- Email: [email protected]
Evaluation of linear and nonlinear sediment transport equations using hillslope morphology
Catena 64 (2005) 272 – 280 www.elsevier.com/locate/catena
Evaluation of linear and nonlinear sediment transport equations using hillslope morphology Francisco J. Jime´nez-Hornero a, Ana Laguna b, Juan V. Gira´ldez a,* b
a Department of Agronomy, E.T.S.I.A.M., University of Cordoba, P.O. Box 3048, 14080 Cordoba, Spain Department of Applied Physics, E.T.S.I.A.M., University of Cordoba, P.O. Box 3048, 14080 Cordoba, Spain
Accepted 12 September 2005
Abstract Although some simple erosive processes like soil creep or tillage redistribution may be satisfactorily described by linear diffusive equations, the complexity of erosion phenomena requires the use of more complete nonlinear equations. Roering et al. [Roering, J.J., Kichner, J.W., Dietrich, W.E., 1999. Evidence for nonlinear, diffusive transport on hillslopes and implications for landscape morphology. Water Resour. Res., 35 853–870.] have proposed a single test, based on the relationship between the curvature and gradient of a hillslope, rather then using a linear diffusion equation. Nevertheless this test, based on steady state conditions, is not complete as it is shown in this work with a counter-example. The hillslope profile used by Roering et al. [Roering, J.J., Kichner, J.W., Dietrich, W.E., 1999. Evidence for nonlinear, diffusive transport on hillslopes and implications for landscape morphology. Water Resour. Res., 35 853–870.], (Fig. 2) can be generated either by a linear diffusion equation under transient conditions or a nonlinear diffusion equation under steady state conditions. Additional information on the soil profile changes would give a more complete interpretation of the hillslope evolution. D 2005 Elsevier B.V. All rights reserved. Keywords: Linear and nonlinear diffusive sediment transport equations; Steady state conditions
* Corresponding author. Tel.: +34 957 016057; fax: +34 957 016043. E-mail address: [email protected] (J.V. Gira´ldez). 0341-8162/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.catena.2005.09.001
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
273
1. Introduction The need for simple models to describe complex phenomena of soil erosion or landscape evolution has led to the use of geomorphic transport laws that are not well based. Such could be the case of the linear diffusive sediment transport on slopes which has been recently discussed by Dietrich et al. (2003) in a thorough review. Culling (1963) was one of the first erosion modeller adopting the linear dependence of solid particle flux with the slope gradient, for the process of soil creep. Dietrich et al. (2003) quoted two erosion studies where such a relation was observed, in two different soils of contrasting textural classes. Another interesting example of erosion described by a linear law is the mechanical or tillage erosion as reported by Govers et al. (1994). After careful measurements of the movement of metallic tracers located in vertical soil pits along slope transects, the particle displacement along the slope was proportional to the slope gradient. Linear diffusive sediment transport equations have been used in many studies in which their validity was assumed without experimental confirmation. Dietrich et al. (2003) criticized this use on the basis of three main points: i) the lack of physical bases of the equation; ii) its narrow field of applicability; and iii) the misfit between the conditions of validity and spatial scale of the models where they have been used. The second and third points are specially important. There are many erosion processes that cannot be properly described by linear diffusive law, as Howard (1994), among other authors, recognized. However the first point requires a closer look. In a previous paper of the authors (Roering et al., 1999) the consequences of linear diffusive sediment transport on the convexity of hillslopes were analysed. In addition to a simple test to assess the feasibility of both, linear and nonlinear equations were proposed. The purpose of this work is the discussion of the conditions under which such a proposal can be accepted. After posing the linear and nonlinear sediment transport equations proposed by Roering et al. (1999), it will be shown that the test for choosing between these equations depends strongly on whether the hillslopes can be assumed to be in steady state or not. Here we show that a linear transport law applied to a transient hillslope can produce similar profiles to the nonlinear law for the steady state case.
