River channels are among the most common landscape features on Earth. An essential characteristic of channels is sinuosity: their tendency to take a circuitous path, which is quantified as along-stream length divided by straight-line length. River sinuosity is interpreted as a characteristic that either forms randomly at channel inception or develops over time as meander bends migrate. Studies tend to assume the latter and thus have used river sinuosity as a proxy for both modern and ancient environmental factors including climate, tectonics, vegetation, and geologic structure. But no quantitative criterion for planform expression has distinguished between random, initial sinuosity and that developed by ordered growth through channel migration. This ambiguity calls into question the utility of river sinuosity for understanding Earth's history. We propose a quantitative framework to reconcile these competing explanations for river sinuosity. Using a coupled analysis of modeled and natural channels, we show that while a majority of observed sinuosity is consistent with randomness and limited channel migration, rivers with sinuosity ≥1.5 likely formed their geometry through sustained, ordered growth due to channel migration. This criterion frames a null hypothesis for river sinuosity that can be applied to evaluate the significance of environmental interpretations in landscapes shaped by rivers. The quantitative link between sinuosity and channel migration further informs strategies for preservation and restoration of riparian habitat and guides predictions of fluvial deposits in the rock record and in remotely sensed environments from the seafloor to planetary surfaces.Single-thread channels abound on planetary surfaces with varied fluids and substrates (Komatsu and Baker, 1996; Karlstrom et al., 2013; Allen and Pavelsky, 2018; Fig. 1). All natural channels, including rivers, invariably deviate from straight-line paths, with typical sinuosity values (ratio of along-channel to straight-line distance) up to ∼3 (Leopold and Wolman, 1957; Howard and Hemberger, 1991). Sinuosity is the most widely used statistic to describe river planform geometry and has been interpreted as a proxy for environmental processes including climate (Stark et al., 2010), the stabilizing effects of vegetation and fine sediment (Braudrick et al., 2009; Davies and Gibling, 2010; Lapôtre et al., 2019), tectonics, geologic structure, and lithology (Harden, 1990; Johnson and Finnegan, 2015).However, to provide a window into past landscape conditions, sinuosity must respond to those conditions over time. Some rivers show abundant evidence for sinuosity evolution as recorded by direct observations or geologic markers like cutoffs (Fig. 1A), scroll bars, terraces, and eroded valley margins (Hooke, 2013). Yet many rivers lack these indicators of channel migration (Fig. 1B), suggesting that river sinuosity may, to some degree, form randomly at channel inception in response to intrinsic irregularities of topography (Mueller, 1968). Extreme examples of channelization also demonstrate extrinsic controls on sinuosity such as crevasses that steer supraglacial meanders (Fig. 1C) and sinuous volcanic channels that show little evidence of migration (Fig. 1D). Ambiguity in the origin of sinuosity calls into question its utility for interpreting environmental history. For example, a classic debate regarding bedrock-bound meanders centers on whether high sinuosity due to channel migration is a relic of previous alluvial states (Winslow, 1893). We consider a broader question: for any channel, under what conditions is sinuosity a diagnostic of channel migration? We propose that the magnitude of sinuosity can distinguish whether the channel form is consistent with randomness or has instead developed mainly through ordered growth by channel migration.For single-thread rivers, theoretical models describe two end-member scenarios for the origin of sinuosity, which we define as random models and migration models, respectively. Random models treat the path of the centerline (the line equidistant between each of the river banks) as a random walk (Langbein and Leopold, 1966). In contrast, migration models explain sinuosity development by explicitly considering hydrodynamic effects that erode and construct river banks and grow meander bends (Camporeale et al., 2007). Neither class of models fully captures river forms. Geometric analyses show that compared to natural channels, centerlines made by pure random walks change direction too quickly and form tangled loops rather than alternating bends (Ferguson, 1976); channel-migration models generate bends, but they are overly regular and sinuous (Howard and Hemberger, 1991). Therefore, to better mimic the geometry of natural channels, a model requires spatial