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Effect of macrophyte spatial variability on channel resistance
Advances in Water Resources 29 (2006) 426–438 www.elsevier.com/locate/advwatres
Effect of macrophyte spatial variability on channel resistance Julian C. Green
*
Department of Geography, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK Received 10 January 2005 Available online 19 July 2005
Abstract Aquatic macrophytes can severely retard flow rates in the river channels that they occupy. Consequently, there is a need to improve our ability to model vegetation resistance, to aid flood prediction and allow for better-informed channel management. An empirical model is developed to calculate flow resistance (ManningÕs resistance coefficient) of channels containing the submergent macrophyte Ranunculus (water-crowfoot). Blockage factors (the proportion of a cross-section blocked by vegetation) were determined for up to nine cross-sections at each of 35 river sites. These were used to create blockage-factor percentiles, which were regressed against vegetation resistance. An exponential best-fit relation involving the 69th blockage-factor percentile gave the best results. A parameter relating the length of the vegetated/solid boundary in contact with the open channel to the length of the conventionally-defined wetted perimeter improved the model fit by acting as a pseudo-measure of the turbulent-energy losses generated within the unvegetated stream by the macrophytes. The model was tested on three additional sites containing different macrophyte species and much higher vegetation blockages, and was found to work well. 2005 Elsevier Ltd. All rights reserved. Keywords: Aquatic macrophytes; Channel resistance; Blockage factor; Spatial variability; Ranunculus; ManningÕs n
1. Introduction Lowland rivers often contain extensive stands of aquatic macrophytes [25], which can dominate local hydraulics. As obstructions to flow, macrophytes increase channel resistance and water stage, while reducing average flow velocities [24]. Flow resistance of vegetated channels can typically be an order of magnitude greater than that of unvegetated channels, but can rise to two orders under exceptional conditions [12]. This has led to the claim that macrophytes contribute to local flooding [24], and they are routinely removed to reduce this perceived risk. In addition to the questionable benefits [7,18] and considerable expense incurred [9], such operations are detrimental to the wider riverine ecosystem [19,21]. Thus, there is a need to
*
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0309-1708/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2005.05.010
understand how aquatic macrophytes affect flow resistance to allow channel management to be adjusted to maintain both hydraulic efficiency and biotic diversity. Several empirical studies have attempted to relate the quantity of aquatic macrophytes in a river to their hydraulic resistance [3,4,8,10,29–31]. Of the parameters employed, the vegetated proportion of several cross-sections (termed the multi-cross-sectional blockage factor) has been demonstrated to be the most appropriate predictor of channel resistance on both theoretical and empirical grounds [14,16]. The blockage factor for a single cross-section is defined as B¼
Av A
ð1Þ
where B is the cross-sectional blockage factor; Av is the area of the wetted cross-section blocked by macrophytes (m2); A is the area of the wetted cross-section (m2). B can be given as a proportion or percentage, but the latter will be used in this paper.
J.C. Green / Advances in Water Resources 29 (2006) 426–438
Green [16] presents a regression equation between this parameter and the vegetation component of resistance for sites containing the trailing submergent macrophyte Ranunculus subgenus Batrachium (Gray) (watercrowfoot): n4 ¼ 0.0043BWM 0.0497
ð2Þ
where BWM is the weighted median of several cross-sectional blockage factors measured at each site; and n4 is defined as n4 ¼ ntot nb
ð3Þ
where n4 is the vegetation component of ManningÕs n resistance coefficient; ntot is the total value of ManningÕs n; nb is the base value of n for a straight, uniform, smooth channel in natural materials (boundary resistance) (all dimensionless) (adapted from [5,1]). While Eq. (2) gives respectable results (R2 value of 62.2% for the regression between predicted and calculated values of ntot), it only accounts for the most significant factor affecting resistance: the exclusion of most of the flow from within macrophyte stands [14]. Other studies have suggested that the spatial location of plants within a channel also affects flow resistance [23,11], with stand arrangements influencing the generation of turbulence in the free stream [13]. However, no study has included this effect in a resistance model. This paper develops a vegetation-resistance model using hydraulic and vegetation data collected at 35 river sites whose floras were dominated by Ranunculus species. The model is developed in two stages: firstly, accounting for variation in macrophyte distribution between cross-sections; and, secondly, accounting for variations within cross-sections. The resulting model will be tested on three additional sites dominated by other macrophyte species and whose values of channel resistance were outside the calibration range of the model.
2. Method Below is an outline of the data collection procedures used in this study. A more detailed description of methods is given in [15]. 2.1. Species selection The main sites investigated were dominated by one of three Ranunculus species: R. peltatus (pond watercrowfoot), favouring slow flow velocities; R. penicillatus subspecies pseudofluitans (stream water-crowfoot), favouring medium flow rates; and R. fluitans (river watercrowfoot), favouring the fastest flow rates [27] (see Table 1). All three are trailing submergents that have a clumped structure with high shooting density, and typically form monospecific vegetation stands. Ranunculus fluitans and
427
R. penicillatus both have capillary aquatic leaves, while R. peltatus has laminar terrestrial leaves. Green [15] demonstrated that it was acceptable to develop a single resistance equation encompassing these species, as the structural differences occur at the comparatively small scale of the leaves and individual shoots (leaf scale), while resistance is usually controlled at the larger scale of entire plants or stands of several plants (stand scale) [13]. 2.2. Field sites Sites were chosen from four lowland, nutrient-rich river networks in southern England: (Bristol) Avon, (Salisbury) Avon, Stour and Thames (see Table 1 for site details). This ensured that the resistance model was not site specific and would be applicable over a wide range of hydraulic conditions. All sites had predominately gravel beds with sufficient stability to permit the establishment of plants. While each site was dominated by a single plant species (excepting Sites E2 and E3), a total of 26 species was recorded across the sites. These additional species were generally limited to small quantities at the channel margins, though some submergent species, such as Oenanthe fluviatilis (river water-dropwort), were subdominant in association with Ranunculus. As these submergents have a similar structure to Ranunculus spp., at the stand scale of resistance, they were not deemed to affect the results significantly. 2.3. Measuring total resistance Total resistance at each site was calculated from the Manning Equation: ntot ¼
K R2=3 S e1=2 V
ð4Þ
where K is a factor to keep the equation dimensionally correct (1 m1/3 s1); R is the hydraulic radius (m); Se is the slope of the energy grade line (dimensionless); V is the mean channel velocity (m s1) (rearranged from Yalin [33]). Values of ManningÕs n and other hydraulic parameters are given in Table 2. 2.3.1. Hydraulic radius and area Hydraulic radius was calculated using the equation: R¼
A WP
ð5Þ
where WP is the wetted perimeter of the channel boundary (m). R was determined for between five and nine cross-sections (five sections for Sites 1–5; seven for Sites 6–19 and E1; nine for Sites 20–35 and E2-3) within each 25-m long reach using a quasi-stratified sampling procedure [15], where the cross-sections were positioned randomly, but with a minimum distance between sections
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J.C. Green / Advances in Water Resources 29 (2006) 426–438
Table 1 Details of vegetation-resistance sites Site
Location
River
UK grid reference
Date
Dominant macrophyte
b50 (mm)
(c3/ab)91 (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Tellisford Freshford Crockerton Kingston Deverill Tellisford Hill Deverill Manor Hincks Mill Farm Upavon Tellisford East Kennet Stoney Littleton Gillingham Tellisford Tellisford Stitchcombe Marnhull Heytesbury Boynton Tellisford Heytesbury Longbridge Deverill Longbridge Deverill Longbridge Deverill Longbridge Deverill Longbridge Deverill Longbridge Deverill Kingston Deverill Steeple Langford Steeple Langford Boynton Boynton Boynton Heytesbury Steeple Langford West Swainford Fm
Frome Frome Wylye Wylye Frome Wylye Shreen Water Avon Frome Kennet Wellow Brook Shreen Water Frome Frome Kennet Stour Wylye Wylye Frome Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Shreen Water
ST 806557 ST 791600 ST 873431 ST 844372 ST 806577 ST 868401 ST 812304 SU 136550 ST 806557 SU 115677 ST 734568 ST 807278 ST 806557 ST 806577 SU 225694 ST 767202 ST 928425 ST 953398 ST 806557 ST 928425 ST 871407 ST 871407 ST 871407 ST 871407 ST 871407 ST 871407 ST 844372 SU 036373 SU 036373 ST 957396 ST 958396 ST 951398 ST 928425 SU 036373 ST 813309
14/5/96 23/5/96 28/5/96 4/6/96 5/6/96 12/6/96 16/6/96 18/6/96 19/6/96 23/6/96 12/7/96 18/7/96 19/7/96 21/7/96 23/7/96 25/7/96 26/7/96 14/8/96 16/8/96 13/5/97 14/5/97 16/5/97 16/5/97 16/5/97 18/5/97 20/5/97 23/5/97 5/6/97 7/6/97 10/6/97 11/6/97 13/6/97 14/6/97 24/6/97 7/7/97
F F Pf Pf F Pf Pf F F P Pf Pf F F Pf Pf Pf Pf F Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf
39 39 7 6 43 10 16 10 39 2 67 32 39 43 38 14 18 18 39 2 26 26 26 26 26 26 2 15 7 10 25 27 2 7 2
14.6 22.6 6.2 8.0 13.6 6.3 10.1 18.5 14.6 2.9 27.3 16.6 14.6 13.6 24.9 10.6 10.8 11.2 14.6 17.1 17.1 17.1 17.1 17.1 17.1 17.1 9.5 9.3 4.8 10.8 20.0 17.3 17.1 4.8 4.3
E1 E2 E3
Marlborough Kingston Deverill Kingston Deverill
Kennet Wylye Wylye
SU 198693 ST 844372 ST 844372
16/7/96 10/7/97 24/8/97
C E E
2 14 14
6.4 9.4 9.4
a = long axis of a grain (mm); b = intermediate axis of a grain (mm); c = short axis of a grain (mm); (c3/(ab))91 = the 91st percentile of the c3/(ab) combination axis. Codes for macrophyte species: F = Ranunculus fluitans; Pf = R. penicillatus pseudofluitans; P = R. peltatus; C = Callitriche stagnalis; E = emergent species.
