Turbulence in the Red Sea outflow plume in the western Gulf of Aden was observed with an upward-looking, five-beam, 600-kHz acoustic Doppler current profiler (ADCP). The “Bottom Lander” ADCP was deployed on the seafloor in two narrow, topographically confined outflow channels south of Bab el Mandeb for periods of 18–40 h at three locations at 376-, 496-, and 772-m depths. Two deployments were taken during the winter season of maximum outflow from the Red Sea and two in the summer season of minimum outflow. These short-term observations exhibit red velocity spectra with high-frequency fluctuations of typically a few centimeters per second RMS velocity during strong plume flow as well as strong subtidal variations. In one winter season event, the plume flow was reduced by a factor of 4 over an 18-h time span. In variance-preserving form, velocity spectra show a separation at frequencies of 0.3–3 cycles per hour between low-frequency and high-frequency signals. The latter show significant coherence between horizontal and vertical velocity components; hence they carried turbulent stress. Based on a comparison with velocity spectra from atmospheric mixed-layer observations, the authors argue that large variance at frequencies of the order of 1 cph was possibly associated with bottom-generated, upward-propagating internal waves. One coherent feature that matched such waves was observed directly. Higher frequencies correspond to turbulent motions of energy-carrying scales. The turbulent Reynolds stress at heights above the bottom between 4 and 30–40 m was computed for most of the ADCP observations. Near the bottom, the streamwise turbulent stress and the streamwise velocity followed a quadratic drag law with drag coefficients ranging from 0.002 to 0.008. There was also significant spanwise stress, hinting at the three-dimensional nature of the boundary layer flow. The time–height variations of the stress and its spectrum proved to be complex, one of its most striking features being angles of up to ∼40° between the direction of the stress and that of the low-frequency flow. The turbulent shear production and eddy viscosity were also examined. On the technical side, the paper discusses the role of the fifth, center-beam velocity measurements in correcting for instrument tilt along with the effect of beam spreading in the 30° Janus configuration of the “regular” four ADCP beams. Instrumental noise and detection limits for the stress are also established.
Ventilation of the deep sea is a crucial element of global climate and climate change. Part of the ventilation results from the outflow of heavy waters from marginal seas into the World Ocean (e.g., Price and O’Neil Baringer 1994), such as from the Mediterranean, the Nordic and the Antarctic seas, and the Red Sea, the topic of this paper. The gravity currents of such outflows entrain the overlying water such that potential density decreases along the path of the plume and transport increases. The terminal depth of outflowing plumes and their terminal transport depend on the characteristics of their source water, on the strength of the entrainment, and on the characteristics of the entrained water along the plume path. Hence, turbulent mixing crucially determines the overall global effect of outflows.
The turbulence in outflow plumes has two energy sources: the work of the bottom shear stress and shear flow instability in the interfacial layer on the upper side of the plumes. Correspondingly, two retarding terms in the momentum balance are the entrainment stress and the bottom stress (Baringer and Price 1997). Thus, understanding the dynamics of outflow plumes requires knowledge of the bottom shear stress (τb). Even though this has long been understood, measurements of the stress have been sparse owing to their difficulty. The easiest, although more expensive, method is to deploy expendable current profilers (XCP) and to infer τb from fitting the logarithmic velocity layer in the lowest few meters above the seafloor. Such observations were carried out, for example, in the Mediterranean Outflow (Johnson et al. 1994a, b) and in the Denmark Strait Overflow (Girton and Sanford 2003). Johnson et al. (1994b) also inferred the bottom shear stress from measurements of the turbulent dissipation rate (ɛ) with expendable dissipation profilers. They found substantial differences between dissipation-based stress data, log-layer-based stress values, and estimates from the bulk momentum balance of the plume. These differences are a consequence of the fact that all listed methods of estimating τb are indirect. For example, while log-layer fits will likely reflect form drag if it is present, form drag does not affect near-bottom dissipation rates.
This paper investigates the turbulent stress and the kinematic and dynamic structure of turbulence in the bottom layer of the Red Sea Outflow plume. The observations analyzed herein were obtained as part of the 2001 Red Sea Outflow Experiment (REDSOX). The objectives of REDSOX are to describe the pathways and downstream evolution of the descending outflow plumes of Red Sea water in the western Gulf of Aden, to quantify the processes that control the final depth of the equilibrated Red Sea Outflow Water (RSOW), and to identify the transport processes and mechanisms that advect RSOW and its properties through the Gulf of Aden and into the Indian Ocean. REDSOX is thus intended to provide the first comprehensive description of the pathways, structure, and variability of the descending outflow plumes from the Red Sea and the equilibrated Red Sea water mass as it enters the western Indian Ocean. Observations of flow and stratification were taken during two cruises in 2001; a winter cruise during the season of maximum outflow from the Red Sea (Murray and Johns 1997) took place from 5 February to 15 March 2001 and a summer cruise during minimum outflow lasted from 21 August to 12 September 2001. We henceforth refer to the first and second cruises as “RSX1” and “RSX2,” respectively. Relatively little being known about the Red Sea Outflow prior to REDSOX, both cruises surveyed the “near field” of the descending plume as well as the “far field” of the further spreading of the equilibrated plume throughout the Gulf of Aden. The latter was also observed with neutrally buoyant RAFOS floats ballasted for the mean depth of the RSOW of 650 m (Bower et al. 2005; Fratantoni et al. 2006). A number of floats were deployed on the seafloor and released at later times.
To date, results from REDSOX have addressed the lateral stirring of the equilibrated RSOW by mesoscale eddies in the far field of the Gulf of Aden (Bower et al. 2002, 2005; Fratantoni et al. 2006) and the equilibration process itself at the juncture between the far field and the near field (Bower et al. 2005). The plume structure and the mixing and entrainment in the active outflow plume flow were analyzed by Peters et al. (2005) and Peters and Johns (2005). Herein, we continue their work with a detailed look at the turbulence in the bottom layer of the outflow plume with focus on the bottom shear stress as an important variable of the plume dynamics.
In reference to the vertical plume structure, Peters et al. (2005) and Peters and Johns (2005) distinguish between a mixed or weakly stratified bottom layer (BL) and a strongly stratified and sheared interfacial layer (IL). They further note that the top of the BL usually coincided with the current maximum, that the RSOW was diluted much less in the bottom layer than in the interfacial layer, and that the IL was thicker than the BL and carried a large fraction of the plume transport, which increased with distance downstream. Hence, the IL was the primary site of entrainment, with weaker entrainment into the BL. This finding implies a degree of decoupling of turbulence in the BL, presumably driven by work of the bottom shear stress, from turbulence in the IL for which there is good evidence that it was generated by local shear instability. Using the observations of bottom turbulence analyzed in detail herein, Peters and Johns (2005) compare the relative magnitudes of bottom stress and interfacial stress. For further information on the REDSOX observations and results on the Red Sea Outflow in general we refer the reader to the above-cited papers.