memory at relatively small length scales but irregularity at larger scales.Many natural time series, such as climatological data, can likewise be described by a combination of inertia and randomness. A first-order autoregressive process (AR-1) is recognized as the simplest possible model—and the default expectation—for any geophysical process that is both dependent on previous states and subject to random disturbances (Roe, 2009). Ferguson (1976) showed that compared to an AR-1 model, a second-order autoregressive model (AR-2) tempers high-frequency variations while still allowing overall bend shapes to vary. That means the AR-2 model represents the simplest model for sinuosity that yields shapes comparable to natural channels. The AR-2 model is strictly geometric and is therefore sufficiently generic to describe channels whose formation processes may differ in detail. For natural channels that lack indicators of channel migration, we propose that spatially correlated randomness as expressed by the AR-2 model represents the null hypothesis for river sinuosity and can be used to test the significance of environmental interpretations.For a global data set of natural channels, we calculated sinuosity and fit AR-2 model parameters to constrain their ranges (Fig. S1 in the Supplemental Material). We then used a suite of model runs to estimate the magnitude of sinuosity (S) that can be explained with Equation 1 using a systematic parameter sweep for b1, b2, and σ. For each parameter combination, we analyzed 100 replicates with different disturbance series (ϵ) to statistically characterize model randomness (Figs. S2 and S3). Because in some model runs the channel crossed itself, we treated each self-intersection as a cutoff loop and removed it to maintain a simplified, continuous centerline. For each set of replicates, we calculated the mean and standard deviation of sinuosity of the simplified centerlines and the average length of the original centerlines bound in self-intersecting loops.The suite of model runs shows that both sinuosity and the degree of self-intersection vary systematically with b1, b2, and σ (Fig. 2A). Higher values of b1 and σ increase sinuosity, with mean values typically <1.5. The inherent randomness in the model causes sinuosity to vary across the replicate simulations (Fig. 2B). For a fraction of the replicates, S is >1.5, particularly where self-intersection is common or σ 0.3 (Fig. 2C); the latter rarely occurs in nature (Fig. S1). These model results imply that randomly generated sinuosity rarely exceeds 1.5 for channels that are both geometrically realistic and have not migrated. We hypothesize that a channel typically only reaches S > 1.5 by migrating.The migration model further shows that bends in initially random channels with different sinuosity follow offset trajectories for sinuosity growth through channel migration (Fig. 3C). These cases arrive, at different dimensionless times, at a critical sinuosity that suggests channel migration (Sc = 1.5). As expected, the dimensionless time to reach the critical sinuosity (tc*) generally decreases with increasing initial sinuosity (Fig. 3D). Moreover, tc* is <15 in all cases, which suggests an upper bound on the amount of channel migration that must occur before the centerline conclusively reflects channel migration. In other words, the initial, random sinuosity of the channel is overwritten over a finite time interval.For a global data set of river planforms, sinuosity mostly varies from 1 to 2.5 (Fig. 4A). We fit the AR-2 model to each observed planform to find the corresponding distribution of sinuosity values predicted by the model. For model cases with no self-intersection in the original centerline, the probability of the random sinuosity exceeding the observed sinuosity (Sobs) is significant for Sobs < 1.2 but declines sharply with increasing Sobs (Fig. 4B). If cases with self-intersection in the original centerline are considered, the random model accounts for a greater range of Sobs but is unlikely to account for Sobs > 2 (Fig. 4C). A separate data set of channels with documented migration (Fig. 4D) shows that migration favors higher sinuosity values, including S > 1.5 for all cases in the Andean foreland with especially high migration rates (>10% wc per year; Sylvester et al., 2019). Substituting tc* = 15 and these observed migration rates in Equation 2, for the Andean foreland rivers we estimate that the time scale to critical sinuosity (Sc = 1.5)—the clear distinction from randomness in the planform—is tc ≈ 150 years.We used a null-hypothesis test to estimate the role of spatially correlated randomness in river channel sinuosity (S) and show that a random model typically produces S < 1.5 (Fig. 2A). The low probability of exceeding this critical value (Sc = 