Enhancement of turbulence through lateral spreading in a stratified-shear flow: Development and assessment of a conceptual model

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011jc007484, 2012 Enhancement of turbulence through lateral spreading in a stratified-shear flow: Development and assessment of a conceptual model Daniel G. MacDonald 1 and Fei Chen 1,2 Received 1 August 2011; revised 19 March 2012; accepted 27 March 2012; published 16 May [1] A conceptual model for the impact of lateral spreading processes on local stratified-shear turbulence is presented and evaluated using field data from the rapidly spreading near-field region of the Merrimack River plume. The conceptual model addresses increases in turbulent kinetic energy associated with the stretching of a Kelvin-Helmholz billow along its rotational axis. A non-dimensional relationship is derived with two fit parameters, including a parameter representing the magnitude of mixing under the no spreading case, and a non-dimensional time scale. For the model to be viable, this time scale, representing the period of time that a Kelvin-Helmholz billow is subject to stretching should be of the same order as the evolution time scale for the billow. Observational data from field experiments are used to evaluate the two fit parameters, indicating a stretching time scale of the same order as evolution time scales reported in the literature. In addition, application of the model to the field data reduces observed variability. We conclude that spreading is an important mechanism capable of enhancing local stratified-shear turbulence. Citation: MacDonald, D. G., and F. Chen (2012), Enhancement of turbulence through lateral spreading in a stratified-shear flow: Development and assessment of a conceptual model, J. Geophys. Res., 117,, doi: /2011jc Introduction [2] Stratified-shear flows are ubiquitous in nature, providing the context for much of the important mixing in the natural environment, and particularly in the ocean environment. Our understanding of these flows has increased dramatically over the last several decades, and a variety of mechanisms for the generation of turbulence in these environments have been identified. Among these, Kelvin- Helmholtz [e.g., Thorpe, 1973; Gregg, 1987] instabilities are perhaps the most studied and significant mechanism for a majority of geophysical flows. Kelvin-Helmholtz (K-H) instabilities take the form of a series of rolled vortices separated by thin braids. As these coherent billows grow, they are eventually arrested by stratification, and subsequently break down to form turbulent patches. Thorpe [1971, 1973] performed the first laboratory visualizations of K-H billows, and defined much of the essential dynamics of the K-H mechanism, which is believed to be the primary 1 Department of Estuarine and Ocean Sciences, School for Marine Science and Technology, University of Massachusetts Dartmouth, Fairhaven, Massachusetts, USA. 2 Now at ExxonMobil Upstream Research Company, Houston, Texas, USA. Corresponding author: D. G. Macdonald, Department of Estuarine and Ocean Sciences, School for Marine Science and Technology, University of Massachusetts Dartmouth, Fairhaven, MA 02719, USA. (dmacdonald@umassd.edu) Copyright 2012 by the American Geophysical Union /12/2011JC mechanism of generation for most of the turbulence observed in the oceanic thermocline [Gregg, 1987]. Smyth et al. [2001] obtained results from numerical model and microstructure observations, lending strong support to the hypothesis that K-H billows represent a valid model for observed turbulent patches. [3] Shear-stratified flows are usually considered in a two dimensional context, defined by the vertical and mean flow directions. However, in many geophysical environments, such flows may also be impacted by lateral forcing mechanisms, resulting in a flow that is spreading laterally as it propagates forward in the direction of mean flow. The impacts of such lateral forcing on the dynamics of stratifiedshear turbulence has not been addressed, although there has been some speculation [MacDonald et al., 2007] that turbulent mixing may be enhanced in regions of significant spreading. Near-field river plumes represent one such region, where lateral spreading and mixing are arguably the two most important processes affecting plume evolution [Hetland, 2010]. [4] The spreading rate in a near-field plume is proportional to the local internal long wave speed (c i ¼ ffiffiffiffiffiffi g h, p where g = gdr/r is the local reduced gravity, Dr is the density difference between the plume and ambient waters, r is a reference density, and h represents the plume thickness) due to strong horizontal density gradients along the edges of the plume [Wright and Coleman, 1971; Hetland and MacDonald, 2008]. Hetland and MacDonald [2008] and Chen et al. [2009] also discuss the important role of mixing in modifying the plume density, resulting in reduced 1of12

