A Framework for Quantifying Reach-Scale Hydraulic Roughness in Mountain Headwater Streams

By inergency On Feb 22, 2024

1. Introduction

Mountain headwater streams are the smallest reaches at the highest end of a mountain landscape and are often referred to as zero-order rather than first-order streams. Headwater streams strongly influence downstream river networks. In South Korea, headwater streams play a vital role in mitigating floods and debris flow disasters, and they represent a significant portion of the total channel length in mountainous regions. Mountain headwater streams typically refer to small, undulating channels in mountainous forests with relatively shallow flow depths and varying flow velocities. Headwater streams differ from large rivers since they exhibit small depth-to-width ratios, large relative roughness, steep channel gradients, and different channel morphology [1,2]. Although differences in the process characteristics between mountain streams and larger flat rivers have been extensively studied, relatively little is known about the hydraulic and morphological processes of mountain headwater streams.

Determining the flow velocity in a stream has been the main focus of many hydrological and geomorphological studies [2]. Different approaches exist for quantifying the flow characteristics of torrent channels. The most widely used method for measuring the flow velocity in natural channels is the point-velocity method using a current meter [3]. The direct measurement of the instantaneous flow velocity at the location of interest has several advantages over other measurement methods. These include ease of use and high accuracy when using a relatively small number of measurements in regular and steady flows. However, the point-velocity technique can be impractical or yield significant errors in torrent channels owing to various bedforms, steep slopes, and isolated large particles [4,5].

An alternative method for obtaining flow velocity is the water-tracing approach. This approach has been increasingly used in mountain hydrology over the past two decades, as it enables the measurement of the reach-averaged velocity for low-depth flows and/or rapidly varying flows in mountain gravel-bed channels [6]. The average velocity is derived by measuring the travel time of solutes in water over a channel section of a certain length using several tracers, including salt, buoyant particles, fluorescent dyes, and thermal tracers. Salt and dye tracers are commonly used in water-tracing due to their ease of use and other desirable features [4,7]. Salt tracers tend to be more widely used for flow measurements than dye tracers, but their use under high-flow conditions is limited due to measurement uncertainty [4,8]. Dye tracers, which are more expensive than salt ones, enable the measurement of higher flows even in small quantities because the dye can be detected at much lower concentrations than salt [4]. Rhodamine WT is preferred in most dye fluorometry applications because of its ease of use, relatively low cost, low absorptive tendency, strong fluorescence, and chemical stability [7,9]. Compared with other ambient water substances, Rhodamine WT has distinct spectral features, making it suitable for use as a water tracer [7,8].

The flow resistance is an essential determinant of the hydraulic and sedimentary processes in mountain channels. It reflects the processes by which the physical shape and bed roughness of a channel influence the flow depth and mean flow velocity in steep channels [10]. Flow resistance caused by objects protruding into the water is quantified by the hydraulic roughness [11]. The hydraulic roughness in mountain streams with large rough elements is particularly difficult to quantify because the large grains in the bed typically protruding above the water surface at low flows can have heights similar to the flow depth during high flows [12,13]. The hydraulic roughness at channel beds can be explicitly associated with the presence of bedform roughness, which is empirically expressed as a log-law formula structure, such as the Manning formula. This formula performs well under deep and uniform flow conditions with relatively small roughness, such as in plain rivers, but its use in mountain streams is still limited owing to the shallow water depth, steep slopes, wide grain size distribution, different types of bed structures, and sharp variations in the flow path direction [13,14].

Mountain torrents typically feature large grains randomly distributed in channels. The bed materials in mountain channels are very coarse; therefore, they protrude well into or completely through the flow. The hydraulic roughness in mountain channels is significantly increased by the presence of larger bed materials such as boulders or rocks [1], while the relative roughness, roughness shape, size distribution, and spacing of its elements may influence the resistance flow [12]. Ferguson [15] and Rickenmann and Recking [16] selected the characteristic grain size (D₈₄) to describe the channel roughness, whereas Aberle and Smart [5] and Lee and Ferguson [17] argued that the grain size might not be an appropriate roughness measure for steep streams. As an alternative, they derived roughness measures from the standard deviation of the bed elevation.

Efficient and accurate field measurements enable all researchers to explore the hydraulic relation between flow resistance and streambed characteristics. This could lead to a better understanding of the influences of bed gradient and roughness on flow behavior in mountain channels. The presence of macrorough elements, both across and down the reach, makes it difficult to obtain a reliable measure of hydraulic variables.