(4 m when five cross-sections were used; 2.1 m when nine sections were sampled). 2.3.2. Slope of energy grade line Se was calculated using the equation: Se ¼
Dh Dhv þ l L
ð6Þ
where Dh is the difference in water surface elevation (m); l is the length of river channel reach (= 25 m); L is the distance between the first and last cross-sections (625 m); Dhv is the upstream velocity head minus downstream velocity head (m); hv is the velocity head (m), defined as ¼
aV 2 2g
ð7Þ
where a is a constant, generally considered equal to 1; g is acceleration due to gravity (9.81 m s2) (adapted from Barnes [2]). Dh was measured over the 25 m reach using the conventional U-tube method [23]. Meanwhile, the difference between the velocity heads was measured over the potentially shorter distance between the first and last cross-sections sampled. 2.3.3. Velocity Velocity was measured using a two-dimensional Valeport electromagnetic current meter, recording at 10 Hz for 30 s at each point. Since the conventional 0.6 or 0.2/0.8 sampling methods are inappropriate to determine mean velocity of a vegetated profile [14], the veloc-
J.C. Green / Advances in Water Resources 29 (2006) 426–438
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Table 2 Site hydraulic data Site
V (m s1)
R (m)
Se
ntot
n4
B50 (%)
B69 (%)
T50
T16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0.360 0.480 0.162 0.082 0.248 0.086 0.258 0.160 0.348 0.069 0.142 0.361 0.187 0.223 0.209 0.186 0.137 0.133 0.286 0.117 0.171 0.194 0.162 0.185 0.190 0.148 0.128 0.131 0.093 0.088 0.088 0.148 0.116 0.137 0.094
0.176 0.329 0.390 0.114 0.295 0.193 0.198 0.499 0.151 0.262 0.265 0.118 0.126 0.235 0.404 0.290 0.377 0.367 0.099 0.572 0.220 0.189 0.212 0.219 0.228 0.252 0.159 0.284 0.363 0.587 0.540 0.420 0.559 0.383 0.325
0.0066 0.0092 0.0006 0.0078 0.0076 0.0031 0.0090 0.0011 0.0062 0.0004 0.0010 0.0090 0.0070 0.0101 0.0029 0.0014 0.0012 0.0009 0.0069 0.0019 0.0028 0.0043 0.0036 0.0017 0.0040 0.0034 0.0090 0.0030 0.0014 0.0024 0.0026 0.0058 0.0027 0.0018 0.0026
0.071 0.094 0.082 0.254 0.152 0.216 0.122 0.130 0.065 0.123 0.089 0.063 0.113 0.171 0.141 0.089 0.131 0.115 0.062 0.258 0.114 0.111 0.133 0.083 0.124 0.156 0.219 0.180 0.205 0.391 0.381 0.289 0.302 0.163 0.257
0.015 0.040 0.051 0.206 0.107 0.179 0.079 0.089 0.006 0.097 0.027 0.006 0.050 0.124 0.092 0.050 0.094 0.079 0.000 0.216 0.079 0.077 0.097 0.050 0.090 0.122 0.175 0.143 0.176 0.356 0.336 0.244 0.276 0.140 0.229
11.3 15.5 17.5 61.7 32.6 60.8 50.8 18.9 21.3 41.7 30.4 12.1 39.4 55.7 35.0 27.7 35.9 27.9 20.1 51.2 35.6 30.6 41.0 12.1 35.5 41.9 54.6 62.8 55.6 56.9 57.8 63.1 63.2 64.4 58.4
14.1 18.3 27.5 68.2 39.4 61.0 52.5 21.1 24.1 45.4 30.8 14.9 42.7 56.9 37.0 28.2 36.9 34.0 22.5 52.4 40.9 34.6 42.7 24.2 38.4 49.7 57.2 66.3 60.5 58.8 58.4 66.6 66.6 68.4 63.8
1.20 1.07 1.21 0.67 1.45 1.08 1.42 1.51 1.27 1.76 1.27 1.13 1.00 1.50 2.13 1.52 1.50 1.61 0.96 1.55 1.42 1.42 1.27 1.28 1.60 1.51 1.10 0.89 1.20 1.79 2.10 1.74 1.76 1.04 1.20
1.04 1.03 1.05 0.60 1.25 0.94 1.24 1.35 1.12 1.41 1.16 1.06 0.96 1.02 1.95 1.38 1.34 1.36 0.93 1.36 1.13 1.20 1.17 1.13 1.32 1.16 0.94 0.67 0.97 1.69 1.88 1.50 1.41 0.71 1.01
E1 E2 E3
0.172 0.078 0.066
0.404 0.223 0.230
0.0003 0.0108 0.0098
0.051 0.490 0.559
0.021 0.448 0.532
18.6 89.4 91.0
20.4 93.5 93.7
1.30 0.48 0.41
1.14 0.30 0.27
V = mean flow velocity; R = hydraulic radius; Se = slope of energy grade line; ntot = total value of ManningÕs resistance coefficient; n4 = vegetation component of ManningÕs n; B50/B69 = 50th and 69th percentiles of the cross-sectional blockage factor; T50/T16 = 50th and 16th percentiles of the vegetation spatial-variability parameter.
ity of as many as five points in each profile was measured, using Table 2 in Green [15] as a guide. The number of profiles used to take velocity readings varied with the distribution of vegetation within a cross-section; sites containing a high degree of fragmentation (and hence a complex distribution of velocities) had the greatest number of profiles measured, with 27 profiles sampled at Site 15. When the velocity of each profile had been determined, the Mean Section Method [26] was used to calculate channel discharge. Finally, using the weighted-median area of cross-sections sampled at a reach (weighting dependant upon spacing between cross-sections), the weighted median velocity of the 25 m section was determined using the equation:
Q A where Q is discharge (m3 s1).
V ¼
ð8Þ
2.4. Calculation of the vegetation-resistance component Following Eq. (3), a value for the boundary component of resistance, nb, was subtracted from total flow resistance to obtain the vegetation component of resistance, n4. nb was calculated using the equation (adapted from Task Force on Friction in Open Channels [28]): nb ¼
R1=6
aR 17.984 log ks
ð9Þ
J.C. Green / Advances in Water Resources 29 (2006) 426–438
where a is the channel shape correction factor (dimensionless), defined as .314 R a ¼ 11.1 ð10Þ dm where dm is the maximum flow depth (m) [19]; ks is the roughness height (m), calculated from the equation: 3 c log k s ¼ 1.54 þ 1.02. log ð11Þ ab 91 where a is the long axis of a grain (mm); b is the intermediate axis of a grain (mm); c is the short axis of a grain (mm); (c3/(ab))91 is the 91st percentile of the c3/(ab) combination axis [12]. A 100-grain sample was taken from each site using the grid-by-number technique [32] (see Green [16] for a description of measurement procedure). Values of b50 and (c3/(ab))91 are given in Table 1. 2.5. Vegetation blockage factor The cross-sectional vegetation blockage factor was determined from Eq. (1). In order to assess the extent of vegetation present in a cross-section, a series of measurement points was positioned across a section, and, at each location, the following variables were recorded: water depth; the species of macrophyte present; height of the top of the macrophyte above the bed; height of the underside of the macrophyte above the bed; and whether that macrophyte was present between that profile and the next. When the spacing between shoots was less than 5 cm in the horizontal or 2 cm in the vertical, the strands were regarded as a part of a blocked section, while an open section was recorded if the shoot spacing was greater than these distances. It was also sometimes necessary to record the presence of a second vegetation layer in the same profile. The profiles were placed evenly across the width where conditions were uniform, though they were also positioned at the edges of macrophyte stands, at sharp gradients in the amount of vegetation present, and where there was a change of macrophyte species. As a result of the movement of macrophytes in the flow and other errors, the width was recorded to a resolution of 5 cm. However, there was less movement of the plants in the vertical, so depths could be resolved to 1 cm or 0.5 cm depending on the turbulence of the water surface.
3. Vegetation spatial variability between cross-sections
vegetation within cross-sections that are only a few metres apart. At the 35 Ranunculus sites, the mean standard deviation of B at a site was as high as 10%, with a maximum range in B between sections recorded at Site 4 (20–74%). This high variability of vegetation distribution in a channel explains why it was necessary to sample so many sections, and why each reach was restricted to only 25 m in length; a distance greater than this would demand sampling even more cross-sections and would be more vulnerable to breaching uniform-flow requirements. It is possible that the average of individual cross-sectional blockage factors may not be the best parameter to reflect the resistance effect of the macrophytes. The general consensus in the boundary-resistance literature is that grain-size percentiles higher than the median size, D50 (where Dp means that p% of the area of the channel bed is covered by grains that have a smaller or equal diameter to the D value [21]), are most appropriate predictors of resistance, due mainly to the fact these percentiles project further into the flow (for a review of the literature see [34]), as well as because lower sampling errors are associated with these percentiles [12]. Similar logic can be applied to vegetation resistance: it can be hypothesised that the resistance of a channel reach will be most affected by the cross-section with the highest vegetation blockage. Moreover, the median value of individual cross-sectional blockage factors, B50, may not be the most precise percentile to sample. In order to test these hypotheses, each study reach was split into 100 units (each 25 cm long). The value of B for each sampled cross-section was assigned to the closest unit. The remaining units were filled by linear interpolation between the measured points. Upstream of the first sampled cross-section and downstream of the last cross-section, the units were given the same value as the nearest measured point (an example is given in Fig. 1). The units were ranked to produce blockage-
70 60
Blockage factor (%)
430
50 40 30 20 10 0
Trailing submerged macrophytes, such as members of Ranunculus subgenus Batrachium (water-crowfoots), form randomly-distributed clumps within river channels [6]. This can result in a high variation in the amount of
0
5
10
15
20
25
Distance (m) Fig. 1. Measured (diamonds) and interpolated values of the crosssectional blockage factor with distance downstream at Site 27.
J.C. Green / Advances in Water Resources 29 (2006) 426–438
factor percentiles, where Bp means that p% of the crosssections in a reach are vegetated by less than or equal to the B value. Fig. 2 shows cumulative percentiles for three sites: one site with uniformly low cross-sectional blockages; one site with consistently high vegetation blockages; and one site with both low and high blockages. Sampling-probability theory (outlined in [12]) was used to determine the precision of sampling different blockage-factor percentiles for the 35 sites (Fig. 3). Percentiles below B10 or above B90 will not be analysed, due to limitations associated with the fact that only between five and nine sections were sampled. For the remaining percentiles, the largest sampling errors were associated with the smaller percentiles (Fig. 3). There was a dramatic improvement in sampling precision between percentiles 25 and 28, after which there was a more gradual improvement in precision. With the exception
60 50
B (%)
40 30 20 Site 31
10
Site 27 Site 8
0 0
20
40
60 Percentile
80
100
of the very highest percentiles, the most precise percentile to sample was the 71st. Although the errors associated with this percentile are not significantly lower than larger percentiles, StudentÕs t tests show that they are lower than the errors for B40–51 at the 0.05-significance level and for B10–39 at the 0.01-significance level. These tests indicate that blockage-factor percentiles above the median value are the most precise to sample, and, thus, may be the most appropriate percentiles to use in a vegetation-resistance model. Linear regressions were determined between each blockage-factor percentile and calculated values of vegetation resistance, n4, determined at each of the 35 sites. Comparison of the R2 values for each equation between a blockage-factor percentile and n4 demonstrates that the percentiles differ in their ability to predict vegetation resistance (Fig. 4). The highest R2 value was 65.2% for the equation involving B60 (Fig. 5a). StudentÕs t tests show that the equations involving B58–62 gave significantly higher R2 values (at the 0.01 level) than those involving B48–52. Since Green [14] presented partial theory suggesting a non-linear relation between B and n, exponential best-fit lines were also fitted to the relations by adding 0.05 to n4 to remove negative values. For these (Fig. 4), the highest R2 value (74.4%) was for the equation involving B69 (Fig. 5b), with the R2 values for equations involving B67–71 also significantly higher at the 0.01 level than those involving B48–52. These results demonstrate that cross-sections with higher than the median vegetation blockage are the best predictors of vegetation resistance. While the exponential curves gave consistently higher R2 values than linear best fits (Fig. 4), the root mean square (RMS) error of the predicted values of n4 compared to the calculated values was lower for the linear model involving B60 (at 0.053) than the exponential model involving B69 (at 0.055). Thus, the competing merits of the linear and exponential models will be compared fol-
75
9 8 7 6
70
R2 value (%).
