Evaluation of hillslope storage with variable width under temporally varied rainfall recharge
 ^{1}Department of Soil and Water Conservation, National Chung Hsing University, Taichung 40227, Taiwan
 ^{2}Department of Water Resources, Taoyuan City Government, Taoyuan 330, Taiwan
 ^{1}Department of Soil and Water Conservation, National Chung Hsing University, Taichung 40227, Taiwan
 ^{2}Department of Water Resources, Taoyuan City Government, Taoyuan 330, Taiwan
Abstract. This study discussed water storage in aquifers of hillslopes under temporally varied rainfall recharge by employing a hillslopestorage equation to simulate groundwater flow. The hillslope width was assumed to vary exponentially to denote the following complex hillslope types: uniform, convergent, and divergent. Both analytical and numerical solutions were acquired for the storage equation with a recharge source. The analytical solution was obtained using an integral transform technique. The numerical solution was obtained using a finite difference method in which the upwind scheme was used for space derivatives and the thirdorder Runge–Kutta scheme was used for time discretization. The results revealed that hillslope type significantly influences the drains of hillslope storage. Drainage was the fastest for divergent hillslopes and the slowest for convergent hillslopes. The results obtained from analytical solutions require the tuning of a fitting parameter to better describe the groundwater flow. However, a gap existed between the analytical and numerical solutions under the same scenario owing to the different versions of the hillslopestorage equation. The study findings implied that numerical solutions are superior to analytical solutions for the nonlinear hillslopestorage equation, whereas the analytical solutions are better for the linearized hillslopestorage equation. The findings thus can benefit research on and have application in soil and water conservation.

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PingCheng Hsieh and TzuTing Huang
Interactive discussion
Status: closed

RC1: 'Comment on hess202150', Anonymous Referee #1, 01 Mar 2021
In this manuscript, a new analytical solution to the Boussinesq equation for variable widths and recharge rates is presented and analyzed.
I am in favor of the idea of the paper, and the paper is reasonable well written. However, there are a number of issues:
 Huyck et al. (2005) presented an analytical solution to the Boussinesq equation (in a different form) for variable widths and recharge rates. This is highly relevant work and has not been discussed. For example, equation 6 implies that the recharge is constant within each time step, which is exactly the same approach as Huyck et al. (2005).
 It should be clarified how the results from Troch et al. (2004) were obtained. Were these provided by any of the authors of that paper? Lines 216 and further indicate that the authors did not code these analytical solutions. Then how were these results obtained.
 The statement on line 222 is problematic: Verhoest and Troch (2000) do not state anywhere that they require 999 terms. They only state that after so many terms the residuals become insignificant, but they never performed an analysis on this. Usually, with these solutions, the results become stable after less than 100 summations.
 My major concern is the statement on line 247: the analytical solution is supposed to be highly sensitive to the fitting parameter b. When comparing a numerical solution to an analytical solution, the results should ALWAYS be equal, regardless of the parameters that are used. The only exception is when oscillations are obtained, but then either the temporal or spatial discretization should be modified. Looking at figures 8 through 15, it is clear that the discrepancies are too large, and something must be wrong. I did not check the mathematical solution, but either there is an issue there, and/or there is something wrong in the coding, and/or the numerical solution has issues. This is something that must be corrected before the paper can be accepted.
 Line 304 states that the results from Troch et al. (2003) were obtained by solving their equation numerically. Line 230231 states that the numerical solutions of Troch et al. (2003) matches the newly developed numerical solution well. This supports my suspicion that something is not right with the new analytical solution.
 There are too many figures in the paper. Something like 12 figures for a paper of this length should be the maximum. For example, I do not think that figure 2 is needed. The comparison with Troch et al. should be presented in less figures, as well as the comparison between the numerical and analytical solutions.
Based on these comments, I do not that that the paper is acceptable at this point.