2. Methods 2.1. Linear and nonlinear transport equations The description of hillslope evolution implies the consideration of several processes (e.g. Tucker and Slingerland, 1994, Table 1), affecting to the rock and to the sediment, regolith or soil. Furbish and Fagherazzi (2001) have reduced these equations to the conservation of mass for rock and soil. Nevertheless a simple equation like the one proposed by Dietrich et al. (2003) may be used for the purposes of the present work, written in terms of solid mass as, qs
Bz Bh ¼ qr u p q s s p þ q s ¼ qr up qs rd q˜ s Bt Bt
ð1Þ
274
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
where z is the elevation of the ground, [L], t is time, [T], u p is the uplift rate, [LT 1], s p is the sediment or soil production rate, [LT 1] (e.g. Heimsath et al., 1997), h is the soil thickness, [L], q r and q s are the respective bulk density of the rock and soil or sediment, [ML 3], and q s is the sediment flow rate, [L 2T 1]. The combination of the two last terms in the intermediate expression follows from Dietrich et al. (2003, Eq. (2)). The sediment transport rate may be written either as linearly dependent on the slope qs ¼ klin rz
ð2Þ
where k lin is a diffusion coefficient, [L 2T 1], or as a more complex function of the slope, qs ¼ f ðrzÞ
ð3Þ
The linear diffusion equation is obtained as, Bz ¼ U Dlin r2 z Bt
ð4Þ
with U = u pq r / q s as the uplift term, [LT 1], and D = k lin / q s as a diffusion coefficient, [LT 1]. Given the reduced values assumed by the uplift rate some authors (e.g. Fernandes and Dietrich, 1997, Eq. (4)) usually neglect this term. This equation describes fairly well most of the mechanical erosion experiments found in the literature (Lobb et al., 1999). Roering et al. (1999) presented a nonlinear diffusion equation for sediment transport on slopes based on a general expression for q s, relating it to the velocity of the downslope movement, m, [LT 1] q˜ s ¼
V m˜ A
ð5Þ
with V / A as the volume of mobile sediment per unit area along the slope, [L]. From this expression they deduced a final form for Eq. (3), q˜ s ¼
knol rz 2 1 jrzjSc1
ð6Þ
with k nol as the nonlinear diffusivity, [L 2T 1], and S c as a threshold slope. A similar form of this equation was adopted by Howard (1994). In order to compare Eqs. (2) and (6), Roering et al. (1999) assumed that the rock uplift rate may be similar to the surface erosion rate in (1), leading to a steady state, the dynamic equilibrium of Gilbert. In such a case (1) becomes
qr up ¼ rd q˜ s qs
ð7Þ
For the linear case they derived an equation
qr up ¼ r2 z qs klin
ð8Þ
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
275
where j 2z is taken as the curvature of the slope (e.g. Farin, 1997, § 11.3), whereas the nonlinear case leads to
qr up rz ¼ rd 2 qs knol 1 jrzjS 1
ð9Þ
c
Eqs. (8) and (9) are the alternative linear and nonlinear diffusive erosion equations for Roering et al. (1999), hereafter named RKD model. Note that both of them are deduced assuming steady state conditions. Simplifying the equations to a two-dimensional case in the plane x–z, with x as the distance allows an easy integration of the equations. The results are shown in Fig. 1. If the value of the slope ratio, S / S c, is small the nonlinearity disappears as the authors commented. The distinction between both models is mainly based on the relation between the second derivative of the elevation with distance and the gradient. Roering et al., (1999; Fig. 2) compare the gradient and curvature with data of a real slope measured in the Oregon Coastal Range. 2.2. Transient linear diffusive transport equations The linear diffusive sediment transport equation, (4), may be integrated for the simple case of an initially uniform slope, z = z 0(1 x / L), with a fixed elevation at the bottom of the slope, and no flow conditions at the divide. Armstrong (1987) discussed several possible lower boundary conditions. The condition adopted here implies a balance between inflow and outflow of solid particles at such a point.
Fig. 1. Different behaviour of linear and nonlinear diffusive models according to Roering et al. (1999).
276
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
The initial elevation is z 0, and total distance is L. The boundary conditions may be written as Bz ¼0 Bx z¼0
x¼0
ð10aÞ
t N0
x¼L
ð10bÞ
t N0
The variables are normalized by L. xN ¼
x L
zN ¼
z L
tN ¼
tDlin L2
ð11Þ
Fig. 2. Relationships between elevation and distance, left, and curvature and gradient for the linear diffusive model, right, with free flow condition downstream at a given time.
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
277
The analytical solution is given by Carslaw and Jaeger (1959, § 3.3.14) with an additional term accounting for the uplift rate assumed to be constant (e.g. Hirano, 1968). " 2 # l 8 X 1 2n þ 1 2n þ 1 p px exp t zN ¼ U N t n þ 2 ð12Þ cos N N p n¼0 ð2n þ 1Þ2 2 2 where the normalized uplift rate, U N, is, UN ¼
L U Dlin
ð13Þ
The relationships between elevation, distance, gradient and curvature are shown in Fig. 2 as an example. The linear diffusive sediment transport equation does not necessarily yield a straight line in the gradient–curvature plot as implied in the RKD model.