In the following, observations, instrumentation, and seafloor topography are introduced first, followed by an introduction of methods of extracting velocity and stress from the bottom-mounted acoustic Doppler current profiler (ADCP). An investigation of the time variability of the turbulent bottom layer flow reveals a separation between low-frequency subtidal variations and turbulent fluctuations. A switch to the spectral domain and an analysis of the turbulent Reynolds stress reveals complex flows with variable near-bottom quadratic drag coefficients and significant cross-flow stress. In the summary and discussion section we attempt to elucidate the flow physics, which relate the turbulence and the lower-frequency flow that generates it. Detail of instrumentation, processing and analysis methods, and estimates of the accuracy and noise limits of the observations are contained in an appendix.
a. Regional topography
The outflow from the Red Sea originates at the 150-km-long Strait of Bab el Mandeb. Its northern end at the Hanish Islands has a sill of ∼160-m depth, while the southern exit is marked by narrows of ∼20-km width and ∼200-m depth at Perim Island (see Fig. 1). The strait has a maximum width close to 100 km, but the deep channel along its axis is much narrower. From the Perim Narrows, the warm, salty, and heavy Red Sea Outflow water flows into the western Gulf of Aden through two channels: the narrow, ∼140-km-long Northern Channel (NC) and the shorter and wider Southern Channel (SC) [indicated in Fig. 1; see also Peters et al. (2005)]. The topographic confinement is extreme in the NC with a typical width of only a few kilometers. The two outflow channels end at the abrupt edge of the tectonically active, up to 1800 m deep, Tadjura Rift at depths of 400–600 m at the SC and about 800 m at the NC.
The plume flow turned out to be strongly influenced by the local seafloor topography; thus we briefly describe the geographical vicinity of the sites where turbulence observations were made. “Bottom Lander” deployments are further introduced below and indicated in Fig. 1. The deployments B1 and B3 were located near the axis of the Southern Channel in a slightly curved, widening segment just downstream of a shallow depression (Fig. 2a). The local width of the channel at an elevation of 40 m above its deepest point is about 2.5 km. The slope on the sides of the channel are fairly gentle, especially on the southwest side. In contrast, the channel sides are steep at the B2 site near the lower end of the NC (Fig. 2b). The site is located just upstream of a significant depression and in a significantly curved channel segment. The channel is only about 1 km wide, as marked by elevations of 40 m above the channel axis. The Lander was deployed about 200 m off this axis at the foot of the steep northern slope. Deployment B4 took place farther upstream in the NC in a comparatively wide and straight stretch of the channel (not shown).
The core of the observations analyzed herein, measurements of the bottom layer turbulence in the Red Sea Outflow plume, were taken with an RD Instruments broadband, 600-kHz, five-beam, upward-looking ADCP, deployed on the seafloor as part of the Bottom Lander. Deployment locations are indicated in Fig. 1, deployments lasting from 18–40 h as detailed in Table 1. With 1-m vertical bins, the ADCP velocity measurements covered a range of heights above bottom (h) between 3.9 and 35–50 m. The ADCP employed during REDSOX has four “regular” beams in a 30° Janus configuration and an additional fifth beam aligned with the instrument axis, and thus pointing up. The ADCP was operated in the low-noise “mode 4.” Following Peters et al. (2005), and as described in more detail below, much of the Lander height range was located within the mixed or weakly stratified bottom layer and below the interfacial layer. Our measurements differ from measurements by bottom tripods and expendable current profilers by excluding most of the constant stress layer in the lowest few meters above the bottom. Thus we did not observe the bottom shear stress proper. However, at heights above 3.9 m and up to 30–40 m, the Bottom Lander ADCP velocity data allow direct computations of the turbulent Reynolds stress. Because of a combination of seasonally weak flow (Peters et al. 2005), weak, marginally resolved turbulence and technical problems, this paper focuses on deployments B1 and B2.
In addition to the Bottom Lander observations, we also utilize stratification and current profiles from a SeaBird 911Plus conductivity–temperature–depth probe (CTD) and a lowered acoustic Doppler current profiler (LADCP) (Peters et al. 2005; Peters and Johns 2005). CTD–LADCP station locations are depicted in Fig. 1.
3. Methods: Reynolds stress and TKE
If the turbulent length scale is sufficiently large relative to the ADCP bin length and if the turbulence is strong enough relative to instrumental noise, ADCPs with diverging acoustic beams in Janus configuration can measure the Reynolds stress as well as the turbulent kinetic energy (TKE), as first discussed by Lohrmann et al. (1990). The approach, the “variance method,” appears to have been reinvented more than once in field and laboratory experiments (Lhermitte 1968; Tropea 1983). Since having been applied to oceanographic ADCP measurements by Lohrmann et al. (1990), it has been utilized by Stacey et al. (1999a, b), Lu and Lueck (1999b), Rippeth et al. (2003), and Peters et al. (2006, manuscript submitted to J. Geophys. Res., hereinafter PET). The beam spreading is not a problem for the variance method; the turbulent velocity field only needs to be statistically homogeneous horizontally over the array of the beams. The determination of the Reynolds stress from the velocity variances of the four ADCP beam velocities is greatly aided by the fact that the instrumental noise cancels in the stress.
The following assumes an upward-looking ADCP with beams 1 and 2, and 3 and 4, respectively, in two perpendicular planes. The velocity vector in (x, y, z) direction is (u, υ, w) with z positive upward. The Reynolds decomposition is applied such that, e.g., u = u + u′. Geometrical considerations lead to a relationship between the velocity variances along the beams to those in Cartesian coordinates. For simplicity we further assume that beams 1 and 2 lie in the x, z plane. From
it follows that
It can readily be seen that the instrumental noise cancels in (2) provided that it is the same in all beams. The preceding assumes symmetry of the beams, that is, a perfectly vertical orientation of the ADCP. Given a five-beam system, small constant roll and pitch angles in radians, α (in the 1–2 beam plane) and β, respectively, can be corrected following Lu and Lueck (1999b):
Pitch and roll angles are combined into instrument tilt in Table 1. The vertical turbulent velocity variance w′2 is given directly by the center beam except for a small error caused by the tilt of the Lander. At low frequencies we can reliably correct w5 for the tilt, the result being w5c. In the second term on the rhs of (3) noise has to be subtracted from the horizontal and vertical turbulent velocity variances. Even with a five-beam system the horizontal Reynolds stress term, , can only be resolved at frequencies lower than a few cph (see the appendix and Fig. A1); thus we neglect the last term in (3). However, a tilt sensitivity test is performed in the appendix by estimating . The y component of the turbulent stress, , is computed similarly to (3).
The shear production of TKE
from Reynolds stress and shear. The vertical turbulent velocity variance is directly given by the center beam . The tilt of the Lander (Table 1) causes only small errors in w5—neglected herein except at low frequencies, where we rely on w5c. Further neglecting terms related to pitch and roll, we follow Lu and Lueck (1999b) in defining a term related to the sum of the beam velocity variances, S2 ≡ (4 sin2θ)−1 Σ4i=1. The horizontal turbulent velocity variance then follows as
ADCPs with a center beam do not allow the determination of the horizontal directional distribution of turbulent velocity variance. However, if the beam pairs 1/2 and 3/4 velocity variances, and , differ, we can conclude that the turbulent velocity field was horizontally anisotropic.
Results from (3), (4), and (5) are sensitive to the averaging time interval chosen in the Reynolds decomposition. Below, we generally use an averaging interval of 30 min, a choice justified by the examination of spectra of the Reynolds stress. These reveal that comparatively low frequencies contribute to the stress. Noting that Fourier transforms are linear, power spectra of beam velocity data can be used in (3) and (5) instead of beam velocity variances in order to obtain frequency spectra of the Reynolds stress and the horizontal turbulent velocity variance.
4. Time–depth characteristics of the flow in the bottom layer
Before discussing the velocity measurements, we introduce a small number of CTD–LADCP casts taken at Lander sites during or just before Lander deployments; their times are listed in Table 2. These casts are the only observations of the stratification available for the interpretation of the Lander velocity profiles. The stratification is important for our analysis because of its effect on the kinematics and dynamics of turbulence. We discuss stratification in terms of T, S, potential temperature Θ, potential density σΘ, and buoyancy frequency N.
