Transverse and longitudinal mixing in real emergent vegetation at low velocities

Understanding solute mixing within real vegetation is critical to predicting and evaluating the performance of engineered natural systems such as storm water ponds. For the first time, mixing has been quantified through simultaneous laboratory measurements of transverse and longitudinal dispersion within artificial and real emergent vegetation. Dispersion coefficients derived from a routing solution to the 2‐D Advection Dispersion Equation (ADE) are presented that compare the effects of vegetation type (artificial, Typha latifolia or Carex acutiformis) and growth season (winter or summer). The new experimental dispersion coefficients are plotted with the experimental values from other studies and used to review existing mixing models for emergent vegetation. The existing mixing models fail to predict the observed mixing within natural vegetation, particularly for transverse dispersion, reflecting the complexity of processes associated with the heterogeneous nature of real vegetation. Observed stem diameter distributions are utilized to highlight the sensitivity of existing models to this key length‐scale descriptor, leading to a recommendation that future models intended for application to real vegetation should be based on probabilistic descriptions of both stem diameters and stem spacings.


Introduction
Natural and engineered environmental systems such as wetlands or storm water ponds typically contain vegetation and are often designed specifically to treat pollutants [Shilton, 2000]. To protect natural systems and design effective engineered systems, it is necessary to quantify the impact of aquatic vegetation on solute mixing. Understanding mixing in these flows is relevant to a number of other environmental applications, e.g., seed dispersal [Merritt and Wohl, 2002] or nutrient transport [Nishihara and Terada, 2010].
Quantifying the impact of vegetation on hydroenvironmental systems is a challenging task. Vegetation is characterized by small leaves and slightly larger stems, located in bodies of water much larger than a single plant, resulting in a multiscale problem that cannot easily be directly modeled. An approach that spatially averages the effects of the vegetation, i.e., a bulk mixing characterization, is generally preferred [e.g., Nepf, 1999].
The mixing of a solute can be described by the temporally and spatially averaged advection-dispersion equation (ADE): where C is the concentration of a conservative solute (e.g., a pollutant or tracer), u, v, and w are the velocity in the x (stream-wise), y (transverse), and z (vertical) directions, t is time, and D x , D y , and D z are the dispersion coefficients in each direction [Fischer et al., 1979]. The velocity terms describe the advection of the solute. The dispersion coefficients aggregate all the mixing processes that may affect the solute within the volume of interest. These include instantaneous velocity fluctuations, leading to turbulent diffusion, and velocity field heterogeneity, producing mechanical dispersion.
While the data sets of these studies provide valuable insights into dispersion within vegetation, the specific physical characteristics of individual plant species imply that different dispersion processes will dominate, and that the values of longitudinal and transverse dispersion parameters obtained in previous investigations may not be transferable to different plant species. The physical characteristics of the plants are also expected to change over time as a result of their seasonal growth cycles. The focus of the current research is on characterizing dispersion for vegetation types typically associated with storm water ponds, specifically Carex acutiformis and Typha latifolia, with the Typha being considered at two different growth stages. As velocities within storm water ponds are typically very low, the present study focuses on obtaining experimental data at velocities 0:02 m/s. Generalization of the observed dispersion characteristics may be achieved through the development of process-based conceptual models that aim to predict dispersion coefficients from measurable plant physical characteristics. Relevant modeling approaches are described below. Nepf et al. [1997b] and Nepf [1999] proposed a model to estimate transverse dispersion within emergent vegetation due to two processes. First, energy dissipation from the stem drag force, due to the wake structure formation, was assumed to appear as turbulent kinetic energy and termed ''turbulent'' diffusion. The validity of this assumption is discussed in more detail in Nikora [2000] and in the subsequent reply by Nepf [2000]. Second, the individual flow paths imposed by the tortuosity caused by the physical obstruction of the stems, was termed ''mechanical'' diffusion. Nepf [1999] proposed that D y may be determined from:

