Interactive comment on “ Notes on the estimation of resistance to flow during flood wave propagation ”

General comments: This manuscript raises the important and relevant discussion of the use of friction coefficients during unsteady flow. The research has been carried out and reported thoroughly and interesting for the HESS audience. However, the structure of the manuscript can be improved and the novelty can be emphasized. The introduction and title suggest that the manuscript is about the difference between friction velocity and roughness coefficients. This is only a very small part of the manuscript, the majority deals with determining the friction velocity itself. This shows exactly the issue in dealing with friction velocity which is not stated in the conclusions: friction velocity is highly uncertain and difficult to determine. However, the authors still suggest to use friction velocity. This requires at least a thorough discussion after section 3 or in the concluding remarks. Secondly, the structure of the manuscript can be improved.


Introduction
Resistance is one of the most important factors affecting the flow in open channels.In simple terms it is the effect of water viscosity and the roughness of the channel boundary which result in friction forces that retard the flow.The largest input into the resistance is attributed to water-bed interactions.
Many authors argue that the description of resistance to flow is unsatisfactory (Beecham et al., 2005;Chaudhry, 2011;Knight, 2013a For the above reasons, it seems more meaningful to consider friction force, rather than resistance coefficients, as a basic term expressing resistance to flow.In this respect, resistance is represented by variability :: is ::::::::::::: straightforward : -:::: they :::: rise :::: with ::::: rising :::::::: resistance :: to ::::: flow. The proper definition and understanding of shear stress and friction velocity is of great importance, since shear stress is an intrinsic variable in a number of hydrological problems, such as bed load transport, rate of erosion and contaminants transport (Garcia, 2007;Julien, 2010) ; (Kalinowska and Rowiński, 2012; van Rijn, 1993) : (G Boundary shear stress is expressed on a range of spatial scales from a point value to a global one (Yen, 2002).The following types of boundary shear stress are defined: local bed shear stress (Khodashenas et al., 2008), average bed shear stress; average wall shear stress (Khiadani et al., 2005); and finally average boundary shear stress, i.e. averaged over a wetted perimeter (Khiadani et al., 2005).It should be noted that the nomenclature is inconsistent, and other authors may use different terminology (Ansari et al., 2011;Khiadani et al., 2005;Khodashenas et al., 2008;Knight et al., 1994).Moreover, a number of definitions of friction velocity exist (Pokrajac et al., 2006).Hence, for clarity a reference to a definition is necessary in each study.
It is difficult to measure bed shear stress directly.The direct method, which uses a floating element balance type device, enables the measurement of the force acting tangentially on a bed, and is used in both field (Gmeiner et al., 2012) and laboratory studies (Kaczmarek and Ostrowski, 1995); however, the results are prone to high uncertainty.The majority of methods measure bed shear stress indirectly, e.g. using hot wire and hot film anemometry (Albayrak and Lemmin, 2011;Nezu et al., 1997), a Preston tube (Molinas et al., 1998;Mohajeri et al., 2012), methods that take advantage of theoretical relations between shear stress and the horizontal velocity distribution (Graf and Song, 1995;Khiadani et al., 2005;Sime et al., 2007;Yen, 2002), methods based on Reynolds shear stress (Biron et al., 2004;Campbell et al., 2005;Czernuszenko and Rowiński, 2008;Dey and Barbhuiya, 2005;Dey and Lambert, 2005;Dey et al., 2011;Graf and Song, 1995;Nezu et al., 1997;Nikora and Goring, 2000) or turbulent kinetic energy (Galperin et al., 1988;Kim et al., 2000;Pope et al., 2006), or methods that incorporate double-averaged momentum equation (Pokrajac et al., 2006).Despite the fact that there is a variety of methods, a handful of them are feasible for application in unsteady flow conditions.