In CTD–LADCP casts taken with B1 stratification is seen to increase gradually with height in the BL, with minimum buoyancy frequencies (N) of about 2 cph, and thus fairly strong stratification even at heights of 10–15 m (Fig. 3). Even though cast 55 appears to show a better developed, nearly 40 m thick, nearly mixed layer, N 2 was still in the range of 5–8 (×10−6 s−2), corresponding to N ≈ l.5 cph.
Peters et al. (2005) and Peters and Johns (2005) note that the plume flow was especially swift toward the lower end of the NC during RSX1, the BL was exceptionally thick, and entrainment was strong. This is the site of Lander deployment B2, which, at a water depth of 772 m, took place in an environment of weaker stratification than the shallower B1 site. The two CTD–LADCP profiles depicted in Fig. 4 show a mixed layer about 30 m thick, with unstable stratification in its lower part. We define N as N ≡ sgn(N 2)|N 2| to retain the sign of stratification. Owing to the great thickness of the BL, the B2 Lander velocity covered only its lower part. LADCP cast 101 (Figs. 4a,b) was taken at a 200-m distance from the location of B2; thus the differences between the Lander and LADCP velocity is not surprising. Otherwise velocity profiles from LADCP and Lander matched quite well in the casts shown in Fig. 3. Velocity vectors are displayed as magnitude (V) and direction (Φ) rather than u and υ.
b. Velocity variability
Following is an outline of general time-domain characteristics of the plume flow in the two Bottom Lander deployments of RSX1, which are our focus. The discussion is based on velocity time series (Figs. 5b,c,e,f) and mean profiles (Figs. 6 and 7). The velocity time series exhibit a wide range of time scales from high-frequency, turbulent fluctuations to slow variations on time scales of the order of the record lengths. Figures 5, 6 and 7 also depict turbulence variables, which will only be further discussed below. We begin the discussion with the comparatively simple flow structure of deployment B1.
The direction of the plume flow (ΦV) at B1 followed the local topography with ΦV ≈ 150° (Fig. 6b, see also Fig. 2a), being unidirectional over at least 35 m above the bottom. The mean speed (V) profile displays a linear height dependence above 15-m height with a more loglike decrease below. This pattern is also present in instantaneous profiles depicted in Fig. 3. The height range of 40 m shown in Fig. 6 does not include the velocity maximum at the top of the BL (cf. Fig. 3). Accommodating the flow direction of B1, we rotate the horizontal coordinates 30° counterclockwise to x′ y′ such that the plume flow is in the −y′ direction (as it is in B2 as shown below). The x′ y′ coordinates are used for B1 throughout the paper. Velocity time series at 14-m height (Fig. 5f) depict a plume flow going through a minimum during the record from an initially strong speed of about 0.5 m s−1. It is visually obvious that high-frequency velocity fluctuations u′, υ′, as well as w′, varied in accord with the low-frequency flow, also going through a minimum during the measurements. The analysis of velocity spectra below shows that instrumental noise did not significantly contribute to the signals as shown in Figs. 5b,c.
The flow of B2 data was faster overall than that of B1 (Figs. 5c,f, 6a and 7a) and more complex. High-frequency vertical and horizontal velocity fluctuations were also larger in B2 than in B1, more so than in proportion to the stronger plume flow of B2 relative to B1 as discussed below. Typical magnitudes of the turbulent velocity components were a few centimeters per second. Note that w is plotted on the same scale as u and υ in Figs. 5b,c. The direction of the mean plume flow of B2 was almost due south (ΦV ≈ 180°, Fig. 7b; see also Fig. 2b). This direction deviates from a channel direction, in a 2–4-km vicinity of the B2 site, of 140°–160° (Fig. 2b). This direction mismatch between the topography and the flow hints at the three-dimensional nature of the velocity field at B2. As in B1, the mean velocity profile of B2 depicted in Fig. 7a does not reach the velocity maximum.
Over the course of deployment B2, the speed of the plume decreased from about 0.6 to 0.2 m s−1 and later began to increase again. This event has already been noted by Peters et al. (2005), who argue that its time scale was too large to be compatible with tidal fluctuations. Furthermore, tides observed in 1995–97 moorings in the NC near the B4 site and in the SC near the B1 site reveal velocity amplitudes of only ≲ 0.05 m s−1 from semidiurnal and diurnal periods combined (Matt and Johns 2006). On this basis, we expect only small tidal signals at B2. Large low-frequency velocity fluctuations, such as in the B2 record, imply similar variations in the plume transport. We suspect that the low-frequency velocity dip at B2 was related to changes in the overflow through Bab el Mandeb, which we did not monitor, however.
The nature of the flow at B2 is further revealed when the time–depth variability of the low-passed υ, u, w is analyzed. Contours for u, υ, w and of the horizontal flow direction (ΦV) presented in Fig. 8 show low-frequency fluctuations not only of υ and u but also of w, with alternating upward and downward flow of up to ±0.015 m s−1. Similar variations of up to ±15° appear in ΦV. The variations in u, w and ΦV are highly correlated; the correlation coefficient of u and w at h = 13.9 m, for example, is 0.91. Hence, eastward and upward flow aternate with westward and downward flow. This is consistent with the notion that, within the local topography, the u velocity component is essentially cross channel. We also note that B2 was located on the lower eastern slopes of the NC such that eastward velocities imply uphill flow, as indeed shown in Fig. 8. The dominant variation in υ, its dip in the last third of the record, has no parallel in u or υ, but we find below that fluctuations in υ and w were also correlated. Nevertheless, Fig. 8 shows that cross-channel flow oscillations were a prominent signal at B2. Toward the end of the B2 record, when the plume flow was comparatively weak, the cross-channel oscillations show an approximately 12-h period with superimposed strong fluctuations of time scales of a few hours. As the first half of the record showed variations on the time scale of about 24 h, it is unclear if the cross-channel signal was influenced by the tides. We add in passing that the described variations of the vertical velocity are significant and reliable even though there is a systematic uncertainty in w on the order of 0.01 m s−1 as discussed in the appendix.
5. Spectral characteristics
a. Velocity power spectra
Among our observations, the plume flow and the turbulence were strongest in B2, and this site shows the best-resolved flow spectra. Even though the B2 site is topographically and hydrodynamically peculiar, as discussed above, its flow spectra are similar to the other sites. Figure 9 depicts power- and cross-spectral variables from a height of h = 13.9 m (i.e., spectra from the time series shown in Figs. 5b,c). The power spectra of u and υ are “red” with a shoulder at cyclic frequencies (σ) of a few cycles per hour (cph) and, at their highest frequencies, with slopes between σ−1 and σ−5/3. The high-frequency end of the spectra is reminiscent of an inertial subrange even though spectral slopes differ somewhat from −5/3 and there is no exact isotropy (see Tennekes and Lumley 1972, p. 254). The streamwise (υ) variance is larger than the spanwise (u) variance, and the horizontal variance exceeds the vertical velocity variance. Spectra of w are less red overall than those of u and υ. The spectra shown in Fig. 9 are representative from other heights except that near the seafloor vertical motions are suppressed. The detection of the inertial subrange is affected by the fairly low Nyquist frequency of 1/(50 s) of the data.
