Relating River Plume Structure to Vertical Mixing

SEPTEMBER 2005 H E T LAND 1667

ROBERT D. HETLAND Department of Oceanography, Texas A&M University, College Station, Texas

(Manuscript received 30 March 2004, in final form 24 February 2005)

ABSTRACT

The structure of a river plume is related to the vertical mixing using an isohaline-based coordinate system. Salinity coordinates offer the advantage of translating with the plume as it moves or expanding as the plume grows. This coordinate system is used to compare the relative importance of different dynamical processes acting within the plume and to describe the effect each process has on the structure of the plume. Vertical mixing due to inertial shear in the outflow of a narrow estuary and wind mixing are examined using a numerical model of a wind-forced river plume. Vertical mixing, and the corresponding entrainment of background waters, is greatest near the estuary mouth where inertial shear mixing is large. This region is defined as the near field, with the more saline, far-field plume beyond. Wind mixing increases the mixing throughout the plume but has the greatest effect on plume structure at salinity ranges just beyond the near field. Wind mixing is weaker at high salinity classes that have already been mixed to a critical thickness, a point where turbulent mixing of the upper layer by the wind is reduced, protecting these portions of the plume from further wind mixing. The work done by mixing on the plume is of similar magnitude in both the near and far fields.

1. Introduction and Geyer 2001; García Berdeal et al. 2002; Hetland and Signell 2005). However, analysis of the plume is River plumes are central to a number of important difficult, particularly interpreting observations, because societal oceanographic problems. For example, a toxic of the changing position of the plume. In the case of dinoflagellate, Alexandrium spp., is associated with the wind forcing, the plume may change position so much the Kennebec–Penobscot River plume in the Gulf of that, at many points, the plume may be only occasion- Maine (Franks and Anderson 1992). Stratification ally present. Also, even when the plume is present, dif- caused by Mississippi–Atchafalaya outflow prevents ferent regimes of the plume may be measured, for ex- ventilation of lower-layer waters, allowing hypoxic con- ample, frontal regions versus the core of the plume. ditions to develop on the continental shelf (Rabalais et Many of these difficulties stem from a Cartesian, or al. 1999). Nearly one-half of all oceanic carbon burial Eulerian, view of the plume. occurs in large river deltas (Hedges and Keil 1995). This paper examines the plume in salinity coordi- Many papers have reported on the various features nates, a natural coordinate system for the plume. Al- of river plumes, particularly a recirculating bulge that though this approach is not Lagrangian, in that the forms in the vicinity of the outflow (e.g., Garvine 1987; plume may be steady in salinity space even while water O’Donnell 1990; Yankovsky and Chapman 1997; Fong flows through it, salinity coordinates offer the advan- 1998; Nof and Pichevin 2001; Garvine 2001; Yankovsky tage of translating with the plume as it moves or ex- et al. 2001) and the on- and offshore motion of the panding as the plume grows. For example, the addition plume in response to upwelling and downwelling wind of wind causes the plume to mix as well as to change stresses (Fong et al. 1997; Pullen and Allen 2000; Fong horizontal position. Salinity coordinates are used to here to examine changes in vertical mixing in isolation by following the plume as it is shifted by currents. The focus in this paper is buoyancy-forced flow from Corresponding author address: Robert D. Hetland, 3146 TAMU, Department of Oceanography, College Station, TX narrow estuaries, where the local internal deformation 77843-3146. radius is larger than the width of estuary mouth. For E-mail: [email protected] the narrow-estuary case, as water leaves the estuary, it

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FIG. 1. This conceptual model of river plume anatomy shows the major regions and indicates the dominant mixing mechanisms. spreads and shoals, becoming supercritical. Here, the stress and by relating the size of the plume to different estuary mouth acts as a hydraulic constriction for the vertical mixing processes. upper layer (Armi and Farmer 1986). The accelerating flow soon becomes unstable, and strong shear mixing 2. Salinity coordinates occurs in the near field (Wright and Coleman 1971; Interpreting measurements of a river plume in Car- MacDonald 2003). Beyond this region, mixing is pri- tesian space may be difficult—for instance, the plume marily caused by wind stress through a mechanism de- may be only occasionally present at certain locations. scribed by Fong and Geyer (2001). Ekman transport in The analysis methods presented below are less sensitive the upper layer may become large enough that shear to the motions of the plume, because these methods instability is induced. At this point, the plume will mix consider the water mass structure of the plume as a and thicken until the local Richardson number is again whole using a coordinate system based on salinity. This above the critical value. This model of wind mixing salinity-based coordinate system follows the plume as it considers only the local wind stress and stratification. It moves and allows the freshwater introduced into the is not yet clear how this balance affects, and is affected domain to be followed as it is mixed with the back- by, the horizontal plume structure. A cartoon of the ground waters. The analysis presented below is based various dynamical regions within the plume shown in on the approach of MacCready et al. (2002), who ex- Fig. 1 demonstrates the mixing history of a water parcel amine long-term estuarine salt balances by calculating as it leaves the river/estuary and eventually becomes salt fluxes across isohalines. The derivation below ex- part of the background waters. tends MacCready et al.’s analysis by demonstrating The goal of this paper is to relate vertical mixing in how changes in isohaline surface area can be used to different dynamical regions of the plume to changes in estimate salt flux at particular salinity classes within the plume structure. In particular, this paper will compare plume, rather than across the entire isohaline surface. the relative importance of different dynamical pro- Here, we will consider a volume V bounded by the cesses acting within different parts of the plume, the sea surface and ocean floor, a face within the river structure of the plume, and the role of wind mixing in where s ϭ 0, and on the seaward edge by an isohaline, determining that structure. Garvine (1999) notes that s (the shaded area in Fig. 2). A portion of the bound- the steady-state alongshore scale of a river plume with- A ing area A defined by the isohaline surface s com- out wind forcing depends on, among other things, the A pletely divides fresher plume water (s Ͻ s ) from the value of background mixing, the minimum value for A rest of the ocean. There is a net freshwater flux of Q diffusivity, and viscosity in a turbulence closure scheme. R across the face of the volume within the river. Garvine’s results show that increasing the background The three-dimensional salt balance equation, mixing decreases the alongshore scale of the plume (roughly related to the total area of the plume). This Ѩs f, ͑1͒ · su͒ ϭϪ١͑ · ١ ϩ paper expands Garvine’s basic result by including wind Ѩt

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FIG. 2. The volume V, bounded on the seaward edge by area A, is shaded gray; the area A ϭ is defined by isohaline s sA. A freshwater flux, QR, is input into the volume on the opposite Ϫ ␦ Ͻ face of V. The second isohaline, sA s sA, is used in an example in the text. The difference Ϫ ␦ ␦ in isohaline area between sA s and sA is A.