AC1: 'Reply on RC1', PingCheng Hsieh, 08 Mar 2021
In this manuscript, a new analytical solution to the Boussinesq equation for variable widths and recharge rates is presented and analyzed.
I am in favor of the idea of the paper, and the paper is reasonable well written. However, there are a number of issues:
 Huyck et al. (2005) presented an analytical solution to the Boussinesq equation (in a different form) for variable widths and recharge rates. This is highly relevant work and has not been discussed. For example, equation 6 implies that the recharge is constant within each time step, which is exactly the same approach as Huyck et al. (2005).
Thank you very much for your comment. After examining the study of Huyck et al. (2005), we could find that in their study the Boussinesq eq. is in a different form, and they derived the analytical solutions by the Laplace transform method for different time steps and then took a summation of all the solutions. The derivation process of analytical solutions is clear but a little complicated when compared with ours. Although the concept of our Eq. (6), meaning that the recharge is constant within each time step, is the same as Huyck et al. (2005), the expressions are different. Don’t you think our expression is more concise and neater? We will add the citation and discussion of Huyck et al. (2005) to the article. Thank you again.
 It should be clarified how the results from Troch et al. (2004) were obtained. Were these provided by any of the authors of that paper? Lines 216 and further indicate that the authors did not code these analytical solutions. Then how were these results obtained.
Thank you very much for your comment. The results obtained from Troch et al. (2004) were not provided by any of the authors of that paper. Because the analytical solutions are given as Eqs. (33) and (35) of the paper of Troch et al. (2004), anyone can use them to reproduce the results in the paper. We will revise the description as follows:
“Both figures reveal that our results using the generalized integral transform technique agree well with the analytical solutions derived by the Laplace transform method, i.e. Eqs. (33) and (35) in Troch et al. (2004), thus validating our analytical solutions.”
 The statement on line 222 is problematic: Verhoest and Troch (2000) do not state anywhere that they require 999 terms. They only state that after so many terms the residuals become insignificant, but they never performed an analysis on this. Usually, with these solutions, the results become stable after less than 100 summations.
Thank you very much for your comment. We mistook the meaning “summation of the first 999 terms” from Verhoest and Troch (2000), and we will revise the description as follows:
“As stated in Verhoest and Troch (2000), after solution summation of the first 999 terms, namely O(10^{3}), the residuals become insignificant. In the present study, the solution summation of the first O(10^{2}) terms, usually less than 50, could reach convergence. The convergence of the present solution is better.”
 My major concern is the statement on line 247: the analytical solution is supposed to be highly sensitive to the fitting parameter b. When comparing a numerical solution to an analytical solution, the results should ALWAYS be equal, regardless of the parameters that are used. The only exception is when oscillations are obtained, but then either the temporal or spatial discretization should be modified. Looking at figures 8 through 15, it is clear that the discrepancies are too large, and something must be wrong. I did not check the mathematical solution, but either there is an issue there, and/or there is something wrong in the coding, and/or the numerical solution has issues. This is something that must be corrected before the paper can be accepted.
Thank you very much for your comment. The reviewer said “When comparing a numerical solution to an analytical solution, the results should ALWAYS be equal, regardless of the parameters that are used.” I totally agree with this point when both solutions are derived from the same governing equations, initial/boundary conditions and input parameters. However, an analytical solution to a LINEARIZED governing equation is possibly not equal to a numerical solution to a NONLINEAR governing equation. This present analytical solution is obtained for a linearized equation, but the present numerical solution is for a nonlinear equation which was described on Line 177 (original version of MS) “a numerical model was developed to solve the original nonlinear equation, Eq. (4)”. Both solutions are to different governing equations, so there are discrepancies in between. For the numerical solution by a finite difference method (F.D.M.) to the same LINEARIZED equation, the results are given below:
It shows that the numerical solutions are equal to the analytical solutions based on the same governing equations and same scenarios, thus justifying that the present analytical solutions are correct.
 Line 304 states that the results from Troch et al. (2003) were obtained by solving their equation numerically. Line 230231 states that the numerical solutions of Troch et al. (2003) matches the newly developed numerical solution well. This supports my suspicion that something is not right with the new analytical solution.
Thank you very much for your comment. The present numerical solutions are for the nonlinear governing equation in our study. In Troch et al. (2003), they derived a numerical solution by finite difference for the same nonlinear equation, Equation (6) in their study. Both results match each other, and this justifies the present numerical solutions in our study are correct. The present analytical solutions to the linearized equation have been justified correct as shown in the response of last comment.
 There are too many figures in the paper. Something like 12 figures for a paper of this length should be the maximum. For example, I do not think that figure 2 is needed. The comparison with Troch et al. should be presented in less figures, as well as the comparison between the numerical and analytical solutions.
Thank you very much for your comment. Original Figures 2, 5, 8, 11, and 14 are deleted now.
Thank you very much for all of your precious comments and suggestions.