1.5; Fig. 2C) indicates that for channels with S > Sc, the null hypothesis can be rejected, and that non-random, ordered channel migration is a necessary condition for the sinuosity observed. The converse (S < Sc)—not rejecting the null hypothesis—does not imply that randomness fully explains the observed sinuosity. Migrating channels with S < Sc are relatively common (Fig. 4D); relatively low sinuosity in migrating channels could result from slow channel migration and/or limited evolution time (Equation 2) or extrinsic constraints. Thus, although the sinuosity criterion cannot rule out migration if S < Sc, the metric does identify cases in which channel migration is definitive (S > Sc).A quantitative signature of migration in river sinuosity lends this common characteristic further significance for interpreting landscape history. Sinuosity in natural channels reflects neither pure randomness nor universal channel migration but rather exists on a continuum between the two. To borrow from Shakespeare's Twelfth Night, some channels are born sinuous (Fig. 1D); some achieve further sinuosity, through channel migration (Fig. 1A); and some have sinuosity thrust upon them, by faulting, valley shape, or other non-random topographic control (Fig. 1C). Random, initial sinuosity is embellished to a degree determined by the rate and duration of channel migration (Fig. 3C). Spatial correlation should be considered for initial planform geometry in numerical models of channel migration, as this geometry and its persistence through time (Fig. 3D) are more realistic than the uncorrelated, ephemeral noise that is most often used (e.g., Frascati and Lanzoni, 2010). At quasi-steady-state, such channel-migration models can develop sinuosity (S > 3.5) that exceeds typical observations (Fig. 4A; Howard, 1996; Frascati and Lanzoni, 2010). Therefore, in the absence of evidence indicating channel migration, spatially correlated randomness should represent the null hypothesis for sinuosity.In rivers characterized by channel migration and non-local avulsions, a given channel may express the two regimes of the randomness-order continuum both in space and over time (Valenza et al., 2020). When a river avulses to create a diversion across its own floodplain, it may reoccupy relict channels and/or cut new ones. An avulsing channel may therefore simultaneously form new reaches guided by random landscape disturbances and also inherit the bends of an antecedent, migrating channel. As the planform of the avulsion channel evolves with bend migration, indicators of its randomness are overwritten.Our findings can inform interpretations of environmental forcing from the sinuosity of natural channels. For example, Stark et al. (2010) interpreted climate forcing from a spatial trend in sinuosity for mixed bedrock-alluvial rivers with reported sinuosity S < 1.5, which is the critical value that we identify. Our analysis suggests that using sinuosity as a proxy for the degree of channel migration should be done with caution for rivers with sinuosity below the critical value; future analyses could account for the role of random sinuosity as a null hypothesis. Our results suggest that river sinuosity could, however, be used to infer alluvial sediment deposited by channel migration in a channel belt many times wider than the river itself (Jobe et al., 2016). This approach could inform predictions for stratigraphic architecture in cases where channel belts are not directly observable as in some seismic images of marine sediments (Pirmez and Imran, 2003) and on planetary surfaces (Burr et al., 2013). Finally, these results may apply to optimizing river restoration projects, which are commonly designed to match idealized meandering forms (Kondolf, 2006). Our analysis shows that for cases where migration by the original channel exhibited minimal migration, restored channels should have low sinuosity (S < 1.5). Meanwhile, natural, rapidly migrating channels drive forest disturbance and primary succession that may contribute to biodiversity (Salo et al., 1986). These observations suggest that high-sinuosity (S >>2) river corridors, which are likely dominated by migration and relatively rare (Figs. 4A and Fig. 4D), should be prioritized for conservation.We thank Samuel Johnstone and an anonymous reviewer for helpful comments, and Roman DiBiase, Douglas Edmonds, and Chris Paola for discussions. The U.S. National Science Foundation (grants EAR-1823530 and EAR-1246761), the British Society for Geomorphology, and Los Alamos National Laboratory (LDRD-20170668PRD1) supported this work. Data and code from this study are available at the University of Virginia's (USA) data repository LibraData (https://doi.org/10.18130/V3/TRTTIS) and at Github (https://github.com/alimaye/AR2-sinuosity), respectively.