Mixing Models Within Emergent Vegetation 2.2.1. Transverse Mixing Models
where U is the spatial and temporal mean longitudinal (i.e., stream-wise) velocity, d is stem diameter, C D is mean bulk drag coefficient, a is frontal facing area per unit volume, and a and b are first-order scaling coefficients. ad is a measure of density. The first term on the right represents mixing resulting from drag force energy dissipation, and the second term flow path tortuosity. Nepf [1999] identified a best fit value of a5 0:8160:09 for experimental results and assumed a value of b 5 1.  Serra et al. [2004] proposed a modification of equation (2). Instead of ad they related mixing due to drag force energy dissipation and flow path tortuosity to the ratio of stem diameter to mean spacing between stems, s, giving: Based on relating drag force to a velocity deficit that encourages transverse mixing, Serra et al. [2004] suggested that: where b s is another dimensionless constant. Assuming cylindrical stems, Serra et al. [2004] derived ad5G ðd=sÞ 2 for a random array of vegetation, with G % 3:4. It is evident that equations (2) and (3) are functionally equivalent, although a and b will have different absolute values. Serra et al. [2004] obtained values of b s C D between 1 and 4, and this range of values provides comparable predictions to equation (2). Tanino and Nepf [2008a] introduced a new model to estimate transverse dispersion based on Nepf [1999] to better describe the transition between drag force energy dissipation and flow path tortuosity dominated dispersion: where / is the solid volume fraction, k t is the turbulent kinetic energy, P snc>rÃ is the probability the nearest stem is further than rÃ; rÃ is the minimum distance between stems necessary to contribute to lateral mixing (assumed to be equivalent to 0:5d), s n is the distance between stems, s nc is the center-to-center distance between stems, P snc<5d is the probability the nearest stem is located within 5d, k ? is the permeability in the direction of flow, and c 1 and c 2 are scaling constants with values of 4.0 and 0.34 as determined by least squares fitting to laboratory data. Again, the first term on the right represents mixing due to drag force energy dissipation and the second flow path tortuosity. Tanino and Nepf [2008a] provided expressions for all of the parameters in terms of /, assuming a random array of cylinders. Therefore, the Nepf [1999] and Tanino and Nepf [2008a] models are effectively dependent on the same term (/ or ad). Both models also share the same semiempirical limitation of requiring best fit values (a, b, c 1 , and c 2 ). It should also be noted that these models all assume a single, mean, value of stem diameter.
Nepf [2012] suggested that the models estimating mixing due to drag force energy dissipation and flow path tortuosity may be simplified depending on solid volume fraction, /. When / < 0:1 drag force energy dissipation dominates and D y Ud ð Þ 21 50:2, and when / ! 0:1 flow path tortuosity dominates and D y Ud ð Þ 21 5ad54p 21 / % /. White and Nepf [2003] suggested that longitudinal dispersion in emergent vegetation is primarily due to vortex trapping (D x;t ) and stem-scale secondary wake dispersion (D x;s ). In vortex trapping, particles are entrained temporarily in the eddies formed behind stems. In secondary wake dispersion, particles travel between areas of lower velocity behind stems and higher velocity between stems, resulting in differential longitudinal advection. The expressions for dispersion due to vortex trapping and secondary wake dispersion are:

Longitudinal Mixing Models
and D where b w is a constant of proportionality describing the residence time in stem wakes (dependent on stem Reynolds number, Re d ), j w describes the wake-trapping zone (fraction of the volume with stem wakes, also Water Resources Research 10.1002/2016WR019937 dependent on Re d ), St is the Strouhal number describing the oscillation of the vortices, c w is an incomplete gamma function, Re t 5Ud m t 1m ð Þ 21 is the turbulent Reynolds number, m t is the turbulent viscosity, m is the molecular viscosity, and Sc t is the turbulent Schmidt number. Lightbody and Nepf [2006b] simplified the secondary wake dispersion model (Equation 7) for / < 0:1 to: and for / ! 0:1, Murphy [2006] proposed: Nepf et al. [1997a] and Nepf [2006a, 2006b] relate longitudinal dispersion to vertical dispersion, following the classic triple integral [Fischer et al., 1979]. Nepf [2006a, 2006b] suggest that vertical variations in longitudinal velocity, generated by vertically heterogeneous plant morphology, are responsible for a large portion of longitudinal dispersion within vegetation.

