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Home , Internal wave, Stokes drift, Thermocline

JUNE 2016 H E N D E R S O N 1947

Upslope Internal- , and Compensating Downslope Eulerian Mean Currents, Observed above a Lakebed

STEPHEN M. HENDERSON School of the Environment, Washington State University, Vancouver, Washington

(Manuscript received 26 June 2015, in final form 14 January 2016)

ABSTRACT

In a small , where flows were dominated by internal with 10–32-h period, slow but persistent mean transport of water over many wave periods was examined. Acoustic Doppler profilers (ADPs) and a vertical string of loggers were deployed where the lower intersected the sloping lakebed. Near (,1 m above) the bed, internal waves, coherent with a lakewide , propagated upslope at 2 2 ;0.023 m s 1. Near-bed wave-induced water fluctuations had a standard deviation of ,0.02 m s 1. Near the surface, velocity fluctuations had similar magnitude, but lateral was unclear. 2 Averaged over many wave periods, the near-bed Eulerian velocity flowed downslope at ;0.01 m s 1, and was roughly cancelled by an upslope internal-wave Stokes drift (estimated by assuming that weakly nonlinear waves propagated without change of form). To examine net transport, while relaxing approximations used to estimate the Stokes drift, the observed temperature range (98–258C) was divided into 0.58C increments, and the depth-integrated, wave-averaged flux of water in each temperature class was calculated. The coldest (near-bed) water was slowly transported onshore, opposite the Eulerian mean velocity. Onshore flux of warm near-surface water was comparable to an Eulerian-mean flux, indicating minimal near-surface Stokes drift. Intermediate water, from the middle of the and the outer boundary layer, was transported offshore by an offshore Stokes drift. The downslope near-bed Eulerian mean velocity, together with in- tensification of mean stratification within 0.4 m of the bed, may enhance boundary layer mixing.

1. Introduction internal waves sometimes advect plankton considerable distances in the coastal (Pineda 1999). However, Ecosystems in , estuaries, and are shaped in shallow environments, individual tracer patches are by water flows that advect heat, salinity, sediments, nu- often followed for only a limited period, making quan- trients, pollutants, and organisms. Advection is some- tification of long-term mean advection difficult. This complicated by flow reversals associated with paper presents observations collected continuously over internal waves. This paper is devoted to analysis of mean two months, to identify persistent advection in a strati- advection in the presence of internal waves. fied lake over many periods. Advection has been quantified by tracing the move- Observations were collected using stationary instru- ment of plankton, drifters, dye, or other chemicals. Such ments, which are convenient for prolonged deployments. tracer observations have yielded valuable insights, such Unfortunately, when averaged over internal wave pe- as quantifying mixing rates in lakes, estuaries, and riods, the Eulerian velocity at a fixed point, which is oceans (Ledwell et al. 1993; Goudsmit et al. 1997; Inall easily calculated from such observations, differs from 2009; Wain and Rehmann 2010) and showing that the Lagrangian velocity experienced by moving water particles. The difference between wave-averaged Eulerian and Lagrangian is called the Stokes drift (Phillips Denotes Open Access content. 1977, his section 3.3). Theoretical expressions for the Stokes drift have been derived for a variety of nondissipa- tive internal waves, including wave modes propa- Corresponding author address: Stephen M. Henderson, School of the Environment, Washington State University, 14204 NE gating normal to a sloping bed (Wunsch 1971), oblique Salmon Creek Ave., Vancouver, WA 98686. planar inertio-gravity waves reflecting from a sloping bed E-mail: [email protected] (Thorpe 1997), wave beams reflecting from the ocean

DOI: 10.1175/JPO-D-15-0114.1

Ó 2016 American Meteorological Society 1948 JOURNAL OF PHYSICAL VOLUME 46 surface (Tabaei et al. 2005; Zhou and Diamessis 2015), and However, it is departures from perfect cancellation that equatorial Kelvin waves (Weber et al. 2014). Here, two represent persistent advection. If net advection were different techniques are used to quantify transport in the calculated as a small difference between large Eulerian presence of substantial internal-wave Stokes drift. The and Stokes transports then severe errors could result, field site and instrumentation are described in section 2, particularly given the small- approximation and the methods used to examine persistent transport used to estimate the Stokes drift. Furthermore, the cal- are explained in sections 3 and 4. Significant internal culation of the Stokes drift discussed above required an wave Stokes drift has previously been measured over the assumption of steady (upslope) wave propagation, but (Huthnance 1995; Inall et al. 2001; clear wave propagation was observed only near the bed. Zhang et al. 2015). However, results presented here are Therefore, net transport was estimated using a , novel in resolving Stokes drift through the internal wave alternative, approach resembling analysis previously bottom boundary layer (BBL), and in extending Stokes used to identify -averaged transport in - drift measurement to lake environments. Additionally, coordinate ocean models (McDougall and McIntosh these measurements highlight an interaction between 2001) and tidally averaged transport in field observa- the Stokes drift and the Eulerian mean flow near the tions [e.g., MacDonald 2006; MacCready (2011) calcu- sloping bed, discussed below. lated transport in estuaries, and Bryden et al. (1994) The waves considered here were coherent with a estimated transport through the Strait of Gibraltar]. lakewide internal seiche. Near the sloping lakebed, the Persistent advection was separated from wave-induced waves propagated upslope as pitched-forward internal fluctuations by partitioning water into temperature bores (section 5a). Farther above the lakebed, waves classes and calculating the depth-integrated transport of were less borelike, and lateral propagation was unclear. water for each class, averaged over several wave periods. The Stokes drift was estimated throughout the BBL, Under certain approximations, the resulting ‘‘iso- under assumptions of weak nonlinearity and steady thermal mean’’ transport is proportional to the mean upslope wave propagation, from vertical profiles of ve- particle velocity (section 4). As expected given near- locity measured using fixed acoustic Doppler profilers cancellation of Eulerian and Stokes transports, mean (ADPs). BBL Stokes drift was directed upslope and was transport of the coolest (near bed) temperature classes partially canceled by a downslope Eulerian mean flow was smaller than would occur if particles simply moved (section 5b). Models that neglect mixing provide a with the Eulerian-mean velocity (section 5c). simple explanation for this observation: since persistent Mean advection was evaluated throughout the water upslope (downslope) particle would advect column. The discussion (section 6) focuses on near-bed steadily cooler (warmer) water past a fixed location, advection, because models suggest that such advection Eulerian and Stokes transports must cancel in the steady might influence the effectiveness of BBL mixing. In the state (Wunsch 1971; Ou and Maas 1986; Thorpe 1997). absence of waves, steady-state models that account for Upslope wave propagation is common in lakes (Imberger BBL mixing predict upslope transport of the densest 1998) and oceans (Klymak and Moum 2003; Hosegood water in the inner boundary layer and downslope and van Haren 2004; Bluteau et al. 2011; Walter et al. transport of lighter water in the outer boundary layer 2014), and upslope near-bed Stokes drift is a general (Phillips et al. 1986; Garrett et al. 1993). BBL models property of upslope propagation (section 3), so the be- that simulate wave-induced velocity fluctuations, but havior measured here might occur in other environments. neglect the Stokes drift, yield similar predictions for the Direct field measurement of the partial cancelation of mean velocity profile (Umlauf and Burchard 2011). Eulerian and Stokes transports is novel, although obser- Simulated advection is vital to continued mixing: vations of internal tidal rectification over continental sheared BBL currents tilt isotherms to maintain strati- slopes have yielded potentially analogous results. Spe- fication, and therefore turbulent fluxes, in the cifically, some near-bed observations (Butman 1988; presence of ongoing mixing. However, given a mixing- Pingree and Le Cann 1989) have revealed downslope induced reduction of inner-BBL stratification, simulated Eulerian-mean velocities and upslope Eulerian-mean flows also advect buoyancy countergradient (when in- wave-generated density fluxes (such fluxes, equal to the tegrated across a horizontal plane enclosing the BBL), covariance between tidal- fluctuations in den- partially canceling turbulent mixing (Garrett et al. sity and upslope velocity, may be associated with the 1993). These predictions, obtained using models that Stokes drift; Middleton and Loder 1989). neglect wave-induced fluxes, are compared with BBL The approximate cancellation between estimated Stokes advection and stratification observed here in the pres- and Eulerian transports presented here provides a simple ence of wave-induced fluxes. Potential implications for explanation of leading-order mean velocities in the BBL. BBL mixing and sediment transport are noted. JUNE 2016 H E N D E R S O N 1949

