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Mathematical Analysis of a Model of River Channel Formation


Díaz Díaz, Jesús Ildefonso and Fowler, A.C. and Muñoz, Ana Isabel and Schiavi, E. (2008) Mathematical Analysis of a Model of River Channel Formation. Pure and Applied Geophysics, 165 (8). pp. 1663-1682. ISSN 0033-4553

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The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin.

Item Type:Article
Uncontrolled Keywords:drainage-basin evolution; driven theory; stability; inception; equation; river models; landscape evolution; nonlinear parabolic equations; free boundaries; singular free boundary flux
Subjects:Sciences > Mathematics > Differential equations
ID Code:15177

BENILAN, P. (1978), Operateurs accretifs et semigroups dans les espaces Lp, Functional Analysis and Numerical Analysis, France-Japan Seminar (H. Fujita, ed.), Japan Society for the Promotion of Science, Tokio, pp. 15–53.

BIRNIR, B., SMITH, T. R., and MERCHANT, G. E. (2001), The scaling of fluvial landscapes. Comput. Geosci. 27, 1189–1216.

CAFFARELLI, L. A., LEDERMAN, C., and WOLANSKI, N. (1997), Pointwise and viscosity solutions for the limit of a two-phase parabolic singular perturbation problem, Indiana Univ. Math. J. 46(3), 719–740.

CAFFARELLI, L. A. and VÁZQUEZ, J. L. (1995), A free-boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc. 347(2), 411–441.

DÍAZ, J. I., FOWLER, A. C., MUÑOZ, A. I., and SCHIAVI, E., Article in preparation.

DÍAZ, J. I., PADIAL, J. F., and RAKOTOSON, J. M. (2007), On some Bernouilli free boundary type problems for general elliptic operators, Proc. Roy. Soc. Edimburgh 137A, 895–911.

DÍAZ, J. I. and VRABIE, I. (1989), Proprietés de compacité de l’opérateur de Green généralisé pour l’équation des milieux poreux, Comptes Rendus Acad. Sciences, París 309, Série I, 221–223.

EVANS, L. C. and GARIEPY, R. F., Measure Theory and Fine Properties of Functions (Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992).

FOWLER, A. C., KOPTEVA, N., and OAKLEY, C. (2007), The formation of river channels, SIAM J. Appl. Math. 67, 1016–1040.

HOWARD, A. D. (1994), A detachment-limited model of drainage basin evolution, Water Resour. Res. 30, 2261–2285.

IZUMI, N. and PARKER, G. (1995), Inception and channellization and drainage basin formation: Upstream-driven theory, J. Fluid Mech. 283, 341–363.

IZUMI, N. and PARKER, G. (2000), Linear stability analysis of channel inception: Downstream-driven theory, J. Fluid Mech. 419, 239–262.

KALASHNIKOV, A. S. (1987), Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk 42, 135–176.

KRAMER, S. and MARDER, M. (1992), Evolution of river networks, Phys. Rev. Lett. 68, 205–208.

LOEWENHERZ, D. S. (1991), Stability and the initiation of channelized surface drainage: A reassessment of the short wavelength limit, J. Geophys. Res. 96, 8453–8464.

LOEWENHERZ-LAWRENCE, D. S. (1994), Hydrodynamic description for advective sediment transport processes and rill initiation, Water Resour. Res. 30, 3203–3212.

NAZARET, B. (2001), Heat flow for extremal functions in some subcritical Sobolev inequalities, Appl. Anal. 80, 95–105.

MEYER-PETER, E. and MU¨ LLER, R. (1948), Formulas for bed-load transport, Proc. Int. Assoc. Hydraul. Res., 3rd Annual Conference, Stockholm, 39–64.

PARKER, G. (1978), Self-formed straight rivers with equilibrium banks and mobile bed, Part 1. The sand-silt river, J. Fluid Mech. 89, 109–125.

SAMARSKI, A. A., GALAKTIONOV, V. A., KURDYUMOV, S. P., and MIKHAILOV, A. P., Blow-up in quasilinear parabolic equations (Walter de Gruyter, Berlin, 1995).

SMITH, T. R., BIRNIR, B., and MERCHANT, G. E. (1997), Towards an elementary theory of drainage basin evolution: II. A computational evaluation, Comput. Geosci. 23, 823–849.

SMITH, T. R. and BRETHERTON, F. P. (1972), Stability and the conservation of mass in drainage basin evolution, Water Resour. Res. 8, 11, 1506–1529.

TUCKER, G. E. and SLINGERLAND, R. L. (1994), Erosional dynamics, flexural isostasy, and long-lived escarpments: A numerical modeling study, J. Geophys. Res. 99, 12.229–12.243.

VRABIE, I. I., Compactness Methods for Nonlinear Evolutions (Pitman Longman, London, 1987).

WILLGOOSE, G., BRAS, R. L., and RODRÍGUEZ-ITURBE, I. (1991), A coupled channel network growth and hillslope evolution model: I. Theory, Water Resour. Res. 27, 1671–1684.

Deposited On:10 May 2012 08:28
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