Stratification Effects in a Bottom Ekman Layer · 2019. 1. 2. · Ekman transport is generally...

Stratification Effects in a Bottom Ekman Layer

JOHN R. TAYLOR AND SUTANU SARKAR

Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California

(Manuscript received 29 October 2007, in final form 8 July 2008)

ABSTRACT

A stratified bottom Ekman layer over a nonsloping, rough surface is studied using a three-dimensionalunsteady large eddy simulation to examine the effects of an outer layer stratification on the boundary layerstructure. When the flow field is initialized with a linear temperature profile, a three-layer structure de-velops with a mixed layer near the wall separated from a uniformly stratified outer layer by a pycnocline.With the free-stream velocity fixed, the wall stress increases slightly with the imposed stratification, but theprimary role of stratification is to limit the boundary layer height. Ekman transport is generally confined tothe mixed layer, which leads to larger cross-stream velocities and a larger surface veering angle when theflow is stratified. The rate of turning in the mixed layer is nearly independent of stratification, so that whenstratification is large and the boundary layer thickness is reduced, the rate of veering in the pycnoclinebecomes very large. In the pycnocline, the mean shear is larger than observed in an unstratified boundarylayer, which is explained using a buoyancy length scale, u*/N(z). This length scale leads to an explicitbuoyancy-related modification to the log law for the mean velocity profile. A new method for deducing thewall stress based on observed mean velocity and density profiles is proposed and shows significant im-provement compared to the standard profile method. A streamwise jet is observed near the center of thepycnocline, and the shear at the top of the jet leads to local shear instabilities and enhanced mixing in thatregion, despite the fact that the Richardson number formed using the mean density and shear profiles islarger than unity.

1. Introduction

The surface wind-driven Ekman layer forms whenfrictional terms contribute to the leading-order momen-tum balance, leading to an ageostrophic flow. Becausethe velocity gradients, and hence the size of the viscousstresses, are strongest at the sea surface, the ageo-strophic component of the flow decreases with depth,leading to a turning of the mean velocity profile knownas the Ekman spiral. The bottom Ekman layer, formedas a mean current flow over the seafloor, is directlyanalogous to the surface wind-driven Ekman layer. In asurface Ekman layer, the Ekman transport (the verticalintegral of the Ekman layer velocity) is directed 90° tothe right of the wind stress in the Northern Hemi-sphere. In a bottom Ekman layer, the Ekman transportis also directed to the right of the bottom stress. How-ever, because the stress at the seafloor is in the opposite

direction of the geostrophic current, the Ekman trans-port in the bottom boundary layer is 90° to the left ofthe geostrophic current, and the Ekman spiral turnscounterclockwise with increasing depth in the NorthernHemisphere. Also like the surface mixed layer, the bot-tom boundary layer is a site of intense turbulent dissi-pation and mixing (Garrett et al. 1993; Thorpe 2005).Together with the surface mixed layer, the bottomboundary layer is a major “hotspot” for diapycnal mix-ing in the ocean (Thorpe 2005). It has long been hy-pothesized that mixing of the density field in bottomboundary layers may be important to abyssal mixingthrough along-isopycnal advection out of the boundarylayer and boundary layer detachment (Munk 1966;Armi 1978). In addition, through frictional loss and Ek-man pumping, the bottom boundary layer provides animportant momentum sink for deep currents and me-soscale eddies.

Three-dimensional numerical simulations of strati-fied bottom Ekman layers have been described previ-ously by several authors but, in all cases, stratificationwas applied with a cooling heat flux at the lower wall,akin to the stable atmospheric boundary layer. Cole-

Corresponding author address: Sutanu Sarkar, Department ofMechanical and Aerospace Engineering, University of California,San Diego, La Jolla, CA 92093.E-mail: [email protected]

NOVEMBER 2008 T A Y L O R A N D S A R K A R 2535

DOI: 10.1175/2008JPO3942.1

© 2008 American Meteorological Society

JPO3942

man et al. (1992) performed a direct numerical simula-tion (DNS) of a turbulent Ekman layer with a frictionReynolds number of Re* ! 340 and a constant heatflux at a smooth, no-slip lower wall. They compared thestratified Ekman layer to a previous study of an un-stratified Ekman layer (Coleman et al. 1990) and foundthat the surface heat flux limits the transport of turbu-lent kinetic energy into the outer layer and broadensthe Ekman spiral. Shingai and Kawamura (2002) con-sidered the same flow at a friction Reynolds numberRe* ! 428.6. They found that the boundary layer thick-ness defined in terms of either the momentum or thebuoyancy flux decreases sharply with the application ofa surface heat flux. In general, these studies imply thatwhen a strong, stable stratification is applied to the wallunder an Ekman layer, stratification acts to suppressthe turbulent production in the boundary layer, increas-ing the turning angle and decreasing the boundary layerheight.

In the ocean, with the exception of isolated hotspots,the seafloor can be assumed to be adiabatic. Changes indensity then affect the Ekman layer through the strati-fication associated with the ambient water. Because amixed layer can be found near the seafloor throughoutmost of the ocean, stratification can be expected toaffect the boundary layer in a much different mannerthan in a typical stable atmospheric boundary layer,where a heat flux is often present at the ground. Bycomparing the expected unstratified turbulent Ekmanlayer depth to the thickness of bottom mixed layers infield data, it is apparent that stratification often limitsthe boundary layer height. In an unstratified Ekmanlayer, the boundary layer height is expected to be ap-proximately h ! 0.5 u*/f, where u* is the friction ve-locity and f is the Coriolis parameter. Typical midlati-tude values, say u* ! 1 cm s

"1 and f ! 10"4 s"1, implyan unstratified Ekman layer depth of about h ! 50 m.

In many cases, especially in a coastal environmentwhere the stratification is typically large, the observedmixed layer heights are often much smaller than 50 m.For example, Perlin et al. (2007) observed a mixed layerthickness of about 10 m in an Ekman layer over theOregon shelf. It is not fully clear how the Ekman layerstructure changes when the Ekman layer height is lim-ited by stratification, and this will be one focus of thepresent study.

The bottom boundary layer plays an important rolein the drag induced on mean currents and mesoscaleeddies. To obtain an accurate prediction of the oceanstate, numerical ocean models must represent this lossof momentum. However, due to computational restric-tions on the grid size, this is not straightforward. Toaccurately apply the no-slip boundary condition at the

seafloor, a numerical model must resolve the viscoussublayer. Because the viscous sublayer in the ocean isthin, O(0.1–10 cm) (Caldwell and Chriss 1979), numeri-cal models are clearly unable to resolve this region, andan approximate boundary condition must be used. It iscommon practice to model the seafloor stress, whichthen provides a Neumann boundary condition for thehorizontal momentum equations. For example, the Re-gional Ocean Modeling System (ROMS) provides threemethods for modeling the bottom stress based on thevelocity at the lowermost grid cell: linear and quadraticdrag coefficients and a law-of-the-wall using a specifiedbottom roughness. A general form for the bottom stressusing the linear drag coefficient #1 and the quadraticdrag coefficient #2 is

!w,x"0

! $#1 % #2&u2 % $2'u,

!w,y"0

! $#1 % #2&u2 % $2'$, $1'

where (w,x and (w,y are the zonal and meridional com-ponents of the bottom stress, respectively. Typical val-ues of the bottom stress coefficients are #1 ! 2 ) 10

"4

m s"1 and #2 ! 0 for linear bottom drag and #1 ! 0 and#2 ! 2 ) 10

"3 for quadratic bottom drag (Haidvogeland Beckmann 1999).

