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==Notes== | ==Notes== | ||
The model incorporates both detachment-limited erosion on slopes and steep headwater channels as well as sediment transport in alluvial channels. The channels can be classified as nonalluvial and alluvial channels. The nonalluvial channel is defined as one in which the bed load sediment flux is less than a capacity load. | The model incorporates both detachment-limited erosion on slopes and steep headwater channels as well as sediment transport in alluvial channels. The channels can be classified as nonalluvial and alluvial channels. The nonalluvial channel is defined as one in which the bed load sediment flux is less than a capacity load. |
Revision as of 11:53, 24 August 2011
MARSSIM
The MARSSIM model is a landform evolution model primarily focuses on relatively long temporal scales (relative to the timescale for noticeable landform change) through fluvial and mass wasting processes.
Model introduction
The program is designed be computationally efficient such that individual runs can be done on a modern microcomputer in no more than a few tens of hours. The more recent additions to the model have focused on processes relevant to planetary landscapes, including lava flows, groundwater seepage and sapping, impact cratering, surface-normal accretion and ablation, and volatile redistribution by radiation-induced sublimation and recondensation. Individual process formulations vary from completely heuristic to modestly mechanistic. Important limitations for some potential applications are the assumption of a single representative bed material grain size in the fluvial system and no tracking of internal stratigraphy of sedimentary deposits.
Model parameters
Uses ports
This will be something that the CSDMS facility will add
Provides ports
This will be something that the CSDMS facility will add
Main equations
- Local vertical rate of land surface elevation change
[math]\displaystyle{ {\frac{\partial y}{\partial t}} - U = {\frac{\partial z}{\partial t}} = - \nabla \ast q }[/math] (1)
- Weathering, Rain Slash, and Mass Movement
1) Potential erosion or deposition due to rain splash and regolith mass movement
[math]\displaystyle{ {\frac{\partial z}{\partial t}}|_{m} = -\nabla \ast q_{m} }[/math] (2)
2) The rate of movement
[math]\displaystyle{ q_{m} = [K_{s} \wp \left (S\right ) + K_{f}{\frac{1}{\left (1 - K_{x}|S|^a\right )}}] s }[/math] (3)
[math]\displaystyle{ \wp \left (S\right ) = sin \theta }[/math] (4)
A linear dependency
[math]\displaystyle{ \wp \left (S\right ) = tan \theta = S }[/math] (5)
- Fluvial Processes
1) Potential channel deposition or erosion
[math]\displaystyle{ {\frac{\partial z}{\partial t}}|_{c} = - \nabla \ast q_{s} = - {\frac{\partial q_{s}}{\partial x}} }[/math] (6)
2) Nonalluvial channels (A) Detachment capacity
[math]\displaystyle{ -{\frac{\partial z}{\partial t}}|_{c} = {\frac{\partial q_{s}}{\partial x}} = C_{d} = K_{t} \left (\tau - \tau_{c}\right ) }[/math] (7)
(B) equations for steady, uniform flow
[math]\displaystyle{ \tau = \gamma RS }[/math] (8)
[math]\displaystyle{ V = K_{n}R^\left ({\frac{2}{3}}\right ) S^\left ({\frac{1}{2}}\right ) / N }[/math] (9)
[math]\displaystyle{ Q_{w} = K_{p} RWV }[/math] (10)
(C) Simple power law equations
[math]\displaystyle{ Q_{w} = K_{a} A^e }[/math] (11)
[math]\displaystyle{ W = K_{w}Q_{w}^b = K_{w} K_{a}^b A^ \left (be\right ) }[/math] (12)
(D)
[math]\displaystyle{ {\frac{\partial z}{\partial t}}|_{c} = - K_{t} \left (K_{z} A^g S^h - \tau_{c}\right ) }[/math] (13)
[math]\displaystyle{ g = 0.6 e \left (1 - b \right ) }[/math] (14)
[math]\displaystyle{ h = 0.7 }[/math] (15)
[math]\displaystyle{ K_{z} = \gamma \{{\frac{N K_{a}^\left (1 - b\right )}{K_{p} K_{w} K_{n}}}^ \left ({\frac{3}{5}}\right ) }[/math] (16)
(E) Channel elevation change (a) The net elevation change of the channel when sufficiently large amount of regolith is delivered to the cell (the channel never erodes bedrock)
[math]\displaystyle{ {\frac{\partial z}{\partial t}} = {\frac{\partial z}{\partial t}}|_{m} + F \xi {\frac{\partial z}{\partial t}}|_{c} }[/math] (17)
[math]\displaystyle{ \xi = W / \delta }[/math] (18)
(b) The channel erosion when the channel is capable of eroding all regolith delivered to it
[math]\displaystyle{ \epsilon = {\frac{\partial z}{\partial t}}|_{m} \delta^2 - {\frac{\partial z}{\partial t}}\delta \left ( \delta - W \right ) }[/math] (19)
The fraction of time the channel spends eroding regolith
