Transport by an intrusion generated by boundary mixing in a lake

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Transport by an intrusion generated by boundary mixing in a lake
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  C+", C%$()'*c)%$ a$d E$+'%$#e$)a" E$g$ee'$gP*b"ca)%$(C+", C%$()'*c)%$ a$d E$+'%$#e$)a" E$g$ee'$g8-2010 Transport by an intrusion generated by boundary mixing in a lake Danielle J. Wain  Iowa State University  , da$e""e.a$@g#a".c%# Chris R. Rehmann  Iowa State University  , 'eh#a$$@a()a)e.ed* F%""% )h( a$d add)%$a" %'!( a):h&://"b.d'.a()a)e.ed*/ccee_&*b(Pa') %f )heE$+'%$#e$)a" E$g$ee'$g C%##%$(    , a$d )heHd'a*"c E$g$ee'$g C%##%$(e c%#&"e)e bb"%g'a&hc $f%'#a)%$ f%' )h( )e# ca$ be f%*$d a)h&://"b.d'.a()a)e.ed*/ccee_&*b(/12. F%' $f%'#a)%$ %$ h% )% c)e )h( )e#, &"ea(e +()h&://"b.d'.a()a)e.ed*/h%)%c)e.h)#". ( A')c"e ( b'%*gh) )% %* f%' f'ee a$d %&e$ acce(( b )he C+", C%$()'*c)%$ a$d E$+'%$#e$)a" E$g$ee'$g a) Dg)a" Re&%()%' @ I%a S)a)eU$+e'(). I) ha( bee$ acce&)ed f%' $c"*(%$ $ C+", C%$()'*c)%$ a$d E$+'%$#e$)a" E$g$ee'$g P*b"ca)%$( b a$ a*)h%'ed ad#$()'a)%' %f Dg)a" Re&%()%' @ I%a S)a)e U$+e'(). F%' #%'e $f%'#a)%$, &"ea(e c%$)ac)dg'e&@a()a)e.ed*.  Transport by an intrusion generated by boundary mixing in a lake Danielle J. Wain 1,2 and Chris R. Rehmann 1 Received 14 July 2009; revised 7 January 2010; accepted 16 March 2010; published 10 August 2010. [ 1 ]  A dye study was conducted to track an intrusion generated at the boundary of a smalllake. Persistent turbid layers offshore presented evidence of possible intrusions from boundarymixing.Afterhighwinds,astreakofRhodamineWTwasinjectedattheboundaryofthelakewheretheslopewasbetween5%and10%.Bothverticalprofilesandlongitudinaltransects of the dye concentration were measured. The three ‐ dimensional dye mappingshowedadistinctdyeintrusion,rangingbetween0.5and1mthick,over200minhorizontalextent offshore, 1 day after the injection. Profiles of temperature microstructure measuredsoon after the injection both at the injection site and offshore showed an elevated eddydiffusivity near the boundary where the dye was injected, indicating that the intrusionresulted from boundary mixing. Because the dominant internal waves are subcritical, themixing is most likely due to seiching currents interacting with the boundary. The propagation characteristics of the intrusion were predicted most closely by a formulation for a radially spreading intrusion governed by a balance between buoyancy and inertia. Theseresults show that intrusion generation and propagation may be a significant process for mass transport in stratified lakes and reservoirs. Citation:  Wain, D. J., and C. R. Rehmann (2010), Transport by an intrusion generated by boundary mixing in a lake,  Water  Resour. Res. ,  46  , W08517, doi:10.1029/2009WR008391. 1. Introduction [ 2 ] Understanding the transport of dissolved substancessuch as oxygen, nutrients, microorganisms, and plankton isessential for managing water quality in lakes and reservoirs.The ability of stratification, which is caused by temperature,salinity, suspended sediment, or dissolved gases, to restrict vertical mixing and control the spatial variability of nutrientsand other substances affects the distribution of dissolvedoxygen in the water column [e.g.,  Rao et al. , 2008], theavailability of nutrients to phytoplankton [e.g.,  MacIntyreet al. , 1999], and transport of pollutants between the hypo-limnion and epilimnion [e.g.,  Morillo et al. , 2008]. The cur-rentmodelinoceanandlakemixingisthatturbulencecreatedat the boundaries by internal waves and currents causes most of the mixing [e.g.,  Gregg  , 1998;  Ledwell et al. , 2000; WunschandFerrari ,2004].Whilemuchworkhasbeendoneto investigate mixing at boundaries in lakes and the ocean,less attention has been paid to the fate of the mixed fluid. One possible outcome, investigated in this study, is that intrusionstransport mixed fluid into the interior.[ 3 ]  Munk   [1966] suggested that the mixing required toexplain historic temperature profiles in the ocean might arisefrom boundaries rather than the ocean interior. Pursuing thissuggestion,  Armi  [1978] proposed that turbulence generated by shear from currents on the ocean bottom was responsiblefor the mixing; he envisioned fluid being mixed on thesloping bottom and advected along isopycnal surfaces intothe ocean interior.  