The friction velocity u* ! &(w/*0 is often used inscaling arguments for both bulk and turbulent proper-ties, such as the boundary layer height, turbulent dissi-pation, Reynolds stresses, etc. Despite its first-orderrelevance, there remains some uncertainty about howto estimate the wall stress from observational data, es-pecially when the velocity profile is affected by wallroughness and stratification. A common method usedto evaluate the friction velocity is the so-called profilemethod (Johnson et al. 1994), which utilizes the classi-cal law-of-the-wall (see, e.g., Pope 2000). Using thismethod, the friction velocity and wall roughness z0 aredetermined by fitting the observed velocity profile tothe logarithmic profile

|u| !u*%

ln! zz0", $2'where |u| is the horizontal velocity magnitude and + !0.41 is the von Kármán constant.

Johnson et al. (1994) applied this method to estimatethe friction velocity at the bottom of the Mediterraneanoutfall plume. They found that estimates of the frictionvelocity depended on how far away from the bottomthe fit was applied, even when restricted to the bottommixed layer. This implies that the standard law-of-the-

2536 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 38

wall was not valid throughout the bottom mixed layer.Dewey et al. (1988) compared several methods for es-timating the bottom stress using microstructure profilesover a continental shelf. They found that the wall stressestimated using the profile method was a factor of 4.5larger than that estimated using the dissipation method,with the friction velocity given by

u* ! "!"z#1#3, "3#

where again $ ! 0.41 is the von Kármán constant. Theauthors speculated that this discrepancy may be theresult of form drag induced by local bedforms. Stahrand Sanford (1999) obtained velocity and dissipationmeasurements in the North Atlantic deep westernboundary current and similarly found that the wallstress estimated from the profile method was 3 timeslarger than the wall stress estimated using the dissipa-tion method.

Perlin et al. (2005) also found that the profile methodgave large values of the wall stress compared to thedissipation method and proposed that the elevatedmean shear could be explained by the influence ofstratification on the boundary layer structure. Theyproposed that when the local stratification is suffi-ciently large, stratification, not distance from the wall,limits the size of the largest turbulent eddies. To quan-tify this hypothesis, they used an empirical function in-volving the Ozmidov scale LOz ! (%/N

3)1/2, where % isthe turbulent dissipation rate. Estimates of the wallstress using this method, which will be referred to hereas the “modified law-of-the-wall,” gave much betteragreement with the dissipation method. Because directfield observations of dissipation require instruments ca-pable of capturing finescale velocity fluctuations, itwould be desirable to have an accurate method for es-timating the wall stress from more commonly observedquantities such as the velocity and density profiles.

Weatherly and Martin (1978) considered a stratifiedEkman layer using a one-dimensional numerical modeland the Mellor–Yamada level-II closure to parameter-ize turbulent mixing. They assumed that the flow out-side the boundary layer was uniformly stratified,steady, and in geostrophic balance; the same assump-tions that will be made in the present study. Near thewall, a mixed layer formed, which was separated fromthe outer layer by a strongly stable pycnocline. Theyfound that the thickness of the bottom Ekman layerwas strongly limited by the presence of a stable strati-fication outside the boundary layer. When the outerlayer buoyancy frequency was N&/f ! 200, they foundthat the angle made by the surface stress relative to thefree-stream velocity was '0 ! 27°, nearly twice the

value in an unstratified boundary layer of '0 ! 15°.Most of the change in the turning angle occurred in thepycnocline, and the flow in the mixed layer was nearlyunidirectional. Perlin et al. (2007) observed a lowerturning angle of 15° ( 5° in a stratified bottom Ekmanlayer over the Oregon shelf. In agreement with thesimulations by Weatherly and Martin (1978), theyfound that nearly all of the Ekman transport occurs inthe relatively thin bottom mixed layer.

Weatherly and Martin (1978) proposed a scalingfor the height of a stratified turbulent Ekman layergiven by

hWM ! Au*f !1 ) N$2f 2 "*1#4, "4#

where A # 1.3 as determined empirically by the one-dimensional model results. In the limit of an unstrati-fied Ekman layer, Eq. (4) gives a significantly largerheight than the conventional value of h ! 0.4 * 0.5+.This is a result of the choice by Weatherly and Martinto use the criteria that the turbulent kinetic energy be-comes zero at the top of the boundary layer. Choosingthe location where the velocity magnitude becomesequal to the free stream would have yielded a boundarylayer height nearly a factor of 3 smaller than that ob-tained using the turbulent kinetic energy (TKE) criteriawhen the outer flow is unstratified. A somewhat differ-ent scaling law was proposed by Zilitinkevich and Esau(2002):

hZE ! CRu*f !1 ) CR2 CuNCS2 N$f "

*1#2

, "5#

where the constants CR ! 0.5 and CuN/C2S ! 0.6 were

determined by fitting to data from a large eddy simu-lation (LES). The functional form in Eq. (5) was foundto be an adequate fit to field observations by Zil-itinkevich and Baklanov (2002).

When a stable stratification is found outside of a tur-bulent well-mixed region, internal waves generated bythe interaction between the turbulent eddies and thestratification are possible. Turbulence-generated inter-nal waves have been found in laboratory experimentsand numerical simulations in a wide variety of flows:shear layers, gravity currents, boundary layers, etc. Tay-lor and Sarkar (2007a) observed turbulence-generatedinternal waves in numerical simulations of a stratifiedEkman layer over a smooth wall at a modest Reynoldsnumber. They found that the vertical energy flux asso-ciated with the upward-propagating internal waves wassmall compared to the integrated boundary layer dissi-pation but was of the same order as the integratedbuoyancy flux. Because the buoyancy flux is respon-


sible for the transfer of turbulent kinetic energy to thepotential energy field, this implies that the energy ra-diated by internal waves is comparable to that used tomix the background density field. As in many previousstudies, they found that the waves propagating throughthe outer layer were associated with a relatively narrowband of frequencies leading to vertical propagationangles between 30° and 60°. Because the waves aregenerated in a turbulent region with a wide range ofspatial and temporal scales, this result is remarkable.The authors described a model based on viscous raytracing that was used to predict the decay in amplitudeof a wave packet after it had traveled a given distancefrom the source. They found that this relatively simplelinear model was able to capture many characteristicsof the observed frequency spectrum of the internalwaves in the outer layer, including the range of domi-nant propagation angles.

This paper will be organized as follows: the govern-ing equations and physical approximations are dis-cussed in section 2, and the numerical method used toevolve the governing equations is presented in section3. Results from the simulations will be separated intoseveral sections: evolution of the mean density and ve-locity profiles will be discussed in section 4, boundarylayer turbulence will be discussed in section 5, methodsfor estimating the friction velocity from field data willbe evaluated in section 6, and turbulence-generated in-ternal waves will be considered in section 7.

