landlab.components.erosion_deposition.shared_stream_power

class SharedStreamPower[source]

Bases: ErosionDeposition

Shared Stream Power Model in the style of Hergarten (2021).

Implements the Shared Stream Power Model in the style of Hergarten (2021). Designed to simultaneously model river incision and sediment transport in order to seamlessly transition between detachment limited to transport limited erosion. Mathematically equivalent to the linear decline model from Davy and Lague (2009), which is used by the base class, ErosionDeposition. In addition, this component is designed to work with varying runoff rates, and can update the discharge and other parameters effected by discharge with each timestep.

Here is the equation for erosion without a threshold:

E = k_bedrock * A**m_sp * S**n_sp - k_bedrock / k_transport * Qs / A

where Q is water discharge, Qs is sediment flux, S is slope, m_sp and n_sp are scaling exponents, coefficient k_bedrock is the erodibility of the bedrock and coefficient k_transport is the ability to transport sediment.

The first term, k_bedrock * A**m_sp * S**n_sp, is the incision term, and the second term, k_bedrock / k_transport * Qs / A, is the transport term. Note that k_bedrock / k_transport determines the relative amount of incision and sediment transport. k_bedrock modifies the incision term.

The equivalent equation used by ErosionDeposition from Davy & Lague (2009) is:

E = K * q**m_sp * S**n_sp - v_s * Qs / q

where K is sediment erodibility, v_s is the settling velocity for sediment, and q is the water discharge.

To translate the shared stream power input for ErosionDeposition, we use the equations:

q = Ar
K = k_bedrock / r**m_sp
v_s = k_bedrock / k_transport

It is important to note that the second two equations were derived only for calibrating the model, and do not necessarily correlate to landscape evolution in nature.

To write the final equation we define the incision term as omega:

omega = k_bedrock * A**m_sp * S**n_sp

and incorporate sp_crit, the critical stream power needed to erode bedrock, giving:

E = omega * (1 - exp(omega / sp_crit) ) - k_bedrock / k_transport * Qs / A

Written by A. Thompson.

References

Required Software Citation(s) Specific to this Component

Hergarten, S. (2021). The influence of sediment transport on stationary and mobile knickpoints in river profiles. Journal of Geophysical Research: Earth Surface, 126, e2021JF006218. https://doi.org/10.1029/2021JF006218

Additional References

Barnhart, K., Glade, R., Shobe, C., Tucker, G. (2019). Terrainbento 1.0: a Python package for multi-model analysis in long-term drainage basin evolution. Geoscientific Model Development 12(4), 1267–1297. https://dx.doi.org/10.5194/gmd-12-1267-2019

Davy, P., Lague, D. (2009). Fluvial erosion/transport equation of landscape evolution models revisited Journal of Geophysical Research 114(F3), F03007. https://dx.doi.org/10.1029/2008jf001146

Examples

>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> from landlab.components import DepressionFinderAndRouter
>>> from landlab.components import ErosionDeposition
>>> from landlab.components import FastscapeEroder
>>> np.random.seed(seed=5000)

Define grid and initial topography:

  • 5x5 grid with baselevel in the lower left corner

  • All other boundary nodes closed

  • Initial topography is plane tilted up to the upper right + noise

>>> grid = RasterModelGrid((5, 5), xy_spacing=10.0)
>>> grid.at_node["topographic__elevation"] = (
...     grid.y_of_node / 10
...     + grid.x_of_node / 10
...     + np.random.rand(grid.number_of_nodes) / 10
... )
>>> grid.set_closed_boundaries_at_grid_edges(
...     bottom_is_closed=True,
...     left_is_closed=True,
...     right_is_closed=True,
...     top_is_closed=True,
... )
>>> grid.set_watershed_boundary_condition_outlet_id(
...     0, grid.at_node["topographic__elevation"], -9999.0
... )
>>> fsc_dt = 100.0
>>> ed_dt = 1.0

Check initial topography

>>> grid.at_node["topographic__elevation"].reshape(grid.shape)
array([[0.02290479, 1.03606698, 2.0727653 , 3.01126678, 4.06077707],
       [1.08157495, 2.09812694, 3.00637448, 4.07999597, 5.00969486],
       [2.04008677, 3.06621577, 4.09655859, 5.04809001, 6.02641123],
       [3.05874171, 4.00585786, 5.0595697 , 6.04425233, 7.05334077],
       [4.05922478, 5.0409473 , 6.07035008, 7.0038935 , 8.01034357]])

Instantiate Fastscape eroder, flow router, and depression finder

>>> fr = FlowAccumulator(grid, flow_director="D8")
>>> df = DepressionFinderAndRouter(grid)
>>> fsc = FastscapeEroder(grid, K_sp=0.001, m_sp=0.5, n_sp=1)

Burn in an initial drainage network using the Fastscape eroder:

>>> for _ in range(100):
...     fr.run_one_step()
...     df.map_depressions()
...     flooded = np.where(df.flood_status == 3)[0]
...     fsc.run_one_step(dt=fsc_dt)
...     grid.at_node["topographic__elevation"][0] -= 0.001  # uplift
...