中文翻译：

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河流曲折描述了随机性和有序增长之间的连续性

河道是地球上最常见的景观特征之一。渠道的一个基本特征是曲折性：它们倾向于采用迂回路径，这被量化为沿流长度除以直线长度。河流曲折被解释为一种特征，它要么在河道开始时随机形成，要么随着曲折迁移而随着时间的推移而发展。研究倾向于假设后者，因此使用河流曲折度作为现代和古代环境因素的代理，包括气候、构造、植被和地质结构。但是，平面表达的定量标准没有区分随机的初始曲折度和通过通道迁移的有序增长形成的曲折度。这种模糊性使人们对河流曲折度对理解地球历史的效用提出质疑。我们提出了一个定量框架来协调这些对河流曲折的相互竞争的解释。使用对模拟和自然通道的耦合分析，我们表明，虽然大多数观察到的曲率与随机性和有限的通道迁移一致，但曲率 ≥1.5 的河流可能通过通道迁移导致的持续有序增长形成其几何形状。该标准构建了河流曲折度的零假设，可用于评估河流塑造的景观中环境解释的重要性。曲折度和河道迁移之间的定量联系进一步为河岸栖息地的保护和恢复策略提供了信息，并指导了岩石记录和从海底到行星表面的遥感环境中河流沉积物的预测。单线通道在具有各种流体和基质的行星表面上比比皆是（Komatsu 和 Baker，1996 年；Karlstrom 等人，2013 年；Allen 和 Pavelsky，2018 年；图 1）。所有自然河道，包括河流，总是偏离直线路径，典型的曲率值（沿河道与直线距离的比率）高达 ~3（Leopold 和 Wolman，1957 年；Howard 和 Hemberger，1991 年）。曲率是描述河流平面几何形状的最广泛使用的统计数据，并被解释为环境过程的代表，包括气候（Stark 等，2010）、植被和细粒沉积物的稳定作用（Braudrick 等，2009；Davies和 Gibling，2010 年；Lapôtre 等人，2019 年），构造、地质结构和岩性（Harden，1990 年；Johnson 和 Finnegan，2015 年）。但是，为了提供一个了解过去景观条件的窗口，曲折必须随着时间的推移对这些条件做出反应。一些河流显示出丰富的曲折演化证据，如直接观察或地质标记（如边界（图 1A）、滚动条、阶地和侵蚀谷边缘）所记录的那样（胡克，2013 年）。然而，许多河流缺乏这些渠道迁移的指标（图 1B），这表明河流曲折可能在某种程度上在渠道开始时随机形成，以响应地形的内在不规则性（Mueller，1968）。通道化的极端例子也证明了对曲折度的外在控制，例如引导冰上曲流的裂缝（图 1C）和几乎没有迁移证据的曲折火山通道（图 1D）。曲折起源的模糊性使其在解释环境历史方面的效用受到质疑。例如，关于基岩曲折的经典辩论集中在由河道迁移引起的高曲折度是否是先前冲积状态的遗迹（Winslow，1893 年）。我们考虑一个更广泛的问题：对于任何渠道，在什么条件下曲折是渠道迁移的诊断？我们提出，曲折度的大小可以区分河道形态是与随机性一致还是主要通过河道迁移的有序增长而发展。对于单线河流，理论模型描述了曲折起源的两种端元情景，即我们分别定义为随机模型和迁移模型。随机模型将中心线的路径（每条河岸之间的等距线）视为随机游走（Langbein 和 Leopold，1966 年）。相比之下，迁移模型通过明确考虑侵蚀和建造河岸以及形成曲折的水动力效应来解释曲折的发展（Camporeale 等人，2007 年）。两类模型都不能完全捕捉河流形态。几何分析表明，与自然通道相比，由纯随机游走形成的中心线改变方向太快并形成缠结的环而不是交替的弯曲（Ferguson，1976）；通道迁移模型生成弯曲，但它们过于规则和曲折（Howard 和 Hemberger，1991）。