5 ps2l h i cz 2 x/ N , (1)

where s is the wave frequency, lx 5 3000 m is the hor- izontal at the depth of the thermocline, the buoyancy frequency hNi 5 [2(g/hri)›hri/›z]1/2, the water density r is calculated from water temperature, and angle brackets hi denote an average over sufficient to remove wave fluctuations (taken above as the entire deployment duration, and taken in formulas in- troduced below as either two days or the entire de- ployment duration). The is absent from (1) because the narrowness of Lacamas Lake ensures dominance of along-lake velocities. The force required to prevent Coriolis from deflecting flow away from the along-lake axis is provided by across-lake baroclinic gradients (i.e., an approximate across-lake thermal balance is observed; Henderson and Deemer 2012). Within about 1 m of the sloping bed of Lacamas Lake, 18–38C seiche-induced temperature fluctuations are characterized by long intervals of slow warming, punc- tuated by short intervals of rapid cooling. The sudden FIG. 1. Lacamas Lake and locations of tripod (cir- cooling events mark arrival of the pitched-forward front cle), deep water temperature profile (square), and additional Hobo faces of near-bed internal bores (Deemer et al. 2015). temperature loggers (triangle, diamond, and star). Arrows show Farther above the bed, waves are more sinusoidal directions of along-lake coordinate x and velocity u, together with across-lake coordinate y and velocity y. Northing and easting (Henderson and Deemer 2012; see also section 5a measured from 45.609778N, 122.436678W. herein). Since bores are confined to the near-bed region, they likely result from internal and boundary layer processes (Thorpe 1992, 1999; Hosegood 2. Field site and instrumentation and van Haren 2004; Boegman et al. 2005a) rather than Measurements were collected using fixed instruments from the midbasin nonlinear steepening (Thorpe 1971, deployed in Lacamas Lake, Washington (Fig. 1), from 1974; Boegman et al. 2005b) that is often important for 18 May to 18 July 2012 (Deemer et al. 2015). Quality larger-amplitude (Horn et al. 2001). Since bore control led to elimination of some velocity data from thickness is comparable to BBL thickness, turbulent early and late in the deployment, so velocities from mixing and dissipation may influence bore dynamics. 24 May to 11 July were selected for presentation here. Bores are often observed near sloping beds of lakes Velocities in this lake are dominated by fluctuations of a (Imberger 1998; Thorpe and Lemmin 1999; Cossu and few centimeters per second, with periods on the order Wells 2013) and oceans (Bluteau et al. 2011; Walter of 1 day. These velocity fluctuations result from in- et al. 2014). ternal waves resembling high-vertical-mode internal Above the turbulent BBL, which extends about 0.4– seiches; horizontal are about twice the 1 m above the bed of Lacamas Lake (Deemer et al. 2015; lake length, and vertical wavelengths are less than the Henderson 2016), stratification strongly inhibits mixing water depth (Henderson and Deemer 2012). Wave (gradient Richardson numbers are consistently above 1; phases propagate upward and energy propagates see section 5a). Within the BBL, sheared upslope and downward, owing to wave dissipation in the BBL downslope flows induce periodic fluctuations in stratifi- (Henderson and Deemer 2012; Henderson 2016). This cation by tilting isotherms. This ‘‘shear-induced periodic vertical seiche propagation contrasts with the verti- stratification’’ (Lorke et al. 2005) elevates vertical gra- cally standing modes that result when energy is dients in temperature (and in of eco- 2 strongly reflected from the surface and the lakebed logically important chemicals such as NO3 and N2O) [for a review of internal waves in lakes, including during downslope flow, and reduces gradients during nondissipative seiches, see Boegman (2009)]. Near the upslope flow (Deemer et al. 2015). thermocline, the wave’s vertical phase speed cz fits the Velocity was measured by three 2-MHz Nortek linear theory prediction Aquadopp ADPs mounted on an aluminum tripod 1950 JOURNAL OF VOLUME 46

removed, and remaining data were averaged to calculate hourly mean velocities at every elevation throughout the 48-day deployment. The two downward-looking ADPs