integrated over volume V is is the freshwater content within V, relative to sA.IfEq. (4) is divided by s , an intuitive freshwater balance is Ѩ A ͵ sdVϪ s ͵ u · dA ϩ s ͵ u · dA formed—the freshwater from the river, QR, must either Ѩt A A A V A A increase VfA in time or be compensated by a freshwater flux, f/sA, across A. ϭϪ͵ ͑ ͒ By knowing Q and the change in freshwater content f · dA, 2 R A over time, the average salt flux, f across A may be estimated. However, it is expected that within the river where u is the three-dimensional flow vector, uA is the Ϫ plume the flux will change at different points within the normal velocity of the surface A itself (such that u uA is the flow through A), and f is the diffusive salt flux. plume, so that an area average of the salt flux over a The generalized Leibnitz theorem (Kundu 1990, p. 75) large isohaline may be difficult to interpret: are changes is used to take the time derivative outside the integral in average flux due to intense localized mixing or in the first term. The advective and diffusive salt fluxes broad-scale changes? through the faces of V are nonzero only on the isoha- Assuming a thin pycnocline, the salt flux across two isohalines within the pycnocline will be similar. This line surface A. Because sA is defined to be constant may be used to derive an estimate of the salt flux as at along A, sA may be taken outside the area integrals. A statement of mass balance within the volume, again a particular salinity class, instead of as an average derived by integrating over V,is across an entire isohaline surface. For example, assume that the plume is approximated as a single, active layer ѨV with horizontally varying salinity overlying a quiescent Ϫ ͵ · d ϩ ͵ · d ϭ Q . ͑3͒ Ѩ uA A u A R layer with a uniform, background salinity of s . The t A A 0 area A may now be related exactly to upper-layer sa- The mass and salt balance equations are combined to linity, sl, alone: form Ѩ Aͩs ͪ ϭ ͵͵ dAͯ ͑ ͒ ϭ ϩ ͵ ͑ ͒ l . 6 sA VfA sAQR f · dA, 4 Ѩ sϽsl zϭ␩ t A where The integral over A may be converted to an integral s d . Differentiatingץ/Aץ over s by converting dA to Ϫ l l sl sA s V ͑s ͒ ϵ ͵ dV ͑5͒ Eq. (4) with respect to sA, noting for the case of a thin fA A s ϭ V A pycnocline that sA sl in the area integral, gives

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Ѩ Ѩ ѨA rameterized by specifying a constant vertical diffusivity/ ͑ ͒ ϭ ϩ ͑ ͒ Ϫ Ϫ Ѩ Ѩ sAVfA QR fA Ѩ , 7 viscosity of 1 ϫ 10 4 m2 s 1, so that the length scale of t sA sl the salt intrusion is not significantly greater than the f where A is the average vertical salt flux associated with channel length, and the time scale of estuarine adjust- s salinity class A. If Eq. (7) is differentiated with respect ment is rapid enough to come into a steady state within s local to A, it becomes a statement of the freshwater approximately one upwelling/downwelling period (Het- s balance at salinity class A, just as Eq. (5) becomes a land and Geyer 2004). The coastal ocean has a 10-m global statement of the freshwater flux after dividing wall along the coast, and the bottom has a uniform s by A. slope of 1/1500, resulting in a maximum depth at the A steady-state version of Eq. (7) may be derived eastern boundary of about 70 m. Horizontal resolution heuristically by taking the difference of Eq. (4) evalu- of 500-m resolution in the immediate vicinity of the s s Ϫ ␦ ated at two neighboring isohalines, A and A s. Be- outflow results in three grid points across the estuary cause the pycnocline is thin, the salt flux across the mouth. f overlapping area is identical. Here, A is the average salt Austin and Lentz (2002) show how the Ekman layer ␦A flux over , the difference in the two areas considered may be shut down on a stratified shelf under wind forc- ␦ → (see Fig. 2). In the limit where s 0, the steady-state ing when the near-shore water becomes vertically well form of Eq. (7) is recovered. mixed, trapping water in the shallow region near the coast. In other simulations in which the topography was 3. Numerical setup kept very shallow along the coast with a very weak bottom slope (not shown here), this caused a portion of Model configuration the plume to be trapped near the coast, even when the The archetype for this configuration used in this pa- plume was upwelled. Regions with stronger near-shore per is the Kennebec River plume in the Gulf of Maine, bottom slopes, such as in the vicinity of the Kennebec which has been the subject of a number of observa- River plume, are not affected by this process. Also, in tional and numerical studies (Fong et al. 1997; Hetland this idealized study, maintaining simply connected iso- and Signell 2005; Geyer et al. 2004). The simulations haline surfaces is important in these first attempts at employ version 2.1 of the Regional Ocean Modeling isohaline analysis. Lentz and Helfrich (2002) show how System (ROMS; Haidvogel et al. 2000). ROMS is a the propagation speed of a coastally trapped buoyant free-surface, hydrostatic, primitive equation ocean jet depends on the bottom slope, and it is plausible that model that uses stretched, terrain-following coordi- this parameter is important in wind-driven plumes as nates in the vertical and orthogonal curvilinear coordi- well. The present study considers only the steep slope nates in the horizontal. The code design is modular, so limit, as topographic dependence is not the focus of this that different choices for advection and mixing, for ex- study. ample, may be applied by simply modifying preproces- Garvine (2001) suggests that model configuration is sor flags. ROMS is open source and freely available. important in influencing some of the aspects of simu- The numerical domain is a narrow estuary attached lated river plumes. In particular, he notes that main- to a uniformly sloping shelf with a straight coastline taining a very shallow depth at the coast prevents the (Fig. 3). The oceanic part of the domain is approxi- formation of a backward-propagating (against the mately 250 km long and 80 km wide, with variable reso- Kelvin wave propagation direction) bulge at the estuary lution concentrated near the estuarine outflow region mouth. Practically, the shallowest possible coastal wall and along the coast. Resolution is decreased near the depths are a few tens of centimeters in models like edges to inhibit small-scale alongshore variability at the ROMS that do not support wetting and drying. In this northern and southern boundaries. The model has 20 study, even in the cases with no wind and no back- vertical s layers, with resolution focused near the sur- ground flow (not shown here), a backward-propagating ϭ ␪ ϭ face (s-coordinate parameters used are hc 10 m, s bulge was not a problem. ␪ ϭ 5.0, and b 0.01). This is equivalent to better than 1-m The model is initiated with no flow and a flat sea vertical resolution in the upper5mofthewater column surface. The initial tracer distribution is uniform back- over the entire domain. Conservative splines are used ground salinity of 32 psu, with vertical temperature to estimate vertical gradients. stratification typical for an east coast continental shelf The estuary is 10 m deep, approximately 20 km long, in summer (a 10-m homogeneous mixed layer above and 1.5 km wide. Freshwater is introduced as a bound- exponential stratification with a 20-m decay scale, rang- ary condition on the westward end. Tides are not ex- ing from 20°C at the surface to 5°C at depth). The plicitly modeled. Tidal mixing within the estuary is pa- plume never interacts directly with the thermocline.

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FIG. 3. The model domain includes a flat-bottomed, prismatic estuary attached to a uniformly sloping coast. The shallowest depths are 10 m in the estuary and at the coast, and 70 m along the seaward edge. (upper right) A detail of the grid shows the model grid focused near the estuary mouth, with 500-m resolution in this region. Moderate resolution increases gradu- ally away from this point to 3 km, until very near the edges where it is more telescoped to much coarser resolution at the boundaries. This was done to increase the domain size and reduce grid-scale noise at the boundaries.

