Observations of the three-dimensional structure and evolution of a thermohaline intrusion in a wide, deep fjord are presented. In an intensive two-ship study centered on an acoustically tracked neutrally buoyant float, a cold, fresh, low-oxygen tongue of water moving southward at about 0.03 m s−1 out of Possession Sound, Washington, was observed. The feature lay across isopycnal surfaces in a 50–80-m depth range. The large-scale structures of temperature, salinity, velocity, dissolved oxygen, and chlorophyll were mapped with a towed, depth-cycling instrument from one ship while the other ship measured turbulence close to the float with loosely tethered microstructure profilers. Observations from both ships were expressed in a float-relative (Lagrangian) reference frame, minimizing advection effects. A float deployed at the tongue’s leading edge warmed 0.2°C in 24 h, which the authors argue resulted from mixing. Diapycnal heat fluxes inferred from microstructure were 1–2 orders of magnitude too small to account for the observed warming. Instead, lateral stirring along isopycnals appears responsible, implying isopycnal diffusivities O(1 m2 s−1). These are consistent with estimates, using measured temperature microstructure, from an extension of the Osborn–Cox model that allows for lateral gradients. Horizontal structures with scales O(100 m) are seen in time series and spatial maps, supporting this interpretation.
The dynamics, structure, and time evolution of the ocean at horizontal scales smaller than 10 km are still rather poorly understood. It will be some years before large-scale models can hope to resolve these scales. Therefore, understanding the relevant physics is vital to developing better parameterizations for their improved skill.
One example of such a phenomenon is thermohaline intrusions, or water of different potential temperature (θ) and salinity (S) properties injected laterally into a vertical profile. These are ubiquitous in the ocean and occur by a variety of processes, but often occur most strongly near ocean fronts. Though often assumed to be formed by double-diffusive flux divergences, a number of processes can generate the lateral pressure gradients needed for interleaving, including differential mixing of θ/S (Hebert 1999), symmetric instability (Banks and Richards 2002), or gravity–current dynamics. They are of interest because of their potential to drive strong stirring along isopycnals, and may even be the cause of thermohaline staircases such as those occurring near Bermuda (Merryfield 2000). In addition, they can lie across isopycnals (Gregg and McKenzie 1979), potentially leading to large diapycnal fluxes as well. Yet, in spite of their importance, neither the morphology, dominant generation mechanisms, nor fluxes of intrusions are well understood. Recent reviews of the theory and observations of intrusions may be found in Ruddick and Kerr (2003) and Ruddick and Richards (2003).
Intrusions are one element of the more general problem of understanding lateral diffusion on 0.1–10-km scales. Though an empirical scale relation by Okubo (1971) appears to accurately predict the magnitude of isopycnal diffusivity as a function of scale, the mechanisms that lead to the stirring and subsequent mixing are still unclear. Yet, none of Okubo’s data were in stratified environments. In one of the few stratified field studies, Sundermeyer and Ledwell (2001) used dye to measure horizontal diffusivities of order 1 m2 s−1, in line with Okubo’s prediction for 1-km scales (the same as in the present study). They argued that shear dispersion (Taylor 1921; Saffman 1962; Young et al. 1982), an interaction between vertical mixing and shear that leads to enhanced horizontal spreading, could not produce the observed spreading. Instead, they suggested that mixing by collapsing turbulent patches was responsible, but their data were insufficient to say more.
The primary impediments to progress in both problems have been technological. Gregg (1980) mapped a 3D intrusion and presented schematics of its structure, but noisy Loran C navigation limited resolution, and evolution was not measured. With P-code or differential GPS navigation and acoustic tracking, it is now possible to conduct measurements with absolute position uncertainties of tens of meters. Still, it is difficult to cover the needed spatial scales in the time needed to avoid temporal aliasing. For example, one can tow around a 10-km butterfly pattern at 5 kt in 5 h. In that time, water parcels may have changed their properties diabatically via mixing. Far more serious, advection will have scrambled the large-scale structure during the time of the survey. The next pattern will likely be of completely different water.
In May 2002, we attempted to surmount these difficulties by centering a two-ship study on an acoustically tracked Lagrangian float (D’Asaro et al. 1996; D’Asaro 2003). As a step toward an eventual open-ocean study, we conducted our measurements in Puget Sound, Washington, where the cost and difficulty of a two-ship study are lower. We focused our study on a cold, fresh, low-oxygen tongue that we observed emanating at about 0.03 m s−1 from the northwest, indicated schematically in Fig. 1; a preview of the 3D observational map is shown in Fig. 2. One ship dedicated itself to mapping the three-dimensional structure with a towed, cycling CTD/ADCP. Simultaneously, the other ship measured microstructure time series close to the float. Our goals were
to measure the 3D structure of the intrusion in an isopycnal-following, float-relative frame,
to measure and understand the time evolution in the float-following reference frame, and
to use the spatial maps, the Lagrangian temperature record, and the microstructure data to determine the diapycnal and isopycnal heat fluxes at the intrusion’s edges.
Goals 1 and 2 are relatively straightforward, but we wish to be specific regarding what we mean by goal 3. As shown by McDougall (1984) and appendix A, the potential temperature θ measured on an error-free isopycnal float can change only by 1) diapycnal mixing by turbulence, 2) isopycnal mixing, 3) double-diffusive flux convergences, or 4) effects related to nonlinearities in the equation of state. In the case of constant stratification, the temperature equation takes a particularly simple and intuitive form:
where ξ and ζ are the isopycnal and diapycnal coordinates, FDD is the flux due to double diffusion, and N.L. represents effects of equation-of-state nonlinearities such as cabelling and thermobaricity. The diapycnal diffusivity is traditionally defined by 〈w′θ′〉 = −Kρ(∂θ/∂ζ), and the isopycnal diffusivity by 〈u′θ′〉 = −Ki(∂θ/∂ξ), where angle brackets represent averaging over some appropriate temporal interval.