Laboratory Setup
Controlled laboratory tests were conducted to simulate low velocity flow through uniform emergent vegetation in a 24 m long by 1 m wide recirculating flume with a horizontal bed. The experimental setup is shown in Figure 1. Channel discharge was maintained by a pump that fed water directly into the channel inlet after a flow expander. Two sets of flow straighteners, a plate drilled with an array of 0.05 m holes and an array of 0.005 m diameter tubes 0.05 m long, were positioned 0.2 and 0.5 m downstream of the inlet aligned in the direction of flow. Flow depth was controlled with a tail-gate located at the channel outlet and measured using a Vernier gauge, maintaining a fixed depth of 0.15 m within the test section.
Rhodamine 6G fluorescent dye was used to investigate solute transport through vegetation. Vertical line source injections were made at the channel center 12 m downstream from the flow straighteners using a copper pipe with 0.0008 m diameter holes at 0.01 m intervals. The pipe was connected to a constant-head tank located directly above the flume.
The flume was fitted with a 2-D laser induced fluorescence (LIF) system to measure temporal and transverse variations in dye concentration at two locations 1 and 2 m downstream from the injection point. To fluoresce the dye, 200 mW green lasers (k 5 532 nm) were installed on the side of the flume perpendicular to the flow at middepth. Observation windows (each 0.1 m wide) were installed across the vegetation bed at the measurement locations and digital cameras fitted below the flume looking up and capturing the full width of the channel and laser beam. Concentration records were captured at 5 Hz as a sequence of 1280 3 1024 8-bit grey-scale images. The entire flume was covered in black-painted wooden boards to minimize background interference from ambient lighting.
A calibration between pixel intensity and concentration was made simultaneously for both measurement locations. A short section of the flume containing the laser windows was isolated from the rest of the flume with baffles and a range of measurements of known Rhodamine 6G concentrations was taken with known laser powers. The calibration measurements were used in accordance with the Beer-Lambert law to obtain best fit values of the attenuation (a w ) and extinction ( 0 ) coefficients of Rhodamine 6G, as well as best fit values n and f describing the relationship between pixel intensity and concentration [West, 2016]. To calibrate a recorded profile, a pixel-by-pixel decay model [Ferrier et al., 1993] was then applied, in which concentration measured at a pixel depends on both pixel intensity and concentration of the previous pixel: where P is the laser power before entering the flume, P n11 is the power in the pixel at location n 1 1 from the laser, C i is the concentration in the pixel at location i, C i 5ðI i P 21 i21 2fÞn 21 , I i is the pixel intensity at i, P i21 is the laser power at the previous pixel, and Dy is the width of a pixel. At i 5 0, P i21 5P.
The artificial vegetation was installed between 10 and 15.2 m downstream from the flow straighteners. Real vegetation was installed between 7 and 17 m. The vegetation was omitted above each observation window to ensure a clear image and for taking measurements of velocity. The real vegetation was cropped to a depth of 0.25 m, permitting access to the flow and to allow the laser blackout covers to be fitted. Profiles of instantaneous vertical and transverse velocity were measured with Met-Flow UVP-DUO velocity probes to verify uniform velocity fields. Velocity data were despiked according to Goring and Nikora [2002].
Two injection methods were employed. Primarily, 5 s pulse injections were made, with 10 repetitions to capture the spread of the data. Second, continuous dye injections were made for 10 min to simulate steady state conditions for one artificial and two real vegetation experiments. The vertical line source eliminated the observational effects of vertical dispersion. The continuous injection eliminated the observational effects of longitudinal dispersion, while the pulse injection allowed both transverse and longitudinal mixing to be observed. The experimental setup is explained in more detail in West [2016]. This experimental setup allowed D x and D y to be determined simultaneously from a single pulse injection for the first time. Simultaneous measurements simplify the practical constraints of measuring dispersion, reducing flume sump dye saturation due to a continuous injection and time spent on experimental procedures, and also ensure that the two measurements are linked to the exact same flow conditions and vegetation arrangement.

Experimental Configurations
Experimental dispersion data were collected for two densities of artificial vegetation (AV) and three types of real vegetation (Carex acutiformis, winter Typha latifolia, and summer Typha latifolia), shown in Figure 2 [ Hart et al., 2016]. Figure 2c also shows the laser line sources used to illuminate the flow at the two measurement windows. The artificial vegetation consisted of 0.004 m diameter drinking straws at two densities of a repeating regular hexagonal grid. The high-density variant had a density of 1594 stems=m 2 with a centerto-center stem spacing of 0.025 m transversely and 0.0125 m longitudinally, with every other row being offset by one stem (inset Figure 2a). The low-density artificial vegetation was generated by removing 3 out of 4 straws, thereby multiplying these dimensions by two.
Vegetation characteristics are presented in Table 2. Stem density was measured by counting the number of stems in a sample area of 0.5 m 2 . This was repeated at up to 10 different locations. At each location, stem diameter was measured at the channel middepth for at least 40 randomly chosen stems, using digital Vernier gauge callipers. As Carex leaves are blade shaped, both width and thickness were recorded. Assuming orientation could be in any direction to the flow, the average of width and thickness was used for diameter. Both live and dead stems were included in the vegetation characterization. Solid volume fraction was estimated as /5p4 21 Nd 2 and frontal facing area as a 5 Nd. Large uncertainties in / are therefore due to the large variations in N and uncertainties in d.
Stem diameter and frontal facing area were also measured from images of the vegetation taken in front of a clear white board, with each cross-sectional image capturing 0.2 m longitudinally of vegetation for Typha and 0.05 m of vegetation for Carex. Separate estimates of d, a, and / are presented based on the image processing. Unless otherwise specified, values of d obtained by calliper (and estimates of a and / based on them) have been used in the data analysis presented here to remain consistent with other studies (Table 1).