The estuary was initialized with a vertically uniform sional experiment has no mass flow into the domain along-channel salt gradient, linearly transitioning from from the river, but identical wind stress. The domain is oceanic salinity to freshwater with a 50-km length scale, identical, except that it is periodic in the north–south in order to decrease the estuarine adjustment time. The direction, with the same cross-shore topography as the model is forced with freshwater at the river end of the three-dimensional simulations. The flow is initially at estuary and spatially uniform but temporally oscillating rest, with no sea surface height anomalies. The eastern north–south wind stress; both are ramped over 1 day. boundary transport is again set to the Ekman transport. The depth-integrated flow at the eastern boundary is A mean alongshore background flow is added to the set equal to the Ekman transport. two-dimensional results before they are applied to the The northern boundary (the upstream boundary in three-dimensional model. the Kelvin wave sense) depth-integrated flow is relaxed In the three-dimensional simulations, mass is con- to results from a two-dimensional experiment to pre- served except for the gain and loss of mass due to the vent drift in the alongshore transport through the do- Ekman transport through the eastern open boundary main. In experiments without this boundary condition, by requiring the northern and southern boundaries the transport averaged over an upwelling/downwelling carry the same alongshore transport. This is accom- Ϫ cycle tended to drift O(0.10 m s 1). The two-dimen- plished by integrating the flow along these two bound-

Unauthenticated | Downloaded 09/29/21 02:36 PM UTC 1672 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 aries, then applying a small transport along all open ous freshwater source may be calculated by equating it boundaries to correct any mass transport imbalances. to the spurious, salty bottom water. All of the simulations have a freshwater flux of 1000 In simulations (not shown) with poor resolution or m3 sϪ1 applied at the river end of the estuary, ramped using low-order advection schemes, the spurious fresh- up over a day to prevent numerical shocks from form- water source may reach 10% of the real, specified fresh- ing. For the cases with wind, a spatially uniform, oscil- water source. The spurious freshwater source can be latory in time, alongshore wind stress with an amplitude reduced by applying a moderate horizontal mixing. Us- of 0.5 ϫ 10Ϫ4 m2 sϪ2 and a period of 4 days was used. A ing higher resolution and the Mellor–Yamada scheme mean background current, flowing in the Kelvin wave (which produces the smoothest fields), even when using propagation direction, of Ϫ0.05 m sϪ1 is specified at the very weak horizontal mixing, reduces the spurious northern boundary (in addition to the two-dimensional freshwater flux to essentially nil. The spurious freshwa- wind-driven transport). The main effect of the back- ter flux is 0.5% of the specified freshwater flux over the ground current is to increase the alongshore freshwater continental shelf region with no wind forcing and flux in the wind-driven simulations. 0.05% with wind forcing. The numerical configuration uses fourth-order hori- Two common turbulence closure schemes are used to ⑀ zontal advection for tracers with a grid-scaled horizon- calculate vertical mixing: Mellor–Yamada and k– tal diffusivity equivalent to 10.0 m2 sϪ1 fora1km2 grid (Mellor and Yamada 1974; Umlauf and Burchard cell (ranging from 5.0 to 50.0 m2 sϪ1 in the resolved 2003). The differences in plume water mass structure portion of the domain). Many numerical models are between different grid resolutions and advection run with a much smaller horizontal diffusivity, under schemes (assuming at least three grid points across the estuary mouth, and at least third-order advection) are the assumption that horizontal mixing should be kept as dwarfed by the differences in water mass structure us- low as possible. However, in earlier simulations there ing different vertical closure schemes. Because the was considerable numerical noise near the estuary structure of vertical mixing is an important theme in mouth in salinity that considerably affected the water this paper, numerical results both schemes have been mass structure of the plume, in some cases causing spu- included to gain a basic appreciation of the sensitivity rious numerical mixing comparable to the vertical mix- of the results presented to the choice of closure scheme. ing calculated within the turbulent closure scheme. It The background, or minimum, mixing used was identi- should be noted that the Mellor–Yamada scheme had Ϫ Ϫ cal for both closures: 5 ϫ 10 6 m2 s 1 for both momen- very little noise even when using a horizontal diffusion tum and tracers. These and other parameters used by nearly two orders of magnitude smaller. A third-order, the closure schemes were the default parameters for upwind scheme is used for horizontal momentum ad- ROMS version 2.1. Both shear and stratification were vection, with no explicit horizontal viscosity applied. averaged horizontally before mixing rates were calcu- Plume dimensions, such as bulge diameter, are sur- lated. The Kantha–Clayson stability function formula- prisingly insensitive to advection scheme, providing the tion was used for the Mellor–Yamada scheme (Kantha domain has adequate resolution near the estuary mouth and Clayson 1994), the Galperin stability function for- (approximately three to five grid points across the mulation was used for the k–⑀ scheme (Galperin et al. mouth). The qualitative structure of the solution is al- 1988). ways the same, even using poor resolution and a low- The configuration used in this paper was chosen to order advection scheme. All of the simulations without balance numerical accuracy with computational speed. wind have a bulge forming directly downstream of the This type of configuration could be applied to a realistic estuary outflow. The largest difference between advec- river plume simulation without significant modification schemes is in the formation of spurious fresh and tions to the standard code, and with reasonable inte- dense water. Numerical over and undershoots in the gration time on modern computers. It takes about9hof vicinity of the front would create slightly saltier and wall clock time to integrate the simulation for a month fresher water on both sides of the front. The freshwater on a single 2.3-GHz Intel Pentium-4 processor. is lighter than the surrounding water and remains at the surface. However, the saltier water is denser than the 4. Results surrounding water and sinks. This artificial unmixing a. Plume structure in physical space creates a pool of salty water along the seafloor, as well as a spurious source of freshwater near the surface. In 1) WIND-FORCED CASE the simulations presented in this paper, since back- The plume moves off- and onshore in response to ground salinity is constant, the magnitude of the spuri- upwelling and downwelling wind stresses, respectively.