As shown later, the latter two terms can be neglected for our case. Then, (1) states that the temperature of a water parcel changes because of the divergence of diapycnal and isopycnal temperature fluxes 〈w′θ′〉, and 〈u′θ′〉, which are parameterized by eddy coefficients multiplied by mean gradients. We directly measure only the diapycnal diffusivity, but wish to take advantage of our spatial information to determine the relative strengths of the diapycnal and isopycnal terms on measured temperature evolution in our data.
In section 4 we attempt to use this equation to relate dθ/dt measured on the float to diapycnal and isopycnal fluxes estimated from microstructure measurements and our observations of gradient and curvature. We find that 1) a float placed near the intrusion’s edge (Fig. 1) warmed significantly in 24 h, while another placed inside the intrusion’s core did not; 2) microstructure measurements indicate that Kρ is elevated at the top and bottom edges, but is much too small to account for the warming at the edge; and 3) isopycnal diffusivity estimated from (1) and a 3D extension of the Osborn–Cox method both yield O(1 m2 s−1), that expected from Okubo’s empirical relation for our 1-km scales.
Based on these observations, we conclude that isopycnal mixing appeared responsible for warming the float near the intrusion’s edge. We observed marginally resolved temperature structures with horizontal scales of hundreds of meters, that appear able to produce the 〈u′θ′〉 needed for horizontal fluxes. We were, however, unable to determine the mechanism responsible for them.
a. Study region
Our measurements were conducted in the “Triple junction” region of the Main Basin of Puget Sound, a wide, deep fjord inland of Juan De Fuca Strait (Figs. 3, 4a). Tidal flows in and out of Possession Sound merge with Main Basin flows in the region south of Whidbey Bank, leading to complicated and strongly horizontally strained flow (Ebbesmeyer et al. 2001).
In contrast to Puget Sound’s constrictions such as Admiralty Inlet, where barotropic flows can exceed 1.5 m s−1, the flows in the basins are weaker (≈0.2 m s−1) and more baroclinic. In fact, shear-wavenumber spectra (not shown) resemble the Garrett–Munk spectrum (Garrett and Munk 1975) typical of open-ocean conditions. During our study period, barotropic and baroclinic components of velocity were comparable (Figs. 4b–d), increasing noticeably toward the end of our experiment as spring tide was approached.
Stratification, below a fresh river-runoff cap above 20–30 m, is also comparable to ocean values (Figs. 4e, 5d). While stratification is usually dominated by salinity, at times strong temperature inversions occur. These appear to occur, at least at times, as dense water flowing over Admiralty Inlet Sill detrains at middepth (Mickett et al. 2004), leading to inversions in both temperature and salinity. The generation mechanism of the present temperature inversion (Fig. 5a) is unknown, but it emanates from Possession Sound.
Data were collected from several instrument systems, deployed from two ships in a coordinated, 4-day experiment during May 2002. Each ship was equipped with a 150-KHz broadband ADCP (RD Instruments, Inc.). Lagrangian floats (D’Asaro et al. 1996) were deployed from R/V Barnes. Via acoustic tracking, the Barnes was maintained within 500 m of the float at nearly all times, while conducting microstructure measurements for 12 h day−1. Tracking and velocity measurements continued during the night.
Simultaneously, R/V Miller conducted larger-scale surveys with its ADCP and our depth-cycling towed body, SWIMS-II. These patterns were centered on the instantaneous float position as monitored by the Barnes. They were continually adjusted based on real-time data, which were monitored and shared between the two ships via a wireless network link.
SWIMS-II is a 300-kg depth-cycling towed body equipped with two Sea-Bird CTDs, up- and down-looking 300-KHz ADCPs, and Sea-Bird dissolved oxygen (DO) and Seapoint chlorophyll sensors. Unlike SeaSoar, it is winched in and out, enabling tight sawtooths. Towing at 4–6 kt, profiles from 5- to 180-m depth were taken every O(200 m). Only the CTD and DO data are discussed here.
Microstructure was measured with the Modular Microstructure Profiler (MMP). This is a loosely tethered instrument equipped with a Sea-Bird CTD, dissolved oxygen sensor, a fast-tip FP07 thermistor, and two airfoil probes for measuring velocity microstructure. Profiles were taken from the surface to 10 m off the bottom every 15–20 min for about 12 h day−1, for a total of 180 profiles (Figs. 3, 4).
where Γ = 0.2 is the mixing efficiency.
A “deep Lagrangian float” (D’Asaro et al. 1996; D’Asaro 2003) was deployed to track the motion of the water. The float was acoustically tracked and thus enabled 1) minimization of advective smearing by conducting our surveys in the moving reference frame of the large-scale flow, and 2) measurement of the Lagrangian evolution of temperature at different points within the intrusion.
After conducting the initial survey and identifying the intrusive feature, the float was ballasted for the potential density surface σ = 22.93 kg m−3, near the depth of the temperature minimum. Details of float operation, performance, and errors are given in appendix B.
Seven deployments were conducted during the 4 days (Fig. 4, black lines). Of these, the first several were short deployments to verify correct ballasting. We will focus on the final two (Figs. 3 and 6), which were of 24-h duration. One was released in the high-horizontal-gradient region near the intrusion’s leading edge, and the other, immediately after recovering the first, was placed farther back inside the more homogenous core.