3. Results and discussion A simple optimisation technique allows the identification of the linear diffusive model parameters, z 0, L, U N and normalized time, t N, through a good fit of (12) to the topographic data of Roering et al. (1999). The fit shown in Fig. 3 corresponds to the parameters elevation, z 0 = 26.13 m, distance, L = 36.41 m, and normalized uplift rate U N = 1.95 d 10 2. The fit has an efficiency index of Nash and Sutcliffe (e.g. Beven, 2000) of 0.99850. The
Fig. 3. Fit of the linear diffusive model to the Roering et al. (1999, Fig. 2) data. The maximum elevation is 26.13 m, the maximum length is 36.52 m, the normalized uplift rate is 0.0195 and the corresponding normalized time, t N = 0.0374. The downslope elevation was kept constant.
278
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
normalized time t N = 0.0400 implies a real time in the range 10.0–16.6 ka (thousand of years) if the value of D lin is taken between 5.3 to 3.2 d 10 3 m2 yr 1 as Roering et al. (1999) proposed later in their paper. These values for D lin are within the estimates gathered by Fernandes and Dietrich (1997, Table 1). The absolute uplift rate is between 2.84 d 10 3 and 1.75 d 10 3 mm yr 1 lower than the 0.1–0.2 mm yr 1 range suggested by Roering et al. (1999) for the area. Neglecting the uplift term in Eq. (4) leads to similar results. The normalized values are, for this case, elevation, z 0 = 25.96 m, distance, L = 36.51 m, and time t N = 0.0376. The efficiency index of this fit is slightly better, 0.99852. The results indicate that the linear diffusive sediment transport equation cannot be discarded by the curvature–gradient test suggested by Roering et al. (1999). The differences between the proposal of Roering et al. (1999) and the results shown here lay on the consideration of the steady state conditions. As Howard (1988) indicated, the concept of dynamic equilibrium is difficult to apply to natural landscapes. Fernandes and Dietrich (1997) estimated that the equilibrium conditions were approximately reached when the total sediment flux attained 90% of its value at the steady state condition, at what they called relaxation time. They concluded that most of the actual complex hillslopes may not have reached the equilibrium state. In the slope analysed above the relaxation time may be about t NR i 1, higher than the found value t N = 0.0376 as shown in Fig. 4. Therefore if steady state conditions do not apply, the curvature–gradient test may not be valid as an indicator of linear or nonlinear diffusive sediment transport equation.
Fig. 4. Normalized sediment flux at the bottom of the hillslope for the case of Fig. 3. The dashed line indicates the 90% reduction of the initial value. The relaxation time may be estimated from the intersection of the two curves as t NR = 1.
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
279
The main consequence of the results obtained here, the doubtful usefulness of the slope geometry test alone for identifying the nature of the erosive processes has been commented before by Armstrong (1987) and Furbish (2003). Armstrong (1987) pointed out that slope system may converge to similar characteristic forms as a result of different processes. However it might be possible to identify the different slope evolution processes obtaining additional information like the changes of the depth of the soil profile along the slope. Similar suggestion was formulated by Furbish (2003) in his analysis of the co-evolution of hillslope and soil. Furbish (2003) stated two important points: i) one cannot determine whether a land-surface profile represents steady or transient state based on geometry of the slope as the unique piece of information; and ii) it is not possible either to discriminate between linear or nonlinear sediment transport as the responsible processes for the evolution of the hillslope without additional information. Roering et al. (1999) used estimations of the denudation, uplift and soil production rates to support the statement that their landscape was possibly in approximate equilibrium. Therefore their analysis of the evolution of the slopes of the Oregon Coastal Range was correct. Nevertheless without this information, which is not always available, the geometric test alone is not sufficient to discard the linear sediment transport laws in the study of hillslopes. As Dietrich et al. (2003) pointed out the spatial variability in the slopes may obscure the linearity, with rock outcrops, tree mounds (Jime´nez-Hornero et al., 2004), soil variability and even the mixing and size-separation of granular material in coarse textured soils (e.g. Jaeger et al., 1996). Many erosion processes are better described by nonlinear diffusive equations (e.g. Kirkby, 1971; Howard, 1994). Nevertheless as the linear equation is simpler, and as far as it can give a perspective of the erosion problem it could not be simply rejected.
