Analysis of Stratified Flow and Separation Over Complex Bathymetry in a Field-Scale Estuarine Model

Oliver B. Fringer and Bing Wang Environmental Fluid Mechanics Laboratory, Department of Civil and Environmental Engineering,

Stanford University, Stanford, CA {fringer, bingwang}@stanford.edu

Abstract

Tidal flows in estuaries produce surface-coherent structures, such as eddies, when the flow interacts with complex topography. These coherent structures, in turn, can be used to infer the properties of the flow, such as speed and direction, turbulent intensity, stratification, suspended sediment concentration, and bathymetry. While simulation of coherent structures is straightforward in smaller domains with idealized geometries, it is extremely difficult to accurately simulate them in real field-scale domains. Accurate simulation of small-scale features requires that the model first accurately simulate the large-scale, tidally-induced flow, which depends to great extent on accurate simulation of salinity and the complex flow features over mudflats that can become exposed during low tides. In this paper, we will analyze the stratified tidal flow over a sill in the Snohomish River estuary and the associated eddies that are produced. We will first discuss model details that are required to simulate the tidal-scale hydrodynamics in the domain that is O(10 km) in extent, followed by the flow physics of the eddies with length scales of O(1 m). 1. Introduction Surface-coherent structures can be thought of as any flow structure on the surface that is coherent, such as boils or eddies. Of significant interest to the Office Naval Research (ONR) is the ability to use remote sensing of surface-coherent structures in rivers and estuaries to infer properties of the flow, such as speed, salinity, sediment concentration, and depth. As an example, infrared images of the free surface in the ocean yield estimates of the sea-surface temperature variability, which can in turn provide estimates of surface winds and heat flux (Wick et al., 1996). In this paper, we study coherent structures induced by tidal flow over a sill in an estuary using high-resolution numerical simulations.

The hydrodynamics of estuaries is difficult to simulate numerically because of the disparate range of length scales involved. At the largest scales, the tides induce currents which depend strongly on the large-scale geometry of the estuary, which is typically O(10 km)-O(100 km) in extent. Superimposed over the tidal currents are the density-induced currents at the interface of the salty ocean and the freshwater inflow, where the salty, dense ocean water tends to flow beneath the fresh, lighter river water in what is termed baroclinic circulation. This exchange flow involves turbulent eddies which are typically O(0.1 m)-O(1 m). Typical numerical simulations of the flow in estuaries seek to capture the tidal-scale hydrodynamics, which requires grid resolutions of O(10 m)-O(100 m) (Wang et al., 2009), and the shorter length-scale processes are captured with turbulence models (Warner et al., 2005). When shorter length-scale motions are to be resolved, such as eddies resulting from flow separation over complex topography, significant computing power is necessary, since resolutions of O(1 m) are required in domains of O(105 m). In addition to the obvious cost related to grid resolution, high-resolution simulations incur significant constraints on the time-step size. The time step size required to resolve eddies on a 1 m grid in a fast-moving, 2 m s−1 current is 0.5 s, which then requires 172,800 time steps to simulate the separation dynamics over one tidal cycle (defined here as 24 hours). Although this constraint is highly limiting, it is necessary on the grounds of simply resolving the time scale of interest. However, further limiting the time step size in estuarine simulations is the wetting and drying of tidal mudflats. Estuaries are typically composed of shallow regions known as mudflats incised by a network of channels. These mudflats are inundated during high tides and are typically exposed during low tides. Numerical simulation of the wetting and drying of tidal mudflats imposes a significant time step limitation on the governing equations because the time step is limited by the requirement that the volumetric

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DOI 10.1109/HPCMP-UGC.2010.14