10.1002/2016WR019937
Data were collected at flow rates of Q51.0, 1.5, 2.0, 2.5, and 3.0 L s 21 with a flow depth of 0.15 m. A single 1 m long patch of vegetation was tested with 10 repeat injections at each flow rate. Although tests on additional vegetation patches would be desirable to reflect field scale variation, this was not feasible due to practical constraints. Within vegetated mixing, the relevant length scale for Reynolds number is typically stem diameter, Re d 5Udm 21 . The collected data covers stem Reynolds numbers of 27 Re d 524, with the majority of tests falling below Re d 5 200, i.e., in the nominally laminar and transitional zones [Nepf, 1999]. Uncertainties in data from the studies presented in Table 1 have been included where possible.

Calculating Dispersion Coefficients
Similar to the Fischer et al. [1979] routing solution to the one-dimensional ADE, Baek et al. [2006] presented a routing solution for the two-dimensional ADE, varying in time longitudinally and transversely:   whereĈ is the routed or predicted concentration profile, t is the mean travel time, and s and k are the integration variables. In a routing or convolution solution to the ADE, each discretization of the upstream concentration profile is independently advected and dispersed downstream. The downstream concentration profile is the sum or superposition of the resulting individual profiles. This allows a prediction of the downstream concentration profile based on an upstream profile provided that U, D x , D y , and t are known. The routing solution uses the ''frozen-cloud approximation'' (x 5 Ut) in the longitudinal direction, i.e., assuming that no dispersion occurs during measurement or that Cðx i ; tÞ ffi Cðx; t i Þ [Fischer et al., 1979].
An optimization approach has been taken to simultaneously obtain best fit values of transverse and longitudinal dispersion to the experimental data using the MATLAB nonlinear optimization function fmincon [The MathWorks Inc., 2015]. This maximizes the goodness of fit between a downstream record and a downstream prediction by simultaneously varying D x , D y , U, and V (the temporal mean, transverse velocity). When optimizing, the transverse mean velocity V has been added (replacing y in the right-hand side of equation (11) with y2V t) as a dummy variable to account for any 3-D flow field effects and to ensure that the spread of solute is accurately reflected. t was taken as the distance between measurement points (1 m) over longitudinal velocity, i.e., t5U 21 . Goodness of fit was determined using a 2-D formulation of the Young et al.
[1980] R 2 t : R 2 t has been shown to be a robust means of evaluating the similarity of two temporal concentration profiles in optimization [Sonnenwald et al., 2013]. Error in optimized D x and D y was estimated from the mean increase or decrease in value needed to cause a 5% reduction in R 2 t [after Lightbody and Nepf, 2006b]. However, it is worth noting that R 2 t is relatively insensitive to changes in either transverse or longitudinal dispersion coefficient.
The continuous injection data have been used to validate the 2-D pulse data collection and the 2-D optimization approach. The continuous injection data were temporally averaged and mass-balanced, resulting in two transverse concentration distributions. A downstream prediction was then optimized using a transverse version of the 1-D routing solution [Fischer et al., 1979]. Figure 3 shows an example upstream, downstream, and routed concentration distribution for one artificial and one real emergent vegetation test case. In the artificial vegetation, both the upstream and downstream distributions are close to Gaussian and there is relatively little spread (allowing both profiles to be shown on the same axes). In contrast, the real vegetation has upstream and downstream profiles that are asymmetric, reflecting the naturally more random spatial variation of the real vegetation. These morphological differences in the vegetation make it more difficult to characterize mixing, as evidenced by the lower R 2 t in the real vegetation case. Mean R 2 t was found to be 0.9346 with a standard deviation of 0.0598 in the artificial vegetation, compared with 0.7427 and 0.1151 in the real vegetation. The lower quality of the fit of the tail for the real vegetation, e.g., Figure 3c, suggests that the Gaussian transfer function assumption implicit in the use the ADE equation may not be entirely suitable.