Relative standard error (%).
Fig. 2. Cumulative frequency of cross-sectional blockage factors at three sites.
431
5 4 3 2 1 0
65 60 exponential model linear model
55
0
20
40 60 B percentile
80
100 50 0
Fig. 3. The median (solid line) and lower and upper quartile (dashed lines) relative standard errors of sampling different blockage-factor, B, percentiles from the 35 Ranunculus sites, using probability theory outlined in [12].
20
40
60
80
100
B percentile
Fig. 4. The R2 values for using different blockage-factor percentiles in the vegetation-resistance model.
432
J.C. Green / Advances in Water Resources 29 (2006) 426–438 0.55
0.45
n4
0.35 y = 0.0043x - 0.0597
0.25
2
R = 0.6515
0.15 0.05 -0.05 0
20
40
a
60 B(60) (%)
80
100
80
100
0.6
n4+0.05
0.5 y = 0.0432e
0.4
0.0281x
2
R = 0.744
0.3 0.2 0.1 0.0 0
b
20
40
60 B(69) (%)
Fig. 5. (a) Linear; and (b) exponential relations between B and n4. Diamonds represent sites dominated by Ranunculus species; triangles represent site dominated by Callitriche stagnalis; crosses represent sites dominated by emergent species.
lowing the inclusion of a parameter to account for the resistance effect of vegetation spatial variability within cross-sections.
4. Vegetation spatial variability within cross-sections The above blockage factor represents the displacement of the flow from within plant stands. Macrophytes also retard flow rates in the adjacent unvegetated zones due to: the plants isolating areas from the main flow, such as immediately downstream of plant stands; and because turbulence is generated at the shear layers between the comparatively slow velocities inside and higher flow rates outside the plants [14]. While it is acknowledged that turbulence is generated three dimensionally by the macrophytes, it is suggested that the degree of turbulent generation is related to the twodimensional length of the vegetated boundary in a cross-section: increasing the length of the boundary between the plants and the unrestricted flow is likely to increase turbulence [14]. When macrophytes are located in large clumps with a relatively short boundary between themselves and the unvegetated stream, there will be minimal turbulence generation. Meanwhile, if the same amount of vegetation is distributed into widely scattered individual plants, the relative surface area of the macro-
phytes will be higher, forcing the flow to weave between the stands, increasing turbulence losses. The position of the macrophytes relative to the stream bed should also be significant to resistance: where the solid boundary is covered by vegetation, it will not affect the flow in the unvegetated channel, while it will have minimal impact on the already slow velocities within the plant body. This cross-sectional measure should be related indirectly to downstream turbulence losses. Where the vegetation is clumped within a cross-section with a short vegetated boundary, it is likely that the plants will also be clumped in the downstream direction. Likewise, if the vegetation is fragmented within a cross-section with a relatively long boundary, it is probable that the individual stands will be relatively small and the flow will be regularly forced from side to side, losing energy. Since it is the relative rather than absolute length of the plant boundary that is significant to resistance, a spatial-variability parameter will be used relating the effective wetted perimeter, WPe, to the conventional wetted perimeter, WP: T ¼
WP e WP
ð12Þ
where T is the vegetation cross-sectional spatial-variability parameter (dimensionless); WP is the length of the solid boundary of the cross-section (conventional wetted perimeter; m); and WPe is the effective wetted perimeter, defined as the sum of the lengths of the vegetation and solid boundaries, but excluding the lengths where the solid and vegetation boundaries meet, and omitting where the vegetation is in contact with the water surface (m) (a comparison of the definitions of WP and WPe is given in Fig. 6). The parameter T acknowledges that channel resistance is proportional to the combined length of the solid and vegetation boundaries in contact with the unvegetated stream, though it is limited by an inability to distinguish between the differing roughnesses of the two boundaries. Where the vegetation is widely scattered throughout a cross-section, the effective wetted perime-
Fig. 6. Definition of: (a) conventionally defined wetted perimeter; and (b) effective wetted perimeter. Thick line is wetted perimeter used for each parameter; thin line is water surface; shading represents area occupied by aquatic macrophytes.
J.C. Green / Advances in Water Resources 29 (2006) 426–438
ter will be higher than the conventional wetted perimeter, so T will be above 1 (for example, the cross-sections at Site 15; Fig. 7). However, when the vegetation is clumped in a cross-section and covers the solid boundary, T may drop below 1 (for example, most of the cross-sections at Site 28; Fig. 8). Under such a situation, the boundary is isolated from the flow and channel resistance will be less than would otherwise occur. In reality, this is a reduction in boundary resistance, but the decrease is caused by the plants. The longer the effective boundary compared to the conventional boundary, the greater the potential for turbulence generation, suggesting that T is positively related to vegetation resistance. When vegetation blockage is zero, T must equal 1, while it will equal zero when there is a total blockage. There are no other fixed points, and there is no theoretical upper limit to T, though, probabilistically, the greatest vegetation boundary length, and hence T, will occur when half the crosssection is blocked. The minimum value of T trends from 1 for unvegetated sections to zero for completely blocked sections (solid line in Fig. 9). However, T values can occur below this line when the proportion of the
433
solid boundary covered by vegetation is greater than the proportion of the cross-sectional area blocked by macrophytes. The average value of T for the 267 cross-sections collected at the 35 sites was 1.38, indicating that, on average, WPe was longer than WP, increasing turbulent energy losses; these values are plotted against the corresponding value of B in Fig. 9. Of the Ranunculus sites, the lowest T value was 0.53 for a cross-section with a blockage of 69.7% at Site 28 (Fig. 8), while Site 4 had the lowest median value, T50 (percentiles calculated in the same manner as for the blockage factors), of only 0.67 for a B50 of 61.6%. Meanwhile, the highest value of T was 2.37 for a cross-section with a blockage of 54.9% at Site 33, while Site 15 had the highest value of T50, at 2.13 for a B50 of 35.0% (Fig. 7). In common with the vegetation blockage factor, the T values for individual cross-sections were used to create T percentiles for each site to determine if a percentile other than the median was most appropriate in the resistance model. Probability analysis (detailed in [12]) was performed to determine the relative precision of sampling each T percentile across the 35 sites (Fig. 10).
Fig. 7. Site 15. (a) Photograph looking downstream; (b) non-dimensionalised representation of successive cross-sections downstream. Black shading represents areas blocked by vegetation. Values of T and B are given alongside each cross-section.
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J.C. Green / Advances in Water Resources 29 (2006) 426–438
Fig. 8. Site 28. (a) Photograph looking downstream; (b) non-dimensionalised representation of successive cross-sections downstream. Black shading represents areas blocked by vegetation. Values of T and B are given alongside each cross-section.
Excluding the extreme percentiles (due to the previously mentioned limitations of this technique) the most precise percentile to sample was the 67th, it being more precise to sample than percentiles above T79 and below T44, at the 0.05-significance level, and percentiles below T38, at the 0.01 level. This suggests that a percentile within the range T44–79 should be the most appropriate for use in a vegetation-resistance model. Residuals between predicted and calculated values of n4 were determined for both the linear and exponential models in Fig. 5. These residuals were then regressed against each percentile of T (Fig. 11). Both models give a similar pattern, with the exponential model producing consistently higher R2 values than the linear model. The R2 values peaked at 37.8% for T12 in the linear model (Fig. 12a), and at 53.1% for T16 in the exponential model (Fig. 12b).
For the exponential model, a StudentÕs t test indicates that the R2 values for the percentiles in the range T14–18 were higher than those in the ranges T9–13 and T19–23, at the 0.05-significance level. The fact that these low percentiles give the best model fit despite their relative imprecision of sampling indicates that it is important for them to be used in the vegetation model. As to why this is the case is uncertain. The regression equations in Fig. 12 are both significant at the 0.01 level and thus are able to account for some of the variation in the residuals in both the linear and exponential models of Fig. 5. Subtracting the fitted resistance residuals (calculated using the equations in Fig. 12) from the fitted values of n4 (calculated using the equations in Fig. 5) improves the ability to model vegetation resistance; regression between the calculated and improved-predicted values of n4 give R2 values of
J.C. Green / Advances in Water Resources 29 (2006) 426–438
435
0.15
2.5
0.10 2.0
Residuals
0.05
T
1.5
0.00 0.0 -0.05
0.5
1.0
1.5
2.0
T (12)
1.0
-0.10 y = -0.1114x + 0.1292
0.5
2
-0.15
R = 0.3777
a -0.20 0.0
0
20
40
60
80
100
0.15
B (%)
0.10
Relative standard error (%) .
0.00 0.0 -0.05
0.5
1.0
1.5
2.0
T (16)
-0.10 -0.15
y = -0.1361x + 0.155 2 R = 0.5316
b -0.20
4
Fig. 12. (a) Plot of T12 against the residuals in the linear relation of B60 against n4; (b) plot of T16 against the residuals in the exponential relation of B69 against n4. Diamonds represent sites dominated by Ranunculus species; triangles represent site dominated by Callitriche stagnalis; crosses represent sites dominated by emergent species.
3
2 1 0 0
20
40 60 T percentile
80
100
Fig. 10. The median (solid line) and lower and upper quartile (dashed lines) relative standard errors of sampling different T percentiles from the 35 sites, using probability theory outlined in [12].
55
exponential model linear model
50 45
R2 value (%) .
0.05 Residuals
Fig. 9. Plot of B against T for individual cross-sections. Diamonds represent cross-sections dominated by Ranunculus species; triangles represent cross-sections dominated by Callitriche stagnalis; crosses represent cross-sections dominated by emergent species. Dashed line: upper envelope for approximately 95% of data points; solid line: theoretical lower bound of envelope, excepting cases where the percentage of the solid boundary covered by vegetation is greater than B.
40
78.4% and 83.0% for the linear and exponential models, respectively. Adding the calculated value for boundary resistance, nb (Eqs. (9)–(11)), to the predicted values of n4 gives estimates of total ManningÕs n. For the linear model, the regression of predicted versus calculated total resistance gives an R2 value of 76.1% (Fig. 13a), with an RMS error of 0.042 (30%; Table 3). Meanwhile, for the exponential model, the regression of predicted versus calculated total resistance gives an R2 value of 81.2% (Fig. 13b), with an RMS error of 0.037 (24%; Table 3). Thus, the higher R2 value and lower RMS error indicate that the exponential model is a better predictor of resistance in channels dominated by Ranunculus species. Combining the regression equations in Figs. 5b and 12b, gives the following equation for the vegetation component of ManningÕs n: n ¼ 0.0432e0.0281B69 þ 0.1361T 0.205 ð13Þ 4
35
16
30
5. Application of the model to different macrophyte species
25 20
0
20
40
60
80
100
T percentile Fig. 11. The R2 values for using different T percentiles in the linear and non-linear models.