Garrett   [1979, 1990] challenged this sce-nario with arguments about the efficiency of the bottommixing: if the bottom turbulence merely stirs fluid that isalready mixed, then the efficiency and the resulting fluxes tothe interior cannot be large. Nevertheless,  Garrett   [1979]admitted the possibility that boundary processes may con-trol vertical mixing in the ocean.[ 4 ] Laboratory experiments on mixing on a sloping boundary suggest that secondary circulation can restratify thefluid near a sloping boundary and possibly increase the effi-ciency of boundary mixing [  Phillips et al. , 1986]. Becauseisopycnals must bend to satisfy the no ‐ flux condition at a sloping boundary, buoyancy forces drive transport both upand down the slope [  Phillips , 1970;  Wunsch , 1970]. Thus,while  Garrett   [1979] raised concerns about boundary pro-cesses merely mixing mixed fluid, the secondary circulation provides a mechanism for boundary mixing to act on strati-fied fluid. However,  Garrett   [1990] argued that the net effect of this secondary circulation on overall mixing is ambiguoussince the restratification is accompanied by a countergradient vertical advective flux. By comparing the vertical diffusiveflux and total flux for spatially variable mixing,  Garrett et al. [1993] defined a mixing effectiveness, by which the basin ‐ averaged mixing rate must be reduced and speculated that it may be small.[ 5 ] Some of the questions about how boundary processescontrol mixing in stratified water bodies can be addressedmoreeasilyinlakesthanintheocean.Twomainmechanismsgenerate mixing at a sloping boundary in a lake (Figure 1).Wind acting on the surface of a stratified lake can set upseiches, which generate currents along the boundaries, as inthe model of   Armi  [1978]. The friction with the bottomgenerates turbulence that mixes the water locally to create a turbulent bottom boundary layer [e.g.,  Gloor et al. , 2000].The seiching motions can also degenerate into higher  ‐ 1 Department of Civil, Construction, and Environmental Engineering, Iowa State University, Ames, Iowa, USA. 2 Now at Applied Physics Laboratory, University of Washington,Seattle, Washington, USA. Copyright 2010 by the American Geophysical Union.0043 ‐ 1397/10/2009WR008391 WATER RESOURCES RESEARCH, VOL. 46, W08517, doi:10.1029/2009WR008391, 2010 W08517  1 of   11  frequency waves. Some of these waves propagate toward the boundarywithacriticalfrequencysetbythestratificationandthe slope of the boundary. When the critical waves approachthe boundary, their energy reflects along the slope andenergizes a turbulent bottom boundary layer [e.g.,  Eriksen ,1998;  MacIntyre et al. , 1999]. Once a turbulent boundarylayer is created on the slope, the mixed boundary layer fluid becomes gravitationally unstable with respect to the stratifiedwater adjacent to it, and the mixed patch collapses and formsan intrusion. The intrusion can then transport fluid and other constituents along an isopycnal [e.g.,  Thorpe , 1998].[ 6 ] Several studies provide evidence for intrusions from boundary processes.  Caldwell et al.  [1978] observed stepped profiles near sloping boundaries in a lake that they attributedto intrusions generated by boundary mixing. Dye injectedinto the water column above a slope in a fjord was eventuallyentrained into a turbulent boundary layer generated by thesemidiurnal tide moving over rough topography, and then it entered the interior as intrusions [  Inall  , 2009]. Intrusions canalso explain turbid layers observed in the interior of a water  body.  Dickson and McCave  [1986] and  Thorpe and White [1988] proposed that nepheloid layers along a continentalslope resulted from boundary mixing due to the semidiurnalM2 tide reflecting critically. Such observations in lakes arerare:  Marti and Imberger   [2008] observed a well ‐ definedturbid layer that they concluded had been advected offshore by currents in the metalimnion that resulted from a secondvertical mode seiche.[ 7 ] Observations of intrusions help to assess the role of  boundary processes in basin scale mixing.  Gloor etal.  [2000]noted a spatial variation in the thickness of bottom mixedlayers: in the deepest part of thelake, well ‐ mixed layers of 4  –  5 m developed, and they disappeared over 10  –  20 days after strong seiching stopped. On the slopes, however, the mixedlayers were thinner and more intermittent, and mixed water masses extended up to several hundred meters into the lakeinterior.Fromtheseobservationsandthetheoryof   Barenblatt  [1978],  Gloor et al.  [2000] suggested that intrusions carrymixed fluid away from the boundaries. Near the bottom, theobservations are consistent with the prediction of   Garrett  [1979] that the efficiency of boundary mixing would below because only mixed fluid is mixed, but on the slopes,intrusions provide a mechanism for removing the mixed fluid[ Gloor et al. , 2000].[ 8 ] While several field experiments have provided evi-dence of intrusions from boundary mixing, intrusions re-sulting from the collapse of turbulent regions have beentracked from the boundary into the interior mostly in labo-ratory experiments. Several experiments have investigatedthe vertical mixing from breaking internal waves andobserved intrusions [e.g.,  Cacchione and Wunsch , 1974;  Iveyand Nokes , 1989], but fewer have quantified the intrusion properties.  De Silva et al.  [1997] and  McPhee ‐ Shaw and  Kunze  [2002] measured propagation of intrusions resultingfrom breaking internal waves on a slope and related theintrusion speed to the energy of the incident internal waves.The former observed no change in the background densitystratification due to the intrusion, whereas the latter observed persistent steps; however, as  De Silva et al.  [1997] noted,eveniftheintrusiondoesnotchangethethermalstructureofa lake, dissolved substances can still be transported offshore.The experiments of   Wells and Helfrich  [2004] provideinformation on the three ‐ dimensional behavior of an intru-sion generated at the boundary; in those, effects of rotationlimited the propagation of the intrusion.[ 9 ] Intrusions generated by boundary mixing may be animportant mechanism to redistribute mixed fluid away fromthe boundaries and into the lake interior. While observationalevidence of intrusions from boundary mixing exists, most studies of the phenomenon have occurred in the laboratory. Figure 1.  Schematic of physical processes leading to boundary mixing and intrusions in a lake. Wind cancreateseichesandcriticalinternalwaves.Turbulencefromseichingcurrentsorwavebreakingcanmixfluidin the bottom boundary layer, and gravitational readjustment of this mixed layer can lead to intrusions of mixed fluid propagating into the interior. WAIN AND REHMANN: INTRUSIONS FROM BOUNDARY MIXING IN A LAKE  W08517W08517 2 of 11  With the exception of   Inall   [2009], no field experiments exist where intrusions have been tracked from the boundary intotheinterior,asisdoneinthelaboratory.Toinvestigatethefateof mixed fluid in a lake, we used a tracer to track an intrusiongenerated at the boundary and conducted simultaneousturbulence measurements. Section 2 describes the lake, themeasurements, and the analysis. In section 3, we present measurements of the wind, eddy diffusivity, and dye con-centrations. In section 4, we compare our measurements to previous work on intrusions and discuss the source of theintrusion, the force balance that drives its propagation, andthe implications for water quality. 2. Experiment [ 10 ] An experiment that combined microstructure mea-surements with a dye release was conducted at Ada HaydenLakeinAmes,IA,USA(Figure2).Dyewasinjected1hafter a storm front passed on 20 July 2005. An atmospheric grav-ity wave produced strong winds that reversed direction by180° as the wave passed. Temperature microstructure wasmeasured 6 h after the dye injection, and the dye cloudwas mapped 1 day after the injection. Another atmosphericgravity wave passed over the lake 17 h after the injection but  before the mapping.[ 11 ] Ada Hayden Lake is an abandoned rock quarry that isused as a secondary water supply for Ames. It consists of two basins, and the experiment was performed in the larger,deeper south basin, which has a surface area of about 0.3 km 2 andamaximumdepthofabout17m.Thefetchintheprimarywind direction is approximately 700 m. The lake is stronglystratified in the summer, with a well ‐ mixed epilimnion lyingover a thick, strongly stratified metalimnion and a weaklystratifiedhypolimnion(Figure3a).Waterentersthelakefromgroundwater and surface water runoff, which is filteredthroughwetlands.Thetwobasinsareseparatedbya3mdeepsill. Because the sill is shallower than the summer thermo-cline, exchange between the two basins most likely consistsonly of epilimnetic waters. Stirring from boat traffic is small because motorized boats are prohibited on the lake. The lakehas steep sides except for a few areas; the northeast corner,where dye was injected, has a more moderate slope ranging between 5% and 10%.