[math]\displaystyle{ \eta = {\frac{\epsilon}{F \xi \delta^2 {\frac{\partial z}{\partial t}}|_{c}}} }[/math] (20)
The net erosion into bedrock for the channel
[math]\displaystyle{ {\frac{\partial z}{\partial t}} = \left ( 1 - \eta \right ) {\frac{\partial z}{\partial c}}|_{c} = \beta \{{\frac{\partial z}{\partial t}}|_{m} + F \xi {\frac{\partial z}{\partial t}}|_{c} \} }[/math] (21)
[math]\displaystyle{ \beta = {\frac{1}{F \xi + 1 - \xi}} }[/math] (22)
(c) Channel erosion when the volume of regolith delivered to the cell per unit time is negative (the removal of regolith by channel erosion pertains only to the volume surrounding the stream and below the level of removal by slope erosion)
[math]\displaystyle{ \xi = \{{\frac{\partial z}{\partial t}}|_{m} - {\frac{\partial z}{\partial t}}\} \delta \left (\delta - W \right ) }[/math] (23)
The net erosion into bedrock
[math]\displaystyle{ \{{\frac{\partial z}{\partial t}}|_{m} - {\frac{\partial z}{\partial t}}\} = \left ( 1 - \eta \right ) {\frac{\partial z}{\partial t}}|_{c} }[/math] (24)
3) Alluvial channels (A) Potential rate of fluvial erosion
[math]\displaystyle{ {\frac{\partial z}{\partial t}}|_{c} = - \nabla \ast q_{sb} }[/math] (25)
[math]\displaystyle{ \phi = \Im \left ({\frac{1}{\Psi}}\right ) = {\frac{q_{sb}}{\omega d \left ( 1 - \mu \right )}} }[/math] (26)
[math]\displaystyle{ {\frac{1}{\Psi}} = {\frac{\tau}{\left ( \gamma_{s} - \gamma \right ) d }} }[/math] (27)
(B) Bed load or total bedload formulas
[math]\displaystyle{ \phi = K_{e} \{{\frac{1}{\Psi}} - {\frac{1}{\Psi_{c}}}\} ^p }[/math] (28)
(C) Total bed sediment discharge
[math]\displaystyle{ Q_{sb} = K_{q} A^ \left (eb \right ) [K_{v} A^ \left (0.6 e \left (1 - b \right ) \right ) S^ \left (0.7\right ) - 1/ \Psi_{c}]^ p }[/math] (29)
[math]\displaystyle{ K_{q} = {\frac{K_{e} \omega \left (1 - \mu \right ) K_{w} K_{a}^b}{\left ( \gamma_{s} - \gamma \right )^p d^ \left ( p - 1 \right )}} }[/math] (30)
[math]\displaystyle{ K_{v} = \gamma \{{\frac{N K_{a}^ \left (1-b\right) }{K_{w} K_{n} K_{p}}}\} ^ \left (0.6\right ) }[/math] (31)
(D) The net erosion of channel
[math]\displaystyle{ {\frac{\partial z}{\partial t}}|_{c} = {\frac{\left (Q_{sbi} - Q_{sbo}\right )}{LW}} }[/math] (32)
(4) Overall volume V of the alluvial channel between the nonalluvial upstream node at elevation Z0 and the downstream unmodified node with elevation En+1
[math]\displaystyle{ V = \delta^2 [\left (n + {\frac{1}{2}} \right ) Z_{0} - n S_{0} l_{0} - \Sigma_{i = 1} ^ \left ( n-1\right ) S_{i} l_{i} + {\frac{1}{2}}Z_{n+1} ] }[/math] (33)
The net deposited volume of sediment for an individual nonalluvial upstream contribution during the time step
[math]\displaystyle{ Q_{sj} \Delta t = v - V }[/math] (34)
Upstream gradient of the alluvial channel node
[math]\displaystyle{ \Im_{0} = S_{0} - [\Sigma_{i=1} ^ \left (n-1 \right ) \left ( n - i \right ) \left ( S_{i} - \Im_{i} \right ) l_{i} + \left (Q_{sj} \Delta t \right ) / \delta^2] / [n l_{0}] }[/math] (35)
Gradient of the alluvial channel node n
[math]\displaystyle{ \Im_{n} = [\left (Z_{0} - Z_{n+1} \right ) - \Im_{0}l_{0} - \Sigma_{i=1}^ \left (n-1\right ) \Im_{i}l_{i}] }[/math] (36)
The new elevations of the alluvial channel point i when all of the bed sediment supplied form upstream is deposited
[math]\displaystyle{ \Im_{i} = Z_{0} - \Im_{0}l_{0} - \Sigma_{k=1}^ \left (i-1 \right ) \Im_{k}l_{k} }[/math] (37)
The new elevations of the alluvial channel point i when location n is a fixed boundary point with elevation Zf
[math]\displaystyle{ \Im_{i} = Z_{f} + \Sigma_{k = n-1}^ \left (i+1\right ) \Im_{k}l_{k} }[/math] (38)
Symbol | Description | Unit |
---|---|---|
U | local tectonic uplift rate | - |
q | spatial divergence of the vector of eroded material flux | - |
y | local land surface elevation | - |
z | land surface elevation caused by fluvial erosion relative to a bedrock-fixed reference frame | - |
m | Potential erosion or deposition due to rain splash and regolith mass movement | - |
qm | spatial divergence of the vector rate of movement | - |
\wp (S) | an increasing function of slope gradient | - |
s | the unit vector in the direction of S | - |
S | local slope gradient | - |
Ks, Kx, Kf | constants assumed to be spatially and temporally invariant | - |
a | exponent assumed to be spatially and temporally invariant | - |
θ | slope angle | - |
x | downstream direction of coordinate | - |
c | potential channel deposition or erosion | - |
