diff --git a/docs/model_docs/lateral/kinwave.qmd b/docs/model_docs/lateral/kinwave.qmd index 119b56148..defd201dd 100644 --- a/docs/model_docs/lateral/kinwave.qmd +++ b/docs/model_docs/lateral/kinwave.qmd @@ -8,8 +8,10 @@ channel and overland flow, assuming that water flow is mostly controlled by topo kinematic wave equations are (Chow, 1988): $$ - \dfrac{\partial Q}{\partial x} + \dfrac{\partial A}{\partial t} = q, \\~\\ +\begin{gathered} + \dfrac{\partial Q}{\partial x} + \dfrac{\partial A}{\partial t} = q,\\ A = \alpha Q^{\beta}. +\end{gathered} $$ These equations can then be combined as a function of streamflow only: diff --git a/docs/model_docs/lateral/sediment_flux.qmd b/docs/model_docs/lateral/sediment_flux.qmd index 754f2cbfe..576f0b835 100644 --- a/docs/model_docs/lateral/sediment_flux.qmd +++ b/docs/model_docs/lateral/sediment_flux.qmd @@ -28,6 +28,7 @@ mobilize 5 classes of sediment: - Large aggregates (mean diameter of $\SI{50}{\mu m}$). $$ +\begin{gathered} \mathrm{PSA} = \mathrm{SAN} (1-\mathrm{CLA})^{2.4} \\ \mathrm{PSI} = 0.13\mathrm{SIL}\\ \mathrm{PCL} = 0.20\mathrm{CLA} \\ @@ -42,21 +43,22 @@ $$ \end{align*} \\ \mathrm{LAG} = 1 - \mathrm{PSA} - \mathrm{PSI} - \mathrm{PCL} - \mathrm{SAG} -``` +\end{gathered} +$$ -where ``\mathrm{CLA}``, ``\mathrm{SIL}`` and ``\mathrm{SAN}`` are the primary clay, silt, sand fractions of the topsoil -and ``\mathrm{PCL}``, ``\mathrm{PSI}``, ``\mathrm{PSA}``, ``\mathrm{SAG}`` and ``\mathrm{LAG}`` are the clay, silt, sand, small and large +where $\mathrm{CLA}$, $\mathrm{SIL}$ and $\mathrm{SAN}$ are the primary clay, silt, sand fractions of the topsoil +and $\mathrm{PCL}$, $\mathrm{PSI}$, $\mathrm{PSA}$, $\mathrm{SAG}$ and $\mathrm{LAG}$ are the clay, silt, sand, small and large aggregates fractions of the detached sediment respectively. The transport capacity of the flow using Yalin's equation with particle differentiation, developed by Foster (1982), is: $$ - \mathbf{TC}_i = (P_e)_i (S_g)_i \, \rho_w \, g \, d_i V_* -``` -where ``\mathbf{TC}_i`` is the transport capacity of the flow for the particle class ``i``, -``(P_e)_i`` is the effective number of particles of class ``i``, ``\SIb{(S_g)_i}{kg m^{-3}}`` is the -specific gravity for the particle class ``i``, ``\SIb{\rho_w}{kg m^{-3}}`` is the mass density -of the fluid, ``\SIb{g}{m s^{-2}}`` is the acceleration due to gravity, -``\SIb{d_i}{m}`` is the diameter of the particle of class ``i`` and ``V_* = \SIb{(g R S)^{0.5}}{m s^{-1}}`` is the -shear velocity of the flow with ``S`` the slope gradient and ``\SIb{R}{m}`` the + \mathrm{TC}_i = (P_e)_i (S_g)_i \, \rho_w \, g \, d_i V_* +$$ +where $\mathrm{TC}_i$ is the transport capacity of the flow for the particle class $i$, +$(P_e)_i$ is the effective number of particles of class $i$, $\SIb{(S_g)_i}{kg m^{-3}}$ is the +specific gravity for the particle class $i$, $\SIb{\rho_w}{kg m^{-3}}$ is the mass density +of the fluid, $\SIb{g}{m s^{-2}}$ is the