Journal of Theoretical Biology 369 (2015) 59–66

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The effect of bottom roughness on scalar transport in aquatic ecosystems: Implications for reproduction and recruitment in the benthos Noel P. Quinn, Josef D. Ackerman n Department of Integrative Biology, University of Guelph, Guelph, ON, Canada N1G 2W1

H I G H L I G H T S

    

The effect of roughness geometry on scalar (i.e., gametes) transport was examined. Height (k), spacing (λ) and shape (round, triangular, and square bars) were examined. Relative scalar transport (RT) was measured to assess retention vs. downstream transport. Flow matched the prediction for λ/k; RT was determined and differed for round shapes. Spatial configuration determines if the scalar is retained or transported downstream.

art ic l e i nf o

a b s t r a c t

Article history: Received 11 July 2014 Received in revised form 6 January 2015 Accepted 7 January 2015 Available online 14 January 2015

Bottom roughness can influence gamete and larval transport in benthic organisms. For example the ratio of the roughness spacing (λ) and roughness height (k) determines the type of roughness flow regime created in two dimensional (2D) flows: λ/ko 8 results in skimming flow; λ/k  8 results in wake interference flow; and λ/k 48 results in isolated roughness flow. Computational fluid dynamic modeling (COMSOL K–ε) was used to examine the effect of roughness geometry (e.g., a gradient in angularity provided by square, triangular and round 2D bottom roughness elements) on the prediction of roughness flow regime using biologically relevant λ/k ratios. In addition, a continuously released scalar (a proxy for gametes and larvae) in a coupled convection-diffusion model was used to determine the relationship among roughness geometry, λ/k ratios, and scalar transport (relative scalar transport, RT¼ ratio of scalar measured downstream in a series of roughness elements placed in tandem). The modeled roughness flow regimes fit closely with theoretical predictions using the square and triangle geometries, but the round geometry required a lower λ/k ratio than expected for skimming flow. Relative transport of the scalar was consistent with the modeled flow regimes, however significant differences in RT were found among the roughness flows for each geometry, and significantly lower RT values were observed for skimming flow in the round geometry. The λ/k ratio provides an accurate means of classifying flow in and around the roughness elements, whereas RT indicates the nature of scalar transport and retention. These results indicate that the spatial configuration of bottom roughness is an important determinant of gamete/larval transport in terms of whether the scalar will be retained among roughness elements or transported downstream. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Computational fluid dynamics Physical ecology Broadcast spawning Gamete release

1. Introduction For many benthic organisms, hydrodynamic conditions are critical for the release of gametes and settlement of larvae in the near-bed

n

Corresponding author. E-mail address: [email protected] (J.D. Ackerman).

http://dx.doi.org/10.1016/j.jtbi.2015.01.007 0022-5193/& 2015 Elsevier Ltd. All rights reserved.

region as well as their dispersal in the water column, because most gametes/larvae act as passive particles entrained in environmental flows (Abelson and Denny, 1997; Pineda et al., 2007; Nishihara and Ackerman, 2013). Much research has focused on the role of turbulence in gamete/larval transport, as turbulence facilitates the encounter between eggs and sperm, and the dispersal and/or settlement of larvae onto the bed (Denny and Shibata, 1989; Crimaldi et al., 2002; Quinn and Ackerman, 2012, 2014). Turbulence caused by bed