were in excellent agreement. Let [u1(t), y1(t)] and [u2(t), y2(t)] be hourly mean velocity time series measured at the same elevation by the two ADPs. At every elevation between 0.135 and 0.8 m, linear regression yielded ic u1 1 iy1 5 Ae 1 (u2 1 iy2) 1 b with 0.997 , A , 1.012, 24 21 18,c1 , 38, jbj , 2.2 3 10 ms , and correlation coefficient r2 . 0.99. Nearer the instruments, agreement degraded a few percent, perhaps owing to disturbance of flow by the instruments (A remained within 2% of 1, r2 remained greater than 0.99, and jbj remained ,3 3 2 2 10 4 ms 1). The third, upward looking, ADP recorded one-minute-averaged velocity every minute throughout the deployment, sampling in standard (not pulse- coherent) mode. From the high-frequency noise floor in velocity spectra, noise in minutely velocity for the 2 upward ADP was estimated as 0.03 m s 1, roughly con- sistent with manufacturer’s estimates. Scatter in time- averaged ADP velocity errors scales with (averaging 2 time) 1/2, so random errors in the hourly, 48-hourly, and 48-day means considered below are respectively 4 3 2 2 2 2 10 3,63 10 4, and 1 3 10 4 ms 1 for the upward ADP. These errors will prove insignificant compared with FIG. 2. Instrument configuration on tripod. Gray lines indicate aluminum tripod, unfilled squares indicate RBR temperature log- observed velocities. Owing to more accurate pulse- gers, and black bars labeled 1–3 respectively indicate downward coherent sampling (Lohrmann et al. 1990), random er- ADPs 1–2 and upward ADP 3. Red dotted-dashed, green short- rors for the downward ADP were smaller. Bias, as might dashed, and blue long-dashed lines respectively indicate the for example result from misalignment of acoustic beams, acoustic beams of these three ADPs. was likely a more significant source of error but cannot be estimated from scatter in measurements from a single (Fig. 2). The tripod was deployed in about 8-m depth on instrument. The regression noted above between re- the sloping lakebed (;0.868 slope, up toward southeast) dundant downward ADP measurements suggests that 2 2 at 45.615158N, 122.422688W (circle, Fig. 1). Two ADPs bias for these instruments was ,2.2 3 10 4 ms 1, which were mounted at elevation z ’ 1.3 m above the bed, will prove small compared with observed velocities. pointed downward to measure velocity every 0.015 m At the tripod deployment site, temperature was from z 5 0.135 m to z 5 1.14 m (Fig. 2). At z , 0.135 m, measured using Richard Brancker Research Ltd. (RBR) velocity measurements may have been contaminated by XR-1060 fast-response loggers at z 5 0.2, 0.4 m acoustic sidelobe reflections from the bed, and were (mounted on tripod leg, recording every 2 s, error , neglected. The third ADP was deployed at z ’ 1.6 m, 0.018C), RBR XR-420 CT loggers at z 5 0.9, 1.3 m pointed upward to measure velocity every 0.33 m along a (mounted on tripod leg, recording every 10 s, error , vertical profile extending from z ’ 2.23 m to the water 0.018C), and Onset Hobo Pro V2 loggers at z 5 2.1, 4.3, surface. Velocities measured within 0.33 m of the surface 5.8, and 7.3 m (mounted on cable extending up from were neglected. tripod, recording every 4.25 min, error , 0.28C). The two downward-looking ADPs operated in pulse- Temperature was measured at four additional loca- coherent mode (Lohrmann et al. 1990), with a long tions in the lake. One Hobo was deployed 16 June– (6.8 m) pulse-to-pulse distance yielding high-resolution, 18 July, about 3 m above the bed in about 10-m depth low-noise measurements. One ADP, used for all results near the lake’s northwest end (triangle, Fig. 1). Two presented in section 5, recorded one 1-s burst of two near-bed Hobos (z ’ 0.3 m) were deployed 16 June– 0.5-s-averaged measurements every 12 s. The second 18 July, downslope (45.616008N, 122.423508W; diamond, downward-looking ADP, used here only to assess Fig. 1) and upslope (45.614288N, 122.420808W; star, measurement accuracy, sampled a single 0.5-s-averaged Fig. 1) of the tripod. About 100 m from the lake’s velocity every 30 s. Low (,75%) correlation data were deepest point (45.620038N, 122.430238W; square, Fig. 1), JUNE 2016 H E N D E R S O N 1951

Du be the difference between u for the moving particle

and u at (x0, z0). Now Du fluctuates through each wave cycle, but its persistent mean value hDui 5 uST is the Stokes drift. For weakly nonlinear, statistically steady waves, given mean velocity gradients not greatly ex- ceeding gradients in wave-frequency velocity, it can be shown (see the appendix) that

›hu0Z0i u ’ , (2) ST ›z

where Z0 is the particle’s boundary-normal displacement

from z0 (throughout, primes indicate departures from wave-averaged values). Interpretation of (2) is discussed in section 4b. FIG. 3. Definition sketch showing coordinates (x, z) and path For field application of (2), profiles of u can be mea- (gray ellipse) of a water particle oscillating about wave-averaged sured using ADPs. It remains to estimate Z0. In principle, location (x , z ). Black dot shows an example of instantaneous 0 0 leading-order Z0 can be calculated by time-integrating particle location, with associated velocity (u, w) and displacement 0 0 (X0, Z0). Closed particle orbit (ellipse) drawn for simplicity; actual ADP measurements of w ,butw maybeunmeasur- particle motion may progress to right if Lagrangian-mean velocity ably small. An alternative approach, used here, is to h i 1 . h i 1 , u uST 0, or left if u uST 0. assume that waves propagated upslope at phase speed c (c , 0 for downslope propagation) while changing 2 in about 17-m depth, temperature profiles were mea- their form only slowly, so u is locally a function of x ct, sured on 18 May and 11 July by lowering a Hach DS5X leading to Sonde (error , 0.158C). ð 1 z Z0(z) ’ u0 dz~, (3) c z~50 3. Stokes drift which can be evaluated from ADP profiles. To derive 2 Previous theoretical expressions for the Stokes drift (3), note that ›u/›x ’2c 1›u/›t for steadily propa- for nondissipative internal waves (Wunsch 1971; Thorpe gating waves, so the mass conservation equation ›u0/›x 1 2 1997; Tabaei et al. 2005; Weber et al. 2014) are not valid ›w0/›z 5 0becomes›w/›z ’ c 1›u/›t. Integrating over in the dissipative BBL. Next, expressions are presented time yields ›Z0/›z ’ u/c at leading order, and integrating for calculation of the BBL Stokes drift from observed vertically from the bed yields (3).Forz , 0.135 m, where velocity profiles (section 3a). The following is based on ADP measurements were unreliable, velocity was esti- previous analysis of general (Middleton and Loder mated by assuming a logarithmic profile, that is, 1989) and surface-wave BBL (Longuet-Higgins 1953) cases, and also has similarities to Lamb (1997). Ex- log(z/z^) u(z) 5 uj , (4) pressions are presented in a form that highlights simple z50:135 m log(0:135/z^) qualitative properties of the internal wave Stokes drift j 5 (section 3b) while facilitating comparison with alterna- where u z50:135 m denotes u evaluated at z 0.135 m, and 2 tive isothermal-mean analysis (section 4). the roughness length z^ 5 10 3 m (results were in- 2 2 sensitive to z^ between 10 5 and 10 1 m). Each hour, (3) a. Quantitative evaluation of the Stokes drift was evaluated by analytic integration of (4) from z^ to Let (x, z) be upslope and bed-normal coordinates, 0.135 m, and trapezoidal integration of ADP profiles with z 5 0 at the bed (Fig. 3). Consider a water particle above 0.135 m. moving back and forth with each wave, with a wave- b. Qualitative properties of the Stokes drift averaged location (x0, z0). Let (u, w)be(x, z) compo- nents of velocity. For simplicity, velocity perpendicular Consider a rigid-lid internal wave propagating in to (u, w) will be neglected in all wave-averaged transport constant depth h. Now the depth-integrated Stokes drift 0 calculations. This assumption is motivated by the dom- equals zero [becauseÐ Z 5 0 at the surface and at the bed, h 5 h 0ij 2 h 0ij inance of along-lake velocities in Lacamas Lake, but and from (2), 0 uST dz uZ z5h uZ z50; see also may be a poor approximation in basins characterized by Weber et al. (2014)]. Therefore, reversals in Stokes drift three-dimensional (e.g., Poincare modes). Let along vertical profiles, noted by many authors (Thorpe 1952 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46