Then, the relations : In :::: this ::::: paper ::: we ::::: apply :::::::: formulae derived from flow equations may be a good water ::::: stage.They have been claimed to be reasonable means of friction velocity assessment in unsteady flow by a number of authors, e.g.Afzalimehr and Anctil (2000); De Sutter et al. (2001); Ghimire andDeng (2011, 2013); Graf and Song (1995); Guney et al. (2013); Rowiński et al. (2000); Shen and Diplas (2010); Tu and Graf (1993); nonetheless, in-depth analysis is still needed ::::::: because uncertainty :: of ::::::::: resistance :: to :::: flow :::::::: evaluated :: by ::::::::::: relationships derived from flow equationsto simplified forms.Simplified methods are welcome, especially for practitioners.However, they must be justified properly, and there seem to be a gap here.This paper discusses the following aspects: simplification of formulae for friction velocity due to type of wave; methods of evaluation of flow depth gradient; impact of channel geometry on friction velocity evaluation; and evaluation of the uncertainty of input variables.The paper presents a methodology which can be used to choose an appropriate method of friction velocity evaluation in a case under consideration.The discussion is illustrated by the analysis and application of friction velocity formulae to experimental field data.Moreover, , ::: Sect :: 5 :::::::::: conclusions ::: are :::::::: provided.: The problem presented herein has been partially considered in the unpublished Ph.D. thesis of the first author of this paper (Mrokowska, 2013).
In the study, two cross-sections, denoted in Fig. 3 : 1 as CS1 and CS2, are considered.Crosssection CS1 was located about 200m ::: 200 :: m from the dam, and cross-section CS2 about 1600m :::: 1600 :: m from it.The shape of the cross-sections is presented in the bottom panel of Fig. 3  water : stage (H) may be observed in Fig. 5 : 4. From the figure it can be seen that the rating curves are not closed for Ol-1, Ol-2 and Ol-3, which is probably caused by too short series of measurement data.
Data set Ol-1 was applied in (Mrokowska and Rowiński, 2012;Rowiński et al., 2000), and data set Ol-4 in (Mrokowska et al., 2013), and to the authors knowledge, none of the data sets have been utilised elsewhere in the context of the evaluation of friction velocity.
The assumption about the type of a flood wave affects the form of friction velocity relations to a great extent.This may be demonstrated by analysing the ::: The : St. Venant model for a rectangular channel which comprises Eqs. ( 3) and ( 4) : :: is ::: the :::: most ::::::::: frequently :::: used ::::::::::: mathematical ::::: model :: to :::::: derive ::::::: formulae ::: on ::::::::: resistance: where g -gravity acceleration , h -flow depth , 3) is the continuity equation and Eq. ( 4) is the momentum balance equation which the terms represent as follows: the gradient of flow depth (hydrostatic pressure term), advective acceleration, local acceleration, friction slope and bed slope.Further on, derivatives will be denoted by Geek letters to stress that they are treated as variables, namely . The friction velocityderived from the model represents the value averaged over a wetted perimeter: the bulk variable.If the channel width is much larger than the flow depth, the mean cross-sectional velocity U is equivalent to the depth-averaged velocity above any location of the bed, and the hydraulic radius R may be substituted by the flow depth h.Consequently, the bulk friction velocity is equivalent to the bed friction velocity.for :::::::::::: practitioners.:::::::: However, :::: they ::::: must :: be :::::::: justified :::::::: properly, ::: and ::::: there :::::: seems :: to ::: be : a :::: gap ::::: here.:: It  4) are negligible, they may be eliminated, and the model for a diffusive wave is obtained.Further omission of the hydrostatic pressure term leads to the kinematic wave model, in which only the term responsible for gravitational force is kept.The simplifications of the St. Venant model have been investigated in many papers in the context of flood wave modelling (Aricó et al., 2009;Dooge and Napiórkowski, 1987;Moussa and Bocquillion, 1996;Yen and Tsai, 2001).Some authors have concluded that the diffusive approximation is satisfactory in the majority of cases (Ghimire and Deng, 2011;Moussa and Bocquillion, 1996;Yen and Tsai, 2001), especially for lowland rivers.However, according to Gosh (2014); Dooge and Napiórkowski (1987); Julien (2002), in the case of upland rivers, i.e. for average bed slopes, it could be necessary to apply the full set of St.
