landlab.components.depth_dependent_taylor_soil_creep.hillslope_depth_dependent_taylor_flux¶

DepthDependentTaylorNonLinearDiffuser Component.

@author: R Glade @author: K Barnhart @author: G Tucker

class DepthDependentTaylorDiffuser[source]¶

Bases: Component

This component implements a depth-dependent Taylor series diffusion rule, combining concepts of Ganti et al. (2012) and Johnstone and Hilley (2014).

Hillslope sediment flux uses a Taylor Series expansion of the Andrews- Bucknam formulation of nonlinear hillslope flux derived following following Ganti et al., 2012 with a depth dependent component inspired Johnstone and Hilley (2014). The flux \(q_s\) is given as:

\[q_s = - K H_* \nabla \eta ( 1 + (S/S_c)^2 + (S/S_c)^4 + .. + (S/S_c)^2(n-1) ) (1 - exp( - H / H_*)\]

where \(K\) is a transport velocity coefficient, \(\eta\) is land surface elevation, \(S\) is the slope gradient (defined as positive downward), \(S_c\) is the critical slope, \(n\) is the number of terms, \(H\) is the soil depth on links, and \(H_*\) is the soil transport decay depth.

The default behavior uses two terms to produce a slope dependence as described by Equation 6 of Ganti et al. (2012).

This component will ignore soil thickness located at non-core nodes.

Examples

First lets make a simple example with flat topography.

>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> from landlab.components import ExponentialWeatherer
>>> from landlab.components import DepthDependentTaylorDiffuser
>>> mg = RasterModelGrid((5, 5))
>>> soilTh = mg.add_zeros("soil__depth", at="node")
>>> z = mg.add_zeros("topographic__elevation", at="node")
>>> BRz = mg.add_zeros("bedrock__elevation", at="node")
>>> expweath = ExponentialWeatherer(mg)
>>> DDdiff = DepthDependentTaylorDiffuser(mg)
>>> expweath.calc_soil_prod_rate()
>>> np.allclose(mg.at_node["soil_production__rate"][mg.core_nodes], 1.0)
True
>>> DDdiff.run_one_step(2.0)
>>> np.allclose(mg.at_node["topographic__elevation"][mg.core_nodes], 0.0)
True
>>> np.allclose(mg.at_node["bedrock__elevation"][mg.core_nodes], -2.0)
True
>>> np.allclose(mg.at_node["soil__depth"][mg.core_nodes], 2.0)
True

Now a more complicated example with a slope.

>>> mg = RasterModelGrid((3, 5))
>>> soilTh = mg.add_zeros("soil__depth", at="node")
>>> z = mg.add_zeros("topographic__elevation", at="node")
>>> BRz = mg.add_zeros("bedrock__elevation", at="node")
>>> z += mg.node_x.copy()
>>> BRz += mg.node_x / 2.0
>>> soilTh[:] = z - BRz
>>> expweath = ExponentialWeatherer(mg)
>>> DDdiff = DepthDependentTaylorDiffuser(mg)
>>> expweath.calc_soil_prod_rate()
>>> np.allclose(
...     mg.at_node["soil_production__rate"][mg.core_nodes],
...     np.array([0.60653066, 0.36787944, 0.22313016]),
... )
True
>>> DDdiff.run_one_step(0.1)
>>> np.allclose(
...     mg.at_node["topographic__elevation"][mg.core_nodes],
...     np.array([1.04773024, 2.02894986, 3.01755898]),
... )
True
>>> np.allclose(
...     mg.at_node["bedrock__elevation"][mg.core_nodes],
...     np.array([0.43934693, 0.96321206, 1.47768698]),
... )
True
>>> np.allclose(mg.at_node["soil__depth"], z - BRz)
True

The DepthDependentTaylorDiffuser makes and moves soil at a rate proportional to slope, this means that there is a characteristic time scale for soil transport and an associated stability criteria for the timestep. The maximum characteristic time scale, \(De_{max}\), is given as a function of the hillslope diffustivity, \(D\), the maximum slope, \(S_{max}\), and the critical slope \(S_c\).

\[De_{max} = D \left( 1 + \left( \frac{S_{max}{S_c}\right )^2 + \left( \frac{S_{max}{S_c}\right )^4 + \dots + \left( \frac{S_{max}{S_c}\right )^{( 2 * ( n - 1 ))} \right)\]

The maximum stable time step is given by

\[dtmax = courant_factor * dx * dx / Demax\]

Where the courant factor is a user defined scale (default is 0.2), and dx is the length of the shortest link in the grid.

