggeststhat turbulence parameters may
vary with acceleration
in tidal flows.Becausea tidal signalis
not obvious in our current records, we looked for a corre-
lation between •i+ and the tidal phase determinedfrom
bottom pressuremeasurements.We found no correlation, nor
did we,findanyrelationship
between
•i+ andt•U/t?tfor the
studied intervals, or between •i+ and the interval length.
During severalhoursin which the flow directionvaried by less
than 8ø,•i+ variedfrom 11 to 18,indicatingthat •i+ variability
is not necessarilyassociatedwith major changesin flow direction.
CouldSmall-Scale
Roughness
Influencec5
+?
Three laboratory studies[Antonia and Luxton, 1972; Mulbeam, 1978; Andreopoulos
and Wood, 1982] have investigated
various aspectsof the responseof turbulent boundary layers
to rough-to-smoothtransitions.Details and some of the findings differ, but the following results are common to all: even
though the flow in the internal boundary layer after the
roughnesschange was hydrodynamically smooth, it did not
have the universal smooth-wall structure characterized by
to
I
I0
!fi•
t.
I0•
lfi*
Fig. 9. Nondimensionalplot of the velocitydata from 11 different
laboratory investigationsof turbulent boundary layers over smooth
walls.(Reprintedfrom Monin and YaglomI-1971].)
U+= 2.5 In z + + 5.1. Instead,the constantB in (10) was
significantlylarger than 5.1 (as large as 15 near the roughness
change), and decreased downstream. In the Antonia and
Luxton [1972] and Andreopoulos
and Wood [1982] studies,B
6410
CHRISS AND CALDWELL' UNIVERSAL SIMILARITY AND VISCOUS SUBLAYER
Error
aO
_u.
•./ ....
a{i•-z)-•-/
.........
J
18'BL
In $
__1
/
I
/
of the internal boundary layer disagree with the figures and
appear to be in error. We have taken the data from the figures.)
Although the scalinghelps to collapsethesedata, estimates
of B and 5mL/Zo from Mulhearn [1978] are significantly
largerat equivalentx/(v/u,)positions.Becausethe upstream
stress distributions were very different in the Antonia and
Luxton [1972] and Mulhearn [1978] experiments,Mulhearn
emphasizes
that the two experimentsshouldnot be considered
equivalentfor the purposeof modelinggeophysicalflows. Because of this, and because the differencesin flow conditions in
the Antonia and Luxton [1972] and Andreopoulosand Wood
[1982] studieswere not sufficientlylarge to constitutean ideal
test of the scaling,the following estimatesof equivalent distances should be treated with caution.
Assuming
that downstream
distances
canbe scaledby v/u,,
U(z)
andusingthev/u, rangeof ourstudytogether
withthoseof
the laboratory experiments,we concludethat the distancesin
Fig. 11. Theoretical mean velocity profile for the case in which
form drag influencesflow abovean internalboundarylayer of thick-
the threestudiescouldrepresentdistances
4-30 timeslargerin
layer velocitiesoutsidethe internal boundarylayer.
influenced the near-bed flow we measured.
the ocean. These estimates,and the fact that the laboratory
ness61B,•.
Hereu, is thefrictionvelocitydetermined
fromthesublayer flowsat the last measuringstations(1.2-2.3 m from the roughvelocityprofile and is the appropriate quantity for use in the internal
boundarylayer, while U, is a frictionvelocitywhichreflectsthe nesschanges)had not yet adjustedto the universalsmoothwall velocity profile, show that roughness changes several
influenceof form drag on the overlyingflow. The error shownis that
which results when sublayer measurementsare used to predict log
meters to tens of meters upstream could have significantly
decreasedto 7 at the last measuring stations, 16 and 55
boundarylayer thicknesses
(respectively)downstream.In Mulhearn's study, B was approximately 9 at the last station. By
using (11) (assuming A = 2.5) to convert the respectiveB
valuesto 5+, we seethat the equivalentvariationin 5+ is from
23 near the roughnesschange to 13-16 at the last stations.