因此，为了更好地模拟自然通道的几何形状，一个模型在相对较小的长度尺度上需要空间记忆，但在较大尺度上需要不规则。许多自然时间序列，如气候数据，同样可以用惯性和随机性的组合来描述。一阶自回归过程 (AR-1) 被认为是最简单的可能模型 - 也是默认期望 - 对于任何既依赖于先前状态又受随机干扰影响的地球物理过程（Roe，2009）。Ferguson (1976) 表明，与 AR-1 模型相比，二阶自回归模型 (AR-2) 可以缓和高频变化，同时仍允许整体弯曲形状发生变化。这意味着 AR-2 模型代表了最简单的曲率模型，其产生的形状可与自然通道相媲美。AR-2 模型是严格的几何模型，因此足够通用来描述其形成过程可能在细节上有所不同的通道。对于缺乏渠道迁移指标的自然渠道，我们提出 AR-2 模型表示的空间相关随机性代表河流曲折度的零假设，可用于测试环境解释的重要性。通道，我们计算了曲率并拟合了 AR-2 模型参数以限制它们的范围（补充材料中的图 S1）。然后，我们使用一套模型运行来估计曲率 (S) 的幅度，可以使用方程 1 对 b1、b2 和 σ 进行系统参数扫描来解释该幅度。对于每个参数组合，我们分析了具有不同干扰序列 (ϵ) 的 100 个重复，以统计表征模型随机性（图 S2 和 S3）。因为在某些模型运行中，通道与自身相交，我们将每个自相交视为一个截止环并将其删除以保持简化的连续中心线。对于每组重复，我们计算了简化中心线的弯曲度的平均值和标准差以及自相交环中约束的原始中心线的平均长度。 模型运行套件表明，弯曲度和自相交程度随 b1、b2 和 σ 系统地变化（图 2A）。b1 和 σ 的较高值会增加曲率，平均值通常 <1.5。模型中固有的随机性导致重复模拟中的曲率变化（图 2B）。对于一部分重复，S > 1.5，特别是在自相交常见或 σ 0.3 的情况下（图 2C）；后者在自然界中很少发生（图 S1）。这些模型结果意味着对于几何上真实且未迁移的通道，随机生成的曲率很少超过 1.5。我们假设通道通常仅通过迁移达到 S > 1.5。迁移模型进一步表明，具有不同曲率的初始随机通道中的弯曲遵循通过通道迁移进行曲率增长的偏移轨迹（图 3C）。这些情况在不同的无量纲时间到达表明通道迁移的临界曲率（Sc = 1.5）。正如预期的那样，达到临界曲率 (tc*) 的无量纲时间通常随着初始曲率的增加而减少（图 3D）。此外，tc* 在所有情况下都 <15，这表明在中心线最终反映通道迁移之前必须发生的通道迁移量的上限。换句话说，通道的初始随机曲率在有限的时间间隔内被覆盖。对于河流地表的全球数据集，曲率主要在 1 到 2.5 之间变化（图 4A）。我们将 AR-2 模型拟合到每个观察到的平面上，以找到模型预测的曲率值的相应分布。对于在原始中心线没有自相交的模型情况，随机曲率超过观察到的曲率 (Sobs) 的概率对于 Sob < 1.2 是显着的，但随着 Sobs 的增加而急剧下降（图 4B）。如果考虑原始中心线自相交的情况，随机模型考虑了更大范围的 Sob，但不太可能考虑到 Sob > 2（图 4C）。具有记录迁移的单独通道数据集（图 4D）显示迁移有利于更高的曲率值，包括安第斯前陆所有情况下的 S > 1.5，迁移率特别高（每年 > 10% wc；Sylvester 等人, 2019)。将 tc* = 15 和方程 2 中这些观察到的迁移率代入，对于安第斯前陆河流，我们估计临界曲率 (Sc = 1.5) 的时间尺度——与平面随机性的明显区别——是 tc ≈ 150 年。我们使用零假设检验来估计空间相关随机性在河道曲折度 (S) 中的作用，并表明随机模型通常会产生 S < 1.5（图 2A）。