To determine if the model can be applied to other macrophyte species and also to higher vegetation blockages (and hence higher vegetation resistance) than was encountered with Ranunculus species, three additional
436
J.C. Green / Advances in Water Resources 29 (2006) 426–438 0.6
0.5
Calculated n
0.4
0.3
0.2
0.1
0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Predicted n
a 0.6
0.5
Calculated n
0.4
0.3
0.2
0.1
0.0 0.0
b
0.1
0.2
0.3
0.4
0.5
0.6
Predicted n
Fig. 13. Predicted versus calculated values of total resistance, n (a) linear model; (b) exponential model. Dashed line equals 1:1 line. Diamonds represent sites dominated by Ranunculus species; triangles represent site dominated by Callitriche stagnalis; crosses represent sites dominated by emergent species.
sites were studied dominated by different species. Site E1, on the River Kennet, was dominated by Callitriche stagnalis (common water-starwort) and did not contain any Ranunculus species. Like Ranunculus, Callitriche species are clumped-trailing submergents and were often found as subdominant species alongside or inter-mingled with Ranunculus at the main study sites. The site at Kingston Deverill, at the headwaters of the River Wylye, was dominated by Ranunculus penicillatus pseudofluitans during the spring and early summer, and was used in the main vegetation-resistance model on 4/6/96 (Site 4) and 23/5/97 (Site 27). However, later in
the season, the branching emergents Rorippa nasturtium-aquaticum (watercress) and Apium nodiflorum (foolÕs watercress) dominated the aquatic flora, and the location was studied for this additional analysis on 10/ 7/97 (Site E2) and 24/8/97 (Site E3). Both the linear and exponential vegetation-resistance models were applied to the extra sites without recalibration. The site containing Callitriche stagnalis (Site E1) had a lower value of total resistance than any of the Ranunculus sites (0.051; Table 2), but generally plotted within the scatter of Ranunculus sites in Figs. 5, 9, 12, 13 and Eq. (13) gave a predicted ntot that was only 9% from the measured value (Table 3). Callitriche and Ranunculus species have similar morphology in that they are clumped-trailing submergents. Therefore, it is likely that a vegetation-resistance model should be applicable to both, providing that resistance is controlled at the scale of individual or groups of plants, (low blockages), rather than at the scale of shoots and leaves (high blockages); Green [13] suggested that the differing porosities between Callitriche and Ranunculus species, produced by differing leaf morphology, would only have a significant effect on channel resistance for blockages above 50% when the channel slope was 1 · 104, increasing to 85% for slopes of 1 · 102. Meanwhile, Sites E2 and E3 were not only dominated by branching emergents that are structurally different to Ranunculus species but the extreme blockages present suggest that the leaf scale of resistance would be dominant at these sites [13]. Thus, a resistance model developed for Ranunculus species may not be able to predict the resistance at Sites E2 and E3, especially given the required extrapolation of the model to higher B and lower T values. However, the exponential best-fit curve between B69 and n4 + 0.05 for the Ranunculus sites plots close to these additional sites (Fig. 5b). This validates the exponential curve, since the inclusion of the additional sites had effectively no effect on the RMS errors for this curve, while it increased the RMS errors from 0.053 to 0.063 for the linear relationship, with the line significantly under predicting resistance at these high blockages. Moreover, Sites E2 and E3 plot as extreme outliers in the plot between the resistance residuals and T12 in the linear model (Fig. 12a), but are much closer to the extrapolated line in the exponential model (Fig. 12b). Consequently, the errors for predicted total ManningÕs n using the exponential model (Eq. 13) are only 3% and 18% for Sites E2 and E3 respectively, while they are 44% and 54% respectively for the linear model (Table 3).
6. Discussion and conclusions This paper supports theory presented in Green [13] that the relation between the vegetation blockage fac-
J.C. Green / Advances in Water Resources 29 (2006) 426–438
437
Table 3 Percentage errors for the four vegetation-resistance models Site
Measured ntot
B60 linear
B69 non-linear
B60, T12 linear
B69, T16 non-linear
Predicted ntot
Difference (%)
Predicted ntot
Difference (%)
Predicted ntot
Difference (%)
Predicted ntot
Difference (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0.071 0.094 0.082 0.254 0.152 0.216 0.122 0.130 0.065 0.123 0.089 0.063 0.113 0.171 0.141 0.089 0.131 0.115 0.062 0.258 0.114 0.111 0.133 0.083 0.124 0.156 0.219 0.180 0.205 0.391 0.381 0.289 0.302 0.163 0.257
0.051 0.069 0.063 0.270 0.144 0.239 0.205 0.067 0.102 0.153 0.133 0.068 0.179 0.230 0.145 0.100 0.134 0.117 0.095 0.206 0.140 0.115 0.157 0.052 0.128 0.164 0.226 0.255 0.220 0.224 0.236 0.266 0.249 0.250 0.231
28 26 23 7 5 11 68 48 58 24 50 9 59 34 3 13 2 2 52 20 23 3 18 38 3 5 3 42 7 43 38 8 17 53 10
0.070 0.076 0.074 0.291 0.125 0.227 0.182 0.069 0.094 0.131 0.114 0.084 0.156 0.211 0.122 0.085 0.109 0.099 0.094 0.180 0.122 0.098 0.130 0.069 0.111 0.158 0.209 0.265 0.216 0.211 0.218 0.275 0.256 0.269 0.238
2 19 9 15 18 5 49 47 45 6 29 34 38 23 14 5 17 14 51 30 7 12 2 18 11 2 4 47 5 46 43 5 15 64 8
0.034 0.048 0.041 0.203 0.149 0.208 0.208 0.085 0.090 0.171 0.118 0.053 0.152 0.205 0.224 0.121 0.148 0.134 0.064 0.216 0.127 0.110 0.153 0.043 0.139 0.160 0.191 0.191 0.192 0.275 0.309 0.300 0.269 0.193 0.200
53 48 49 20 2 4 71 35 39 38 33 16 35 20 58 36 12 16 3 16 11 2 15 48 12 3 13 6 7 30 19 4 11 18 22
0.057 0.061 0.062 0.218 0.140 0.199 0.196 0.098 0.092 0.168 0.116 0.074 0.132 0.195 0.232 0.118 0.137 0.129 0.065 0.210 0.121 0.106 0.134 0.067 0.136 0.162 0.182 0.202 0.193 0.286 0.320 0.324 0.293 0.210 0.220
20 35 24 14 8 8 61 25 42 36 31 17 17 14 64 33 4 12 5 19 6 5 1 19 10 4 17 12 6 27 16 12 3 29 14
E1 E2 E3
0.051 0.490 0.559
0.051 0.375 0.366
0 23 35
0.056 0.590 0.579
10 20 4
0.039 0.276 0.259
25 44 54
0.056 0.476 0.461
9 3 18
RMS of Ranunculus sites
31
tor and channel resistance is non-linear, with successive increases in the blockage factor producing an everincreasing effect on resistance. The nature of this relation may also explain why a blockage factor higher than the median gives the best-fit relation with resistance. Moreover, while the two-dimensional parameter T cannot represent the full three-dimensional resistance effect of macrophyte spatial variability, this model is the first to attempt to account for the turbulent energy losses generated in the unvegetated areas adjacent to trailing submergent macrophytes such as Ranunculus species. While this model has been exclusively developed for Ranunculus, the three additional sites studied indicate that it may be applicable to morphologically different species and for higher blockages than found at the main
28
30
24
sites. Further research into other species is required, though, and this model is unlikely to hold for macrophyte species possessing much higher porosities than Ranunculus, such as Sparganium emersum (unbranched bur-reed). The model presented here represents a significant advance in our ability to determine vegetative channel resistance given the quantity and distribution of vegetation at a given moment in time. Admittedly, taking measurements for Eq. (13) is more time consuming than a direct measurement of resistance. However, resistance is temporally variable, due to its stage dependence. Research is ongoing to combine this model with formulae to predict the flow-induced bending of the plants, in order to produce a model of how vegetation-resistance changes with discharge.
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Acknowledgements Appreciation goes to Dr and Mrs R.G.C. Green for sponsoring this project and assisting in the field; Mr A. Hayes for helping with field equipment; and various landowners who gave permission to work at these sites. Dr P.J. Wood commented on an earlier version of this paper.
References [1] Arcement GJ, Schneider VR. Guide for selecting ManningÕs roughness coefficients for natural channels and flood plains. United States Geological Survey Water Supply Paper 2339, 1989. [2] Barnes HH. Roughness characteristics of natural channels. United States Geological Survey Water Supply Paper 1849. US Government Printing Office, Washington, 1967. [3] Bakry MF, Gates TK, Khattab AF. Field-measured hydraulic resistance characteristics in vegetation-infested canals. J Irrigation Drainage Eng 1992;118(2):256–74. [4] Champion PD, Tanner CC. Seasonality of macrophytes and interaction with flow in a New Zealand lowland stream. Hydrobiologia 2000;441:1–12. [5] Cowan WL. Estimating hydraulic roughness coefficients. Agric Eng 1956;37(7):473–5. [6] Dawson FH, Robinson WN. Submerged macrophytes and the hydraulic roughness of a lowland chalkstream. Ver Int Vereinigung Theor Agwandte Limnol 1984;22:1944–8. [7] Dawson FH. Aquatic plant management in semi-natural streams: the role of marginal vegetation. J Environ Manage 1978;6(3): 213–21. [8] Dawson FH. The seasonal effects of aquatic plant growth on the flow of water in a stream. In: Proceedings, 5th international symposium on aquatic weeds, European Weed Research Society, 1978. p. 71–8. [9] Dawson FH. Ecology and management of water plants in lowland streams. Report of the Freshwater Biological Association, No. 57, 1989, p. 43–60. [10] Fisher K. The hydraulic roughness of vegetated channels, Report SR 305, Hydraulics Research Ltd., Wallingford, 1992. [11] Fisher KR, Reeve CE. Impact of aquatic vegetation cutting. In: White WR, Watts J, editors. 2nd International conference on river flood hydraulics, 1994, p. 439–47.