(Antonia and Luxton suggestthat k is significantlylarger than
0.41 immediately downstream of the roughnesschange.If so,
Although we have referredto the Andreopoulosand Wood
experimentas a study of a rough-to-smoothtransition,it actually studiedthe responseof an initially smoothboundarylayer
flow to a very narrow strip of roughnessfollowed again by a
smooth surface.Although the roughnessstrip extendedonly
15 cm in the flow direction, the flow in the internal boundary
layer had not adjustedto the "universal"smooth-wallprofile
235 cm downstream.Thus relatively small patchesof surface
roughnesslocated significant distances upstream may have
the 5 + = 23 estimatemay be 10% too high.)Thus upstream beenresponsible
for our observationof 5+ valueslargerthan
roughnesschangescould be responsiblefor the large valuesof
11.1. However'
5 + we observe.
1. If upstream roughnesschanges were present, why is
there agreement(Figure 6) betweenthe velocitiesdetermined
by the rotor and those extrapolated upward from the heated
Do the downstream distancesin these laboratory studies
represent significant distancesin our experiment? One approach (following Andreopoulosand Wood [1982] and Mulhearn [1978]) is to nondimensionalizethe distance downstream from the roughnesschange x by a length Zo which
characterizesthe downstream smooth surface. Since Zo de-
thermistormeasurements
assuming
constantu, ?
2. Why does 5 + vary during periodsof nearly constant
flow direction ?
It is clear from Figure 11 that the answer to the first
pendsstronglyon 5 + (whichvariessignificantlydownstream), questiondependson the height of any internal boundary layer
there is some question about which Zo value to choose.An- presentand on the differencein u, betweenthe internal
dreopoulos
and Woodusethe Zoat theirlaststa[ion(ef- boundarylayer and the flow above.The data of Andreopoulos
fectivelyassuming5+= 13), whereasMulhearn uses a Zo and Wood [1982] and Antonia and Luxton [1972] indicate
value calculated assuming a smooth plate unperturbed by that at xu,/v = 24,250,5mL/Zowas approximately55. Mulroughness
changes(5+ • 11).Becausezo computedassuming hearn'sresultssuggest
a value6 timeslarger.Assumingu, 6 + = 11 is almosttwicethat computedassuming6 + = 13 (see 0.25 cm/s (typical in our study),the equivalentdistancefrom a
roughnesschangewould have been approximately 14 m in our
experiment,and 5mEcould have been 55 Z o (or more). In an
for the downstreamsurface.This is equivalent to nondimen- earlier experiment [Chriss and Caldwell, 1982], we observed
sionalizing
x by v/u,. Scalingx by v/u, and scalingthe inter- Zo values up to 1.4 cm on the Oregon shelf (but interpreted
nal boundary layer thickness,5•BL, by the roughnesslength, the associatedvelocity profiles in terms of form drag due to
Z o, for the rough surfaceresultsin the nondimensionalthick- distributedbiogenicroughnesselements),so 5mEcould be apness(6mL/Zo) of the internal boundary layer at Antonia and proximately77 cm.
Although the existenceof a 77-cm-thick internal boundary
Luxton's last station agreeing(to within 30%) with 5mL/Zo at
the equivalentx/(v/u,) position in the Andreopoulosand layer would not degrade the velocity predictions at 59 cm
Wood experiment.Scalingx in this manner, the valuesof B at (Figure 11), errors in U(59) are expectedfor 5mLlessthan 59
equivalent nondimensionaldownstreamdistancesin the two cm. Figure 11 indicatesthat the error in predicting U(59) is
experiments agree very well, except close to the roughness given by
change.(In both the Antonia and Luxton paper and the Andreopoulosand Wood paper, equationsdescribingthe growth
error= [(U, - ,,)/k]In (59/t$mL)
(15)
equation (13)), it seemsreasonableto eliminate this variability
by fixingthe 5 + valueusedto computethe "characteristic"
Zo
CHRISS AND CALDWELL' UNIVERSAL SIMILARITY AND VISCOUS SUBLAYER
Using(15) with t•iB
L = 30 cm and u, = 0.25 cm/s,the error is
only0.4 cm/sevenif the frictionvelocityU, outsidethe internal boundarylayer were twice u,. Thus the agreementbe-
20
tween the rotor velocitiesand the velocitiespredictedby sublayer data (Figure 6) cannot be used to argue against upstreamroughnesseffectsas causesof the observed5 + variability. Changesin flow direction and site during the various
deploymentscould have causedthe distancebetweenthe sensors and upstream "roughnesspatches" to vary, thereby re-
18
sultingin 5 + changes.