qs | volumetric sediment transport rate per unit channel width | - |
qsw | wash load (a part of sediment discharge, which is assumed to never be redeposited except in depressions) | - |
qsb | bed load (including bed load and suspended load), which is carried in capacity amounts if an alluvial bed is present | - |
τ | shear stress | - |
τc | critical shear stress | - |
C | actual detachment rate | - |
Cd | intrinsic detachment capacity | - |
γ | unit weight of water | - |
R | the hydraulic radius | - |
N | Manning's resistance coefficient | - |
Kp | a form factor close to unity | - |
Qw | hydraulic geometry for dominant discharge (Assumed to be equal for both bed erosion and sediment transport) | - |
W | channel width | - |
Kt | this constant includes both effects of substrate erodibility as well as magnitude of the dominant discharge | - |
τc | critical Shields number for the onset of sediment motion | - |
ψs | the fraction of bed shear stress | - |
qt * | Einstein number | - |
qt | volume sediment transport rate per unit width | L2 / T |
If | flood intermittency | - |
tf | cumulative time the river has been in flood | T |
Gt | the annual sediment yield | M / T |
ta | the number of seconds in a year | - |
Qf | sediment transport rate during flood discharge | L2 / T |
αt | dimensionless coefficient in the sediment transport equation, equals to 8 | - |
nt | exponent in sediment transport relation, equals to 1.5 | - |
τc * | reference Shields number in sediment transport relation, equals to 0.047 | |
CZ | dimensionless Chezy resistance coefficient. | |
Sl | initial bed slope of the river | - |
ηi | initial bed elevation | L |
τ | shear stress on bed surface | - |
qb | bed material load | M / T |
Δx | spatial step length, equals to L / M | L |
Qw | flood discharge | L3 / T |
Δt | time step | T |
Ntoprint | number of time steps to printout | - |
Nprint | number of printouts | - |
aU | upwinding coefficient (1=full upwind, 0.5=central difference) | - |
αs | coefficient in sediment transport relation | - |
u* | shear velocity | L / T |
αr | coefficient in Manning-Strickler resistance relation | - |
τb * | non-dimensional total shear stress | - |
Notes
The model incorporates both detachment-limited erosion on slopes and steep headwater channels as well as sediment transport in alluvial channels. The channels can be classified as nonalluvial and alluvial channels. The nonalluvial channel is defined as one in which the bed load sediment flux is less than a capacity load.
The simple power law equations of hydraulic geometry for dominant discharge Qw (equation 11 ) is used for runoff on slopes as well as in channels implies that little regolith or depression storage of precipitation occurs during erosion events, since runoff production is assumed to be areally uniform. On badland slopes this is probably a reasonable approximation. For landscapes with appreciable infiltration capacity, particularly where vegetated, the assumption is made that most runoff erosion on slopes occurs during infrequent, very intense rainfall events due to saturateion overland flow or shallow interflow through large macropores.
Channel maybe eroding both regolith material which is delivered into the channel by mass wasting as well as uneroded bedrock. Regolith material is assumed to be more erodible than bedrock by a factor F, where F >= 1. In landscapes with a shallow weathering regolith, such as in badlands and steep mountain slopes, bedrock is commonly exposed in headwater rills and channels, and a large ratio of bedrock to regolith erodibility (F >> 1) would be appropriate. However, in landscapes in deeply saprolitized bedrock, till, or uncemented alluvium, the parent material and the surface soil involved in mass wasting may have nearly equivalent erodibility (F ~ 1). In calculating net erosion in matrix cells containing nonalluvial channels, during each time step, the channel is assumed to erode first regolith delivered to the channel by slope erosion and then, if it is capable of eroding all regolith, bedrock. The fraction of time during each time step that the channel is eroding
Examples
An example run with input parameters, BLD files, as well as a figure / movie of the output
Follow the next steps to include images / movies of simulations:
- Upload file: http://csdms.colorado.edu/wiki/Special:Upload
- Create link to the file on your page: [[Image:<file name>]].
See also: Help:Images or Help:Movies
Developer(s)
References
Key papers