Representation of flowing water in landscape evolution models
(LEMs) is often simplified compared to hydrodynamic models, as LEMs make
assumptions reducing physical complexity in favor of computational
efficiency. The Landlab modeling framework can be used to bridge the divide
between complex runoff models and more traditional LEMs, creating a new type
of framework not commonly used in the geomorphology or hydrology communities.
Landlab is a Python-language library that includes tools and process
components that can be used to create models of Earth-surface dynamics over a
range of temporal and spatial scales. The Landlab OverlandFlow component is
based on a simplified inertial approximation of the shallow water equations,
following the solution of

Numerical models of overland flow have a variety of applications. Examples
include mapping urban flooding events

The simplifying assumption of steady-state discharge is made for two reasons:
there can be significant differences between hydrologic timescales for
individual flood and storm events (minutes to days) and geomorphic timescales
of rock uplift and landscape evolution (thousands to millions of years) that
may be complex to resolve. Additionally, computational power is often a
limiting factor, as some processes in LEMs do not lend themselves to
parallelization, so making assumptions about how water fluxes are calculated
(e.g., Eq.

Whereas many LEMs generalize surface water flow using steady-state
assumptions, most physical models of runoff production simulate changing
surface water discharge through time, capturing the spatial and temporal
variability of flowing water across a modeled landscape

Image illustrating the differences between steady-state and
non-steady hydrology and incision at a single point within a watershed. In
this schematic, the effective precipitation rate (

Sample workflow for the Landlab OverlandFlow component. Users create or adapt a pre-developed model driver, where the grid, components, and model utilities are imported and instantiated. The time loop is set in the driver, and at each time step the component methods are called and the data structures are updated.

The assumption of steady-state discharge in LEMs is not always reasonable.
Steady-state hydrologic conditions are rarely achieved in larger catchments
with long flow paths or in landscapes dominated by short-duration
precipitation events. Additionally, the traditional steady-state model
(Eq.

Adding hydrologic variability to LEMs has also been shown to impact watershed
morphology and landscape evolution. Previous work coupling spatially variable
rainfall models with steady-state discharge in erosion models has illustrated
impacts on landform morphology, including relief and drainage network
organization

To investigate the role of non-steady flow routing on landform evolution, a
hydrodynamic model has been incorporated into the Landlab modeling toolkit.
In this paper, we describe the fundamentals of the Landlab modeling
framework, the theoretical background of the Landlab OverlandFlow component,
based on a two-dimensional flood inundation model

Landlab is a Python-language, open-source modeling framework, developed as a
highly flexible and interdisciplinary library of tools that can be used to
address a range of hypotheses in Earth-surface dynamics

Landlab offers several different grid types. However, because the core
algorithm in the OverlandFlow component can only be applied to structured
grids, only the RasterModelGrid class is described here. The RasterModelGrid
class can build both square (

Example of the Landlab structured grid type with key topological
elements shown. In the Landlab OverlandFlow component, RasterModelGrid class
stores data at both nodes and links. Links denoted as west (

Model data are stored on these grid elements using Landlab data fields. The
data fields are NumPy array structures that contain data associated with a
given grid element. To store and access data on these fields, data are
assigned using a string keyword and are accessed using Python's mutable
dictionary data structure. Data are attached to the grid instance using these
fields and can be accessed using the string name keyword and updated by
multiple Landlab components. For example, a field of values representing
water depth at a grid node can be accessed using the following syntax:

Simple example of Landlab RasterModelGrid, demonstrating both node and link boundary conditions. The OverlandFlow class calculates fluxes at active links, and can update the surrounding fixed links according to these fluxes. No fluxes are calculated at inactive links. Water depth is updated at core and open boundary nodes. No calculations are performed on closed or fixed gradient boundaries. Note that RasterModelGrid cell elements and link directionalities are not illustrated here.

Model boundary conditions are set within a Landlab grid object. Boundary
conditions are set on nodes and links (Fig.

List of variables used in the OverlandFlow and DetachmentLtdErosion. For each variable, the name, grid element, and units are given.