A TrackPoint II acoustic tracking system was used to monitor the float’s position and to maintain R/V Barnes close to it. Range estimates are accurate to <1 m. However, angle uncertainties reduce the accuracy of ship-relative position fixes to O(10 m). Uncertainty in ship heading further increases the uncertainty of absolute position to ±≈50 m, as evident in Fig. 6 (red lines). This error dominates the uncertainty in differential GPS measurements [O(10 m)]. To minimize it, fixes during ship turns were eliminated from the float time series.
Each deployment traces out two M2 cycles, as seen in plan view (Figs. 3) and in time series (Fig. 6). In the second release, which is closer to spring tide (Fig. 4), the float traces O(30 m) tidal depth excursions, which are mostly absent in the first. In both releases, higher-frequency excursions of O(10–20 m) are common, which will be examined in more detail later.
To assess the float’s lateral water tracking, ADCP velocities from R/V Barnes were interpolated onto the instantaneous depth of the float, time integrated, and compared with the float trajectories (Figs. 6a,b). In the second deployment, along-channel errors are <100 m, consistent with our error estimate of <50 m (appendix B). However, differences are greater (several hundred meters) in the cross-channel direction, especially in the second deployment. These errors accumulate when the ship is not directly over the float, and horizontal shears cause the ADCP to measure a different velocity. Horizontal shears from SWIMS-II transects are O(0.05 m s−1 km−1), which would lead to errors of this magnitude. The larger errors in the cross-channel direction and closer to spring tide, which show more irregular flow and stronger shear, support this interpretation. Thus the ADCP/float differences indicate ship/float offsets in the ambient horizontal shear rather than imperfect float performance. They do, however, preclude confirmation of our predicted upperbound lateral float drift (<50 m), and limit the resolution of our Lagrangian-frame maps to several hundred meters (section 3).
In the vertical, estimates from MMP of the depth and temperature of the target isopycnal surface are overplotted in Figs. 6c and 6d. During some rapid vertical excursions, the MMP and float estimates differ, which we again attribute to horizontal offsets between the ship and the float (evidence for structures with horizontal scales of a few hundred meters are presented in the next section). At all other times, the float remains within several meters of the target. Note that off-isopycnal excursions are expected for a Lagrangian water parcel in the presence of turbulence (appendix B), and so again are not evidence for float errors. After excursions, the float’s ballasting guarantees that it always return to the target after a time O(N−1), which is evidenced by the continued agreement between the float depth and the target isopycnal depth. We conclude that measurement errors owing to horizontal offsets are too large to confirm our upper-bound estimates for lateral and vertical float uncertainty, but also find no evidence to invalidate them.
3. The intrusion: Structure
A minimum in temperature and dissolved oxygen dominates typical profiles during our experiment (Fig. 5). [Oxygen saturation (not shown) is about 70%. Furthermore, saturation decreases with increasing temperature. Therefore, the oxygen minimum is not simply a saturation effect.] The cruise-mean detided velocity (Fig. 5c) indicates that the layer is flowing toward the southwest, out of Possession Sound, sandwiched between northeastward flow above and below. (Velocity will be discussed in more detail below.) We will refer to this feature as “the intrusion.”
The colder, fresher water in the intrusion core overlies warmer, saltier water below, giving rise to the possibility of double-diffusive convection. The density ratio,
where α ≡ − (1/ρ)(∂ρ/∂T) and β ≡ (1/ρ)(∂ρ/∂S), governs the potential for double-diffusive effects, with negative values implying double-diffusive stability, values between 0 and 1 indicating double-diffusive convection and values above two allowing salt fingering. Owing to the strong salinity stratification, salinity does not invert as is often the case in intrusions. Consequently, there is no regime favorable for salt fingering on the top of the intrusion (Fig. 5b). Observed values of Rρ ≈ 0.5 on the intrusion’s underside lie in the diffusive convection regime.
b. Lagrangian reference frame
We wish to combine all of our measurements over the four days to 1) present a three-dimensional view of the intrusion and 2) detect its evolution. However, the vertical heaving of isopycnals by internal waves [η(z, t) = O(20–30 m)], and horizontal tidal motions [O(1–3 km)], would cause smearing in simple averaged quantities. We wish to take advantage of the water-following float to minimize the effects of horizontal and vertical advection in our interpretations. Formally, we seek a transformation from the Eulerian reference frame, in which the measurements are conducted, to a Lagrangian reference frame:
where the vector displacement of each Lagrangian tag,
is given by the integral of its velocity.
We perform the transformation in two steps, first addressing vertical motions and then horizontal. First, a set of isopycnal surfaces spaced 1 m apart is generated, based on the cruise-mean density profile. Then, data are interpolated onto these density surfaces and plotted versus the mean depth of each isopycnal, ζ. This so-called semi-Lagrangian transformation (Sherman 1989) minimizes smearing by vertical advection accompanying vertical isopycnal displacements η. To be precise, the physical vertical coordinate z = ζ + η, where all quantities are positive upward.
The horizontal motion of the float gives the horizontal portion of the transformation. Specifically, the transformed coordinates equal the measurement location minus the vector position of the float relative to a fixed reference, taken here to be the first 24-h release location. When the float is not in the water, we use the integral of the ADCP velocity at the depth of the float as a proxy. As seen in Fig. 6, this introduces errors of up to several hundred meters.
In the absence of measurement errors, (4) is exact at the float location. However, the presence of vertical and/or horizontal shear causes distortion. Horizontal shear causes the ship to measure the wrong velocity except when directly over the float. Consequently, errors grow with distance from the float. Vertical shear causes time-dependent distortion as features are advected past the float. In theory, the latter could be partly accounted for by using the depth-dependent ADCP velocities to do the transformation, but we have not attempted this. Instead, we prefer to present quantities in the depth-independent frame, but always simultaneously present mean velocity over the time of measurements to give an indication of shear distortion.
