Landscapes evolve towards surfaces with complex networks of channels and ridges in response to climatic and tectonic forcing. Here, we analyse variational principles giving rise to minimalist models of landscape evolution as a system of partial differential equations that capture the essential dynamics of sediment and water balances. Our results show that in the absence of diffusive soil transport the steady-state surface extremizes the average domain elevation. Depending on the exponent m of the specific drainage area in the erosion term, the critical surfaces are either minima (0 < m < 1) or maxima (m > 1), with m = 1 corresponding to a saddle point. We establish a connection between landscape evolution models and optimal channel networks and elucidate the role of diffusion in the governing variational principles.

中文翻译：

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景观高程和排水网络的变化分析


响应气候和构造强迫，景观向具有复杂的河道和山脊网络的表面演化。在这里，我们分析了变分原理，产生了景观演化的简约模型，作为一个偏微分方程系统，捕捉了沉积物和水平衡的基本动态。我们的结果表明，在没有扩散土壤传输的情况下，稳态表面使平均域高程达到极值。根据侵蚀项中特定流域面积的指数 m，临界表面要么是最小值 (0 < m < 1)，要么是最大值 (m > 1)，其中 m = 1 对应于鞍点。我们在景观演化模型和最优通道网络之间建立了联系，并阐明了扩散在控制变分原理中的作用。