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flow out of a given cell over one time step cannot exceed the volume in that cell. Since the volume in the cells over mudflats becomes very small as the mudflats become dry, the time-step size is severely limited to O(0.1 s) with grid-resolutions of O(1 m). When the horizontal scales of interest are long relative to the depth, the flow is hydrostatic, whereby vertical accelerations are insignificant and so do not affect the pressure field. As a result, the pressure field results only from the weight of the fluid above a parcel of fluid, and so it can be computed directly from the depth beneath the free surface and the density field. For most ocean and estuarine models, this is a reasonable approximation. However, when the horizontal scale of the motion is on the order of the depth, the nonhydrostatic pressure, or the hydrodynamic pressure which arises from forces due to fluid motion, becomes important. Unlike the hydrostatic pressure, the nonhydrostatic pressure requires solution of a three-dimensional (3D), elliptic equation. Therefore, high-resolution simulations of estuaries not only incur computational expense due to high grid resolution and small time steps, but the short length-scales of interest relative to the depth require the nonhydrostatic pressure, the solution of which can increase the computational expense by as much as a factor of 10. In this paper, we perform high-resolution simulations of the flow in the Snohomish River Estuary, near the city of Everett, WA. The Snohomish River discharges into Possession Sound, which is one of the basins in Puget Sound (Figure 1). The mean depth in the estuary is 3 to 5 m at the center of the river channel, and much of the estuary consists of mudflats which become dry during low tide. Figure 2a shows the bathymetry relative to low tide; negative depth values are always submerged, while positive depths are exposed, or dry, during low tide. The lower mainstem of the estuary is defined by Jetty Island to the west and the city of Everett to the east. To the north of Jetty Island, there is a bypass consisting of extensive intertidal mudflats (indicated by positive depth values). The bypass connects the main channel to Possession Sound during high tide and is disconnected during low tide. At the tip of Jetty Island, there is an abrupt sill extending over roughly one-third of the channel width. The river bed has been considerably scoured by disturbed flow around this sill, as illustrated by the bathymetry in Figure 2b. We focus on resolving the eddies in the lee of this sill in order to understand the physics of flow separation in the presence of a highly energetic, oscillatory, tidal flow. Although simulations of stratified flow and separation of tidal flow over idealized topography have been performed (e.g., Klymak and Gregg, 2003), no simulations of separation over realistic topography in a real, three-dimensional estuarine model, particularly one that simulates a domain as complex as that of the Snohomish, exist in the literature.

Figure 1. The Snohomish River estuary

Figure 2. Bathymetry (m above low tide) of the estuary (a)

and around the sill at the tip of Jetty Island (b). Flow details along transects E-E’ and S-S’ are discussed in this paper.

2. Methodology We implement the parallel, unstructured-grid, finite-volume ocean model SUNTANS (Fringer et al., 2006) to simulate the hydrodynamics in the Snohomish River estuary. The SUNTANS model solves the Navier-Stokes equation under the Boussinesq approximation on an unstructured, prismatic grid that is triangular in the horizontal and structured in the vertical. The flexibility of local refinement in the horizontal provides a great advantage in simulating multiscale processes. SUNTANS is written in the C programming language with Message Passing Interface (MPI) for parallelization, and the ParMETIS package is used for parallel graph partitioning and reordering. A major addition to what is described in Fringer et al. (2006) is the semi-Lagrangian method for advection of momentum, which ensures stability in the presence of severe wetting and drying conditions in which the thickness of the cell near the free surface may become exceedingly small. The model predicts the inundation of intertidal mudflats reasonably well, which is important given the importance of wetting and drying in the hydrodynamics of shallow estuaries (Zheng et al., 2003).

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The simulations in this paper build upon the simulations presented in Wang et al. (2009, 2010), which demonstrate that the model accurately predicts tidal-scale currents and salinity dynamics, as exemplified by favorable comparison to measurements. Details regarding model validation, boundary conditions, and turbulence parameterizations can also be found in those papers. The simulation setup for the results in this paper is similar, although higher-resolution (by a factor of 8) is employed. The computational domain for the high-resolution simulations presented in this paper is sized to capture the effects of the tides in Possession Sound and the currents they produce to transport the salinity front in and out of the Snohomish River. The unstructured grid in this domain is depicted in Figure 3, which consists of roughly 500,000 horizontal grid cells with resolution varying from 1 m in the vicinity of the sill of interest to 300 m in Possession Sound. Over 70% of the resolution is confined to within 1000 m of the sill. The vertical resolution is 0.2 to 0.3 m in the upper 20 m of the water column, and the vertical grid-spacing stretches to 30 m in the deepest portion of the domain in the Sound. In total, the 3D domain is discretized with 12 million computational cells, which requires roughly 4GB RAM. For the present simulations, we ignore the nonhydrostatic pressure, as it does not influence the separation dynamics considerably. The bathymetry shown in Figure 2 is what has been resolved by this grid, which shows that the sill is reasonably well-represented. In the present simulations, the time-step size is 0.1 s and is restricted by the horizontal Courant number in the finest cells due to wetting and drying. In this paper, we report on the results over a 24-hour tidal cycle during spring (strong) tides. This requires 864,000 time-steps, which consumes 48,000 CPU hours using 200 processors on the MJM machine at the US Army Research Laboratory DoD Supercomputing Resource Center (ARL DSRC) (3.0GHz Intel Woodcrest processors).