Instantiate the SharedStreamPower component:

>>> ssp = SharedStreamPower(
...     grid,
...     k_bedrock=0.00001,
...     k_transport=0.001,
...     m_sp=0.5,
...     n_sp=1.0,
...     sp_crit=0,
... )

Now run the E/D component for 2000 short timesteps:

>>> for _ in range(2000):  # E/D component loop
...     fr.run_one_step()
...     df.map_depressions()
...     ssp.run_one_step(dt=ed_dt)
...     grid.at_node["topographic__elevation"][0] -= 2e-4 * ed_dt
...

Now we test to see if topography is right:

>>> np.around(grid.at_node["topographic__elevation"], decimals=3).reshape(
...     grid.shape
... )
array([[-0.477,  1.036,  2.073,  3.011,  4.061],
       [ 1.082, -0.08 , -0.065, -0.054,  5.01 ],
       [ 2.04 , -0.065, -0.065, -0.053,  6.026],
       [ 3.059, -0.054, -0.053, -0.035,  7.053],
       [ 4.059,  5.041,  6.07 ,  7.004,  8.01 ]])

Initialize the Shared Stream Power model.

Parameters:

  • grid (ModelGrid) – Landlab ModelGrid object

  • k_bedrock (str, or array_like, optional) – Erodibility for bedrock (units vary).

  • k_transport (str, or array_like, optional) – Ability to transport sediment (units vary).

  • runoff_rate (float, optional) – Runoff rate. Scales Q = Ar. [m/yr]

  • m_sp (float, optional) – Discharge exponent (units vary).

  • n_sp (float, optional) – Slope exponent (units vary).

  • sp_crit (str or array_like) – Critical stream power to erode substrate [E/(TL^2)]

  • F_f (float, optional) – Fraction of eroded material that turns into “fines” that do not contribute to (coarse) sediment load.

  • discharge_field (str or array_like, optional) – Discharge [L^2/T]. The default is to use the grid field "surface_water__discharge", which is simply drainage area multiplied by the default rainfall rate (1 m/yr). To use custom spatially/temporally varying rainfall, use ‘water__unit_flux_in’ to specify water input to the FlowAccumulator.

  • solver ({"basic", "adaptive"}, optional) –

    Solver to use. Options at present include:

    1. "basic" (default): explicit forward-time extrapolation. Simple but will become unstable if time step is too large.

    2. "adaptive": adaptive time-step solver that estimates a stable step size based on the shortest time to “flattening” among all upstream-downstream node pairs.

property K

Erodibility of substrate (units depend on m_sp).

__init__(grid, k_bedrock=0.001, k_transport=0.001, runoff_rate=1.0, m_sp=0.5, n_sp=1.0, sp_crit=0.0, F_f=0.0, discharge_field='surface_water__discharge', solver='basic')[source]

Initialize the Shared Stream Power model.

Parameters:

  • grid (ModelGrid) – Landlab ModelGrid object

  • k_bedrock (str, or array_like, optional) – Erodibility for bedrock (units vary).

  • k_transport (str, or array_like, optional) – Ability to transport sediment (units vary).

  • runoff_rate (float, optional) – Runoff rate. Scales Q = Ar. [m/yr]

  • m_sp (float, optional) – Discharge exponent (units vary).

  • n_sp (float, optional) – Slope exponent (units vary).

  • sp_crit (str or array_like) – Critical stream power to erode substrate [E/(TL^2)]

  • F_f (float, optional) – Fraction of eroded material that turns into “fines” that do not contribute to (coarse) sediment load.

  • discharge_field (str or array_like, optional) – Discharge [L^2/T]. The default is to use the grid field "surface_water__discharge", which is simply drainage area multiplied by the default rainfall rate (1 m/yr). To use custom spatially/temporally varying rainfall, use ‘water__unit_flux_in’ to specify water input to the FlowAccumulator.

  • solver ({"basic", "adaptive"}, optional) –

    Solver to use. Options at present include:

    1. "basic" (default): explicit forward-time extrapolation. Simple but will become unstable if time step is too large.

    2. "adaptive": adaptive time-step solver that estimates a stable step size based on the shortest time to “flattening” among all upstream-downstream node pairs.

static __new__(cls, *args, **kwds)
cite_as = '\n    @article{barnhart2019terrain,\n      author = {Barnhart, Katherine R and Glade, Rachel C and Shobe, Charles M\n                and Tucker, Gregory E},\n      title = {{Terrainbento 1.0: a Python package for multi-model analysis in\n                long-term drainage basin evolution}},\n      doi = {10.5194/gmd-12-1267-2019},\n      pages = {1267---1297},\n      number = {4},\n      volume = {12},\n      journal = {Geoscientific Model Development},\n      year = {2019},\n    }\n    '
property coords

Return the coordinates of nodes on grid attached to the component.

property current_time

Current time.