1997; Weber et al. 2014), are expected. For the case of a a. Definition of isothermal-mean transport nondissipative mode-1 internal wave propagating up- Divide the range of measured into slope, Stokes drift is directed toward shallow water near closely spaced increments centered on T , T , T , ... the bed and near the surface, with a reversal in the mid- 1 2 3 such that T 1 5 T 1DT. Let Dq(T ) be the depth- water column (Wunsch 1971; see also section 4b herein). j 1 j j integrated, wave-averaged volume flux of all water with The near-bed Stokes drift transports water in the direc- temperatures between T 2DT/2 and T 1DT/2. Since tion of wave propagation, as can be proved by substituting j j Dq(Tj) is proportional to DT, it will be convenient to (3) into (2) to yield 2 2 work with q(T ) 5Dq(T )/DT (with units m2 s 1 8C 1), ð j j ›u0 1 z hu02i which equals u 5 u0 dz~ 1 . (5) ST › ð z c z~50 c h 5 F q(Tj) (Tj, z) dz, (6) The second term of (5) always contributes drift in the z50 direction of wave propagation. Through much of the BBL, where jujÐincreases with z,and›u/›z tends to have the same sign z ~ as ~5 udz (because boundary layer phase shifts are usu- F 5 hd 2 i z 0 (Tj, z) [T(t, z) Tj]u(t, z) , (7) ally less than 908), so the first term reinforces the second. From the above discussion, the Stokes drift of prop- and the time average in (7) is effectively limited to the agating internal waves flows in the direction of wave jth temperature class by the delta function propagation near the bed, with at least one reversal ( higher in the water column. DT21 jT(t, z) 2 T j ,DT/2 d 2 5 j [T(t, z) Tj] . (8) 0 otherwise 4. Estimation of transport in isothermal The function F (T, z) partitions flux into temperature coordinates and elevation classes. Integrating F (T, z) over z gives In this section, an alternative isothermal-mean view of depth-integrated transport in a temperature class [(6);it advection is outlined. Following MacDonald (2006), can also be shown that integrating F (T, z) over T gives

McDougall and McIntosh (2001), and many others, an the Eulerian mean velocity at fixed z]. For given Tj, the isothermal-mean transport q(T) is defined, representing range of depths in which d[T(t, z) 2 Tj] is nonzero moves the wave-averaged lateral flux of water with tempera- up and down as isotherms are displaced by internal tures near T (section 4a). For cases of negligible mixing, waves, but the horizontal location at which the integral (and under simplifying assumptions), proportionality (6) is evaluated is fixed, and does not move laterally with between q(T) and particle (Eulerian plus Stokes) ve- water particles. locity is noted (section 4b). As clarified in section 4b, this Isothermal-mean fluxes were evaluated at the tripod analysis uses temperature to trace vertical motion (cf. location using ADPs and the vertical string of temper- Pinkel et al. 2012), allowing relaxation of the assump- ature gauges. At each temperature gauge elevation, tions (steady propagation and weak nonlinearity) that d[T(t, z) 2 Tj] was evaluated from hourly mean tempera- were required in section 3. Even in cases with rapid ture and (8) with temperature resolution DT 5 0.58C, turbulent mixing, for which temperature is not a tracer and u(z, t) was evaluated by linearly interpolating (in z) of wave-induced motion, boundary layer–averaged ADP hourly mean velocity profiles. (Tj, 0) was set to Lagrangian-mean velocities can be recovered if mixing zero, and (Tj, h) was set equal to the value estimated at is confined to the BBL (section 4b). the highest temperature gauge. Vertical trapezoidal in-

Measured hourly mean velocities are expected to be tegration of the F (Tj, z) values estimated at tempera- predominantly bed-parallel near the bed, and surface- ture logger locations was then used to evaluate (6). parallel near the water surface. For the analysis of near- b. Relation between isothermal-mean transport and bed flows in section 3, coordinates (x, z) were defined as particle velocity bed-parallel and bed-normal. Given the small bed slope 8 5 h i 1 0 (0.86 ), bed-parallel and bed-normal coordinates Let ZT ZT ZT be the elevation of the temperature- may be indistinguishable from horizontal and vertical T isotherm [following McDougall and McIntosh (2001), coordinates when analyzing meter data. The ZT is set to 0 (or h) whenever T is colder (or warmer) distinction between coordinates is neglected in the than all temperatures observed in the water column]. following analysis, which addresses only the dominant, A vertical profile of thickness-weighted isothermal- lateral component of the flow. mean transport velocity uT is defined such that the JUNE 2016 H E N D E R S O N 1953

FIG. 4. Stokes drift for a nondissipative linear internal wave with zero Eulerian mean flow (amplitude exaggerated). (a) The nonbolus contribution to Stokes drift unb. Cross section shows a snapshot of temperature (color), isotherms (gray lines), and horizontal velocity (arrows) for a mode-1 wave of wavelength L propagating in direction 1x with phase speed c and maximum velocity amplitude u0.Atx/L 5 0 0.25, shear is negative (›u /›z , 0), so upward displacement (Z1 . 0) carries intermediate-temperature water (dotted gray line) into 0 a region of leftward flow (Du1 , 0). At x/L 5 0.75, both shear and displacement reverse (›u /›z . 0, Z2 , 0), so intermediate-temperature water is again in a region of leftward flow (Du2 , 0). The velocity averaged along the moving isotherm is hui 1 unb, which equals unb in this h i 5 5 h › 0 › 0 i , case because the Eulerian-mean velocity u 0. Approximating velocity with a linear function of depth yields unb ( u / z)ZT 0. For 0 › 0 › near-bed and near-surface waters (solid and dashed gray lines), unb is small, because ZT and u / z are small. (b) The bolus velocity ub.A cold near-bed water layer, bounded by solid gray isotherms, is thicker at x/L 5 0.25, where flow is to right, than at x/L 5 0.75, where flow is to left (i.e., DZ1 .DZ2). A rightward ‘‘bolus flux’’ results. Letting DZ 5 mean thickness, the instantaneous thickness is approximately D 1 › 0 › D h 0› 0 › i . 5 h 0› 0 › i . D h i 5 Z(1 ZT / z), the mean bolus flux is Z u ZT / z 0, and the mean bolus velocity is ub u ZT / z 0 (the Eulerian flux Z u 0 in this case). A warm near-surface layer (bounded by dashed lines) experiences a similar bolus velocity. An intermediate layer (bounded by dotted lines) has the same thickness at x/L 5 0.25 as at x/L 5 0.75, and experiences no bolus velocity. (c) Vertical profile unb (dashed gray), ub (solid gray), and uST 5 unb 1 ub (black line).