However,this hypothesis
doesnot explain the variationof 5+ duringperiodswhenthe flow direc-
'
I
'
6411
I
I ß '
I
ß
16
l•+
ß
fi
+
14
tion was nearly constant.There seemsto be somedependence
of 5+ on u, duringtheseperiods(Figure12).Perhapsu,
changescould have caused5+ to vary. Analysisof Antonia
12
Could Distributed RoughnessElements
Have Caused the 6+ Variation?
103 ø
O
99 ø
1-1 96 ø
+
I0
0.20
ationof 5+ withu, is not largeenoughto explainthevariability in 5+ we observedfor nearlyconstantflow direction.
+
ß 97ø
and Luxton's data from two locations (x = 30 cm, 117 cm)
indicatesthat whenu, wasincreased
by a factorof approximately 1.6, 5+ did increase,but by lessthan 13%. This vari-
'
•
I
0.22
•
I
I
0.24
ß 95ø
i
I
0.26
i
I
0.28
&
89*
ß
84*
I
0.:•0
u•
Fig. 12. Nondimensionalsublayerthicknessversusfriction velocity for deployment 1 data intervals with flow from similar directions.
The mean flow directionfor eachinterval is givenwith respectto the
tripod coordinatesystem.
In another experiment [Chriss and Caldwell, 1982] on the
Oregon shelf in October 1978, the heated thermistorsprofiled
from the viscous sublayer to 20 cm above the sediment-water
In view of theseresults,it is possiblethat the variability of
interface.The velocity profilesexhibited "kinds" located 11-15 5+ withu, for nearlyconstant
flowdirection(Figure12)may
cm above the sediment-water interface. We concluded that the
primarily reflect changesin the nondimensionalroughness
flow above the kinks was influenced by form drag due to heights of distributed roughnesselementsdue to changesin
small-scaletopography,and that the regionbelowthem repre- u,. Sinceit is unlikelythat the roughness
geometryand the
sented an internal boundary layer between roughnessele- position of our sensorrelative to roughnesselementswould be
ments.Valuesof 5+, corresponding
to the z0 valuesreported the same for all flow directions, one should not expect a
for the internal boundarylayer, rangefrom 7.8 to 16.1,with 4 strongcorrelation
of 5+ and u, whendata for manyflow
of 5 lying between13.3 and 16.1. Our attemptsto obtain directionsare plotted together(as in Figure 4). However, there
stereophotographsof the area failed,but bottom photographs shouldbe a strongrelationship
between
5+ andu, for interalong the sameisobath,65 km south,frequentlycontainedsea vals when flow is from the same direction. In addition to the
urchins(6-8 cm in diameter) and biogenicsedimentmounds data shown in Figure 12, there were three other instancesof
up to 15 cm high. We concludedthat thesefeaturescould have more than one flow measurementfrom approximately the
samedirection. Although in each instancethere are only two
beenresponsiblefor the form drag.
Chenand Roberson[1974] studiedthe responseof the near- measurements(Table 1), each pair comesfrom a different dewall air flow in a smoothpipe to low concentrations
of equi- ploymentusingdifferentsensors.In all cases,5 + is larger at
distanthemisphericalroughnesselements.The roughnesscon- the larger friction velocity.