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roughness is one of the primary factors controlling flow and transport near the bottom (Nowell and Jumars, 1984; Crimaldi et al., 2002; Hendriks et al., 2006) and the importance of local small-scale hydrodynamics to these processes has been recognized (Yund et al., 2007; Reidenbach et al., 2009). Mathematical and conceptual models have reached contradictory conclusions concerning the effect of bottom roughness on larval settlement. Specifically, Eckman’s (1990) one-dimensional model theorized that larval flux and settlement would increase with the density of roughness elements, whereas Crimaldi et al. (2002) predicted that increased density of roughness elements would reduce larval settlement. Quinn and Ackerman (2012, 2014) suggested from empirical results that simple metrics, such as roughness density, are not sufficient to characterize near-bed flows, rather the spatial configuration of the roughness elements are required to understand the hydrodynamics and transport in these complex near-bed regions. The relationships between the spatial configuration of bottom roughness and the nature of near-bed flows are extremely complex but can be idealized in two-dimensions (2D) for simple bedforms such as transverse bars (reviewed in Schindler and Ackerman, 2010). These near-bed flow regimes can be predicted by the roughness height (k), water depth (d), the longitudinal distance between roughness elements, or roughness spacing (λ; i.e., wavelength), and roughness groove width (j; the space between the roughness elements) using the approach developed for pipe roughness by Morris (1955); Fig. 1, which has been applied to streambed roughness previously (Young, 1992; Davis and Barmuta, 1989; reviewed in Schindler and Ackerman, 2010). Roughness spacing (λ) and roughness height (k) are of particular importance in 2D transverse bars where λ/k (Roughness Index) can be used to characterize three types of near-bed ‘roughness flow regimes’ (Morris, 1955; Fig. 1). For flows where λ/ko8, the fluid in the spaces between the roughness elements are disconnected from the faster moving flow above them resulting in (1) skimming flow (Leonardi et al., 2003, 2004). With flows of λ/k 8, the wakes downstream of each roughness element interact with each other, resulting in (2) wake-interference flow. Lastly, for flows with λ/k48, the wakes downstream of the roughness elements are intermittent and dissipate before reaching the next roughness elements

downstream, leading to (3) isolated-roughness flow (Djenidi et al., 2008; Schindler and Ackerman, 2010). Whereas the hydrodynamics of near-bed regions caused by these idealized 2D roughness elements can be characterized, the determination of scalar transport in them remains to be determined. The purpose of this study is, therefore, to examine the λ/k prediction for flow over different types of roughness geometry and to examine the downstream transport and retention of a released scalar under these flow regimes. The released scalar is used to model the transport of gametes or larvae from a benthic population in a near-bed environment. By relating gamete/larval transport to bottom roughness, the role of physical parameters on biological processes involving benthic populations can be better understood.

2. Methods 2.1. Flow environment COMSOL multiphysics (version 3.4, COMSOL Inc.) computational fluid dynamic modeling program was used to examine how different bottom roughness parameters influence benthic hydrodynamics and subsequent scalar transport (a proxy for sperm and larval transport). Specifically, we used a 2D K–ε model, a type of Reynolds-averaged Navier–Stokes (RANS) model based on Reynolds decomposition (i.e., time average versus fluctuations; Davidson, 2004), to model the turbulent flow conditions given by    μ ρðu  ∇ÞK ¼ P K  ρε þ ∇  μ þ t ∇K ð1Þ

σK

ε ε2 ρðu  ∇Þε ¼ C ε1 P K  C ε2 ρ þ∇  K

K



μþ



μt ∇ε σε

 ð2Þ

where ρ is the density, u is the velocity vector, PK is the production of turbulent kinetic energy, μ is the dynamic viscosity, ∇ is the Laplacian operator, and the unknowns are K the turbulent kinetic

1

Height above bottom, z (m)

d 0

k λ

j

1

0 1

0

Downstream distance, x Fig. 1. COMSOL streamline plots illustrating the three main flow regime types over square 2D transverse roughness elements: (A) isolated roughness over λ/k ¼12; (B) skimming flow over λ/k¼ 3.3; and (C) wake interference flow over λ/k¼ 8.3. Also shown on (A) are the roughness parameters of roughness height (k), water depth (d), the longitudinal distance between roughness elements or roughness spacing (λ; wavelength), and roughness groove width (j; space between the roughness elements). Note that the panels represent a portion of the model domain that illustrates the streamlines around the roughness element rather than the entire domain.