Despite these shortcomings, we will refer to the fully back-advected frame as Lagrangian, to distinguish from the semi-Lagrangian frame, which only accounted for vertical advection. As will be seen, the Lagrangian maps are sharpened somewhat relative to the semi-Lagrangian counterparts, but are qualitatively similar. (The semi-Lagrangian transformation, by contrast, is essential to forming averages; isopycnal heaving is sufficient to completely smear out features averaged in depth coordinates.)
An instantaneous section of θ, S, and dissolved oxygen taken by towing SWIMS-II close to the blue track in Fig. 3 is shown in Figs. 7a–c. Profiles were taken every 200 m. A cold, low-oxygen tongue is seen emanating from Possession Sound. Owing to the strong salinity stratification, salinity does not invert, as discussed above. This section was taken shortly after the release of the float for the first 24-h deployment. The section passed within 100 m of the float; in fact, the line had to be diverted slightly to avoid it and the Barnes. The float lies slightly below the temperature minimum, near the region of maximum horizontal gradient. Isopycnal surfaces (gray) slope upward to the northeast at the float depth, and downward below.
The small-scale structure near the intrusion nose is better seen by plotting temperature profiles along the same line (Fig. 8a). Temperature decreases moving toward the northeast (bluer profiles). The location of the float is indicated. Small interleaving structures are visible in the high-horizontal-gradient region. Upon transformation to isopycnal coordinates (Fig. 8b), it is clear that the structures are quasi-coherent for several hundred meters along isopycnal surfaces—for example, the feature near ζ = 78 m.
Returning to Fig. 7, we next present corresponding slices (green line, Fig. 3) through our 4-day mean fields, constructed by averaging all MMP and SWIMS-II measurements on density surfaces, and binning horizontally in both semi-Lagrangian (vertically advected but horizontally unadvected, middle panels) and fully Lagrangian coordinates (bottom). To emphasize horizontal rather than vertical salinity differences, the southernmost salinity profile is subtracted from each, unlike the top panel.
In the semi-Lagrangian frame, the relative absence of vertical motion by internal waves is evident. The vertical position of the float (black and gray lines) now indicates the isopycnal for which it is ballasted. However, the horizontal motion of the float, dominated by tidal motions, indicates the degree to which these fields are advectively smeared.
In the Lagrangian frame, the float trajectories are condensed, indicating the placement of the first float near the edge of the intrusion (as seen in the SWIMS-II cuts, top), and the second float farther back inside the intrusion core. Scatter in the Lagrangian float position (black and gray) indicates the overall accuracy of the transformation. Instead of single points, as they would be in the absence of error, integration of velocity errors leads to error in xR and thus their Lagrangian position. Regardless, sharpening of the salinity anomaly field in the Lagrangian frame is evident, as is resolution of a temperature saddle near latitude 47.84°. Elsewhere, the fields are qualitatively similar and demonstrate the nose of the intrusion as it emanates from Possession Sound.
Time-mean velocities measured from the Barnes ADCP (Figs. 5c and 7, arrows) are computed by first rotating by 45°, to highlight motion into (Uin, dark line) and across (Ucross, light line) Possession Sound, and time averaging over an integral number of M2 periods. Error bars, shaded, indicate the combined effects of 1) random measurement errors of ≈0.13 cm s−1, computed assuming that each of the 223 20-min velocity estimates composing the mean is independent; and 2) a deterministic error resulting from the short averaging period. Though an integral number of M2 periods are spanned, an additional error of ≈0.13 cm s−1 results from the incomplete K1 contribution. The relative phasing of M2 and K1 determines the sign. This error source is overwhelmingly barotropic and thus affects the mean and not the profile shape.
Unfortunately, the requirement of averaging out the tides precludes transforming velocity into semi-Lagrangian coordinates, since we only have density information for part of the time. (Since the Barnes is always near the float, the horizontal transformation would have negligible effect.) Nevertheless, it is apparent in the depth-coordinate averages that water at the approximate depth of the feature is moving outward from Possession Sound. This is more evident in Fig. 7, where Ucross vectors are overlain. Water above the intrusion is moving into Possession, in response to strong southerly winds persisting for several days before the cruise (not shown). [This is the reverse of the normal surface flow, which is the southward fresh river output from the Skagit River (Ebbesmeyer et al. 2001).] Water below is also moving into Possession in the usual sense for the deep exchange flow (Ebbesmeyer et al. 2001). The intrusion is emanating from the northwest at about 0.03 m s−1, sandwiched between two northbound layers.
e. Plan-view maps
The intrusion’s lateral structure is examined next (Fig. 9), by constructing plan-view maps of T, S, and dissolved oxygen averaged over the density range spanned by the intrusion (Fig. 7; ζ = 50–86 m). The intrusion is evident in all maps as a tongue of cold, fresh, low-oxygen water extending southwestward from Possession Sound. Top panels represent the semi-Lagrangian measurements, while the lower panels represent the fully Lagrangian frame. Aside from a slight sharpening evident in salinity, few qualitative differences are apparent between the two reference frames. The collapsing of the float trajectories (white and black paths) onto much smaller loci in the latter reference frame is apparent. Again, the first float was located near the edge of the intrusion, and the second farther within the core.
f. Cross-isopycnal structure
The temperature minimum weakens and occurs at successively shallower isopycnals toward the southwest along the instantaneous cut (Fig. 8). In addition to the instantaneous fields, this shoaling is evident in analogous plots made along the same cut through the mean fields as before (Fig. 10a). It is also evident in salinity and DO anomaly fields (Figs. 10b,c). One explanation is that, as the intrusion formed, it experienced a buoyancy flux convergence that made it lighter as it extended farther from Possession Sound. In double-diffusively driven intrusions, which are also often observed to slope across isopycnals (Gregg and McKenzie 1979), this may result because salt fingering transports heat and salt differentially. Here, the possibility of double-diffusive convection at the bottom but not at the top, or stronger mixing at the top or bottom, could have led to the imbalance. Unfortunately, since we only observe its final state, these processes occurred before our experiment and cannot be verified. An alternate explanation results from the observation (McDougall and Giles 1987) that property maxima spanning isopycnal surfaces do not alone imply divergent buoyancy fluxes: depth-dependent lateral fluxes can also be responsible.