[ 12 ] During July, the mean winds at Ada Hayden Lakeare 2  –  3 m/s SSE. Stronger winds (>5 m/s) are typicallyassociated with storms, most often from the south. Windmeasurements come from the Ames Municipal Airport,approximately 8 km south of the lake. Comparing thesemeasurements to those from several meteorological stationsin communities surrounding the lake suggests that windspeeds measured at the airport represent the conditions at thelake. The wind was measured every minute at the airport, andthen a moving average over a 15 min interval was computed.The strength of the wind, which can cause the stable densitystructure to overturn, was compared to the strength of thestratification, which resists overturning, with the Lake number [  Imberger and Patterson , 1990],  L  N  ¼  gS  t   1    z  T =  z  s ð Þ  s u 2 * A 3 = 2s  1    z  v =  z  s ð Þ ;  ð 1 Þ where  g   is the acceleration of gravity,  z  T  is the center of themetalimnion,  z  s  is the height of the water surface,  r  s  is thedensity at the surface,  u *  is the shear velocity in the water,  A s is the area of the surface,  z  v  is the height of the center of volume, and  S  t   is the stability of the water body defined as S  t   ¼ Z   z  s 0  z  v    z  ð Þ   z  ð Þ  A z  ð Þ d  z  ;  ð 2 Þ where  A (  z  )isthesurfaceareaasafunctionofdepth.Theshear velocity in the water was computed as u*  ¼   a   s   1 = 2 u* a  ;  ð 3 Þ Figure 2.  Map of the south basin of Ada Hayden Lake.Depth contours are marked in meters. Dye was injected at Station A, and microstructure was measured near Stations Aand B. Solid circles indicate where dye profiles were mea-sured for the transect series in Figure 7. Figure 3.  Conditions during the experiment: (a) averagetemperature profile, (b) buoyancy frequency  N  , and (c) tur- bidity profile before the dye injection. To remove internalwave effects, turbidity as a function of temperature was con-verted to a profile of turbidity as a function of depth using themean temperature profile. WAIN AND REHMANN: INTRUSIONS FROM BOUNDARY MIXING IN A LAKE  W08517W08517 3 of 11  where  r  a   is the air density and  u * a   is the shear velocity of thewind,calculatedwiththeformulasforweakand strongwindsin the work of   Wüest and Lorke  [2003]. Low Lake numbersindicate that the surface force from wind stress can tilt the isotherms, and a Lake number of 1 implies upwelling[  Imberger and Patterson , 1990].[ 13 ] Temperature profiles were measured with the tem- perature sensor on a Self  ‐ Contained Underwater FluorescenceApparatus (SCUFA) from Turner Designs, which was sam- pled simultaneously at 1 Hz with an SBE 50 Digital Ocean-ographic Pressure Sensor from Sea  ‐ Bird Electronics. Theinstrumentswereloweredbyhandatapproximately0.25m/s,and 76 vertical profiles were measured. The first 42 of these profiles were measured over 5 h on 20 July 2005, before andimmediately after the dye injection. The last 34 profiles weremeasured over 4 h on 21 July 2005. The water column wasdivided into 1 m bins, and all the measurements in the depth bin were averaged to produce a mean temperature profile for the experiment, with the mean value associated with thecenter of the bin. The equation of state of   Chen and Millero [1977] was used to compute the density profile from themean temperature profile. This density profile was linearlyinterpolated onto a 0.25 m grid, and the buoyancy frequencywas computed using a centered difference (Figure 3b). Tur- bidity was also measured with the SCUFA.[ 14 ] Temperature microstructure was measured at anonshore site and an offshore site (sites A and B, respectively,in Figure 2) with a Self  ‐ Contained Autonomous MicroPro-filer (SCAMP) manufactured by Precision Measurement Engineering. Ten profiles were measured at each site. TheSCAMP measures small ‐ scale temperature fluctuations withThermometrics FP07 thermistors, which have a nominalresponse time of 7 ms, though the actual response dependson probe speed and sensor construction [ Gregg  , 1999]. Thefall rate of the SCAMP was approximately 0.1 m/s, andtemperatures were recorded at 100 Hz. Analog signal pro-cessors in the SCAMP computed the time derivative of thevoltage signals from the thermistors before the signals weredigitized,andprofilesoftemperaturegradientwerecomputedwith Taylor  ’ s hypothesis.