There are three link boundary statuses: active, inactive, and fixed. Link
boundary status is tied to the neighboring nodes. Once boundary conditions
are set on the nodes, link boundary conditions are automatically updated.
Active links occur where fluxes are calculated and are found in two cases:
(1) between two core nodes or (2) between one core node and one open boundary
node. Fixed links can be assigned a value that can be set or updated during
the model run and are located between a fixed gradient node and a core node.
Fluxes are not calculated on inactive links, which occur in two cases:
(1) between a closed boundary and a core node or (2) between any pair of
boundary nodes of any type (Fig.

Solving explicit two-dimensional hydraulic formulations can be
computationally challenging. For example, the 1-D shallow water equation
includes four terms:

In the

The Landlab OverlandFlow component adapts a two-dimensional hydrodynamic
algorithm to simulate flow at all points across the gridded domain. This
algorithm, developed for the LISFLOOD-FP model, was incorporated into Landlab
for modeling overland flow. Similar to the diffusive approximation, the
LISFLOOD-FP algorithm assumes a negligible contribution from the advection
term of the shallow water equations

To start the model, a stable time step is calculated. Stable time steps are
set according to the Courant–Friedrichs–Lewy criteria which evaluate the
ratio of time step size to grid resolution. If large time steps are used,
areas of high slope are prone to wave oscillations, leading to a spatial
“checkerboard” pattern of water depths. If time steps are very small, there
are significant impacts on the computational performance of a model. To
maximize the trade-off between computational efficiency and stability of the

To calculate water discharge at all grid locations,

List of parameters used in the OverlandFlow and DetachmentLtdErosion. For each variable, the name and units are given.

Water depth is calculated on nodes and updated at each time step as a
function of the surrounding volumetric water fluxes (

By default, this model assumes that all rainfall is spatially uniform and temporally constant, and all rainfall is converted to surface runoff. No infiltration or subsurface flow is considered within the model equations; however, the OverlandFlow component could be easily coupled with an infiltration component. Spatially or temporally variable rainfall could be generated by another process component or set manually by the user in a driver file. Effective rainfall depths are applied over the basin and added to the surface water depths at each time step.

The

If the

To illustrate the flexibility of the OverlandFlow component, we present an
example in Sect. 7, in which water discharge calculated by the OverlandFlow
component is used in the erosion component. Specifically, we explore a case
where incision rate is solved explicitly and depends on local water
discharge and water surface gradient

To use the coupled Landlab OverlandFlow and DetachmentLtdErosion model, the
user interacts with a driver file (Fig.

To set up a grid instance, the user can create a rectangular grid by passing
the number of rows, number of columns, and grid resolution (

An alternative method is to read in gridded terrain data from other file
types. The original intent of

Node boundary conditions are set throughout the grid in a Landlab
OverlandFlow model to delineate the modeling domain
(Algorithm 1, Line 4). For flow to move
out of a watershed or system, an open boundary must be set at the outlet(s).
If the node location of the outlet is unknown, there is a utility within the
grid (

Grid characteristics and parameters for analytical solution tests.

The

Landlab components have a standard initialization signature and take the grid
instance as the first keyword (Algorithm 1,
Lines 6–8). Any default parameters are also in the
component signature and can be updated when the component is called. These
parameters can be adjusted according to the physical nature of the landscape
being tested. For the OverlandFlow component, Eq. (

To couple the OverlandFlow and DetachmentLtdErosion components, values for
water discharge (

To validate the OverlandFlow component, we compared model output against an
analytical solution for wave propagation on a flat surface, following

Solving for the leftmost boundary of the modeling domain (

Sensitivity of the Landlab OverlandFlow component to changes in grid
resolution, tested against the analytical solution. Panel

All analytical solution tests were modeled across a rectangular
RasterModelGrid instance with dimensions of 800 m by 6000 m. The water
depth boundary condition (Eq.

Following

The smallest time step over the duration of the

In all grid resolution tests, the OverlandFlow predicted wave fronts closely
approximate the analytical solution
(Fig.