- Rowiński et al. (2000), and next Shen and Diplas (2010) applied the formula derived from the St. Vernant equations: ::: set :: of ::::::::: equations: - Tu and Graf (1993) derived the equation from the St. Venant momentum balance equation: where - Dey and Lambert (2005) derived the formula from the 2D Reynolds equations which incorporated data on bed roughness.To see the equation please refer to (Dey and Lambert, 2005).

Evaluation of the gradient of flow depth ϑ
The gradient of flow depth ϑ = ∂h ∂x is a significant variable in both dynamic (Eqs.( 5), ( 6), ( 15) :: 5, :: 6, :: 15) and diffusive (Eq.( 8) : 8) friction velocity formulae.Moreover, the evaluation of ϑ is widely discussed in hydrological studies on flow modelling and rating curve assessment (Dottori et al., 2009;Perumal et al., 2004;Schmidt and Yen, 2008).The gradient of flow depth is evaluated based on flow depth measurements at one or a few gauging stations.Due to the practical problems with performing the measurements, usually only one or two cross-sections are used.This constitutes one crucial obstacle when seeking friction velocity.

Kinematic wave concept
According to the kinematic wave concept, the gradient of flow depth is evaluated implicitly based on measurements in one cross-section by Eqs. ( 16) or ( 17) (Graf and Song, 1995;Perumal et al., 2004).This approach is encountered in friction velocity assessment studies (De Sutter et al., 2001;Graf and Song, 1995;Ghimire andDeng, 2011, 2013;Tu and Graf, 1993).However, this method has been challenged in rating-curve studies (Dottori et al., 2009;Perumal et al., 2004;Schmidt and Yen, 2008) due to its theoretical inconsistency.As Perumal et al. (2004) presented, Jones introduced the concept in 1915 in order to overcome the problem of ∂h ∂x evaluation in reference to looped rating curves, i.e. non-kinematic waves.The looped shape of non-kinematic waves results from the acceleration of flow and the gradient of flow depth (Henderson, 1963;Silvio, 1969).The kinematic wave, on the other hand, has a one-to-one relationship between the water level and discharge :::: stage :::: and :::: flow ::: rate, which is equivalent to a steady flow rating curve.Both rating curves are illustrated in the upper panel of Fig. 2 : 5 after (Henderson, 1963).The kinematic wave concept results in the following approximation: the Jones formula : ::::: Jones ::::::: formula ::::: which :: is :::::: applied :: in :::: this ::::: study: : Furthermore, ϑ may be expressed by the temporal variation of the discharge :::: flow ::: rate : instead of the flow depth (Ghimire and Deng, 2011;Julien, 2002), which leads to the following approximation: Both approximations, Eqs. ( 16) and ( 17), affect the time instant at which ∂h ∂x = 0.As shown in the upper panel of Fig. 2 : 5, in the case of a non-kinematic subsiding wave, the peak of the flow rate ∂Q ∂t = 0 in a considered cross-section is followed by the temporal peak of the flow depth ∂h ∂t = 0, while the spatial peak of the flow depth ∂h ∂x = 0 is the final one.The bottom panel of Fig. 2 : 5 presents schematically the true arrival time of ∂h ∂x = 0 for the non-kinematic wave, and the arrival time approximated by the kinematic wave assumption in the form of Eq. ( 16) and Eq. ( 17).Both formulae underestimate the time instant at which ∂h ∂x = 0.As a matter of fact, from the practical point of view, the evaluation of the friction velocity is exceptionally important in this region, as intensified transport processes may occur just before the wave peak (Bombar et al., 2011;De Sutter et al., 2001;Lee et al., 2004).Consequently, it seems that the admissibility of the kinematic wave assumption should be thoroughly verified for a wave under consideration.