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Results for the case with wind forcing is shown in Fig. u Fr ϭ , ͑8͒ 4. For this figure, the Mellor–Yamada mixing scheme is ͌gЈh used for this particular simulation; the results from the k–⑀ closure are qualitatively similar. The upper panels where show sea surface salinity with surface currents overlaid, ␩ ⌬␳ and lower panels show the freshwater thickness for the u ϭ ͵ |u| dz, gЈ ϭ g , and Ϫ ␳ final four days of the simulation. Freshwater thickness, h 0 ␩ ␩ relative to a reference salinity s0, is defined as the ver- Ϫ ⌬␳ ϭ ␳ Ϫ ͵ ␳|u| dzͲ͵ |u| dz, ͑9͒ tical integral of the salinity anomaly, (s0 s)/s0. The 0 Ϫh Ϫh plume changes position considerably over one cycle of the wind stress forcing. During upwelling, the plume and the upper-layer thickness h is defined as the point ␳ ϭ 1 ␳ Ϫ ␳ ⌬␳ nearly loses contact with the coast, a trait that becomes where ⁄2( 0 min). Note that this definition of more pronounced as the duration of upwelling in- is used only for the Froude number calculations. The creases. During downwelling, the plume is pressed depth-dependent flow speed in the upper layer is |u|; against the coast, developing a strong coastal current. the depth-dependent density is ␳. The integrations are The plume’s response to upwelling and downwelling is over the upper layer, between the free surface and the not symmetric: during upwelling the plume is blown interface defined by h. The upper-layer density is offshore, during downwelling the plume is pushed weighted by the flow speed to be consistent with a layer alongshore. This causes most of the alongshore fresh- model. water flux to occur during downwelling conditions; The numerically simulated near-field outflow region alongshore freshwater flux is essentially halted during has a similar structure to that described by Wright and upwelling. Wind also affects the thickness of the plume Coleman (1971), where the flow from South Pass is by stretching the plume out as it moves away from the shown to rapidly shoal at the mouth of the pass, with shore during upwelling, and pressing plume to the coast Froude numbers over 2 just past the point where the during downwelling. This affect is seen clearly in the pycnocline shallows during ebb tide. Beyond this point, snapshots of freshwater thickness (Fig. 4), where the the flow decelerates and becomes saltier due to entrain- freshwater thickness is least after upwelling, and great- ment of denser, sluggish background waters. In ap- est after downwelling. proximately 8–10 channel widths (depending on the phase of the tide) the South Pass outflow has entrained enough background water so that the Froude number is 2) NO-WIND CASE below 1. Note that it is the decrease in momentum, Figure 5 shows the properties of the plume with no rather than the increase in density, that is responsible wind forcing on day 16 of the simulation. A bulge has for the decrease in the Froude number due to entrain- formed near the outflow, with a recirculating gyre. The ment. bulge grows in time; as noted by Fong (1998), only a In the simulations presented here, the estuary mouth portion of the freshwater introduced continues down- acts as a constriction. The results of Armi and Farmer coast as a coastal current. In the case presented here, (1986) show how flow through a constriction must be about half of the freshwater input into the domain is supercritical, even when the estuarine exchange is not carried away by the coastal current. Because of this, maximal (Stommel and Farmer 1953). The simulated freshwater accumulates within this bulge and the bulge estuarine exchange is not maximal. Hetland and Geyer expands and thickens. The vertical salt flux along the (2004) argue this is not expected for a prismatic estuary | ץ ץ␬ surface defining the upper layer ( s/ z zϭϪh, discussed channel), but the simulated upper-layer outflow is su- in more detail below) is plotted in the third panel of Fig. percritical near the mouth of the estuary. Thus, the 5 and is used to estimate the regions where vertical simulated estuarine outflow is similar to flow through a mixing is strong. There is a region of strong vertical constriction in the case where only the upper-layer flow mixing near the estuary outflow about two orders of becomes supercritical (e.g., the bottom three panels in magnitude larger than the vertical mixing found in the Fig. 2 of Armi and Farmer 1986). One notable differ- rest of the plume. When the distribution of maximum ence is that Armi and Farmer (1986) show a relatively vertical salt flux is compared with the Froude number gradual transition to higher Froude numbers, with a (the fourth panel in Fig. 5), it is apparent that the region rapid transition back to subcritical flow in the form of a of high mixing is associated with supercritical flow. The hydraulic jump. In the numerical simulations presented Froude number in the upper layer is defined operation- here, as well as in the results of Wright and Coleman ally here as (1971), the transition to supercritical flow by shoaling

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FIG. 4. (top) Sea surface salinity and (bottom) freshwater thickness are plotted on a color scale for the final 4 days of the simulation with wind using the Mellor–Yamada closure. Instantaneous surface current vectors are overlaid in the upper panels. Contours in the lower panels are sea surface salinity. The yellow arrow indicates the direction of the wind during maximum upwelling and downwelling; the yellow dot indicates zero wind stress at that instant.

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FIG. 5. Four properties of the plume are shown on day 16 of the simulation without wind: sea surface salinity (with surface current | ץ ץ␬ vectors overlaid), freshwater thickness, a weighted average of the vertical salt flux [log( s/ z) zϭϪh], and the Froude number. Contours in the three right panels are sea surface salinity. Property definitions are given in the text. of the pycnocline is relatively rapid (occurring over less outflow, creating a thick, homogeneous mass of water than 1 km) in relation to the gradual transition back to within the bulge. subcritical flow through mixing (occurring over approximately 5 km). This difference is due to the pres- b. Plume structure in salinity space ence of mixing within the model and actual plumes that 1) FRESHWATER BUDGET is absent in the inviscid solutions of Armi and Farmer. Freshwater volume within different salinity classes is used in order to examine changes in whole plume struc- 3) PLUME REGIONS ture. The freshwater volume, relative to the reference

Approximate salinity ranges may be estimated for salinity, s0, is defined as the integral of the freshwater the boundaries between the estuary, near-field, and far- fraction field regions. The salinity ranges of the three plume s Ϫ s ͑ ͒ ϭ ͵͵͵ 0 ͑ ͒ regions are slightly different, depending on if the plume Vf sA dV, 10 Ͻ s is considered in three-dimensional space, or as a single s sA 0 active layer (see the appendix for more exact defini- where the volume integral is bounded by the isohaline tions of each view of the plume). These slight differ- sA, such that all of the water fresher than sA is contained ences are to be expected, given the different definitions in the integral. To determine the distribution of fresh- ץ ץ of plume salinity. However, the three regions are al- water as a function of salinity class, Vf / sA is plotted ץ ways discernible and the salinity ranges are similar for for the two turbulent closure schemes. Integrating Vf / ץ both views of the plume. The estuarine outflow surface sA over a range of salinities will give the total fresh- salinity (s|z ϭ ␩) ranges from 15 to 18 psu, with higher water contained within those salinity classes. The inte- salinity outflow during downwelling. The upper-layer gral over the entire range is identical for all cases, since salinity (sl) leaving the estuary ranges from 18 to 20 psu, the each case has the same freshwater input. also with higher salinity during downwelling. The salin- As mixing increases, freshwater will generally be ity range of the near field is seen most clearly in the moved toward higher salinity classes. Freshwater distri- cases without wind. Water leaving the near field has an butions are shown for the two turbulent mixing closures ϭ upper-layer salinity of approximately sl 25–26 psu. used in this paper (Fig. 6, top). Although the qualitative This water recirculates within the bulge that forms just structure is similar between the different closure downstream, in the Kelvin wave sense, of the estuarine schemes, there are significant differences in the actual

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ץ ץ FIG. 6. (top) The distribution of freshwater in salinity space, Vf / sA, compared for the two different mixing schemes with and without wind forcing. (bottom) Time-dependent anomalies

in Vf, plotted as percent deviation from mean values with a contour interval of 10%, for the Mellor–Yamada case with wind.