flux (hZT1dTi 2 hZTi)uT attributed to elevations be- and the bolus velocity tween hZTi and hZT1dTi equals the flux q(T)dT of 5 h 0 › 0 › i water with temperatures between T and T 1 dT, ub u ( ZT / z) . (13) leading to The origins of unb and ub are illustrated in Figs. 4a and 4b for the mode-1 wave u 5 u cos[2p(x 2 ct)/L] cos(pz/h) q(T) 0 u 5 . (9) with u c. Averaging at the moving elevation of an T ›hZ i/›T 0 T isotherm (Pinkel and Anderson 1997; Alford 2001; Pinkel 2008) yields a mean velocity approximately hui 1 To separate wave-induced temperature fluctuations u ,sou accounts for the difference between the Eu- from seasonal warming, angle brackets here represent nb nb lerian mean and the mean on the isotherm (Fig. 4a; only short (2 day) averages, rather than averages over isothermal-coordinate models, which begin by averag- the entire deployment duration. ing on isotherms, do not require u ). The transport For statistically steady, weakly nonlinear waves, the nb velocity u 5 hui 1 u 1u 5 (mean velocity on moving velocity u , which represents water transport near Z ,is T nb b T T isotherm) 1 (bolus velocity). The bolus velocity ac- related to velocity at hZ i by T counts for fluctuations in layer thickness (Fig. 4b), so ’ h j i 1 j uT is a ‘‘thickness-weighted mean’’ in the terminology uT u z5hZ i uID z5hZ i , (10) T T of Young (2012) [the term ‘‘bolus velocity,’’ as used here and by other researchers (e.g., Dukowicz and where the velocity calculated from isotherm displace- Greatbatch 1999), does not imply the existence of ment is ‘‘turbulent surges’’ (Helfrich 1992)or‘‘trappedcores’’ ›hu0Z0 i with u . c (Flierl 1981)]. For the wave of Fig. 4, nonbolus u 5 T (11) ID ›z velocity is opposite wave propagation and maximum at 52 2 2 p intermediate depths [unb (u0/2c)sin ( z/h)], whereas [from (4b) of McDougall and McIntosh (2001),simplified bolus velocity is in the wave propagation direction as noted between our (A1) and (A2)]. To interpret (11), and maximum near the bed and near the surface 5 1 5 2 2 p first write uID unb ub, where the non-bolus velocity [ub (u0/2c)cos ( z/h); Fig. 4c]. To illustrate (6)–(9), consider the flux of water with 5 h › 0 › 0 i unb ( u / z)ZT , (12) temperatures between 108 and 10.58C, that is, q(10.258C) 1954 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46 with DT 5 0.58C. Let the lower and upper solid gray lines in Fig. 4b respectively be the 108 and 10.58C isotherms. 21 Now d[T(t, z) 2 Tj] equals DT between these iso- therms, and equals zero everywhere else. From (6) and (7), q(10.258C) is the depth-integrated, wave-averaged transport of the water between the two isotherms nor- 21 malized by DT , and from (9) uT is the mean velocity of particles in this layer. Expression (2) for the Stokes drift resembles (11) and has a similar interpretation. The distinction is the pos- sible difference between the particle displacement Z0 0 and the isotherm displacement ZT . The Stokes drift can be broken into nonbolus and bolus components by re- 0 0 placing ZT with Z in (12) and (13). If mixing is negli- gible, given weakly nonlinear waves and near-vertical 0 5 0 temperature gradients, ZT Z , so from (2) and (11),

u 5 hui 1 u ; (14) FIG. 5. Temperature profiles measured on 18 May (gray) and T ST 11 July (black) near the lake’s deepest point (solid lines) and at tripod location (pluses). that is, the thickness-weighted isothermal-mean velocity u equals the mean particle velocity [the approximations T ð stated above may be important to (14), as is discussed 1 z1 u 5 (u 2 hui) dz~, (15) further in section 6]. ST T z1 z~50 Both bolus and nonbolus terms contribute to the Stokes drift throughout the BBL. However, their values where z1 was set to 1.1 m. are modified by BBL friction [see discussion following (5)] and may differ from the frictionless case illustrated in Fig. 4. For a steadily propagating wave, the bolus 5. Results velocity [second term on right of (5)] remains in the a. Description of internal waves direction of wave propagation throughout the BBL. The nonbolus velocity [first term on right of (5)]isalsoin Stratification in Lacamas Lake strengthened as the the direction of wave propagation within some neigh- lake warmed during summer (Fig. 5). At the tripod lo- borhood of the bed. Therefore, the near-bed nonbolus cation, between 18 May and 11 July, water temperature velocity is opposite the above-BBL nonbolus velocity measured near the bed (elevation z 5 0.2 m) and near (this reversal occurs because the no-slip condition en- the surface (z 5 7.3 m) respectively increased by 1.18 and sures that ›u0/›z and Z0 have the same sign near the bed, 5.08C. At the deep water location, for depths exceeding in contrast to the opposite signs for the above-BBL about 3 m, the temperature gradient decreased with in- motion of Fig. 4). creasing depth. Similarly, at the tripod location, the 0 6¼ 0 Given substantial mixing, ZT Z and uT differs from strongest temperature gradients were measured in the the mean particle velocity. However, if mixing within a upper water column (between z 5 4.3 and 7.3 m, 24-h- single wave period is substantial but limited to the BBL averaged temperature gradients measured on 18 May 2 2 (as may be a good approximation given sufficiently large and 11 July were dhTi/dz 5 1.58Cm 1 and 2.78Cm 1), gradient Richardson number above the BBL), then with weaker gradients at greater depths (between z 5 equality between thickness-weighted isothermal-mean 1.3 and 4.3 m, average temperature gradients measured 2 2 velocity and particle velocity still holds in a boundary on 18 May and 11 July were 0.548Cm 1 and 0.538Cm 1). layer-integrated sense. To see this, let z1 beÐ a fixed el- However, at the tripod, this trend reversed in the BBL, z1 5 evation above the boundaryÐ layer, note that 0 uID dz where elevated mean temperature gradients were ob- h 0 0 ij z1 5 h 0 0ij 5 u ZT z5z1 [from (11)], 0 uST dz u Z z5z1 [from served (between z 0.2 m and 1.3 m, average temper- 0 j 5 0j (2)], and ZT z5z1 Z z5z1 (given negligible mixing ature gradients measured on 18 May and 11 July were 2 2 above the BBL, and other assumptions previously 0.818Cm 1 and 1.18Cm 1). In contrast to the expecta- noted above). In this case, isothermal-mean analysis tion that mixing causes weak near-bed mean stratifica- yields an estimate of the boundary layer-averaged tion, the increase in BBL stratification was most marked Stokes drift: nearest the bed (between z 5 0.2 m and 0.4 m, average JUNE 2016 H E N D E R S O N 1955

