The Impact of Hydrologic Characteristics of Mountain Watersheds on Geometric and Hydraulic Parameters of Natural Torrent Beds

The aim of the paper is to determine the geometric and hydraulic characteristics of natural torrent beds and to propose a regional hydraulic geometry equations of mountain watersheds for the High Tatras region. The research was conducted in 26 natural torrents and their watersheds on the reference sections and profiles under the sediment source zones. Two different regression equations to determine the relationships of hydraulic geometry (hitherto used without asymptote and newly proposed with asymptote) were compared. The analyses showed a strong correlation relationship between watershed area Aw (km 2) and bankfull geometric characteristics of natural crosssections: width of the bed inside the banks Bbf (m), mean depth of the bed Hbf (m), channel cross-section area Abf (m2) and hydraulic characteristic – bankfull discharge Qbf (m 3.s-1). The results were tested by t-test and ShapiroWilk test. The determination coefficient (R2) for the relationships without asymptote ranged between R2 = 0.919 and R2 = 0.972; p – values from Shapiro – Wilk test ranged between p = 0.0359 and p = 0.8027. The determination coefficient (R2) for the relationships with asymptote ranged between R2 = 0.952 and R2 = 0.974; p – values from Shapiro – Wilk test ranged between p = 0.0221 and p = 0.8617. Based on the analysis we found that the new equation with the asymptote provides very good results.


INTRODUCTION
Out of the total length of the watercourses in the Slovak Republic (SR) stretching at 61,147 km, approximately 24,000 km (39.25%) are of a torrent nature.In the SR, the torrent watersheds are located in the protected areas (national parks, protected landscape areas).This fact must be taken into account in all interventions into torrent beds.Each torrent and watershed have a specific set of characteristics.There are several reasons for the acquisition of scientific knowledge in morphogenesis of torrent channels and regional hydraulic geometry of mountain watercourses and watersheds.The first is the possibility to use the results in practice (ecological torrent control, flood and erosion control, torrent revitalization, natural torrents management etc.).Developing hydraulic geometry relationships is of great importance for different applications of stream restoration design and hydrological modeling [Osati et al. 2016].The second reason is the fact that this issue has not been adequately solved in the SR.Authors [Powel et al., 2004] indicate that in the USA are used in designing and revitalization of watercourses the regional equations of hydraulic geometry which are several decades old and are unsuited for existing conditions of water flows and watersheds.Author [Howell, 2009] suggests that regional hydraulic curves are especially useful in stream restoration projects where the stream is so degraded that natural bankfull channel geometry can no longer be determined and a reference reach is unavailable.Regional curves can also be used in project such as road, bridge and culvert construction.Another author [Wilkerson, 2008] states that this knowledge is essential for planners, engineers, geomorphologists, environmentalists, agricultural interests, developments situated on flood prone lands and other stakeholders in flooding and flood management.Regional curves should be applied only in projects within the same region or in the region with scientific evidence showing that it follows the same hydraulic geometry curves [Howell, 2009].Physiographic region is a region with all parts similar in geologic structure and climate and with a unified geomorphic history [Wilkerson, 2008].Its relief features differ significantly from those of adjacent regions.Another author [Leopold et al., 1992] states that there are two types of channel hydraulic geometry: at -astation and downstream.Regional equations represents the relationships between geomorphologic and hydrologic characteristics of watershed and geometric and hydraulic characteristics of the channel profile.An important summary of regional hydraulic geometry research in the USA was published by another authors [Bieger et al., 2015] and [Blackburn -Lynch et al., 2017].These authors dealed with the relations between the watershed area A w (km 2 ) and bankfull geometric characteristics of natural cross-sections: the width of the bed inside the banks B bf (m), the mean depth of the bed H bf (m), the channel cross-section area A bf (m 2 ) and hydraulic characteristic -the bankfull discharge Q bf (m 3 .s - ).We point to few authors [Jakubis, 2008, Pšida, 2014] who were involved into the problem of relations between basic hydrologic characteristics of the watershed and the bankfull regional characteristics of natural torrent beds.This article discusses a new approach to the regional hydraulic geometry issues.The superstructure and the new current direction in the research of regional hydraulic geometry is the research of relations between the basic hydrologic characteristics of the watersheds: the mean longterm annual precipitation depth in the watershed P (mm), the mean long-term annual runoff depth in the watershed Q (mm) and the mean long-term annual climatic evaporation depth in the watershed E (mm) and the above mentioned geometric and hydraulic characteristics of the natural watercourse channels.he runoff generative process is highly variable and dependent on hydrologic and climatic characteristics of a watershed.The relations of runoff formation in small mountain watersheds were analyzed by the following authors in detail [Tani, 1997, Gallart et al., 2002, Hrnčíř et al., 2010].Precipitation is a prerequisite for runoff formation which depends on rainfall characteristics, initial moisture conditions, soil and geological conditions, vegetation, and topographic features [Han et al., 2012].Surface runoff and later discharge in torrent channels has a significant impact on the long -term morphogenesis of torrent beds.Geometric characteristics of watercourse channels are created during longterm morphogenesis.In general, higher long-term precipitation sums in the watershed are a prerequisite for the higher runoff and consequently the for the higher discharge in torrent beds.In different natural conditions of the watersheds (hydrologic, climatic, geological, pedological, vegetation, morphology), characteristic of different physical geographic regions, exist in torrent beds specific geometric and hydraulic characteristics exist in torrent beds.There are certain regional specificities that manifest themselves in geometric and hydraulic characteristics of channels, their relations and ratios.