amount of freshwater found at different salinity classes. shown for cases with wind and no wind. Values of Al, Notably, the k–⑀ closure seems to mix more than the averaged over one upwelling/downwelling cycle (the Mellor–Yamada closure for the case with wind. Be- period of time between days 16 and 20), are shown in cause of the large differences between these two upper panel of Fig. 7. The results show that, between schemes, results that depend on the turbulence closure the two turbulent closure schemes, the largest differ- are presented using both schemes. The local maximum ence in Al is within the near field, with surface salinities ץ ץ in Vf / sA for the no-wind case shows the buildup of of approximately 20–26 psu, both with and without Ͻ water within the recirculating bulge at approximately wind. Within the estuary (sl 20 psu) and in the far 24 psu. Ͼ field (sl 26 psu) the both schemes produce similar For the wind-driven cases, the largest changes in values for Al. freshwater enclosed by an isohaline, sA, are at the Wind affects both the near field and far field, with the boundary between the near and far fields (s ϳ 26 psu). A largest difference in Al at the interface between the The lower panel of Fig. 6 shows the time-dependent near and far fields. Wind has very little affect on Al anomalies of Vf as a percent of the mean. Freshwater is within the estuary and at very high salinity values. An lost from the region of high variability (24 psu Ͻ s Ͻ A example of time-dependent changes in Al due to wind 28 psu) during upwelling, and the freshwater is replen- forcing is shown in the lower panel of Fig. 7, where ished during downwelling. This variability is due en- percent changes in Al are are plotted as a function of sl tirely to changes in mixing within the plume and the and time. These results are based on the Mellor– corresponding flux of freshwater across isohaline sur- Yamada run with wind; results for the k–⑀ scheme were faces, since s otherwise follows the plume as it is A qualitatively similar. The largest anomalies in Al form moved adiabatically. at salinity ranges between the near and far fields just after downwelling, similar to the wind-driven anomalies 2) HORIZONTAL PLUME STRUCTURE in Vf. This high anomaly propagates toward higher sa-

The area Al, enclosed by the upper-layer salinity con- linity values during upwelling, again suggesting water tour sl, is calculated as a function of sl for the four moves from the near field to the far field in a coherent model runs. Both turbulence closure schemes are pulse during upwelling.

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FIG. 7. (top) The area, Al, averaged over one upwelling/downwelling wind stress cycle (16–20 day) enclosed by the upper-layer salinity contour, sl. (bottom) Time-dependent anomalies, plotted as percent deviation from mean values with a contour interval of 25%, for the Mellor–Yamada case with wind.

The variability in plume area contains both adiabatic done with the layer model where upper-layer thickness plume motions, increases and decreases in area due was averaged over salinity classes (red lines in Fig. 8). only to the plume being stretched by upwelling and Again, the square root of the area covered by each compressed by downwelling, and nonadiabatic changes salinity range was integrated and summed along salinity to plume structure due to mixing within the plume. class to estimate an along-plume coordinate. Based on the freshwater anomalies shown in Fig. 6 The two cases without wind are very similar. The (bottom), which are entirely nonadiabatic, approxi- halocline remains at a relatively constant depth with a mately one-half of the variability in wind-forced plume slight decrease in depth at the end of the near-field area is due to adiabatic motions, the other half to nona- region. Both the Mellor–Yamada and k–⑀ closure pro- diabatic mixing processes. duce a similar vertical structure at all portions of the plume. The horizontal changes discussed above are apparent, with the k–⑀ scheme producing a larger near- 3) VERTICAL PLUME STRUCTURE field region. An average salinity profile was calculated as a func- In salinity space, the simulations with wind are, on tion of sea surface salinity by finding the average depth average, thinner in the near field and thicker in the far of each isohaline underneath a particular range of sea field when compared with the no-wind cases. The k–⑀ surface salinity. The area covered by each sea surface simulations were 20% thinner in the near field and 20% salinity range is integrated, and an along-plume coor- thicker in the far field after the inclusion of wind. Wind dinate with units of distance is calculated by taking the caused the Mellor–Yamada simulations to be 10% thin- square root of this quantity. The isohaline depth and ner in the near field and 60% thicker in the far field. along-plume distance calculated from each of the four The wind-driven surface layer is less stratified in the standard cases were averaged over one upwelling/ Mellor–Yamada case, and the halocline for the k–⑀ downwelling period (from days 16 to 20). The result is case is shallower than the Mellor–Yamada case every- an idealized average cross section of the plume in sa- where. Both closure schemes show that, beyond the linity space. The results are shown as a salinity space estuary, wind causes the plume to thicken; as the plume profile (black lines in Fig. 8). A similar calculation was gets saltier, the decrease in plume thickness at the end

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FIG. 8. Ideal cross sections of the salinity structure. Thin lines show isohalines with an interval of 1 psu. Thick lines are drawn every 5 psu (e.g., 20, 25, and 30 psu). The rightmost thick line shows the 30-psu isohaline in all four panels.

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Fig 8 live 4/C SEPTEMBER 2005 H E T LAND 1679