2 internal wave speeds and a slowing of the spreading rate. Thus, it is clear that mixing directly impacts lateral spreading. An understanding of the impacts of lateral spreading on the local turbulent field would provide an understanding of processes acting in the opposite direction, and yield a much clearer understanding of the dynamics of developing stratified-shear layers. [5] The dynamics of stratified-shear turbulence has been the focus of significant research effort over the last several decades, particularly within the estuarine and ocean science communities. Early studies on stratified entrainment [e.g., Ellison and Turner, 1959] laid the groundwork for Richardson number (Ri) dependence, and the advancement of observational [e.g., Gregg, 1987] and numerical techniques [e.g., Smyth et al., 2001] have yielded details of the dynamics of stratified turbulence at small scales. Recently, acoustic and microstructure observations of a stratified shear layer in an estuary have provided insight into the selfsimilar organization of turbulent structures at high Reynolds numbers [Geyer et al., 2010]. Here, this significant base of knowledge in stratified-shear turbulence is extended, as we propose a mechanism for lateral spreading to enhance the available turbulent kinetic energy (TKE) through the stretching of developing K-H billows. [6] We begin with an assessment of the important variables in a stratified-shear flow, and the creation of appropriate non-dimensional parameters to represent the dynamics. Here we assume the flow exhibits values of ɛ/nn 2 sufficiently high to insure isotropic turbulence. Following MacDonald and Geyer [2004] and Imberger and Ivey [1991] the important variables in a two-layer stratified-shear environment, considered in the classic two-dimensional context are the velocity difference, Du, the reduced gravity, g, across the shear layer, the shear layer thickness, h i, and the mean TKE dissipation rate per unit mass within the layer, ɛ. Although other scaling approaches can be envisioned, the use of Du and g are straightforward, and easy to obtain from traditional oceanographic measurement techniques. [7] From these variables, two independent non-dimensional parameters can be created: a bulk Richardson number, Ri B = g h i /(Du) 2, and a mixing parameter, ɛ/g Du. The mixing parameter represents the efficiency with which mean flow energy is converted to turbulent energy, and can be Ri B 1 Ri f ɛ Dug, related to an interfacial drag coefficient, C Di ¼ ð Þ where Ri f is the flux Richardson number. Given only two independent dimensionless parameters, the mixing parameter must be a function solely of Ri B. MacDonald and Geyer [2004] suggested a constant value for this mixing parameter on the order of for values of Ri B on the order of [8] Following additional observations in the Merrimack River plume (Massachusetts), extending further into the near-field plume region, MacDonald et al. [2007] report that the skill of this scaling appears to decrease with increasing distance from the estuary mouth. In MacDonald et al. [2007], ɛ is surrendered as the turbulent variable in place of the buoyancy flux, B ¼ ðg=rþr w. Not only is B a more direct indicator of mixing, but it is also more readily calculated with the control volume technique employed [MacDonald and Geyer, 2004]. Assuming a constant flux Richardson number, Ri f = BP 1, where P is the shear production of TKE, and homogenous and stationary turbulence (P = B + ɛ), B and ɛ are directly proportional. They further define the mixing parameter as x ¼ B g Du. Here, we also use B instead of ɛ, and focus on the mixing parameter x. The results from MacDonald et al. [2007] as well as additional data from May 2007 are shown in Figure 1. The Merrimack River data sets include multiple estimates of B from a number of longitudinal passes through a highly energetic discharge plume. In general, B, Du, and g were all found to decay with distance from the river mouth. However, B was found to decrease at a faster rate than would be predicted by the assumption of a constant value of x, but little correlation of x with Ri B was observed. The MacDonald and Geyer [2004] scaling was derived from data in the Fraser River plume just seaward of the river mouth and the bottom attached salt front, where the outgoing river water lifts off from the bottom. This scaling was found to be most effective in a similar lift-off location with regards to the Merrimack River data set. MacDonald et al. [2007] speculated that the decrease of x with seaward distance may be related to lateral spreading, as spreading is maximal near the river mouth and decreases as the plume propagates seaward due to the integrated effect of mixing decreasing the density difference between the plume and ambient waters. [9] In this manuscript, we extend the scaling of MacDonald and Geyer [2004] to include the effects of lateral spreading, and develop an analytical model to explain the relationship between lateral spreading and turbulent mixing for a stratified-shear flow, testing the relationship with data collected from the near-field of the Merrimack River plume (MA). [10] In the next section we will describe the theoretical scaling followed by development of the analytical model in section 3. Section 4 describes the field experiments, with a discussion in section 5, and a summary in section Theoretical Scaling [11] The parameters in the upper (unshaded) portion of Figure 2 represent the scaling associated with the two-layer stratified-shear problem within the traditional two dimensional context, as discussed in section 1. Several reasons for the failure of this scaling through the evolution of the nearfield plume, as shown in Figure 1, can be postulated. One possibility is that the mixing parameter, x, decreases as a function of Ri B [e.g., Ellison and Turner, 1959; Christodoulou, 1986]. However, this is not supported by the calculated values of Ri B, as shown in Figure 3. Values of Ri B range from approximately 0.05 to 0.8. However, no clear relationship is observed between x and Ri B across this range. [12] Alternatively, it could be argued that the observed decrease in x with distance from the mouth could be driven by acceleration in the upper layer. Kelvin-Helmholz instabilities are driven by shear strong enough to overcome stratification and create a coherent overturning structure [e.g., Thorpe, 1987]. The overturn structure grows in size as it evolves within the sheared environment, before it eventually breaks down into three-dimensional turbulence. Acceleration in the upper layer resulting in a more strongly sheared environment at the moment of initiation of a K-H billow would be accounted for in the local value of Ri B,as discussed above, which does not appear to be correlated with changes in x. However, significant acceleration in the 2of12