2. FormulationThe turbulent Ekman layer considered here is

formed when a steady flow in geostrophic balance en-counters a nonsloping, adiabatic lower wall, as illus-trated in Fig. 1. The primary objective of this study is toconsider a controlled environment in which we can ex-amine the influence of the outer layer stratification ona turbulent Ekman layer. Therefore, we make severalsimplifying approximations. The free stream is assumedto be in geostrophic balance and aligned with the x axis.The seafloor is represented by a nonsloping, rough sur-face. The roughness elements are too small to be re-solved directly by the grid, but their effect is param-eterized through a near-wall model. The lateral bound-aries are periodic, which is consistent with theassumption that the flow is statistically homogeneous inthe horizontal plane and that the domain is largeenough so that the flow is decorrelated over a distanceequal to the domain size. In outer units, D ! U"/f, thedomain size is 0.108D # 0.108D # 0.0405D in the x, y,and z directions, respectively. If we assume that U" !0.0674 m s$1 and f ! 10$4 rad s$1, then D ! 674 m andthe domain size is 72.8 m # 72.8 m # 27.3 m. As a result

of the horizontal periodicity, the mean vertical velocitymust be zero. Therefore, the features of oceanic bound-ary layers owing to Ekman pumping/suction driven bylarge-scale horizontal gradients in the outer flow willnot be present here.

The goal of an LES is to accurately solve a low-pass-filtered version of the Navier–Stokes equations. Be-cause the filtered version of the nonlinear advectionterms involves the total and filtered velocity fields, weare left with fewer equations than unknowns, the well-known turbulence closure problem. A model is thennecessary to write the residual stresses in terms of thefiltered quantities. After normalizing with the free-stream velocity U", the length scale, D ! U"/f, and theouter layer density gradient, d%/dz", the LES-filteredgoverning equations can be written as

!u!t

& u " !u ! $1#0

!p$ & f k̂ # 'ı̂ $ u (

$ Ri%#$

#0k̂ &

1Re%

!2 u $ ! " &, '6(

!#$

!t& u " !#$ !

1Re%Pr

!2#$ $ ! " ", '7(

! " u ! 0 '8(

where the Reynolds number, Richardson number, andPrandtl number are defined by

Re% !U%D

', Ri% ! $

g#0

d#dz%

D2

U%2 , Pr !

'

(, '9(

%0 is the constant density used to define the momentumusing the Boussinesq approximation, ) is the molecularkinematic viscosity, * is the molecular diffusivity, and #and " are the subgrid-scale stress and density flux, re-

FIG. 1. Schematic of computational model. Dimensional param-eters can be obtained by assuming U" ! 0.0674 m s

$1 and f !10$4 rad s$1. The domain size is 72.8 m # 72.8 m # 27.3 m. Threevalues of outer layer stratification are considered: N"/f ! 0, 31.6,and 75.


spectively. The density and pressure have been decom-posed into a plane average plus a fluctuation; that is,! " #!$ % !&. The hydrostatic pressure gradient and theplane-averaged buoyancy force are in balance and donot appear in Eq. (6). An alternative normalization canbe carried out using the friction velocity u* " '(w/!0.This leads to the friction Reynolds and Richardsonnumbers:

Re* "u*!

", Ri* " )

g#0

d#dz$

!2

u2*"

N$2

f 2, ! "

u*f

.

*10+

Relevant input and output nondimensional parametersare listed in Table 1. The dimensional parameters U,,N,, f, -, and the roughness length scale z0 are inputs tothe simulations. Note that Re, is the same for eachsimulation, but the drag coefficient depends on theouter layer stratification, and hence the friction Reyn-olds number varies between each case. We have per-formed simulations at three different values of Ri*,equivalent to changing the free-stream density gradi-ent. For comparison with oceanographic conditions, ob-servations of the bottom boundary layer over the Or-egon shelf by Perlin et al. (2007) provide estimates ofRe* " 4 . 10

4 ) 8 . 105 and N,/f " 75. Therefore,both the Reynolds numbers and stratification levelsconsidered in the present study are comparable withthe field data of Perlin et al. (2007). To provide dimen-sional scalings for our simulations, we will use U, "0.0674 m s)1, f " 10)4 s)1, and - " 10)6 m2 s)1, which,assuming that u*/U, " 0.049, yields / " u*/f 0 33 m.The applied roughness length scale z0 " 0.16 cm isconsistent with observations by Perlin et al. (2005), whofound that z0 " 0.05–2 cm, depending on the methodused to infer the wall stress.

3. Numerical methods

Simulations have been performed using a computa-tional fluid dynamics solver developed at the Universityof California, San Diego. The algorithm and numericalmethod are described in detail in Bewley (2008). Be-cause periodic boundary conditions are applied in thehorizontal directions, derivatives in these directions are

computed using a pseudospectral method, while deriva-tives in the vertical direction are computed with sec-ond-order finite differences. Time stepping is accom-plished with a mixed explicit/implicit scheme usingthird-order Runge–Kutta and Crank–Nicolson. It canbe shown that the numerical scheme ensures the dis-crete conservation of mass, momentum, and energy. Toprevent spurious aliasing due to nonlinear interactionsbetween wavenumbers, the largest 1/3 of the horizontalwavenumbers are set to zero, the so-called 2/3 de-aliasing rule (Orszag 1971).

To prevent the formation of spurious energy near thegrid scale, a low-pass spatial filter is applied to the ve-locity and temperature fields. A fourth-order compactfilter with a sharp wavenumber cutoff (Lele 1992) isapplied in the vertical direction every 10 time steps. Anopen boundary condition is implemented at the top ofthe computational domain to prevent spurious reflec-tions of upward-propagating internal gravity waves.The combination of a radiation boundary condition(Durran 1999) and a sponge-damping region that isused here was also used by Taylor and Sarkar (2007a),who found that only 6% of the wave energy was re-flected back from the open boundary.

The subgrid-scale stress tensor ! and the subgrid-scale density flux " in Eqs. (6) and (7) are evaluatedusing the dynamic Smagorinsky model. The Smagorin-sky coefficients, C and C!, are evaluated using the dy-namic procedure as formulated by Germano et al.(1991). As in Germano et al. (1991), the Smagorinskycoefficients have been averaged over horizontal planes.After averaging, if either of the Smagorinsky coeffi-cients is negative, it is truncated to zero. This effectivelyprevents backscatter of energy from the subgrid scalesto the resolved scales and helps to ensure numericalstability (Armenio and Sarkar 2002). While it is morecomputationally intensive than other methods, the dy-namic Smagorinsky model has been chosen here be-cause it avoids the empirical specification of the Smag-orinsky coefficient and has been shown to perform wellfor wall-bounded and density-stratified flows (e.g., Ar-menio and Sarkar 2002; Taylor et al. 2005).

To ensure accuracy of the solution obtained usingLES, it is generally necessary to resolve the turbulentscales responsible for a substantial portion, say, 150%,

TABLE 1. Relevant physical parameters. Re,, N,/f, Pr, and z0 are input parameters and the other parameters are outputs of thenumerical model.

