The basic equations governing the evolution of the surface in SPMODEL evolution are as follows. Default parameters are listed in Table 1.
Shortrange transport is considered as a linear diffusive process
1)
where k_{s} is
the shortrange material transport coefficent (or effective hillslope diffusivity)
and can be specified by the following xmlfile parameters:
<param name="bedRockDiffusivity"> 2.0e1 </param>
<param
name="sedimentDiffusivity"> 1.0e2</param>
At depths below sealevel we apply this formulation to simulate downslope transport of submarine sediment, specified by the following xmlfile 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 shortrange transport. This is viable since the longrange transport algorithm typically provides a much more stingent constraint on timestepping.
Longrange transport is modelled as a firstorder kinetic reaction
based on the local fluvial carrying capacity. The fluvial carrying capacity,
q, is defined :
where k_{f} is the longrange
(or fluvial) material transport coefficient used to define the channel erosivity,
sepcified by the
<param name="fluvialConstant"> 3.0e6 </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 longrange 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 heightfield computed at the previous timestep.
Downstream entrainment of sediment is governed by a firstorder kinetic equation
of the type:
where L_{d} defines the characteristic
lengthscale for fluvial entrainment and can be sepcified by the following xmlfile
pramaters:
<param name="bedRockErosionLengthScale"> 10e3 </param>
<param name="sedimentErosionLengthScale"> 1e3</param>
As with the shortrange transport, we computre the log rnage stansport with an explicit differencing scheme. This imposes stringent limits on the timstep (Raq amplify).
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 xmlfile:
<param
name="flexureInterval"> 100 </param>
The &betarelease does not explicitly include lake formation, groundwater interactions (recharge via surface infiltration and or discharge) or evapotranspiration. 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.
This &betarelease
Table 1. User specified parameters
Symbol 
Description 
xml file name 
Units 
Typical value 
k_{s} 
shortrange material 
sedimentdiffusivity 
m^{2}/s 
1.0e1 
k_{f} 
longrange material 
fluvialConstant 

4e3 
L_{d} 
fluvial entrainment 
sedimentErosionLengthScale 
m 
1e3 

characteristic precipitation rate 
oroRate 
m/yr 
0.5 

orographic preciptation 
oroLength 
m 
0 
S_{L} 
sea level 
seaLevel 
m 

F_{u} 
tectonic uplift rate 
upliftRate 
m/yr 
 
&rho_{c} 
upper crustal density 
bedRockDensity 
kg/m^{3} 
3.8e3 
&rho_{m} 
sediment density 
sedimentDensity 
kg/m^{3} 
2.4e3 
&rho_{a} 
mantle desnity 
asthenosphereDensity 
kg/m^{3} 
3.3e3 
D 
lithospheric flexural rigidity 
flexuralRigidity 

3.0e23 





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