2. Methods 2.1 Model Description
We explore the influence of longitudinal rainfall gradients on large transverse rivers first using a simple 1-dimensional river incision model. We model erosion as detachment-limited (Howard, 1994; Roe et al., 2002; Whipple & Tucker, 1999) following a general form of the SPM:
\(E=KA^{m}S^{n},\ \) (1a)\(K=K_{p}{\overset{\overline{}}{P}}^{m},\) (1b)\(E=K_{p}Q^{m}S^{n},\) (1c)
where E is the erosion rate; K andKp are erosional efficiency coefficients;A is upstream drainage area; S is the channel slope;\(\overline{P}\) is the upstream average rainfall rate; Q is water discharge and is calculated as \(\overline{P}\)A , which assumes that all rainfall is converted to runoff; and m andn are positive constant exponents (Table 1). We use n = 2 and m = 1 for all model runs as values of n > 1 appear more appropriate in many settings (e.g., Adams et al., 2020; Harel et al., 2016; Lague, 2014). First-order results do not rely on choices of m or n providing the ratio between the two is approximately maintained, but the nonlinear dependence of erosion rate on slope (i.e., n = 2) affects details of the transient behavior. Also, because m = 1, K is directly proportional to both\(\overline{P}\) and Q . We explicitly treat the influence of climate on erosional efficiency (e.g., Adams et al., 2020; Roe et al., 2002) such that Kp is independent of rainfall, but still encapsulates a number of factors including rock properties and details of erosional processes (Royden & Perron, 2013; Whipple & Tucker, 1999). Rock uplift rate (U ) and Kpare spatially and temporally uniform and invariant across model runs.
We define drainage area (A ) following Hack, (1957):
\(A=k_{a}x^{h}+A_{c},\) (2)
where x is distance along the channel downstream from the drainage divide, ka and h are constants, and Ac is the upstream drainage area at the channel head – equal to 1 km2. Channel length (L ) and drainage area are fixed and do not evolve over the course of a model run.
For simplicity, we model orographic precipitation as constant gradients in rainfall (i.e., linear changes with distance). Although a constant gradient is a simplification, it is a reasonable approximation to commonly observed orographic rainfall patterns, which can be both top- and bottom-heavy (e.g., Anders et al., 2006; Bookhagen & Burbank, 2010; Bookhagen & Strecker, 2008; Roe, 2005). Further, the framework we develop from these simple rainfall patterns is generally applicable to addressing more complex versions of the fundamental problem we address here – the effects of spatially concentrated rainfall.