4. Conclusions The linear diffusive sediment transport equation is a simple solution to slope erosion problems which may be useful in many cases. The test of linearity of the diffusion equation based on the curvature–gradient diagram only has shown to be no longer valid where transient conditions are applicable. Additional information like the evolution of soil profile along slopes is required.
Acknowledgements The authors acknowledge the suggestions of one anonymous referee which improved this work. This research has been partly supported by the European Union, TERON project, FAIR3-CT96-1478, and the Spanish Government, project CAO 01-001-C4-01. F.J. Jime´nez-Hornero is grateful for the support from the Consejerı´a de Innovacio´n, Ciencia y Empresa, Junta de Andalucı´a (Ayudas para facilitar el Retorno de Investigadores a Centros de Investigacı´o´n y Universidades de Andalucı´a).
280
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
References Armstrong, A.R., 1987. Slopes, boundary conditions in the development of convexo–concave forms — some numerical experiments. Earth Surf. Process. Landf. 12, 17 – 30. Beven, K.J., 2000. Rainfall-Runoff Modeling: The Primer. John Wiley, Colchester. Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solids. Cambridge University Press, Cambridge. Culling, W.E.H., 1963. Soil creep and the development of hillside slopes. J. Geol. 71, 127 – 161. Dietrich, W.E., Bellugi, D.G., Sklar, L.S., Stock, J.D., Heimsath, A.M., Roering, J.J., 2003. Geomorphic transport laws for predicting landscape form and dynamics. In: Wilcock, P.R., Iverson, R.M. (Eds.), Prediction in Geomorphology. Amer. Geophys. Union, Washington, pp. 103 – 132. Farin, G., 1997. Curves and Surfaces for Computer-Aided Geometric Design. Academic Press, San Diego. Fernandes, N.F., Dietrich, W.E., 1997. Hillslope evolution by diffusive processes: the timescale for equilibrium adjustments. Water Resour. Res. 33, 1307 – 1318. Furbish, D.J., 2003. Using the dynamically couple behaviour of land-surface geometry and soil thickness in developing and testing hillslopes evolution model. In: Wilcock, P.R., Iverson, R.M. (Eds.), Prediction in Geomorphology. Amer. Geophys. Union, Washington, pp. 169 – 181. Furbish, D.J., Fagherazzi, S., 2001. Stability of creeping soil and implications for hillslope evolution. Water Resour. Res. 37, 2607 – 2618. Govers, G., Vandaele, K., Desmet, P., Poesen, J., Bunte, K., 1994. The role of tillage in soil redistribution on hillslopes. Eur. J. Soil Sci. 45, 469 – 478. Heimsath, A.M., Dietrich, W.E., Nishiizumi, K., Finkel, R.C., 1997. The soil production function and landscape equilibrium. Nature 388, 358 – 361. Hirano, M., 1968. A mathematical model for slope development. J. Geosci. 11, 13 – 52. Howard, A.D., 1988. Equilibrium models in geomorphology. In: Anderson, M.G. (Ed.), Modelling a Geomorphological System. John Wiley, New York, pp. 49 – 72. Howard, A.D., 1994. A detachment-limited model of drainage basin evolution. Water Resour. Res. 30, 2261 – 2285. Jaeger, H.M., Nagel, S.R., Behringer, R.P., 1996. The physics of granular materials. Phys. Today 49, 32 – 38. Jime´nez-Hornero, F.J., Gira´ldez, J.V., Laguna, A., 2004. Estimation of the role of obstacles in the downslope soil flow with a simple erosion model: the analytical solution and its approximation with the lattice Boltzmann model. Catena 57, 261 – 275. Kirkby, M.J., 1971. Hillslope process-response models based on the continuity equation. In: Brunsden, D. (Ed.), Slopes: Form and Process, Inst. Brit. Geogr. Spec. Pub., vol. 3, pp. 15 – 30. Lobb, D.A., Kachanoski, R.G., Miller, M.H., 1999. Tillage translocation and tillage erosion in the complex upland landscapes of southwestern Ontario. Can. Soil Tillage Res. 51, 189 – 209. Roering, J.J., Kichner, J.W., Dietrich, W.E., 1999. Evidence for nonlinear, diffusive transport on hillslopes and implications for landscape morphology. Water Resour. Res. 35, 853 – 870. Tucker, G.E., Slingerland, R., 1994. Erosional dynamics, flexural isostasy, and long-lived escarpments: A numerical modeling study. J. Geophys. Res. 99 (B6) 12: 229–243.
