GravelBedrockEroder: rock incision, bed-load transport, and downstream abrasion in a river network

Model bedrock incision and gravel transport and abrasion in a network of rivers.

@author: gtucker

class GravelBedrockEroder(*args, **kwds)

Bases: Component

Drainage network evolution of rivers with gravel alluvium overlying bedrock.

Model drainage network evolution for a network of rivers that have a layer of gravel alluvium overlying bedrock.

GravelBedrockEroder is designed to operate together with a flow-routing component such as FlowAccumulator, so that each grid node has a defined flow direction toward one of its neighbor nodes. Each core node is assumed to contain one outgoing fluvial channel, and (depending on the drainage structure) zero, one, or more incoming channels. These channels are treated as effectively sub-grid-scale features that are embedded in valleys that have a width of one grid cell.

As with the GravelRiverTransporter component, the rate of gravel transport out of a given node is calculated as the product of bankfull discharge, channel gradient (to the 7/6 power), a dimensionless transport coefficient, and an intermittency factor that represents the fraction of time that bankfull flow occurs. The derivation of the transport law is given by Wickert & Schildgen (2019), and it derives from the assumption that channels are gravel-bedded and that they “instantaneously” adjust their width such that bankfull bed shear stress is just slightly higher than the threshold for grain motion. The substrate is assumed to consist entirely of gravel-size material with a given bulk porosity. The component calculates the loss of gravel-sized material to abrasion (i.e., conversion to finer sediment, which is not explicitly tracked) as a function of the volumetric transport rate, an abrasion coefficient with units of inverse length, and the local transport distance (for example, if a grid node is carrying a gravel load Qs to a neighboring node dx meters downstream, the rate of gravel loss in volume per time per area at the node will be beta * Qs * dx, where beta is the abrasion coefficient).

Sediment mass conservation is calculated across each entire grid cell. For example, if a cell has surface area A, a total volume influx Qin, and downstream transport rate Qs, the resulting rate of change of alluvium thickness will be (Qin - Qs / (A * (1 - phi)), plus gravel produced by plucking erosion of bedrock (phi is porosity).

Bedrock is eroded by a combination of abrasion and plucking. Abrasion per unit channel length is calculated as the product of volumetric sediment discharge and an abrasion coefficient. Sediment produced by abrasion is assumed to go into wash load that is removed from the model domain. Plucking is calculated using a discharge-slope expression, and a user-defined fraction of plucked material is added to the coarse alluvium.

Parameters:

• grid (ModelGrid) – A Landlab model grid object

• intermittency_factor (float (default 0.01)) – Fraction of time that bankfull flow occurs

• transport_coefficient (float (default 0.041)) – Dimensionless transport efficiency factor (see Wickert & Schildgen 2019)

• abrasion_coefficient (float (default 0.0 1/m)) – Abrasion coefficient with units of inverse length

• sediment_porosity (float (default 0.35)) – Bulk porosity of bed sediment

• depth_decay_scale (float (default 1.0)) – Scale for depth decay in bedrock exposure function

• plucking_coefficient (float or (n_core_nodes,) array of float (default 1.0e-4 1/m)) – Rate coefficient for bedrock erosion by plucking

• coarse_fraction_from_plucking (float or (n_core_nodes,) array of float (default 1.0)) – Fraction of plucked material that becomes part of gravel sediment load

Examples

>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> grid = RasterModelGrid((3, 3), xy_spacing=1000.0)
>>> elev = grid.add_zeros("topographic__elevation", at="node")
>>> sed = grid.add_zeros("soil__depth", at="node")
>>> sed[4] = 300.0
>>> grid.status_at_node[grid.perimeter_nodes] = grid.BC_NODE_IS_CLOSED
>>> grid.status_at_node[5] = grid.BC_NODE_IS_FIXED_VALUE
>>> fa = FlowAccumulator(grid, runoff_rate=10.0)
>>> fa.run_one_step()
>>> eroder = GravelBedrockEroder(grid, abrasion_coefficient=0.0005)
>>> rock_elev = grid.at_node["bedrock__elevation"]
>>> for _ in range(200):
...     rock_elev[grid.core_nodes] += 1.0
...     elev[grid.core_nodes] += 1.0
...     fa.run_one_step()
...     eroder.run_one_step(10000.0)
...
>>> int(elev[4] * 100)
2266

Initialize GravelBedrockEroder.