To assess the effects of beam spreading, we present regular velocity spectra as well as the horizontal power spectrum computed from the beam velocities using (5). The latter is unaffected by beam spreading and has been corrected for instrumental noise. As seen in Fig. 9, the sum of the regular υ and u spectra matches the unbiased spectrum from (5) closely. In this sense the effect of beam spreading is minimal. The power spectrum from w5 is not a true vertical spectrum because of the instrument tilt. The related bias is minor, however, as discussed in the appendix, and it does not affect our discussion. The appendix also discusses noise levels, which are insignificant in Fig. 9.
The red character of the log–log power spectra is interrupted by a shoulder in the vicinity of 1 cph. The structure of the spectra becomes much clearer in their variance-preserving form (Fig. 9b), which demonstrates a break at σ ≈ 0.3 cph between low-frequency and high-frequency regimes. Maxima in the horizontal variance-preserving υ spectra appear near σ = 10 cph, while the the w5 spectrum rises a plateau above 10 cph. Between 1 cph and the Nyquist frequency the cross spectrum of the streamwise υ and w5 shows significant squared coherence (R2) with a phase ranging from −30° to −60° (Figs. 9c,d). In contrast, the spanwise, uw, coherence was smaller with less statistical significance. The cross spectrum indicates the presence of coherent structures carrying stress in the streamwise (y) direction.
The spectra from B2 thus present a structure with a weak gap at ∼0.3 cph, considerable high-frequency variance between this gap and the Nyquist frequency of 36 cph, and significant correlation between streamwise and vertical velocities above the gap frequency. Qualitatively, this structure also occurred in the B1 and B4 observations (not shown). The variance-preserving spectra from B2 demonstrate that we resolved the energetic scales of the turbulence and part of the inertial subrange. In B2 (Fig. 9), as well as in B1 (not shown), the spanwise-vertical coherence increased and became more significant with increasing height above bottom, an indication of coherent, stress-carrying structures also in the spanwise direction.
It is reasonable to assume that, at sufficiently large frequency, any slowly evolving turbulent velocity field would approximately be advected with the mean current as a frozen field. Thus, we have added wavenumber axes to Fig. 9 that correspond to advection with the mean speed V = at the height of the measurements, where u and υ are the arithmetic mean velocity components. The wavenumber axis has to be taken with a grain of salt owing to the significant variability of the low-frequency speed V. Interpreting the power spectra depicted in the Fig. 9 in terms of horizontal wavenumber indicates the occurrence of significant energy at horizontal scales from tens of meters to the order of a kilometer. The principal caveat is that the frozen field assumption may not hold at large periods of the order of an hour.
The further analysis of the flow spectra is based on their full frequency–height variability as contoured for B1 and B2 in Fig. 10, which depicts horizontal energy spectra computed from (3). It is instructive to compare the spectra from the Red Sea Outflow with observations in the atmospheric mixed layer. This approach has an oceanographic precedent in Soulsby (1977). We utilize normalized summary model spectra based on the 1968 Kansas and 1973 Minnesota Experiments (Kaimal and Finnigan 1994, p. 47). The sites of these experiments were of course chosen for their flatness; thus their underlying bottom topography differs radically from the Red Sea Outflow channels. Yet the atmospheric model provides a useful benchmark. Noting that the Monin–Obukhov length is infinite because of the vanishing heat flux at the seafloor, we adopt the atmospheric mixed-layer scaling and mixed-layer spectra. The height of the bottom layer (H) is taken to be the mixed- layer height. Atmospheric model spectra consist of a flat low-frequency portion followed by an inertial subrange where the spectrum drops like k−5/3 as function of horizontal wavenumber k. In variance-preserving form, there thus is a spectral maximum that separates a k+1 region from a k−2/3 region. According to Eqs. (2.32) and (2.33) of Kaimal and Finnigan (1994), the wavelength of the spectral maximum of the horizontal spectra (λm,h) is height independent, while the spectral maximum wavelength of the vertical velocity (λm,w) varies as follows:
Spectral maximum wavelengths are converted to frequencies with the respective mean bottom layer velocities, 0.45 m s−1 for B1 and 0.5 m s−1 for B2. With assumed mean bottom layer heights of 35 m (B1; see Fig. 3) and 60 m (B2; see Fig. 4), the locations of the model spectral maxima following (6) and (7) have been plotted in Fig. 10.
In Fig. 10, features of the horizontal velocity spectra show a tendency to line up parallel to the h axis, while the patterns of the w5 spectra are slanted in h–logσ space. This difference is qualitatively consistent with the difference between λm,h and λm,w given by (6) and (7). At small h/H spectral maxima of w are shifted to larger σ relative to those of u and υ. Near the seafloor the 50-s sampling of the Lander ADCP does not resolve the expected spectral maximum of w, and we generally resolve only part of the inertial subrange. The horizontal velocity spectra from both Lander deployments show large variance at frequencies lower than expected from the atmospheric boundary layer model.
Coherence spectra allow analyzing the typical space and time scales of the velocity variability. For simplicity and brevity, we focus on the vertical velocity and begin with the coherence as function of frequency and vertical separation (Fig. 11). At low frequencies near 0.1 cph, temporal vertical velocity variations were coherent across the height range of the Lander ADCP. In contrast, high-frequency signals became uncorrelated even at small vertical separation. The high vertical coherence at small σ is also visible in the time domain for the vertical as well as the horizontal velocity components in Fig. 8. The coherence spectra allow assigning a typical vertical scale of the turbulent velocity field. Somewhat arbitrarily choosing a squared coherence of 0.5, the associated vertical coherence scale in the 1–10-cph frequency range is 12 m for B2 and 6–12 m for B1. Generally, the high-frequency vertical coherence was significantly larger during B2 than B1, implying larger vertical length scales in B2 than in B1. This result is consistent with finding the turbulence during B2 more energetic than during B1. The coherence spectra contoured in Fig. 11 were computed by averaging all cross- spectra of w5c at the same vertical separation regardless of actual height above bottom. Figure 11a shows peaks in coherence at σ = 1 cph and σ = 2.5 cph. Such peaks imply the existence of coherent processes embedded in a more random field.
The preceding discussion examines the coherence as function of frequency and vertical separation. Similarly, we can compute the coherence as function of vertical wavenumber and separation in time. Basing this exercise on 50-s average vertical profiles of w5, we find no coherence. This implies a typical time scale of the fluctuating turbulent velocity field smaller than 50 s and, by means of the frozen field assumption, a typical horizontal coherence scale smaller than 25 m. These results are consistent with the small vertical scale found above. In contrast, low-frequency motions, shown as 30-min averages in Fig. 8, did exhibit temporal coherence as discussed above.
6. Reynolds stress, TKE, and shear production
The final section of our analysis addresses the turbulent stress in the bottom layer, the most important property of the turbulence with respect to the plume dynamics. Along with the stress, the turbulent kinetic energy (TKE), the shear production (P), and the eddy viscosity (Az) are also computed.
a. Variability of turbulent stress and TKE
In applying (3), we compute the velocity variances from the five beams from 30-min data segments, adding the variance from the 50-s ensembles recorded internally in the ADCP to the explicitly computable variance from the 50-s average velocity data. The choice of 30-min segments is a compromise. On one hand, the “M ≳ 55 criterion” of Lu and Lueck (1999a) discussed in the appendix indicates using 1-h segments. On the other, the spectral gap between high- and low-frequency motions was sometimes at σ ≈ 3 cph, suggesting the use of 20-min segments. Fortunately, section 6d shows that the difference in stress from 30-min and 1-h segments is negligible.