AC1: 'Reply on RC1', PingCheng Hsieh, 08 Mar 2021

RC2: 'Comment on hess202150', Anonymous Referee #2, 16 Apr 2021
Comment on hess202150
This study applied both analytical and numerical approaches to solve hillslope hydrological dynamics equation, and tested (as well as compared) the results in some idealized situations. However, the manuscript was written more like a mathematic article though dealing with a practical problem in hydrology. Thus I think some major revisions are needed to meet the criteria of HESS. Please see my detailed comments as following.
Major comments:
Just as I mentioned, simulating the outlet discharge water of a hillslope is a practical hydrological problem. While many mathematic tools can be employed to solve the problem, the only metric to evaluate them is to compare their outcomes with some real observed measurements. However, this study stopped by testing its framework in some idealized conditions without checking with the real situation and data. On the other side, the topic of explicitly solving hillslope hydrology is not new. In fact, based on my knowledge, some land models have already employed the conception of hillslope and solve its hydrology dynamics explicitly using numerical solutions. These models have been tested and applied at different scales, and the observations are also available at different scales. So at such stage, conducting a similar research but only in idealized conditions is not decent for publication in HESS (maybe more suitable for a journal for applied mathematics). To overcome this shortage, the authors may consider using some real data to configure and evaluate their model, even at a local scale. Thus it can let us see more clearly the ability of each (analytical or numerical) methods and benefit future research. Please note that all required real data must be available as hydrological modelers have already depicted and validated the hillslope from local to global scales. So I see no excuse to refuse this suggestion.
Specific comments:
L41, “by means of isotope study”: Please delete these words.
L77, “The ground surface is vegetation free, …”: Please discuss the potential effects of vegetation.
L98, Equation (6): The n here should not be mixed with the n for drainable porosity.
L102, Equation (7): s/w=nh=bnD, because b<1, so h<D? But D is the average depth, how can h be less than its average everywhere?
L103, “where b is a fitting parameter …”: Please show more detail for the method used in tuning b.
L194, Equation (37): Please show more detail how to use Taylor series expansion to transform the Eq (13) to the Eq(37).
L203232: What is the major difference between this work and Torch et al. (2003, 2004)? The authors should particularly stress it in the manuscript because the similarity is too high in my view based on the current description.
L263: “Theta = 5%”: Is the theta angle of slope? How to understand the symbol of percentage?
 AC2: 'Reply on RC2', PingCheng Hsieh, 27 Apr 2021
Interactive discussion
Status: closed

RC1: 'Comment on hess202150', Anonymous Referee #1, 01 Mar 2021
In this manuscript, a new analytical solution to the Boussinesq equation for variable widths and recharge rates is presented and analyzed.
I am in favor of the idea of the paper, and the paper is reasonable well written. However, there are a number of issues:
 Huyck et al. (2005) presented an analytical solution to the Boussinesq equation (in a different form) for variable widths and recharge rates. This is highly relevant work and has not been discussed. For example, equation 6 implies that the recharge is constant within each time step, which is exactly the same approach as Huyck et al. (2005).
 It should be clarified how the results from Troch et al. (2004) were obtained. Were these provided by any of the authors of that paper? Lines 216 and further indicate that the authors did not code these analytical solutions. Then how were these results obtained.
 The statement on line 222 is problematic: Verhoest and Troch (2000) do not state anywhere that they require 999 terms. They only state that after so many terms the residuals become insignificant, but they never performed an analysis on this. Usually, with these solutions, the results become stable after less than 100 summations.
 My major concern is the statement on line 247: the analytical solution is supposed to be highly sensitive to the fitting parameter b. When comparing a numerical solution to an analytical solution, the results should ALWAYS be equal, regardless of the parameters that are used. The only exception is when oscillations are obtained, but then either the temporal or spatial discretization should be modified. Looking at figures 8 through 15, it is clear that the discrepancies are too large, and something must be wrong. I did not check the mathematical solution, but either there is an issue there, and/or there is something wrong in the coding, and/or the numerical solution has issues. This is something that must be corrected before the paper can be accepted.
 Line 304 states that the results from Troch et al. (2003) were obtained by solving their equation numerically. Line 230231 states that the numerical solutions of Troch et al. (2003) matches the newly developed numerical solution well. This supports my suspicion that something is not right with the new analytical solution.
 There are too many figures in the paper. Something like 12 figures for a paper of this length should be the maximum. For example, I do not think that figure 2 is needed. The comparison with Troch et al. should be presented in less figures, as well as the comparison between the numerical and analytical solutions.
Based on these comments, I do not that that the paper is acceptable at this point.