4. Lagrangian evolution
We now wish to use the float and microstructure measurements to monitor the evolution of the intrusion at its edge (first float release) and in its core (second release). Lagrangian floats are well suited for studying flow evolution, since advective changes, which normally dominate, are minimized. We overcome their complete lack of spatial information by repeated microstructure profiling close by. The aim is to relate observed temperature changes to diapycnal and isopycnal mixing. Specifically, if a float follows the water perfectly, then its temperature evolution is given by (1). We will first present the raw time series; then, in the next subsection we will evaluate these terms for each deployment.
b. Lagrangian time series
Time series of temperature and pressure from Lagrangian floats (Fig. 6) have been presented before (D’Asaro et al. 1996; Barth et al. 2004) and will only be discussed briefly here. The float depth (Figs. 6c,g) is variable on a wide range of time scales; notably the semidiurnal tide. In addition, rapid vertical excursions are present, indicating vertical advection by internal waves and turbulence.
Since adverse effects are negligible (appendix B), temperature evolution (Figs. 6d,h) is only impacted by mixing [(1)]. The occurrence of both temperature increases and decreases, which may at first be surprising, are understandable by thinking of the float as following a small water volume within a complex strain field. The temperature evolution at any given time depends on the temperature of the adjacent water parcels, which can be warmer or cooler than the float (an example of similar Lagrangian scalar evolution during a simulated mixing event is shown in D’Asaro et al. 2004). In the case of the first deployment near the intrusion edge, these conspire to produce a general warming trend (absent in the second) of about 0.2°C over 24 h.
We now examine in detail the time series of Lagrangian-frame MMP profiles for each deployment. Our goals are twofold: to provide further evidence that the float is tracking the water, and to document mixing and profile evolution at the edge and core of the intrusion. Particularly in the first deployment, advective distortion by mean shear is minimized, so the series are representative of the processes leading to mixing.
For each deployment, temperature profiles near the beginning (top) and end (bottom) are plotted versus depth (Figs. 11, 12). They are taken at variable temporal spacings (indicated at lower left in each panel), generally several per hour. Float temperature and pressure over a 5-min period bracketing the profile is plotted in gray. The target isopycnal, and those at mean depths 8 m above and below, are plotted with dotted lines and gray shading. Internal-wave displacements are apparent as these lines rise and fall; strain causes them to squeeze together and pull apart. It is clear that both displacement and strain increase with time during this period, as spring tide approaches. Much of the variability, however, appears to be at much higher than tidal frequencies.
At nearly all times in both deployments, the float lies close to the target, and is always sandwiched between the upper and lower isopycnals. Error is greatest when strain is high, indicating weak stratification and small restoring forces pinning the float to the target isopycnal (e.g., Fig. 12; 17.79 h). Still, the agreement between pressure/temperature pairs from the float and MMP is clear, indicating they are sampling the same water.
The progression from the beginning (green) to end (red) states of each deployment is seen. In the case of the first deployment, the deep temperature minimum erodes with time, contextualizing the time series trend observed in Fig. 6d. Unfortunately, a substantial amount of the decay occurs during the period in each deployment when MMP was not in use. Nevertheless, evolution can be observed. Interleaving structures appear and disappear, particularly near the beginning of the deployment. The specific feature observed in the instantaneous SWIMS-II spatial cut (Figs. 7a–c, 8) occurs in Fig. 11 at time 3.79 h. We speculate that these blobs of warm water from the southwest are stirred along isopycnals by the flow field on several hundred meters’ scale. Scalar fields are strained, increasing gradients and eventually leading to mixing.
As suggested by Fig. 15, evolution of temperature is much less in the second deployment, despite the stronger internal-wave displacements, strain, shear, and diapycnal mixing. As will be seen, this appears to be due to weaker isopycnal temperature gradients and/or curvature in the core of the intrusion.
Temperature during the first float deployment, at the edge of the intrusion, demonstrates a marked trend, increasing 0.2°C in 24 h (Fig. 6, lower left). The second does not (lower right). We wish to use (1), together with our direct microstructure measurements, to evaluate the hypothesis that differences in mixing are responsible for the different temperature histories at the two locations. Specifically, it appears that diapycnal mixing is a factor-of-10 too weak to warm the first float sufficiently. Instead, an isopycnal diffusivity of ≈1 m2 s−1 is sufficient. This value is in line with what is expected for the horizontal length scales (Okubo 1971), and consistent with an estimate made from our measurements of temperature microstructure by applying a three-dimensional generalization of the Osborn–Cox method (next section).
The warming during the first deployment is examined in more detail in Fig. 13. Float depth and temperature from Fig. 6 are replotted at lower right, showing the warming trend. Plotting mean Lagrangian temperature profiles from MMP at the beginning (green) and end (red) of the deployment (left), it is clear that the warming extends over a range of isopycnals bracketing the float. (The corresponding T–S plot, shown below, provides an alternate view.) The water surrounding the intrusion displays a clear temperature minimum of 8.4°C at the beginning of the deployment, in agreement with the float measurements (green vertical bar), slightly deeper than the minimum. By the end, the minimum has been eroded significantly, again in agreement with the float measurements (red vertical bar).