tripod location, upward propagation is less clear. Never- theless, a weak trend can be discerned by comparing temperature fluctuations with the theoretical propagation for the lakewide wave calculated from (1) using the energy-weighted mean frequency s 5 1.5 cycles per day [dashed gray lines, Fig. 6a; propagation toward in- creasing elevation mapped onto propagation toward warmer temperatures using the measured temperature profile hT(z)i]. As expected given horizontal-mode-1 structure of the lakewide internal wave (Henderson and Deemer 2012), temperature fluctuations observed at the tripod (z 5 1.3 m) were coherent with, but opposite to, fluctuations measured near the lake’s northwest end (triangle, Fig. 1; for the band 0.75–2.5 cycles per day, bulk coherence2 5 0.69 and phase 5 1608, not shown; the 208 departure from 1808 phase was consistent with esti- mated upward propagation speed, and the 1-m elevation difference between the two instruments). Comparing near-bed temperature fluctuations mea- sured at the tripod with fluctuations measured upslope and downslope of the tripod (i.e., at locations of star and diamond in Fig. 1) reveals upslope wave propagation (Fig. 6b). In the dominant 0.75–2.5 cycles per day fre- quency band, near-bed temperature fluctuations mea- sured at the tripod lagged those measured 114 m downslope by 1.4 h, yielding an upslope propagation 2 speed of 0.023 m s 1 [for these , cross spectra evaluated with 5 degrees of freedom (not shown) yielded phase 5 (360831.4 h) 3 frequency, with r2 of FIG. 6. Time series of hourly mean temperature and velocity. 0.87]. Outside the 0.75–2.5 cycles per day band, co- (a) Temperature from vertical instrument stack (elevations z 5 0.2, herence between the two temperature time series 0.4, 0.9, 1.4, 2.1, 4.3, 5.8, 7.3 m) at tripod location, each curve dropped markedly (coherence2 exceeded 0.79 for all 14 a single instrument, with successively higher instruments recording higher temperatures. Gray dashed lines indicate theoretical up- frequencies in the 0.75–2.5 cycle per day band, but for ward propagation of lakewide internal wave. (b) Near-bed tem- only four of 85 frequency bins outside this band; peratures measured by RBR temperature loggers mounted on not shown). tripod (solid black), and by nearby deep (thick gray), and shallow Owing to internal bores (section 2), near-bed tem- (dashed black) Hobo loggers. (c) Along- (black) and across-lake perature fluctuations exhibited long intervals of slow (gray) velocity measured 0.15 m above bed at tripod. (d) As in (b), but for near-surface temperatures. (e) As in (c), but for near-sur- warming, punctuated by shorter intervals of rapid face velocity. cooling (Fig. 6b). This asymmetry between warming and cooling phases was greatly weakened above the BBL (Fig. 6a) because bores were confined to the temperature gradients measured on 18 May and 11 July near-bed region. 2 2 were 0.978Cm 1 and 1.58Cm 1). Hourly mean velocities fluctuated with the passage of Internal waves caused the hourly mean temperature each internal wave, with along-lake velocity dominating to fluctuate a few degrees (Fig. 6a). Wave-induced over across-lake velocity [Fig. 6c; here the along-lake temperature fluctuations were dominated by frequen- direction, 1268 clockwise from north (arrows, Fig. 1), was cies between 0.75 and 2.5 cycles per day (after removing chosen to maximize along-lake velocity variance]. seasonal warming by subtracting a best-fit quadratic Dominance of along-lake velocity was consistent with from the 48-day time series measured at z 5 1.3 m, this along-lake, upslope wave propagation. At z 5 0.135 and frequency band was responsible for 78% of remaining 1.0 m, respectively, frequencies between 0.75 and 2.5 temperature variance). Observations from the deepest cycles per day contributed fluctuations with standard 2 part of the lake have shown upward-propagating in- deviations 0.011 and 0.020 m s 1 (only slightly less than 2 ternal waves (Henderson and Deemer 2012). At the the upslope phase speed of 0.023 m s 1), accounting for 1956 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46

70% of detrended velocity variance. Weak upslope flow was associated with cold water, whereas warming oc- curred during downslope flow [compare Figs. 6b and 6c; for an explanation of this trend, see Deemer et al. (2015)]. Arrival of internal bores was associated with sudden transitions from downslope to upslope velocity. Stratification often inhibited mixing (Deemer et al. 2015), as indicated by the Richardson number (Phillips 1977)

g(r 2 r )(z 2 z ) Ri 5 l u u l , (16) r 2 2 1 y 2 y 2 l[(ul uu) ( l u) ]