N.P. Quinn, J.D. Ackerman / Journal of Theoretical Biology 369 (2015) 59–66

61

ε

ð3Þ

where the closure coefficients are Cε1 ¼1.44, Cε2 ¼1.92, and Cμ ¼0.09, and the turbulent diffusion coefficients are σK ¼ 1.0 and σε ¼1.3 (Jones and Launder, 1972) The K–ε model solves for K and ε, with a good convergence rate, relatively low memory requirements, and performs well for external flow problems around complex geometries (COMSOL, 2013). Model solutions (i.e., convergence) were achieved using an iterative method for the numerical solution of a nonsymmetric system of linear equations via the built-in segregated GMRES (generalized minimal residual method; Saad and Schultz, 1986) solver. Subdomain flow field parameters were set to freshwater at 20 1C to simulate lake conditions (density (ρ)¼ 1000 kg m  3 and dynamic viscosity (μ)¼ 1  10  3 Pa s). The modeling environment consisted of a spatial domain 5 m long and 1 m high above the lake bottom. The boundary conditions of the domain were set in COMSOL. The left-most boundary was set to an ‘inlet’ condition by specifying the velocity, U, of the flow and the rightmost boundary was set using the ‘outlet’ condition to allow the flow to exit. The bottom boundary was set to the ‘wall logarithmic boundary layer’ (i.e., no-slip) condition to generate benthic boundary layer flow (e.g., Ackerman, 2014). The top boundary was set to the ‘symmetrical boundary’ condition, which assumes the same hydrodynamic characteristics on either side of the boundary (in this case, in the water column above z¼1 m) and thus restricts the model’s computational domain. In this way, we modeled the flow conditions in the near-bed region where the benthic organisms are found. Given the complexity of near-bed roughness geometries including those created by the organisms (e.g., shells, corals, submerged aquatic vegetation, etc.), we chose to model flow over idealized objects in order to elucidate their effect on fluid flow and scalar transport (e.g., Schindler and Ackerman, 2010). In this case, we chose three roughness element geometries representing a gradient in angularity (or sphericity; e.g., Gordon et al., 2004) including: (a) square; (b) triangular; and (c) round (hemispherical) transverse bars (see Figs. 1, 2 and 3). Roughness elements were placed on the bottom of the domain at various λ apart and with various k, to obtain the different values of λ/k under consideration (described below). The influence of the number of bottom roughness elements on the flow regime was examined using 3, 4, and 5 elements in tandem. The first roughness element was always a minimum of 2 m downstream of the inlet to allow for adequate flow development, evaluated through visual observations of the modeled velocity gradient, which indicated a logarithmic boundary layer. The spatial domain was modeled using a free mesh with 751 mesh elements that were generated with COMSOL. The free mesh approach concentrates mesh elements around complex boundaries (Fig. 2). Five velocities (U¼ 10, 20, 30, 40, and 50 cm s  1) were examined that covered the range of fully turbulent flow conditions found in freshwater mussel habitats in the Laurentian Great Lakes (U¼10 and 20 cm s  1; Murthy and Schertzer, 1994; Ackerman et al. 2001), as well as higher velocities seen in marine mussel environments (U¼ 30, 40, and 50 cm s  1; Ackerman and Nishizaki, 2004; Marchinko and Palmer, 2003) included for comparison. 2.2. Bottom roughness λ/k values The square bottom roughness geometry was the primary geometry examined, as the square form has been used to examine different flow regimes in the past to model angular benthic substrates (Davis and Barmuta, 1989; Young, 1992). In terms of the roughness spacing, λ, seven values at 10-cm intervals from 20 to 80 cm were used, which

Downstream distance, x (m) Fig. 2. COMSOL plot of the modeling environment for round 2D transverse roughness elements of λ/k¼ 6.3 and the model free mesh structure, which adds more mesh elements around a complex boundary, indicated by the smaller mesh elements seen around the round roughness elements and the location of scalar release (x¼ 1.75 m; i.e., 1 m upstream of the first roughness element). Flow is through the meshed region from left (inlet at x¼ 0 m) to right (outlet at x ¼5 m). A wall logarithmic boundary layer (i.e., no-slip) condition was set at z¼ 0 m, whereas a symmetrical boundary condition was used at z¼ 1 m. The latter assumes the same hydrodynamic characteristics on either side of z¼ 1 m, which effectively extends the flow depth, but restricts the computational domain.

