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.