FIG. 7. Near-bed profile of along-lake Eulerian velocity (thin where ra, ua, and ya are density, across-lake velocity, black line) and internal-wave Stokes drift (thick gray line) aver- and along-lake velocity at elevations za. Above the BBL aged over the 48-day deployment (positive velocities indicate 5 (zl, zu 2.1, 4.3 m, respectively, with velocities esti- upslope flow). mated by linear interpolation of ADP data), Ri ex- ceeded 0.25, 1, and 2 respectively in 99.9%, 94%, and 85% of cases, indicating consistent suppression of mix- drift (Garrett et al. 1993) predict mean warming at rate ing by stratification (not shown). Nearer the bed, (zl, ›T/›t ; hui(›T/›z)sinu, where u 5 bed slope. Setting 21 21 zu 5 0.2 m, 0.4 m), Ri exceeded 0.25, 1, and 2 re- hui 5 0.01 m s , ›T/›z 5 18Cm , and bed slope 2 spectively in 91%, 48%, and 24% of cases, indicating u 5 0.878 yields mean warming rate of 1.38C day 1,or less consistent inhibition of BBL mixing. 628C over the course of the 48-day deployment. Such In contrast to the upslope propagation evident near rapid warming was not observed (observed near-bed 2 the bed (Fig. 6b), no clear propagation is discernible warming was 0.028C day 1; Fig. 5). A possible explana- from near-surface temperatures (Fig. 6d; cross-spectral tion is that an upslope near-bed Stokes drift, predicted in analysis between laterally displaced temperature gauges section 5a, opposed downslope particle advection. revealed no clear trend, with an r2 between frequency Stokes drift estimation from (2) and (3) requires an and cross-spectral phase of 0.04). Relationships between assumption of steady upslope wave propagation, but near-surface velocity and temperature were also unclear coherent propagation was observed only in the domi- (Figs. 6d,e), for reasons that are unknown (forcing by nant frequency band 0.75–2.5 cycles per day (section 5a). fluctuating provides one possible explanation). When u0 was calculated as the velocity contributed by Theoretically, the observed upslope propagation of this frequency band, the Stokes drift averaged over the near-bed internal waves implies onshore near-bed 48-day deployment was consistent with partial cancel- Stokes drift, with a reversal to offshore drift higher in lation of the near-bed Eulerian mean flow by the Stokes the water column (section 3b). The near-surface Stokes drift (thick gray line opposite thin black line, Fig. 7). drift is difficult to predict, partly owing to the lack of Boundary layer-averaged estimated Stokes drift ex- clear wave propagation near the surface. ceeded the Eulerian mean flow by 30%, a difference that increases if u0 is not limited to the 0.75–2.5 cycles per b. Eulerian mean velocity and Stokes drift day band (including all frequencies gives an estimated Steady wave propagation was assumed to derive ex- mean boundary layer–averaged Stokes drift double pressions for the Stokes drift (section 3). Since clear the mean Eulerian velocity, although this calculation propagation was observed only near the bed, this section is unjustified because coherence was limited at higher presents results only for transport within 1.1 m of the frequencies). bed, measured by the downward-looking ADPs. Dis- Fluctuations in Stokes drift observed during the 48- cussion of transport through the remainder of the water day deployment tended to oppose fluctuations in the column is deferred to section 5c. Eulerian-mean velocity. Averaged over a 2-day running Averaged over the 48-day deployment, the near-bed mean and from the bed to z 5 1.1 m, fluctuations in 2 Eulerian velocity flowed downslope at about 0.01 m s 1 Eulerian-mean velocity resembled fluctuations in Stokes (thin black line, Fig. 7). If water particles were on av- drift, although scatter was considerable (compare thin erage transported downslope at this speed, then advec- blackandthickgraylinesinFig. 8; by regression, tion of warmer water from upslope locations would boundary layer-averaged uST 521.4hui 2 9 3 2 2 cause rapid warming. BBL models neglecting Stokes 10 4 ms 1 with r2 5 0.46). JUNE 2016 H E N D E R S O N 1957

FIG. 8. Time series of wave-averaged (2-day running mean), vertically averaged (from bed to z 5 1.1 m) downslope Eulerian velocity (thin black line) and upslope internal-wave Stokes drift. Stokes drift estimates based on weak nonlinearity and steady wave propagation [(2) and (3), thick gray line] and based on isothermal- FIG. 9. Upslope isothermal transport averaged over 48 h on 4–5 mean analysis [(15), circles]. June (thin black line) and 8–9 July (thick gray line). c. Isothermal-mean transport water classes, was much smaller than expected given the During selected 2-day intervals early and late in the wave-averaged downslope Eulerian velocity, and trans- deployment, the isothermal-mean flux q(T)[(6)] in- port remained consistently small despite substantial dicates onshore transport of the warmest (near surface) fluctuations in Eulerian velocity. The Stokes drift esti- water and offshore transport of intermediate-temperature mated from isothermal-mean analysis, uST, cancels the water (Fig. 9). Early in the deployment, mean onshore Eulerian mean flow more exactly than the Stokes drift transport of the coolest (near bed) water was also ob- estimated in section 5b, possibly owing to errors in served [q(T) . 0 for T , 128C; thin black line, Fig. 9]. steady-propagation and weak-nonlinearity assumptions. Early in the deployment, water temperatures were rel- At middepths, the thickness-weighted isothermal-mean atively cool, so no transport of the warmest water classes velocity was offshore of Eulerian mean, indicating re- was measured [q(T) 5 0 for T . 188C; thin black line, versal of the Stokes drift (Fig. 10; such a reversal is ex- Fig. 9]. As the lake warmed, warmer temperature classes pected from theory; section 3). Near the surface, where were transported onshore, and the transport of the along-lake wave propagation was unclear (section 5a), coolest temperature classes reduced to zero [q(T) 5 isothermal- and Eulerian-mean velocities did not differ 0 for T , 118C; thick gray line, Fig. 9]. greatly (indicating minimal Stokes drift), and a shore- To facilitate comparison with section 5b, the thickness- ward Eulerian mean flow transported warm water weighted isothermal-mean velocity uT [(9)] was com- onshore. pared with the Eulerian-mean velocity (Fig. 10; the near-bed part of Eulerian velocity profile was previously 6. Discussion shown in Fig. 7). Recall that, under simplifying assump- tions and neglecting mixing, the thickness-weighted iso- The near-bed Stokes drift of internal waves propa- thermal mean velocity equals the mean particle velocity gating up a sloping lakebed has been found to oppose a [i.e., equals Eulerian-mean velocity plus Stokes drift; downslope Eulerian-mean velocity, as has been pre- (14)]. Near the bed, the thickness-weighted isothermal- dicted on theoretical grounds (Wunsch 1971; Ou and mean velocity was much less than the Eulerian mean, as Maas 1986; Garrett et al. 1993). Opposition of BBL- expected given the tendency for cancellation of down- integrated Stokes and Eulerian transports results from slope Eulerian and upslope Stokes transports found in the kinematics of density conservation, and is in- section 5b. This result, obtained by averaging over the dependent of the responsible for generating the entire experiment, was supported by approximate can- Eulerian-mean flow. Indeed, the observations presented celation of BBL-averaged, 2-day-running-mean hui and here do not resolve mean forcing terms, which might uST [compare thin black line with circles, Fig. 8; uST de- include isopycnal setup (Umeyama and Shintani 2014), fined by (15)]. This finding can be restated as follows: wave-induced momentum fluxes (Longuet-Higgins 1953; the wave-averaged transport, integrated over the coolest Bordes et al. 2012; Grisouard and Bühler 2012; Pinkel 1958 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46