3 Figure 1. Plot of x = B/(g Du) divided by C = [e.g., MacDonald and Geyer, 2004]. This plot shows a similar trend to Figure 11 from MacDonald et al. [2007], with data from both May 2006 and May Error bars represent a subset of potential errors in the control volume method [MacDonald et al., 2007], particularly those errors which represent relative uncertainties between separate data points, and not errors expected to be correlated across the entire range of data. The bold solid line is a best fit (least squares) to the data, shown with 90% confidence limits (bold dashed lines). In general, the skill of the assumption x = C is shown to decrease with seaward distance in the plume. upper layer at time scales consistent with the evolution of an individual K-H billow, could conceivably affect the growth of the instability, resulting in a profile of x that is not correlated with mean values of Ri B. To test this hypothesis, we can estimate a scaling for the evolution of a K-H billow based on direct numerical simulations (DNS) performed by Smyth et al. [2001]. The time required for the evolution of their K-H billow from conception to decay (based on their Figure 1) is approximately 4000 s. Normalizing this value by the local shear present in the DNS model yields a non-dimensional time scale of approximately 140 (i.e., t = T KH ( u/ z) 140, where t and T KH represent the non-dimensional, and dimensional KH billow evolution timescales, respectively). The results of Thorpe [1973], as shown in his Figure 1, and reiterated in Thorpe [2005, equation (3.10)], suggest similarly that t , with the lower limit representing the disappearance of visually coherent billows and the upper limit representing the complete collapse of turbulence. Similarly, we can normalize a time scale for spatial acceleration in the Merrimack nearfield plume, ( u/ x) 1, by the shear, ( u/ z), yielding values ranging from 10 3 to 10 4 or higher. Thus, there is a clear disparity between the K-H evolution timescale and the Figure 2. turbulence. Summary of dimensional and independent non-dimensional variables for shear-stratified 3of12

4 Figure 3. Plot of x versus Ri B for data from 2006 and 2007 passes. Error bars similar to Figure 1. acceleration timescale, and it is unlikely that these two processes significantly interact. [13] We should remark here that we use the shear time scale, ( u/ z) 1 h i /Du, as a representative time scale for turbulent processes. Other logical turbulence time scales might include N 1 [Gregg, 1987] and Du/g [Thorpe, 1973; Koop and Browand, 1979], both of which are related as a function of Ri B (e.g., h i /Du= N 1 Ri 1/2 B =(Du/g )Ri B ). Thus, under the assumption of constant Ri B, all of these time scales are dynamically similar, varying only by a constant. The KH billow evolution time scales from Thorpe [1973, 2005] presented above, were originally reported as a function of Du/g, and have been transformed based on reported Ri B values (0.06 from Thorpe [1971, 2005] and from Thorpe [1973]). [14] Finally, we consider the direct impact of spreading on the developing K-H billows by developing a method to parameterize spreading and its effects. To accomplish this, we enhance the two-dimensional scaling approach of MacDonald and Geyer [2004] and Imberger and Ivey [1991] by the inclusion of an additional variable, d, to represent lateral spreading, as shown in the lower portion of Figure 2. The variable d = v/ y, where v and y are cross-plume velocity and distance, respectively, represents the lateral strain rate of the plume as proposed by Hetland and MacDonald [2008], who also showed empirically that d = DW W u r Dr, where W represents plume width, u r represents the radial velocity within the plume, and Dr represents an increment of radial (i.e., along plume) distance, across which the change in width, DW, is observed. This expression yields a time scale that can be interpreted as the time required for the plume to double in width. To non-dimensionalize this new spreading variable, we choose to normalize by our turbulent time scale, the reciprocal of the vertical shear (Du/h i ), yielding the spreading parameter, f, shown in Figure 2. [15] Thus, the five variables in Figure 2 yield three independent dimensionless parameters, denoted as x, f, and Ri B. The variable x represents the efficiency with which energy is extracted from the mean flow and converted into turbulent energy, and is similar to the ratio for which MacDonald and Geyer [2004] proposed a constant value of The MacDonald and Geyer [2004] value is based on the maximum value of ɛ in a given vertical profile. In the present definition of x, the vertical mean of B for a given profile should be used, as suggested in MacDonald et al. [2007]. They suggest that the mean value should scale as approximately 0.6 times the maximum value. Given this adjustment, and assuming a constant Ri f = 0.18, would yield a constant value of x based on the MacDonald and Geyer [2004] conclusions of approximately , which is used to scale the data in Figure 1. [16] The non-dimensional quantity, f, represents plume spreading, which can be interpreted as a ratio of the turbulent time scale to the lateral expansion time scale. Reanalysis of the MacDonald and Geyer [2004] data suggests an average value of f for their data of approximately [17] Within this new framework, it is implied that x = f(f, Ri B ), so that under constant Ri B, spreading alone controls the value of the mixing parameter. In the next section, a conceptual and analytical model is developed to define the form of the x = f(f, Ri B ) relationship. 3. Conceptual Model [18] Consider a turbulent field characterized by independent and isolated turbulent patches, presumably arising from K-H billows. The prevalence of such instabilities has been observed in similar environments [e.g., Geyer and Farmer, 1989; Tedford et al., 2009] and in direct numerical simulations [e.g., Smyth et al., 2001]. In addition, recent work based on the same data set used later in this analysis 4of12

