An understanding of step-pool geometry has important practical applications in ecological restoration, erosion control and hazard assessment in mountain streams. However, published analysis is insufficient: (1) fully to identify the controls on step-pool geometry, and; (2) to allow inference to be made as to the origin of step-pools and the hydraulic conditions during step-pool formation. The choice of length scales used for normalisation in previous studies has often been ad hoc. In addition, formative discharge is likely to be an important morphological control variable, and field investigations are hampered by a lack of direct knowledge of it. This investigation therefore synthesises data from experimental studies of alluvial step-pools, where formative conditions are known, and uses rigorously identified dimensionless variables to identify: (1) the magnitude and mechanism of the stabilising effect of step-pools; (2) the controls on step-pool geometry; (3) the mechanism(s) responsible for step-pool formation and; (4) the formative flow regime(s) and hydraulic domain of step-pool formation.

Bed mobility in formative flows is controlled by the quantity D₁₀₀/h_c, where D₁₀₀ is the diameter of the largest grains in the sediment mixture and h_c is the formative critical flow depth, for which we introduce the term “formative roughness”. Formative flows are competent to move all grain sizes for D₁₀₀/h_c ≤ 1.3. There is a limiting channel slope that can be stabilised by step-pools, the central tendency of which is given by the Shields-type relationship S_lim = 0.06(D₁₀₀/h_c). This limiting slope indicates that the stabilising effect of step-pools is comparable to that achieved by replacing the bulk sediment mixture in a plane-bed channel with uniform grains of intermediate diameter D₁₀₀/2. The stabilising effect appears to be due the grains being arranged in stable configurations rather than the energy dissipation (form roughness) of step-pools.

Step height H in experimental step-pools primarily is controlled by the size of the largest transported grains D_100t, such that H ≈ D_100t. Step spacing L appears to scale with h_c. Dimensionless step spacing data L/h_c are collapsed well by the formative flow final Froude number Fr.

The relationship between L/h_c and Fr shows the existence of two regimes: L/h_c is greater than minimum antidune spacing but decreases rapidly with increasing Fr for Fr < 0.9 (region 1), but is a weak function of Fr and close to the minimum antidune spacing for Fr > 0.9 (region 2). We argue that step formation via substrate-based mechanisms is consistent with step spacing in region 1, and therefore while substrate-based step formation mechanisms alone are active in the low Fr part of region 1, these mechanisms are progressively replaced by the antidune mechanism as Fr increases, with the antidune mechanism being dominant throughout region 2. Thus, we provide a unifying theory for substrate-based mechanisms and the antidune mechanism.

Dimensionless step spacing data do not support the jammed state hypothesis (Church and Zimmerman, 2007), the simplified cascade model (Allen, 1983), the upstream-forced cascade model (Marion et al., 2004; Comiti et al., 2005) or the maximum flow resistance model (Abrahams et al., 1995) of step formation.

There is an associated transition with increasing Fr in the dominant flow regime during step formation from tumbling flow to standing waves. However, even where standing waves are dominant during step formation, the final flow regime over the stabilised bed depends on dimensionless step height H/h_c: for H/h_c > 1.1 tumbling flow develops from the standing waves, while for H/h_c < 1.1 tumbling flow is drowned out and standing waves persist as the final flow regime. There is no lower formative discharge limit to step-pools; rather, step-pools grade into cascades as Fr decreases. The upper formative discharge limit for step-pools is in the range 0.7 < (D₁₀₀/h_c)_lim < 1.0. We suggest that well developed step-pools are anchored and immobilised antidunes, while at formative roughness D₁₀₀/h_c < (D₁₀₀/h_c)_lim, antidunes remain unanchored and mobile.

对阶梯池几何形状的理解在山间溪流的生态恢复、侵蚀控制和危害评估方面具有重要的实际应用。然而，已发表的分析是不够的：（1）完全识别阶梯池几何的控制，以及；(2) 允许推断阶梯水池的起源和阶梯水池形成过程中的水力条件。在以前的研究中用于标准化的长度尺度的选择通常是临时的。此外，形成性排放很可能是一个重要的形态控制变量，并且由于缺乏对它的直接了解而阻碍了实地调查。因此，本研究综合了冲积阶池实验研究的数据，其中形成条件已知，并使用严格识别的无量纲变量来识别：(1)阶梯池稳定作用的大小和机制；(2) 阶梯池几何的控制；(3) 负责阶梯池形成的机制；(4) 阶梯水池形成的形成流态和水力域。

地层流中的床流度受D ₁₀₀ / h _{c 的控制}，其中D ₁₀₀是沉积物混合物中最大颗粒的直径，h _c是地层临界流深度，为此我们引入了术语“地层粗糙度” . 对于D ₁₀₀ / h _c  ≤ 1.3，形成流能够移动所有粒度。存在一个可以通过阶梯池稳定的极限通道坡度，其中心趋势由 Shields 型关系给出S _lim  = 0.06( D ₁₀₀ / h _c）。这个极限斜率表明阶梯水池的稳定效果与用中等直径D ₁₀₀ /2 的均匀颗粒代替平面床通道中的大量沉积物混合物所达到的效果相当。稳定效应似乎是由于晶粒以稳定的配置排列，而不是阶梯池的能量耗散（形状粗糙度）。

实验阶梯池中的阶梯高度H主要由最大输送颗粒的尺寸D _{100 t 控制}，因此H  ≈  D _{100 t}。步距L似乎与h _c成比例。无量纲步距数据L / h _c被形成流最终 Froude 数Fr很好地压缩。

L / h _c和Fr之间的关系表明存在两种状态：L / h _c大于最小反沙丘间距，但随着Fr的增加而迅速减小，对于Fr  < 0.9（区域 1），但它是Fr的弱函数并且接近到Fr  > 0.9（区域 2）的最小反沙丘间距。我们认为通过基于衬底的机制形成的台阶与区域 1 中的台阶间距一致，因此，虽然基于衬底的台阶形成机制在低Fr 中是活跃的在区域 1 的一部分，随着Fr 的增加，这些机制逐渐被反沙丘机制取代，反沙丘机制在整个区域 2 中占主导地位。因此，我们为基于底物的机制和反沙丘机制提供了统一的理论。

无量纲步距数据不支持阻塞状态假设（Church 和 Zimmerman，2007 年）、简化的级联模型（Allen，1983 年）、上游强制级联模型（Marion 等人，2004 年；Comiti 等人，2005 年）或阶梯形成的最大流动阻力模型（Abrahams 等，1995）。

在从翻滚流到驻波的阶梯形成过程中，随着主要流态中Fr的增加，存在相关的转变。然而，即使在台阶形成过程中驻波占主导地位，稳定床的最终流动状态取决于无量纲台阶高度H / h _c：对于H / h _c  > 1.1 翻滚流从驻波发展而来，而对于H / h _c  < 1.1 翻滚流被淹没，驻波作为最终流态持续存在。阶梯池没有形成排放下限；相反，阶梯池分级为级联作为Fr减少。阶梯水池的形成排放上限在 0.7 < ( D ₁₀₀ / h _c ) _lim  < 1.0 的范围内。我们建议发育良好的阶梯池是锚定和固定的反沙丘，而在形成粗糙度D ₁₀₀ / h _c  < ( D ₁₀₀ / h _c ) _{lim 时}，反沙丘保持未锚定和移动。