__init__(grid, intermittency_factor=0.01, transport_coefficient=0.041, abrasion_coefficient=0.0, sediment_porosity=0.35, depth_decay_scale=1.0, plucking_coefficient=0.0001, coarse_fraction_from_plucking=1.0)

Initialize GravelBedrockEroder.

calc_abrasion_rate()

Update the volume rate of bedload loss to abrasion, per unit area.

Here we use the average of incoming and outgoing sediment flux to calculate the loss rate to abrasion. The result is stored in the bedload_sediment__rate_of_loss_to_abrasion field.

The factor dx (node spacing) appears in the denominator to represent flow segment length (i.e., length of the link along which water is flowing in the cell) divided by cell area. This would need to be updated to handle non-raster and/or non-uniform grids.

Examples

>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> grid = RasterModelGrid((3, 3), xy_spacing=1000.0)
>>> elev = grid.add_zeros("topographic__elevation", at="node")
>>> elev[3:] = 10.0
>>> sed = grid.add_zeros("soil__depth", at="node")
>>> sed[3:] = 100.0
>>> fa = FlowAccumulator(grid)
>>> fa.run_one_step()
>>> eroder = GravelBedrockEroder(grid, abrasion_coefficient=0.0002)
>>> eroder.calc_transport_rate()
>>> eroder.calc_abrasion_rate()
>>> int(eroder._abrasion[4] * 1e8)
19

calc_bedrock_abrasion_rate()

Update the rate of bedrock abrasion.

Note: assumes _abrasion (of sediment) and _rock_exposure_fraction have already been updated. Like _abrasion, the rate is a length per time (equivalent to rate of lowering of the bedrock surface by abrasion). Result is stored in the field bedrock__abrasion_rate.

>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> grid = RasterModelGrid((3, 4), xy_spacing=100.0)
>>> elev = grid.add_zeros("topographic__elevation", at="node")
>>> elev[:] = 0.01 * grid.x_of_node
>>> sed = grid.add_zeros("soil__depth", at="node")
>>> sed[:] = 1.0
>>> fa = FlowAccumulator(grid)
>>> fa.run_one_step()
>>> eroder = GravelBedrockEroder(grid, abrasion_coefficient=1.0e-4)
>>> eroder.calc_rock_exposure_fraction()
>>> round(eroder._rock_exposure_fraction[6], 4)
0.3679
>>> eroder.calc_transport_rate()
>>> np.round(eroder._sediment_outflux[5:7], 3)
array([ 0.024,  0.012])
>>> eroder.calc_abrasion_rate()
>>> np.round(eroder._abrasion[5:7], 9)
array([  1.20000000e-08,   6.00000000e-09])
>>> eroder.calc_bedrock_abrasion_rate()
>>> np.round(eroder._rock_abrasion_rate[5:7], 10)
array([  4.40000000e-09,   2.20000000e-09])

calc_bedrock_plucking_rate()

Update the rate of bedrock erosion by plucking.

The rate is a volume per area per time [L/T], equivalent to the rate of lowering of the bedrock surface relative to the underlying material as a result of plucking. Result is stored in the field bedrock__plucking_rate.