5 Figure 4. Cartoon of spreading, spinning and squeezing of a K-H billow. b is width of the billow and l is diameter of the billow. compares mixing estimates derived from overturn scales indicative of K-H type instabilities [e.g., Thorpe, 1977], and control volume estimates of mixing, suggesting that the observed overturns are primarily responsible for the observed mixing (D. G. MacDonald et al., manuscript in preparation, 2012). [19] Such vortices may be represented as horizontal cylinders oriented in the cross stream direction (Figure 4), with a diameter, l, representing the outer length scale of the turbulence which should scale with the Ozmidov scale (L o ¼ ɛ 1 2 N 3 2 ), and a velocity scale, u, which is the characteristic turbulent velocity. The cylinders also have a length, b, in the cross-stream direction (note that b need not be the full width of the plume but it is likely that b l), that is increased at the same rate as the overall lateral expansion rate of the plume. This is a critical assumption, which is supported by the modeling study of Hetland and MacDonald [2008], where it was found that most important plume variables, including the spreading rate, showed very little variability in the cross-plume direction, within the core of the plume. We hypothesize here that such stretching of these K-H cylinders, particularly during the pre-turbulent roll-up stage, may have significant impacts on the diameter and velocity scales, thus impacting the energetics of the resulting turbulence. A similar mechanism, albeit with significant differences in scale and generation dynamics, was discussed conceptually by Farmer et al. [2002] to explain an increase in turbulence associated with boils observed in Haro Strait. [20] The rate of energy supply to the large-scale eddies, and thus the TKE dissipation rate, can be scaled [Taylor, 1932] as: ɛ u 3 l [21] In our model, the KH cylinders are stretched at the same rate as the plume when macro-scale horizontal expansion occurs. The volume of an individual cylinder, V, can be assumed constant, and expressed as, V = 0.25pl 2 b. Thus, l b 1 2 ð1þ ð2þ Equation (2) describes changes in the vertical length scale of K-H billows as the instability undergoes lateral stretching. The relationship in (2) does not rely on the absolute size of the instability but indicates only the nature of the proportionality. [22] In order to express (1) in terms of the spreading parameter, f, we apply conservation of angular momentum to the hypothetical cylinders. The angular momentum, ~H, is defined as ~H ¼ R ~r ~udm, where dm is an element of mass, and r is the radius of the vector from the center axis. Applying the appropriate scales associated with the cylindrical KH tubes, including a maximum radii of 0.5l, and the rotation speed at the edge equal to the turbulent velocity, u!, into the conservation of H, then we have ~H ¼ Z l=2 0 r 2u r ð2rpbrþdr ð3þ l As ~H is constant, the integral above leads to u b 1 l 3. With (2) and (1), we obtain ɛ b 2. Since b is increased at the same rate as the overall lateral expansion rate of the plume width, W, we can also argue that ɛ W 2. [23] Note that the validity of the assumption of conserved angular momentum could be compromised by the effect of a baroclinic gradient producing additional torque on the evolving billow. To assess the magnitude of this mechanism, a time scale can be generated by comparing the torque resulting from this mechanism (i.e., R h l=2 l=2 b R i 0 h rgzdz xdx) to the total angular momentum expressed by (3). Substituting reasonable values from the Merrimack plume region yields a time scale on the order of 30 min, an order of magnitude higher than billow evolution times predicted by the results of Smyth et al. [2001] and Thorpe [1973, 2005]. This time scale suggests that baroclinic torque is ineffective at producing significant changes in angular momentum in the near-field plume, and that our assumption of conservation is reasonable. [24] Consider a time period for which a K-H cylinder is subjected to the stretching processes caused by lateral spreading, which could be scaled as T KH = Dr/Du, where Dr represents the distance traveled by the billow during the stretching process, and Du u r assuming no velocity in the ambient ocean water below the shear layer. Here, we assume that a billow may be subject to stretching processes throughout the duration of its evolution, and thus use the evolution time scale, T KH, as the stretching time scale. The change in plume width over this time period can be expressed as DW =(dt KH )W o, where W o is the initial width. Applying the spreading parameter f yields DW W o ¼ fdu h i T KH ð4þ Assuming a constant flux Richardson number throughout the evolution of the billow, we can then derive an expression for changes in x due to lateral spreading: x x o ¼ ɛ ɛ o W 2 ¼ 1 þ DW W o W o 2 ¼ ð1 þ tfþ 2 ð5þ 5of12