Evaluation of linear and nonlinear sediment transport equations using hillslope morphology Francisco J. Jime´nez-Hornero a, Ana Laguna b, Juan V. Gira´ldez a,* b
a Department of Agronomy, E.T.S.I.A.M., University of Cordoba, P.O. Box 3048, 14080 Cordoba, Spain Department of Applied Physics, E.T.S.I.A.M., University of Cordoba, P.O. Box 3048, 14080 Cordoba, Spain
Accepted 12 September 2005
Abstract Although some simple erosive processes like soil creep or tillage redistribution may be satisfactorily described by linear diffusive equations, the complexity of erosion phenomena requires the use of more complete nonlinear equations. Roering et al. [Roering, J.J., Kichner, J.W., Dietrich, W.E., 1999. Evidence for nonlinear, diffusive transport on hillslopes and implications for landscape morphology. Water Resour. Res., 35 853–870.] have proposed a single test, based on the relationship between the curvature and gradient of a hillslope, rather then using a linear diffusion equation. Nevertheless this test, based on steady state conditions, is not complete as it is shown in this work with a counter-example. The hillslope profile used by Roering et al. [Roering, J.J., Kichner, J.W., Dietrich, W.E., 1999. Evidence for nonlinear, diffusive transport on hillslopes and implications for landscape morphology. Water Resour. Res., 35 853–870.], (Fig. 2) can be generated either by a linear diffusion equation under transient conditions or a nonlinear diffusion equation under steady state conditions. Additional information on the soil profile changes would give a more complete interpretation of the hillslope evolution. D 2005 Elsevier B.V. All rights reserved. Keywords: Linear and nonlinear diffusive sediment transport equations; Steady state conditions
* Corresponding author. Tel.: +34 957 016057; fax: +34 957 016043. E-mail address: [email protected] (J.V. Gira´ldez). 0341-8162/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.catena.2005.09.001
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
273
1. Introduction The need for simple models to describe complex phenomena of soil erosion or landscape evolution has led to the use of geomorphic transport laws that are not well based. Such could be the case of the linear diffusive sediment transport on slopes which has been recently discussed by Dietrich et al. (2003) in a thorough review. Culling (1963) was one of the first erosion modeller adopting the linear dependence of solid particle flux with the slope gradient, for the process of soil creep. Dietrich et al. (2003) quoted two erosion studies where such a relation was observed, in two different soils of contrasting textural classes. Another interesting example of erosion described by a linear law is the mechanical or tillage erosion as reported by Govers et al. (1994). After careful measurements of the movement of metallic tracers located in vertical soil pits along slope transects, the particle displacement along the slope was proportional to the slope gradient. Linear diffusive sediment transport equations have been used in many studies in which their validity was assumed without experimental confirmation. Dietrich et al. (2003) criticized this use on the basis of three main points: i) the lack of physical bases of the equation; ii) its narrow field of applicability; and iii) the misfit between the conditions of validity and spatial scale of the models where they have been used. The second and third points are specially important. There are many erosion processes that cannot be properly described by linear diffusive law, as Howard (1994), among other authors, recognized. However the first point requires a closer look. In a previous paper of the authors (Roering et al., 1999) the consequences of linear diffusive sediment transport on the convexity of hillslopes were analysed. In addition to a simple test to assess the feasibility of both, linear and nonlinear equations were proposed. The purpose of this work is the discussion of the conditions under which such a proposal can be accepted. After posing the linear and nonlinear sediment transport equations proposed by Roering et al. (1999), it will be shown that the test for choosing between these equations depends strongly on whether the hillslopes can be assumed to be in steady state or not. Here we show that a linear transport law applied to a transient hillslope can produce similar profiles to the nonlinear law for the steady state case.