Figure 3. Computational grid. For clarity, cell centers are shown in the larger-domain figures and triangles are only

shown in the figure of the smallest domain.

3. Results and Discussion 3.1 Tidal-scale Hydrodynamics The tides in Snohomish River estuary consist of strong diurnal (daily) and semi-diurnal constituents, which, over the course of one-day, lead to a strong flood and ebb tide, followed by a weak flood and ebb tide. Figure 4 illustrates the predicted near-surface salinity distribution on the estuarine scale and Figure 5 shows a vertical salinity profile along transect E-E’ (see Figure 2a). During the strong ebb (falling tide), the salinity front propagates a distance of 10 km in the offshore direction, and sits outside the river mouth during low tide (Figures 4a,b,c and Figures 5a,b,c). During this process, the stratification, as measured by the vertical gradient of salinity, decays as the salt front moves downstream due to mixing induced by bottom-generated turbulence (compare, for example, the stratification at 10 km in Figure 5a to that in Figure 5b). At low tide, the inter-tidal mudflats are exposed (as indicated by the dark red regions in Figure 5c) and the bypass is disconnected. During the subsequent strong flood (Figures 4c,d,e and Figures 5c,d,e), the salinity front moves up-estuary and the water column is relatively well-mixed due to strong bottom-generated turbulence and the effects of strain-induced periodic stratification (SIPS; Simpson et al., 1990). During SIPS, the currents that are further from the bottom, and hence less influenced by the boundary layer, are faster and transport saline or fresh waters more efficiently. During flood tides, this strained velocity field tends to transport salt water over fresh water, while during flood tides, the straining tends to transport fresh water over salt water. As the water level rises during the flood tide, the bypass is inundated and reconnected, and flow through the bypass injects salt water into the main channel (Figures 4d,e). After the tide proceeds to the weak ebb, stable stratification (light, fresh water overlying heavy, salty water) develops upstream where the horizontal density gradient is the greatest as favored by the straining of the flow (Figure 4f and at 12 km in Figure 5f). This stratification develops due to the combined effects of SIPS and gravitational circulation. During the subsequent weak flood tide, the effects of bottom-turbulence are not strong enough to counteract the effects of gravitational circulation and SIPS, leading to a prominent salt-wedge with near-horizontal lines of constant salinity that extends over 10 km at the end of the weak ebb tide (Figures 4g and 5g).

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Figure 4. Near-surface salinity distribution (in psu) at the

estuarine scale at different stages of a tidal cycle. Dark red indicates dry areas and gray indicates where the plotted

layer is below the local depth. The tidal stage is shown at the lower-right corner of each panel.

Figure 5. Salinity (in psu) profile along transect E-E’ (see

Figure 2a) over a tidal cycle. Contours are for every 3 psu. The horizontal axis is the distance along transect from E (left) to E’ (right) in km. The tidal stage is shown at the lower-right

corner of each panel. 3.2 Flow Near the Sill While the model captures the estuarine-scale tidal dynamics well (for comparison to observations, see Wang et al., 2009 and 2010), it also resolves the details of the flow in the vicinity of the sill. Figures 6a and 6b depict the near-surface velocity field during the strong ebb just