Time series of the x- and y-stress components, τx = −ρ and τy = −ρ, respectively, from the energetic B2 and the less energetic B1 at a height of 13.9 m are depicted in Figs. 5a,d. The stress components attained maximum magnitudes of ≳1 Pa in B2 and <0.5 Pa in B1. They show a good deal of short-term scatter and strong variability even in 2.5-h averages. Time series from both deployments further show a slow decrease of the turbulent stress with time in response to the decreasing plume flow velocity. In B2 (Fig. 5a) the stress is seen to decrease all the way toward the end of record even though the velocity had begun to increase again. It appears thus that there was a time delay of stress variations relative to those of the velocity, a property not present in the B1 record.
In the B1 deployment the mean stress decreased with increasing height from 0.18 to 0.08 Pa (Fig. 6c). The direction of the stress was within 10° to the flow direction near the bottom, but attained a 30° angle to the right of the velocity vector higher above the bottom. Hence, the plume flow was three-dimensional with cross-stream stress. Although τ varied vertically, the three measured components of velocity variance show little height variation except in near the bottom (Fig. 6d). The ratio of the Reynolds stress to TKE, written here as Rτ ≡ τ/u′212u′234), characterizes the correlation between vertical and horizontal velocity fluctuations and thus the ability of the turbulence to generate stress. The different depth variations of τ and TKE lead to a profile of Rτ with values of 0.25 near the bottom followed by a decay to about 0.1 near 25-m height (not shown). Figure 6d reiterates the finding from above of larger horizontal than vertical velocity variance.
The complexity of the flow at B2 is further expressed in the relationship of the velocity and stress vectors. The time-mean stress had a maximum of 1 Pa at 5-m height, decreased above at first more strongly and then, above h = 20 m, more gently to about 0.65 Pa at the top of the ADCP range (Fig. 7d). The mean stress vector had an angle of 25° to the left of the velocity vector near the bottom with an increase to approximately 40° at larger h. The “misalignment” between velocity and stress vectors is related to low-frequency contributions to the stress. If the stress is computed only from the internally recorded 50-s ensemble data, the angle between the two vectors near the bottom becomes much smaller, as also shown in Fig. 7b. We have noted above that a significant component of the flow variability at B2 in the 0.1–1 cph frequency band was associated with cross-channel, cross-stream oscillations. The analysis of the Reynolds stress relates these oscillations to cross-channel, cross-stream stress.
As B1, B2 also had larger horizontal than vertical velocity variance components. In B2, the vertical velocity variance varied only weakly with h, while the two horizontal components of the variance showed maxima near 5-m height and an upward decrease in the lowest 15 m above the bottom (Fig. 7d). This behavior leads to an almost constant degree of correlation between vertical and horizontal velocity fluctuations with Rτ ranging from 0.3 to 0.36. The Rτ of B2 was larger than that of B1 and had a different mean profile, an indication of differences in the dynamics of the turbulence. With > , there was anisotropy in the horizontal turbulent flow at B2. As mentioned above, our methods preclude specifications of the anisotropy as, for example, in terms of streamwise and spanwise components.
Combining findings from turbulent stress and TKE, the lowest 15–20 m appear different than the waters above. This is further emphasized in contour plots of the Reynolds stress components as function of time and height (Figs. 12a,b). Both components exhibit visually different variability patterns in the lowest ∼15 m in comparison with the layers above. Close to the bottom the stress became small in the last third of the record when the flow became slow. In contrast, above ∼15 m, τx and τy exhibit large and highly variable values between hours 36 and 44 in Fig. 12. In either layer, above and below 15-m height, variations of τx and τy show strong vertical coherence in the form of pulses with large vertical extent relative to the vertical range of the data.
Given the significant low-frequency variations in the low-frequency flow pointed out above, one may expect that these variations caused the pronounced fluctuations in the Reynolds stress shown in Figs. 12a,b. This expectation is substantiated by a correlation analysis of stress, τ, versus shear, Vz ≡ (∂u/∂z)2 + (∂υ/∂z)2. We analyze 30-min average data from B2 further averaged in the vertical from 15- to 40-m height. The 95% correlation coefficient of τ and Vz thus defined is 0.55, while the correlation coefficients of τ and the mean ∂u/∂z and ∂υ/∂z are 0.43 and −0.53, respectively. The number of independent data is 80; hence, all correlations are statistically significant. The correlations between Reynolds stress and the three measures of shear explain only about one-quarter of the variance, or less, owing to substantial scatter. The scatter is partly due to sharp peaks in τ that have no parallel in shear. Nevertheless, it is the low-frequency shear that allows the turbulence to extract energy from the low-frequency flow: thus the preceding correlation exercise provides evidence that the variability of the turbulence was caused by low-frequency fluctuations of the streamwise as well as spanwise velocity.
Contours of the turbulent stress from B1 (Figs. 12c,d) reflect the much smaller stress when compared with B2. They show fairly small vertical variations throughout their range and, as B2, exhibit vertical coherence of temporal fluctuations. Without showing graphics, we note that the Reynolds stress observations from the B4 deployment show a weak mean stress of 0.1 Pa through the lowest 10 m above the bottom. There are no valid stress data above 10 m. Here B4 was located in a more uniform, wider, and straighter channel section than either B2 or B1, and the velocity and stress vectors were aligned within 10° of each other throughout the range of valid data.
b. Log-layer, quadratic drag law
Simple relationships between stress and velocity prevail in the constant stress layer near the seafloor. Two aspects of these relationships are the logarithmic height dependence of the velocity profile and the quadratic scaling of the stress as a function of the near-bottom velocity. Below, we also scale turbulence variables accordingly and exchangeably refer to the scaling as “constant stress” or “law of the wall.” The substantial height variability of the turbulent stress depicted in Figs. 6c and 7c as well as the finding of substantial angles between stress and velocity vectors indicate that the conditions during some of our observations were more complex than the simple constant stress layer scenario. This is further substantiated by the finding that estimates of the bottom stress from logarithmic fits of the observed velocity profiles yield unrealistic results. On the basis of 30-min average velocity profile segments between 3.9 m and either 6.9 or 11.9 m, we find that logarithmic fits yield stress values too large by factors of 4 or more while fitted roughness lengths are too large by orders of magnitude. The log-layer stress estimates increase weakly with increasing profile length.
Conversely, we can use the observed near-bottom Reynolds stress τ from h = 3.9 m to scale the observed velocity profiles. In the logarithmic velocity profile of the law of the wall the shear is Vzl = u*/(κ h), where von Kármán’s constant is κ, and u* = τ/ρ is the friction velocity. We normalize the observed shear by Vzl, taking τ from the observed Reynolds stress at 3.9-m height. On average we find that the normalized shear, Vzn ≡ Vz/Vzl ranged from 1.25 to 1.45 in B2 with little height dependence (Fig. 7a). Hence, the B2 mean shear profile deviates little from the law of the wall model. On the contrary, the B1 normalized shear increased from 1.3 at 3.9 m to about 4 near 30-m height (Fig. 6a). The B1 observed velocity profiles thus differed substantially from log-layer behavior. Even though the flow at B1 and B2 did not follow the law of the wall, the application of law of the wall scaling highlights dynamically important differences in the mean flow profiles.
Despite the preceding complications we are interested in examining quadratic drag laws because they represent the almost universal parameterization of bottom stress in flow models. Furthermore, finding even approximate quadratic drag behavior would lend credence to our Reynolds stress data. Owing to the complications indicated above, we examine regressions of the components of the observed stress, τx versus the drag-law counterpart τbx = ρcduV and similarly τy versus τby = ρcdυV. We also examine the standard approach of relating τ to τb = ρcdV2. Here, cd is the drag coefficient, ρ is a reference density, and V = (u2 + υ2) is the speed. Note that τb = τ 2bx + τ 2by as required. All data are taken from the lowest ADCP bin at 3.9-m height above the seafloor. The results for the successful Bottom Lander deployments with reliable stress data are listed in Table 3. The appendix explains the availability of valid stress data.