MATERIALS AND METHODS
Analyzed torrents and watersheds are located in the Tatras National Park (TNP) in the geomorphological unit of Tatry, the subunit of Východné Tatry, the part of High Tatras and the geomorphological unit of Podtatranská kotlina, the subunit of Tatranské predhorie (Figure 1) at altitudes from 860 to 2654 amsl.High Tatras are the highest mountains in the SR.
Between 2007 and 2017, twenty-six torrents with enclosing channel profiles near the southern border of TNP (near the northern border of the geomorphological unit of Podtatranská kotlina) were analyzed.Analyzed torrents are situated in two main river watersheds: Váh and Poprad.The watershed areas of 26 analyzed torrents range from S p = 0.25 km 2 (Važecký) to S p = 19.34km 2 (Poprad) with the median S p = 3.21 km 2 .The discharges Q 100 range from Q 100 = 1.5 m 3 .s - (Važecký) to Q 100 = 35.7 m 3 .s - (Biely Váh) with the median Q 100 = 10.25 m 3 .s - .The torrents of High Tatras are typical mountain torrents with frequent and rapid changes in discharges, sediments formation, transport of these sediments and their accumulation.Geological structure of analyzed area is complicated.In the High Tatras can be distinguished three main building blocks: the crystalline core, mountain ranges and sedimentary cover and flysch fill adjacent depressions.Crystalline core occurs on a large part of the ridge and on the southern slopes of the mountains.In the higher parts of watersheds occurs migmatites -porphyritic granites and granodiorites.In the eastern part of the High Tatras, from the highest to middle position in a continuous strip occur porphyric granodiorites to granites, hercynian.In lower parts of analyzed watersheds forms the bedrock flysch -sandstones and calcareous claystones.From soil types are dominant in higher elevations Lithic Leptosols and others From a methodological point of view is particurlary important the selection of reference sections and flow profiles and the determination of its geometric and hydraulic characteristics.In this regard, we follow the articles of authors [Page, 1988] and [Rosgen and Silvey, 1996].In terrain under the source zones of torrents [Schumm, 1977, Lehotský andNovotný, 2004] were estabilished reference longitudinal sections (RLS) with reference cross sections (RCS).On the RCS were by leveling determined their geometric characteristic.Longitudinal slope was measured by levelung on RLS.On the RCS the sample of sediments was taken with weight at least 50 kg for granulometric analysis to determination of median grain diameter.Wetted perimeter and hydraulic radius were determined within the office processing.
To calculate bankfull discharge Q bf (m 3 .s - ), we used the Equation according to Parker [2004]: We used the following regression Equations for analysis: • and with asymptote: Equation ( 2) is generally used by many authors to estabilish hydraulic geometry relations.Equation ( 3) is a newly suggested means to the same end.Geometric and hydraulic characteristics of RCS are shown in Table 1.
In determination of basic elements of the water balance equation for all of the 26 analyzed In this connection, our research proceeded from these basic assumptions: • The basic hydrological characteristics of the watershed indentified and evaluated in the long run; • development of torrent beds (torrent morphogenesis) is a long-time process of hundreds to thousands of years; • component ΔS in equation ( 4) is not considered in analyses, change of water retention in the watershed is negligible in a long-time process; • from precipitation in the watershed is formed runoff and subsequently discharge in watercourse beds which has a dominant effect on their development (morphogenesis), • long-term runoff Q expresses the runoff from the watershed in all forms; • similar natural conditions in specific regions, mostly soil and geological, vegetation, topographic features, create similar conditions for the morphogenesis (long-term development) of torrent beds; • regional hydraulic geometry equations can be developed from these results, having a broad practical application.To determine the elements of the equation ( 4), relationships was derived by authors [Szolgay et al., 1997].The authors drew on the knowledge that the mean annual climatic evaporation depth E (mm) can be determined as a function of potential evaporation index EP i and of the mean longterm annual precipitation depth P (mm): On the basis of the relation ( 4) and ( 5) and the data measured at 54 meteorological stations in the SR, authors [Szolgay et al., 1997] derived for the conditions of SR the following empirical relationships: and The potential evaporation index EPi was calculated using the relation: The values of the mean long-term annual precipitation depth P (mm) and the mean long-term annual temperature T (ºC) for all of the analyzed watersheds were derived from the data measured at 8 meteorological stations of the High Tatras (Table 2), Figures (2)  Using the results of equations ( 4), ( 6), ( 7) and ( 8), respectively, the coefficient QA w as: = ( .  ).1000 −1 was calculated.Hy- drological characteristics of reference watersheds are shown in Table 3.