To test the Landlab OverlandFlow component with different roughness and
resolution characteristics, a RasterModelGrid instance with dimensions of
800 m by 6000 m was initialized with a resolution of

The smallest time step over the duration of the low friction model run (

In all velocity–roughness conditions, the wave fronts predicted by the
Landlab OverlandFlow component correlate well with the analytical solution
defined using Eq. (

Sensitivity of the Landlab OverlandFlow component with a changing
Manning's

The Landlab OverlandFlow component can be used in hydrology applications,
routing precipitation across a real landscape DEM and estimating runoff for
every point within a discrete RasterModelGrid instance. Discharge values can
be calculated at every point in the watershed and updated at each time step.
Updated water depths, driven by changing discharge, can be used to calculate
shear stress following the depth–slope product:

Equation (

Results from the real landscape example. Panel

Here, we illustrate a single storm routed across a DEM. In addition to water discharge, water depth and bed shear stress are calculated by the model at all grid locations. This implementation of the OverlandFlow component illustrates how hydrologists can use Landlab as a simplified distributed runoff model to estimate the flow of water and sediment resulting from a single storm on a real landscape.

To route runoff across a real landscape, a DEM can be read into Landlab and
converted easily into a RasterModelGrid instance. The Spring Creek watershed
is used in this example, as a preprocessed DEM for the watershed has been
used before in Landlab applications

The DEM was preprocessed using the Landlab SinkFiller component to ensure all surface water flow can be removed from the domain. This component fills pits in the DEM in a D4 routing scheme, where all nodes have at least one downstream neighbor in one of the four cardinal directions (Algorithm 1, Lines 8–9). If this step were to be skipped, flow may pond in “lakes” or “pits” in the domain, where flow cannot travel out of a given node location until the water surface elevation of the lake exceeds the bed elevation of one of the four neighboring nodes.

Precipitation parameters for the three storm cases routed across the test basins.

To initiate flow across the domain, a single storm was routed across the
watershed. A theoretical “base storm” (Table

In order to illustrate the downstream movement of the flood wave, hydrographs
were plotted at three locations within the channel. The three hydrographs
correspond to the three starred locations on the watershed DEM in
Fig.

Water depths are variable at each point throughout the model run, changing as
a function of discharge inputs, outputs, and effective rainfall rate at each
time step (Eq.

In this example, we illustrate hydrographs across a real landscape and the resulting shear stress values. These results can be used to explore the processes controlling overland flow in a gauged landscape. Shear stress values can be used to estimate sediment transport rates and make interpretations about spatial patterns of erosion and deposition, as well as total sediment yields for particular storm events. These data can be used to explore landscape sensitivity to different rainfall events and runoff conditions.

The implementation of the OverlandFlow component in Landlab allows us to investigate the impact of storm characteristics on the resulting hydrograph and how these hydrographs drive erosion processes throughout the basin. Here, we demonstrate the abilities of this new component, how the component resolves the details of the storm hydrograph, and how these hydrographs compare to the traditional steady-state method used in LEMs. Additionally, in coupling this new component with the Landlab DetachmentLtdErosion component, these model results illustrate the erosion magnitudes and patterns in response to a hydrograph and allow us to make inferences about how this type of hydrodynamic model could impact long-term geomorphic evolution of similar watersheds.

Two test basins evolved using the Landlab FlowRouter and
StreamPowerEroder components

To test the new Landlab OverlandFlow component, two synthetic watersheds were
generated using the Landlab FlowRouter and StreamPowerEroder components

To initiate flow and incision, three precipitation events were modeled across
both watersheds. These storms were represented as spatially uniform across
the model domain, and intensities were constant for the given storm duration.
No infiltration or subsurface flow was modeled in these test cases. The base
storm, following the example in the real landscape, has a rainfall intensity
of 5 mm h

Discharge was calculated at all grid locations during each model run. To capture the entire overland flow event, all simulations were run for 24 modeled hours, although flow had nearly stopped after 12 h of modeled time. A single base storm on the square watershed run for 24 modeled hours took approximately 80 s on a 2014 iMac with 4 GHz Intel Core i7 processors.

The OverlandFlow results from the two test basins were coupled with the
DetachmentLtdErosion component in Landlab to test the impact of non-steady
hydrology on erosional patterns. At each time step, the DetachmentLtdErosion
component calculated total incision depth at all points in the grid using
Eq. (

OverlandFlow output for all storms described in
Table

The hydrographs measured at the outlet of both the square and long basins are
compared with the steady-state hydrographs (Fig.

As expected, the OverlandFlow component is also sensitive to changes in
rainfall characteristics in both test basins. In the square basin, extending
the duration of the storm (green line; Fig.