Deployment B4 was taken in a comparatively wide, straight, and uniform section of the NC. The flow and the stress were almost due east; especially there was no significant spanwise stress (not shown). The observations from 3.9-m height closely follow quadratic drag law behavior with a rather “standard” drag coefficient of 0.022. In contrast, B2 had spanwise stress about half as large as the streamwise stress, while there was little spanwise low-frequency velocity. As a consequence, the drag coefficients related to regressing the streamwise τy versus τby differs from the cd obtained from the standard approach of regressing τ versus τb. With values of 0.008–0.009, both estimates of cd are much larger than standard values. With respect to plume dynamics and modeling plume flows it is important to note the variability range of 0.002–0.008 of drag coefficients in the Red Sea Outflow. We note that large cd near 0.01 have been observed elsewhere (e.g., Stacey et al. 1999a). The discussion of the bottom stress is continued below.
c. Shear production and eddy viscosity
As the shear production P is the product of the Reynolds stress and the squared mean shear, (4), and both shear and stress were large near the bottom during B1 and B2, P was strongly bottom-intensified as seen in contours of P (Fig. 13) and in mean profiles of P (Figs. 6e and 7e). Contours of P from B2 show especially large values near the seafloor during the first ∼24 h of the deployment when the plume flow was strong. The time variability of P during B2 (Fig. 13) was dominated by pulses similar to those observed in the Reynolds stress. The observed vertical structure of P is qualitatively expected from law-of-the-wall scaling, Pl = u3*/(κh). In both B1 and B2 mean profiles the bottom intensification of P was even stronger than predicted by Pl (Figs. 6e and 7e). But, while P from B2 matched Pl above the 20-m height, P stayed above Pl throughout the B1 profile.
The observed P of the order of magnitude of 10−5 W kg−1 are compatible with, but smaller than, dissipation rates observed by Johnson et al. (1994b) in the Mediterranean Outflow with a faster plume flow than during B2. Our observed P are also compatible with measurements in tidal flows with speeds roughly comparable to B2 by Lu and Lueck (1999b) and Rippeth et al. (2002).
d. Spectrum of the Reynolds stress
For single-point velocity measurements, the spectra of the components of the Reynolds stress are given by the cospectrum of u and w and υ and w, respectively. For ADCPs, this is only true at low frequencies because of beam spreading. However, because of the linearity of the Fourier transform, the spectrum of the Reynolds stress can be computed from (3) by replacing the beam velocity variances with the corresponding power spectra of the beam velocities, an approach already taken by Lu and Lueck (1999b). The power spectrum thus obtained is unaffected by beam spreading because (3) is valid as long as the turbulence is horizontally homogeneous. Spectra of the Reynolds stress components, regular cospectra scaled in terms of stress, and integrals of the Reynolds stress spectra are shown in Fig. 14 (B2) and Fig. 15 (B1).
During B2 and B1, motions in the energy-containing range identified in section 5a, frequencies above 1 cph and below the Nyquist frequency of 36 cph carried large Reynolds stress variance relative to the total stress (Figs. 14 and 15). Hence the spectra of the observed Reynolds stress show large contributions from low-frequency motions. Moreover, in the spanwise direction the directly resolved stress variance at σ < 36 cph had the opposite sign of the variance at σ > 36 cph in both B1 and B2. Frequencies of σ > 36 cph correspond to horizontal wavelengths smaller than 50 m, scales of the order of and smaller than the typical bottom layer thickness. If the frozen field assumption holds, the low-σ, 1-cph, end of the energy-containing range corresponds to a horizontal scale of about 1 km. The spectra of the Reynolds stress thus indicate that, especially in the spanwise direction relative to the plume flow, small-scale, super-Nyquist, motions, roughly in the inertial subrange of the turbulence, and low-frequency, larger-scale motions had rather different effects on the stress.
7. Summary and discussion
Turbulence in the bottom layer of the Red Sea Outflow plume was observed with a bottom-mounted, upward-looking, five-beam, 600-kHz ADCP. It was deployed in the two outflow channels in the western Gulf of Aden south of Bab el Mandeb at depths of 376, 496, and 772 m. Two deployments took place in the winter season of maximum outflow and two in the summer season of minimum outflow, these short-term current observations lasting between 18 and 40 h. Measured horizontal and vertical velocities covered the water column at heights above the seafloor from ∼4 to 30–50 m. They exhibit strong subtidal flow variations as well as high-frequency fluctuations of typically a few centimeters per second rms velocity during strong low-frequency plume flow.
Velocity spectra are red with only minor maxima within their frequency range. In variance-preserving form, spectra from all deployments reveal a separation at frequencies of 0.3–3 cph between a low-frequency and high-frequency component. The high-frequency band shows an excess of horizontal over vertical kinetic energy and anisotropy in the horizontal. This band also displays significant coherence between the local horizontal and vertical velocity components. The coherence extends across the height range of the measurements. A comparison with spectra from atmospheric mixed-layer observations indicates that, at a 50-s sampling interval, our measurements cover the energy-containing range of the turbulence and extend only partly into the inertial subrange. Differences in the height–frequency structure of vertical and horizontal velocity spectra are qualitatively consistent with atmospheric mixed-layer spectra. However, large variance appeared at frequencies well below the expected spectral maximum, a finding discussed below. The temporal and high-frequency vertical coherence scales of the observed velocity field are small while low-frequency signals at σ ≲ 1 cph show high vertical coherence.
Most of the observations resolve the Reynolds stress throughout the height range of the ADCP in strong plume flow and at least near the bottom in weaker flow. Close to the seafloor the stress exhibits a quadratic drag law behavior even though the stress vector shows substantial angles with the velocity vector near the bottom. Drag coefficients range from 0.002 to 0.009 depending on location. Resolved mean stress values range from a detection threshold of ∼0.05 Pa to maxima above 2 Pa. In two deployments with well-resolved turbulent stress (B1 and B2), the stress decreases from high values near the bottom by 30%–50% toward the top of the 30–40-m range of the ADCP. Well above the bottom, the stress acts at angles of 30°–40° relative to the flow vector in both deployments. The angle between stress and velocity is smaller near the bottom but still 25° in one especially curved and topographically complex channel section (B2). At this site, the turbulent stress shows different magnitudes as well as different variability patterns in lowest 15–20 m above the bottom when compared with higher layers.
Rates of shear production P range from ∼5 × 10−5 W kg−1 near the bottom in strong flow to a detection threshold of ∼3 × 10−7 W kg−1. Instances of P < 0 are restricted to weak flow where the turbulence was unresolved. Eddy viscosity data computed from Reynolds stress and vertical shear increase with height above bottom and vary greatly with values up to 0.25 m2 s−1 at h = 40 m in the strong flow of the B2 deployment. When the scaling of a constant stress layer, or of the law of the wall, is applied to the observations qualitative and quantitative variations in the velocity and turbulence profiles emerge between the B1 and B2 Lander deployments. We attribute these differences in local flow dynamics to the seafloor topography around the deployment sites.
