1. Introduction
Flows over complex terrain create variable stresses on the boundary and the nonlinear generation of turbulence that can occur at multiple temporal and spatial scales. For example, flow within coral reef systems is complex because they have irregular, branching morphologies with reef topography varying at scales ranging from centimeters to kilometers (Rosman and Hench 2011). In shallow nearshore regions with rough surfaces such as reefs or rocky shorelines, bottom stress is often a significant term in the momentum balance and the primary form of dissipation loss. Thus, correct parameterization of the bottom stress is essential to understanding these flows (Monismith 2007). Bottom stress is typically represented using a bottom drag coefficient CD. The bulk (or total) bottom drag CD,B is a combination of two components, skin friction drag CD,τ resulting from tangential stresses and pressure form drag CD,p resulting from nonuniform pressure distribution on the boundary (Kundu and Cohen 2008).
Stresses on the bottom lead to the nonlinear generation of turbulence, which is ubiquitous in flows with high Reynolds numbers (Re). Turbulence causes not only mixing of tracers of environmental interest but also mixing of momentum and the formation of a boundary layer. For a well-developed turbulent boundary layer, an inertial sublayer region exists where mean velocities exhibit a logarithmic profile. Within this region, the mean velocity profile is related to the generation of turbulence by shear at the bed; the boundary layer follows the classical law of the wall, and roughness is often expressed as a hydrodynamic roughness scale z0 (e.g., Reidenbach et al. 2006; Kundu and Cohen 2008). However, on surfaces where the roughness elements take up a significant fraction of the water column, such as shallow coral reefs, it is unclear if these assumptions hold in these conditions (Lentz et al. 2017).
For flow over complex surfaces, at least three distinct types of roughness configurations exist, each with their own scaling law (Wooding et al. 1973). The first, k-type roughness (where k is a roughness height), is typical of sand grain roughness studies (Nikuradse 1933), where the horizontal spacing is typically somewhat greater than the element height. In this regime, unstable eddies form behind roughness elements that are shed into the flow above the boundary, resulting in a turbulent structure scaled to the height of the roughness elements (Grant and Madsen 1982; Perry et al. 1969; Jimenez 2004). The second type, δ-type roughness (where δ is the boundary layer thickness), results in the formation of stable eddies between roughness elements, which are typically spaced closer than their height. This causes “skimming flow” along the top of the roughness elements, and its turbulent structure is nearly independent of the roughness height (Wooding et al. 1973; Perry et al. 1969; Jimenez 2004). A third class, k–δ-type roughness, is characterized by the formation of eddies in the lee of the roughness elements and the reattachment of flow behind the elements, which requires a relatively small concentration of elements (Wooding et al. 1973). This type of roughness scales with both the height of the roughness elements, and their concentration is defined as the ratio of the frontal area of each element to the average horizontal surface area (Grant and Madsen 1982). Flow over wavy surfaces such as sand bed ripples, as well as over many coral reef forms (as will be demonstrated in the results), is of the k–δ type.
In circulation models, turbulence is often parameterized in Reynolds-averaged Navier–Stokes (RANS) approaches, which require a parameterization of the bottom roughness. New field-scale technology to measure bottom bathymetry is advancing rapidly with increasing resolution, including sonar, hyperspectral, lidar, and fluid-lensing techniques. Measured variability in bathymetry is now often available at scales much smaller than the resolution of the computational grid. Typically, the mean depth value is taken to represent depth in the grid cell, and the remainder of the data is not used.
Significant work has been conducted to measure site-specific hydrodynamic roughness values on reefs (Rosman and Hench 2011; Reidenbach et al. 2006; Rogers et al. 2017, 2016, 2015; Lentz et al. 2016; Hearn 1999; Symonds et al. 1995; Lowe et al. 2005; McDonald et al. 2006; Thomas and Atkinson 1997; Falter et al. 2004), but a clear connection to the reef surface complexity is lacking. Several studies have suggested characterizing a reef surface based on fractal dimension (Hearn 2011a,b), or spectral energy of rugosity (Nunes and Pawlak 2008; Jaramillo and Pawlak 2011), but they lack a connection to measured hydrodynamic roughness. Thus, the state of current practice for computing bottom stress is to estimate a hydrodynamic roughness value based on general site parameters, based on literature from similar systems, or calibrated from field measurements. To our knowledge, there is yet no unified theory for how turbulent flow over complex terrain connects to hydrodynamic roughness or how multiple spatial terrain scales interact to create the total drag.