The DepthDependentTaylorDiffuser has a boolean flag that permits a user to be warned if timesteps are too large for the slopes in the model grid (if_unstable = ‘warn’) and a boolean flag that turns on dynamic timestepping (dynamic_dt = False).

>>> DDdiff = DepthDependentTaylorDiffuser(mg, if_unstable="warn")
>>> DDdiff.run_one_step(2.0)
Topographic slopes are high enough such that the Courant condition is
exceeded AND you have not selected dynamic timestepping with
dynamic_dt=True. This may lead to infinite and/or nan values for slope,
elevation, and soil depth. Consider using a smaller time step or dynamic
timestepping. The Courant condition recommends a timestep of
0.09534076073069653 or smaller.

Alternatively you can specify if_unstable=’raise’, and a Runtime Error will be raised if this condition is not met.

Next, lets do an example with dynamic timestepping.

>>> mg = RasterModelGrid((3, 5))
>>> soilTh = mg.add_zeros("soil__depth", at="node")
>>> z = mg.add_zeros("topographic__elevation", at="node")
>>> BRz = mg.add_zeros("bedrock__elevation", at="node")

We’ll use a steep slope and very little soil.

>>> z += mg.node_x.copy() ** 2
>>> BRz = z.copy() - 1.0
>>> soilTh[:] = z - BRz
>>> expweath = ExponentialWeatherer(mg)

Lets try to move the soil with a large timestep. Without dynamic time steps, this gives a warning that we’ve exceeded the dynamic timestep size and should use a smaller timestep. We could either use the smaller timestep, or specify that we want to use a dynamic timestep.

>>> DDdiff = DepthDependentTaylorDiffuser(mg, if_unstable="warn", dynamic_dt=False)
>>> expweath.calc_soil_prod_rate()
>>> DDdiff.run_one_step(10)
Topographic slopes are high enough such that the Courant condition is
exceeded AND you have not selected dynamic timestepping with
dynamic_dt=True. This may lead to infinite and/or nan values for slope,
elevation, and soil depth. Consider using a smaller time step or dynamic
timestepping. The Courant condition recommends a timestep of
0.004 or smaller.

Now, we’ll re-build the grid and do the same example with dynamic timesteps.

>>> mg = RasterModelGrid((3, 5))
>>> soilTh = mg.add_zeros("soil__depth", at="node")
>>> z = mg.add_zeros("topographic__elevation", at="node")
>>> BRz = mg.add_zeros("bedrock__elevation", at="node")
>>> z += mg.node_x.copy() ** 2
>>> BRz = z.copy() - 1.0
>>> soilTh[:] = z - BRz
>>> expweath = ExponentialWeatherer(mg)
>>> DDdiff = DepthDependentTaylorDiffuser(mg, if_unstable="warn", dynamic_dt=True)
>>> expweath.calc_soil_prod_rate()
>>> DDdiff.run_one_step(10)
>>> np.any(np.isnan(z))
False

Now, we’ll test that changing the transport decay depth behaves as expected.

>>> mg = RasterModelGrid((3, 5))
>>> soilTh = mg.add_zeros("soil__depth", at="node")
>>> z = mg.add_zeros("topographic__elevation", at="node")
>>> BRz = mg.add_zeros("bedrock__elevation", at="node")
>>> z += mg.node_x.copy() ** 0.5
>>> BRz = z.copy() - 1.0
>>> soilTh[:] = z - BRz
>>> expweath = ExponentialWeatherer(mg)
>>> DDdiff = DepthDependentTaylorDiffuser(mg, soil_transport_decay_depth=0.1)
>>> DDdiff.run_one_step(1)
>>> soil_decay_depth_point1 = mg.at_node["topographic__elevation"][mg.core_nodes]
>>> z[:] = 0
>>> z += mg.node_x.copy() ** 0.5
>>> BRz = z.copy() - 1.0
>>> soilTh[:] = z - BRz
>>> DDdiff = DepthDependentTaylorDiffuser(mg, soil_transport_decay_depth=1.0)
>>> DDdiff.run_one_step(1)
>>> soil_decay_depth_1 = mg.at_node["topographic__elevation"][mg.core_nodes]
>>> np.greater(soil_decay_depth_1[1], soil_decay_depth_point1[1])
False

References

Required Software Citation(s) Specific to this Component

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

Additional References

Ganti, V., Passalacqua, P., Foufoula-Georgiou, E. (2012). A sub-grid scale closure for nonlinear hillslope sediment transport models Journal of Geophysical Research: Earth Surface 117(F2). https://dx.doi.org/10.1029/2011jf002181

Johnstone, S., Hilley, G. (2015). Lithologic control on the form of soil-mantled hillslopes Geology 43(1), 83-86. https://doi.org/10.1130/G36052.1

Initialize the DepthDependentTaylorDiffuser.