ϭ of the near-field region seen in the no-wind case is ab- is the freshwater thickness. Solving for h hc, the criti- ϭ sent. cal upper-layer thickness when Ri Ric, gives The largest time-dependent changes in wind-forced 2␶ Ri Fr plume thickness are at the interface between the near ϭ ͱ c ϭ ͌ ϭ d hc ␳ ⌬␳ ր␳ 2hf Ric Frd 2hf , and far fields, at the same point in salinity space as the 0 f hf g f 0 Frc maximum percent variability in plume area and fresh- ͑12͒ water volume. The plume thins during upwelling, and thickens during downwelling, as expected. Time- stating that the final upper-layer thickness depends dependent changes in plume thickness due to wind only on the initial conditions, the value of Ric, and the stress (not shown) indicate that the plume thickness magnitude of the (maximum) wind stress. This equa- changes by approximately 60% at the interface be- tion may be cast in terms of an upper-layer, freshwater ϭ ϭ ␶ ␳ ϭ tween the near and far fields, but only about 15% at Froude number, Fd uf /cf, where uf /f 0 hf and cf Ͼ ͌h g⌬␳ ␳ both higher salinity classes (sl 30) and lower salinity f f / 0. This upper-layer, freshwater Froude Ͻ Ͻ ⑀ classes (20 sl 22). The Mellor–Yamada and k– number is similar to the densimetric Froude number cases are similar, with the principal differences being often used in studies of estuarine circulation (e.g., the same as discussed above for the average profiles. Hansen and Rattray 1966; MacCready 1999). If the critical Richardson number is written as a critical ϭ 2 c. Far-field wind mixing Froude number, Ric 1/Frc, the critical thickness may Away from the mouth of the estuary, mixing in the be simply calculated as 2 times the freshwater thickness plume is due primarily to surface wind stress. The basic times a ratio of the freshwater Froude number to the mechanism was described by Fong and Geyer (2001), critical Froude number. who describe a one-dimensional model in which a Equation (12) may be converted to a critical salinity buoyant layer is mixed by shear mixing. The shear be- by conserving the total freshwater in the water column ϭ ⌬ tween the upper and lower layers is created by the Ek- (hf hc s/s0)toget man transport of the upper layer. If the Ekman trans- h Fr ϭ ͩ Ϫ f ͪ ϭ ͩ Ϫ c ͪ ͑ ͒ port is large enough to induce shear instability in the sc s0 1 s0 1 . 13 hc 2Frd upper layer, it will mix, entraining lower-layer water until the Richardson number rises above the critical If the local salinity in the plume is less than sc, mixing value. Fong and Geyer calculated the depth at which will occur, if the local salinity is greater than sc, the this criteria would be reached, based on a density dif- water column is stable. ference between the upper and lower layers and a Equation (13) was derived assuming a slab-like upper specified wind stress. In their calculation, the upper- layer, with constant velocity and density. If the layer is layer density was approximated as constant. considered to have uniform gradients, as in Fong and Ϫ1/2 This theory can be extended to include density Geyer (2001), there will be an additional 2 factor on changes in the upper layer due to entrainment of lower- the right-hand side. However, changes such as this are layer water. Consider a layer of purely freshwater, with equivalent to changes in the critical bulk Richardson ␳ ␳ number, and do not change the dynamical meaning of density f and hf thick, overlaying denser ( 0) ocean water, infinitely deep. A wind stress, ␶, blows over the the equation. water causing and Ekman transport in the upper layer After the plume leaves the near-field region, mixing ϭ ␶ ␳ due to the inertia of the estuarine outflow is suppressed of uh / 0 f, where u is the velocity in the upper layer and h is the upper-layer thickness. The upper layer may and wind mixing dominates. The critical thickness of undergo shear mixing, and entrain water from the un- the plume as a function of freshwater thickness a wind derlying ocean when the bulk Richardson number, Ri stress [Eq. (12)] suggests that wind mixing may act to ϭ⌬␳ ␳ ⌬ 2 ⌬␳ stabilize the water mass structure of the plume. Even if gh/ 0 u (where is the density difference, and ⌬u is the velocity difference between the two layers), the upper layer is mixed by some other mechanism before the maximum in wind stress occurs, wind mixing becomes lower than a critical value Ric. Now, allow the density of the upper layer to decrease as deeper water will still mix the upper layer to the same thickness, so ⌬␳ ϭ ⌬␳ ⌬␳ ϭ ␳ Ϫ ␳ long as h is larger than the thickness of the upper layer is entrained, so that hf f h , where f ( f 0) c is the density difference between freshwater and the after the initial mixing. However, this assumes that the reference state, and horizontal distribution of freshwater is given. It is likely that this distribution is in some ways related to the ␩ s Ϫ s ϵ ͵ 0 ͑ ͒ water mass structure itself. For instance, vertical thick- hf dz 11 ϪH s0 ness of the mixed layer will determine the speed of the

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Ekman transport in the upper layer, and density ѨA Ϫ1 ϭ ͩ ͪ ͑ ͒ anomalies between the plume and the background flow fA QR Ѩ , 14 sA will determine the propagation speed of the coastal jet. If, on the other hand, the plume has already been mixed where A is the average area contained within isohaline to or beyond the critical thickness, the plume will not sA over one upwelling/downwelling cycle. mix further. This mixing could be caused by a previous A direct estimate of the salt flux is calculated by large wind event, for example. In this case, the theory interpolating the salt flux to the depth of the upper- suggests that the plume will be protected from further layer thickness wind mixing; the water mass structure of the plume will Ѩs be determined by the largest previous wind event. ϵ ␬ ͑ ͒ f␬ ͩ Ѩ ͪ , 15 To examine how well this theory can predict the sa- z zϭϪh linity structure of a river plume, the plume sea surface again averaged over one upwelling/downwelling cycle. salinity was compared to the estimated critical salinity Here, ␬ is the turbulent diffusivity calculated within the in the upper layer. In Fig. 9, the simulated sea surface model. A comparison between f and f␬ for the two salinity is compared to the critical upper-layer salinity A turbulence closure schemes is shown in Fig. 10. The two calculated from Eq. (12) (noting that h S ϭ h⌬S) using f 0 estimates of the salt flux show the same structure for the local freshwater thickness and Ri ϭ 3.0. During c each turbulence closure, with high salt flux in the near- maximum upwelling, there is a band of sea surface sa- field where shear mixing is strong, decreasing by nearly linity associated with high mixing that close to the pre- an order of magnitude at higher salinities, where wind dicted critical upper-layer salinity. These parts of the mixing is the dominant entrainment process. The two plume are being mixed by the wind, such that the sa- closures, however, are distinct. The Mellor–Yamada linity is not greater than the critical salinity. Locations scheme has a much higher salt flux in the near field as in the plume that have a salinity greater than the critical compared with the k–⑀ scheme. salinity are associated with lower mixing. These parts of The assumption of a steady state appears to be valid the plume are not as affected by wind mixing because for salinity classes that have a time scale smaller than the salinity there is greater than the critical salinity. the period of forcing, here about 4 days. The upper This suggests that these portions of the plume with sa- panel of Fig. 11 shows the plume time scales at different linity higher than the critical salinity are protected from salinity classes by integrating the freshwater volume further wind mixing. During downwelling, nearly all of contained within an isohaline and dividing this quantity the plume has subcritical values of surface salinity, and by the freshwater flux. This provides a filling time, the strong vertical mixing is confined to fresher waters in time it would take the freshwater flux, Q , to replace the the near-field region, where mixing always occurs re- f freshwater volume, V , for each salinity class. gardless of the phase of the wind stress. This is in agree- f Different portions of the plume have different time ment with the freshwater volume analysis above, which scales with respect to both wind and freshwater forcing suggests that freshwater is moved from lower to higher due to the differences in the volume of water at differ- salinity classes during upwelling. ent salinity classes. In the above estimate of salt flux, a The value of Ri used is only an effective value, since c steady state is assumed. The validity of this assumption actual mixing in the plume is controlled not only by may be estimated by taking a ratio of the time-depen- shear mixing of the upper-layer Ekman flow, but also dent term to the freshwater forcing term in equation. shear caused by geostrophic or inertial flow in addition Integrating over one upwelling/downwelling cycle of to the Ekman flow. The enhanced shear in the upper period T gives layer requires that the effective critical Richardson number used in the theory here be larger than typically Change in fresh water content Ѩ tϩT used, since the theory of critical thickness only includes ϭ ͫ ͑s V ͒ͬ Fresh water flux input Ѩs A fA Ekman induced shear. A t ϫ ͑ ͒Ϫ1 ͑ ͒ TQf . 16 d. Average salt flux over one upwelling/ The ratio of these terms is plotted in the lower panel of downwelling cycle Fig. 11. The steady-state assumption is valid for lower- After the plume has reached a quasi-steady state, the salinity classes lower than a certain value. This value average entrainment into the plume over an upwelling/ changes, depending on whether wind is included, but it downwelling cycle can be calculated from the steady seems to correspond to salinity classes that have a fill- Ϫ1 state form of Eq. (7), ing times, Vf Qf , of about 3–5 days.