We show in appendix A that temperature at the float should evolve according to (1). However, differential advection also influences the temperature evolution of water above and below, potentially distorting the structures plotted in Fig. 13. To assess this, the mean velocity averaged over the two M2 periods spanning the deployment is plotted at right. The corresponding distance a particle would travel in 24 h, indicated at top, varies by <100 m from about 70 to 90 m. We therefore expect minimal shear distortion in the 20-m range bracketing the float. Above the temperature minimum (70 m), relative horizontal displacements O(200 m) may play a greater role.
Directly measured ε and Kρ over the beginning, end, and their mean (top-middle panels) show clear enhancement at the top and bottom edges of the intrusion. However, the measured diffusivity is 1–2 orders of magnitude weaker than that required (vertical lines) if only diapycnal mixing is responsible. These values are estimated next from (1).
It is straightforward to estimate the diapycnal diffusivity required to effect the observed warming in the first deployment by integrating (1) in time for Ki = 0. That is, we solve the diffusion equation with different values of Kρ to determine that which best matches the observations. In Fig. 14a the observed temperatures before/after deployment 1 are replotted (heavy black/gray), together with a Gaussian fit to the before profile (thin), and solutions to the diffusion equation for (constant) Kobsρ = 1.5 ± 0.3 × 10−4 m2 s−1 and KGaussρ = 80 × 10−4 m2 s−1 (Table 1). The former value, an upper bound on observed diffusivity (Fig. 13c), appears far too weak to effect the observed warming (right dotted line). (Given the weaker mixing in the intrusion core, using a nonconstant Kρ, not shown, only strengthens this result.) The latter value, some 40 times larger than observed, produces the observed warming (left dotted line). However, the vertical spreading associated with a diffusivity this strong was not observed. Instead, the net warming from 50 to 85 m appears more consistent with lateral processes.
Salt fingering is not possible, but fluxes due to double-diffusive convection on the lower edge could also warm the float. However, estimates of the heat flux by Huppert (1971) using the observed gradients yield values similarly too small. In addition, no step-like structures, signatures of this type of convection, are present. We conclude that double-diffusive fluxes are not responsible for the warming.
Since double-diffusive and nonlinear effects are negligible, divergent isopycnal flux is the only remaining candidate. To estimate the isopycnal diffusivity required, we use the analogous solution of (1) for Kρ = 0, constant Ki, and perform a half-Gaussian fit to the Lagrangian-frame mean temperature along the float’s isopycnal (Fig. 14b). In the horizontal case, data were not sufficient to determine before/after profiles, so we estimate Ki = 3.3 ± 2 m2 s−1 from the best-fit Gaussian horizontal structure and the observed rate of temperature change.
To investigate the effects of departures from Gaussian shape, the dia- and isopycnal diffusivities required to produce the observed warming were also computed more directly from the curvature of the mean fields. (The difference in the two estimates also provides a measure of the error in the calculations.) From (1), we have Kρ = TtT−1ζζ, Ki = TtT−1ξξ, for constant diffusivities. From the mean fields, we estimate Tζζ = (2.5 ± 1) × 10−3 °C m−2, and Tξξ = (5 ± 3) × 10−7 °C m−2, yielding values of Kρ = 2.5 ± 1 × 10−3 m2 s−1 and Ki = 3 ± 2 m2 s−1.
For comparison, Okubo’s relation for these scales (1 km) yields Ki ≈ 1 m2 s−1. These values, summarized in Table 1, indicate that the observed diapycnal diffusivity was too weak, but that the two methods of determining the required isopycnal diffusivity yielded similar values. In the next section we compare these isopycnal diffusivities with an observational estimate from a three-dimensional extension of the Osborn–Cox method, which also yields a comparable value.
By contrast, the second deployment (Fig. 15) did not display a trend over the 24 h, despite the stronger diapycnal mixing during that period. This is consistent with the observed lack of strong lateral gradient and curvature inside the intrusion core. In addition, shear is greater during the second deployment, potentially causing greater distortion of profiles by differential advection.
5. Osborn–Cox in a horizontal gradient
The Osborn–Cox relation (Osborn and Cox 1972) has long been used to estimate diapycnal diffusivity from temperature microstructure by assuming that temperature variance is produced by turbulent vertical fluxes and destroyed by diffusive smoothing. However, it has long been known that the method fails in intrusive and frontal regions because of the presence of lateral gradients (Gregg 1975; Joyce 1977; Alford and Pinkel 2000). Here we present a reformulation of Osborn–Cox allowing lateral gradients, assuming that the lateral fluxes may be parameterized in terms of an isopycnal diffusivity. The two conclusions are 1) Osborn–Cox grossly overestimates the true diffusivity when lateral gradients are present, and 2) the isopycnal diffusivity may be estimated observationally.
where ζ and ξ are the diapycnal and isopycnal coordinates, and χ ≡ 2κ is the rate of diffusive dissipation of temperature variance. The balance between the first and third terms gives the standard Osborn–Cox expression. Now, diffusive smoothing of temperature variance is still replaced by turbulent transport down mean gradients, but the gradients are not vertical as assumed by Osborn–Cox. [Winters and D’Asaro (1996) present an alternative and less restrictive derivation of the same result.]
To proceed further, we parameterize the transports with the eddy coefficients,
Substituting into (6) we finally get
which differs from the standard Osborn–Cox expression,
Thus, standard application of Osborn–Cox overestimates Kρ if there is an isopycnal temperature gradient, since χ is balanced by lateral as well as vertical convergence of θ′2. As shown next, the overestimation can be quite severe with modest temperature gradients and typical isopycnal diffusivity.
