class BasicDdRt

Model BasicDdRt

terrainbento BasicDdRt model program.

Erosion model program using linear diffusion, stream power with stream power with a smoothed threshold that increases with incision depth and spatially varying erodibility based on two bedrock units, and discharge proportional to drainage area.

Landlab components used:

1. FlowAccumulator

2. DepressionFinderAndRouter (optional)

3. StreamPowerSmoothThresholdEroder

4. LinearDiffuser

class BasicDdRt(clock, grid, water_erosion_rule__threshold=0.01, water_erosion_rule__thresh_depth_derivative=0.0, **kwargs)

Bases: terrainbento.base_class.two_lithology_erosion_model.TwoLithologyErosionModel

BasicDdRt model program.

This model program combines the BasicRt and BasicDd programs by allowing for two lithologies, an “upper” layer and a “lower” layer, and permitting the use of an smooth erosion threshold that increases with erosion depth. Given a spatially varying contact zone elevation, \(\eta_C(x,y))\), model BasicDdRt evolves a topographic surface described by \(\eta\) with the following governing equations:

\[ \begin{align}\begin{aligned}\frac{\partial \eta}{\partial t} = -\left[\omega - \omega_{ct} (1 - e^{-\omega /\omega_{ct}}) \right] + D\nabla^2 \eta,\\\omega = K(\eta, \eta_C) Q^{m} S^{n}\\K(\eta, \eta_C ) = w K_1 + (1 - w) K_2\\\omega_{ct}(x,y,t) = \max(\omega_c + b D_I(x,y,t)\\w = \frac{1}{1+\exp \left( -\frac{(\eta -\eta_C )}{W_c}\right)}\end{aligned}\end{align} \]

where \(Q\) is the local stream discharge, \(S\) is the local slope, \(m\) and \(n\) are the discharge and slope exponent parameters, \(W_c\) is the contact-zone width, \(K_1\) and \(K_2\) are the erodabilities of the upper and lower lithologies, \(\omega_{c}\) is the in initial erosion threshold (for both lithologies) and \(b\) is the rate of change of threshold with increasing cumulative incision \(D_I(x,y,t)\), and \(D\) is the regolith transport parameter, \(w\) is a weight used to calculate the effective erodibility \(K(\eta, \eta_C)\) based on the depth to the contact zone and the width of the contact zone. \(\omega\) is the erosion rate that would be calculated without the use of a threshold and as the threshold increases the erosion rate smoothly transitions between zero and \(\omega\).

The weight \(w\) promotes smoothness in the solution of erodibility at a given point. When the surface elevation is at the contact elevation, the erodibility is the average of \(K_1\) and \(K_2\); above and below the contact, the erodibility approaches the value of \(K_1\) and \(K_2\) at a rate related to the contact zone width. Thus, to make a very sharp transition, use a small value for the contact zone width.

Refer to Barnhart et al. (2019) Table 5 for full list of parameter symbols, names, and dimensions.

The following at-node fields must be specified in the grid:

• topographic__elevation

• lithology_contact__elevation

__init__(clock, grid, water_erosion_rule__threshold=0.01, water_erosion_rule__thresh_depth_derivative=0.0, **kwargs)

Parameters

• clock (terrainbento Clock instance) –

• grid (landlab model grid instance) – The grid must have all required fields.

• m_sp (float, optional) – Drainage area exponent (\(m\)). Default is 0.5.

• n_sp (float, optional) – Slope exponent (\(n\)). Default is 1.0.

• water_erodibility_upper (float, optional) – Water erodibility of the upper layer (\(K_{1}\)). Default is 0.001.

• water_erodibility_lower (float, optional) – Water erodibility of the upper layer (\(K_{2}\)). Default is 0.0001.

• contact_zone__width (float, optional) – Thickness of the contact zone (\(W_c\)). Default is 1.

• regolith_transport_parameter (float, optional) – Regolith transport efficiency (\(D\)). Default is 0.1.

• water_erosion_rule__threshold (float, optional) – Erosion rule threshold when no erosion has occured (\(\omega_c\)). Default is 0.01.

• water_erosion_rule__thresh_depth_derivative (float, optional) – Rate of increase of water erosion threshold as increased incision occurs (\(b\)). Default is 0.0.

• **kwargs – Keyword arguments to pass to TwoLithologyErosionModel. Importantly these arguments specify the precipitator and the runoff generator that control the generation of surface water discharge (\(Q\)).

Returns

BasicDdRt

Return type

model object

Examples

This is a minimal example to demonstrate how to construct an instance of model BasicDdRt. For more detailed examples, including steady- state steady-state test examples, see the terrainbento tutorials.

To begin, import the model class.

>>> from landlab import RasterModelGrid
>>> from landlab.values import random, constant
>>> from terrainbento import Clock, BasicDdRt
>>> clock = Clock(start=0, stop=100, step=1)
>>> grid = RasterModelGrid((5,5))
>>> _ = random(grid, "topographic__elevation")
>>> _ = constant(grid, "lithology_contact__elevation", value=-10.)

Construct the model.

>>> model = BasicDdRt(clock, grid)

Running the model with model.run() would create output, so here we will just run it one step.

>>> model.run_one_step(1.)
>>> model.model_time
1.0

run_one_step(step)

Advance model BasicDdRt for one time-step of duration step.

The run_one_step method does the following:

1. Creates rain and runoff, then directs and accumulates flow.

2. Assesses the location, if any, of flooded nodes where erosion should not occur.

3. Assesses if a PrecipChanger is an active boundary handler and if so, uses it to modify the erodibility by water.

4. Updates the spatially variable erodibility value based on the relative distance between the topographic surface and the lithology contact.

5. Updates the threshold value based on the cumulative amount of erosion that has occured since model run onset.

6. Calculates detachment-limited erosion by water.

7. Calculates topographic change by linear diffusion.

8. Finalizes the step using the ErosionModel base class function finalize__run_one_step. This function updates all boundary handlers handlers by step and increments model time by step.

Parameters

step (float) – Increment of time for which the model is run.

main()

Executes model.