6 which can be rewritten as ¼ Dr h i x ¼ x o ð1 þ tfþ 2 ð6þ where t ¼ DuT KH h i is a non-dimensional time scale of K-H billow evolution as discussed in section 2, and x o is an initial, or non-spreading, value of x. [25] Note that we have assumed constant values of g and Du, reflecting the fact that the model follows a single roll-up or instability, created under specific conditions at a single point in time. Equation (6) then describes the evolution of that roll-up due to the nature of its laterally spreading environment. [26] A key aspect of (6) is the two fit parameters, x o, and t, both of which have a distinct physical meaning. The value of x o in (6) represents the initial value of x, or the value of x when there is no lateral spreading, (i.e., f = 0). This can be interpreted as representing the turbulent potential of the rollup at the moment of its conception, which is then further modified through the stretching effects of lateral spreading (i.e., the (1 + tf) 2 term) during the course of its evolution. Note that the lateral stretching affects all billows similarly regardless of initial size, so that (6) can be used to characterize the mean dissipation across a region of constant f as effectively as it characterizes a single roll-up. The nonspreading case should be comparable to the pioneering laboratory studies of two-layer entrainment by Ellison and Turner [1959] and other entrainment studies as summarized by Christodoulou [1986]. Unfortunately, the nature of these entrainment experiments, compared to the buoyancy flux dynamics discussed here, make it difficult to use the laboratory results to propose reasonable values of x o.in addition, there are significant differences between the laboratory and geophysical scales, likely resulting in vastly different Reynolds numbers, and potentially different turbulent dynamics [e.g., Geyer et al., 2010]. [27] The second coefficient in (6) is the value t, which is a normalized time scale for K-H billow evolution, assuming that spreading impacts would affect a K-H billow through the majority of its development, from initial generation, through overturning, to ultimate turbulent decay. Thus, our value of t must be of the same order as the previous estimates of the K-H evolution timescale for this conceptual spreading model to be viable. As discussed above, based on the results of Smyth et al. [2001] and Thorpe [1973], estimates for the KH evolution time scale (normalized by the shear) are on the order of 80 to 250. [28] In the following section, we use data collected from the Merrimack River near-field plume to evaluate the fit parameters, x o and t. 4. Field Experiments and Results [29] Data collected in the near-field region of the Merrimack River plume (Figure 5) are used to estimate the fit parameters of (6), and to validate its effectiveness. The Merrimack River mouth is constrained by two jetties approximately 300 m wide, with a shallow bar (approximately 3 m deep at low tide) located about 500 m beyond the end of the jetties. During high flow periods, the river discharge creates an energetic and rapidly expanding buoyant jet in the coastal ocean, as described by MacDonald et al. [2007] and Chen et al. [2009], with a bottom attached salt front anchored on the seaward edge of the bar during ebb tides. The resulting flow is strongly sheared and stratified, with negligible influence from bottom friction, given that an essentially stagnant ambient ocean layer m thick sits below the plume. Our data (MacDonald et al., manuscript in preparation, 2012) indicates an overturn scale [e.g., Thorpe, 1977] on the order of 10 cm, consistent with estimates of the Ozmidov scale, suggesting that mixing between the plume and ambient water is driven primarily by K-H type instabilities which are forming overturns on the order of the Ozmidov scale before decaying to turbulence. This turbulence quickly establishes a two-layer system, characterized by a shear-stratified flow rather than a discrete interface, with values of Ri B of order 0.05 to 0.8. [30] The data used in this study were collected during three days of field observations: May 21, 2006, and May 9 and 11, The 2006 measurements are the same measurements reported in MacDonald et al. [2007], and occurred during unusually high discharge conditions of approximately 1,300 m 3 /s. The 2007 measurements occurred during more typical freshet conditions, with an outflow of approximately 300 m 3 /s. Mean velocity profiles were recorded with two RD Instruments 1200 khz acoustic Doppler current profilers (ADCPs). One of the ADCPs was in a downward looking position, mounted off the side of the research vessel. The other mounted on an Acrobat tow body (Sea Sciences, Inc.) in an upward looking position, towed at approximately 4 m depth. Data from the two ADCPs provides a full velocity profile, which extends to the very near surface. This is crucial to fully characterize the shallow buoyant plume. Velocity profiles were collected at a rate of approximately 1 Hz, using 0.25 m vertical bins. Ship velocity was removed via bottom tracking from the downward mounted profiler. Pressure, temperature, and conductivity were recorded using Micro-CTD (Applied Microsystems Ltd.) and XR-620 (RBR Ltd.) profilers attached to a frame towed by the research vessel, undulating between the surface and approximately 4 m depth. [31] Hydrographic data were collected along passes in the along-stream direction, as shown in Figure 5. All passes occurred during the second half of the ebb tide, with a bottom attached salt front established at the bar. On both sampling days, the weather was calm, with winds generally less than 5 m/s as measured at the NDBC Isles of Shoals buoy, approximately 25 km northeast of the field site. Winds on May and May were generally westerly, and directed offshore, while winds on May were from the south. [32] Figure 6 depicts the local radial velocity, u r (assuming a radial coordinate system centered at the river mouth), and density fields from pass 3 on May The saltdominated density stratification is stable, with warmer, fresher water over cooler, saltier water. Velocity differences between plume water and the surrounding water are of order 1 m/s, with maximum differences in density of approximately 15 s T units. The density and velocity structures of the near-field region are characterized by a rapidly thinning and mixing fresh water layer seaward of the bar, resulting in high shear and stratification. 6of12