[12] Green JC. An assessment of the precision of sampling different grain size percentiles. Earth Surf Process Landform 2003;23(9): 979–91. [13] Green JC. Modelling channel resistance in vegetated streams: review and development of new theory. Hydrol Process 2005; 19(6):1245–59. [14] Green JC. Velocity and turbulence distribution around lotic macrophytes. Aquatic Ecol 2005;39(1):1–10. [15] Green JC. Comparison of blockage factors in modelling the channel resistance of vegetated rivers. River Res Appl 2005;21: 1–16. [16] Green, JC. Choice of percentiles and axes to determine grain resistance. J Hydraul Eng, ASCE, in press. [18] Hearne JW, Armitage PD. Implications of the annual macrophyte growth cycle on habitat in rivers. Regul Rivers: Res Manage 1993;8:313–22. [19] Hey RD. Bar form resistance in gravel-bed rivers. J Hydraul Eng, Am Soc Civil Eng 1988;114(12):1498–508. [21] Kellerhals R. Stable channels with gravel-paved beds. Waterways Harbours Div, Am Soc Civil Eng 1967;93(WW1):63–84. [23] Petts GE. Rivers. London: Butterworths; 1983. [24] Pitlo RH, Dawson FH. Flow resistance of aquatic weeds. In: Pieterse AH, Murphy KJ, editors. Aquatic weeds: the ecology and management of nuisance aquatic vegetation. Oxford: Oxford University Press; 1990. p. 74–84. [25] Preston CD, Croft JM. Aquatic plants in Britain and Ireland. Colchester: Harley Books; 1997. [26] Shaw EM. Hydrology in practice. London: Chapman and Hall; 1988. [27] Spink AJ, Murphy KJ, Westlake DF. Distribution and environmental regulation of species of Ranunculus subgenus Batrachium in British rivers. Arch Hydrobiol 1997;139(4):509–25. [28] Task Force on Friction in Open Channels. Friction factors in open channels. J Hydraul Div, Am Soc Civil Eng 1963;89(HY2): 97–143. [29] Watson D. The effects of aquatic macrophytes on channel roughness and flow parameters. PhD Thesis, University of Southampton, 1986. [30] Watson D. Hydraulic effects of aquatic weeds in UK rivers. Regul Rivers: Res Manage 1987;1(3):211–27. [31] Watts JF, Watts GD. Seasonal change in aquatic vegetation and its effect on river channel flow. In: Thornes JB, editor. Vegetation and erosion: processes and environments. Chichester: John Wiley & Sons Ltd.; 1990. p. 257–67. [32] Wolman MG. A method of sampling coarse river-bed material. Am Geophys Union Trans 1954;35(6):951–6. [33] Yalin MS. River mechanics. Oxford: Pergamon Press; 1992. [34] Yen BC. Open channel flow resistance. J Hydraul Eng, Am Soc Civil Eng 2002;128(1):20–39.
Effect of macrophyte spatial variability on channel resistance Julian C. Green
*
Department of Geography, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK Received 10 January 2005 Available online 19 July 2005
Abstract Aquatic macrophytes can severely retard flow rates in the river channels that they occupy. Consequently, there is a need to improve our ability to model vegetation resistance, to aid flood prediction and allow for better-informed channel management. An empirical model is developed to calculate flow resistance (ManningÕs resistance coefficient) of channels containing the submergent macrophyte Ranunculus (water-crowfoot). Blockage factors (the proportion of a cross-section blocked by vegetation) were determined for up to nine cross-sections at each of 35 river sites. These were used to create blockage-factor percentiles, which were regressed against vegetation resistance. An exponential best-fit relation involving the 69th blockage-factor percentile gave the best results. A parameter relating the length of the vegetated/solid boundary in contact with the open channel to the length of the conventionally-defined wetted perimeter improved the model fit by acting as a pseudo-measure of the turbulent-energy losses generated within the unvegetated stream by the macrophytes. The model was tested on three additional sites containing different macrophyte species and much higher vegetation blockages, and was found to work well. 2005 Elsevier Ltd. All rights reserved. Keywords: Aquatic macrophytes; Channel resistance; Blockage factor; Spatial variability; Ranunculus; ManningÕs n
1. Introduction Lowland rivers often contain extensive stands of aquatic macrophytes [25], which can dominate local hydraulics. As obstructions to flow, macrophytes increase channel resistance and water stage, while reducing average flow velocities [24]. Flow resistance of vegetated channels can typically be an order of magnitude greater than that of unvegetated channels, but can rise to two orders under exceptional conditions [12]. This has led to the claim that macrophytes contribute to local flooding [24], and they are routinely removed to reduce this perceived risk. In addition to the questionable benefits [7,18] and considerable expense incurred [9], such operations are detrimental to the wider riverine ecosystem [19,21]. Thus, there is a need to
*
Tel.: +44 1509 222751/222794; fax: +44 1509 223930. E-mail address: [email protected]
0309-1708/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2005.05.010
understand how aquatic macrophytes affect flow resistance to allow channel management to be adjusted to maintain both hydraulic efficiency and biotic diversity. Several empirical studies have attempted to relate the quantity of aquatic macrophytes in a river to their hydraulic resistance [3,4,8,10,29–31]. Of the parameters employed, the vegetated proportion of several cross-sections (termed the multi-cross-sectional blockage factor) has been demonstrated to be the most appropriate predictor of channel resistance on both theoretical and empirical grounds [14,16]. The blockage factor for a single cross-section is defined as B¼
Av A
ð1Þ
where B is the cross-sectional blockage factor; Av is the area of the wetted cross-section blocked by macrophytes (m2); A is the area of the wetted cross-section (m2). B can be given as a proportion or percentage, but the latter will be used in this paper.
J.C. Green / Advances in Water Resources 29 (2006) 426–438
Green [16] presents a regression equation between this parameter and the vegetation component of resistance for sites containing the trailing submergent macrophyte Ranunculus subgenus Batrachium (Gray) (watercrowfoot): n4 ¼ 0.0043BWM 0.0497
ð2Þ
where BWM is the weighted median of several cross-sectional blockage factors measured at each site; and n4 is defined as n4 ¼ ntot nb
ð3Þ
where n4 is the vegetation component of ManningÕs n resistance coefficient; ntot is the total value of ManningÕs n; nb is the base value of n for a straight, uniform, smooth channel in natural materials (boundary resistance) (all dimensionless) (adapted from [5,1]). While Eq. (2) gives respectable results (R2 value of 62.2% for the regression between predicted and calculated values of ntot), it only accounts for the most significant factor affecting resistance: the exclusion of most of the flow from within macrophyte stands [14]. Other studies have suggested that the spatial location of plants within a channel also affects flow resistance [23,11], with stand arrangements influencing the generation of turbulence in the free stream [13]. However, no study has included this effect in a resistance model. This paper develops a vegetation-resistance model using hydraulic and vegetation data collected at 35 river sites whose floras were dominated by Ranunculus species. The model is developed in two stages: firstly, accounting for variation in macrophyte distribution between cross-sections; and, secondly, accounting for variations within cross-sections. The resulting model will be tested on three additional sites dominated by other macrophyte species and whose values of channel resistance were outside the calibration range of the model.
2. Method Below is an outline of the data collection procedures used in this study. A more detailed description of methods is given in [15]. 2.1. Species selection The main sites investigated were dominated by one of three Ranunculus species: R. peltatus (pond watercrowfoot), favouring slow flow velocities; R. penicillatus subspecies pseudofluitans (stream water-crowfoot), favouring medium flow rates; and R. fluitans (river watercrowfoot), favouring the fastest flow rates [27] (see Table 1). All three are trailing submergents that have a clumped structure with high shooting density, and typically form monospecific vegetation stands. Ranunculus fluitans and
427
R. penicillatus both have capillary aquatic leaves, while R. peltatus has laminar terrestrial leaves. Green [15] demonstrated that it was acceptable to develop a single resistance equation encompassing these species, as the structural differences occur at the comparatively small scale of the leaves and individual shoots (leaf scale), while resistance is usually controlled at the larger scale of entire plants or stands of several plants (stand scale) [13]. 2.2. Field sites Sites were chosen from four lowland, nutrient-rich river networks in southern England: (Bristol) Avon, (Salisbury) Avon, Stour and Thames (see Table 1 for site details). This ensured that the resistance model was not site specific and would be applicable over a wide range of hydraulic conditions. All sites had predominately gravel beds with sufficient stability to permit the establishment of plants. While each site was dominated by a single plant species (excepting Sites E2 and E3), a total of 26 species was recorded across the sites. These additional species were generally limited to small quantities at the channel margins, though some submergent species, such as Oenanthe fluviatilis (river water-dropwort), were subdominant in association with Ranunculus. As these submergents have a similar structure to Ranunculus spp., at the stand scale of resistance, they were not deemed to affect the results significantly. 2.3. Measuring total resistance Total resistance at each site was calculated from the Manning Equation: ntot ¼
K R2=3 S e1=2 V
ð4Þ
where K is a factor to keep the equation dimensionally correct (1 m1/3 s1); R is the hydraulic radius (m); Se is the slope of the energy grade line (dimensionless); V is the mean channel velocity (m s1) (rearranged from Yalin [33]). Values of ManningÕs n and other hydraulic parameters are given in Table 2. 2.3.1. Hydraulic radius and area Hydraulic radius was calculated using the equation: R¼
A WP
ð5Þ
where WP is the wetted perimeter of the channel boundary (m). R was determined for between five and nine cross-sections (five sections for Sites 1–5; seven for Sites 6–19 and E1; nine for Sites 20–35 and E2-3) within each 25-m long reach using a quasi-stratified sampling procedure [15], where the cross-sections were positioned randomly, but with a minimum distance between sections
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Table 1 Details of vegetation-resistance sites Site
Location
River
UK grid reference
Date
Dominant macrophyte
b50 (mm)
(c3/ab)91 (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Tellisford Freshford Crockerton Kingston Deverill Tellisford Hill Deverill Manor Hincks Mill Farm Upavon Tellisford East Kennet Stoney Littleton Gillingham Tellisford Tellisford Stitchcombe Marnhull Heytesbury Boynton Tellisford Heytesbury Longbridge Deverill Longbridge Deverill Longbridge Deverill Longbridge Deverill Longbridge Deverill Longbridge Deverill Kingston Deverill Steeple Langford Steeple Langford Boynton Boynton Boynton Heytesbury Steeple Langford West Swainford Fm
Frome Frome Wylye Wylye Frome Wylye Shreen Water Avon Frome Kennet Wellow Brook Shreen Water Frome Frome Kennet Stour Wylye Wylye Frome Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Wylye Shreen Water
ST 806557 ST 791600 ST 873431 ST 844372 ST 806577 ST 868401 ST 812304 SU 136550 ST 806557 SU 115677 ST 734568 ST 807278 ST 806557 ST 806577 SU 225694 ST 767202 ST 928425 ST 953398 ST 806557 ST 928425 ST 871407 ST 871407 ST 871407 ST 871407 ST 871407 ST 871407 ST 844372 SU 036373 SU 036373 ST 957396 ST 958396 ST 951398 ST 928425 SU 036373 ST 813309
14/5/96 23/5/96 28/5/96 4/6/96 5/6/96 12/6/96 16/6/96 18/6/96 19/6/96 23/6/96 12/7/96 18/7/96 19/7/96 21/7/96 23/7/96 25/7/96 26/7/96 14/8/96 16/8/96 13/5/97 14/5/97 16/5/97 16/5/97 16/5/97 18/5/97 20/5/97 23/5/97 5/6/97 7/6/97 10/6/97 11/6/97 13/6/97 14/6/97 24/6/97 7/7/97
F F Pf Pf F Pf Pf F F P Pf Pf F F Pf Pf Pf Pf F Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf Pf
39 39 7 6 43 10 16 10 39 2 67 32 39 43 38 14 18 18 39 2 26 26 26 26 26 26 2 15 7 10 25 27 2 7 2
14.6 22.6 6.2 8.0 13.6 6.3 10.1 18.5 14.6 2.9 27.3 16.6 14.6 13.6 24.9 10.6 10.8 11.2 14.6 17.1 17.1 17.1 17.1 17.1 17.1 17.1 9.5 9.3 4.8 10.8 20.0 17.3 17.1 4.8 4.3
E1 E2 E3
Marlborough Kingston Deverill Kingston Deverill
Kennet Wylye Wylye
SU 198693 ST 844372 ST 844372
16/7/96 10/7/97 24/8/97
C E E
2 14 14
6.4 9.4 9.4
a = long axis of a grain (mm); b = intermediate axis of a grain (mm); c = short axis of a grain (mm); (c3/(ab))91 = the 91st percentile of the c3/(ab) combination axis. Codes for macrophyte species: F = Ranunculus fluitans; Pf = R. penicillatus pseudofluitans; P = R. peltatus; C = Callitriche stagnalis; E = emergent species.