Re, Re* Ri* N,/f Pr u*/U, z0// 20

4.55 . 107 1.08 . 105 0 0 0.0488 4.80 . 10)5 15.4°4.55 . 107 1.09 . 105 1000 31.6 5–10 0.0490 4.78 . 10)5 18.9°4.55 . 107 1.12 . 105 5625 75 0.0497 4.71 . 10)5 24.8°


of the turbulent kinetic energy. Near walls, this crite-rion becomes increasingly stringent as the Reynoldsnumber increases. To avoid the need to resolve the verysmall turbulent motions near the lower wall, we haveused a near-wall model to estimate the wall stress basedon the resolved velocity at the first grid point. We haveused a model proposed by Marusic et al. (2001), whichStoll and Porte Agel (2006) found works well for highReynolds number, rough wall boundary layers. To com-pensate for known deficiencies in the Smagorinsky LESmodel at very large Reynolds numbers, the near-wallmodel was augmented by a novel adaptive stochasticforcing procedure. The forcing amplitude is small, lessthan 6% of !w/!t, and limited to the first nine gridpoints near the wall. This technique is described in de-tail and validated in Taylor and Sarkar (2007b).

The flow has been discretized using 128 grid points inthe horizontal directions and 201 points in the verticaldirection. The horizontal grid spacing in wall units was"#x $ "

#y $ 1856 before dealiasing. The minimum ver-

tical grid spacing, which occurs at the lower wall, was"z# (1) $ 121, while the maximum grid spacing is "#z $1075, or "z /D $ 4.86 % 10

&4. The first grid point, wherethe near-wall model is applied, was located 60.7 wallunits away from the wall, which is within the expectedlogarithmic region. The grid spacing at the top of themixed layer is always sufficient to resolve the local El-lison scale, defined as

LE $'!"2(1#2

d)!*#dz. '11(

It has been found by Taylor and Sarkar (2008) that theEllison scale provides a reasonable estimate for thescale of the turbulent eddies that are primarily respon-sible for entrainment in to the bottom mixed layer. Re-solving the Ellison scale at the top of the boundarylayer is therefore necessary in order to capture thegrowth of the mixed layer due to turbulent entrainment(Taylor and Sarkar 2008). The vertical grid spacing andthe Ellison scale for each of the stratified simulationspresented here are shown in Fig. 2. The asterisks markthe location where the local Reynolds stress is 10% ofits maximum value, which provides a good estimate forthe boundary layer height.

To obtain an initial condition for the velocity field, alow Reynolds number unstratified simulation was con-ducted until the flow reached a steady state. The veloc-ity field from this simulation was then interpolated ontoa finer grid and a simulation at a higher Reynolds num-ber was continued until all transients had decayed. Thestratified simulations were initialized using the steady-

state unstratified velocity field and an undisturbed,piecewise-linear temperature profile. The initial tem-perature profile has a relatively thin mixed layer with athickness of about z ! 0.04+. This was done so that thestochastic forcing, mentioned above, would not pro-duce spurious internal waves by forcing a stratified re-gion. For z , 0.08+, the temperature profile was set to- $ zd-/dz|., and for 0.04 / z/+ / 0.08, a linear profilewas used to make the integrated heat content over thedomain the same as if - $ zd-/dz|. everywhere. Eachstratified simulation was run for about tf $ 1.5 to allowthe unstratified turbulence levels to adjust to the im-posed stratification. This creates a better initial velocityfield, and the temperature field was then reset to thepiecewise-linear profile. The time at which the tem-perature field was reset will be referred to as t $ 0.

4. Mean boundary layer structure

The time history of the plane-averaged temperaturegradient is shown in Fig. 3. After the flow is initialized,the mixed layer near the wall grows rapidly. As the flowdevelops, a strongly stratified pycnocline forms abovethe mixed layer. It can be shown that if there is no netheat flux into the domain (which is a good approxima-tion given that the molecular heat flux through the topof the domain is small), then a pycnocline is necessaryto maintain heat conservation after the formation of amixed layer. At the start of each stratified simulation,two distinct pycnoclines are visible for a short time, oneabove the bottom mixed layer and another above a

FIG. 2. Vertical grid spacing and the Ellison scale. Asterisksmark the top of the boundary layer, defined as the location wherethe local Reynolds stress is 10% of its maximum value.


region where the residual motions from the initial ve-locity field mix the temperature field. Eventually thesepycnoclines merge, and the resulting single pycnoclinegradually moves away from the wall as the mixed layergrows. In the case with N!/f " 75, the pycnocline has alarger stratification, 4 times the outer layer value, com-pared to the case with N!/f " 31.6. After about tf " 6,the temperature fields in both stratified simulationsreach a quasi-steady state where the temperature gra-dient in the pycnocline does not grow further and themixed layer growth is relatively slow.

Two averaging methods will be used for velocity- andtemperature-dependent fields. The Reynolds average,denoted by angle brackets, will be taken as an averageover a horizontal plane and in time. To remove any biasdue to inertial oscillations, averages are taken over oneinertial period after the flow has reached a quasi-steadystate. However, because the mixed layer thickness con-tinues to increase in time, albeit slowly, temporal aver-ages are not appropriate for the thermal fields andwould, for example, lead to a smearing out of the pyc-nocline. Temperature will therefore not be averaged intime, but the plane average of the temperature-depen-dent fields will be taken at the time corresponding tothe center of the Reynolds average window, unless oth-erwise noted.

The plane-averaged temperature profiles after theflow that have reached quasi-steady state are shown inFig. 4a. Both profiles correspond to a time t " 9.4/f after

initializing the temperature field. The components ofthe Reynolds-averaged horizontal velocity are shown inFig. 4b. Compared to the temperature profiles, it isapparent that most of the Ekman transport is confinedto the mixed layer. The Ekman layer height is reducedby the outer layer stratification at a level that is quan-titatively consistent with the scaling law of Zilitinkevichand Esau (2002), given in Eq. (5). The magnitude of thecross-stream velocity increases when the outer layer isstratified, resulting in a broadening of the Ekman spi-ral, as shown in Fig. 5. It is interesting to note thatalthough the density gradient is zero near the lowerwall, the surface turning angle #0 " tan

$1(%y/%x) in-creases with the outer layer stratification, as listed inTable 1. The angle of Ekman veering, defined by

! " tan$1! &"'&u'", (12)is shown as a function of z/* in Fig. 6. When the flow isunstratified, the turning of the mean velocity occursgradually throughout the boundary layer. The rate ofturning in the mixed layer does not depend strongly onthe outer layer stratification, and when the boundarylayer height decreases significantly, the rate of turning,d#/dz, becomes very large in the pycnocline.

Despite the fact that the boundary layer heightchanges significantly, the Ekman transport is nearly in-dependent of the outer layer stratification. An expres-

FIG. 3. Evolution of the plane-averaged temperature gradient.


sion for the Ekman transport can be found by integrat-ing the mean streamwise momentum equation. Assum-ing that the mean flow is steady, and neglecting themolecular viscosity,

!0

!

!"" dz #u2*f

cos$#0%. $13%

As seen in Table 1, both u*/U&, and '0 increase withN&/f. Their effect partially cancels, so that the Ekmantransport normalized by U& and f is nearly independentof N&. Specifically, evaluation of the right-hand side ofEq. (13) yields a transport of 0.107 m2 s(1, 0.105 m2 s(1,and 0.101 m2 s(1 for N&/f # 0, 31.6, and 75, respectively,where the transport has been made dimensional by tak-ing U& # 0.0674 m s

(1 and f # 10(4 rad s(1. The sameresult could be obtained by numerically integrating thecross-stream velocity profiles.