2.5

1.0

2.0

Height above bottom, z (m)

K2

μt ¼ C μ ρ

Height above bottom, z (m)

energy and ε the rate of dissipation of kinetic energy; μt is the eddy viscosity given by

0.8 1.5

0.6

1.0 0.5

0.4

0.0

0.2 -0.5 -1.0

0.0

-1.5 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Downstream distance, x (m) Fig. 3. COMSOL plot of the released scalar concentration over triangular 2D transverse roughness elements. Scalar was released at a point 1 m upstream of the first roughness element and at the same height from a 5-cm high flat interior boundary perpendicular to the flow. The λ/k ratio is 6.25, corresponding to a predicted flow regime of skimming flow, which was confirmed by the scalar carried downstream with limited fluid entering the spaces between roughness elements. The white dashed line indicates the location of the RT value transect.

were combined with seven values of the roughness height, k, at 2-cm intervals from 2 to 14 cm, to provide a total of 49 combinations of λ/k ranging from 1.43 to 40. This range of parameters was used to match the biologically-relevant roughness spacings and heights that have been observed for freshwater dreissenid mussels (Quinn and Ackerman, 2011) and would also apply to other benthic organisms occupying the space between roughness elements or attached to their surface. A subset of 16 out of the 49 λ/k values was examined for the triangle and round (hemispherical) roughness geometries in order to determine the influence of angularity of roughness shape. In this case, four λ values at 10-cm intervals from 20 to 50 cm and four k values at 2 cm from 2 to 8 cm were used.

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2.3. Scalar environment A 2D steady state convection and diffusion (i.e., mass balance) model given by u  ∇c ¼ ∇  ðD∇cÞ þR

ð4Þ

where c is the concentration of the scalar, u is the velocity field generated in the K–ε model (see above), D is the diffusion coefficient, and R (¼0) is a reaction, was applied to the modeled hydrodynamic solution provided from the 2D K–ε turbulence models described above to examine the transport of a continuously released scalar over the roughness elements. In this case, the modeled scalar was released continuously over 200 s from a point 1 m upstream of the upstream-most roughness element at the same k as the roughness element under examination (i.e., the scalar was released at z ¼2 cm at x ¼1.75 m for a roughness element k ¼2 cm high located at x¼ 2.75 m). The location of scalar release was a 5-cm long flat boundary extending from the top of the roughness element and perpendicular to the flow (see Fig. 3). A relative scalar concentration of 1 mol m  3 was used with a diffusion coefficient (D) of 10  11 m2 s  1, which is the value that has been reported for sperm in broadcast spawners (Reidel et al., 2005; Inamdar et al., 2007). Note that this is lower than the D for dissolved gases in water (i.e., D  10  9 m2 s  1) and the molecular diffusion of momentum (i.e., kinematic viscosity, ν 10  6 m2 s  1). The scalar concentration for a particular trial was determined by recording the modeled results at five points along a downstream transect that bisected each roughness element and taking an average of those five points (Fig. 3). The relative scalar transport (RT) for each trial was determined through the following expression,    C1 1 RT ¼ C 0 C1 C4 where C0 is the initial scalar concentration at the point of release, C1 is the scalar concentration between the first and second roughness elements, and C4 is the scalar concentration between the fourth and fifth roughness elements. RT was developed to provide an indication of the relative amount of scalar that was retained upstream versus transported downstream, which is why we focused on locations C1 and C4, respectively. A high RT (44) indicates that the majority of the scalar was carried downstream (i.e., skimming flow) whereas a low RT (  1) indicates that the scalar was trapped within the spaces of the roughness elements (i. e., isolated roughness). An intermediate RT (  2) would indicate wake interference flow, as the majority of scalar was still trapped, but a relatively larger portion would escape the wake region and be carried downstream. RT values were compared among the different velocities and roughness geometries through one and two-way ANOVAs, and the relationship between λ/k and RT was determined through linear regression, using the statistical program STATISTICA version 6.0 (StatSoft Inc., Tulsa, OK, USA).

3. Results 3.1. Roughness flow regime The number of roughness elements in tandem did not appear to affect the nature of the flow regime, which behaved in a similar manner regardless of whether there were 3, 4, or 5 elements present (Fig. 4). Nonetheless, 5 roughness elements were used for all remaining trials. Increasing velocity did not change the qualitative aspect of the roughness flow regime (e.g., skimming, wake interference or isolated roughness flow), as the flow regimes were similar based on visual observations of model results from models run from