To our knowledge, this paper describes the first deep turbulence observations with an ADCP. Shallow-water, estuarine, and coastal applications include Stacey et al. (1999a, b), Lu and Lueck (1999b), and Rippeth et al. (2003). These observations focus on the Reynolds stress but do not investigate the velocity spectrum. Numerous traditional turbulence measurements with current meters on bottom tripods have concentrated on the turbulence in the lowest few meters above the bottom, a zone excluded in our observations, and have focused on the inertial subrange of the turbulence spectrum. Conversely, our focus is on the energy-carrying scales in consequence of the limitations of ADCP measurements. The instrumentation and methods employed herein were also used in an analysis of turbulence in the wintertime, strongly atmospherically forced Adriatic Sea by PET. Further, McPhee (2004) analyzed energy-carrying turbulence scales under the pack ice. His approach of relating spectral maxima to turbulent fluxes cannot be applied here because our measurements do not sufficiently resolve the inertial subrange.
The primary purpose of the Bottom Lander observations was to measure the bottom stress. While the measurements were successful, they raise a number of problems. One important conclusion from our study is that caution is advised when “standard” drag coefficients are used to estimate the bottom shear stress. A standard value of 0.003 applied to the Red Sea Outflow results in both over- and underestimates in the bottom stress by factors of 1.5–3. Moreover, is the observed 3.9-m stress representative of the “effective” bottom stress felt by the plume flow? The observed strong vertical decay of the Reynolds stress with increasing height in the lowest 30–40 m above the seafloor suggests caution. In the complex topography of the Red Sea Outflow channels the plume flow may well have been subject to form drag, a process associated with hard-to-measure lateral pressure gradients and not reflected in the turbulent Reynolds stress. The vertical structure of flow and stress in B2 and the large angles between stress and velocity in B2 and B1 strongly suggest that the local momentum balance in the bottom plume layer was three-dimensional, involving spanwise, secondary flows. Such three-dimensionality would probably not be resolved even in fairly high-resolution numerical circulation models. We hypothesize that the complexity of the flow in the bottom layer of the descending Red Sea water was induced by the local seafloor topography. One could argue that the sites of the B2 and B1 deployments are “anomalous” to varying degrees. However, one may counter with the question what fraction of the seafloor is hydrodynamically “normal.”
The occurrence of significant velocity and stress variance at low frequencies is especially interesting to us. We remind the reader that our observations show an excess of low-frequency, and hence presumably large-scale, motions relative to the atmospheric mixed layer as a model. Our question is whether some of the low-frequency variability was related to internal waves. This question is important because the presence of internal waves would be an important component of the flow in the bottom layer as well as the interfacial layer above, with ramifications concerning turbulent mixing in BL and IL. When gravity currents flow over “bumpy” topography, such as in the vicinity of our Bottom Lander deployment sites, they likely excite internal lee waves, which are free to propagate upward carrying momentum and energy (e.g., Bell 1978). We note that internal waves can transport mean momentum in addition to, and quite differently than turbulence (e.g., Dillon et al. 1989). If such waves have zero or small horizontal phase speed relative to ground, they likely encounter critical layers higher up in the water column. Waves would thus add shear variance to the interfacial plume layer and contribute to shear instability. Future plume observations might consider this scenario.
In further discussion of internal waves, we reiterate that significant stable stratification existed within the bottom layer at times, and, of course, always in the interfacial layer. Hence the BL was able to support vertically propagating internal waves much of time. It also has to be kept in mind that horizontal velocity signals from sufficiently large-scale internal waves will extend through even an unstratified bottom layer. We indeed observed one event during B1 that we interpret as the occurrence of upward-propagating, bottom-generated internal waves. Figure 16 depicts contours of bandpassed υ-velocity data generated by subtracting a 1-h running mean from the recorded 50-s velocity averages. For a short time during decelerating plume flow, a coherent train of velocity variations alternating in sign and with downward phase propagation appeared in the along-channel velocity component. Applying the dispersion characteristics of free, linear internal waves in a back-of-the-envelope manner, we assume an effective N 2 = 2 × 10−5 s−2 and an effective advective horizontal flow speed of 0.3 m s−2. From Fig. 16 we read off a period of 30 min and a vertical phase speed of −35 m/30 min. One solution of the dispersion relationship consistent with these numbers is a wave arrested relative to ground by the southeastward plume flow but propagating vertically. Its horizontal wavelength is 580 m and its vertical wavelength 340 m, while the period of 30 min corresponds to its intrinsic frequency.
This paper demonstrates that bottom-mounted ADCPs can be a powerful tool for examining the turbulent flow structure of descending gravity currents and measuring their near-bottom stress. During reasonable sea states, our Bottom Lander is simple to deploy even at great depth. Unless more complicated, gimbaled platforms guaranteeing minimal tilt are used, a fifth center beam is essential. With respect to future investigations, we see a need to extend the vertical range of the velocity measurements into the interfacial layer such that the presence of internal waves can be detected. We also see a need for bottom-mounted profiling CTD measurements to complement bottom-mounted ADCPs. Unfortunately, autonomously profiling a CTD in strong currents remains a significant engineering challenge.
The Red Sea Outflow Experiment (REDSOX) was funded by the National Science Foundation under Contracts OCE-9819506, OCE-9818464, and OCE-0351116. Additional support was provided to the “Climate Process Team Gravity Currents” under OCE-0336799. The Bottom Lander and the lowered CTD–ADCP package used during REDSOX were acquired with funds from the ONR/DURIP program under N00014-98-1-0328. We are grateful for the good work of the Masters and crews of the R/V Knorr and the R/V Maurice Ewing. The REDSOX PIs also include Amy Bower and Dave Fratantoni. Numerous members of the science parties on the RSX1 and RSX2 cruises greatly contributed to the work reported herein, especially RSMAS technicians, Mark Graham and Robert Jones. The multibeam echosounder data from the Gulf of Aden were worked up by Stephen Swift. The soundings from the R/V L’Atalante were generously provided by Philippe Huchon. We benefited from discussions with Rolf Lueck and Youyu Lu, comments by two anonymous reviewers, and suggestions by RD Instruments field engineers. Clearance to work in the territorial waters of Djibouti, Eritrea, and Yemen is gratefully acknowledged.
This section contains technical notes on detailed aspects of instrumentation and methods. It discusses the effects of beam spreading, the instrumental noise floor, and detection limits in stress and shear production.
During RSX1 (RSX2) the Bottom Lander ADCP recorded 100-ping (50 ping) ensembles in beam coordinates every 50 s (25 s) in the deployments listed in Table 1. The ensembles contained not only velocities, echo amplitudes, and correlation magnitudes, but also sums of velocities, sums of squared velocities, and number of pings, so that velocity variances and the Reynolds stress can be computed as discussed above. We deployed the Bottom Lander by wire, lowering it down to a few meters above the seafloor, firing an acoustic release, and letting the Lander fall the rest of the way. While this method is simple, efficient and robust, it affords little control over the final orientation of the instrument, which is not gimbal mounted. Thus the ADCP had some tilt during all deployments as listed in Table 1. A firmware error required discarding some of the velocity variance data above the first bin. The corresponding data loss was minor in RSX1 but severe in RSX2. Velocity variance data from the last third of B1 and those from B3 are unusable. During deployments B1 and B4 the ADCP compass became frozen upon landing of the Lander on the bottom. In these cases, the instrument heading was reconstructed with the aid of LADCP velocity profiles taken at the Lander deployment sites.
