EmiliaJarochowska
commented
10 months ago Thanks for this, @xyl96. Is it correct that it is the updated version after our discussion on Tuesday?
In the first equations: What is $k_v$? Over what area is $S$ calculated?
Can $P - I$ be simplified? By keeping both we introduce two new unknowns into the model. $P$ may seem easy to obtain, but will be hard to get for 1 ky timescales and I is even less constrained: can be near 0 to 100% $P$ depending on rock type. Instead of varying these two parameters we could just use a parameter that corresponds to the surface discharge?
This could have the advantage that if $P$ and I don’t vary between cells so they don’t need to be calculated per each cell?
However, it is still questionable whether the run-off generated from a single cell could produce enough shear stress to detach lithified carbonate rocks.
What are the critical values of shear stress to disintegrate lithified carbonates? Perhaps we will never reach them so we won’t need to worry about the second part?
Note that whether the coupling of chemical dissolution should be in the loop or not should be discussed.
Whatever is disintegrated will be subject to chemical erosion so I think it should be coupled. But another question is how this relates to chemical erosion of lithified rock: should it be applied first, and then mechanical erosion?
I cannot tell if the CA approach is not too complicated. The one you described first, based on discharge, seems simple and probably sufficient. The CA would add quite some extra computations. What did @jhidding say about that?
What we gonna do now
There are some things in this specification that you left open. So perhaps we should tie these up before you start coding?
- "The infiltration is difficult to estimate, worthy to have a discussion" - See my comment above, is it possible to simplify $P - I$?
- "kv = 0.23m/y in the paper, and the 'resistant rocks' are 0.023m/y. This is a very debatable constant! Worthy have a discussion." This is a parameter, not part of the algorithm, so I believe we can always modify it for a given run, so the discussion is not needed for the implementation.
- "We would run this simple model with time step of second (or if easier, in day?, worthy have a discussion)" I didn't get whether you want to apply it only if lithified rock is disintegrated into transportable sediment, because then maybe we never reach that stage?
We can also start quantifying the chemical dissolution of rocks (https://github.com/MindTheGap-ERC/CarboKitten.jl/issues/5).
Unless @jhidding disagrees, this is perhaps what you can indeed start while we clarify the points above?
NiklasHohmann
xyl96
Aims
The sub-aerial exposure of the carbonate platform could remove significant amount of strata previously deposited. For example, in Natih formation, ancient channels are observed (Grélaud et al., 2016). More recently, 35Cl data is deployed on recently exposed carbonate terrace and demonstrated that the denudations are indeed happening and under the situation of monsoon tropical climate, the rate is estimated to be around 20mm/kyr (Chauveau et al., 2021) However, most carbonate architecture models did not take this into account due to the complexity of denudation.
Herein, we have two ways to quantify the sub-aerial denudation: 1) empirical relationship derived from 35Cl data and 2) conducting metamodels from landscape evolution models (e.g., WEPP, GOLIME, CAESAR, etc.). The first approach solely considers the remove of materials in a certain time period while the second approach could consder both removal of material and redeposition. We could compare the outcomes of the two different approaches with the original CarbonKitten (i.e., no denudation) and/or with the constant denudation rates. Ideally, we could utilise mathmatical tools to quantitatively tell the differences among them.
Empirical relationship derived from 35Cl data
Research based on the karst region and carbonate platform terrace suggested that the denudation rates are mainly controlled by precipitation and slopes, although the debates about which factor is more important is still ongoing (Ryb et al., 2014, Thomas et al., 2018). In general, the precipitation mainly controls the chemical dissolution while the slopes mainly controls the physical ersions. In addition, the type of carbonates may also play an important role (Kirklec et al., 2022), but given this feature is not studied well we will ditch it for now. Yang et al., 2020 have compiled the denudation rates (mm/kyr) along with precipitation and slopes and therefore we can use their data to create a function relates denudation rates (mm/kyr) to precipitation and slopes (their data is here). This is an empirical relationship and have a relatively large uncertainty in terms of fitting.
Fig 1. The relationship between MAP (mean precipitation per year, mm/y) and denudation rates (mm/ky)
Fig 2. The relationship between the curvature (i.e., slope or height) and the denudation rates (mm/ky)
We can see that both the slope and precipitation could increase the denudation rates, and reaches a 'steady state' after a certain point.
In terms of impletementation, we could simply feed the relationship here with kyr-averaged precipitation data, and the slope data. However, this emperical method may have issues in estimating sediment traps.
Forward modelling
This approach consists of two parts:
and the total denudation = chemical weathering + physical erosion.