10 to 50 cm s  1 (data not provided). Subsequent trials were, therefore, run at U¼ 10 cm s  1, corresponding to turbulent Reynolds numbers (Re) ranging from 2000 to 14,000 based on Re¼ Ul/ν where l¼length scale (in this case k) and ν ¼kinematic viscosity. The modeled flow regimes for the square geometry were compared visually to the predicted flow regimes based on the roughness parameters (Fig. 5). Skimming flow predominated and the highestvelocity skimming flows (i.e., the flow in the space was more isolated from the flow above the roughness) were found in the bottom right corner of Fig. 5. Isolated roughness flow was the next most frequent type observed on the left side of Fig. 5, and there was a narrow ‘band’ of wake-interference flow around λ/k 8, which indicated the transition between skimming and isolated-roughness flows. Generally, the model results followed the predicted flow regimes closely (Schindler and Ackerman, 2010). Results from the three roughness geometries were generally similar to the predictions of λ/k for the separation of isolatedroughness flows and wake-interference flows at λ/k ranging from 8 to 9 (Fig. 6). Skimming flows were found for λ/k up to 7 or 8, wake interference flows between λ/k of 7 to 9, and isolated flows for λ/k greater than 8 to 9 for the square and triangular geometries. The round roughness element geometries did not, however, follow this pattern for skimming flow regimes, which were observed for λ/k up to 6 (see Fig. 7).

3.2. Scalar transport Velocity did not have a significant effect on scalar RT values (Fig. 8) using the 9 λ/k ratios (i.e., the lowest values of λ and k, ranging from 3.33 to 20) per velocity. No significant difference was found for the RT values among the five velocities under a two-way ANOVA (F4, 224 ¼ 0.58, P¼0.93), but there was a significant difference among RT values across the three flow regimes (F2, 224 ¼ 7.28; Po0.0001). No significant interaction terms (velocity  flow regime) were detected. RT values for scalar transport over the square roughness geometry were contoured in a similar manner as the flow regime classification. In this case, there was a separation between skimming and the other two flow regimes (Fig. 9). Skimming flow was found predominantly on the right half of the RT¼16 contour, which occurred at λ/k values o8 criterion for the flow regime. The pattern was not linear, rather the RT¼16 contour overlapped the λ/k¼8 criterion at λ  77 cm and k 9.6 cm but diverged to λ/k¼ 3.3 at λ  20 cm and k6 cm. The 1

Height above bottom, z (m)

62

0 1

0 1

0 0

1

2

3

4

5

Downstream distance, x (m) Fig. 4. COMSOL streamline plot comparing the number of 2D bottom roughness elements ((A) three elements, (B) four elements, and (C) five elements) in tandem to the nature of the roughness flow regime. The parameters shown are λ ¼70 and k ¼6, giving a λ/k ratio of 11.7 for all three plots. All plots indicate isolated roughness flow as predicted from the λ/k ratio.

N.P. Quinn, J.D. Ackerman / Journal of Theoretical Biology 369 (2015) 59–66

1

80

Isolated roughness flow

8

Height above bottom, z (m)

λ/k = 8

70

Roughness spacing, λ (cm)

63

60

Wake interference flow

50

0 1

0 1

0

40

0

1

2

3

4

5

Downstream distance, x (m) Fig. 7. COMSOL streamline plots for: (A) round 2D transverse roughness elements at λ/k¼ 6.25 illustrating wake interference flow; and (B) square 2D transverse roughness elements at λ/k ¼6.25 illustrating skimming flow; and (C) round 2D transverse roughness elements at λ/k ¼ 2.5 illustrating skimming flow.

30

Skimming flow

8

20 2

4

6

8

10

12

14

Roughness height,k (cm)

5

Height above bottom, z (m)

1

0

10 cm s 20 cm s 30 cm s 40 cm s 50 cm s

4

Relative transport

Fig. 5. The flow regimes generated from the ratio of roughness spacing (λ) to roughness height (k) in square 2D transverse bars. Isolated roughness flows are found to the left of the λ/k ratio¼ 8, skimming flows are found to the right of that line, whereas wake interference flows are found around the line λ/k ¼ 8.

3

2

1

1 0 skimming

wake interference isolated roughness

Flow regime classification 0

Fig. 8. The relationship between velocity and scalar relative transport within three roughness flow regimes detected over square 2D transverse roughness elements: skimming flow (λ/k o7); wake interference flow (7 o λ/k o 9); and isolated roughness flow (λ/k 49). Values are means 7SE, N ¼ at least 5.