AC1: 'Reply on RC1', PingCheng Hsieh, 08 Mar 2021
In this manuscript, a new analytical solution to the Boussinesq equation for variable widths and recharge rates is presented and analyzed.
I am in favor of the idea of the paper, and the paper is reasonable well written. However, there are a number of issues:
 Huyck et al. (2005) presented an analytical solution to the Boussinesq equation (in a different form) for variable widths and recharge rates. This is highly relevant work and has not been discussed. For example, equation 6 implies that the recharge is constant within each time step, which is exactly the same approach as Huyck et al. (2005).
Thank you very much for your comment. After examining the study of Huyck et al. (2005), we could find that in their study the Boussinesq eq. is in a different form, and they derived the analytical solutions by the Laplace transform method for different time steps and then took a summation of all the solutions. The derivation process of analytical solutions is clear but a little complicated when compared with ours. Although the concept of our Eq. (6), meaning that the recharge is constant within each time step, is the same as Huyck et al. (2005), the expressions are different. Don’t you think our expression is more concise and neater? We will add the citation and discussion of Huyck et al. (2005) to the article. Thank you again.
 It should be clarified how the results from Troch et al. (2004) were obtained. Were these provided by any of the authors of that paper? Lines 216 and further indicate that the authors did not code these analytical solutions. Then how were these results obtained.
Thank you very much for your comment. The results obtained from Troch et al. (2004) were not provided by any of the authors of that paper. Because the analytical solutions are given as Eqs. (33) and (35) of the paper of Troch et al. (2004), anyone can use them to reproduce the results in the paper. We will revise the description as follows:
“Both figures reveal that our results using the generalized integral transform technique agree well with the analytical solutions derived by the Laplace transform method, i.e. Eqs. (33) and (35) in Troch et al. (2004), thus validating our analytical solutions.”
 The statement on line 222 is problematic: Verhoest and Troch (2000) do not state anywhere that they require 999 terms. They only state that after so many terms the residuals become insignificant, but they never performed an analysis on this. Usually, with these solutions, the results become stable after less than 100 summations.
Thank you very much for your comment. We mistook the meaning “summation of the first 999 terms” from Verhoest and Troch (2000), and we will revise the description as follows:
“As stated in Verhoest and Troch (2000), after solution summation of the first 999 terms, namely O(10^{3}), the residuals become insignificant. In the present study, the solution summation of the first O(10^{2}) terms, usually less than 50, could reach convergence. The convergence of the present solution is better.”
 My major concern is the statement on line 247: the analytical solution is supposed to be highly sensitive to the fitting parameter b. When comparing a numerical solution to an analytical solution, the results should ALWAYS be equal, regardless of the parameters that are used. The only exception is when oscillations are obtained, but then either the temporal or spatial discretization should be modified. Looking at figures 8 through 15, it is clear that the discrepancies are too large, and something must be wrong. I did not check the mathematical solution, but either there is an issue there, and/or there is something wrong in the coding, and/or the numerical solution has issues. This is something that must be corrected before the paper can be accepted.
Thank you very much for your comment. The reviewer said “When comparing a numerical solution to an analytical solution, the results should ALWAYS be equal, regardless of the parameters that are used.” I totally agree with this point when both solutions are derived from the same governing equations, initial/boundary conditions and input parameters. However, an analytical solution to a LINEARIZED governing equation is possibly not equal to a numerical solution to a NONLINEAR governing equation. This present analytical solution is obtained for a linearized equation, but the present numerical solution is for a nonlinear equation which was described on Line 177 (original version of MS) “a numerical model was developed to solve the original nonlinear equation, Eq. (4)”. Both solutions are to different governing equations, so there are discrepancies in between. For the numerical solution by a finite difference method (F.D.M.) to the same LINEARIZED equation, the results are given below:
It shows that the numerical solutions are equal to the analytical solutions based on the same governing equations and same scenarios, thus justifying that the present analytical solutions are correct.
 Line 304 states that the results from Troch et al. (2003) were obtained by solving their equation numerically. Line 230231 states that the numerical solutions of Troch et al. (2003) matches the newly developed numerical solution well. This supports my suspicion that something is not right with the new analytical solution.
Thank you very much for your comment. The present numerical solutions are for the nonlinear governing equation in our study. In Troch et al. (2003), they derived a numerical solution by finite difference for the same nonlinear equation, Equation (6) in their study. Both results match each other, and this justifies the present numerical solutions in our study are correct. The present analytical solutions to the linearized equation have been justified correct as shown in the response of last comment.
 There are too many figures in the paper. Something like 12 figures for a paper of this length should be the maximum. For example, I do not think that figure 2 is needed. The comparison with Troch et al. should be presented in less figures, as well as the comparison between the numerical and analytical solutions.
Thank you very much for your comment. Original Figures 2, 5, 8, 11, and 14 are deleted now.
Thank you very much for all of your precious comments and suggestions.