超过此临界值的低概率（Sc = 1.5；图 2C）表明对于 S > Sc 的通道，可以拒绝零假设，并且非随机、有序的通道迁移是观察到的曲率的必要条件. 相反的 (S < Sc)——不拒绝零假设——并不意味着随机性完全解释了观察到的曲率。S < Sc 的迁移通道相对常见（图 4D）；迁移通道中相对较低的曲率可能是由于通道迁移缓慢和/或演化时间有限（方程 2）或外在约束。因此，尽管曲率标准不能排除 S < Sc 时的迁移，但该度量确实识别了通道迁移是确定的 (S > Sc) 的情况。河流曲折迁移的定量特征为解释景观历史赋予了这一共同特征进一步的意义。自然通道的曲折既不是纯粹的随机性也不是普遍的通道迁移，而是存在于两者之间的连续体上。借用莎士比亚的《第十二夜》，有些频道天生就是曲折的（图 1D）；有些通过通道迁移实现了进一步的弯曲（图 1A）；通过断层、山谷形状或其他非随机地形控制（图 1C），有些具有曲折性。随机的初始曲折被修饰到由通道迁移的速率和持续时间决定的程度（图 3C）。在通道迁移的数值模型中，初始平面几何形状应考虑空间相关性，因为这种几何形状及其随时间的持续性（图 1）。3D) 比最常用的不相关的、短暂的噪声更真实（例如，Frascati 和 Lanzoni，2010）。在准稳态下，这种通道迁移模型可以产生超过典型观测值的曲率 (S > 3.5)（图 4A；Howard，1996；Frascati 和 Lanzoni，2010）。因此，在没有证据表明河道迁移的情况下，空间相关的随机性应该代表曲率的零假设。空间和时间（Valenza 等人，2020 年）。当一条河流在其自身的洪泛区形成分流时，它可能会重新占用废弃的河道和/或切断新的河道。因此，一个撕脱的通道可能同时在随机景观干扰的引导下形成新的河段，并且还继承了先行迁移通道的弯曲。随着撕脱通道的平面形状随着弯曲迁移而演变，其随机性指标被覆盖。我们的研究结果可以从自然通道的曲折性解释环境强迫。例如，斯塔克等人。(2010) 根据报告的曲率 S < 1.5 的混合基岩冲积河流的曲率空间趋势解释气候强迫，这是我们确定的临界值。我们的分析表明，对于曲率低于临界值的河流，应谨慎使用曲率作为渠道迁移程度的代理；未来的分析可以解释随机曲率作为零假设的作用。然而，我们的结果表明，河流曲折度可用于推断由河道迁移沉积在比河流本身宽许多倍的河道带中的冲积沉积物（Jobe 等，2016）。在某些海洋沉积物的地震图像（Pirmez 和 Imran，2003 年）和行星表面（Burr 等人，2013 年）无法直接观察到河道带的情况下，这种方法可以为地层结构的预测提供信息。最后，这些结果可能适用于优化河流修复项目，这些项目通常被设计为匹配理想化的蜿蜒形式（Kondolf，2006）。我们的分析表明，对于原始通道的迁移表现出最小迁移的情况，恢复的通道应具有低曲率（S < 1.5）。同时，自然，快速迁移的渠道推动了森林干扰和原始演替，这可能有助于生物多样性（Salo 等人，1986 年）。这些观察结果表明，高弯曲度 (S >>2) 河流廊道可能以迁徙为主且相对稀少（图 4A 和图 4D），应优先保护。我们感谢塞缪尔·约翰斯通 (Samuel Johnstone) 和匿名审阅者有用的评论，以及 Roman DiBiase、Douglas Edmonds 和 Chris Paola 的讨论。美国国家科学基金会（EAR-1823530 和 EAR-1246761）、英国地貌学会和洛斯阿拉莫斯国家实验室 (LDRD-20170668PRD1) 支持这项工作。这项研究的数据和代码可在弗吉尼亚大学（美国）的数据存储库 LibraData (https://doi.org/10.18130/V3/TRTTIS) 和 Github (https://github.