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FIG. 9. (top) The maximum vertical salt flux for extreme upwelling and downwelling winds in the upper two panels, and the corresponding critical salinity vs (bottom) the actual plume surface salinity. In all, the logarithmic color scale represents a weighted | ץ ץ␬ average of the vertical salt flux ( s/ z zϭϪh) for a particular horizontal point in the domain. The lower panels are based on Eq. (13). ϭ ϭ Ϫ The solid line shows the relation sc smodel; the dashed line shows a 5-psu offset, defined by sc smodel 5.

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Fig 9 live 4/C 1682 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35

| ץ ץ␬ FIG. 10. Two estimates of fA, the vertical salt flux at salinity class sA, are shown. Red lines show s/ z zϭϪh, the vertical salt flux calculated from the model at the base of the upper layer (z ϭϪh). The blue lines are calculated from Eq. (14), assuming a steady state. Both estimates have been filtered so that the resolution in salinity space is approximately 1 psu. Thin gray lines in the upper panel represent the colored lines in the bottom panel, and vice versa..

e. Work done by vertical mixing surface mixed layer) when calculating the density. Po- tential energy changes due to changes in sea surface Turbulent mixing does work against buoyancy by height, adiabatic changes in plume structure, and raising the center of mass of the water column. The rate changes due to mixing of the thermocline were all large of work, dW/dt, done by vertical mixing on the density compared to potential energy changes due to salt flux, structure of the plume, equal to the rate of nonadiabatic and have been excluded from this calculation. changes in potential energy due to vertical mixing, is Wind increases the total rate of mixing work. Figure calculated as g times a volume integral of the vertical 12 (top) shows time series of the rate of mixing work for salt flux, the four standard cases. Work done by mixing is highest dW Ѩ␳ when the wind stress is greatest. Wind increases the ϵ ͵ g␬ dV, ͑17͒ dt Ѩz work done by mixing for both closure schemes, with the V total rate of mixing work using the Mellor–Yamada referred to simply as the rate of mixing work. In the scheme greater than the k–⑀ scheme. The asymmetry in calculations here, only density changes due to salt flux rate of mixing work due to the wind is also greater in were considered; this was accomplished by holding the Mellor–Yamada scheme, with most work done dur- temperature constant (the mean temperature of the ing periods of upwelling wind. For the case without

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Fig 10 live 4/C SEPTEMBER 2005 H E T LAND 1683

FIG. 11. (top) Flushing time scales (volume of freshwater over the freshwater flux) of plume

waters within isohaline surfaces sA. The freshwater volume relative to sA is averaged for days 16–20, over one upwelling/downwelling cycle. Plume time scales range from just over 1 day at the start of the near field to 18 days at the highest salinity classes (the average total time of integration for this calculation). (bottom) The relative importance of the time-dependent term in Eq. (7).

⑀ wind, the rate of mixing work calculated using the k– where Al is the area defined by upper-layer salinities closure is 60% larger than that calculated using the less than sl. The rate of work done by mixing as a func- Mellor–Yamada scheme, excluding mixing in the estu- tion of salinity class is averaged over one upwelling/ ary. Mixing within the estuary is due only changes in the downwelling cycle (from days 16 to 20), and the results density of water entering the estuary, since mixing is are plotted in the middle panel of Fig. 12. The rate of Ͻ held constant within estuary. mixing work within the estuary (sl 20 psu) is nearly The combined rate of mixing work done in the estu- identical at all times and for both closure schemes, as ary and near field is comparable to the rate of mixing expected. For the no-wind case, the rate of work done ϳ work done in the far field, and the rates of work done by mixing is higher both in the near field (at sl 25 psu) Ͼ in the estuary and near field are similar. The additional as well as at higher salinity classes (at sl 28 psu) for rate of mixing work added by the wind, calculated as the k–⑀ scheme. Wind significantly increases the rate of the rate of work in the case with no wind subtracted work done by mixing at higher salinity classes. For the from the case with wind, is about 10% less than the rate k–⑀ scheme, wind decreases the rate of work done by of work in the no wind case (again, excluding mixing in mixing in the near field for the k–⑀ scheme, and shifts the estuary) for the k–⑀ closure. However, the addi- the mixing to lower salinity classes. Wind increases the tional rate of work from including wind is almost 3 rate of work done by mixing at all salinity classes for the times larger, excluding the estuary, than the no wind Mellor–Yamada scheme. case for the Mellor–Yamada closure. Mixing rates (e.g., Fig. 10) are related to work per The two closure schemes mix different salinity classes unit area rather than total work in a given salinity class. within the plume at different rates. The rate of work The lower panel of Fig. 12 shows the rate of work done by mixing per unit area as a function of salinity class. done by mixing as a function of upper-layer salinity, sl, was calculated as The total increase rate of work done by the wind mixing is very large within the far field; however, because the area of the plume is large at these salinity classes, the Ѩ dW Ѩ ␩ Ѩ␳ ͩ ͪ ϵ ͵͵ ͩ͵ g dzͪ dA, ͑18͒ work per unit area in this region remains small. Al- Ѩs dt Ѩs Ѩz l l Al ϪH though wind forcing raises the rate of work per unit

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FIG. 12. (top) A time series of total work done by vertical mixing for different simulations. (middle) The work done by mixing averaged over days 1–20, one upwelling/downwelling cycle, plotted as a function of upper-layer salinity. (bottom) The average work done by mixing per unit area as a function of upper-layer salinity.

area done in the far field, rate of work per unit area rate of work done per unit area in the near filed is done in the near field remains about an order of mag- smaller for the k–⑀ scheme. This is consistent with the nitude larger. Also, although the total rate of work k–⑀ simulations having weaker mixing and larger values done in the near field is the same for both schemes, the of Al in the near field.

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Fig 12 live 4/C SEPTEMBER 2005 H E T LAND 1685