2. Methods 2.1. Linear and nonlinear transport equations The description of hillslope evolution implies the consideration of several processes (e.g. Tucker and Slingerland, 1994, Table 1), affecting to the rock and to the sediment, regolith or soil. Furbish and Fagherazzi (2001) have reduced these equations to the conservation of mass for rock and soil. Nevertheless a simple equation like the one proposed by Dietrich et al. (2003) may be used for the purposes of the present work, written in terms of solid mass as, qs
Bz Bh ¼ qr u p q s s p þ q s ¼ qr up qs rd q˜ s Bt Bt
ð1Þ
274
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
where z is the elevation of the ground, [L], t is time, [T], u p is the uplift rate, [LT 1], s p is the sediment or soil production rate, [LT 1] (e.g. Heimsath et al., 1997), h is the soil thickness, [L], q r and q s are the respective bulk density of the rock and soil or sediment, [ML 3], and q s is the sediment flow rate, [L 2T 1]. The combination of the two last terms in the intermediate expression follows from Dietrich et al. (2003, Eq. (2)). The sediment transport rate may be written either as linearly dependent on the slope qs ¼ klin rz
ð2Þ
where k lin is a diffusion coefficient, [L 2T 1], or as a more complex function of the slope, qs ¼ f ðrzÞ
ð3Þ
The linear diffusion equation is obtained as, Bz ¼ U Dlin r2 z Bt
ð4Þ
with U = u pq r / q s as the uplift term, [LT 1], and D = k lin / q s as a diffusion coefficient, [LT 1]. Given the reduced values assumed by the uplift rate some authors (e.g. Fernandes and Dietrich, 1997, Eq. (4)) usually neglect this term. This equation describes fairly well most of the mechanical erosion experiments found in the literature (Lobb et al., 1999). Roering et al. (1999) presented a nonlinear diffusion equation for sediment transport on slopes based on a general expression for q s, relating it to the velocity of the downslope movement, m, [LT 1] q˜ s ¼
V m˜ A
ð5Þ
with V / A as the volume of mobile sediment per unit area along the slope, [L]. From this expression they deduced a final form for Eq. (3), q˜ s ¼
knol rz 2 1 jrzjSc1
ð6Þ
with k nol as the nonlinear diffusivity, [L 2T 1], and S c as a threshold slope. A similar form of this equation was adopted by Howard (1994). In order to compare Eqs. (2) and (6), Roering et al. (1999) assumed that the rock uplift rate may be similar to the surface erosion rate in (1), leading to a steady state, the dynamic equilibrium of Gilbert. In such a case (1) becomes
qr up ¼ rd q˜ s qs
ð7Þ
For the linear case they derived an equation
qr up ¼ r2 z qs klin
ð8Þ
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
275
where j 2z is taken as the curvature of the slope (e.g. Farin, 1997, § 11.3), whereas the nonlinear case leads to
qr up rz ¼ rd 2 qs knol 1 jrzjS 1
ð9Þ
c
Eqs. (8) and (9) are the alternative linear and nonlinear diffusive erosion equations for Roering et al. (1999), hereafter named RKD model. Note that both of them are deduced assuming steady state conditions. Simplifying the equations to a two-dimensional case in the plane x–z, with x as the distance allows an easy integration of the equations. The results are shown in Fig. 1. If the value of the slope ratio, S / S c, is small the nonlinearity disappears as the authors commented. The distinction between both models is mainly based on the relation between the second derivative of the elevation with distance and the gradient. Roering et al., (1999; Fig. 2) compare the gradient and curvature with data of a real slope measured in the Oregon Coastal Range. 2.2. Transient linear diffusive transport equations The linear diffusive sediment transport equation, (4), may be integrated for the simple case of an initially uniform slope, z = z 0(1 x / L), with a fixed elevation at the bottom of the slope, and no flow conditions at the divide. Armstrong (1987) discussed several possible lower boundary conditions. The condition adopted here implies a balance between inflow and outflow of solid particles at such a point.
Fig. 1. Different behaviour of linear and nonlinear diffusive models according to Roering et al. (1999).
276
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
The initial elevation is z 0, and total distance is L. The boundary conditions may be written as Bz ¼0 Bx z¼0
x¼0
ð10aÞ
t N0
x¼L
ð10bÞ
t N0
The variables are normalized by L. xN ¼
x L
zN ¼
z L
tN ¼
tDlin L2
ð11Þ
Fig. 2. Relationships between elevation and distance, left, and curvature and gradient for the linear diffusive model, right, with free flow condition downstream at a given time.
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
277
The analytical solution is given by Carslaw and Jaeger (1959, § 3.3.14) with an additional term accounting for the uplift rate assumed to be constant (e.g. Hirano, 1968). " 2 # l 8 X 1 2n þ 1 2n þ 1 p px exp t zN ¼ U N t n þ 2 ð12Þ cos N N p n¼0 ð2n þ 1Þ2 2 2 where the normalized uplift rate, U N, is, UN ¼
L U Dlin
ð13Þ
The relationships between elevation, distance, gradient and curvature are shown in Fig. 2 as an example. The linear diffusive sediment transport equation does not necessarily yield a straight line in the gradient–curvature plot as implied in the RKD model.