There is strong observational motivation for (7). That is, it is well established that overturns working against a mean stratification produce the fluctuations. Though the mechanisms that produce the horizontal stirring structures are not understood, dye studies indicate that their time-integrated effects (over many eddy periods) can indeed be represented by a diffusivity. If one accepts the parameterization (8) in terms of an isopycnal diffusivity, its calculation is straightforward:
This equation can be used, with independent measurements of Kρ (such as from airfoils), to estimate Ki, if the isopycnal temperature gradient is known.
We demonstrate these ideas in Fig. 16. Using standard Osborn–Cox [(11)], Kρ is estimated from measured χ (Fig. 16a) and temperature gradient (Fig. 16c) during the first (Figs. 16a–e) and second (Figs. 16f–j) deployments, and plotted in Fig. 16e, (thin) versus Kρ computed from shear probes using Osborn’s method [(2), thick]. In each case, the two methods yield similar values over portions of the water column, but substantial differences result between the two estimates when the horizontal gradient is large, as in the intrusion.
The isopycnal diffusivity could in theory be obtained by subtraction using (11), but as we have just seen, the overestimation appears to be so large that this would be impractical. Instead, we evaluated Kρ from (9) with χ = 0 and different values of Ki to determine the overestimation due to lateral fluxes. The horizontal gradient (Fig. 16d) is computed from the 3D fully Lagrangian fields through the cut indicated in Fig. 3. For Ki = (0.5–3) m2 s−1, the predicted overestimation agrees well with that observed, with Ki = 1 m2 s−1 (dashed line) giving the best agreement. These calculations show that 1) lateral gradients contaminated the Osborn–Cox estimates as predicted by (6) and 2) Ki ≈ 1 m2 s−1, agreeing well with Okubo’s value for these scales and the float temperature budget estimates (Table 1).
We have presented intensive observations of the structure and evolution of a 1-km-scale cold/fresh water mass. The combination of Lagrangian, microstructure, and towed techniques proved an effective combination, enabling a calculation of the temperature budget of a water parcel near the mass’s edge using (1). With towed surveys on the large scale, we mapped the water mass’s scalar and velocity fields, indicating that it 1) was flowing out of Possession Sound at 0.03 m s−1, 2) crossed isopycnals, and 3) displayed small-scale temperature horizontal structures at its edge, indicative of isopycnal stirring. Lagrangian float and microstructure measurements in the water mass’s reference frame indicated 1) divergent heat fluxes were sufficient at the intrusion’s edge to warm water parcels there 0.2°C in 24 h, 2) diapycnal fluxes were at least an order of magnitude too small to accomplish this, and 3) isopycnal diffusivities Ki ≈ 1 m2 s−1 inferred from the warming matched those expected from Okubo (1971) and estimated from the Osborn–Cox method when modified to include lateral fluxes.
The Osborn–Cox method has been used for decades and is already known to be suspect in intrusive regions, but we have pointed out that given the size of typical isopycnal diffusivities and horizontal gradients, contamination can be orders of magnitude larger than the signal. This cautions, in particular, against the estimation of mixing efficiency Γ, for example, from simultaneous measurements of χ and ε. Perceived variations in Γ may result from lateral contributions. On the other hand, we have seen that both quantities, together with velocity microstructure and lateral gradients, can yield estimates of the isopycnal diffusivity, Ki.
In our specific case, we observed horizontal temperature structures of several hundred meters’ horizontal scale, supporting our interpretation that isopycnal fluxes were important. Scale analysis indicates that features of this size would only have to have O(0.01 m s−1) to effect the estimated Ki ≈ 1 m2 s−1, as also concluded by Ruddick and Hebert (1988). Given the strong tidal straining in our exact region (Ebbesmeyer et al. 2001), there is plenty of energy available for this variability.
In general, the processes that lead to Ki are not known. Shear dispersion (Young et al. 1982) and vortical-mode structures from collapsing mixed patches (Sundermeyer and Ledwell 2001) are two candidates, but observations to-date are too sparse for further comment. We can speculate, however, that since both of these mechanisms are tied intimately to diapycnal mixing, it might be expected that Ki on these scales might be related to Kρ. If so, the isopycnal diffusivity may someday be predictable in terms of the diapycnal diffusivity or, better yet, the internal-wave energy level. Many more careful studies, preferably with dye, of the processes leading to isopycnal spreading are required.
The authors thank the captains of R/V Miller and R/V Barnes for their excellent seamanship and Paul Aguilar, Steve Bayer, Glenn Carter, Earl Krause, Jack Miller, John Mickett, and Dave Winkel for their help in collecting the data. This work was supported by NSF Grant OCE-0095382.
Scalar Evolution in a Lagrangian Reference Frame
Consider an “isopycnal” float—that is, a float that moves vertically to remain on a surface of constant potential density referenced to the surface, σ(S, T, P), and that moves horizontally with the velocity of the water. As shown in the text and appendix B, the Lagrangian floats used in this experiment have these characteristics to an extremely high degree of accuracy. The equation for a scalar C measured on the float is (McDougall 1984)
where ζ and ξ are diapycnal and isopycnal coordinates and ζ is positive upward.
This is similar to the familiar equation governing evolution in a fixed reference frame, with the addition of the term including the diapycnal velocity w∗. This, quite rigorously, is the flow of water through isopycnals, and results from diapycnal mixing (Pedlosky 1996; De Szoeke and Bennett 1993). Its action on scalar evolution is through simple advection of diapycnal gradients.