The aim of this paper is to address the fundamental question of how complex terrain is related to the hydrodynamic roughness, including what horizontal and vertical length scales are important. To address this question, we conducted an extensive field campaign to measure flow on a coral reef and developed a nonhydrostatic hydrodynamic model of the reef based on detailed bathymetric measurements using fluid-lensing techniques (section 2). The field observations are used to compute the spatially varying drag coefficients (section 3). Based on the validated model, we present a simple model to predict total drag on complex terrain using statistics of the surface (section 4). We conclude with an overview discussion (section 5) and summary (section 6).
2. Methods
a. Hydrodynamic measurements and data analysis
Ofu Island, American Samoa, lies in the southwest Pacific Ocean (14.2°S, 169.7°W) (Fig. 1a). Because of the tidal range and shallow reef crest, the onshore wave-driven flow is forced by breaking waves and modulated by the tides (Koweek et al. 2015). The focus of this study is on pool 400 located on the south shore of Ofu Island within the National Park of American Samoa (Fig. 1b).
Field experiment overview, Ofu Reef, American Samoa. (a) Ofu and Olosega Islands, location of pool 400 experiment, and weather station (WS), and (b) pool 400 experiment instrumentation layout showing drifter tracks from subset of drifter releases (black lines) with start (◊) and finish (∆) and approximate wave-driven flow direction over reef crest (black arrows). Local coordinates of AS and CS directions. Green squares denote velocity, pressure, and temperature measurements, red circles denote high-resolution pressure and temperature measurements, and yellow circles denote low-resolution pressure and temperature measurements (not all shown). Map data: Google Earth, CNES/Airbus.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
The field experiment consisted of an array of velocity, pressure, and temperature sensors deployed from 10 to 28 March 2017 designed to characterize the waves and hydrodynamics of pool 400 (Fig. 1b; appendix). Measurements of velocity data are taken in east, north, and up coordinates and transformed to a local coordinate system with x and y in the alongshore (AS) and cross-shore (CS) directions and z upward from mean sea level (MSL) (Table 1). The instantaneous measurements are time averaged (15 min) to give Eulerian velocity uE(uE, υE, wE), free surface deviation from MSL η, and wave statistics. Because of very small waves in pool 400 (Stokes drift ≪ uE), we assume the mean Lagrangian velocity u = uE. The Lagrangian depth-averaged mean velocity U(U, V, W) is calculated by combining data at a given location assuming uE = 0 at the bottom and averaging over the depth. Additional details are in the appendix.
Notation.
b. Bathymetric UAV measurements
An airborne survey was conducted over pool 400 in July 2016 using unmanned aerial vehicles (UAV) and fluid-lensing computational methods (Chirayath and Earle 2016; Suosaari et al. 2016; Chirayath 2016) to resolve centimeter-scale bathymetry of the reef (Fig. S1 in the supplemental material). A UAV electric quadcopter platform was custom built to host a nadir-pointing high-frame-rate video camera, relay synchronized position data and survey a region with sequential flights, each up to 20 min in duration. Videos frames were sorted into 120-frame bins and processed using the experimental fluid-lensing algorithm (Chirayath and Earle 2016) to remove refractive distortions caused by ambient surface waves. The corrected images and UAV position data were used as input frames for structure from motion to produce 2D, centimeter-scale orthophotos and a dense 3D bathymetry model. Calibration targets were distributed at varying water depths for georeferencing and bathymetry validation. Finally, terrestrial and millimeter-scale underwater gigapixel photogrammetry was performed to calibrate and verify 2D fluid-lensing reconstructions from airborne data, perform georectification, and validate derived 3D bathymetry. Where high-frequency Gaussian noise was present in the raw bathymetry as a result of resolution limitations, depth grids were low-pass filtered to 10- to 30-cm grid scale.