Some components may keep track of the current time. In this case, the current_time attribute is incremented. Otherwise it is set to None.

Return type:

current_time

definitions = (('flow__link_to_receiver_node', 'ID of link downstream of each node, which carries the discharge'), ('flow__receiver_node', 'Node array of receivers (node that receives flow from current node)'), ('flow__upstream_node_order', 'Node array containing downstream-to-upstream ordered list of node IDs'), ('sediment__influx', 'Sediment flux (volume per unit time of sediment entering each node)'), ('sediment__outflux', 'Sediment flux (volume per unit time of sediment leaving each node)'), ('surface_water__discharge', 'Volumetric discharge of surface water'), ('topographic__elevation', 'Land surface topographic elevation'), ('topographic__steepest_slope', 'The steepest *downhill* slope'))
classmethod from_path(grid, path)

Create a component from an input file.

Parameters:

  • grid (ModelGrid) – A landlab grid.

  • path (str or file_like) – Path to a parameter file, contents of a parameter file, or a file-like object.

Returns:

A newly-created component.

Return type:

Component

property grid

Return the grid attached to the component.

initialize_optional_output_fields()

Create fields for a component based on its optional field outputs, if declared in _optional_var_names.

This method will create new fields (without overwrite) for any fields output by the component as optional. New fields are initialized to zero. New fields are created as arrays of floats, unless the component also contains the specifying property _var_type.

initialize_output_fields(values_per_element=None)

Create fields for a component based on its input and output var names.

This method will create new fields (without overwrite) for any fields output by, but not supplied to, the component. New fields are initialized to zero. Ignores optional fields. New fields are created as arrays of floats, unless the component specifies the variable type.

Parameters:

values_per_element (int (optional)) – On occasion, it is necessary to create a field that is of size (n_grid_elements, values_per_element) instead of the default size (n_grid_elements,). Use this keyword argument to acomplish this task.

input_var_names = ('flow__link_to_receiver_node', 'flow__receiver_node', 'flow__upstream_node_order', 'surface_water__discharge', 'topographic__elevation', 'topographic__steepest_slope')
property k_bedrock

Erodibility for bedrock (units vary).

property k_transport

Ability to transport sediment (units vary).

property m_sp

Discharge exponent (units vary).

property n_sp

Slope exponent (units vary).

name = 'SharedStreamPower'
optional_var_names = ()
output_var_names = ('sediment__influx', 'sediment__outflux', 'topographic__elevation')
run_one_step_basic(dt=1.0)

Calculate change in rock and alluvium thickness for a time period ‘dt’.

Parameters:

dt (float) – Model timestep [T]

run_with_adaptive_time_step_solver(dt=1.0)

CHILD-like solver that adjusts time steps to prevent slope flattening.

Parameters:

dt (float) – Model timestep [T]

property runoff_rate

Runoff rate. Scales Q = Ar. [m/yr]

property sediment_influx

Volumetric sediment influx to each node.

property shape

Return the grid shape attached to the component, if defined.

property sp_crit

Critical stream power to erode substrate [E/(TL^2)]

unit_agnostic = True
units = (('flow__link_to_receiver_node', '-'), ('flow__receiver_node', '-'), ('flow__upstream_node_order', '-'), ('sediment__influx', 'm3/s'), ('sediment__outflux', 'm3/s'), ('surface_water__discharge', 'm**2/s'), ('topographic__elevation', 'm'), ('topographic__steepest_slope', '-'))
update_runoff(new_runoff=1.0)[source]

Update runoff variables.

Updates runoff_rate, K, v_s, and "water__unit_flux_in" for a new runoff rate. Works only if discharge field is set to "water__unit_flux_in".

Parameters:

new_runoff (str or array_like) – New runoff rate.

property v_s

Effective settling velocity for chosen grain size metric [L/T].

classmethod var_definition(name)

Get a description of a particular field.

Parameters:

name (str) – A field name.

Returns:

A description of each field.

Return type:

tuple of (name, *description*)

classmethod var_help(name)

Print a help message for a particular field.

Parameters:

name (str) – A field name.

classmethod var_loc(name)

Location where a particular variable is defined.

Parameters:

name (str) – A field name.

Returns:

The location (‘node’, ‘link’, etc.) where a variable is defined.

Return type:

str

var_mapping = (('flow__link_to_receiver_node', 'node'), ('flow__receiver_node', 'node'), ('flow__upstream_node_order', 'node'), ('sediment__influx', 'node'), ('sediment__outflux', 'node'), ('surface_water__discharge', 'node'), ('topographic__elevation', 'node'), ('topographic__steepest_slope', 'node'))
classmethod var_type(name)

Returns the dtype of a field (float, int, bool, str…).

Parameters:

name (str) – A field name.

Returns:

The dtype of the field.

Return type:

dtype

classmethod var_units(name)

Get the units of a particular field.

Parameters:

name (str) – A field name.

Returns:

Units for the given field.

Return type:

str