In order to apply the kinematic wave approximation, the wave celerity must be evaluated.Usually, celerity is :::::: Celerity :::: can ::: be assessed by the formula for a wide rectangular channel derived from the Chezy equation (Eq.18) (Henderson, 1963;Julien, 2002)or the Manning equation (Eq.??) (Ghimire and Deng, 2011;Julien, 2002) : , ::: and :: it :: is :::::: applied :: in :::: this ::::: study.Tu (1991); Tu and Graf (1993) proposed another method for evaluating C: However, we would like to highlight the fact that in Eq. ( 19) ∂h ∂t is in the denominator, which constrains the application of the method.As a result, a discontinuity occurs for the time instant at which ∂h ∂t = 0.When the results of Eq. ( 19) are applied in Eq. ( 16), the discontinuity of ϑ as a function of time occurs at the time instant at which C = 0, which is between t( ∂U ∂t = 0) and t( ∂h ∂t = 0).This effect is illustrated in the section on field data application (Sect ?? :: 4.1).

Linear approximation based on two cross-sections
Because of the drawbacks of kinematic wave approximation, it is recommended to evaluate the gradient of the flow depth based on data from two cross-sections (Aricó et al., 2008(Aricó et al., , 2009;;Dottori et al., 2009;Julien, 2002;Warmink et al., 2013), which is, in fact, a two-point difference quotient :::::::: (backward :: or ::::::: forward).Nonetheless, a number of problematic aspects of this approach have been pointed out.
Firstly, Koussis (2010) has stressed the fact that flow depth is highly affected by local geometry; hence, the proper location of the cross-sections is a difficult task.Moreover, Aricó et al. (2008) have pointed that lateral inflow may affect the evaluation of the gradient of flow depth, and for this reason the cross-sections should be located close enough to each other to allow the assumption of negligible lateral inflow.On the other hand, the authors have claimed that the distance between cross-sections should be large enough to perform a robust evaluation of the flow depth gradient.The impact of distance between cross-sections on the gradient of flow depth has been studied in (Mrokowska et al., 2015) with reference to dynamic waves generated in a laboratory flume.The results have shown that with a too long distance, the gradient in the region of the wave peak is misestimated due to the linear character of approximation.On the other hand, with a too short distance, the results may be affected by fluctuations of the water surface which :: in :::: such :::: case : are large relative to the distance between cross-sections.
Another drawback of the method is the availability of data.Very often, data originate from measurements which have been performed for some other purpose.Consequently, the location of gauging stations and data frequency acquisition do not meet the requirements of the evaluation of the gradient of flow depth (Aricó et al., 2009).The latter problem applies to the case studied in this paper.
3.3.3Methods based on higher :::::: Higher : order approximation Mathematically, the gradient of flow depth represents the local value, by the definition of the derivative.

Uncertainty of input data and the results
The friction velocity, as with other physical variables, should be given alongside the level of uncertainty of the results (Fornasini, 2008).The uncertainty of results depends on the evaluation method and the quality of the data.As shown in the proceeding sections, neither of these is perfect when a friction velocity assessment is performed.For this reason, an appropriate method of uncertainty evaluation must be chosen in order to obtain information about the quality of the result.Friction velocity is usually applied to further calculations, and for this reason information about the uncertainty of results is of high importance.In the case of unrepeatable experiments Mrokowska et al. (2013) have suggested applying ::::::::::: deterministic :::::::: approach : -: the law of propagation of uncertainty (Holman, 2001;Fornasini, 2008), which for Eq. ( 15) takes the form of Eq. ( 22)and represents the maximum In ::: this ::::::: method ::: the :::::: methodology The preceding sections have demonstrated that the application of friction velocity formulae requires a thorough analysis of flow conditions and available methods.To sum up, the following issues should be considered during the evaluation of friction velocity :::::::: resistance :: to :::: flow :::: from :::: flow :::::::: equations: 1. What is the shape of the channel -is simplification of the channel geometry applicable?