The chemical weathering part has been documented in issue #5. While the physical erosion is listed below:
The presence of fluvial systems are debatable in carbonate platfrom tops due to highly permeable features of carbonate, but local redistribution of detached sediments by water are typically found (e.g., in Bahamas). That is to say, the platforms have limited long-range transportations for eroded carbonate sediments, but may still have transportation of materials to the neighboring place. Herein, I propose we could couple the cellular models proposed by Tucker et al., 1998 and van de Wiel et al., 2006.
In a nutshell, we first calculate how much material would be eroded in a cell based on the following equation:
Q is surficial run-off, S is gradient, m=1/3, n=2/3. The hb is the height. Q = (P-I) A, where P is precipitation, I is infiltration, and A is the area of a catchment. We can simply just apply an infiltration factor to P, and Q = P (1 - infiltration_factor), where this factor varies between 0 and 1. As we do not consider the fluvial systems, A = the area of one cell in this context. That is to say, only the run-off produced within the cell will be considered and by doing so we can avoid creating fluvial systems as we only consider the short-range erosion. As chemical weathering part suggested, The infiltration is difficult to estimate, worthy to have a discussion. Q is the max discharge (A P). Therefore Q/Q* is a rate depicting 'the fullness' of the run-off. kv = 0.23m/y in the paper, and the 'resistant rocks' are 0.023m/y. This is a very debatable constant! Worthy have a discussion. S is slope and the original paper defines it as the tan of slope angle Howard 1994. Herein, as only short-ranged water-based transportation is considered, we could only consider the slopes between two neigboring cells.
This equation comes from incision of fluivial systems into bedrocks. Thus, compared to equations desgined to describe eroding 'loose sediments', this is more appropriate to be used in carbonate platform where carbonate sediments are lithified quickly. Applying this equation does not mean we try to model the evolution of fluvial systems, instead, we only use this equation to calculate how much material is removed in a single cell.
Can we apply a equation designed to model river erosion in to a single cell? I think we can do here. This is because this equation derives from equations descrbing detaching rocks bu shear stress, and this equation has nothing to do with river 'morphology':
where the shear stress $\tau$ depends on Q (discharge) and slope (S). This makes sense: higher discharge and higher slope would allow higher water velocity and higher erosive ability. However, it is still questionable whether the run-off generated from a single cell could produce enough shear stress to detach lithified carbonate rocks.
Such short-ranged physical erosion is very similar to soil creeping and "diffusion' Kaufman et al., 2001:
where kd is a constant descrbing diffusivity. The diffusion equation is developed based on regolith mass-wasting and it is unclear whether we can apply it to bedrock-like carbonate rocks. However, we could still have a try and compare them.
The deposition of the eroded material is distributed to the neighboring 8 cells according to the differences in height van de Wiel et al., 2006.
Herein, the Vi is the amount of eroded material waiting to be transported to the neighboring lower cells = $\Delta$ hb * A (cell area). Vi,j is the amount of material that are recieved in each cell.
So We would run this simple model with time step of second (or if easier, in day?, worthy have a discussion), and utilise the meterologic data of Bahamas, and run it for 1kyr. The workflow would be in the follwoing pic:
Note that whether the coupling of chemical dissolution should be in the loop or not should be discussed.
The initial topography could be randomly generated and/or dervied from CarboKitten.
Ideally, we could get a relationship between the topography and denudation rates in kyr scale:
Then we can apply this 'relationship' into the CarboKitten. Just like the production function, it will be a function of $\Delta$ h.
Alternatively, we can get a rule similar to CellularAutomata: we consider 9 cells (or 16 cells), and we derive a rule of the amount of erosion and deposition from the model described above: for a single cell, calculating how many cells are topographically higher than the target cell and how many are lower. We can assigned a weighted contribution factor based on the $\Delta$ h of the surrounding cells (differences in height), and caculate whether this cell would receive or erode (see the following diagram). Such weighted contribution factor is the cellular automata rule we are trying to derive from the model described above. This is slightly different from the cellular automata we used for the production, as the cellular automata rules we considered herein not only think about how many cells are neighboring, but also to what extent they are different.
For example, each cell has a vector to describe its topograhy data. the first number is it's initial topography at a certain time, the second is the eroded amount in one time step and the third is the deposted amount in one time step. The amount of materials being eroded and deposited depends on the topographic maps. By doing so, we not only extend the timescale from landscape evolution model from seconds or days to 1kyr, but also significantly reduce the calculations within the CarboKitten model by replacing the process-based model with simpler few math equations or Cellular automata rules.
Note this means that this relationship only would work under a certain climate conditions, for example, in this case we use Tropical Monson scenario (Bahamas).
What we gonna do now
We can start with the emperical data and try to use the emperical relationship to quantify the erosion rates based on slope and precipitation.
We can also start quantifying the chemical dissolution of rocks (#5).
We can begin to write codes of calculating gradient matrix.