Examples

>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> grid_res = 100.0
>>> grid = RasterModelGrid((3, 3), xy_spacing=grid_res)
>>> elev = grid.add_zeros("topographic__elevation", at="node")
>>> elev[4] = 1.0
>>> sed = grid.add_zeros("soil__depth", at="node")
>>> fa = FlowAccumulator(grid)
>>> fa.run_one_step()
>>> eroder = GravelBedrockEroder(grid)
>>> eroder.calc_rock_exposure_fraction()
>>> eroder.calc_bedrock_plucking_rate()
>>> predicted_plucking_rate = 1.0e-6 * 1.0e4 * 0.01 ** (7.0 / 6.0) / grid_res
>>> round(predicted_plucking_rate, 9)  # Kp Q S^(7/6)
4.64e-07
>>> int(round(eroder._pluck_rate[4] * 1e9))
464

calc_implied_depth(grain_diameter=0.01)

Utility function that calculates and returns water depth implied by slope and grain diameter, using Wickert & Schildgen (2019) equation 8.

The equation is:

h = ((rho_s - rho / rho)) * (1 + epsilon) * tau_c * (D / S)

where the factors on the right are sediment and water density, excess shear-stress factor, critical Shields stress, grain diameter, and slope gradient. Here the prefactor on D/S assumes sediment density of 2650 kg/m3, water density of 1000 kg/m3, shear-stress factor of 0.2, and critical Shields stress of 0.0495, giving a value of 0.09801.

Examples

>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> grid = RasterModelGrid((3, 3), xy_spacing=1000.0)
>>> elev = grid.add_zeros("topographic__elevation", at="node")
>>> elev[3:] = 10.0
>>> sed = grid.add_zeros("soil__depth", at="node")
>>> sed[3:] = 100.0
>>> fa = FlowAccumulator(grid)
>>> fa.run_one_step()
>>> eroder = GravelBedrockEroder(grid)
>>> water_depth = eroder.calc_implied_depth()
>>> int(water_depth[4] * 1000)
98

calc_implied_width(grain_diameter=0.01, time_unit='y')

Utility function that calculates and returns channel width implied by discharge, slope, and grain diameter, using Wickert & Schildgen (2019) equation 16.

The equation is:

b = kb * Q * S**(7/6) / D**(3/2)

where the dimensional prefactor, which includes sediment and water density, gravitational acceleration, critical Shields stress, and the transport factor epsilon, is:

kb = 0.17 g**(-1/2) (((rho_s - rho) / rho) (1 + eps) tau_c*)**(-5/3)

Using g = 9.8 m/s2, rho_s = 2650 (quartz), rho = 1000 kg/m3, eps = 0.2, and tau_c* = 0.0495, kb ~ 2.61 s/m**(1/2). Converting to years, kb = 8.26e-8.

Examples

>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> grid = RasterModelGrid((3, 3), xy_spacing=10000.0)
>>> elev = grid.add_zeros("topographic__elevation", at="node")
>>> elev[3:] = 100.0
>>> sed = grid.add_zeros("soil__depth", at="node")
>>> sed[3:] = 100.0
>>> fa = FlowAccumulator(grid)
>>> fa.run_one_step()
>>> eroder = GravelBedrockEroder(grid)
>>> chan_width = eroder.calc_implied_width()
>>> int(chan_width[4] * 100)
3833
>>> grid.at_node["surface_water__discharge"] *= 1.0 / (3600 * 24 * 365.25)
>>> chan_width = eroder.calc_implied_width(time_unit="s")
>>> int(chan_width[4] * 100)
3838

calc_rock_exposure_fraction()

Update the bedrock exposure fraction.

The result is stored in the bedrock__exposure_fraction field.

Examples

>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> grid = RasterModelGrid((3, 4), xy_spacing=100.0)
>>> elev = grid.add_zeros("topographic__elevation", at="node")
>>> sed = grid.add_zeros("soil__depth", at="node")
>>> sed[4] = 1000.0
>>> sed[5] = 0.0
>>> fa = FlowAccumulator(grid)
>>> fa.run_one_step()
>>> eroder = GravelBedrockEroder(grid)
>>> eroder.calc_rock_exposure_fraction()
>>> eroder._rock_exposure_fraction[4:6]
array([ 0.,  1.])
>>> sed[4] = 1.0  # exposure frac should be 1/e ~ 0.3679
>>> sed[5] = 2.0  # exposure frac should be 1/e^2 ~ 0.1353
>>> eroder.calc_rock_exposure_fraction()
>>> np.round(eroder._rock_exposure_fraction[4:6], 4)
array([ 0.3679,  0.1353])

calc_sediment_influx()

Update the volume influx at each node.