Parameters:

grid (ModelGrid) – Landlab ModelGrid object
linear_diffusivity (float, optional, DEPRECATED) – Hillslope diffusivity / decay depth, m/yr Default = 1.0
slope_crit (float, optional) – Critical gradient parameter, m/m Default = 1.0
soil_transport_decay_depth (float, optional) – characteristic transport soil depth, m Default = 1.0
nterms (int, optional. default = 2) – number of terms in the Taylor expansion. Two terms (default) gives the behavior described in Ganti et al. (2012).
dynamic_dt (bool, optional, default = False) – Whether internal timestepping is used.
if_unstable (str, optional, default = "pass") – What to do if unstable (options are “pass”, “raise”, “warn”)
courant_factor (float, optional, default = 0.2) – Courant factor for timestep calculation.
soil_transport_velocity (float, optional, default = 1.0) – Velocity parameter for soil transport, m/yr. Diffusivity is the product of this parameter and soil_transport_decay_depth.

__init__(grid, linear_diffusivity=None, slope_crit=1.0, soil_transport_decay_depth=1.0, nterms=2, dynamic_dt=False, if_unstable='pass', courant_factor=0.2, soil_transport_velocity=1.0)[source]¶

Initialize the DepthDependentTaylorDiffuser.

Parameters:

grid (ModelGrid) – Landlab ModelGrid object
linear_diffusivity (float, optional, DEPRECATED) – Hillslope diffusivity / decay depth, m/yr Default = 1.0
slope_crit (float, optional) – Critical gradient parameter, m/m Default = 1.0
soil_transport_decay_depth (float, optional) – characteristic transport soil depth, m Default = 1.0
nterms (int, optional. default = 2) – number of terms in the Taylor expansion. Two terms (default) gives the behavior described in Ganti et al. (2012).
dynamic_dt (bool, optional, default = False) – Whether internal timestepping is used.
if_unstable (str, optional, default = "pass") – What to do if unstable (options are “pass”, “raise”, “warn”)
courant_factor (float, optional, default = 0.2) – Courant factor for timestep calculation.
soil_transport_velocity (float, optional, default = 1.0) – Velocity parameter for soil transport, m/yr. Diffusivity is the product of this parameter and soil_transport_decay_depth.

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 = (('bedrock__elevation', 'elevation of the bedrock surface'), ('soil__depth', 'Depth of soil or weathered bedrock'), ('soil__flux', 'flux of soil in direction of link'), ('soil_production__rate', 'rate of soil production at nodes'), ('topographic__elevation', 'Land surface topographic elevation'), ('topographic__slope', 'gradient of the ground surface'))¶

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 = ('soil__depth', 'soil_production__rate', 'topographic__elevation')¶

name = 'DepthDependentTaylorDiffuser'¶

optional_var_names = ()¶

output_var_names = ('bedrock__elevation', 'soil__depth', 'soil__flux', 'topographic__elevation', 'topographic__slope')¶

run_one_step(dt)[source]¶

Parameters:: dt (float (time)) – The imposed timestep.

property shape¶: Return the grid shape attached to the component, if defined.

soilflux(dt)[source]¶

Calculate soil flux for a time period ‘dt’.

Parameters:: dt (float (time)) – The imposed timestep.

unit_agnostic = True¶

units = (('bedrock__elevation', 'm'), ('soil__depth', 'm'), ('soil__flux', 'm^2/yr'), ('soil_production__rate', 'm/yr'), ('topographic__elevation', 'm'), ('topographic__slope', 'm/m'))¶

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 = (('bedrock__elevation', 'node'), ('soil__depth', 'node'), ('soil__flux', 'link'), ('soil_production__rate', 'node'), ('topographic__elevation', 'node'), ('topographic__slope', 'link'))¶

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