As has been found in previous studies, the large den-sity gradient at the top of the boundary layer coincideswith an increase in the mean shear. The temperaturegradient normalized by the outer layer value is shownin Fig. 7a, and the Reynolds-averaged velocity andshear profiles are shown in Figs. 7b–c. A peculiar fea-ture of the mean velocity profile when N&/f # 75 is that

the mean velocity and the mean shear are maximum atthe same location near the center of the pycnocline. Itcan be shown that this is the result of the rapid rate ofveering that occurs in the pycnocline in this case. Usingthe definition of the Ekman veering angle, the square ofthe mean shear can be written as

"d!u"dz #2 ) "d!""dz #2 # "d!|u|"dz #2 ) !|u|"2"d#dz#2, $14%where |u| # *u2 ) +2. At the center of the pycnocline,the first term on the right-hand side is zero because thevelocity magnitude |u| is maximum at this location.

FIG. 5. Mean velocity hodograph showing the Ekman spiral.

FIG. 4. (a) Instantaneous plane-averaged temperature profiles. Thick lines show the profilesat the center of the velocity- averaging window; thin lines show the profiles at the start and endof the averaging window. (b) Plane and time-averaged horizontal velocity components, aver-aged over one inertial period.


However, because the Ekman veering rate is very largeat this location, the second term leads to a local maxi-mum of the mean shear in the pycnocline. Most of theenhanced shear in the lower portion of the pycnocline isdue to the spanwise shear, while the streamwise sheardominates in the upper portion of the pycnocline.

Coleman (1999) showed that in an unstratified tur-bulent Ekman layer, the mean velocity magnitudeshould follow the classical logarithmic law (or law-of-the-wall). The law-of-the-wall is expected to hold in aregion far enough from the wall where viscosity can beneglected, but near enough to the wall so that theboundary layer depth is not felt. In this case, the onlyrelevant length scale must be the distance from the wall,so that

|d!u"dz | #u*l

, $15%

where l # &z and & is the von Kármán constant that isempirically found to be about 0.41. The Reynolds-averaged horizontal velocity magnitude is plotted inFig. 8 on a semilogarithmic scale. Very near the wall, all

cases are in reasonably good agreement with the un-stratified logarithmic law. Deviations from the logarith-mic velocity profile in the cases when stratification ispresent can be seen clearly by plotting the normalizedvelocity gradient

! #"zu*

|d!u"dz |. $16%The quantity ' can be interpreted as the ratio of theobserved mean velocity gradient to that expected fromthe logarithmic law and is shown in Fig. 9. When theouter layer is stratified, the mean shear in the pycno-cline increases significantly compared to the log-lawvalue. It is worth noting that deviations from the law-of-the-wall begin well within the mixed layer. The con-sequences for this observation will be discussed in sec-tion 7.

5. Boundary layer turbulence

To understand the increase in mean velocity andmean shear in the pycnocline, it is necessary to considerthe influence of stratification on the turbulent eddies.The turbulent kinetic energy in the mixed layer scaleswith the friction velocity u*, which as we have seen doesnot change significantly with the addition of an outerlayer stratification. In addition, the density gradient inthe pycnocline appears to scale with the outer layerstratification. Therefore, as the stratification in theouter layer increases, the kinetic energy associated witheddies at the top of the mixed layer remains about thesame while the potential energy required for an eddy tooverturn increases. Therefore, when the stratificationbecomes large enough, turbulence is inhibited at thetop of the boundary layer, and as a result, the growth ofthe mixed layer is strongly limited.

One consequence of the damping of turbulence bystratification is a decrease in the turbulent stresses,!u(w(" and !) (w(". The decrease in boundary layerheight with increasing stratification is very apparentfrom the Reynolds stress profiles shown in Fig. 10.Here, the Reynolds stress includes both the resolvedand subgrid-scale contributions. Note that above theboundary layer, the Reynolds stress approaches a small,nonzero value owing to the presence of turbulence-generated internal gravity waves. It is evident from Fig.10 that the Reynolds stresses change more rapidly withheight when the outer layer stratification is stronger.Changes in the Reynolds stress with height result in amomentum flux that must be balanced by other termsat a steady state. The steady-state plane-averaged hori-zontal momentum equations can be written as

FIG. 6. Ekman veering angle.


!!u"!t

# 0 # $!!u"w""

!z% f!#" % $

!2!u"

dz2, &17'

!!#"!t

# 0 # $!!#"w""

!z$ f&!u" $ U%' % $

!2!#"

dz2. &18'

We have found that the Reynolds-averaged eddy vis-cosity is cogradient. Hence, in the upper portion of theEkman layer where d!("/dz ) 0, the spanwise Reynoldsstress is positive as evidenced in Fig. 10. Because bothcomponents of the Reynolds stress become nearly zeroin the outer layer, there must be a region near the topof the Ekman layer where d!(*w*"/dz ) 0. When theleading-order y-momentum balance is between theReynolds stress and Coriolis terms, the region withd!(*w*"/dz + 0 leads to !u" + U,. Therefore, a zonal jetis an inherent feature of a steady-state turbulent Ek-man layer.

The turbulent kinetic energy budget can be found bydotting u* into the perturbation momentum equationsand taking the plane average, to give

!k!t

# $12

!

!z!w"u"iu"i" $

!

!z1&0

!w"p"" %1

Re*

!2k

!z2

$ !Sij"!u"iu"j" $1

Re*!!u"i!xj !u"i!xj" $ Ri*!w"&""

$!

!z!u"i'31" % !'ji !u"i!xj", &19'

where k # !u*i u*i "/2 is the turbulent kinetic energy.Reading from left to right, the terms on the right-handside of Eq. (19) can be identified as the turbulent trans-port, pressure transport, viscous diffusion, production,dissipation, buoyancy flux, subgrid transport, and sub-grid dissipation, respectively. The leading terms in theturbulent kinetic energy budget for N,/f # 75 areshown in Fig. 11. Near the wall, the leading-order bal-ance is between production and dissipation and doesnot differ significantly from the unstratified case asshown in the inset. In the upper portion of the mixed

FIG. 8. Reynolds-averaged horizontal velocity magnitude.

FIG. 7. (a) Temperature gradient, (b) horizontal velocity magnitude, and (c) mean shear.


layer, the turbulent transport appears as a source termrepresenting the advection of turbulent eddies towardthe pycnocline. Pressure transport, buoyancy flux, anddissipation act as energy sinks in the upper mixed layer.In the pycnocline, starting at about z/! " 0.15, the tur-bulent transport and buoyancy flux decrease as strati-fication suppresses turbulent motion while pressuretransport becomes the dominant source term. Whenthe pressure transport is positive the vertical energyflux, #p$w$% increases with height, consistent with aninternal wave field that is gaining energy. This is directevidence of the generation of internal waves by theinteraction between boundary layer turbulence and astable stratification. The properties of these waves willbe discussed in section 6.

The instantaneous temperature field is shown forN&/f " 75 in Fig. 12 in an x–z plane at t " 11.2/f. Figure12b shows an enlarged version of the box drawn in Fig.12a. Isotherms are drawn every 0.025!d#'%/dz&. Themixed layer is very homogeneous with disturbancesrarely exceeding the contour level. Turbulence-gener-ated internal waves are visible as disturbances of theisotherms in the outer layer, which is generally stati-cally stable with the notable exception of the region justabove the pycnocline. Density overturns are visibleboth below and above the pycnocline. As highlighted inFig. 12b, the overturns above the pycnocline are remi-niscent of Kelvin–Helmholtz billows associated with anegative mean shear. The irreversible mixing resultingfrom these overturns may be responsible for the re-duced temperature gradient above the pycnocline asseen in Fig. 7a.