Validation of the 2-D Routing Equation
Continuous injection data for the Typha and high-density artificial vegetation were used to validate the application of the 2-D routing equation for obtaining dispersion coefficients from pulse injection data. Figure 4 presents a comparison of D y optimized from both the continuous and pulse injection data. As it is not possible to optimize for longitudinal velocity from transverse continuous concentration records, an estimate of velocity based on number of stems and stem diameter, U5QA 21 ½ð12 ffiffiffi ffi N p d1Nd 2 Þ=ð12 ffiffiffi ffi N p dÞ, where A is empty flume cross-sectional area, has been used instead. Ideally, recorded velocity data would be used here, but it was not of sufficient quality in this case due to difficulties with the probes at low velocities. The velocities differ due to the continuous injection velocity estimate not accounting for vegetation layout and For both the high-density artificial vegetation and the winter Typha vegetation, there is good agreement between D y from the continuous and the pulse injections. Agreement for the summer Typha is not as good.

10.1002/2016WR019937
The difference between the pulse and continuous optimized results is likely to be the result of a more complex interaction between D x and D y that is captured only in a full 2-D routing optimization. The large error bars on the continuous injection values are the result of taking an optimized value from a single injection. Overall, the comparison confirms that the 2-D ADE routing equation can be used to form reasonable bulk scale estimates of dispersion coefficients for emergent vegetation. Figure 5 shows the mean optimized transverse and longitudinal dispersion coefficients obtained from 2-D optimization of the pulse injection data. The expected trend of increasing dispersion with velocity is clearly shown [Fischer et al., 1979]. Figures 5a and 5b show only a slight difference in dispersion for the two densities of artificial vegetation. Longitudinal dispersion is approximately two-and-a-half times greater than transverse dispersion. The lowdensity artificial vegetation shows slightly greater transverse dispersion compared with the high-density artificial vegetation, whereas longitudinal dispersion within the two is nearly identical. Dimensionless dispersion, D Ud ð Þ 21 (Figures 5c and 5d) is constant within error bars for both D x and D y , confirming a linear relationship with U. The similarity of dispersion coefficients for the two types of artificial vegetation suggests that the similar pattern of the stems at the two densities is playing an important role in controlling dispersion.

Results From the 2-D Routing Analysis
Figures 5e and 5f show dispersion within the real vegetation, which is greater than that within the artificial vegetation by an order of magnitude. Carex and summer Typha have very different physical characterizations, yet show similar values of transverse dispersion. Winter Typha, which is similar in characterization to summer Typha, shows much greater transverse dispersion. Longitudinal dispersion varies between all three vegetation types, increasing with solid volume fraction. The greater values and variability of dispersion for the real vegetation reflect the increased complexity of the flow field. It may be seen from Figure 2 that winter Typha is less uniform with greater stem spacings than summer Typha which also has larger stems (d 5 0.010, cf. d 5 0.019 m) and is denser (/50:047, cf. /50:013 m). Carex appears to be much more uniform than the Typha and is also more dense with /50:077. Winter and summer Typha both show optimized mean longitudinal velocities for the same flow rate consistent with the increase in solid volume fraction between the two (visible as spread on x axis coordinates of Figures 5e-5h). Carex, despite having a solid volume fraction greater than summer Typha, has a lower optimized mean velocity. This may be due to the high number of stems (N 5 3388) increasing the length of the flow path.    Figure 6 compares the new dispersion data with the data outlined in Table 1, plotted as a function of solid volume fraction, /. Figure 6 demonstrates both the breadth of the available data and the variability in results. Dimensionless longitudinal dispersion is typically an order of magnitude greater than transverse dispersion. The transverse dispersion data does not suggest a dependency on /, whereas there is some indication of an inverse trend between longitudinal dispersion and solid volume fraction.  Table 1.