1

0 0

1

2

3

4

5

Downstream distance, x (m) Fig. 6. COMSOL streamline plots for (A) square, (B) triangular, and (C) round 2D transverse roughness element geometries for a λ/k ratio¼ 10. Isolated roughness flow was observed in each geometry, as indicated by the small flow recirculation regions behind the elements that dissipate before the next element downstream.

separation between wake interference and isolated roughness flows was less obvious, a result of the relatively similar RT values between the two flow regimes. This similarity was also evident in the comparison of RT values among the flow regimes (Fig. 10). A significant difference was observed for the square roughness geometry RT values among the three flow regimes under a one-way ANOVA (F2, 244 ¼79, Po0.0001). Pairwise differences were found between all three flow types (Fig. 10). There was also a minor significant negative association between the λ/k and RT (R2 ¼0.23, Po0.0001). The magnitude of RT value for skimming flow was high in the full 49 λ/ κ ratio data set for square geometries (see Fig. 9), but the results were similar for other flow regimes. RT values for the triangular and round geometries were compared for the different flow regimes (Fig. 11A and B, respectively). Results for the triangular geometry were similar to the round geometry including a separation between skimming flow and the other two flow regimes

at RT¼6; this occurred at λ/k values o8 criterion, and diverged to λ/ k 3 at the lowest λ and k examined (Fig. 11A). The pattern for scalar transport over the round geometry followed a similar pattern but the RT¼6 contour was somewhat parallel to the λ/k¼ 6 criterion determined above, and overlapped λ/k4 (Fig. 11B). The difference in RT values between wake interference and isolated roughness flows was not large, and the separation between the two flows regimes was not clear. However, a larger region of isolated roughness and wake interference flow was evident in the contour plots, extending past λ/k¼8, with skimming flow confined to the bottom right corner of the contour plot. This indicates that skimming flow occurs at a lower λ/k for round versus triangular or square geometries. Significant differences in RT values were found among the flow regimes for the round and triangular geometries under one-way ANOVAs (F2, 79 ¼ 19.8, Po0.0001; and F2, 79 ¼ 12.6, Po0.0001, respectively). Pairwise differences were detected between each combination of flow regimes (Fig. 10). A two-way ANOVA of the RT values comparing roughness geometry and flow regime revealed a significant difference in RT values among geometries (F2, 239 ¼141, Po0.0001) and among the flow regimes (F15, 239 ¼515, Po0.0001). There was no significant geometry  flow regime interaction. Pairwise differences were seen between the round and triangular geometries (P¼0.014) and between the round and square geometries (P¼0.03).

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80

Roughness spacing, λ (cm)

70 λ /κ = 8

Relative transport (RT)

RT = 16

60

0 10 20 30 40 50 60 70

50

40

30

20 2

4

6

8

10

12

14

Roughness height, κ (cm) Fig. 9. RT contours for the ratios of bottom roughness spacing (λ) to roughness height (k) over square 2D transverse roughness elements. The lighter contours indicate higher RT values. The dashed white line represents the region of predicted wake interference flow (i.e., λ/k ¼8), with skimming flow to the right of the solid line and isolated roughness to the left. The solid white line represents the RT¼ 16 threshold value observed for skimming flow in the scalar. The white dotted lines in the bottom left corner indicate the subsection of λ/k ratios and contour plots used for the triangular and round roughness geometries. 60 round triangle square - 16 ratio set square - full data set

Relative transport

50

40

30

20

10

0 skimming

wake inference

isolated roughness

Flow regime classification Fig. 10. The relationship between roughness element geometry and relative scalar transport for skimming flow (λ/ko 7), wake interference flow (7 o λ/ko 9) and isolated roughness flow (λ/k4 9). Note the difference in the RT value for skimming flow in the full 49 λ/k ratio data set obtained for square geometries (see Fig. 9). Values are means 7SE, N ¼ at least 10.