c. 3D nonhydrostatic model
The Stanford Unstructured Nonhydrostatic Terrain-following Adaptive Navier–Stokes Simulator (SUNTANS) modeling framework is a three-dimensional nonhydrostatic RANS model under the Boussinesq approximation (Fringer et al. 2006). The model grid is implemented in a rectangular horizontal grid with z-level vertical coordinate with domain length L, width W, average depth 〈D〉, and model grid spacing ΔxG, ΔyG, ΔzG. To compute the turbulent viscosity νT, a 3D k–ϵ turbulence model is employed within the generic length scale framework (Warner et al. 2005) with molecular viscosity ν = 1 × 10−6 m2 s−1. Time stepping is third-order Adams–Bashforth, and nonlinear momentum advection is central differencing. Initial conditions are zero velocity with uniform density. To achieve an equilibrium boundary layer, boundary conditions are periodic in the streamwise (x) direction and free slip on the lateral boundaries and free surface. The bottom boundary condition is a quadratic drag law with roughness height z0,ur, imposed on the bottom grid cell (Fringer et al. 2006). For reef simulations, we assume z0,ur = 0.033 cm, a value typically used for sand. The model is forced with a uniform pressure gradient, which is adjusted in time to achieve a desired domain-averaged velocity UB (Nelson and Fringer 2017). Model simulations are run to steady state, typically at least five flow-through periods. The bathymetry for the reef simulations is linearly detrended in x and y, to minimize edge effects, and is smoothed along the periodic edges using 
3. Field measurement results
a. Hydrodynamic conditions
During the observation period, tidal variation of the free surface η is ±0.5 m, winds are light at less than 5 m s−1, and wave forcing on the forereef consists of several long-period swell events with Hrms exceeding 1 m that propagate toward the reef crest and break; and thus Hrms in the lagoon is quite small (Figs. 2a–d). The effect of tidally modulated wave forcing creates a pressure gradient that drives a tidally varying flow in the pool center (Fig. 2e). The drifter tracks show the large-scale flow field from the reef crest exits through low areas in the reef crest and channels (predominantly near H0), and the general flow patterns are consistent over tidal phase on multiple days (Fig. 1b).
Oceanic forcing conditions from March 2017 on Ofu Reef. (a) Tidal level η; (b) wind velocity U10 in AS (solid) and CS (dashed) directions; (c) wave power spectral density Sηη (m2 Hz−1) as a function of period T on forereef FR5; (d) rms wave height Hrms on forereef FR5 (solid), reef crest D-5 (dashed), and middle of pool 400 D0 (dotted–dashed); and (e) observed depth average velocities at site D0 in alongshore x (solid) and cross-shore y (dashed) directions.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
b. Momentum balance
The mean water surface offset η0 between the offshore and each site in the lagoon is based on solving (3) for low forcing conditions (see appendix) and shows a steep increase from the shallow outlet at K0 and a more gradual increase within the pool from site G0 to A0 (Fig. 3a). The time-averaged flow magnitude |q| generally increases along the pool to a maximum at H0 where much of the flow exits the pool (Figs. 1b and 3b).
Mean water level setup and drag results along centerline of pool 400. (a) Mean water level setup η0 relative to offshore; (b) mean flow |q|; (c) drag coefficient CD from spatially averaged CD from momentum balance in (3) (black x), local CD from least squares fit of (4) (red dash), and local CD from Reynolds stress in (A2) (blue triangle); (d) hydrodynamic roughness z0, with colors as in (c); (e) magnitude of terms in momentum balance in (3), US (cyan x), NL (blue square), PG (black triangle), RSG (orange circle), BT (red diamond), and ST (yellow asterisk); and (f) depth below mean surface z showing measurement stations A0–K0. Error bars are 90% confidence intervals of the mean.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
The results for the spatially averaged 〈CD〉 from the momentum balance vary along the transect between O(0.01) and O(0.1) (Fig. 3c). Comparing to the results using a logarithmic fit to the velocity profile in (4) and the Reynold’s stress method (appendix, Eq. A2), 〈CD〉 are similar magnitude but fall outside the 95% confidence limits (Fig. 3c). The likely explanation for this is that 〈CD〉 is averaged over O(100) m, while the log fit and Reynold’s stress methods are local measurements. The spatially averaged roughness scale z0 from the momentum balance varies from O(1) to O(10) cm along the transect, and results from the local log fit are similar (Fig. 3d). Time-average velocity profiles normalized by the depth-averaged velocity show a logarithmic profile within the midwater column (Fig. S2). This confirms the applicability of (4) for determining a local roughness value. At the two deeper sites (B0, D0) the profile is logarithmic over the measured depth, but at the two shallower sites (F0,H0) the profile deviates from a logarithmic profile below 0.3 and above 0.6 
4. Theory and model results
a. Theoretical approach
b. Model validation and results
While RANS models have shown effectiveness in computing hydrodynamic flows and drag coefficients at a range of scales, previous studies note some limitations in computing the details of turbulence and eddy formation especially for bluff bodies where the details of flow separation are important (Rodi 1997; Lübcke et al. 2001). To validate the hydrodynamic model, a series of three idealized simulations on a cylinder, cube, and sinusoidal bottom are conducted with known experimental results (Table 2) in addition to validation on the Ofu reef presented next (Table 3).