Currently, there is no universal consensus on quantifying the hydraulic roughness in shallow mountain channels. This is mainly due to limited data on the flow and associated roughness measurements in the field, making accurate field measurements essential for elucidating the relationship between the flow characteristics and bed microtopography at a reach scale. The aim of this study was to propose a framework for quantifying the reach-average hydraulic roughness from limited local data in mountain headwater streams. To achieve this objective, dye fluorometry was used to estimate the reach-average flow velocity under various hydraulic conditions. Between-site hydraulic parameters were obtained from LiDAR surveys and field measurements.

4. Discussion

4.1. Comparison of Point Velocity and Reach-Average Velocity

Accurate measurements of the flow velocity are necessary to understand the hydrologic, hydraulic, and sedimentary processes in mountain channels. The point-velocity method using a current meter has been widely used to quantify the flow velocity under both laboratory and field conditions. However, the presence of turbulence at very high flows can result in significant velocity measurement errors. In some special cases, this method underestimates the flow characteristics with errors greater than 50% [30]. Furthermore, the macroroughness of the streambed hinders the use of a current meter under low-flow conditions.

The conventional point-velocity method can measure the velocity at points of interest but is not suitable for measuring the reach-average velocity along streams. Thus, it is advisable to use alternative techniques for measuring the flow characteristics in mountain streams, which are characterized by coarse bed materials, steep slopes, and low depths. The dye trace method has shown favorable results in terms of accuracy when compared to current metering as a method for measuring flow in mountain channels, with its accuracy increasing in rivers with unsteady flow conditions and irregular streambed geometry. In any case, both methods can be complementary.

Each method has advantages and disadvantages, which limit their use. Given these considerations, it is necessary to select the appropriate type of measurement technique used to quantify water characteristics depending on the hydraulic and geomorphological conditions. Current meters are not suitable for extremely low or high flows in mountainous regions. The use of dye tracers in mountain streams is preferred, provided that tracer injection and detection are available.

4.2. Use of Dye Tracer

The use of artificial tracers for flow measurements is convenient due to their low cost, easy handling, low impact, and satisfactory results in mountain streams. The selection of the most suitable tracer depends on several factors, including the water quality, amount and type of suspended sediment, distance between the release and detection locations, and flow characteristics [31]. Before their use in the field experiments, water tracers, including a fluorescent (Rhodamine WT) and a chemical tracer (NaCl), were preliminarily compared to evaluate the differences in their behavior. Sodium chloride (NaCl) is the most frequently used tracer, providing the best results with a low environmental impact. However, its low detectability under high-flow conditions is a major drawback [4,30]. In contrast, even small quantities of Rhodamine WT are readily detectable, enabling good measurements to be acquired under very high-flow conditions. Given the chemical and spectral characteristics of the tracers, Rhodamine WT is highly suitable for use as a water tracer in steep, small mountain streams.

The field experiments also indicated that measurement errors can be mainly attributed to dye losses between the two measurement locations, resulting in poor measurement accuracy. Dye losses occur when excessively long reaches are involved or when finely suspended sediment exists, particularly clay or organic flocculant particles. Water-tracing dyes tend to adhere to the surfaces of landform materials such as stone, rock, and aquatic plants [32]. Previous studies have indicated that significant losses of Rhodamine WT may occur when diluted concentrations are exposed to direct sunlight for extended periods, owing to the photochemical decay of the dye [30]. This is a potential source of error that must be considered when performing dye-fluorometric studies. In this study, most exposure times for the dye experiments were short enough that the sorption and photochemical decay losses would be negligible.

4.3. Quantifying Hydraulic Roughness

Mountain streams have complex and rough bed morphology with immobile boulders, bedrock constrictions, or large woody debris. The streambed has long been regarded as the self-adjusted hydraulic system that produces to maximize flow resistance to maintain greater bed stability [13,16,21,22]. Bed stability is closely related to the grain sizes that can be entrained or deposited under various bed gradients and flow conditions. The grain size distribution (D₅₀, D₈₄) of the streambed is widely used to represent the flow resistance in natural streams due to its simplicity [1]. However, the field estimation of the grain size distribution in mountain headwater streams can result in large uncertainties due to operational bias, limited sample size, and spatially heterogeneous grain size distribution [22]. Alternatively, roughness height measures (

I P R_{90}, S T D_{z}

) are preferable to quantify the flow resistance in steep mountain streams [19,30].