The stability of the shear above the pycnocline can be

examined using the gradient Richardson number. Themean gradient Richardson number, defined as

#Rig% "()g!"0*d##%!dz

)d#u%!dz*2 + )d#$%!dz*2"

#N2%

#S2%, )20*

FIG. 9. Nondimensional velocity gradient, , " (-z/u*) d#u%/dz.FIG. 10. Reynolds stress profiles.

FIG. 11. TKE budget at the top of the mixed layer and pycnoclinefor N&/f " 75. Inset shows the TKE budget near the wall.


as shown in Fig. 13a. Heaving of the pycnocline, whichis visible in Fig. 12, causes significant variations in thepycnocline height. Because we are interested in theshear and stratification a small distance above thepycnocline, averaging over a constant height could beproblematic.

Therefore, the average operator in Eq. (20) has beentaken with respect to a constant distance from the cen-ter of the pycnocline (identified by the maximum tem-perature gradient). It has been shown that a stratifiedflow can develop linear shear instabilities if Rig is lessthan 0.25 somewhere in the flow (Miles 1961; Howard1961). In our simulations !Rig" # 0.25 in the mixedlayer, while !Rig" becomes very large in the outer layerwhere the stratification is large compared to the meanshear. Because !Rig" $ 1 above the pycnocline in bothcases, it is perhaps surprising that local occurrences ofRig # 0.25 are not uncommon even above the pycno-cline. Figure 13b shows the probability of the localRig # 0.25. In the mixed layer (where the distance fromthe pycnocline is large and negative), the probability ofRig # 0.25 is very high, as expected. The probability of

Rig # 0.25 drops to nearly zero in the pycnocline, but alocal maximum in the probability occurs above the pyc-nocline. A coincident local maximum in the probabilityof local density overturns can also be seen above thepycnocline (not shown). About 70% of the overturnsoccurring above the pycnocline are associated with alocal shear that is larger in magnitude than the plane-averaged value. This suggests that while the mean shearis not unstable based on the typical gradient Richard-son number criteria, local shear instabilities may drivethe overturns observed in this region.

To illustrate the appearance of overturns and un-stable shear profiles in the region above the pycnocline,Fig. 14 shows an instantaneous x%z slice through asmall section of the flow when N&/f ' 75. The shadingindicates du(/dz, lines show isotherms at an interval of0.01)d!*"/dz&, and circles indicate locations whereRig # 0.25. The perturbation shear appears to beclosely associated with undulations in the pycnoclineheight. As we have seen, occurrences of Rig # 0.25 arecommon above and below the pycnocline. Most of theregions with an unstable shear above the pycnocline are

FIG. 12. Isotherms projected onto an x–z plane for the case with N&/f ' 75 (a) fullcomputational domain and (b) zoom of boxed region near the pycnocline.


associated with a negative streamwise shear perturba-tion. It also appears that radiated internal waves (vis-ible by phase lines of du!/dz that slope up and to theleft) are sometimes associated with Rig " 0.25, but thisonly occurs near the pycnocline; in the outer layer, Rigis always large.

Because changes in the local shear and the localstratification can cause variations in Rig, it is of interest

to examine the distribution of low Rig events. Figure 15shows a scatterplot of the deviation of the local shearand buoyancy frequency from the background values,for events with 0 " Rig " 0.25. Only one height isshown for clarity, z/# $ 0.195, which corresponds to thesecondary peak in Rig above the pynocline, as shown inFig. 13b. At this location, the mean buoyancy frequencyis 73f, and the mean gradient Richardson number is

FIG. 14. Instantaneous streamwise shear, du!/dz, with overlaid isopycnals for N%/f $ 75.Circles show where Rig " 0.25 locally.

FIG. 13. (a) Mean gradient Richardson number, &Rig', and (b) probability of occurrences ofthe local Rig " 0.25. Vertical profiles have been averaged in terms of the distance from themaximum temperature gradient.


Rig ! "N2#/"S#2 ! 3.15. Because the mean shear is domi-

nated by the streamwise component at this location,only this component is considered. It is clear from Fig.15 that most of the occurrences of Rig $ 0.25 are whendu%/dz $ 0 and nearly all occurrences of Rig $ 0.25 arewhen the stratification is less than its mean value. Fewexceptions occur when the shear is very large and nega-tive, with Rig $ 0.25 even though the local stratificationis high.

6. Turbulence-generated internal waves

In the outer layer above the pycnocline, verticallypropagating internal waves can be observed. Thesewaves, which are generated as turbulent eddies interactwith the stratified ambient, were also observed at a lowReynolds number and are described in detail by Taylorand Sarkar (2007a). The importance of the energy ra-diated by turbulence-generated internal waves to themixed layer growth has been discussed in several pre-vious studies (Linden 1975; E and Hopfinger 1986). An

examination of the steady-state turbulent kinetic en-ergy equation, integrated through the boundary layer,reveals that the turbulent production can be balancedby three sink terms: the integrated dissipation rate, theintegrated buoyancy flux, and the vertical energy fluxassociated with the internal wave field. To compare therelative sizes of the three sink terms, Fig. 16 shows thevertical energy flux, "p%w%#, normalized by the inte-grated dissipation and the integrated buoyancy flux. Toleading order, the turbulent production is balanced bythe dissipation, indicating that the bulk mixing effi-ciency is very small. This is expected given that most ofthe dissipation occurs in the unstratified region near theseafloor. Of the two remaining sink terms, the verticalenergy flux is comparable to the integrated buoyancyflux.

Taylor and Sarkar (2007a) made a similar analysis ofthe boundary layer energetics for a lower Reynoldsnumber boundary layer and also found that the verticalenergy flux was much smaller than the integrated dis-sipation but of the same order as the integrated buoy-ancy flux. Specifically, they found that the ratio of thevertical energy flux to the integrated dissipation at thetop of the pynocline was between 0.01 and 0.03, whilethe ratio of the vertical energy flux to the integratedbuoyancy flux at the same location was between 0.5 and1. The present results are generally consistent withthose of Taylor and Sarkar (2007a), but the verticalenergy flux is about a factor of 2 smaller in the presentstudy.