acceleration due to gravity, +$\SIb{d_i}{m}$ is the diameter of the particle of class $i$ and $V_* = \SIb{(g R S)^{0.5}}{m s^{-1}}$ is the +shear velocity of the flow with $S$ the slope gradient and $\SIb{R}{m}$ the hydraulic radius of the flow. The detached sediment are then routed down slope until the river network using the `accucapacityflux`, `accupacitystate` functions depending on the transport capacity from Yalin. @@ -192,7 +194,7 @@ $$ \log\left(C_{ppm}\right) = 5.435 - 0.286\log\left(\frac{\omega_{s,50}D_{50}}{\nu}\right)-0.457\log\left(\frac{u_*}{\omega_{s,50}}\right) \\ +\left(1.799-0.409\log\left(\frac{\omega_{s,50}D_{50}}{\nu}\right)-0.314\log\left(\frac{u_*}{\omega_{s,50}}\right)\right)\log\left(\frac{uS}{\omega_{s,50}}-\frac{u_{cr}S}{\omega_{s,50}}\right) $$ -And the gravel equation (``\SI{2.0}{mm} \leq D_{50} < \SI{10.0}{mm}``) is: +And the gravel equation ($\SI{2.0}{mm} \leq D_{50} < \SI{10.0}{mm}$) is: $$ \log\left(C_{ppm}\right) = 6.681 - 0.633\log\left(\frac{\omega_{s,50}D_{50}}{\nu}\right)-4.816\log\left(\frac{u_*}{\omega_{s,50}}\right) \\ +\left(2.784-0.305\log\left(\frac{\omega_{s,50}D_{50}}{\nu}\right)-0.282\log\left(\frac{u_*}{\omega_{s,50}}\right)\right)\log\left(\frac{uS}{\omega_{s,50}}-\frac{u_{cr}S}{\omega_{s,50}}\right) @@ -236,8 +238,10 @@ their material ``E_{R,\mathrm{bed}}`` for the bed and ``E_{R,\mathrm{bank}}`` fo rectangular channel, assuming it is meandering and thus only one bank is prone to erosion, they are calculated from the equations (Neitsch et al, 2011): $$ - E_{R,\mathrm{bed}} = k_{d,\mathrm{bed}} \left( \tau_{e,\mathrm{bed}} - \tau_{cr,\mathrm{bed}} \right) 10^{-6} L W \rho_{b, \mathrm{bed}} \Delta t \\~\\ +\begin{gathered} + E_{R,\mathrm{bed}} = k_{d,\mathrm{bed}} \left( \tau_{e,\mathrm{bed}} - \tau_{cr,\mathrm{bed}} \right) 10^{-6} L W \rho_{b, \mathrm{bed}} \Delta t \\ E_{R,\mathrm{bank}} = k_{d,\mathrm{bank}} \left( \tau_{e,\mathrm{bank}} - \tau_{cr,\mathrm{bank}} \right) 10^{-6} L h \rho_{b, \mathrm{bank}} \Delta t +\end{gathered} $$ where $\SIb{E_R}{ton}$ is the potential bed/bank erosion rates, $\SIb{k_d}{cm^3 N^{-1}, s^{-1}}$ is the erodibility of the bed/bank material, $\SIb{\tau_e}{N m^{-2}}$ is the effective @@ -271,7 +275,7 @@ order, the smaller the diameter is. As the median diameter is only used in wflow for the estimation of the river bed/bank sediment composition, this supposition should be enough. Actual refined data or calibration may however be needed if the median diameter is also required for the transport formula. In a similar way, the bulk densities of river bed -and bank are also just assumed to be of respectively 1.5 and 1.4 g cm$^{-3}$. +and bank are also just assumed to be of respectively $\SI{1.5}{g cm^{-3}}$ and $\SI{1.4}{g cm^{-3}}$. Table: Classical values of the channel cover vegetation coefficient (Julian and Torres, 2006) @@ -296,8 +300,10 @@ Then, the repartition of the flow shear stress is refined into the