The scattering observations in Figure 6 suggested that

I P R_{90}^{L}

may not entirely capture the flow roughness in steep channels. It can be influenced by other factors, such as the grain and spill resistances [2,13]. Although the empirical Equation (8) is analogous to previous studies, it is helpful in cases of steep and rough channels in South Korea. Nonetheless, this study mainly contributes an alternative hydraulic relation that enhances our understanding of estimating the mean flow velocity in mountain streams. LiDAR is commonly used to collect high-resolution terrain data that capture the variation in geomorphologic characteristics, not only at a specific location but also along stream reach. A LiDAR survey provides new opportunities for accurately obtaining detailed topographic information and streambed roughness height measures, such as the interpercentile range

I P R_{90} and the standard deviation S T D_{z}

. Moreover, there are time and costs saving advantages for LiDAR surveying applications. However, LiDAR imagery can often underestimate the depths of pockets between macroroughness elements due to shadow effects. Recently, an algorithm has been developed to reduce the shadow errors in mapping rough grain surface, but it is not considered in this study [33].

4.4. Limitations of the Study

In this study, the hydraulic roughness was quantified from field observations in mountain streams. However, this study has potential limitations. First, the hydraulic roughness is regarded as a constant parameter over time. However, every stream has a self-organized fluvial process to adjust the bed morphology to maximize the flow resistance [13,22]. Although the flow rate or water depth is assumed to be unchanged over the reach, the presence of protruding objects and large boulders can lead to a spatial variation in the bed roughness with the relative submergence of roughness elements (the ratio of flow depth to roughness height) in steep mountain channels [2,5,22]. This study does not deal with the hydraulic behavior of steep and rough mountain streams yet.

Field measurements were conducted in this study for low flows with a discharge per unit width of 0.04–0.43 m²/s. It seems reasonable that within the range, the proposed hydraulic relationships might be valid in mountain headwater streams, such as the Gwanak and Baekun sites. Furthermore, the flow resistance may be influenced by other factors in addition to hydraulic roughness height, including the boulder arrangement and protrusion, energy dissipation, or erosional process. However, we have neglected these factors in this study.

5. Conclusions

This study focused on the influences of macroroughness elements on the flow in steep channels and their incorporation into hydraulic relations, and it examined a research framework for quantifying the hydraulic roughness of mountain streams based on limited field measurements. The hydraulic geometry relationships in mountain headwater streams were developed based on the reach-average velocity and associated hydraulic variables. A dye-tracing technique was used to measure the average velocity over the entire reach length. Various hydraulic roughness heights were expressed in terms of the grain (D₅₀ and D₈₄) and form resistance ( $I P R_{90}, S T D_{z}$ ). Of the various measures for hydraulic roughness in the dimensionless hydraulic relations, $I P R_{90}^{L}$ exhibited the largest $R^{2}$ (=0.68) and the smallest RMSE (=0.07 m/s) across 22 field measurements, implying that $I P R_{90}^{L}$ was superior (in terms of the $R^{2}$ and RMSE) in explaining the variations in the reach-average velocity among mountain headwater streams. Consequently, the dimensionless hydraulic geometry relationship incorporating the discharge ( $Q), bed slope (S)$ , Q, and longitudinal $I P R_{90}$ was developed to accurately estimate the flow characteristics in steep and rough streams.

The flow rate is thought to have a significant influence on the flow resistance. The effect of hydraulic roughness on the flow increased as the relative submergence of the bed elements (roughness height/water depth) decreased, thus the proposed hydraulic relation between the reach-average velocity and hydraulic roughness might not be suitable for other regions. Nevertheless, the research framework for quantifying hydraulic roughness from local data can be applicable in mountain headwater streams.

In mountain streams with small flow depths, large protruding boulder elements and irregular bed morphology cause a high flow resistance, rendering many traditional approaches used for flatter streams and rivers inapplicable. The accuracy of the proposed hydraulic relation equation was evaluated using local data from field measurements; however, it was subject to considerable uncertainty with the varying flow rate. Furthermore, the utility of the proposed equations is constrained by the availability of data, while the flow characteristics in mountain streams may be influenced by factors other than the bed roughness. Further studies considering a wide spectrum of flow characteristics are needed to explore the flow behavior in mountain streams.