Idealized hydrodynamic model simulations. Note that for all simulations ν = 1 × 10−6 m2 s−1 and z0,ur = 0.0001 cm.
Reef hydrodynamic model simulations’ base condition. Note that Var = varies, Reh = 〈hb〉UB/ν, and for all simulations ν = 1 × 10−6 m2 s−1, z0,ur = 0.033 cm, and ΔxG = ΔxB.
The simulations are conducted at Re similar to those expected on the field site, where mean velocities are on average O(0.1) m s−1 (Fig. 2). The horizontal width w of the dominant bottom features are O(1) m (Figs. 4 and 5). Thus, at the site, the average Rew = (UBw)/ν = 1 × 105 but can vary from 6 × 104 to 1 × 106.
Hydrodynamic model results at site D0 with large isolated coral heads showing wake separation and pressure distribution. (a) Top-down view showing mean velocity 
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
Hydrodynamic model results at site F0 with small irregular corals showing wake separation and pressure distribution. Legend and notes are as in Fig. 4. Dashed lines in (a) and (b) represent (y = 12.95 m) location of (c)–(f), and dashed lines in (c)–(f) represent (z = −0.83 m) location of (a). Velocity vector scale in (a), (c), and (d) is 10 cm s−1.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
To compare to results from the literature, CD,B for the cylinder and cube simulations are computed by 
Thus, the model accurately computes the drag coefficients for a range of rounded and sharp surfaces at Re typical of the study site. For forms with rounded features such as a cylinder at Rew greater than 5 × 105, the model may overpredict the drag coefficient. However, the typical Rew on this reef is 1 × 105, below this threshold. In addition, many of the forms on the reef have sharp edges and the drag coefficient is not expected to vary with Re.
The reef surface is modeled using a similar approach as the idealized domains (Table 3). Model results at site D0 with rounded isolated roughness elements show strong horizontal variability in the flow field near the bottom with separated wakes in both the horizontal and vertical (Figs. 4a,d). The aerial image shows the roughness elements at this site are Porites corals surrounded by sand (Fig. 4b). The pressure field shows corresponding high pressures in the front of the roughness elements (some from the free surface 
Model results at site F0, which has more irregular bathymetry, are similar, but the wakes behind the roughness elements and resulting pressure field are larger and dominated by the large-scale features (Figs. 5a,c,d,e,f). The aerial image shows this site has smaller irregular coral shapes with some sand patches (Fig. 5b).
The model results for 〈CD,B〉 generally show good agreement with the field observations, although with a high degree of scatter (Fig. 6). The high degree of scatter is likely due primarily to the spatial scales of the different results. The model simulations are conducted at 20–32-m spatial scale owing to model time and bathymetric limitations. The field results are averaged over O(100) m for the results from the momentum equation, but are local measurements for the logarithmic fit and Reynolds stress methods. Thus, while there is scatter in these results, the validation confirms that this model reasonably computes 〈CD,B〉 on these reef surfaces and can be applied to additional model scenarios and reef locations.
Comparison of model computed total drag coefficient 〈CD,B〉 and field observations for field sites on shallow reef. Black x is 〈CD,B〉 from bulk pressure gradient from (3), red circle is CD from log fit to local velocity profile from (4), and blue triangle is CD from near-bed Reynold’s stress from (A2). Model simulations are run at average depth and flow velocities at each site (Table 3).
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
c. Drag
The model is used to explore the variability of the terms in (13). The Reynolds number Reh is varied over the range of observed values from 104 to 105 by changing UB and, as expected, has very little effect on CD,B (Fig. 7). Since the flow is in a turbulent regime, the irregular forms of the reef create separated boundary layers (Figs. 4 and 5), which are not dependent on Reh.
