6. The transport equations

The basic equations governing the evolution of the surface in SPMODEL evolution are as follows. Default parameters are listed in Table 1.

6.1. Short-range transport

Short-range transport is considered as a linear diffusive process
1)
where ks is the short-range material transport coefficent (or effective hillslope diffusivity) and can be specified by the following xml-file parameters:

<param name="bedRockDiffusivity"> 2.0e1 </param>
<param name="sedimentDiffusivity"> 1.0e2</param>

At depths below sea-level we apply this formulation to simulate down-slope transport of submarine sediment, specified by the following xml-file parameters (see note in Section 1: Background for future plans):

<param name="submarineBedRockDiffusivity"> 1.0e1 </param>
<param name="submarineSedimentDiffusivity"> 3.0e3 </param>

We use an explicit difference method for computation of the short-range transport. This is viable since the long-range transport algorithm typically provides a much more stingent constraint on timestepping.

6.2. Long-range transport

Long-range transport is modelled as a first-order kinetic reaction based on the local fluvial carrying capacity. The fluvial carrying capacity, q, is defined :

where kf is the long-range (or fluvial) material transport coefficient used to define the channel erosivity, sepcified by the

<param name="fluvialConstant"> 3.0e-6 </param>

Q is the local flux of water and S is the local downstream slope. By default, coefficients m and n are not specified (ie, they are implicitly set to unity). However, Section 3: Plugins explains how to provide your own long-range transport law, with an example that specifically incorporates these coefficients.

In equation 2, Q is determined at runtime by routing all precipitation across the modelled landscape using a "bucket passing" algorithm. All precipitation is assumed to traverse the landscape with a charactertic timescale much shorter than the computational timstep. S is derived from the evolved height-field computed at the previous timestep.

Downstream entrainment of sediment is governed by a first-order kinetic equation of the type:

where Ld defines the characteristic length-scale for fluvial entrainment and can be sepcified by the following xml-file pramaters:

<param name="bedRockErosionLengthScale"> 10e3 </param>
<param name="sedimentErosionLengthScale"> 1e3</param>

As with the short-range transport, we computre the log rnage stansport with an explicit differencing scheme. This imposes stringent limits on the timstep (Raq amplify).

6.3. Flexural isostasy

Changes in suface loads induced by movement of material across the surface of the Earth, through fault displacements and through changes in sea level induce a flexural isostatic response. We assume that the domain of interest is characterised by uniform flexural properties (e.g., the flexural rigidity, D) We solve the thin elastic plate force balance to compute the isostatic deflection using a fourier domain method on a 2n x 2n grid overlayed onto the nodal heightfield. Interpolation of the computed flexural deflections onto nodal heightfield can be used via Cubic of linear interpolants via setting the appoririate xml parameter ie:

<param name="interpolation">Cubic</param>

By default, flexure is calculated at every timestep, but less frequent flexural updates can be set by adjusting the flexureInterval parameter in the xml-file:

<param name="flexureInterval"> 100 </param>

6.4. Infiltration, evaporation, transpiration and discharge

The &beta-release does not explicitly include lake formation, groundwater interactions (recharge via surface infiltration and or discharge) or evapo-transpiration. Implicitly all precipitation is converted to overland flow. At pits in the landscape water is assumed to infiltrate/evaporate at a rate greater than influx, so that no standing surface water bodies develop. Future relases will incorporate lake development through explicit definition of infiltration, evaporation, transpiration and/or discharge regimes.

6.5. Precipitation

This &beta-release

6.6. Kinematic faulting

 

Table 1. User specified parameters

Symbol

Description

xml file name

Units

Typical value

ks

short-range material
transport coefficents

sedimentdiffusivity
bedRockDiffusivity
submarineBedRockDiffusivity
submarineSedimentDiffusivity

m2/s

1.0e-1
1.0e-1
1.0e-1
1.0e-3

kf

long-range material
transport coefficients

fluvialConstant

 

4e-3

Ld

fluvial entrainment
length-scales

sedimentErosionLengthScale
bedRockErosionLengthScale

m

1e3
10e3

 

characteristic precipitation rate

oroRate

m/yr

0.5

 

orographic preciptation
length-scale

oroLength

m

0

SL

sea level

seaLevel

m

 

Fu

tectonic uplift rate

upliftRate

m/yr

-

&rhoc

upper crustal density

bedRockDensity

kg/m3

3.8e3

&rhom

sediment density

sedimentDensity

kg/m3

2.4e3

&rhoa

mantle desnity

asthenosphereDensity

kg/m3

3.3e3

D

lithospheric flexural rigidity

flexuralRigidity

 

3.0e23

 

 

 

 

 

 


Notes. See Braun & Sambridge (1977) for a detailed outline of the governing equations