7 Figure 5. (a) Location and (b, c) timing of sampling passes at Merrimack River (Figure 5a). Axes in Figure 5a represent coordinates (in meters) with an origin at the river mouth. The two thin dashed lines in Figure 5a indicate approximate boundaries of the plume core. [33] To estimate the three parameters identified in Figure 2, it was necessary to evaluate a number of parameters directly from the observational data. The base of the shear layer was defined based on the local fresh water flux per unit width, Q f ¼ R Fu r bdz, where b is a unit width, F is the local freshness, F =(S o S)/S o, and S o and S are the ambient ocean and local salinities, respectively. The base of the shear layer was determined by identifying the depth associated with 90% of the total vertically integrated freshwater flux. [34] In many cases, the shear layer extends from the water surface down to its base as defined by the local freshwater flux. However, in some cases, particularly in the lift-off region, the shear layer is overlain by a relatively well-mixed plume layer. To account for this, we define the top of the shear layer as the depth at which the salinity is equal to the surface salinity plus 0.3 psu (although this choice is somewhat arbitrary, it is small compared to a typical DS of 10 to 20 psu, and the results are fairly insensitive to small changes in this value). We then define the shear layer thickness, h i,as the difference between the calculated shear layer base and top depths. The upper and lower limits of the shear layer are shown on the plots in Figure 6, as an example. It should be noted that, due to extremely high stratification, the Ozmidov scale, and the vertical scale of overturns observed in the CTD tow-yo data are much smaller (on the order of 10 cm) than the shear layer thickness, which is on the order of 1 to 2 m. Thus, there is little potential for developing K-H billows to directly interact with the surface boundary, which might have the effect of limiting billow development and altering the premise of the conceptual model developed in section 3. [35] Differential quantities necessary for the analysis were defined based on the upper and lower bounds of the stratified-shear layer. For example, the shear, Du, and the reduced gravity, g, were estimated from the velocity and density differences between the top and base depths of the layer. In this study, we limit our data to the lift-off and nearfield region of the plume, which we constrain to distances from the river mouth between 500 and 3000 m. Beyond this region, resolution issues hamper the ability to accurately estimate h i due to limited plume depth, and an ADCP resolution of 0.25 m. 7of12

8 Figure 6. Contours of (left) density and (right) velocity through the near-field region, representing data from pass 3 on May 11, The dotted and dashed lines represent the top and bottom of the stratified shear layer, respectively, as defined in the text. The interface depth (h i ) is the vertical distance between the dotted and dashed lines. [36] Turbulent mixing within the plume is represented by x. Buoyancy flux, B, was estimated along observational passes on all sampling days using a control volume approach [e.g., MacDonald and Geyer, 2004; MacDonald et al., 2007; Chen and MacDonald, 2006; McCabe et al., 2008]. In this case, the control volume approach utilized data from along plume transects to compare fluxes approximately 400 m apart. Localized spreading rates (i.e., DW) are derived from the difference in vertically integrated freshwater flux. Diapycnal volume and density fluxes are evaluated from differences in vertically integrated horizontal volume and density fluxes above a specified isopycnal. Calculation of these quantities allows the vertical advective density transport to be extracted from the total vertical density flux following a Reynolds decomposition, leading directly to estimates of Reynolds salt flux (i.e., r w ¼ rw rw), and thus of B. Use of successive isopycnals allows a vertical of profile of B to be generated for each control volume. The value of B used in this analysis represents the vertically averaged buoyancy flux observed within a given control volume through the defined upper layer. MacDonald et al. [2007] described the use of the control volume method to measure B within the Merrimack River plume, and compared the results to autonomous underwater vehicle (AUV) microstructure data and Regional Ocean Modeling System (ROMS) numerical model output. The control volume results and numerical model output were in excellent agreement, consistent with the integrative nature of both the observational technique and the closure schemes inherent in the model. The microstructure results, while highly variable, were generally an order of magnitude or more lower than the other estimates, suggesting a high degree of heterogeneity in the turbulent structure, consistent with the assumption of K-H billows as the primary turbulent mechanism in the shear-stratified layer [e.g., Smyth et al., 2001; MacDonald et al., manuscript in preparation, 2012]. This paper and others utilizing variants of the control volume technique [e.g., MacDonald and Geyer, 2004; Chen and MacDonald, 2006; McCabe et al., 2008], provide