Dependence of total 〈CD,B〉, shear 〈CD,τ〉, and form 〈CD,p〉 drag coefficients on Reynolds number Reh = UB〈hb〉/ν for range of observed flows at site D0. Note relative depth 〈D〉/〈hb〉 and average steepness 〈hb/λx〉 are held constant for these simulations.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
The relative depth 〈D〉/〈hb〉 is varied by changing 〈D〉 at sites D0 and F0 over the range of observed values. Both sites show decreasing 〈CD,B〉 with increasing 〈D〉/〈hb〉, consistent with the theory in (5) (Fig. 8). This trend is primarily a result of 〈CD,p〉 changing, while 〈CD,τ〉 remains very small over these simulations. Equation (5) matches the model results well for high relative depth (site D0) while somewhat underpredicting the drag at smaller relative depth (site F0).
Dependence of drag coefficients on relative depth 〈D〉/〈hb〉 at sites D0 and F0, comparing model results for total 〈CD,B〉, shear 〈CD,τ〉, and form 〈CD,p〉 drag, compared to least squares fit of (5) by changing z0. Note for these simulations, steepness 〈hb/λx〉 and Reynolds number Reh are held constant.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
The variation in relative slope 〈hb/λx〉 is achieved by simulating many different model sites, while holding Reh and 〈D〉/〈hb〉 constant (the same as site D0 base case) (Fig. 9). The 〈CD,B〉 increases linearly for 〈hb/λx〉 > 0.05 because of an increase in CD,p. For idealized sinusoidal forms, flow separation and high form drag occur at hb/λx > 0.1 (Salvetti et al. 2001), but the reef model results show a dominance in form drag for 〈hb/λx〉 > 0.05, likely since the irregular reef bedforms create more flow separation at a lower average slope. At very low slopes, 〈CD,p〉 trends toward zero and 〈CD,τ〉 becomes the dominant term in (13). These simulations are of a highly smoothed spatially downsampled reef (ΔxG > 3 m) with the model grid resolution the same as the bathymetry. Thus for this reef, form drag due to pressure 〈CD,p〉 is the dominant term at all sites.
Model results for results for total 〈CD,B〉, shear 〈CD,τ〉, and form 〈CD,p〉 drag as a function of spatially averaged slope 〈hb/λx〉 at multiple sites. Note for these simulations, relative depth 〈D〉/〈hb〉 and Reynolds number Reh are held constant. For very small 〈hb/λx〉 < 0.05, UB and 〈D〉 are held constant with decreasing grid resolution; thus for these simulations 〈D〉/〈hb〉 and Reh are not constant.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
In Fig. 9, there is some variability in 〈hb/λy〉 that cannot be independently varied. But, because most reef surfaces at this scale are symmetric, 〈hb/λx〉 and 〈hb/λy〉 are typically of similar order (Fig. 10a). Thus, 〈hb/λy〉 is not expected to be a significant parameter in predicting 〈CD,B〉, a simplification also used in studies on sediment bedforms (Wooding et al. 1973; Grant and Madsen 1982). However, for forms that are not symmetric it could become an important factor.
Symmetry of reef bedforms and effect of flow direction reef sites. (a) PDF of ratio of spatially averaged slopes in the cross-flow (y) and streamwise (x) directions showing most reef forms are symmetric, and (b) variations in flow direction have little effect on total 〈CD,B〉, shear 〈CD,τ〉, and form 〈CD,p〉 drag at site D0.
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
Finally, in all the simulations we have assumed unidirectional flow, and the factor 〈hb/λx〉 is not dependent on the sign of the flow. However, it is well known that, because of flow separation, a bluff body will have less form drag if it has an abrupt front and gently sloping lee rather than a gentle front and abrupt lee (Kundu and Cohen 2008). We vary the flow direction on a square domain (32 m × 32 m) at site D0. The 〈CD,B〉 varies slightly with flow direction by up to 7% (Fig. 10b). As a result, we consider flow sign a minor effect for these reef surfaces, which are primarily symmetric.
d. Hydrodynamic length scales
An analysis of all model runs on the Ofu reef with a best fit of 41 model simulations to (18) and (19) gives a1 = 0.38 with a coefficient of determination R2 of 0.79 (Fig. 11). The results in Fig. 11 are based on results within 6 × 103 < Reh < 7 × 104, 1.1 < 〈D〉/〈hb〉 < 6.3, and 0.7 < 〈hb/λx〉 < 0.29. However, (18) is likely valid as long as the flow is turbulent; additionally the upper limit of 〈D〉/〈hb〉 approaching a pure boundary layer, and the lower limit of 〈hb/λx〉 approaching a flat bottom are likely not limiting. The results for 〈z0〉 compare well to other studies on ideal sinusoidal forms (Salvetti et al. 2001; Zilker and Hanratty 1979; Buckles et al. 1984; Henn and Sykes 1999) and triangular ripples (a1 = 0.36) (Jonsson and Carlsen 1976) but are somewhat higher than 〈z0〉 for sand ripples (a1 = 0.27) (Nielsen 1992).