4. Discussion The roughness index, λ/k, provides an accurate means of classifying roughness flow regimes as skimming, isolated roughness, or wake interference flows. COMSOL k–ε model results confirmed these flow regime relationships for idealized 2D square and triangular bottom roughness elements, however the results were offset for round roughness geometry perhaps due to the absence of discontinuities (edges; or minimal angularity) where flow separation occurs. To the best of our knowledge, this is the first study to examine scalar transport in the near-bed region as it is influenced by bottom roughness of differing geometries and λ/k ratios. This comparison, which is pertinent to benthic organisms in general, builds on results from models of sperm transport over roughness elements of different roughness height, k (Quinn and Ackerman, 2012). In that case, the results indicate that above a certain k, scalar (sperm) released within the space between two roughness elements was retained within the

Fig. 11. Contours of RT over (A) triangular and (B) round 2D transverse roughness elements for ratios of roughness spacing (λ) and height (k). The lighter contours indicate higher RT values. The dashed white line represents the region of wake interference flow, with skimming flow to the right of the solid line and isolated roughness to the left. The solid white line represents the RT threshold for observed skimming flow of the scalar.

space rather than being transported downstream. The manipulation of both roughness height and spacing, λ/k, in this study helps to generalize that result. Specifically, when the ratio of λ/k is less than 8 for the square and triangular geometries and 6 for the round geometry, skimming flow results, and the fluid between the spaces becomes isolated from the flow above, essentially trapping the released scalar (sperm) within that space. Sperm released above the roughness elements will, however, be transported at higher rates downstream in the skimming flow resulting in relatively higher relative scalar transport (RT). This has implications ecologically for the position and nature of gamete release in broadcast spawners (see Marshall and Bolton, 2007). Under isolated roughness and wake interference flows, scalar released upstream would enter the space between the roughness elements at a higher rate, and the resulting RT values indicate that a relatively higher proportion of scalar was retained versus transported downstream. RT values were lowest in the isolated roughness flows, indicating that most of the scalar remained in the space downstream of the first few roughness elements and was not transported downstream. RT values were slightly, yet significantly,

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higher under wake interference flow, indicating that the majority of scalar remained downstream of the first few roughness elements, but a higher proportion of scalar was transported downstream relative to isolated roughness flow. The segregation of scalar transport by RT into patterns consistent with those provided for flow regimes by λ/k ratio is intuitively satisfying. In this case, RT¼16 for the square or 6 for the triangular and round geometries, respectively provides a quantitative assessment for scalar transport and retention, which might represent different strategies for broadcast spawners (Marshall and Bolton, 2007). It would be valuable to determine how well this pattern applies to the spatial pattern of roughness elements or the positioning of spawning individuals in the field. In terms of gamete release, wake interference flow would be ideal. Such flow would allow gametes to enter and be retained in areas (i.e., spaces between roughness elements) where gametes could become concentrated, as well as allowing some gametes to be transported downstream where there may encounter gametes from other individuals. In the field, however, local populations of dressenid mussels appear to exist under conditions of skimming flow, i.e., k¼10 to 15 cm and λ  30 cm (Quinn and Ackerman, 2011). It remains to be determined whether gamete release in benthic organisms occurs under flow conditions that enhance gamete encounter via wake interference flow (e.g., marine algae; Pearson et al., 1998). As indicated above, the relationship between bottom roughness and larval transport/settlement has led to some contradictory findings. Eckman (1990) predicted an increase in larval settlement with roughness density, whereas Crimaldi et al. (2002) predicted a decrease. When the combined influence of roughness height and spacing on flow regime type is considered (i.e., λ/k ratio), however, their conclusions can be interpreted in a different way. Specifically, Eckman’s (1990) prediction holds true when one considers that increasing the number of roughness elements would increase the likelihood that different roughness flow regimes would be created. In that case, there would be an opportunity for areas of scalar retention in recirculation zones under isolated-roughness and wakeinterference flows, which would enhance settlement relative to a flat-bottom configuration. Conversely, Crimaldi et al. (2002) predicted that larval settlement probability would be lowest at the smallest clam spacing (3.0, 4.4, and 6.1 cm) and highest clam height (0.9 and 1.8 cm), which corresponds to a λ/k ratio¼1.69 where skimming flow and high RT values occur. High relative transport rates would transport larvae downstream before settlement could occur, confirming Crimaldi et al.’s (2002) predictions. However, it should be noted that an alternate choice of λ and k could have resulted in different λ/k ratios leading to isolated-roughness and wake-interference flows, where settlement would likely increase. Quinn and Ackerman (2011, 2012, 2014), found that mussel shells placed uniformly at high density on the bottom of a flow chamber led to skimming flow, in contrast to wake-interference flow observed under a low mussel-density configuration. The bottom roughness geometry that differed most from the predicted flow regimes characterized by λ/k¼8 criterion was the round geometry, which had a λ/k¼6 transition. Another difference was that wake interference flow occurred over a wider range of λ/k values than for the other two geometries, where such ratios would have predicted skimming flow for square and triangular geometries. RANS models of flow over a similar set of roughness geometries found that the flow resistance (i.e., friction factor) was lowest over round bars, intermediate over triangular ones and highest over square bars over a range of Re comparable to our study (Ryu et al., 2007). It appears that flows over discontinuities, such as sharp edges or high angularity (low sphericity), lead to flow separation and areas of recirculation, which can be persistent and restricted to the immediate upstream portion of the sharp edge (Nowell and Jumars, 1984).