3. Results and discussion A simple optimisation technique allows the identification of the linear diffusive model parameters, z 0, L, U N and normalized time, t N, through a good fit of (12) to the topographic data of Roering et al. (1999). The fit shown in Fig. 3 corresponds to the parameters elevation, z 0 = 26.13 m, distance, L = 36.41 m, and normalized uplift rate U N = 1.95 d 10 2. The fit has an efficiency index of Nash and Sutcliffe (e.g. Beven, 2000) of 0.99850. The
Fig. 3. Fit of the linear diffusive model to the Roering et al. (1999, Fig. 2) data. The maximum elevation is 26.13 m, the maximum length is 36.52 m, the normalized uplift rate is 0.0195 and the corresponding normalized time, t N = 0.0374. The downslope elevation was kept constant.
278
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
normalized time t N = 0.0400 implies a real time in the range 10.0–16.6 ka (thousand of years) if the value of D lin is taken between 5.3 to 3.2 d 10 3 m2 yr 1 as Roering et al. (1999) proposed later in their paper. These values for D lin are within the estimates gathered by Fernandes and Dietrich (1997, Table 1). The absolute uplift rate is between 2.84 d 10 3 and 1.75 d 10 3 mm yr 1 lower than the 0.1–0.2 mm yr 1 range suggested by Roering et al. (1999) for the area. Neglecting the uplift term in Eq. (4) leads to similar results. The normalized values are, for this case, elevation, z 0 = 25.96 m, distance, L = 36.51 m, and time t N = 0.0376. The efficiency index of this fit is slightly better, 0.99852. The results indicate that the linear diffusive sediment transport equation cannot be discarded by the curvature–gradient test suggested by Roering et al. (1999). The differences between the proposal of Roering et al. (1999) and the results shown here lay on the consideration of the steady state conditions. As Howard (1988) indicated, the concept of dynamic equilibrium is difficult to apply to natural landscapes. Fernandes and Dietrich (1997) estimated that the equilibrium conditions were approximately reached when the total sediment flux attained 90% of its value at the steady state condition, at what they called relaxation time. They concluded that most of the actual complex hillslopes may not have reached the equilibrium state. In the slope analysed above the relaxation time may be about t NR i 1, higher than the found value t N = 0.0376 as shown in Fig. 4. Therefore if steady state conditions do not apply, the curvature–gradient test may not be valid as an indicator of linear or nonlinear diffusive sediment transport equation.
Fig. 4. Normalized sediment flux at the bottom of the hillslope for the case of Fig. 3. The dashed line indicates the 90% reduction of the initial value. The relaxation time may be estimated from the intersection of the two curves as t NR = 1.
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
279
The main consequence of the results obtained here, the doubtful usefulness of the slope geometry test alone for identifying the nature of the erosive processes has been commented before by Armstrong (1987) and Furbish (2003). Armstrong (1987) pointed out that slope system may converge to similar characteristic forms as a result of different processes. However it might be possible to identify the different slope evolution processes obtaining additional information like the changes of the depth of the soil profile along the slope. Similar suggestion was formulated by Furbish (2003) in his analysis of the co-evolution of hillslope and soil. Furbish (2003) stated two important points: i) one cannot determine whether a land-surface profile represents steady or transient state based on geometry of the slope as the unique piece of information; and ii) it is not possible either to discriminate between linear or nonlinear sediment transport as the responsible processes for the evolution of the hillslope without additional information. Roering et al. (1999) used estimations of the denudation, uplift and soil production rates to support the statement that their landscape was possibly in approximate equilibrium. Therefore their analysis of the evolution of the slopes of the Oregon Coastal Range was correct. Nevertheless without this information, which is not always available, the geometric test alone is not sufficient to discard the linear sediment transport laws in the study of hillslopes. As Dietrich et al. (2003) pointed out the spatial variability in the slopes may obscure the linearity, with rock outcrops, tree mounds (Jime´nez-Hornero et al., 2004), soil variability and even the mixing and size-separation of granular material in coarse textured soils (e.g. Jaeger et al., 1996). Many erosion processes are better described by nonlinear diffusive equations (e.g. Kirkby, 1971; Howard, 1994). Nevertheless as the linear equation is simpler, and as far as it can give a perspective of the erosion problem it could not be simply rejected.