Hydrodynamic roughness 
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
The vertical profiles of horizontal velocity at sites D0 and F0 show high horizontal variability in u but relatively similar profiles in horizontally averaged 〈u〉H when normalized by UB above the mean depth 
Vertical profiles of horizontal velocity. (a) Two example sites showing high spatial variability and self-similarity of horizontally averaged flow 〈u〉H above mean depth 
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
A final question of interest is what horizontal spatial scales are most important on the reef for creating drag. A series of simulations at sites D0 and F0 are conducted by varying the bathymetry grid resolution ΔxB from 10 cm to 24 m, where the 24-m resolution is flat. Model grid resolution ΔxG is held constant at 10 cm, and a cubic spline method is used to interpolate from the coarser bathymetry grid to the finer model grid. The total drag 〈CD,B〉 increases and then levels off with increasing grid resolution with 〈CD,p〉 the primary component of this increase (Figs. 13a,b). At the highest modeled resolution, 〈CD,B〉 approaches the field observed values, but as noted previously, there are inherent differences in the spatial averaging of these different methods. For the modeled bathymetry, the important result is that 〈CD,B〉 is approaching an asymptotic value, where enhanced grid resolution is contributing little to the drag.
Computed drag and bathymetry energy spectra at different spatial resolutions at sites (left) D0 and (right) F0. (a),(b) Total 〈CD,B〉, shear 〈CD,τ〉, and form 〈CD,p〉 drag from the model as a function of bathymetry horizontal grid resolution ΔxB showing 〈CD,B〉 approaching an asymptotic value indicative of dominant horizontal scale. The observational result for 〈CD,B〉 is shown for reference with results from different methods as shown in Fig. 6. Spectral energy density of (c),(d) bottom depth surface 
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
We use Fourier analysis of the bathymetry in the x direction with wavelength λ and the spectral energy density (SED) S(λ)averaged in the y direction applied at different bathymetric grid resolutions ΔxB. SED of the depth 
The mean dynamic pressure field 
Dynamic pressure field 
Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-18-0013.1
5. Discussion
Based on field observations from a shallow reef in American Samoa, we compute spatially averaged drag coefficients at several sites, which are averaged over O(100) m. Based on the momentum balance, we conclude that within the pool the primary terms in the depth-averaged momentum equation are between pressure gradient and bottom stress, a result which is well known in these conditions (Monismith 2007). The pressure gradient develops as a result of breaking waves on the forereef, which are tidally modulated, and drive the unidirectional alongshore flow in the pool. This relatively simple flow in the absence of waves provides the ideal setting to better understand the role of bottom bathymetry of the complex reef in creating hydrodynamic roughness.
Using high-resolution UAV fluid-lensing techniques, we mapped the bottom bathymetry of the pool and used this data to model the reef using the SUNTANS nonhydrostatic hydrodynamic model. The results from the model for bottom drag coefficient 〈CD,B〉 are in excellent agreement with idealized test cases and generally in good agreement with the field observations, with some scatter likely due to variations in spatial averages from the different methods. The choice of using a RANS model allows for modeling field-scale sections of the reef with moderate computational cost but drastically simplifies the turbulent properties of the flow. Work to extend this analysis using a large-eddy simulation (LES) code could address some of these issues and potentially improve the model results but would require very large computational resources. However, for the purposes of this study, primarily to predict total drag, the selected model appears to be adequate.
The modeled hydrodynamic flows on the reef illustrate the complexity of flows over these surfaces. Wakes form in both the horizontal and vertical, which create imbalances in the pressures, and generally reattach behind the larger reef forms, confirming that many reefs can be characterized as having k–δ-type roughness. These pressure differences lead to form drag 〈CD,p〉, which is the primary cause of total drag in the model results. Drag due to shear stress 〈CD,τ〉 is a minor term except for nearly flat model geometries, a condition that is very rare in reef environments.