In the square basin, each storm has a clear hydrograph signature. These
patterns are distinct from the long basin results. In the long basin, all
three storm hydrographs have lower peak discharges than similar storms in the
square basin (Fig.

To understand how non-steady hydrologic methods drive erosion in comparison to
more traditional LEM methods, total incised depths for the three storm cases
can be compared to predicted geomorphic steady-state incised depths after 10
modeled years. This application tests how the different hydrologic methods
(steady vs. non-steady) impact morphology in LEM applications, following the
work of

The pattern of increasing downstream incision is seen in all storm cases
(Fig.

DetachmentLtdErosion output for all storms described in
Table

Overall, these results suggest that, when compared to the OverlandFlow
component, hydrologic steady-state predictions can over- or underestimate the
peak of a hydrograph depending on basin orientation or shape
(Fig.

The patterns of erosion support earlier findings by

The Landlab OverlandFlow model is flexible enough to be used in a number of scientific applications not discussed here. While the model does simulate surface flow over the entire domain, internally it makes no distinction between hillslope or channel processes, which can be problematic, as hillslopes make up the majority of a watershed area and supply sediment to the channels. If coupled with a hillslope sheet-wash component, OverlandFlow could be used to examine how non-steady channel processes interact with hillslope processes to sculpt watersheds across a range of spatial and temporal scales. Furthermore, these hillslope processes can be coupled with a fluvial transport-limited component and applied at event scales to explore sediment delivery from hillslopes to channels and how quickly sediment moves through a watershed. At landscape evolution timescales, evolved topographies resulting from more physically based hydrology and sediment transport components can be compared to traditional models to evaluate how physical parameters within the fluvial and hillslope models impact landscape relief and organization.

Other opportunities include evaluating the impact of spatially variable
parameters on model behavior. Spatial variability in rainfall could be
explored with the development of new components that model orography or
variability in storm cell size. Following the work of

Another potential application is coupling the OverlandFlow component to
Landlab's ecohydrology components

Finally, the applications explored in this paper are on shorter
timescales, ranging from event- to decadal-scale runs. An interesting future
direction is exploring the OverlandFlow component in true landscape evolution
runs (millennia or longer). Preliminary work modeling 10

This paper illustrates the theory behind the OverlandFlow component and how to use it as part of Landlab. Being part of the Landlab modeling framework comes with many advantages. The OverlandFlow component can make use of DEM input and output utilities and be coupled with other process components. Results from the real landscape application demonstrate that the OverlandFlow component can be used to route flow from observed rainfall events across a watershed DEM. This method can be used to estimate the grain sizes moved by real storm events and, in the future, could be coupled with other components and calibrated to understand the erosional response to flooding events.

The OverlandFlow component can also be coupled to the DetachmentLtdErosion component to explore impacts of a hydrograph on erosion on decadal scales. In the synthetic landscapes explored here, the hydrograph results from the OverlandFlow component demonstrate a sensitivity to basin shape, precipitation duration, and intensity. The incision results predicted by using steady-state and non-steady water discharge are distinct in both the patterns and magnitudes of eroded depth and incision rates. Landscape evolution driven by non-steady runoff showed increasing incision rates moving downstream in the modeled watersheds. These results suggest that non-steady runoff could have important implications for predicting watershed relief and hypsometry in landscapes with different rainfall regimes and that choice of runoff method can have implications for both short- and long-term modeling results.

The Landlab OverlandFlow and DetachmentLtdErosion
components are part of Landlab version 1.0.0. Source code for the Landlab
project is housed on GitHub:

The authors declare that they have no conflict of interest.

This research was supported by the National Science Foundation grants ACI-1147519 and ACI-1450338 (PI: Nicole M. Gasparini), ACI-1148305 and ACI-1450412 (PI: Erkan Istanbulluoglu), ACI-1147454 (PI: Gregory E. Tucker) and ACI-1450409 (PI: Gregory E. Tucker; co-PI: Daniel E. J. Hobley), as well as the Tulane University Department of Earth and Environmental Sciences Vokes Fellowship (Jordan M. Adams). The authors are grateful to the topical editor Jeffrey Neal and reviewers Astrid Kerkweg, Katerina Michaelides, and Dapeng Yu, whose comments greatly improved the manuscript. Edited by: J. Neal Reviewed by: K. Michaelides and D. Yu