(4 m when five cross-sections were used; 2.1 m when nine sections were sampled). 2.3.2. Slope of energy grade line Se was calculated using the equation: Se ¼
Dh Dhv þ l L
ð6Þ
where Dh is the difference in water surface elevation (m); l is the length of river channel reach (= 25 m); L is the distance between the first and last cross-sections (625 m); Dhv is the upstream velocity head minus downstream velocity head (m); hv is the velocity head (m), defined as ¼
aV 2 2g
ð7Þ
where a is a constant, generally considered equal to 1; g is acceleration due to gravity (9.81 m s2) (adapted from Barnes [2]). Dh was measured over the 25 m reach using the conventional U-tube method [23]. Meanwhile, the difference between the velocity heads was measured over the potentially shorter distance between the first and last cross-sections sampled. 2.3.3. Velocity Velocity was measured using a two-dimensional Valeport electromagnetic current meter, recording at 10 Hz for 30 s at each point. Since the conventional 0.6 or 0.2/0.8 sampling methods are inappropriate to determine mean velocity of a vegetated profile [14], the veloc-
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429
Table 2 Site hydraulic data Site
V (m s1)
R (m)
Se
ntot
n4
B50 (%)
B69 (%)
T50
T16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0.360 0.480 0.162 0.082 0.248 0.086 0.258 0.160 0.348 0.069 0.142 0.361 0.187 0.223 0.209 0.186 0.137 0.133 0.286 0.117 0.171 0.194 0.162 0.185 0.190 0.148 0.128 0.131 0.093 0.088 0.088 0.148 0.116 0.137 0.094
0.176 0.329 0.390 0.114 0.295 0.193 0.198 0.499 0.151 0.262 0.265 0.118 0.126 0.235 0.404 0.290 0.377 0.367 0.099 0.572 0.220 0.189 0.212 0.219 0.228 0.252 0.159 0.284 0.363 0.587 0.540 0.420 0.559 0.383 0.325
0.0066 0.0092 0.0006 0.0078 0.0076 0.0031 0.0090 0.0011 0.0062 0.0004 0.0010 0.0090 0.0070 0.0101 0.0029 0.0014 0.0012 0.0009 0.0069 0.0019 0.0028 0.0043 0.0036 0.0017 0.0040 0.0034 0.0090 0.0030 0.0014 0.0024 0.0026 0.0058 0.0027 0.0018 0.0026
0.071 0.094 0.082 0.254 0.152 0.216 0.122 0.130 0.065 0.123 0.089 0.063 0.113 0.171 0.141 0.089 0.131 0.115 0.062 0.258 0.114 0.111 0.133 0.083 0.124 0.156 0.219 0.180 0.205 0.391 0.381 0.289 0.302 0.163 0.257
0.015 0.040 0.051 0.206 0.107 0.179 0.079 0.089 0.006 0.097 0.027 0.006 0.050 0.124 0.092 0.050 0.094 0.079 0.000 0.216 0.079 0.077 0.097 0.050 0.090 0.122 0.175 0.143 0.176 0.356 0.336 0.244 0.276 0.140 0.229
11.3 15.5 17.5 61.7 32.6 60.8 50.8 18.9 21.3 41.7 30.4 12.1 39.4 55.7 35.0 27.7 35.9 27.9 20.1 51.2 35.6 30.6 41.0 12.1 35.5 41.9 54.6 62.8 55.6 56.9 57.8 63.1 63.2 64.4 58.4
14.1 18.3 27.5 68.2 39.4 61.0 52.5 21.1 24.1 45.4 30.8 14.9 42.7 56.9 37.0 28.2 36.9 34.0 22.5 52.4 40.9 34.6 42.7 24.2 38.4 49.7 57.2 66.3 60.5 58.8 58.4 66.6 66.6 68.4 63.8
1.20 1.07 1.21 0.67 1.45 1.08 1.42 1.51 1.27 1.76 1.27 1.13 1.00 1.50 2.13 1.52 1.50 1.61 0.96 1.55 1.42 1.42 1.27 1.28 1.60 1.51 1.10 0.89 1.20 1.79 2.10 1.74 1.76 1.04 1.20
1.04 1.03 1.05 0.60 1.25 0.94 1.24 1.35 1.12 1.41 1.16 1.06 0.96 1.02 1.95 1.38 1.34 1.36 0.93 1.36 1.13 1.20 1.17 1.13 1.32 1.16 0.94 0.67 0.97 1.69 1.88 1.50 1.41 0.71 1.01
E1 E2 E3
0.172 0.078 0.066
0.404 0.223 0.230
0.0003 0.0108 0.0098
0.051 0.490 0.559
0.021 0.448 0.532
18.6 89.4 91.0
20.4 93.5 93.7
1.30 0.48 0.41
1.14 0.30 0.27
V = mean flow velocity; R = hydraulic radius; Se = slope of energy grade line; ntot = total value of ManningÕs resistance coefficient; n4 = vegetation component of ManningÕs n; B50/B69 = 50th and 69th percentiles of the cross-sectional blockage factor; T50/T16 = 50th and 16th percentiles of the vegetation spatial-variability parameter.
ity of as many as five points in each profile was measured, using Table 2 in Green [15] as a guide. The number of profiles used to take velocity readings varied with the distribution of vegetation within a cross-section; sites containing a high degree of fragmentation (and hence a complex distribution of velocities) had the greatest number of profiles measured, with 27 profiles sampled at Site 15. When the velocity of each profile had been determined, the Mean Section Method [26] was used to calculate channel discharge. Finally, using the weighted-median area of cross-sections sampled at a reach (weighting dependant upon spacing between cross-sections), the weighted median velocity of the 25 m section was determined using the equation:
Q A where Q is discharge (m3 s1).
V ¼
ð8Þ
2.4. Calculation of the vegetation-resistance component Following Eq. (3), a value for the boundary component of resistance, nb, was subtracted from total flow resistance to obtain the vegetation component of resistance, n4. nb was calculated using the equation (adapted from Task Force on Friction in Open Channels [28]): nb ¼
R1=6
aR 17.984 log ks
ð9Þ
J.C. Green / Advances in Water Resources 29 (2006) 426–438
where a is the channel shape correction factor (dimensionless), defined as .314 R a ¼ 11.1 ð10Þ dm where dm is the maximum flow depth (m) [19]; ks is the roughness height (m), calculated from the equation: 3 c log k s ¼ 1.54 þ 1.02. log ð11Þ ab 91 where a is the long axis of a grain (mm); b is the intermediate axis of a grain (mm); c is the short axis of a grain (mm); (c3/(ab))91 is the 91st percentile of the c3/(ab) combination axis [12]. A 100-grain sample was taken from each site using the grid-by-number technique [32] (see Green [16] for a description of measurement procedure). Values of b50 and (c3/(ab))91 are given in Table 1. 2.5. Vegetation blockage factor The cross-sectional vegetation blockage factor was determined from Eq. (1). In order to assess the extent of vegetation present in a cross-section, a series of measurement points was positioned across a section, and, at each location, the following variables were recorded: water depth; the species of macrophyte present; height of the top of the macrophyte above the bed; height of the underside of the macrophyte above the bed; and whether that macrophyte was present between that profile and the next. When the spacing between shoots was less than 5 cm in the horizontal or 2 cm in the vertical, the strands were regarded as a part of a blocked section, while an open section was recorded if the shoot spacing was greater than these distances. It was also sometimes necessary to record the presence of a second vegetation layer in the same profile. The profiles were placed evenly across the width where conditions were uniform, though they were also positioned at the edges of macrophyte stands, at sharp gradients in the amount of vegetation present, and where there was a change of macrophyte species. As a result of the movement of macrophytes in the flow and other errors, the width was recorded to a resolution of 5 cm. However, there was less movement of the plants in the vertical, so depths could be resolved to 1 cm or 0.5 cm depending on the turbulence of the water surface.
3. Vegetation spatial variability between cross-sections
vegetation within cross-sections that are only a few metres apart. At the 35 Ranunculus sites, the mean standard deviation of B at a site was as high as 10%, with a maximum range in B between sections recorded at Site 4 (20–74%). This high variability of vegetation distribution in a channel explains why it was necessary to sample so many sections, and why each reach was restricted to only 25 m in length; a distance greater than this would demand sampling even more cross-sections and would be more vulnerable to breaching uniform-flow requirements. It is possible that the average of individual cross-sectional blockage factors may not be the best parameter to reflect the resistance effect of the macrophytes. The general consensus in the boundary-resistance literature is that grain-size percentiles higher than the median size, D50 (where Dp means that p% of the area of the channel bed is covered by grains that have a smaller or equal diameter to the D value [21]), are most appropriate predictors of resistance, due mainly to the fact these percentiles project further into the flow (for a review of the literature see [34]), as well as because lower sampling errors are associated with these percentiles [12]. Similar logic can be applied to vegetation resistance: it can be hypothesised that the resistance of a channel reach will be most affected by the cross-section with the highest vegetation blockage. Moreover, the median value of individual cross-sectional blockage factors, B50, may not be the most precise percentile to sample. In order to test these hypotheses, each study reach was split into 100 units (each 25 cm long). The value of B for each sampled cross-section was assigned to the closest unit. The remaining units were filled by linear interpolation between the measured points. Upstream of the first sampled cross-section and downstream of the last cross-section, the units were given the same value as the nearest measured point (an example is given in Fig. 1). The units were ranked to produce blockage-
70 60
Blockage factor (%)
430
50 40 30 20 10 0
Trailing submerged macrophytes, such as members of Ranunculus subgenus Batrachium (water-crowfoots), form randomly-distributed clumps within river channels [6]. This can result in a high variation in the amount of
0
5
10
15
20
25
Distance (m) Fig. 1. Measured (diamonds) and interpolated values of the crosssectional blockage factor with distance downstream at Site 27.