2. What methods of evaluating input variables, especially ϑ = ∂h ∂x , are feasible in the case under study?
4. What is the uncertainty of the input variables, and which of them are most significant?
4 Field data application Although the above considerations seem to be quite universal, their significance will be illustrated based on a set of data from an experiment carried out in natural settings.The detailed analyses shown for these practical cases may provide advise on how to proceed in similar situations.
Moreover, due to the linear character of this method, ϑ lin is unsuitable to express the variability of the flood wave shape.As a result, it overestimates the time instant at which ϑ = 0 when the downstream cross-section is taken into account (as in Ol-1), and underestimates the time instant when the upstream cross-section is used (as in Ol-2, Ol-3, Ol-4).Next, the lateral inflows might have an effect on the flow, and thus the estimation of ϑ by the linear method.When it comes to ϑ Tu&Graf , the results are in line with ϑ kin and ϑ wt except for the region near the peak of the wave where discontinuity occurs.This occurs due to the form of Eq. ( 19), which cannot be applied if ∂h ∂t = 0, as was theo-retically analysed in Sect ?? :::: 3.3.1.Consequently, the method must not be applied in the region of a rising limb in the vicinity of the wave peak and in the peak of the wave itself.negligible.On the other hand, for data set Ol-1, the bed slope and the maximum flow depth gradient are of magnitude 10 −4 .Moreover, the acceleration terms reach the magnitude of 10 −4 along the rising limb.However, the acceleration terms are of opposite signs, and the overall impact of flow acceleration on the results might not be so pronounced.The comparison between Ol-1 and Ol-2, which originate from the same experiment, shows that in cross-section CS1, which is closer to the dam, more terms of the momentum balance equation are significant.From the results for CS2 it may be concluded that the significance of the temporal variability of flow parameters decreases along the channel.
Another method which may be used to identify the type of wave is analysis of the sensitivity of friction velocity to input variables (Mrokowska and Rowiński, 2012;Mrokowska et al., 2013) or a kind of stability analysis in which one observes the impact of a small change in the value of the input variable on the friction velocity value (Mrokowska et al., 2013).

Evaluation of friction velocity
4.3 ::::::::: Evaluation :: of ::::::: friction ::::::: velocity Figure 8 presents the results for the friction velocity evaluated using the formula for the dynamic wave (Eq.( 15) :: 15), using different methods to evaluate ϑ.As can be seen from the figure, u * kin and u * wt agree well with each other.There is also good agreement with u * Tu&Graf along the falling limbs of waves.In Ol-1, Ol-3, and Ol-4 it is observed that the discontinuity occurs between the time instants of maximum U and maximum h, as is noted in the theoretical part of this paper (Sect ?? :::: 3.3.1).The effect of the discontinuity depends on the time step applied in the analysis, and when the step is large enough, as in the case of Ol-2, the discontinuity may be overlooked.When it comes to u * lin , it deviates to high extent from the previous results, and is considered as not reliable due to the comments on ϑ lin presented in Sect ?? :::: 3.3.2.
, ∆I = 0.0001.As can be seen from Fig. 9, the results for friction velocity obtained by the formula for dynamic waves (Eq.15) and the formula for diffusive waves (Eq.8) agree well with each other.The slight difference between the results occurs in data set Ol-1.This is caused by the acceleration terms, which appear to be significant in Ol-1 along the leading edge (Fig. 7).Consequently, in this region, the application of Eq. ( 15) may be considered.However, the results of Eq. ( 8) lie within the uncertainty bounds of the results of Eq. ( 15); hence, the application of the simplified formula :: for :::::::: diffusive :::: wave : is acceptable.