Transverse Dispersion 5.2.1. Artificial Vegetation
The high-density artificial vegetation behaves very similarly to the low-density artificial vegetation, with slightly reduced transverse dispersion (Figure 5a). This is at odds with the Nepf [1999] mixing model. Nepf [2012] suggested that at / < 0:1 transverse mixing is dominated by turbulent processes and so, from the first term of equation (2) and ad % /, the high-density artificial vegetation would be predicted to have 1.6 times greater transverse dispersion than the low-density artificial vegetation. However, in this case, substituting in Dy Ud ð Þ 21 values for high and low-density artificial vegetation respectively, the ratio is instead 0:45=0:6250:73. Guo et al. [2014] created a numerical model for periodic arrays of cylinders and showed, consistent with this result, that longitudinal and transverse dispersion increased with decreasing solid volume fraction in a diamond array of vegetation.
It is likely that periodic artificial vegetation presents a special case compared to random artificial vegetation due to the development of a periodic velocity field [Meftah and Mossa, 2013;West, 2016]. Visual inspection of the dye traces within the artificial vegetation, not shown, show clear repeating variations in dye concentration that meander back and forth. The repeating variations can be shown to correlate with stem spacing, suggesting that stem spacing-rather than stem diameter-is the dominant length scale. If stem-spacing instead of diameter is used to nondimensionalize the artificial vegetation, i.e., D y Us ð Þ 21 , then the ratio of high-density to low-density dispersion becomes 0:07=0:0551:4. This is consistent with Nepf [1999], demonstrating that stem-spacing is the appropriate length scale.
The repeating variations in dye concentration suggest large coherent flow structures. The regularity of the observed flow structures is only possible within periodic arrays of artificial vegetation, since within random vegetation the positioning of stems ensures that such structures are broken down. The stem spacing length scale in the artificial vegetation here is inconsistent with the results of Tanino and Nepf [2008a], who showed that stem spacing as the dominant length scale occurs at much greater solid volume fractions (/ > 0:1) than seen here. It follows that periodic arrays of cylinders are not representative of real vegetation for dispersion analysis. Figure 7 shows the Nepf [1999], Tanino and Nepf [2008a], and Nepf [2012] predictions of dimensionless transverse dispersion against /. A value of C D 5 1 has been assumed within equation (2). The Tanino and  Table 1) with existing models (lines).
It is immediately apparent that none of the models is capable of predicting the high levels of dimensionless transverse dispersion associated with the Wadzuk and Burke [2006] Phragmites data or the new winter Typha or Carex data. With real vegetation, only dispersion for the summer Typha and the Nepf [1999] Spartina fall close to any of the model predictions. Assuming drag force energy dissipation as the dominant mechanism, the enhanced dispersion values in real vegetation suggest that more energy dissipation is occurring than the models predict. Given its high stem density and decreased optimized velocities (increased mean travel times) the Carex may instead be experiencing enhanced flow path tortuosity. The scatter in the data for real vegetation implies that the models all omit some key mixing processes. Burke and Wadzuk [2009] suggests that these processes could be pockets of turbulence and dead zones, and that accounting for these within the Wadzuk and Burke [2006] data makes the data more consistent with the existing models. Dead zone effects could also account for the previously discussed lower optimized mean velocities of the Carex.

Longitudinal Dispersion 5.3.1. Artificial Vegetation
Similar to transverse dispersion, there is almost no difference in longitudinal dispersion between the two densities of artificial vegetation (Figure 5b). However, while a difference in D y would be expected, it is less clear that a difference in D x is expected. White and Nepf [2003] suggested that longitudinal dispersion due to vortex trapping may change with ad, but that vortex trapping is also heavily influenced by terms dependent on Re d , which remains the same between densities. Similarly, according to Lightbody and Nepf [2006b], secondary wake dispersion is dependent on C D , which is again dependent on Re d . Figures 8a and 8b show predictions of the vortex trapping and secondary wake terms of the White and Nepf [2003] longitudinal dispersion model, respectively, compared to a line of equality. For the vortex trapping term, D x;t , the model parameters b w 52:25; j w 51, and St 5 0.2 have been used, based on the values given by White and Nepf [2003]. The simplified secondary wake dispersion term, equation (8), provided by Lightbody and Nepf [2006b] has been used. Typically, drag coefficient can be obtained experimentally by measuring a change in head before and after the vegetation [Nepf, 1999]. However, due to the low velocities within the flume, its horizontal bed, and the resolution of the available instrumentation, it was not possible to measure a change in head. Drag has, therefore, been estimated from the results of  Tanino and Nepf [2008b] and Tinoco and Cowen [2013], who both utilized an expression derived from Ergun [1952]:

