## Abstract

A model associated with the formation of sedimentary ocean deltas is presented. This model is a generalized one-dimensional Stefan problem bounded by two moving boundaries, the shoreline and the alluvial-bedrock transition. The sediment transport is a non-linear diffusive process; the diffusivity modeled as a power law of the fluvial slope. Dimensional analysis shows that the first order behavior of the moving boundaries is determined by the dimensionless parameter 0 ≤ R_{a b} ≤ 1-the ratio of the fluvial slope to bedrock slope at the alluvial-bedrock transition. A similarity form of the governing equations is derived and a solution that tracks the boundaries obtained via the use of a numerical ODE solver; in the cases where the exponent θ in the diffusivity model is zero (linear diffusion) or infinite, closed from solutions are found. For the full range of the diffusivity exponents, 0 ≤ θ → ∞, the similarity solution shows that when R_{a b} < 0.4 there is no distinction in the predicted speeds of the moving boundaries. Further, within the range of physically meaningful values of the diffusivity exponent, i.e., 0 ≤ θ ∼ 2, reasonable agreement in predictions extents up to R_{a b} ∼ 0.7. In addition to the similarity solution a fixed grid enthalpy like solution is also proposed; predictions obtained with this solution closely match those obtained with the similarity solution.

Original language | English (US) |
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Pages (from-to) | 538-549 |

Number of pages | 12 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 366 |

Issue number | 2 |

DOIs | |

State | Published - Jun 15 2010 |

### Bibliographical note

Funding Information:This work was supported by the STC program of the National Science Foundation via the National Center for Earth-surface Dynamics under the agreement Number EAR-0120914. The authors of grateful for fruitful discussion and input from Chris Paola, University of Minnesota; John Swenson, University of Minnesota Duluth; and Wonsuk Kim, University of Texas.

## Keywords

- Dual moving boundaries
- Enthalpy solution
- Sediment delta
- Similarity solution
- Stefan problem