On the other hand, the results obtained by Eq. ( 15) and by formula for steady flow (Eq.( 12) :: 12) differ from each other.For Ol-1, Ol-2 and Ol-4 the results of Eq. ( 12) fall outside the uncertainty bounds of Eq. ( 15) along the substantial part of leading edge of the waves.In data set Ol-4, the time period could be observed in which the uncertainty bounds of Eq. ( 15) and Eq. ( 12) do not overlap.
The significant discrepancies along the leading edge of a flood wave indicate that the application of Eq. ( 12) in this region is incorrect.

Analysis of the Manning coefficient
Manning n is calculated from Eq. ( 1) ::: with :: S ::::::: derived :::::::::: analytically ::::: from ::: the ::: St. :::::: Venant :::::: model for data sets Ol-1, Ol-2, Ol-3 and Ol-4.An intrinsic part of the formula is S -the friction slope.As the resistance equation with the Manning coefficient (Eq.( 1)) has been derived for steady state conditions, and its application in unsteady flow is questionable, it is difficult to decide which way of evaluating S is theoretically meaningful.S may be taken as the friction slope obtained from the momentum balance equation (S from Fig. 7) or as the bed slope (I).When S is obtained from the momentum balance equation, the method of evaluating :: In :::: fact, :::::::: Manning :: n :::: may ::: be :::: also :::::::::: recalculated :::: from ::::::: friction ::::::: velocity ::::::: results.::: All :::::::: analyses :: of ::::::::::::: simplifications ::: and ::::::::: evaluation ::: of ϑ is significant :::::::: presented ::::: above :::::: apply :: to ::::::::: evaluation :: of :::::::: Manning ::: n, :: as :::: well.Figure 10 presents the results of n for ϑ evaluated by the wave translation (n wt ) and linear approximation (n lin ) methods.In addition, n st is evaluated for S = I.Discrepancies between n st and n wt result from the difference between I and S depicted in Fig. 7, and the discrepancies are most pronounced for Ol-1.Moreover, it can be seen that n lin differs considerably from the other results in all cases.This indicates that the method of evaluating ϑ may have a significant effect on n.Note that Manning n st reaches its minimum value at the time instant of maximum U , hence it decreases with increasing velocity.On the other hand, it may not be true for n wt and n lin , besause theit values depend :::::: minor streams in the tables presented in (Chow, 1959).The minimum values of Ol-2 and Ol-3 correspond with "clean straight, full stage, no rifts or deep pools", while the minimum value of Ol-1 does not match n for natural streams presented in the tables.The maximum values may be assigned to "same as above, but more stones and weeds".The minimum value of n for Ol-4 may be assigned to "sluggish reaches, weedy, deep pools" and the maximum value to "vary weedy reaches, deep pools".The higher values of n for data set Ol-4 compared to the other data sets result from the fact that U is smaller than in the other cases (Fig. 4).The Manning n coefficients have been evaluated in a completely different way for the measurement data from this field site by Szkutnicki (1996); Kadłubowski and Szkutnicki (1992).:: In ::: that :::::: study, n was treated as a constant parameter in the St.
Venant model, and its value was assessed by optimising the model performance.The authors have reported that for spring conditions, n ∈ [0.04, 0.09].In this analysis, the results for Ol-1, Ol-2, Ol-3 are smaller, and the results for Ol-4 fall within the mentioned bounds.

Figure 5 .Figure 6 .
Figure5.Comparison between rating curve for flood wave and steady flow with characteristic points, based on(Henderson, 1963) (upper panel), and impact of kinematic wave approximation (Eqs.(16) and (17)) on the assessment of time instant at which ϑ = 0 ::::: ∂h ∂x = 0 (lower panel).The site of the experiment in Olszanka River (upper panel), and the shape of measurement cross-sections CS1 and CS2 (lower panel)).Temporal variability of flow depth h and mean velocity U for experimental flood waves in Olszanka River.Rating curves of experimental flood waves in Olszanka River.

Figure 10 .
Figure 10.Temporal variability of Manning n evaluated for different assumptions about energy ::::: friction : slope S for experimental flood waves in ::the : Olszanka River.