strong support for the consistency of the method with other estimates of turbulence, and we use the method here to provide estimates of both buoyancy flux and plume spreading. [37] The value of f was determined from observations using (4), with calculated width ratios (i.e., DW/W o ) generated from the control volume analysis: f ¼ DW W o h i Dr CV Here, Dr CV and DW represent the along plume length of the control volume, and the change in width across the control volume, respectively. This differs from (4) where the length scale (DuT KH ) and width expansion were defined across a distance representing the evolution of a KH billow. In either case, the expression represents an average value of f over the chosen distance. The use of the control volume method to constrain spreading rates was investigated in Chen et al. [2009], where control volume estimates agreed favorably with both drifter analyses and numerical model output. Note that our calculated values of DW/W o represent a depthaveraged value of the plume spreading rate, which may exhibit significant vertical structure, with enhanced spreading near the surface and less spreading near the base of the plume. It should also be recognized that DW/W o is a nondimensional plume spreading rate, with increase in width normalized by the total plume width. Thus, even with no decrease in the interfacial wave speed, which defines the rate of lateral expansion due to buoyancy differences, f will tend to decrease as the local plume width increases in the seaward direction. [38] Ri B was estimated directly from observations using the equation in Figure 2, with the value of Ri B plotted against the parameters x and f in Figures 3 and 7, respectively. No clear trend between Ri B and the other two dimensionless parameters is apparent on these plots. ð7þ 8of12

9 Figure 7. [39] In Figure 8, x is plotted directly against f. The observations show a clear relationship between the two dimensionless parameters. With distance increasing from the river mouth, the plume data enters the graph in Figure 8 at Plot of f versus Ri B for data from 2006 and 2007 passes. the top right, with strong initial mixing, and high spreading rates, as indicated by high values of both x and f, and then move down and to the left as the plume propagates seaward. Figure 8. Measured lateral spreading parameter, f, versus the mixing parameter, x, based on observational data with error bars (see Figure 1). A best fit curve based on the conceptual mechanism described in the text is shown by the black line. The gray arrow indicates the direction of plume water propagation through the parameter space. Discharged water enters the graph at the top right, and then moves down and to the left as the plume propagates seaward. A point approximating average values from MacDonald and Geyer [2004] is also shown as the filled star at the upper right corner. 9of12

10 Figure 9. Plot of x CV /x S, with x S defined by equation (6) and constants as defined in Figure 8, with best fit line and 90% confidence limits (bold solid and dashed lines, respectively). This plot is similar to Figure 1, with the exception that spreading processes are now considered in the scaled estimate of x. The best fit line and confidence limits from Figure 1 are shown in gray for reference. Although considerable scatter remains in the data, the clear trend seen in Figure 1 of decreasing x with seaward distance in the plume is less apparent, suggesting that this decrease was primarily a function of plume spreading. An approximate marker for the conditions described by MacDonald and Geyer [2004] is also plotted for reference. 5. Discussion [40] The values of f and x in Figure 8, can be used to evaluate the relationship described by (6), by fitting an appropriate quadratic relationship to the data. The best quadratic fit for the combined 2006/2007 data sets is shown in Figure 8, with values of x o = and t = 360, and an R 2 value of The fit is considerably improved by considering the 2007 data set only, yielding x o =5 10 5, t = 386, and an R 2 value of Despite the increase in R 2 observed by excluding the 2006 data set, both fits yield very similar values of x o and t. [41] The value of t 360 best describing the data is within a factor of 1.5 to 4 of the estimates based on Smyth et al. [2001] and Thorpe [1973, 2005], which are on the order of The fact that the time required for the degree of turbulent enhancement seen in the Merrimack near-field plume is of the same order as the plume evolution time scale derived from laboratory and DNS results suggests that the distinct processes of lateral plume spreading, and turbulence generation within the stratified shear layer can and do interact. Our higher observed value of t suggests that developing roll-ups might be subject to enhancement through lateral expansion throughout the majority of their life-cycle, even beyond the loss of visually coherent billow structures. Other explanations for the discrepancy between our value of t and those derived from the literature might include the vertical structure of the spreading profile, which is unaccounted for in our average value of f. Given the quadratic nature of (6), values of f that are enhanced near the surface would result in lower values of t than reported here, depending on the nature of the vertical structure of spreading. Our higher value of t may also be due to unresolved processes related to high Re flows, or to an increased duration of coherency of the billow structure related directly to the presence of lateral spreading. Geyer et al. [2010] have recently shown that the structure of turbulence at high Re values is substantially different than that observed in laboratory experiments and DNS simulations