4. Conclusions The linear diffusive sediment transport equation is a simple solution to slope erosion problems which may be useful in many cases. The test of linearity of the diffusion equation based on the curvature–gradient diagram only has shown to be no longer valid where transient conditions are applicable. Additional information like the evolution of soil profile along slopes is required.
Acknowledgements The authors acknowledge the suggestions of one anonymous referee which improved this work. This research has been partly supported by the European Union, TERON project, FAIR3-CT96-1478, and the Spanish Government, project CAO 01-001-C4-01. F.J. Jime´nez-Hornero is grateful for the support from the Consejerı´a de Innovacio´n, Ciencia y Empresa, Junta de Andalucı´a (Ayudas para facilitar el Retorno de Investigadores a Centros de Investigacı´o´n y Universidades de Andalucı´a).
280
F.J. Jime´nez-Hornero et al. / Catena 64 (2005) 272–280
References Armstrong, A.R., 1987. Slopes, boundary conditions in the development of convexo–concave forms — some numerical experiments. Earth Surf. Process. Landf. 12, 17 – 30. Beven, K.J., 2000. Rainfall-Runoff Modeling: The Primer. John Wiley, Colchester. Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solids. Cambridge University Press, Cambridge. Culling, W.E.H., 1963. Soil creep and the development of hillside slopes. J. Geol. 71, 127 – 161. Dietrich, W.E., Bellugi, D.G., Sklar, L.S., Stock, J.D., Heimsath, A.M., Roering, J.J., 2003. Geomorphic transport laws for predicting landscape form and dynamics. In: Wilcock, P.R., Iverson, R.M. (Eds.), Prediction in Geomorphology. Amer. Geophys. Union, Washington, pp. 103 – 132. Farin, G., 1997. Curves and Surfaces for Computer-Aided Geometric Design. Academic Press, San Diego. Fernandes, N.F., Dietrich, W.E., 1997. Hillslope evolution by diffusive processes: the timescale for equilibrium adjustments. Water Resour. Res. 33, 1307 – 1318. Furbish, D.J., 2003. Using the dynamically couple behaviour of land-surface geometry and soil thickness in developing and testing hillslopes evolution model. In: Wilcock, P.R., Iverson, R.M. (Eds.), Prediction in Geomorphology. Amer. Geophys. Union, Washington, pp. 169 – 181. Furbish, D.J., Fagherazzi, S., 2001. Stability of creeping soil and implications for hillslope evolution. Water Resour. Res. 37, 2607 – 2618. Govers, G., Vandaele, K., Desmet, P., Poesen, J., Bunte, K., 1994. The role of tillage in soil redistribution on hillslopes. Eur. J. Soil Sci. 45, 469 – 478. Heimsath, A.M., Dietrich, W.E., Nishiizumi, K., Finkel, R.C., 1997. The soil production function and landscape equilibrium. Nature 388, 358 – 361. Hirano, M., 1968. A mathematical model for slope development. J. Geosci. 11, 13 – 52. Howard, A.D., 1988. Equilibrium models in geomorphology. In: Anderson, M.G. (Ed.), Modelling a Geomorphological System. John Wiley, New York, pp. 49 – 72. Howard, A.D., 1994. A detachment-limited model of drainage basin evolution. Water Resour. Res. 30, 2261 – 2285. Jaeger, H.M., Nagel, S.R., Behringer, R.P., 1996. The physics of granular materials. Phys. Today 49, 32 – 38. Jime´nez-Hornero, F.J., Gira´ldez, J.V., Laguna, A., 2004. Estimation of the role of obstacles in the downslope soil flow with a simple erosion model: the analytical solution and its approximation with the lattice Boltzmann model. Catena 57, 261 – 275. Kirkby, M.J., 1971. Hillslope process-response models based on the continuity equation. In: Brunsden, D. (Ed.), Slopes: Form and Process, Inst. Brit. Geogr. Spec. Pub., vol. 3, pp. 15 – 30. Lobb, D.A., Kachanoski, R.G., Miller, M.H., 1999. Tillage translocation and tillage erosion in the complex upland landscapes of southwestern Ontario. Can. Soil Tillage Res. 51, 189 – 209. Roering, J.J., Kichner, J.W., Dietrich, W.E., 1999. Evidence for nonlinear, diffusive transport on hillslopes and implications for landscape morphology. Water Resour. Res. 35, 853 – 870. Tucker, G.E., Slingerland, R., 1994. Erosional dynamics, flexural isostasy, and long-lived escarpments: A numerical modeling study. J. Geophys. Res. 99 (B6) 12: 229–243.