Since we are analyzing motions of small scales and small velocity magnitude, we have to properly account for the spreading of the acoustic beams in the ADCP and for its instrumental noise. With 30° beam angles, as in our ADCP, their horizontal spread L is equal to the distance from the transducers. Hence, L is ∼40 m at the edge of the ADCP range and of about the same magnitude as the bottom layer thickness. Owing to the beam spreading, ADCPs measure the true velocity vector only in an asymptotic sense for scales much larger than the spread of the beams, whereas turbulent eddies can be on the order of L or smaller. Following Lu and Lueck (1999a), the parameter relevant for assessing the effects of beam spreading is M = 𝒯V/L, where V is the mean speed and 𝒯 is the averaging interval for computing velocities. Statistical tests of the horizontal homogeneity of the velocity field and of the error velocity lead to a requirement of M ≳ 55 for effects of the beam spreading to become statistically insignificant.
We examine the effect of beam spreading by comparing two measures of the vertical velocity: the center beam velocity corrected for pitch and roll angles of the ADCP (w5c) and the “regular” vertical velocity (w) computed from the four regular beams with standard algorithms (Lu and Lueck 1999a). The directly measured, uncorrected center beam velocity is w5. Figure A1 depicts frequency power spectra (Aw) of w and w5c as well as phase (Ψ) and squared coherence (R2) of the two variables at heights from 3.9 to 33.9 m. The regular and center beam vertical velocities are increasingly less correlated with increasing σ and increasing height. However, the phase spectra stay close to zero at frequencies up to about 20 cph (k ≲ 0.01 cpm) at all heights. The phase is especially important to us because it critically enters covariance terms such as . At h = 24 m, the regular w is attenuated at cyclic frequencies (σ) above 3 cph. This corresponds to cyclic wavenumbers (k) of 0.002 cycles per meter (cpm) under the assumption of advection of a frozen turbulence field with the observed mean flow. Hence, the vertical velocity variance is underestimated in the normal ADCP calculation at wavelengths smaller than about 500 m. While at first sight this value might seem large, it is consistent with the requirement of M ≳ 55 of Lu and Lueck (1999a). With mean speeds of 0.29, 0.5, and 0.53 m s−1, respectively, for the three heights of 4, 24, and 34 m in Fig. A1, M = 55 corresponds to frequencies of 6, 1.4, and 1.06 cph.
In four-beam ADCP systems the solution of (u, υ, w) as function of the beam velocities (u1, u2, u3, u4) is overdetermined. The “standard” transformation matrix [see, e.g., Lu and Lueck (1999a), their Eq. (2)] yields true velocity data only in an asymptotic sense because the beam velocity data are not horizontally homogeneous across the beam array owing to turbulence and instrumental noise. The properties of the standard transformation matrix are such that uncorrelated beam velocity data of variance magnitude 1 are transformed into horizontal variances of magnitude 2 and vertical variances of magnitude 1/3. Hence horizontal signals are exaggerated and vertical signals are damped. Uncorrelated beam data of equal variance represent the limit of isotropic turbulence with coherence scales much smaller than the beam separation. Horizontal boosting and vertical damping assume varying degrees, smaller than indicated above, in more general cases with some degree of coherence across the beam spread. Figure A1 illustrates a case of damping in the vertical velocity variance from standard ADCP processing algorithms. Through a comparison with power spectra of the horizontal flow based on the variance method (5), we can also demonstrate a positive bias in the horizontal variance from standard ADCP algorithms (not shown).
Our Bottom Lander had varying degrees of tilt as indicated in Table 1. The tilt has little effect on the low-frequency horizontal velocity but significant effect on the low-frequency w. Pitch and roll angles are accurate to ±1° according to the manufacturer, a figure consistent with the experience of Lu and Lueck (1999a). This angle of 1° has to be compared with an average slope of the NC and SC of ⅓° (Peters et al. 2005) and a typical slope of the canyon bottom at the B2 site of 1.5° as seen in Fig. 2b. A plume speed of 0.5 m s−1 down a 1° slope results in w ≈ −0.01 m s−1. Hence, we are unable to measure the mean vertical velocity component of the plume flow. As we expect a mean downward flow in the outflow channels, we cannot follow the approach of Lu and Lueck (1999a) who tried to minimize errors in pitch and roll by minimizing w. However, the vertical and temporal variations of w shown in Fig. 8 are well resolved and significant.
Measurements of the Reynolds stress are sensitive to the instrument tilt. We thus reconsider the unobserved and hence neglected second of the two pitch and roll correction terms in (3), the term related to the horizontal flow correlation . This term can be modeled as Ruυ(u′212*, where Ruυ is a correlation coefficient. In a sensitivity test we assume that Ruυ is about as large as the largest correlation observed between w′ and u′ or υ′, which is summarized as Rτ above and shown in Figs. 6c and 7c. We assume Ruυ = 0.4 = const and carry the modeled term through the actual ADCP processing algorithms. We find that τ varies by about ±20% depending on the signs of the horizontal correlation terms in the x and y Reynolds stress equations. The neglect of the correction term with in (3) thus seems to have only modest effects.
Velocity data shown in Fig. 5 and spectra depicted in Fig. A1 are only minimally affected by instrumental noise. We determined the noise level from data taken with the Bottom Lander in a separate deployments in the Adriatic Sea in 2003 (PET), in which every ping was recorded at rates of 0.67–0.8 Hz, such that a flat noise floor emerges at the highest frequencies well beyond the Nyquist frequencies of the REDSOX data. The spectral noise velocity variance was determined to be 5–7 (×10−7 m2 s−2 cph−1), slightly larger in w5 than in the u1–u4 beam velocities owing to a slightly shorter acoustic bin of w5. The noise level appears to increase weakly with increasing distance from the transducers, an effect ignored herein. The total noise variance corresponding to the noise floor given above, assuming a white noise spectrum, is 8 ± 1 × 10−4 m2 s−2 in w5 and 7 ± 1 × 10−4 m2 s−2 in u1 to u4. These values are consistent with those of Lu and Lueck (1999b) and a velocity standard deviation of 2.7 cm s−1 stated by RD Instruments for our setup. Our velocity noise variance is much lower than a value of 3.4 × 10−3 m2 s−2 stated by Stacey et al. (1999a) for a 1200-kHz broadband ADCP with 0.25-m bins.
The drag-law computations discussed above can aid finding the lower limit of detectable Reynolds stress. During the first half of B4 record the plume flow was weak with velocities of at most 0.1 m s−2 (not shown). The drag law indicates a near-bottom stress below 0.01 Pa for this period, while the computed Reynolds stress varies between 0.025 and 0.06 Pa. Thus it appears that, for our ADCP setup, the threshold of detectable stress is on the order of 0.05 Pa. This value is consistent with the uncertainty analysis of Lu and Lueck (1999b). We adapt their 95% confidence bound for true zero stress [their Eqs. (12) and (15)], simplified as Δ95 = 0.4τ3/4. This yields a detection threshold τ = Δ95 of τ = 0.025 Pa.
Figure 12 depicts contours of half-overlapping half-hour estimates of the stress; there is no additional smoothing. The time–depth patterns of τx and τy are coherent and well contourable. Their coherence lends credence to data and methods, as does the variability of P. One of the fundamental aspects of turbulence is that it is dissipative. An approximate production–dissipation balance should hold in the weakly stratified bottom layer of outflow plumes owing to a negligible buoyancy flux. Given the necessity of the shear production to balance dissipation, P has to be positive under most circumstance after appropriate averaging. Most of our computed P are indeed positive except in regions of weak turbulence. Areas of P < 0 occurred, especially during B1 (Fig. 13b), when the flow was weak. Finding most P > 0 supports the validity of our measurements of stress and shear production. Distributions of P from B1 (not shown) indicate a detection threshold with respect to P of a few times 10−7 W kg−1.
Corresponding author address: H. Peters, Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149. Email: email@example.com