Longitudinal Mixing Model Comparisons With the New Real Vegetation Results
where a 0 and a 1 are parameters dependent on the vegetation. Their experimental results combined produce the empirical best fit relationships: a 0 57276:43d123:55 a 1 532:70d13:01/10:42 which have been used in equation (13) to estimate C D . The experimental work did not provide sufficient evidence of vertical heterogeneity to support the application of the Nepf [2006a, 2006b] models. This is not surprising as the shallow flow depth of 0.15 m here is expected to minimize the effects due to vertical shear.
Similar to results presented in White and Nepf [2003], D x;t is several times smaller than D x;s . The vortex trapping mechanism on its own is insufficient to predict longitudinal mixing ( Carex data over-estimated. The latter may reflect the high Reynolds numbers associated with this particular test series, but it should also be noted that the predicted values are highly sensitive to the estimated C D values such that any error in the laboratory quantification of d or / may have a substantial impact on the modeled value of D x;s .

Seasonal Variation in Real Vegetation
Through the growing seasons of spring and summer, vegetation becomes denser and stems thicken before dying back during the winter [O'Hare, 2015]. The winter Typha is irregular, and stems exist at odd angles and in clumps. In comparison, summer Typha has stems approximately twice as thick with three times the solid volume fraction. The Nepf [1999] (equation (2)) and Tanino and Nepf [2008a] (equation (5)) transverse dispersion models suggest that the summer Typha vegetation would result in greater dimensionless transverse dispersion than winter Typha. Figure 5g shows the reverse. Dimensionless longitudinal dispersion (Figure 5h) shows minimal seasonal dependency. Given the differences in vegetation characterization, none of the longitudinal mixing models predict this.
The Carex vegetation is from the summer season. Shucksmith et al. [2010] showed Carex increasing in density with growth (age), but exhibiting relatively constant longitudinal dispersion, suggesting that winter Carex would perform similarly to the results presented for summer Carex. This is consistent with the Typha data presented here. Valiela et al. [1975] showed quite significant variation in biomass through a season for salt marsh vegetation. Karunaratne et al. [2003], however, showed that seasonal variation of vegetation could be minimal in some cases. These observations suggest that additional investigation into seasonal variation is necessary.
The seasonal differences in dimensionless transverse dispersion of Typha are greater than their measured physical characteristics would indicate suggesting that some additional characteristic is of importance. The clustering of stems in winter Typha is potentially affecting dispersion, either causing them to act as single, larger, obstacle, or by trapping flow within the clusters. Burke and Wadzuk [2009]  ignored when nondimensionalizing solely by stem diameter. Ideally tests of additional 1 m patches of winter Typha would be carried out to better isolate any of these effects.