before low tide. During this time, the strength of the currents has subsided so as not to overwhelm the eddies that form over the sill, thus producing a clear picture of the separation patterns. The figure shows the development of complicated flow patterns that result from exposure of mudflats as the water level falls (the dark red regions in Figure 6). At this time, the bypass is disconnected from Possession Sound (Figure 6a), thus causing water in the bypass channel to flow back into the main channel, in a sense that is opposite that during ebb tide, as shown in Figure 6b. The reversing flow in the bypass channel interacts with the strong downstream flow in the main channel, and the low water level exposes the sill crest, leading to separation eddies on both sides of the sill at the surface. It is these eddies that erode the bed and form the pronounced scour holes on the west side of the sill as depicted in Figure 2b. The scour holes form on the western side of the sill because the separation eddies do not form during flood tides. Deeper waters during flood tides inundate the sill and allow water to flow over, rather than around it, thereby decreasing the strength of the separation. The complex flow field around the sill leads to complex salinity structure in its vicinity. However, details of the surface salinity field depend both on the large-scale velocity field as well as the small-scale currents around the sill, and so sill effects on the salinity may only be apparent at specific stages of a tidal cycle. This has important ramifications for the potential to use remote sensing of surface salinity to infer the bottom bathymetry. Figure 7 shows the predicted near-surface salinity distribution near the sill and Figure 8 depicts the vertical salinity profile along transect S-S’ (see Figure 2b) at different stages of a tidal cycle. Note that, unlike the previous figures, the color range in these figures varies in order to highlight the variability close to the sill at each instant in time. Early in the strong ebb, when the water column is strongly stratified (Figures 8b and c), flow over the sill brings water of higher salinity to the free surface, leading to signatures of slightly salty water over the sill in Figures 7b and c. Later in the ebb, when strong turbulent mixing causes stratification to decay, the strength of the surface signatures over the sill diminishes, as shown in Figures 7d and 8d. However, as the water level continues to fall, the sill crest is exposed and the slowly-moving fluid upstream of the sill traps higher- salinity water, thereby creating higher-salinity water upstream of the sill, as shown in Figures 7e,f and 8e,f. During the subsequent strong flood, salinity is transported back into the estuary, but despite having stronger horizontal gradients, the vertical gradients are reduced due to strong mixing. This leads to overall weaker surface signatures induced by the sill and scour holes, with the most prominent variability in the surface salinity caused by the confluence of flow from the bypass into the main channel (Figures 7h,i,j and

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Figures 8h,i,j). The merging of the two water masses of differing salinity produces a strong front that is most pronounced in Figure 7i. This front is discussed in more detail in Wang et al. (2009). After the tide proceeds to weak ebb, the surface signatures remain weak (Figures 7l,m and Figures 8l,m) until the salt wedge propagates past the site, leading to vertical stratification that again leads to a region of higher salinity above the sill, as shown in Figures 7n and 8n.

Figure 6. Near-surface velocity field (a) in the estuary and (b) around the sill at low tide. Arrows show the magnitude and

direction of the flow and the color indicates the magnitude of the northing velocity (m s−1). Arrows in the two panels are of

different scales. Dark red indicates dry areas and gray indicates where the plotted layer is below the local depth.

Figure 7. Near-surface salinity distribution (in psu) around

the sill in the domain shown in Figure 2(b). Dark red indicates dry areas and gray indicates where the plotted

layer is below the local depth. The tidal stage is shown at the lower-right corner of each panel.

Figure 8. Salinity profile (in psu) along transect S-S’ (Figure 2) over a tidal cycle. Contours are for every 3 psu from 3 psu to

27 psu. The horizontal axis is the distance (m) along the transect from S to S’. The tidal stage is shown at the lower-

right corner of each panel. 4. Conclusions We have demonstrated the feasibility of performing simulations that resolve separation and eddies over complex topography in a field-scale estuarine model. In doing so, the simulations resolve length and time scales that span five decades (from tens of kilometers to less than a meter and from one day to less than a second). Although we did not discuss the details of model setup and parameterizations for the large-scale tidal flow due to the length of this contribution, it is important to note that the model we employ has been validated extensively against field observations in order to verify its correctness in predicting the large-scale flow features (Wang et al., 2009 and 2010). Since these drive the small-scale separation over the sill, analysis of the small-scale hydrodynamics rests on the assumption that the driving flow is correct. While further measurements are needed to verify the correctness of the small-scale flow, the simulations qualitatively produce the same sill-scale

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coherent structures that are observed in the actual flow. These separation eddies and the small-scale flow around the sill induces notable perturbations in the surface salinity when the water is vertically stratified, which occurs only at phases in the tide at which the salt wedge is over the sill. When the vertical stratification is weak, variability in the salinity field at the surface is dominated by the influence of horizontal stratification, which interacts with the horizontal flowfield to produce strong lateral salinity fronts, particularly at the confluence of the bypass and main river channel. Acknowledgments The authors gratefully acknowledge the support of ONR grant N00014-05-1-0177 (Scientific officers: Dr. Thomas Drake, Dr. C. Linwood Vincent, and Dr. Terri Paluszkiewicz). We also thank Dr. Robert Street for his numerous inputs on the modeling effort, and we thank our collaborators at Stanford University and the University of Washington who are involved in the Coherent Structures in Rivers and Estuaries Experiment (COHSTREX) project. Simulations were carried out on the MJM cluster at the ARL DSRC as part of a DoD Challenge Grant.

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