at low Re. [42] In the Merrimack plume the shear layer thickness is on the order of 1 m, with plume velocities on the order of 1 m/s (Figure 6), implying a value of T KH of approximately 6 min, based on the calculated value of t = 360. This value suggests that the life cycle of a roll up is on the order of 6 min for an instability to be generated, begin to overturn and transition to turbulence. [43] The best fit value of x o is approximately , indicating an interfacial drag coefficient C Di of the same order (C Di ¼ Ri B Ri f x, where the two Richardson numbers are of the same order). As would be expected, this C Di value is significantly smaller than a bottom drag coefficient, which is typically on the order of MacDonald and Geyer [2004] obtained an interfacial drag coefficient of , larger by an order of magnitude, but consistent with the fact that it was from a region subject to significant lateral spreading, while the value of x o represents the base value for the no-spreading case. The entrainment studies of Ellison and Turner [1959] and Christodoulou [1986] suggest entrainment coefficients on the order of With some assumptions, this value can be converted to a comparable 10 of 12

11 value of x o, which is also on the order of 10 2, two to three orders of magnitude higher than the value obtained here (and in MacDonald and Geyer [2004]). These observed differences may be due to Reynolds number effects, as the entrainment studies were carried out in the laboratory. A full analysis of these discrepancies is beyond the scope of the present manuscript, but the results from this study can provide a benchmark for future studies regarding shearstratified mixing at high Reynolds numbers in non-spreading environments. [44] Finally, we return to the motivation for this study, the observed failure of the two-dimensional scaling for stratified shear turbulence observed by MacDonald et al. [2007] and represented in Figure 1 of this manuscript. To evaluate the effectiveness of our enhanced scaling technique we consider x the ratio, rather than x x o ð1þtfþ 2 C as shown in Figure 1, plotted as a function of distance from the mouth in the Merrimack plume in Figure 9. Although scatter remains in the data shown in Figure 9, a comparison of the data in Figures 1 and 9, along with their respective best fit lines and 90% confidence intervals, suggests that the skill of the scaling approach has been improved by inclusion of the lateral spreading mechanism for turbulent enhancement. For example, the slope of the linear fit through the data in Figure 1 (log(x/c) versus distance) yields a value of approximately 0.24 km 1, and an R 2 value of 0.20, compared to a similarly calculated slope through the data of Figure 9 of 0.08 km 1 (R 2 = 0.03). Thus, the best fit slope has been reduced by a factor of 3. Likewise, the R 2 values should be compared by considering that the values represent the improvement of the fit over the assumption of a constant value. Thus, the minimal R 2 value for the data in Figure 9 suggests that a slope of 0 would be equally appropriate, while the R 2 value for Figure 1 suggests a downward trend, but one weakened by the degree of scatter in the data. [45] The fact that there is considerable scatter remaining in the data displayed on Figure 9, and that R 2 values associated with the fit of (6) as shown in Figure 8 are only on the order of is not surprising and is reflective of the fact that there are a variety of factors playing a role in the determination of mixing. We do not suggest that spreading is the only important process. As discussed above, the effects of vertical structure in the spreading profile may introduce scatter as well as overestimating the value of t. Additionally, variability in mixing efficiency, as quantified by the value of Ri f may also play a role. However, the fact that the value of t derived from the field data matches reasonably well with laboratory and DNS estimates, and that application of the spreading relationship removes the downward trend apparent in Figure 1, strongly suggests that spreading is an important process, with significant impacts on mixing. 6. Summary [46] This manuscript presents estimates of five key variables related to near-field plume structure and evolution, including vertical shear, stratification, turbulent buoyancy flux, lateral strain rate of the plume and the interface depth. Three dimensionless parameters are derived, representing the turbulent mixing, x, the lateral spreading rate, f, and the instability of the flow, Ri B. Evaluation of these parameters using observed field data from the near-field Merrimack River plume suggests that the mixing and spreading parameters are clearly related, but that neither parameter is well correlated with Ri B. A conceptual model based on the longitudinal stretching of individual K-H billows provides a direct mechanism for lateral plume spreading to impact the energetics of the local turbulent field, resulting in a quadratic relationship between the mixing and spreading parameters. The fit of the quadratic relationship to data from the Merrimack River near-field plume explains a majority of the observed variability, and yields two values, the value of x in the nonspreading limit, referred to as x o, and a turbulence evolution time scale, t. The value of x o differs significantly from laboratory-derived values, perhaps suggesting a strong Reynolds number dependence. However, the value of t is reasonably consistent with other estimates of the turbulent evolution time scale based on laboratory and DNS results and lends credence to the viability of the lateral spreading mechanism as a means of turbulent enhancement. 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