centration, ,•, the ratio of the sum of the base areas of the
For comparisonwith our field results,we consideronly the
hemispheresto the total boundary area, was varied from low roughnessconcentration(4 = 0.0046) laboratory results
0.0046 to 0.039 by changingthe radii of the individual hemi- because the 5 + values and the variations in 5 + observed at
spheresfrom 0.55 cm to 1.59 cm. The downstreamand cross- theseconcentrationsare comparableto thosein our data, and
streamspacingremainedfixed at approximately20 cm. Veloc- becauseuseof the low concentrationdata will yield conservageometry.Recallingthat h+ varied
ity profiles were obtained at one position from which the tive estimatesof roughness
roughnesselement had been removed. The nondimensional from 112 to 226 in the • = 0.0046runs,and usingu, = 0.25
distance(xu,/v) from the upstreamroughness
elementvaried cm/s as a typical friction velocity, the equivalent roughness
from 25,600 to 84,300, whereasthe nondimensionalheight of heights
ontheseafloor
wouldbe6-14cm.Assuming
uniformthe roughness
elements
(h+ = hu,/v) variedfrom 112to 433 ly spacedroughnesselementsof 10 cm diameter on an otherduring the experiment.In the internal boundary layer, the wise smooth boundary, the spacing between the elements
profiles indicate that the "constant"B increaseswith increas- would have to be 1.3 m in order to have • = 0.0046. Bottom
ing u, (whichwe computedfrom the slopeof the logarithmic photographs[Chriss and Caldwell, 1982] indicate that these
profile in the internal boundary layer, taking k = 0.4). For
estimatesof roughnesselement height and spacing are not
,•= 0.0046,
B increased
from3.5to 11asu, increased
from31
unreasonable.
to 62 cm/s.The equivalentchangeof 5 + is from 9 to 18. For
• = 0.039,B increased
from 10.0to 17.2(5+ = 17 to 5+ = 25)
as u, increasedfrom 18 to 41 cm/s.For runs at fixed ,•, the
sizeand spacingof the roughnessremainedconstantand only
the flow velocity(and u,) changed.Thusat fixed,• the significantvariableis the nondimensional
roughness
height,h+. For
To what extent would a roughnessgeometrysimilar to that
of Chen and Robersoninfluencethe accuracyof our velocity
predictions at 59 cm? Chen and Roberson found that, with
one exception,a noticeable kink in the velocity profiles oc-
,• = 0.0046, h+ varied from 112 to 226, whereasfor ,• = 0.039,
elements.
In the• = 0.0046run withthelargesth+, U, above
theinternalboundary
layerwas41% largerthanu, insideit.
h + varied from 192 to 433.
curredat the heightof the roughness
elements,
and that U,
above the kink was related to the form drag causedby the
6412
CHRISS AND CALDWELL'
UNIVERSAL SIMILARITY AND VISCOUS SUBLAYER
TABLE 1. Friction Velocitiesand NondimensionalSublayer
Thicknesses for Pairs of Data Intervals With Mean Flow
From Approximatelythe SameDirection
Deployment
Data
Flow
u,,
Number
Interval
Direction*
cm/s
6+
1
1
B
C
200 ø
210 ø
0.24
0.13
15.0
11.6
2
2
C
D
105 ø
107 ø
0.13
0.10
14.4
8.0
3
3
C
G
110 ø
105 ø
0.29
0.21
16.1
10.7
most reasonable causes appear to be upstream changes of
surfaceroughnessor distributedroughnesselements.
5. Although the Savonius rotor data in the logarithmic
layer does not demonstratean influence of form drag on the
near-bottom flow, the accuracyof the rotor is inadequateto
rule out such an influence.
*Tripod coordinate system.
Assumingu, = 0.25 cm/s, U,-
1.41 u,, 14-cm-highrough-
nesselements,and an internal boundary layer of equivalent
thickness,equation (15) indicatesthat the error in predicting
U(59) from the heated thermistor measurementswould be less
than 0.4 cm/s. Therefore roughnesseffectssimilar to those
studied by Chen and Roberson could have caused our ob-
served6+ variability without significantlydegradingthe
agreementbetween measuredand predicted velocitiesat 59
cm (Figure 6).
If internal boundarylayersthinnerthan 59 cm werepresent,
however,our estimateof 0.43 for von Karman'sconstantmay
be too low. Given an extreme case of a 14-era-thick internal
boundarylayer overlainby a regionwith a 41% larger friction
.velocity,the true k could be 0.07 higher than we estimated.
The existenceof k significantlylarger than 0.4, while unexpectedin flow over flat uniform surfaces,could neverthelessbe
consistentwith observationsthat k is not always0.4 in internal boundarylayersassociatedwith roughnesseffectsEAntonia and Luxton, 1972; Paola et al, 1980]. If internal boundary
layerspresentwerean appreciable
portionof 59 cm thick,our
6. In typical flow conditions,it is difficult to use widely
separatedsensorsto test hypothesesabout boundary layer
flow, becausethe differencebetweenthe velocitypredictionsof
two models may be of the same magnitude as the calibration
error of the sensors.Had we profiled our sensorsoccasionally
to 50-100 cm, we could have obtained data more suitable for
hypothesistesting.