Specifically, given a buoyancy flux Jb, the diapycnal velocity
in the absence of laterally divergent buoyancy fluxes, isopycnal slope, and curvature (St. Laurent et al. 2001; Garrett 2001). If we define the diapycnal diffusivity in terms of Jb = −KρN 2, then it is clear that if N 2 is independent of z, then
which can be understood by considering an isopycnal mixed more strongly from below than from above [i.e., (∂Kρ/∂ζ) < 0]. In this case, water at the isopycnal’s depth becomes heavier, displacing the isopycnal upward and causing water to flow downward through the isopycnal (w∗ < 0). In this case of constant N, (A1) reduces to the familiar (1).
In the case of nonconstant N, the effects of w∗ and (∂Kρ/∂ζ) cannot be distinguished (but are nonequal); it is therefore convenient to define
Equation (A6) relates the change in measured temperature on an isopycnal float to the curvature in the T–S relation. Once again, for constant N this reduces to the more familiar (1). Though unintuitive in appearance, (A6) related isopycnal temperature changes to 1) advection of the cross-isopycnal component of temperature gradient, 2) isopycnal mixing, and 3) nonlinear terms. Term 1 is more easily and intuitively expressible in terms of the spice (Flament 2002).
Float Details and Errors
The temperature time series observed from the float will evolve in a manner different from (A1) 1) if the float is not accurately following the water or 2) owing to the offset between the temperature sensor and the center of mass. Since our results depend on these, we provide a more detailed description of the float operation and errors associated with reasons 1 and 2.
The floats consist of a temperature and pressure sensor housed in an aluminum hull whose compressibility is 80% that of seawater. A motor-driven piston is pushed in and out of the bottom of the hull so as to increase the float’s compressibility to 93% of seawater. The difference is equivalent to a density equivalent of 0.003 kg m−3 in 10 m, or a stratification of 0.002 s−1, roughly 2% of the actual stratification. The close match of the compressibility of the float and the water implies that the float will settle to a fixed isopycnal after deployment and remain on that isopycnal despite large vertical excursions of the isopycnal. D’Asaro et al. (1996) compare the variations in temperature relative to pressure for floats deployed in a stratified ocean and find that the motion of a float relative to its isopycnal is less than 2% of the motion of the isopycnal itself.
Some motion of the float relative to the water is caused by its thermal expansion. The float’s thermal expansion 7.3 × 10−5 °C−1 is significantly less than that of water at the operating temperature, 12 × 10−5 °C−1. The difference results in about 0.5 m of downward float displacement relative to an isopycnal for 0.1°C of warming of the water at the same isopycnal. This is approximately 20 times too small to contribute to the observed warming.
In the presence of mixing, the density of a water parcel does not necessarily remain constant; that is, water parcels do not stay on isopycnals. This occurs through intermittent mixing events. These will push the float off its isopycnal. D’Asaro (2003) analyzes the response of a float to such a perturbation. The float will return to its target isopycnal with a time scale of approximately τ = CiwAr/VN, where A is the horizontal area of the float, r is its radius, V is its volume, N is the stratification, and Ciw is the internal wave drag coefficient, about 4. For this float, τ ≈ 1.2N−1, so the float will return to its isopycnal in about a buoyancy period. The amount of displacement will be approximately the overturning scale of the turbulence, roughly 10 cm for this data. During each such excursion, the float will be advected by the ambient shear relative to the isopycnal. This shear is unlikely to be larger than 2N the limit of shear instability. An rms 10-cm diapycnal displacement with this shear leads to an average velocity of 2 mm s−1. The coherence time of this shear is unlikely to be longer than the M2 tidal time scale, TM2 = 6800 s, causing a net displacement of about 14 m. The total displacement during the two tidal periods of a day will be some multiple of this; 14 m if the velocity is sinusoidal, or 50 m if it causes a random walk with coherence time TM2. It is important to note that the water itself undergoes a similar process of shear dispersion (Young et al. 1982), so some of this float displacement mimics that of the water and is not strictly an instrumental error.
There is little reason to doubt that a very small neutrally buoyant float will move horizontally at the same speed as the water in which it is embedded. Except for the buoyancy control, the float is entirely passive and can exert no force to oppose the accelerations imposed upon it and the water by the ambient pressure gradients. However, the finite size of the float causes it to move with some average of the surrounding volume. It is convenient to quantify this in terms of the effective center of the float. This is nominally at the float’s center of transverse area, but could conceivably be displaced from this location, perhaps by up to one-quarter of the float’s length, about 15 cm. The shear at the float depth averaged over the 1-day duration of the float deployment is estimated below at less than 0.2N. This shear, acting on an offset of the float’s depth of 15 cm, results in a net isopycnal displacement of the float of 26 m from the water at its center. Again, this is not strictly an instrumental error, but partially represents the inherent dispersion of the water.
Last, the offset of the temperature sensor causes it to sense water 1/2 float length above the float’s center of mass, potentially leading to a net isopycnal displacement of 52 m. This upper bound is smaller than the horizontal length scales of O(hundreds of meters). Moreover, the water at the top of the float would tend to advect cooler water, counteracting the observed warming. Consequently, we conclude that the observed warming during the first deployment is due to mixing rather than imperfect water following.
Temperature measurements on the float were used to assess the warming of the water by mixing. Several days prior to the experiment, the temperature sensors were calibrated by embedding them and a Sea-bird Electronics reference sensor in a common aluminum block and cycling the temperature of an isothermal bath from 7.0° to 10.0°C over 36 h, in 6-h, 0.5°C increments. The resulting accuracy is <3 × 10−3°C.
Corresponding author address: M. Alford, Applied Physics Laboratory, 1013 E. 40th St., Seattle, WA 98105-6698. Email: email@example.com