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Whereas we are not aware of a distribution of the roughness elements in habitats other than dreissenid mussel habitats in lakes, the type of recirculation zone found immediately downstream of dune crests in rivers can provide some insights as asymmetrical dunes had persistent recirculation zones immediately downstream of the lee (downstream) slope of the dune, whereas symmetrical dunes had intermittent recirculation zones with evidence of vortex shedding (Best and Kostaschuk, 2002). The angle of the lee slope was the main determinant of the recirculation zone created and symmetrical dunes had lower lee slope angles (Best and Kostaschuk, 2002). In the current study, the round roughness geometry has the lowest lee slope angle and would be most analogous to a symmetrical dune with an intermittent recirculation zone. Intermittent recirculation and the shedding of eddies found downstream of round roughness elements, would act more like isolated roughness or wake interference flows than the skimming flow predicted by λ/k. Eventually, increases in k combined with decreases in λ would lead to skimming flow. It would be valuable to examine roughness element geometry in different habitats using the insights developed here. The use of an idealized 2D model environment allows for the relative ease of manipulating the physical model environment, which is an important consideration for relatively complex ecology addressed here. Unfortunately, a comparable and perhaps more realistic 3D κ–ε model would not converge for velocities less than 1 m s  1, which is an unrealistic U value for the natural conditions that the model was designed to simulate. It is relevant to note that a previous study utilizing computational fluid dynamic models to examine the transport of solutes within sediments found that a 3D model only provided marginal improvement over a 2D model, and thus concluded that a 2D model was the optimal balance between model simplicity and predictive capacity (Meysman et al., 2006). It is likely that a 3D model used to extend our results would provide similar results to the 2D model but also new aspects which are not observed in 2D models. The inclusion of roughness height, spacing, and geometry in models of scalar transport provide more realistic depictions of near-bed flow regimes (Young, 1992). Results from this study will prove valuable for understanding transport processes near the bed, which may be relevant to the management and conservation of benthic environments. For example, gamete and larval transport are critical life-history processes for most benthic organisms, so evaluating the impact of bottom roughness on these stages is essential. Results from this study can also be applied to invasive species where control of dispersal is important to limit potentially harmful access to an ecosystem. Moreover, these results may stimulate some interesting field-based hypotheses regarding the location, position, and nature of gamete release (Marshall and Bolton, 2007). The approach used by this study may prove useful in a wide variety of applications related to benthic habitats.

Acknowledgments The authors would like to thank Peter Blouw for his assistance with the COMSOL program. We thank Dr. M.T. Nishizaki for comments on the manuscript. This work was supported by a NSERC Discovery Grant to JDA. References Abelson, A., Denny, M., 1997. Settlement of marine organisms in flow. Annu. Rev. Ecol. Syst. 28, 317–339. Ackerman, J.D., 2014. Role of fluid dynamics in dreissenid mussel biology. In: Nalepa, T.F., Schloesser, D.W. (Eds.), Quagga and Zebra Mussels: Biology, Impact, and Control, second ed. CRC Press, Boca Raton, FL, p. 775 pp (pp. 471-483). Ackerman, J.D., Loewen, M.R., Hamblin, P.F., 2001. Benthic-pelagic coupling over a zebra mussel bed in the western basin of Lake Erie. Limnol. Oceanogr. 46, 892–904.

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The effect of bottom roughness on scalar transport in aquatic ecosystems: implications for reproduction and recruitment in the benthos.

Bottom roughness can influence gamete and larval transport in benthic organisms. For example the ratio of the roughness spacing (λ) and roughness heig...
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