7. Laboratory resultssuggestthat the near-bed flow may
adjust slowly to changesin surfaceroughness,and that even
the flow near the viscous sublayer may be influenced by
roughnesschangeswhich occurred several meters to several
tens of meters upstream. Extremely detailed information on
upstreammicrotopographywill be required for the unequivocal interpretationof near-bedflow measurements.
APPENDIX: THE SAMPLING ERROR ANALYSIS
Errorsin 6+ valuescan occurbecause
profilingsensors
do
not make
continuous
measurements
at each elevation.
The
"sampled"mean velocity at an elevationmay differ from the
"true" mean velocity, which would have been measuredby a
fixed sensorsamplingcontinuously.We evaluated the effect of
suchsamplingerror on 6+ measurements
by assuming
a vis-
coussublayer
with6+ = 11.1anda typicalu, (0.25cm/s),and
determining how sampling errors at each elevation would
changethe 6+ value computedfrom this profile.Assuming
k = 0.4,the errorin 6+ dependsonlyon the errorin determining the linear velocity gradient in the sublayerand on the
error in the mean velocityat the top of the heatedthermistor
profile(in the lower logarithmiclayer).
An ideal samplingerror analysisrequiresthe subsampling
estimateof k = 0.43 may not be significantlyin error. Because of continuous time series from fixed elevations. Data from
we lack the continuousprofilesrequired to evaluatethis influanotherexperimentconductedat the 185-mstationduringthe
enceof internal boundary layers,our estimatemust be viewed
June1979cruiseEChrissand Caldwell,1984] werecollectedby
with caution.
usinga profiler with an extremelyslow vertical motion (2.1
CONCLUSIONS
During this experiment the near-bed flow was hydrodynamicallysmooth. The data have been used to test the
assumptionthat the flow behavedlike a universallysimilar,
neutrally buoyant flow over a smooth wall. We find the following:
x 10-4 cm/sto 1.2 x 10-3 cm/s),sothat selected
shortintervals of the data representcontinuoustime seriesfrom nearly
fixedelevations.The elevationsat the top of the profileduring
that experimentcorrespondedonly to the upper buffer layer
(z+ = 18-29)ratherthan to the lowerlog layer(z+ = 30-40)
but are sufficientlycloseto provideestimatesof potentialsampling errors in data from the lower log layer. It should be
1. A'viscous
sublayer
wasalways
present
anditsthicknessnotedthat the nondimensional
sublayerand bufferlayerspecscaledroughlywith v/u,, but the nondimensional
thickness tra presented in Chriss and Caldwell [-1984] contain less
varied more than in laboratory flows over perfectlysmooth energy than similar nondimensionalspectra from smoothwalls.Thus evenwhena viscoussublayerexists,the boundary walledlaboratory flows and that, in usingthesetime seriesfor
layer flow cannot alwaysbe preciselydescribedby the equa- the samplingerror analysis,we assumethat they are repretions for a universallysimilar, neutrally buoyant boundary sentativeof thosewhich would have beenobtainedduring the
layer over a smooth wall.
experimentreportedin this paper.Consideringthe spatialand
2. Estimatesof von Karman's constantare not appreci- temporal proximity of the two experiments,we feel this asably differentfrom valuesdeterminedin the laboratoryand in sumption is reasonable. We comment, however, that if the
other geophysicalflows.Theseestimatesare not definitive be- {scaled)energyin the very near bottom velocityfluctuations
cause the influence of internal boundary layers on the more closelyresembledthat of smooth-walledlaboratoryexmeasurements cannot be determined.
periments,then the followingestimateswould representlower
3. Measurednondimensionalsublayerthicknesses
are rep- boundson potentialsamplingerrors.
resentativeof flow conditions averaged over at least several
To evaluatepotential errors in the log layer data collected
squaremeters of the seafloor.