AC1: 'Reply on RC1', PingCheng Hsieh, 08 Mar 2021

RC2: 'Comment on hess202150', Anonymous Referee #2, 16 Apr 2021
Comment on hess202150
This study applied both analytical and numerical approaches to solve hillslope hydrological dynamics equation, and tested (as well as compared) the results in some idealized situations. However, the manuscript was written more like a mathematic article though dealing with a practical problem in hydrology. Thus I think some major revisions are needed to meet the criteria of HESS. Please see my detailed comments as following.
Major comments:
Just as I mentioned, simulating the outlet discharge water of a hillslope is a practical hydrological problem. While many mathematic tools can be employed to solve the problem, the only metric to evaluate them is to compare their outcomes with some real observed measurements. However, this study stopped by testing its framework in some idealized conditions without checking with the real situation and data. On the other side, the topic of explicitly solving hillslope hydrology is not new. In fact, based on my knowledge, some land models have already employed the conception of hillslope and solve its hydrology dynamics explicitly using numerical solutions. These models have been tested and applied at different scales, and the observations are also available at different scales. So at such stage, conducting a similar research but only in idealized conditions is not decent for publication in HESS (maybe more suitable for a journal for applied mathematics). To overcome this shortage, the authors may consider using some real data to configure and evaluate their model, even at a local scale. Thus it can let us see more clearly the ability of each (analytical or numerical) methods and benefit future research. Please note that all required real data must be available as hydrological modelers have already depicted and validated the hillslope from local to global scales. So I see no excuse to refuse this suggestion.
Specific comments:
L41, “by means of isotope study”: Please delete these words.
L77, “The ground surface is vegetation free, …”: Please discuss the potential effects of vegetation.
L98, Equation (6): The n here should not be mixed with the n for drainable porosity.
L102, Equation (7): s/w=nh=bnD, because b<1, so h<D? But D is the average depth, how can h be less than its average everywhere?
L103, “where b is a fitting parameter …”: Please show more detail for the method used in tuning b.
L194, Equation (37): Please show more detail how to use Taylor series expansion to transform the Eq (13) to the Eq(37).
L203232: What is the major difference between this work and Torch et al. (2003, 2004)? The authors should particularly stress it in the manuscript because the similarity is too high in my view based on the current description.
L263: “Theta = 5%”: Is the theta angle of slope? How to understand the symbol of percentage?
 AC2: 'Reply on RC2', PingCheng Hsieh, 27 Apr 2021
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