J.C. Green / Advances in Water Resources 29 (2006) 426–438
factor percentiles, where Bp means that p% of the crosssections in a reach are vegetated by less than or equal to the B value. Fig. 2 shows cumulative percentiles for three sites: one site with uniformly low cross-sectional blockages; one site with consistently high vegetation blockages; and one site with both low and high blockages. Sampling-probability theory (outlined in [12]) was used to determine the precision of sampling different blockage-factor percentiles for the 35 sites (Fig. 3). Percentiles below B10 or above B90 will not be analysed, due to limitations associated with the fact that only between five and nine sections were sampled. For the remaining percentiles, the largest sampling errors were associated with the smaller percentiles (Fig. 3). There was a dramatic improvement in sampling precision between percentiles 25 and 28, after which there was a more gradual improvement in precision. With the exception
60 50
B (%)
40 30 20 Site 31
10
Site 27 Site 8
0 0
20
40
60 Percentile
80
100
of the very highest percentiles, the most precise percentile to sample was the 71st. Although the errors associated with this percentile are not significantly lower than larger percentiles, StudentÕs t tests show that they are lower than the errors for B40–51 at the 0.05-significance level and for B10–39 at the 0.01-significance level. These tests indicate that blockage-factor percentiles above the median value are the most precise to sample, and, thus, may be the most appropriate percentiles to use in a vegetation-resistance model. Linear regressions were determined between each blockage-factor percentile and calculated values of vegetation resistance, n4, determined at each of the 35 sites. Comparison of the R2 values for each equation between a blockage-factor percentile and n4 demonstrates that the percentiles differ in their ability to predict vegetation resistance (Fig. 4). The highest R2 value was 65.2% for the equation involving B60 (Fig. 5a). StudentÕs t tests show that the equations involving B58–62 gave significantly higher R2 values (at the 0.01 level) than those involving B48–52. Since Green [14] presented partial theory suggesting a non-linear relation between B and n, exponential best-fit lines were also fitted to the relations by adding 0.05 to n4 to remove negative values. For these (Fig. 4), the highest R2 value (74.4%) was for the equation involving B69 (Fig. 5b), with the R2 values for equations involving B67–71 also significantly higher at the 0.01 level than those involving B48–52. These results demonstrate that cross-sections with higher than the median vegetation blockage are the best predictors of vegetation resistance. While the exponential curves gave consistently higher R2 values than linear best fits (Fig. 4), the root mean square (RMS) error of the predicted values of n4 compared to the calculated values was lower for the linear model involving B60 (at 0.053) than the exponential model involving B69 (at 0.055). Thus, the competing merits of the linear and exponential models will be compared fol-
75
9 8 7 6
70
R2 value (%).
Relative standard error (%).
Fig. 2. Cumulative frequency of cross-sectional blockage factors at three sites.
431
5 4 3 2 1 0
65 60 exponential model linear model
55
0
20
40 60 B percentile
80
100 50 0
Fig. 3. The median (solid line) and lower and upper quartile (dashed lines) relative standard errors of sampling different blockage-factor, B, percentiles from the 35 Ranunculus sites, using probability theory outlined in [12].
20
40
60
80
100
B percentile
Fig. 4. The R2 values for using different blockage-factor percentiles in the vegetation-resistance model.
432
J.C. Green / Advances in Water Resources 29 (2006) 426–438 0.55
0.45
n4
0.35 y = 0.0043x - 0.0597
0.25
2
R = 0.6515
0.15 0.05 -0.05 0
20
40
a
60 B(60) (%)
80
100
80
100
0.6
n4+0.05
0.5 y = 0.0432e
0.4
0.0281x
2
R = 0.744
0.3 0.2 0.1 0.0 0
b
20
40
60 B(69) (%)
Fig. 5. (a) Linear; and (b) exponential relations between B and n4. Diamonds represent sites dominated by Ranunculus species; triangles represent site dominated by Callitriche stagnalis; crosses represent sites dominated by emergent species.
lowing the inclusion of a parameter to account for the resistance effect of vegetation spatial variability within cross-sections.
4. Vegetation spatial variability within cross-sections The above blockage factor represents the displacement of the flow from within plant stands. Macrophytes also retard flow rates in the adjacent unvegetated zones due to: the plants isolating areas from the main flow, such as immediately downstream of plant stands; and because turbulence is generated at the shear layers between the comparatively slow velocities inside and higher flow rates outside the plants [14]. While it is acknowledged that turbulence is generated three dimensionally by the macrophytes, it is suggested that the degree of turbulent generation is related to the twodimensional length of the vegetated boundary in a cross-section: increasing the length of the boundary between the plants and the unrestricted flow is likely to increase turbulence [14]. When macrophytes are located in large clumps with a relatively short boundary between themselves and the unvegetated stream, there will be minimal turbulence generation. Meanwhile, if the same amount of vegetation is distributed into widely scattered individual plants, the relative surface area of the macro-
phytes will be higher, forcing the flow to weave between the stands, increasing turbulence losses. The position of the macrophytes relative to the stream bed should also be significant to resistance: where the solid boundary is covered by vegetation, it will not affect the flow in the unvegetated channel, while it will have minimal impact on the already slow velocities within the plant body. This cross-sectional measure should be related indirectly to downstream turbulence losses. Where the vegetation is clumped within a cross-section with a short vegetated boundary, it is likely that the plants will also be clumped in the downstream direction. Likewise, if the vegetation is fragmented within a cross-section with a relatively long boundary, it is probable that the individual stands will be relatively small and the flow will be regularly forced from side to side, losing energy. Since it is the relative rather than absolute length of the plant boundary that is significant to resistance, a spatial-variability parameter will be used relating the effective wetted perimeter, WPe, to the conventional wetted perimeter, WP: T ¼
WP e WP
ð12Þ
where T is the vegetation cross-sectional spatial-variability parameter (dimensionless); WP is the length of the solid boundary of the cross-section (conventional wetted perimeter; m); and WPe is the effective wetted perimeter, defined as the sum of the lengths of the vegetation and solid boundaries, but excluding the lengths where the solid and vegetation boundaries meet, and omitting where the vegetation is in contact with the water surface (m) (a comparison of the definitions of WP and WPe is given in Fig. 6). The parameter T acknowledges that channel resistance is proportional to the combined length of the solid and vegetation boundaries in contact with the unvegetated stream, though it is limited by an inability to distinguish between the differing roughnesses of the two boundaries. Where the vegetation is widely scattered throughout a cross-section, the effective wetted perime-
Fig. 6. Definition of: (a) conventionally defined wetted perimeter; and (b) effective wetted perimeter. Thick line is wetted perimeter used for each parameter; thin line is water surface; shading represents area occupied by aquatic macrophytes.
J.C. Green / Advances in Water Resources 29 (2006) 426–438
ter will be higher than the conventional wetted perimeter, so T will be above 1 (for example, the cross-sections at Site 15; Fig. 7). However, when the vegetation is clumped in a cross-section and covers the solid boundary, T may drop below 1 (for example, most of the cross-sections at Site 28; Fig. 8). Under such a situation, the boundary is isolated from the flow and channel resistance will be less than would otherwise occur. In reality, this is a reduction in boundary resistance, but the decrease is caused by the plants. The longer the effective boundary compared to the conventional boundary, the greater the potential for turbulence generation, suggesting that T is positively related to vegetation resistance. When vegetation blockage is zero, T must equal 1, while it will equal zero when there is a total blockage. There are no other fixed points, and there is no theoretical upper limit to T, though, probabilistically, the greatest vegetation boundary length, and hence T, will occur when half the crosssection is blocked. The minimum value of T trends from 1 for unvegetated sections to zero for completely blocked sections (solid line in Fig. 9). However, T values can occur below this line when the proportion of the
433
solid boundary covered by vegetation is greater than the proportion of the cross-sectional area blocked by macrophytes. The average value of T for the 267 cross-sections collected at the 35 sites was 1.38, indicating that, on average, WPe was longer than WP, increasing turbulent energy losses; these values are plotted against the corresponding value of B in Fig. 9. Of the Ranunculus sites, the lowest T value was 0.53 for a cross-section with a blockage of 69.7% at Site 28 (Fig. 8), while Site 4 had the lowest median value, T50 (percentiles calculated in the same manner as for the blockage factors), of only 0.67 for a B50 of 61.6%. Meanwhile, the highest value of T was 2.37 for a cross-section with a blockage of 54.9% at Site 33, while Site 15 had the highest value of T50, at 2.13 for a B50 of 35.0% (Fig. 7). In common with the vegetation blockage factor, the T values for individual cross-sections were used to create T percentiles for each site to determine if a percentile other than the median was most appropriate in the resistance model. Probability analysis (detailed in [12]) was performed to determine the relative precision of sampling each T percentile across the 35 sites (Fig. 10).
Fig. 7. Site 15. (a) Photograph looking downstream; (b) non-dimensionalised representation of successive cross-sections downstream. Black shading represents areas blocked by vegetation. Values of T and B are given alongside each cross-section.
434
J.C. Green / Advances in Water Resources 29 (2006) 426–438
Fig. 8. Site 28. (a) Photograph looking downstream; (b) non-dimensionalised representation of successive cross-sections downstream. Black shading represents areas blocked by vegetation. Values of T and B are given alongside each cross-section.
Excluding the extreme percentiles (due to the previously mentioned limitations of this technique) the most precise percentile to sample was the 67th, it being more precise to sample than percentiles above T79 and below T44, at the 0.05-significance level, and percentiles below T38, at the 0.01 level. This suggests that a percentile within the range T44–79 should be the most appropriate for use in a vegetation-resistance model. Residuals between predicted and calculated values of n4 were determined for both the linear and exponential models in Fig. 5. These residuals were then regressed against each percentile of T (Fig. 11). Both models give a similar pattern, with the exponential model producing consistently higher R2 values than the linear model. The R2 values peaked at 37.8% for T12 in the linear model (Fig. 12a), and at 53.1% for T16 in the exponential model (Fig. 12b).
For the exponential model, a StudentÕs t test indicates that the R2 values for the percentiles in the range T14–18 were higher than those in the ranges T9–13 and T19–23, at the 0.05-significance level. The fact that these low percentiles give the best model fit despite their relative imprecision of sampling indicates that it is important for them to be used in the vegetation model. As to why this is the case is uncertain. The regression equations in Fig. 12 are both significant at the 0.01 level and thus are able to account for some of the variation in the residuals in both the linear and exponential models of Fig. 5. Subtracting the fitted resistance residuals (calculated using the equations in Fig. 12) from the fitted values of n4 (calculated using the equations in Fig. 5) improves the ability to model vegetation resistance; regression between the calculated and improved-predicted values of n4 give R2 values of
J.C. Green / Advances in Water Resources 29 (2006) 426–438
435
0.15
2.5
0.10 2.0
Residuals
0.05
T
1.5
0.00 0.0 -0.05
0.5
1.0
1.5
2.0
T (12)
1.0
-0.10 y = -0.1114x + 0.1292
0.5
2
-0.15
R = 0.3777
a -0.20 0.0
0
20
40
60
80
100
0.15
B (%)
0.10
Relative standard error (%) .