Result is stored in the field bedload_sediment__volume_influx.

calc_sediment_rate_of_change()

Update the rate of thickness change of coarse sediment at each core node.

Result is stored in the field sediment__rate_of_change.

Examples

>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> grid = RasterModelGrid((3, 4), xy_spacing=100.0)
>>> elev = grid.add_zeros("topographic__elevation", at="node")
>>> elev[:] = 0.01 * grid.x_of_node
>>> sed = grid.add_zeros("soil__depth", at="node")
>>> sed[:] = 100.0
>>> grid.status_at_node[grid.perimeter_nodes] = grid.BC_NODE_IS_CLOSED
>>> grid.status_at_node[4] = grid.BC_NODE_IS_FIXED_VALUE
>>> fa = FlowAccumulator(grid)
>>> fa.run_one_step()
>>> eroder = GravelBedrockEroder(grid)
>>> eroder.calc_transport_rate()
>>> eroder.calc_sediment_influx()
>>> eroder.calc_sediment_rate_of_change()
>>> np.round(eroder._sediment_outflux[4:7], 3)
array([ 0.   ,  0.038,  0.019])
>>> np.round(eroder._sediment_influx[4:7], 3)
array([ 0.038,  0.019,  0.   ])
>>> np.round(eroder._dHdt[5:7], 8)
array([ -2.93000000e-06,  -2.93000000e-06])

calc_transport_rate()

Calculate and return bed-load transport rate.

Calculation uses Wickert-Schildgen approach, and provides volume per time rate. Transport rate is modulated by available sediment, using the exponential function (1 - exp(-H / Hs)), so that transport rate approaches zero as sediment thickness approaches zero. Rate is a volume per time. The result is stored in the bedload_sediment__volume_outflux field.

Examples

>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> grid = RasterModelGrid((3, 3), xy_spacing=100.0)
>>> elev = grid.add_zeros("topographic__elevation", at="node")
>>> elev[3:] = 1.0
>>> sed = grid.add_zeros("soil__depth", at="node")
>>> sed[3:] = 100.0
>>> fa = FlowAccumulator(grid)
>>> fa.run_one_step()
>>> eroder = GravelBedrockEroder(grid)
>>> eroder.calc_transport_rate()
>>> round(eroder._sediment_outflux[4], 4)
0.019

run_one_step(global_dt)

Advance solution by time interval global_dt, subdividing into sub-steps as needed.

update_rates()

Update rate of sediment thickness change, and rate of bedrock lowering by abrasion and plucking.

Combined rate of rock lowering relative to underlying material is stored in the field bedrock__lowering_rate.

Examples

>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> from landlab.components import FlowAccumulator
>>> grid = RasterModelGrid((3, 4), xy_spacing=100.0)
>>> elev = grid.add_zeros("topographic__elevation", at="node")
>>> elev[:] = 0.01 * grid.x_of_node
>>> sed = grid.add_zeros("soil__depth", at="node")
>>> sed[:] = 1000.0
>>> grid.status_at_node[grid.perimeter_nodes] = grid.BC_NODE_IS_CLOSED
>>> grid.status_at_node[4] = grid.BC_NODE_IS_FIXED_VALUE
>>> fa = FlowAccumulator(grid)
>>> fa.run_one_step()
>>> eroder = GravelBedrockEroder(grid)
>>> eroder.run_one_step(1000.0)
>>> np.round(elev[4:7], 4)
array([ 0.    ,  0.9971,  1.9971])