usingthe 57-s profiler period, portionsof the continuoustime
4. The variability in the nondimensionalsublayerthick- seriesdiscussed
aboveweredividedinto intervals10 min long,
nesscould be causedby a number of phenomena,but the closeto the shortestaveragingtime usedin our study.These
CHRISS AND CALDWELL: UNIVERSAL SIMILARITY AND VISCOUS SUBLAYER
6413
serieswere then subsampledat intervalswhen the 57-s profiler (softwaredevelopment),as well as Ralph Moore and Robert Schum,
would have been within the top 0.1 cm of its profile. The who constructedand helpedcalibratethe sensors.PriscillaNewberger
and Stuart Eide read the manuscriptand furnishedadviceand assistpercent differencebetweenthe mean velocity of the subsam- ance in innumerable other ways. This material is based upon work
pied data and the mean of the entire 10-min serieswas taken supportedby the National ScienceFoundation under grant OCEto be the "sampling error" for the experiment.Based on 55 7918904.
suchsamplingexperiments,the mean value and the standard
deviations of the "sampling error" are 0.17% and 2.61%, reREFERENCES
spectively.Eleven similar experiments using a 566-s profiler
Andreopoulos,
J.,
and
D.
H.
Wood, The responseof a turbulent
period and a 40-min averaging interval yield 0.18% and
boundary layer to a short length of surface roughness,d. Fluid
3.43% for the mean and standard deviation of the sampling
error.
To estimatethe effectof samplingerror on the linear velocity gradient in the viscoussublayer,one would ideally like to
subsamplecontinuousseriesfrom various sublayer locations.
Unfortunately, for most casesduring the very slow profiler
experiment,the elevationscorrespondingto the sublayer occurred in the center of the traverse where the vertical motion
wassufficiently
fast(1.2 x 10-3 cm/s)that it is impossible
to
obtaineven 10 min of sublayerdata at relativelyconstantz+.
We have thus had to restrict our analysis for the sublayer
measurements
to intervalsof low u, in whichthe sublayerwas
sufficientlythick that one sensorremained in the sublayer at
the top of the traverse.Due to this constraint, we were unable
to subsamplecontinuousseriesobtained from positionsbelow
z+= 4. We assume,however,that the samplingerror estimatesobtainedusingdata from the regionz + = 4-9 are not
significantlydifferent from estimateswe would have obtained
at other sublayerpositions.Each samplingexperimentconsisted of subsamplingthe continuousseriesat intervalswhen the
57-s profiler (for example) would have been at given sublayer
positions,and for each position computing the percent difference between the mean velocity of the subsampleddata and
the mean of the entire series.
The total effect of the samplingerrors at all sublayer positions on the computed velocity gradient was determined as
follows:we firstassumed
a knownsublayerprofilewith u, =
0.25 cm/s and then added the percentagesamplingerror for
each elevation to the velocity for the known profile. The re-
sultingvelocitieswere treatedas raw data and a new u,
(which now includedthe effectsof samplingerror) was deter-
minedby linearregression.
Comparisonof the new u, values
with 0.25 cm/syieldedthe percentsamplingerror in u,. On
the basis of 32 such sampling experiments(using both 57-s
and 566-s profiler periods),the mean and standard deviation
of the estimatederrorsin u, were0.39% and 4.21%,respectively.
Given theseestimatesof the samplingerror in the velocity
at the top of the profileand the samplingerror in u,, we can
look at "worstcase"errorsin •5+ whichmight resultif errors
two
standard
deviations
from
the mean
occurred
simulta-
neouslyin both of thesequantities.Given a true •5+ of 11.1,
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are, in fact, uncorrelated, the worst case lower and upper
limitsfor •5+ are 9.3 and 12.9,respectively.
Acknowledgments.Specialthanks are extendedto Mark Matsler
(profilerdevelopment),SteveWilcox (electronicsdesign),Stuart Blood
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6414
CHRISS AND CALDWELL: UNIVERSAL SIMILARITY AND VISCOUS SUBLAYER
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D. R. Caldwell,Schoolof Oceanography,Oregon State University,
Corvallis, OR 97331.
T. M. Chriss,Departmentof Oceanography,DalhousieUniversity,
Halifax, Nova Scotia, CN B3H 4J1.
(ReceivedDecember4, 1981;
revisedFebruary 27, 1984;
acceptedFebruary 27, 1984.)