0.00 0.0 -0.05
0.5
1.0
1.5
2.0
T (16)
-0.10 -0.15
y = -0.1361x + 0.155 2 R = 0.5316
b -0.20
4
Fig. 12. (a) Plot of T12 against the residuals in the linear relation of B60 against n4; (b) plot of T16 against the residuals in the exponential relation of B69 against n4. Diamonds represent sites dominated by Ranunculus species; triangles represent site dominated by Callitriche stagnalis; crosses represent sites dominated by emergent species.
3
2 1 0 0
20
40 60 T percentile
80
100
Fig. 10. The median (solid line) and lower and upper quartile (dashed lines) relative standard errors of sampling different T percentiles from the 35 sites, using probability theory outlined in [12].
55
exponential model linear model
50 45
R2 value (%) .
0.05 Residuals
Fig. 9. Plot of B against T for individual cross-sections. Diamonds represent cross-sections dominated by Ranunculus species; triangles represent cross-sections dominated by Callitriche stagnalis; crosses represent cross-sections dominated by emergent species. Dashed line: upper envelope for approximately 95% of data points; solid line: theoretical lower bound of envelope, excepting cases where the percentage of the solid boundary covered by vegetation is greater than B.
40
78.4% and 83.0% for the linear and exponential models, respectively. Adding the calculated value for boundary resistance, nb (Eqs. (9)–(11)), to the predicted values of n4 gives estimates of total ManningÕs n. For the linear model, the regression of predicted versus calculated total resistance gives an R2 value of 76.1% (Fig. 13a), with an RMS error of 0.042 (30%; Table 3). Meanwhile, for the exponential model, the regression of predicted versus calculated total resistance gives an R2 value of 81.2% (Fig. 13b), with an RMS error of 0.037 (24%; Table 3). Thus, the higher R2 value and lower RMS error indicate that the exponential model is a better predictor of resistance in channels dominated by Ranunculus species. Combining the regression equations in Figs. 5b and 12b, gives the following equation for the vegetation component of ManningÕs n: n ¼ 0.0432e0.0281B69 þ 0.1361T 0.205 ð13Þ 4
35
16
30
5. Application of the model to different macrophyte species
25 20
0
20
40
60
80
100
T percentile Fig. 11. The R2 values for using different T percentiles in the linear and non-linear models.
To determine if the model can be applied to other macrophyte species and also to higher vegetation blockages (and hence higher vegetation resistance) than was encountered with Ranunculus species, three additional
436
J.C. Green / Advances in Water Resources 29 (2006) 426–438 0.6
0.5
Calculated n
0.4
0.3
0.2
0.1
0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Predicted n
a 0.6
0.5
Calculated n
0.4
0.3
0.2
0.1
0.0 0.0
b
0.1
0.2
0.3
0.4
0.5
0.6
Predicted n
Fig. 13. Predicted versus calculated values of total resistance, n (a) linear model; (b) exponential model. Dashed line equals 1:1 line. Diamonds represent sites dominated by Ranunculus species; triangles represent site dominated by Callitriche stagnalis; crosses represent sites dominated by emergent species.
sites were studied dominated by different species. Site E1, on the River Kennet, was dominated by Callitriche stagnalis (common water-starwort) and did not contain any Ranunculus species. Like Ranunculus, Callitriche species are clumped-trailing submergents and were often found as subdominant species alongside or inter-mingled with Ranunculus at the main study sites. The site at Kingston Deverill, at the headwaters of the River Wylye, was dominated by Ranunculus penicillatus pseudofluitans during the spring and early summer, and was used in the main vegetation-resistance model on 4/6/96 (Site 4) and 23/5/97 (Site 27). However, later in
the season, the branching emergents Rorippa nasturtium-aquaticum (watercress) and Apium nodiflorum (foolÕs watercress) dominated the aquatic flora, and the location was studied for this additional analysis on 10/ 7/97 (Site E2) and 24/8/97 (Site E3). Both the linear and exponential vegetation-resistance models were applied to the extra sites without recalibration. The site containing Callitriche stagnalis (Site E1) had a lower value of total resistance than any of the Ranunculus sites (0.051; Table 2), but generally plotted within the scatter of Ranunculus sites in Figs. 5, 9, 12, 13 and Eq. (13) gave a predicted ntot that was only 9% from the measured value (Table 3). Callitriche and Ranunculus species have similar morphology in that they are clumped-trailing submergents. Therefore, it is likely that a vegetation-resistance model should be applicable to both, providing that resistance is controlled at the scale of individual or groups of plants, (low blockages), rather than at the scale of shoots and leaves (high blockages); Green [13] suggested that the differing porosities between Callitriche and Ranunculus species, produced by differing leaf morphology, would only have a significant effect on channel resistance for blockages above 50% when the channel slope was 1 · 104, increasing to 85% for slopes of 1 · 102. Meanwhile, Sites E2 and E3 were not only dominated by branching emergents that are structurally different to Ranunculus species but the extreme blockages present suggest that the leaf scale of resistance would be dominant at these sites [13]. Thus, a resistance model developed for Ranunculus species may not be able to predict the resistance at Sites E2 and E3, especially given the required extrapolation of the model to higher B and lower T values. However, the exponential best-fit curve between B69 and n4 + 0.05 for the Ranunculus sites plots close to these additional sites (Fig. 5b). This validates the exponential curve, since the inclusion of the additional sites had effectively no effect on the RMS errors for this curve, while it increased the RMS errors from 0.053 to 0.063 for the linear relationship, with the line significantly under predicting resistance at these high blockages. Moreover, Sites E2 and E3 plot as extreme outliers in the plot between the resistance residuals and T12 in the linear model (Fig. 12a), but are much closer to the extrapolated line in the exponential model (Fig. 12b). Consequently, the errors for predicted total ManningÕs n using the exponential model (Eq. 13) are only 3% and 18% for Sites E2 and E3 respectively, while they are 44% and 54% respectively for the linear model (Table 3).
6. Discussion and conclusions This paper supports theory presented in Green [13] that the relation between the vegetation blockage fac-
J.C. Green / Advances in Water Resources 29 (2006) 426–438
437
Table 3 Percentage errors for the four vegetation-resistance models Site
Measured ntot
B60 linear
B69 non-linear
B60, T12 linear
B69, T16 non-linear
Predicted ntot
Difference (%)
Predicted ntot
Difference (%)
Predicted ntot
Difference (%)
Predicted ntot
Difference (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0.071 0.094 0.082 0.254 0.152 0.216 0.122 0.130 0.065 0.123 0.089 0.063 0.113 0.171 0.141 0.089 0.131 0.115 0.062 0.258 0.114 0.111 0.133 0.083 0.124 0.156 0.219 0.180 0.205 0.391 0.381 0.289 0.302 0.163 0.257
0.051 0.069 0.063 0.270 0.144 0.239 0.205 0.067 0.102 0.153 0.133 0.068 0.179 0.230 0.145 0.100 0.134 0.117 0.095 0.206 0.140 0.115 0.157 0.052 0.128 0.164 0.226 0.255 0.220 0.224 0.236 0.266 0.249 0.250 0.231
28 26 23 7 5 11 68 48 58 24 50 9 59 34 3 13 2 2 52 20 23 3 18 38 3 5 3 42 7 43 38 8 17 53 10
0.070 0.076 0.074 0.291 0.125 0.227 0.182 0.069 0.094 0.131 0.114 0.084 0.156 0.211 0.122 0.085 0.109 0.099 0.094 0.180 0.122 0.098 0.130 0.069 0.111 0.158 0.209 0.265 0.216 0.211 0.218 0.275 0.256 0.269 0.238
2 19 9 15 18 5 49 47 45 6 29 34 38 23 14 5 17 14 51 30 7 12 2 18 11 2 4 47 5 46 43 5 15 64 8
0.034 0.048 0.041 0.203 0.149 0.208 0.208 0.085 0.090 0.171 0.118 0.053 0.152 0.205 0.224 0.121 0.148 0.134 0.064 0.216 0.127 0.110 0.153 0.043 0.139 0.160 0.191 0.191 0.192 0.275 0.309 0.300 0.269 0.193 0.200
53 48 49 20 2 4 71 35 39 38 33 16 35 20 58 36 12 16 3 16 11 2 15 48 12 3 13 6 7 30 19 4 11 18 22
0.057 0.061 0.062 0.218 0.140 0.199 0.196 0.098 0.092 0.168 0.116 0.074 0.132 0.195 0.232 0.118 0.137 0.129 0.065 0.210 0.121 0.106 0.134 0.067 0.136 0.162 0.182 0.202 0.193 0.286 0.320 0.324 0.293 0.210 0.220
20 35 24 14 8 8 61 25 42 36 31 17 17 14 64 33 4 12 5 19 6 5 1 19 10 4 17 12 6 27 16 12 3 29 14
E1 E2 E3
0.051 0.490 0.559
0.051 0.375 0.366
0 23 35
0.056 0.590 0.579
10 20 4
0.039 0.276 0.259
25 44 54
0.056 0.476 0.461
9 3 18
RMS of Ranunculus sites
31
tor and channel resistance is non-linear, with successive increases in the blockage factor producing an everincreasing effect on resistance. The nature of this relation may also explain why a blockage factor higher than the median gives the best-fit relation with resistance. Moreover, while the two-dimensional parameter T cannot represent the full three-dimensional resistance effect of macrophyte spatial variability, this model is the first to attempt to account for the turbulent energy losses generated in the unvegetated areas adjacent to trailing submergent macrophytes such as Ranunculus species. While this model has been exclusively developed for Ranunculus, the three additional sites studied indicate that it may be applicable to morphologically different species and for higher blockages than found at the main
28
30
24
sites. Further research into other species is required, though, and this model is unlikely to hold for macrophyte species possessing much higher porosities than Ranunculus, such as Sparganium emersum (unbranched bur-reed). The model presented here represents a significant advance in our ability to determine vegetative channel resistance given the quantity and distribution of vegetation at a given moment in time. Admittedly, taking measurements for Eq. (13) is more time consuming than a direct measurement of resistance. However, resistance is temporally variable, due to its stage dependence. Research is ongoing to combine this model with formulae to predict the flow-induced bending of the plants, in order to produce a model of how vegetation-resistance changes with discharge.
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J.C. Green / Advances in Water Resources 29 (2006) 426–438
Acknowledgements Appreciation goes to Dr and Mrs R.G.C. Green for sponsoring this project and assisting in the field; Mr A. Hayes for helping with field equipment; and various landowners who gave permission to work at these sites. Dr P.J. Wood commented on an earlier version of this paper.
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