RESULTS AND DISCUSSION
The analyses showed a strong correlation between the coefficient QA w and the bankfull geometric characteristics of the torrents´ natural cross-sections: the width of the bed inside the banks B bf (m), the mean depth of the bed H bf (m), the channel cross-section area A bf (m 2 ) and hydraulic characteristic -the bankfull discharge Q bf (m 3 .s - ).The results were tested by the t-test and the Shapiro-Wilk test.The determination coefficient (R 2 ) for the relationships without the asymptote ranged between R 2 = 0.921 and R 2 = 0.968; p -values from the Shapiro -Wilk test ranged between p = 0.0080 and p = 0.5379.The determination coefficient (R 2 ) for the relationships  with asymptote ranged between R 2 = 0.953 and R 2 = 0.970; p -values from the Shapiro -Wilk test ranged between p = 0.0016 and p = 0.8625.Derived regional regression equations for analyzed correlation relations are listed in Tables 4 and 5 also contains the absolute and the regression coefficients for particular regional equations.5 contains the statistical characteristics and testing of particular relations.In all cases, the number of root mean standard errors using the equation with the asymptote, is lower than when  the equation without the asymptote was used.The graphic representation of the regional curves are shown in Figures.( 4) - (7).
The graphic presentation of residuals for each relations of the Shapiro-Wilk test are shown in Figures.( 8) - (15).Based on our analysis, we recommend the use of the regression equation ( 3) with theasymptote to determine regional equations of hydraulic geometry.
In relation to bankfull geometric and hydraulic characteristics, coefficient QA w was evaluated for the first time in the SR [Jakubis, 2008] in geomorphological unit Poľana (Central Slovakia, neovolcanites) in 25 torrents with characteristics with the medians A w = 4.47 (km 2 ), B bf = 2.40 (m), H bf = 0.55 (m), A bf = 1.05 (m 2 ), Q bf = 1.14 (m 3 .s - ).The determination coefficients in these relationships ranged between R 2 = 0.907 and R 2 = 0.989.Author [Pšida, 2014] evaluated the relationships between the QA w coefficient and the-above mentioned bankfull characteristics of four torrents in four different geographic regions -geomorphological units in the SR with different bedrocks with characteristics with the medians A w = 18.81 (km 2 ), B bf = 8.10 m, H bf = 0.83 (m), A bf = 4.53 (m 2 ), Q bf = 8.02 (m 3 .s - ).The determination coefficients ranged between R 2 = 0.705 and R 2 = 0.960.Both authors confirmed varying and specific natural geometric and hydraulic characteristics of torrent beds and their long -term morphogenesis in watersheds with different bedrocks.

CONCLUSION
The torrents are often located in protected areas.Therefore, a very sensitive approach to interfering with the torrent, flood and erosion control is needed.If such principles are not respected, various errors in torrent control designing and revitalization may occur, with negative ecological and environmental consequences in the landscape.Derived regional equations and curves make it possible to come up with a valuable input to the ecological designing and dimensioning in torrent control, flood and erosion control and torrent revitalization in the areas from which they were derived and, at the same time, to the gradual creation of hydrologic landscape regions.The results of research in regional hydraulic geometry provides these options.
(nonrendzic) Leptosols, in lower positions Haplic Podzols to Humic Podzols.From the aspect of hydrological efficiency have the soils in the higher positions of watersheds high retention capacity with medium permeability and in the lower positions medium retention capacity and permeability [Composite Authors, 2002].The watersheds (by the climatic regions of SR ) are located in climatic region C -cold subregions: C1 -moderately cool, C2 -cool mountainous and C3 -cold mountainous.The average annual precipitation in the watersheds range from 1037 mm (Päť prameňov) to 1379 mm (Biely Váh).Average annual temperatures range from 0.9 to 3.8 °C.

Figure 1 .
Figure 1.Map of Slovakia research area (High Tatras) T -mean long-term annual temperature in the watershed (ºC) and (3).

Figure 2 .
Figure 2. Relation between φH w and the mean annual precipitation depth P RMSE: root mean square error, SWp: p -value by Shapiro-Wilk test Table

Figure 3 .
Figure 3. Relation between φH w and the mean annual temperature T

Figure 4 .
Figure 4. Relation between QAw and B bf

Table 1 .
Characteristics of RCS Explanatory notes: B bf -width of the RCS inside the banks [m], H bf -mean depth of the RCS [m], A bf -RCS area [m 2 ], S -energy gradient [m/m], R bf -hydraulic radius of RCS [m], D 50 -grain diameter [m], Q bf -bankfull discharge [m 3 /s]

Table 2 .
Mean annual precipitation and temperature in the High Tatras stations

Table 3 .
Hydrological characteristics of experimental watersheds

Table 5 .
Statistic characteristics and testing of analyzed relations Explanatory notes: R -index of correlation, R 2 -index of determination,