2.7 times greater during downslope flow than during upslope flow. Since BBL shear intensifies stratification and chemical gradients during downslope flows (Lorke et al. 2005; Deemer et al. 2015), high speeds coincide with strong gradients, possibly increasing the effective- ness of BBL mixing. As noted in section 1, BBL models that neglect wave- induced fluxes predict upslope transport of the coldest water in the inner boundary layer, and downslope transport of warmer water in the outer boundary layer (Phillips et al. 1986; Garrett et al. 1993; Umlauf and Burchard 2011). Here, in the presence of substantial Stokes drift, the thickness-weighted isothermal-mean velocity profile resembled these predictions in the sense that the coldest water was advected upslope and slightly warmer water was advected downslope (e.g., consider

the bottom meter of the uT profile in Fig. 10). However, FIG. 10. Eulerian-mean (thin black line) and thickness-weighted isothermal-mean (thick gray line) velocity profiles, averaged over in contrast to the mixing-induced reduction in mean the 48-day deployment. Eulerian velocity interpolated between near-bed stratification predicted by previous models, the z 5 1.14 and 2.23 m. To calculate the plotted thickness-weighted observed mean stratification intensified within 0.4 m of isothermal-mean velocity, time series of 2-day running mean of uT the bed. To clarify the potential significance of this re- were calculated using (9), and then averaged over the entire sult, consider two processes that might reduce the im- deployment. portance of BBL mixing to basinwide stratification (Garrett et al. 1993). First, the efficiency of BBL mixing et al. 2012), and turbulent Reynolds stresses. Under- would be low if BBL stratification were weak. Second, standing of this environment may be improved by fur- the partial cancelation of mixing by advective buoyancy ther research into forcing of mean currents. However, fluxes predicted by modeling studies (section 1) would even in other environments subjected to different forc- be absent if stratification were not reduced in the BBL. ing, the kinematic results emphasized here may apply. To address the interaction between BBL mixing and Upslope near-bed Stokes drift is a general property of mean advection, further work to examine mean strat- steady upslope propagation, and upslope-propagating ification very near the bed, and the role of wave non- internal waves are often observed over sloping beds in linearity and time-dependent mixing, may prove lakes (Imberger 1998; Thorpe and Lemmin 1999) and valuable. oceans (Hosegood and van Haren 2004; Bluteau et al. Several assumptions were necessary to derive the 2011; Walter et al. 2014). simple results of sections 3 and 4.Readersseeking Isothermal-mean analysis revealed a residual onshore generalizations are referred to Middleton and Loder flux of warm near-surface water, an offshore flux of (1989) and Plumb (1979) for three-dimensional flows underlying intermediate water, and a weak onshore flux and relaxation of the steady-propagation assumption; to of the coldest water. The implied Stokes drift reversed, Andrews and McIntyre (1978) and Flierl (1981) for from onshore near the bed to offshore in the midwater mean particle motions in strongly nonlinear waves; to column, consistent with theoretical expectations (sec- Walin (1977, 1982), McDougall and McIntosh (2001), tion 3b). Near the surface, wave propagation was un- Young (2012),andGroeskamp et al. (2014) for isopycnal- clear and the Stokes drift was small. mean analysis; and to Giddings et al. (2014) for related Owing to the downslope Eulerian mean flow, near- sigma-coordinate analysis. For isothermal-mean anal- bed downslope flows tended to have higher speeds than ysis (sections 4 and 5c), the neglect of across-lake mo- upslope flows (e.g., thin black line, Fig. 6c). One ex- tions may be particularly significant. Such motions pected consequence is a downslope mean bed stress, introduce a neglected Stokes drift contribution ›hu0Y0i/ which may influence sediment transport (cf. Butman ›y (where y and Y0 are across-lake and dis- 1988). An additional expected consequence is en- placement), which can be interpreted in a manner hancement of during downslope flow. Av- similar to the resolved ›hu0Z0i/›z term (Middleton and eraged over the entire deployment, the cube of near-bed Loder 1989). The Eulerian mean velocity may also vary water speed (i.e., juj3 at z 5 0.135 m), to which turbulent in the across-lake direction (horizontal circulation is dissipation may be proportional (Henderson 2016), was possible). Such neglected across-lake variability is likely JUNE 2016 H E N D E R S O N 1959 smaller than Eulerian and Stokes velocities, as evi- in (3) of McDougall and McIntosh (2001) is third order if denced by their near-cancelation (compare black line (›hui/›x, ›hui/›z) is of order (›u0/›x, ›u0/›z)]. Now av- and circles, Fig. 8). However, since BBL-integrated eraging (A1) gives Eulerian and Stokes velocities almost cancel, across- ›u0 ›u0 lake variability might be substantial compared with the u ’ X0 1 Z0 . (A2) small net transport. The Stokes drift estimates of sec- ST ›x ›z tions 3 and 5b required assumptions of steady propa- gation and weak nonlinearity that were not required for The derivation of (2) from (A2) follows previous more isothermal analysis. Comparison with isothermal anal- general analysis (e.g., Middleton and Loder 1989). ysis suggests that these assumptions introduced errors of Specifically, 30%–40%. h 0 › 0 › i 5 ›h 0 0i › 2 h 0 › 0 › i The analysis applied here required only conventional X ( u / x) X u / x u ( X / x) measurements: velocity profiles for the evaluation of 5 (1/2)›2hX02i/›t›x 2 hu0(›X0/›x)i Stokes drift under steadily propagating waves, and 0 0 0 0 52hu (›X /›x)i 5 hu (›Z /›z)i, combined velocity and temperature profiles for evalua- tion of isothermal-mean transport. Similar analysis can where the second equality follows from u0 ’ ›X0/›t, the be envisaged in other limnologic or oceanographic third from statistically steady flow, and the fourth from cases, although potential difficulties include strong mass conservation. Physically, given statistically steady nonlinearity and departures from steady propagation wave field, the correspondence between the lateral dis- (for Stokes drift estimation), three-dimensionality placement component of the Stokes drift (left of equa- (which complicates both Stokes drift and isothermal- tions) and the bolus velocity (right of final equation) mean analysis), and the possibility that residual means occurs because particles must converge laterally (›X/ may be unmeasurably small. ›x , 0) for a layer to thicken (›Z/›z , 0). Substituting the final form into (A2) yields (2). Acknowledgments. John Harrison and Bridget Deemer contributed to planning and execution of fieldwork, and provided valuable comments on this REFERENCES manuscript. Further fieldwork assistance was provided Alford, M., 2001: Fine-structure contamination: Observations and a by Abby Lunstrum, Homer Adams, Andrew Harwood, model of a simple two-wave case. J. Phys. Oceanogr., 31, 2645– Alyson Day, Joshua Arnold, and Matthew Schultt. 2649, doi:10.1175/1520-0485(2001)031,2645:FSCOAA.2.0.CO;2. Funding was provided by the National Science Foun- Andrews, D., and M. McIntyre, 1978: An exact theory of nonlinear dation Awards 1045286 and 1355211, and by U.S. Geo- waves on a Lagrangian-mean flow. J. Fluid Mech., 89, 609–646, logical Survey Award 06HQGR0126. doi:10.1017/S0022112078002773. Bluteau, C., N. Jones, and G. Ivey, 2011: Dynamics of a tidally- forced stratified shear flow on the continental slope. APPENDIX J. Geophys. Res., 116, C11017, doi:10.1029/2011JC007214. Boegman, L., 2009: Currents in stratified water bodies. 2: Internal waves. Encyclopedia of Inland Waters, L. Boegman, Ed., Ac- Expression for Stokes Drift ademic Press, 539–558. ——, G. Ivey, and J. Imberger, 2005a: The degeneration of internal Approximating the velocity near (x0, z0) as a linear waves in lakes with sloping topography. Limnol. 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