For parameterizing the bathymetry and dimensional analysis, we approximate the reef surface as a superposition of wavy sinusoidal bedforms with potentially four nondimensional parameters governing the flow. These include a roughness Reynolds number 
Roughness on wavy bedforms such as reefs are characterized by k–δ-type roughness, which scales with a roughness height and a roughness concentration. Drawing on previous work on sediment bedforms and idealized studies, we propose a parameterization for the bottom roughness scale 〈z0〉 [(18)], which is a function of the roughness height based on the spatial rms of the surface 〈hb〉, the roughness concentration based on the spatial rms slope 〈hb/λx〉, the coefficient a1, and the unresolved roughness 
The result in (18) is likely applicable for a range turbulent flows over complex terrains with k–δ-type roughness where relative depth 
As 〈hb/λx〉 increases, flow will not reattach between elements, and a larger-scale boundary layer will form dependent only on 〈hb〉, characteristic of 
For geometry that is highly resolved, 
While the local velocity profile on the reef is highly variable, the spatially averaged flow is logarithmic above the mean depth and justifies using a log-layer approximation for modeling larger-scale flows. Additionally, while only the slope in the streamwise direction is considered in (18), the spatial averaging does account for some variations in the spanwise direction, since the spatial average is applied in both horizontal directions over some areas. Waveforms that are parallel to flow would yield 〈hb/λx〉 = 0 and thus z0 = 0, while flow perpendicular to the same bedforms would give nonzero 〈hb/λx〉 and z0.
Finally, we investigate which horizontal spatial scales are relevant to creating total drag on the reef. Based on model simulations where we hold the model resolution constant but change the bathymetry resolution using an average of the high-resolution subgrid-scale bathymetry, we find that the drag approaches an asymptote for small scales, implying that a moderate horizontal length scale dominates the drag. This also implies that the very finescale features do not contribute much to the overall pressure field and thus the total drag, the result of the large-scale features creating wakes that isolate the small features from the main flow. This dominant horizontal spatial scale predicted by energy spectra of the slope showed a maximum around the same horizontal length scale of the large roughness features of the reef. A model resolution of O(10–50) grid cells per this dominant spatial scale is required to resolve the flow separation and accurately model the form drag on the reef.
6. Summary
We conducted extensive field observations on a coral reef and developed a nonhydrostatic hydrodynamic model for flows over the complex terrain. We present a simplified method to estimate the hydrodynamic roughness z0 on a complex surface, with only the rms of the depth and streamwise slope as well as a coefficient a1 required as inputs. Moving forward, the method presented here is likely applicable to other wavy bedforms with k–δ-type roughness. Further work could explore how the coefficient a1 may vary depending on other flow conditions and geometries and the transition to k-type roughness for high slopes. Additionally, surface waves are common in shallow flows; thus, understanding how oscillatory flows interact with the mean hydrodynamic drag over complex surfaces is a logical next step. Finally, while this method well predicts the average stress to model the large-scale flow, it likely does not well approximate the local bed shear stress, which is often used to model sediment transport and should be investigated further.
Acknowledgments
This work was supported by a grant from the National Science Foundation (OCE-1536502; “Collaborative research: Wave driven flow through a shallow, fringing reef”) to SGM, JJA, and CBW. The data from this study are available by request from the corresponding author. Suggestions from two anonymous reviewers, and discussions with Geno Pawlak significantly improved this manuscript. We wish to acknowledge the field team: Annie Adelson, Emma Reid, Benjamin Hefner, Ron Instrella, and Will Roderick. Assistance with modeling was provided by Oliver Fringer, Yun Zhang, Eric Mayer, Kurt Nelson, and Wenhao Chen. This work was conducted under permits from the U.S. Department of the Interior National Park Service, National Park of American Samoa, and the American Samoa Department of Marine and Wildlife Resources.
APPENDIX
Instrumentation, Sampling, and Initial Processing
Pressure, velocity, drifter, conductivity, and temperature measurements were made on the reef, and a weather station was located 1.8 km to the southwest (Table A1). Monitoring stations with larger instruments were attached to polyethylene plates and secured to the reef in areas of dead coral or sand. Monitoring stations with only small sensors were secured directly to dead corals or rubble.
Field experiment instrumentation and sampling rate for each site.
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