Characterizing Real Vegetation
Although several of the evaluated models provide reasonable estimates of dispersion within artificial vegetation, dispersion within real vegetation is generally poorly predicted. There are three main vegetation characteristics used within the models: solid volume fraction, frontal facing area, and stem diameter. The first two are bulk properties. Stem diameter, d, however, is a mean value, which is sensitive to variations in individual stem properties. Measurements of d taken by hand compared to those obtained from image analysis (  Water Resources Research 10.1002/2016WR019937 Figure 9 (left column) shows the distribution of measured stem diameters for each of the three types of real vegetation. In this case, the measurements made using callipers and those derived from photographic image processing have been combined. For all three types of vegetation, the stem diameter distribution is skewed and the mean is larger than the median (d 50 ) of the distribution. The mode (peak) of the distribution for winter Typha is less than the mean, while the mode and the mean are approximately the same for the summer Typha and Carex distributions. The distributions of both winter Typha and Carex exhibit a clear single peak, while the summer Typha appears to have two peaks, one close to the mean diameter at around 0.017 m and a second at around 0.002 m. Figure 9 (right column) shows the effect of nondimensionalizing the observed values of D x and D y using the stem diameter distributions instead of the single mean stem diameter. Here the mean values of optimized D x and D y , as presented in Figure 5, have been used for nondimensionalization. In each case, the distributions are skewed, and show considerable spread around the mean values of nondimensional dispersion presented in Figures 5g and 5h, i.e., D y Ud ð Þ 21 values of $1.0, 1.3, and 0.2 and D x Ud ð Þ 21 values of $18, 3, and 3 for Carex, winter Typha and summer Typha, respectively. The greatest spread is in the dimensionless longitudinal dispersion for Carex (Figure 9b), which varies from approximately zero to more than 30 depending on whether the largest or smallest observed stem diameter is assumed. Figure 9 (right column) also presents dimensionless transverse dispersion as predicted by Tanino and Nepf [2008a] and secondary wake longitudinal dispersion as predicted by White and Nepf [2003]. These values have been presented earlier in this paper, and are all based on the mean stem diameter from calliper measurements. As has already been shown, the Tanino and Nepf [2008a] model provides a good estimate of the nondimensional transverse dispersion for the summer Typha, but significantly underestimates the observed transverse dispersion for Carex and winter Typha. The White and Nepf [2003] model provides a reasonable estimate of nondimensional longitudinal dispersion for both winter and summer Typha, but an underestimate for Carex.
Given the skewed nature of the stem diameter distributions, it may be argued that the peak (mode) or d 50 (median) values should provide a more representative average compared with the geometric mean. Nondimensionalizing the observed values of dispersion using either of these (smaller) stem diameters would, however, act to increase DðUdÞ 21 , tending to move the observed data further away from the existing model predictions. Alternatively, there may be some justification for using d 90 , the value representing the diameters of the largest 10% of stems, as it is often the larger length scales that dominate dispersion. If d 90 > d then d 90 acts to reduce DðUdÞ 21 , in this case by approximately a factor of two. This does not significantly bring the experimental results more in line with model predictions.
These observations suggest either that the existing models are unable to capture the complex mixing processes that characterize real, heterogeneous, vegetation, or that the relevant length scale is not stem diameter. For winter Typha, Figure 2c shows clusters of stems that could effectively be acting as a single larger stem. Conversely, while the modeled dispersion for summer Typha provides a good estimate of the observed data, it is evident that stems with diameters lower than the mean diameter appear with equal frequency to the mean diameter. These smaller diameter stems must therefore have an equal contribution to mixing, but are ignored in any estimates based on mean stem diameter alone. The models for both transverse and longitudinal dispersion under estimate dispersion for Carex. While this is also the case with the winter Typha, it cannot be due to the same reasons as they have dissimilar morphology. While winter Typha is quite patchy, Figure 2b shows the Carex appears quite dense. Visual inspection of dye traces through the Carex show a very long tail indicating significant dye-trapping within the vegetation. This longitudinal dispersion mechanism is not well accounted for in any of the models. Tanino and Nepf [2008a] included the distribution of stem spacings as a key mixing-length control within their model for transverse dispersion. The data presented here suggests that the predictions of both longitudinal and transverse dispersion within real emergent vegetation may require the variability in stem diameter and stem clustering to be considered alongside the variation in stem spacing.

Modeling Recommendations
The new data for dispersion in real emergent vegetation is not predicted satisfactorily by any of the existing models. The scatter and uncertainty in modeling the available data suggests that for design purposes it may be appropriate to adopt the vegetation (and season)-specific parameter values, as presented in Figures Water Resources Research 10.1002/2016WR019937 5e and 5f. O'Hare [2015] has suggested the application of plant morphogroups (plant grouping based on physical characteristics) to hydrodynamic modeling, and this approach could prove suitable for simplifying the task of choosing species and characterizations for modeling in the future.

Conclusions
A new data set has been presented that, for the first time, simultaneously describes transverse and longitudinal dispersion in both artificial and real emergent vegetation. Within periodic artificial vegetation, a regular velocity field develops that make it unsuitable for simulating real vegetation. Dispersion within the real vegetation is complex and shows differences between the two species (Carex acutiformis and Typha latifolia) and two seasons (winter and summer Typha). Dispersion has been nondimensionalized by mean stem diameter, and Carex shows significantly greater dimensionless longitudinal dispersion, compared with the other two (D x Ud ð Þ 21 % 18 versus D x Ud ð Þ 21 % 3). Carex and winter Typha have similar dimensionless transverse dispersion (D y Ud ð Þ 21 % 1:0), while summer Typha shows much less dimensionless transverse dispersion (D y Ud ð Þ 21 % 0:2).
The new emergent vegetation mixing data have been combined with other previously published results and the combined data set compared with models for predicting dispersion in emergent vegetation. The data set shows significant scatter compared to model predictions. The Nepf [2012] transverse mixing model and White and Nepf [2003] longitudinal mixing model currently offer the closest predictions of dispersion coefficient, but much of the new and existing data is not predicted well.
The lack of predictive capability of the existing emergent vegetation dispersion models and the variability of the available dispersion data highlights that real vegetation is a complex and highly variable natural material. The results suggest that relying on mean diameter to describe mixing length scales fails to fully characterize the vegetation, and that future research should focus on developing models that can fully capture the distribution of mixing length scales and the associated mixing processes.