5. Discussion error as long as the critical thickness (or corresponding upper-layer salinity) has not yet been reached. The various regions of the plume are most apparent The fact that wind mixing is reduced after the plume in the cases without wind forcing. In this case, the near reaches its critical thickness may be the reason that the Ͼ field is apparent as a local maximum in the work done, total area of the plume at very high salinity classes (sl ץ ץ a local high in Al/ sl, as well as feeding a local maxi- 30) changes very little with the inclusion of wind. Since ץ ץ mum in Vf / s〈. It is interesting to note that, although the average mixing rate in the far field is still orders of the mixing rates are lower for the k–⑀ closure (Fig. 10), magnitude smaller than in the near field and that rate the amount of mixing is greater (Fig. 12) because the does not change substantially with the inclusion of wind area over which the mixing occurs is larger (Fig. 7). within the far field, the area of the plume at very high The largest changes between the cases with wind and salinity classes may remain the same, since the area is the cases without wind are in the range of salinities inversely related to the mixing across that isohaline and between the near and far field. At these salinity classes, that mixing is near to the background mixing regardless the wind takes water leaving the near field and mixes of the presence of the wind. this water toward higher salinity classes, primarily dur- Clearly the wind will also change the geographical ing upwelling (see Fig. 7). In general, the interface be- position of the plume, stretching it out during upwelling tween the near and far field is apparent in the anomaly (increasing the area) and pressing it against the coast fields of the cases that include wind, but the near and during downwelling (decreasing the area). However, on far field are not apparent when simply looking at time- average, the area of the plume at very high salinity averaged area enclosed by upper-layer salinity con- classes seemed to be relatively insensitive to the pres- tours; the wind forcing tends to erase this boundary in ence of wind. This result would change if the mean of the mean. the wind stress were nonzero. The plume areas were all Wind moves water from near-field to far-field salinity similar (within 10%) with and without wind, except that classes when the wind stress is the highest. Because of the k–⑀ scheme was about 20–30% larger at the highest the asymmetries in mixing with regard to the phase of salinity classes than the other three cases (not shown). the wind stress, the calculations using the Mellor– Changes in the plume area at very high salinity classes Ͼ Yamada scheme tend to move water toward higher sa- (sl 31) are dominated by the increasing freshwater linity classes more during upwelling than downwelling. introduced to the system, since changes in the freshwa- The percent anomalies in freshwater volume (Fig. 6) ter volume contained within the highest salinity class and plume area (Fig. 7) tend to be highest at the inter- are exactly related to the total freshwater introduced face between the near and far fields and become lower into the system. at higher salinity classes. The percent anomaly de- The freshwater volume of the plume is the truest creases because the freshwater volume and plume area representation of the plume in salinity space, since it is at higher salinity classes is larger, resulting in a rela- completely independent of plume position. However, tively smaller percent anomaly. In addition to this, how- because water mass modification is related to both the ever, water will be mixed quickly up to its local critical mixing rate and the local changes plume area [see Eq. salinity [Eq. (13)], after which mixing will be sup- (7)], changes in plume structure do depend indirectly pressed, further reducing wind-forced anomalies in on plume position through the plume area. In this re- freshwater volume and plume area at higher salinity spect, the salinity space view of the plume is incomplete classes. without some understanding of the Cartesian view of Wind stress may act to stabilize the structure of the the plume. plume in salinity space by actively mixing the plume only to a particular point. Equation (12) and the cor- 6. Conclusions responding analysis presented in Fig. 9 suggest that wind mixing will be strong until the plume has reached The water mass structure of an idealized river plume a particular critical thickness, after which mixing will was examined in the context of changing wind stress decrease. Portions of the plume at or above the critical amplitude for two common turbulence closure salinity will be protected from further turbulent mixing schemes, Mellor–Yamada and k–⑀. Wind stress changes by the wind. Given the same horizontal freshwater the position of the plume and increases the mixing thickness distribution, the plume will mix to the same within the plume. The focus of this paper is the rela- end member independent of mixing history, so if poor tionship between plume horizontal dimensions and ver- numerical resolution causes more or less mixing in the tical mixing, so the changing position of the plume is near-field region, wind mixing may compensate for this deemphasized through the use of salinity coordinates.

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This coordinate system allows the size of the plume to tions (providing the measurements resolve the thin sur- be investigated independently from the position of the face plume). Salinity coordinates offer an integrative plume. view of the plume that does not depend on small The plume can be divided into two dynamically dis- changes in frontal position, may therefore be used to tinct regions: the near field, characterized by both in- supplement more conventional model/data compari- ertial shear mixing and wind mixing, and the far field, sons based in geographic space. Many of the results characterized by only wind mixing when present. The discussed in this paper would be difficult to show using near field mixing is localized at lower salinity values in a Cartesian view of the plume. Salinity coordinates are the cases presented here between approximately 20 and extremely useful in examining mixing processes within 26 psu. Wind mixing caused the largest changes in sa- the plume, since adiabatic changes in position are ig- linity space plume structure at salinity ranges bounding nored by the calculation. the near and far field: 24–29 psu in the cases presented Garvine’s (1999) results can now be interpreted using here. Wind mixing did not affect higher salinity classes isohaline coordinates. In a steady state, all of the fresh- because the plume had reached its critical thickness (or water input into the system through the river must pass equivalently, its critical salinity). Turbulent mixing through each isohaline in the plume. In order of isoha- caused by the wind is suppressed after this point, and lines beyond the near-field mixing region to pass this the mixing approaches background levels. The critical freshwater through at background values of diffusivity, thickness depends on the magnitude of the wind stress the surface area of that isohaline must be large. If the and the equivalent freshwater thickness. The mixing at background diffusivity is reduced by one-half, the sur- the highest salinity classes is therefore largely con- face area must double to compensate, so that the fresh- trolled by background mixing. water flux through the surface remains constant. Wind Mixing within the plume, caused by advective shear forcing modifies the structure of the regions that are mixing and wind stress, is related to the surface area of susceptible to wind mixing, that are fresher than the isohalines within the plume. Given the same freshwater critical salinity, by increasing the mixing and reducing flux, strong mixing requires only a small isohaline area, the area proportionally. However, at higher salinity whereas weaker mixing requires a larger isohaline area classes the plume is typically protected from the effects to maintain the same total freshwater flux across the of further wind mixing, the local salinity is above the isohaline. Strong mixing due to inertial shear is con- critical salinity, preventing strong mixing and increasing fined to the near field. Wind mixing is strongest during the plume area. Thus, although differing in details, this upwelling; however, the average rate of mixing in the study confirms Garvine’s basic result that plume struc- far field is much smaller than in the near field. Thus, the ture depends fundamentally on mixing and the mixing far-field plume has a much larger surface area and parameterization. longer time scales of water mass modification. Time- scale analysis shows that the near-field plume is in a Acknowledgments. I thank Parker MacCready, steady balance by the end of the integration time (20 Rocky Geyer, Steve Lentz, and Rich Signell for many days), but the far-field plume is still changing. The helpful comments and suggestions. This project was higher rates of mixing and correspondingly lower areal supported by ONR Grant N00014-03-1-0398. extent in the near field are responsible for the faster adjustment times within the near field. APPENDIX A reasonable approximation to the plume structure Definition of Terms is a single active layer in which salinity may vary horizontally within the layer, but the pycnocline is very thin In this paper, there are two complementary views of everywhere. A single layer approximation may be used the plume within salinity space. The first is a three- to obtain an estimate of the salt flux across the pycno- dimensional view, using an isohaline surface to bound cline that has the same characteristics as a direct esti- different regions of the plume. The critical variables in mate using a weighted average of the vertical salt flux. this view of the plume are the isohaline sA, the area One of the difficulties in comparing numerical simu- defined by this isohaline A, and the volume enclosed by lations of river plumes to observations is that river this isohaline V. Freshwater contained within the iso- plumes change position, so at any given point, the haline surface may be defined in one of two ways. The plume may be present only some of the time. Because freshwater relative to a constant reference salinity, s0,is the analysis presented in this paper focuses on distri- Vf. The freshwater relative to the salinity bounding iso- butions of salinity, the analysis methods may be applied haline is VfA. Derivatives with respect to salinity are to both numerical output and hydrographic observa- defined using sA.

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