[ 15 ] The temperature gradients were then used to computethe dissipation of temperature variance  c T  and eddy diffu-sivity  K  T .Followingproceduressimilartothoseusedby Sogaand Rehmann  [2004], we computed c T  by assuming isotropyand integrating the difference of the observed temperaturegradient spectrum  S  obs , computed in segments of 512 pointsor about 0.5 m, and the noise spectrum  S  n  over the wavenumber   k  1,  T  ¼  6  D T Z  1 0 S  obs  k  1 ð Þ   S  n  k  1 ð Þ½  d k  1 ;  ð 4 Þ where  D T  is the molecular diffusivity of heat. The noisespectrum is an estimate derived from measurements with anFP07 thermistor in quiescent water. The eddy diffusivity wascomputed with the relation from  Osborn and Cox  [1972],  K  T  ¼   T 2  @  T  =@   z    2  :  ð 5 Þ The mean temperature gradient   ∂ T  /  ∂  z   was determined byfitting a line to the temperature in each segment. Values of  ∂ T  /  ∂  z   and c T  were assigned to 0.5 m intervals in the vertical,and profiles of the ensemble averages and their 95% confi-dence limits were computed for each sampling site from 200 bootstrapresampled populations.Thestatistics for  c T andthetemperaturegradientwereusedtocomputeprofilesof   K  T andtheir 95% confidence limits using equation (5).[ 16 ]  Wain and Rehmann  [2005] addressed the uncertaintyin the eddy diffusivity  K  T  computed with the Osborn ‐ Coxmethod, which comes from the fit of the mean temperaturegradient for each segment, the resolution of   c T , the assump-tion of isotropy, and the validity of the Osborn ‐ Cox balance.An analysis of the robust linear fit used to compute thetemperaturegradientyieldedanuncertaintyoflessthan1%in ∂ T  /  ∂  z  , and more than 95% of   c T  was resolved in all spectra  before they met the noise spectrum. The assumption of isot-ropy used in (4) can be questioned in strongly stratified tur- bulence because vertical motions are suppressed relative tothe horizontal motions. Theory for strongly stratified,unsheared turbulence predicts small errors, while theory for strongly stratified, sheared turbulence suggests that (4) willoverpredict   c T  by a factor of 2  –  3 [  Rehmann and Hwang  ,2005]; the latter result is consistent with results from direct numerical simulations [  Itsweire et al. , 1993]. For turbulencein Kelvin ‐ Helmholtz billows with Cox number   K  T /   D T  > 10,as in the results reported below, errors can be up to a factor of 3, but for turbulence in later stages, the errors are less than a factor of 2 [ Smyth and Moum , 2000]. With microstructuremeasurements alone, evaluating the validity of the produc-tion ‐ dissipation balance behind (5) is difficult; however, a similar balance has been assumed in many other measure-ments of turbulence on a sloping boundary in a lake [e.g., Wüest et al. , 1996;  MacIntyre et al. , 1999].[ 17 ] To assess the state of the turbulence in the bottom boundary layer, we computed dimensionless parameters that involve  " , the rate of dissipation of turbulent kinetic energy.The dissipation can be estimated by fitting a theoretical formof the temperature gradient spectrum, srcinally derived by  Batchelor   [1959], to spectra measured with the SCAMP. Theone ‐ dimensional spectrum  S  B ( k  1 ) of the temperature gradient is S  B  k  1 ð Þ ¼  ffiffiffi q 2 r    T  D T k  B   e   2 = 2    Z  1  e   x 2 = 2 d  x 0@1A ;  ð 6 Þ wheretheparameter  q istakentobe3.4[  Ruddicketal. ,2000].The Batchelor wavenumber is  k  B  = ( " /  n   D T2 ) 1/4 , where n   is thekinematic viscosity and    = (2 q ) 1/2 k  1 /  k  B . The spectrum (6)was fit to measured temperature gradient spectra at highwavenumbers by adjusting  k  B , and the dissipation wascomputed from the best fit.[ 18 ] Rhodamine WT was used to track mixed fluid fromthe slope into the interior. The presence of a turbid layer (Figure 3c) and steps in the individual temperature profiles,which indicate mixed layers that may result from intrusions,were used to determine the target depth of 6 m for injection.Thedyewasmixedwithsurfacewatertomatchthedensityonthetarget isotherm. The dye was injected in a 10 m horizontalstreak over 5 min by pumping from the shore through a hoseconnected to a diffuser that spread the dye and reduced theturbulence generated by the injection. The injection deviceincludedtheSCUFAandSBE50sothatthediffusercouldbeset to the target depth. Once the dye mixture was emptiedfrom the vessel in which it was mixed, the remaining dye wasflushed out of the hoses with surface water. WAIN AND REHMANN: INTRUSIONS FROM BOUNDARY MIXING IN A LAKE  W08517W08517 4 of 11
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