Taylor and Sarkar (2007a) found that the internalwave energy spectrum in the outer layer could be ex-plained by viscous damping of the waves based on thewavenumber and vertical propagation speed for a par-ticular frequency. Given a turbulent spectrum that wasassumed to be characteristic of the waves upon genera-tion, they showed that the viscous decay term in thefully nonlinear numerical simulations resulted in outerlayer waves in a relatively narrow frequency range. Vis-cous ray tracing was used to show that the vertical ve-locity amplitude for a specific frequency and wavenum-ber can be written as

A&kh, !, z' ! A0&kh, !, z0'|k0||k| exp! ("!kh&!2 ( f 2'1#2 "z0

z

|k|4&N2 ( !2'(1#2 dz$#, &21'where k0 and A0 are the wavenumber and wave ampli-tude at z ! z0 in the generation region. Because themolecular viscosity and diffusivity used by Taylor andSarkar were two orders of magnitude smaller than the

present values, it is of interest to evaluate their viscousinternal wave model using the present high Reynoldsnumber simulations. The viscous decay model is com-pared to the observed frequency spectra of the outer

FIG. 15. Perturbation streamwise shear and buoyancy frequencyfor events with 0 $ Rig $ 0.25 at z/) ! 0.195. The mean gradientRichardson number, "Rig# ! 3.15.


layer waves in Fig. 17. To compare the predicted waveamplitude to the simulations, it is convenient to con-sider the spectral distribution of dw/dz, which is esti-mated by multiplying A(kh, !, z) by the vertical wave-number

m " #kH!N2 # !2!2 # f 2 "1"2. $22%

Because the subgrid-scale eddy viscosity that is used aspart of the LES model is not negligible compared to themolecular viscosity in the outer layer, we have used & '&sgs in Eq. (21). Figure 17 shows the spectral amplitudesof (w)/(z normalized by * and u*. To show the com-bined contributions of all values of kH, the square rootof the sum of the squared amplitudes of (w)/(z is shownas a function of !/f. The left-hand side shows the ob-served spectra at z " z0, corresponding to a locationjust above the pycnocline (solid line). The spectrum atz " z0 was then smoothed and used as input (dashedline) to the viscous decay model. The viscous decaymodel predicts the amplitude of each frequency andwavenumber component after propagating a distancez # z0. The predicted amplitude from the model iscompared to the observed amplitudes from the numeri-cal simulation on the right-hand side, corresponding toa location at the top of the computational domain. The

qualitative agreement between the observed and pre-dicted wave amplitudes is good; in particular, the de-crease in amplitude of the low-frequency waves is cap-tured well. At high frequencies, as ! → N, the observedwave amplitudes are significantly lower than the modelprediction. It is possible that this is the result of non-linear wave–wave and wave–mean flow interactionsthat are neglected in the linear viscous model.

7. Evaluating methods for estimating the wallstress

Because we have high-resolution velocity and densityprofiles through a steady Ekman layer, we are able toevaluate the performance of several methods for esti-mating the friction velocity from observational data un-der idealized conditions. The profile method estimatesthe friction velocity by assuming that the mean shearfollows the unstratified law-of-the-wall, to give

u*,p " #zd+u,dz

. $23%

When the turbulent dissipation is measured, the frictionvelocity can be estimated by assuming a balance be-tween the turbulent production and dissipation. Asshown in the inset of Fig. 11, the production and dissi-

FIG. 16. Vertical energy flux normalized by (a) the integrated turbulent dissipation and (b)the integrated buoyancy flux.


pation are in approximate balance throughout themixed layer. The so-called balance method assumes aproduction and dissipation balance using the observedmean shear and disspation rate, so that the friction ve-locity is

u*,b !!!"d"u#dz . $24%A third method called the dissipation method can beformed through a combination of the profile and bal-ance methods by using the production and dissipationbalance and assuming that the mean shear follows theunstratified law-of-the-wall. The friction velocity pre-dicted by this method takes the form

u*,! ! "!"z#1#3. $25%

Estimates from these models are compared to the ob-served friction velocity in Fig. 18. To illustrate the tem-poral variability in the friction velocity, &1' (the stan-dard deviation) is also shown. As has been found byprevious studies (Perlin et al. 2005; Johnson et al. 1994;Lien and Sanford 2004), the performance of the frictionvelocity estimates depends on the location of the ob-served shear and on dissipation. Therefore, estimates ofthe friction velocity are shown using the mean shearand/or the dissipation rate at two heights, z/( ! 0.06and z/( ! 0.12, which correspond to about 1.98 and 3.96mab, respectively. When the outer flow is unstratified,

as shown in the left column of Fig. 18, all of the abovemethods provide a reasonable estimate of the frictionvelocity. However, when the outer layer is stratified,the profile method and to a lesser degree the balancemethod do not provide accurate estimates for the fric-tion velocity. The dissipation method appears to be themost accurate of the three methods in the mixed layer.However, because direct measurements of the dissipa-tion rate are not always available from observations, itis desirable to have a method for estimating the frictionvelocity using more commonly measured quantitiessuch as velocity and density profiles.

The error in the profile method is the result of theincrease in the mean shear with stratification as wasseen in Fig. 9, which is not accounted for by the tradi-tional law-of-the-wall. A modified law-of-the-wall thataccounts for the increase in mean shear at the top of astratified boundary layer was proposed by Perlin et al.(2005). They proposed that the mean velocity gradientcould be modeled as

d"u#dz

!u*lp

, $26%

where

lp ! "z$1 ) z#hd%, $27%

and hd is a measure of the boundary layer depth, whichis limited by stratification. To use Eq. (26) to predictthe friction velocity for a given velocity profile, an es-

FIG. 17. Spectra of *w+/*z from simulated results (solid line) and the viscous decaymodel (dashed line) for (a),(b) N, f ! 31.6 and (c),(d) N, f ! 75. Vertical linesshow N/f.


timate for hd must first be obtained. Perlin et al. (2005)proposed using the Ozmidov scale, lOz ! "#/N3, at thetop of the mixed layer to set hd. We have used a similarcriterion to evaluate this method in Fig. 18. Specificallythe mixed layer height d is defined as the locationwhere $%/d ! 0.01d%/dz&. Then hd is set so that lp ! lOzat z ! d.

It is evident from Fig. 18 that the modified law-of-the-wall provides a significant improvement over theprofile method. However, as shown in Fig. 19, theOzmidov scale varies very rapidly between the mixedlayer and the pycnocline, so the estimate of hd dependsstrongly on the definition of the mixed layer depth. Inthe case when N&/f ! 31.6, the rapid change in theOzmidov scale leads to hd ! 0.67', which is significantlylarger than the boundary layer height, h ! 0.215'. As aresult, lp is significantly larger than the observed shearlength scale in the upper half of the mixed layer, asshown in Fig. 20. An alternative method to account forthe decrease in the length scale with stratification wasproposed by Brost and Wyngaard (1978) and can bewritten as

1l

!1

!z(

1lb

, )28*

where lb ! Cb+w ,w ,-1/2/N is a buoyancy length scale.

Nieuwstadt (1984) suggested the value of Cb ! 1.69,which was consistent with his local scaling theory. This

length scale is shown in Fig. 20 and compares favorablyto the observed shear length scale below the center ofthe pycnocline. Practically, it is difficult to measure thevertical velocity, especially in the boundary layer whereit cannot be deduced from isopycnal displacements. Wehave found that most of the decrease in the length scalewith height is due to an increase in the local stratifica-tion rather than to a change in the turbulent velocity.An alternative length scale can then be formed by re-

FIG. 19. Ozmidov scale.

FIG. 18. Estimates of the friction velocity using several different methods at two locations in the mixed layer.Horizontal lines show the friction velocity observed in the simulations and .1/ of the time series. (left to right)Models are the balance method Eq. (24), the dissipation method Eq. (25), the profile method Eq. (23), the modifiedlaw-of-the-wall Eq. (26), and the modified profile method Eq. (30). Note that when the flow is unstratified, themodified law-of-the-wall and the modified profile method are identical to the profile method.


placing the vertical turbulent velocity with the frictionvelocity:

1l

!1

!z"

N#z$Cbu*

. #29$

This simplified form still provides a reasonable estimatefor the shear length scale, as shown in Fig. 20. Thefriction velocity can then be recovered from the meanvelocity and density profile without the need for directturbulence measurements, specifically

u*,m%p ! !z!"#d&u'dz $2 " #d&"'dz $2%1#2 % N#z$&, &30'where we have taken Cb ! 1. In the limit of an unstrati-fied boundary layer, this method becomes equivalent tothe profile method, so we will refer to this as the modi-fied profile (m-p) method. Estimates of the friction ve-locity based on Eq. (30) are shown as triangles in Fig.18. While the estimated friction velocity is somewhatlarge, the modified profile method provides a signifi-cant improvement over the profile method and is com-parable to the modified law-of-the-wall. It is worth not-ing that because stratification effects enter into themodified law-of-the-wall through hd in Eq. (27), whichis independent of z, stratification effects are nonlocal inthis model. By comparison, the modified profile

method is the result of a local balance between shearand stratification.

The length scale given in Eq. (29) can also be used toform a model velocity profile by integrating

dUdz

!u*l

, #31$

to obtain

U#z$u*

!1!

log#z#z0$ "1

Cbu*'

0

z

N#z$$ dz$. #32$

If a constant buoyancy frequency, say, N(, were used inplace of N(z) in Eq. (32), the form of the velocity pro-file would become log linear. The model velocity in Eq.(32) can, therefore, be viewed as an analog to the log-linear profile from Monin–Obukhov theory that is com-monly used in the atmospheric literature, with theObukhov length replaced by u*/N. Figure 21 showsprofiles of the observed mean velocity compared to Eq.(32) for both a constant and a nonconstant N. Tosmooth fluctuations in the instantaneous profile ofN(z), the profiles of N(z) have been averaged over atime window of t ! 1/f. When a depth-dependent buoy-ancy frequency, N(z), is used in Eq. (32), the parameteris set to Cb ! 1 to be consistent with Eq. (30). WhenN(z) is replaced by the constant free-stream buoyancyfrequency, N(, it is found that Cb ! 0.2 provides a good

FIG. 20. Length-scale profile derived from the mean shear from the LES (line with filledcircles) compared to several model profiles. Shaded regions show where d&p'/dz ) d&p'/dz(.


fit to the observed profiles. Using either a constant or anonconstant buoyancy frequency in Eq. (32), the devia-tions from the logarithmic law owing to stratificationare well represented. Note that in the outer layer, thevelocity profiles are affected by the boundary layerheight and Eq. (32) is not expected to hold.

8. Conclusions

We have examined a benthic Ekman layer formedwhen a uniformly stratified, steady geostrophic flow en-counters a flat, adiabatic seafloor. The thermal fieldrapidly develops a three-layer structure with a well-mixed region near the wall separated from the uni-formly stratified outer layer by a pycnocline. The outerlayer is populated by upward-propagating internalwaves that are generated by the boundary layer turbu-lence. After the initial spinup, a quasi-steady state isreached, characterized by a slow mixed layer growthand a nearly constant density gradient in the pycno-cline. When the strength of the outer layer stratificationis increased, the wall stress increases slightly, but theboundary layer thickness decreases significantly. Thestructure of the boundary layer is clearly confined bystratification as evidenced by the Reynolds stress, tur-bulent heat flux, and turning angle, which all nearlyvanish above the pycnocline. Because the Ekman trans-port balances the wall stress in the integrated stream-wise momentum equation, and the wall stress remains

relatively constant, the increase in outer layer stratifi-cation is accompanied by an increase in the magnitudeof the cross-stream velocity.

The increased cross-stream velocity in the mixedlayer leads to a broadening of the Ekman spiral. Therate of veering in the mixed layer is a function of z butdoes not depend strongly on the external stratification.When the stratification is increased and the boundarylayer is thinner and the surface turning angle is larger,the amount of veering that occurs in the pycnoclineincreases. This finding is consistent with the results ofWeatherly and Martin (1978), who found that most ofthe veering occurred in the pycnocline using one-dimensional simulations. We have also found that whenthe outer layer stratification is large, the rapid rate ofturning in the pycnocline causes the mean velocity andthe mean shear to be maximum at the same locationnear the center of the pycnocline.

An interesting feature of the observed Ekman layerstructure is the appearance of local shear instabilitiesabove the pycnocline, despite the fact that the meanshear is stable with respect to the local stratification.Between 10% and 25% of the vertical profiles exhibit alocal gradient Richardson number less than 0.25. Asimilar observation has been made in terms of the oc-currence of overturning events. Mixing during theseevents is significant and appears to cause a local mini-mum in the density gradient above the pycnocline.Analysis of events with Rig ! 0.25 at a height above the

FIG. 21. Simulated velocity profiles compared to predictions by the model profiles of Eq.(32) for a constant and nonconstant N.


pycnocline indicates that these events are more likely tobe associated with a below average density gradientthan an above average shear.

In the outer layer, turbulence-generated internalwaves are observed radiating away from the boundarylayer. The vertical energy flux associated with thesewaves is negligible compared to the turbulent dissipa-tion integrated through the boundary layer. This is notsurprising because most of the dissipation occurs nearthe seafloor where the flow is unstratified. However,the vertical energy flux at the top of the boundary layeris nearly half of the integrated buoyancy flux. This find-ing is consistent with a study at a lower Reynolds num-ber by Taylor and Sarkar (2007a) and implies that theturbulence-generated internal waves may removeenough energy from the boundary layer to affect thegrowth rate of the mixed layer. The viscous internalwave model of Taylor and Sarkar (2007a) has beenapplied using the combined molecular and turbulentviscosity to estimate the decay rate of the turbulence-generated internal waves. The model qualitatively cap-tures the decay of low-frequency waves but overesti-mates the amplitude of high-frequency waves.

As was seen in previous studies (e.g., Perlin et al.2005; Johnson et al. 1994), an increase in the meanshear has been observed at the top of the mixed layer.The increase in mean shear with respect to an unstrati-fied boundary layer can lead to significant errors in thefriction velocity estimated from observed velocity pro-files using the profile method. It has been uncertainwhether this increase in the mean shear could be ex-plained in terms of the local stratification. Because theunstratified logarithmic law appears to hold very closeto the wall, the profile method is adequate, in principle,if the mean velocity very near the wall can be obtained.However, this is often difficult or impossible in practice.We have evaluated the performance of a variety oftechniques for estimating the friction velocity givenmean quantities at various heights in the mixed layer.Because the turbulent production and dissipation arethe dominant terms in the turbulent kinetic energyequation throughout the mixed layer, the dissipationmethod agrees very well with the observed friction ve-locity. The modified law-of-the-wall, proposed by Per-lin et al. (2005), shows considerable improvement overthe standard profile method, especially near the top ofthe mixed layer. However, like the balance and dissi-pation methods, the modified law-of-the-wall requiresknowledge of the turbulent dissipation rate. Direct ob-servation of the dissipation rate is difficult, especially inactive turbulent regions because it involves small-scalevelocity gradients. When such information is not avail-able, it is desirable to have an alternative method for

estimating the friction velocity u*. We have introduceda buoyancy length scale, u*/N(z), which leads to amodification of the well-known log-law by a buoyancy-related augmentation of the mean velocity. Use of thismodified mean velocity profile instead of the unstrati-fied log law leads to a significant improvement in de-ducing the friction velocity.

Acknowledgments. We are grateful for the supportprovided by Grant N00014-05-1-0334 from ONR Physi-cal Oceanography, program manager Scott Harper.

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