effective she and the bed and bank of the river using the equations developed by Knight (1984) for a rectangular channel: $$ - \tau_{e,\mathrm{bed}} = \rho g R_{H} S \left(1 - \dfrac{SF_{\mathrm{bank}}}{100}\right) \left(1+\dfrac{2h}{W}\right) \\~\\ +\begin{gathered} + \tau_{e,\mathrm{bed}} = \rho g R_{H} S \left(1 - \dfrac{SF_{\mathrm{bank}}}{100}\right) \left(1+\dfrac{2h}{W}\right) \\ \tau_{e,\mathrm{bank}} = \rho g R_{H} S \left( SF_{\mathrm{bank}}\right) \left(1+\dfrac{W}{2h}\right) +\end{gathered} $$ where $\rho g$ is the fluid specific weight ($\SI{9800}{N m^{-3}}$ for water), $\SIb{R_H}{m}$ is the hydraulic radius of the channel, $\SIb{h}{m}$ and $\SIb{W}{m}$ are the water level and river width. $SF_{\mathrm{bank}}$ is the proportion of shear stress acting on the bank (%) and is estimated @@ -307,8 +313,10 @@ $$ $$ Finally the relative erosion potential of the bank and bed of the river is calculated by: $$ - \mathrm{RTE}_{\mathrm{bed}} = \dfrac{E_{R,\mathrm{bed}}}{E_{R,\mathrm{bed}}+E_{R,\mathrm{bank}}} \\~\\ +\begin{gathered} + \mathrm{RTE}_{\mathrm{bed}} = \dfrac{E_{R,\mathrm{bed}}}{E_{R,\mathrm{bed}}+E_{R,\mathrm{bank}}} \\ \mathrm{RTE}_{\mathrm{bank}} = 1 - RTE_{\mathrm{bed}} +\end{gathered} $$ And the final actual eroded amount for the bed and bank is the maximum between $\mathrm{RTE} \subtext{\mathrm{sed}}{exeff}$ and the erosion potential $E_R$. Total eroded amount of sediment diff --git a/docs/model_docs/lateral/waterbodies.qmd b/docs/model_docs/lateral/waterbodies.qmd index c60ed4794..d218018c8 100644 --- a/docs/model_docs/lateral/waterbodies.qmd +++ b/docs/model_docs/lateral/waterbodies.qmd @@ -152,7 +152,7 @@ waterlevel = "lake_waterlevel" ### Additional settings Storage and rating curves from field measurement can be supplied to wflow via CSV files supplied in the same folder of the TOML file. Naming of the files uses the ID of the lakes -where data are available and is of the form lake\_sh\_1.csv and lake\_hq\_1.csv for +where data are available and is of the form `lake_sh_1.csv` and `lake_hq_1.csv` for respectively the storage and rating curves of lake with ID 1. The storage curve is stored in a CSV file with lake level $\SIb{}{m}$ in the first column `H` and diff --git a/docs/model_docs/vertical/sbm.qmd b/docs/model_docs/vertical/sbm.qmd index 08b51a1ba..ed3a50e3e 100644 --- a/docs/model_docs/vertical/sbm.qmd +++ b/docs/model_docs/vertical/sbm.qmd @@ -369,8 +369,10 @@ degree of the layer, and a Brooks-Corey power coefficient (parameter $c$) based pore size distribution index $\lambda$ (Brooks and Corey, 1964): $$ - \mathrm{st}=\subtext{K}{sat}\left(\frac{\theta-\theta_r}{\theta_s-\theta_r}\right)^c\\~\\ +\begin{gathered} + \mathrm{st}=\subtext{K}{sat}\left(\frac{\theta-\theta_r}{\theta_s-\theta_r}\right)^c\\ c=\frac{2+3\lambda}{\lambda} +\end{gathered} $$ Here $\SIb{}{mm t^{-1}}$ denotes milimeter per time step. @@ -446,7 +448,7 @@ $$ K_0e^{-fz} & \text{if $z < \subtext{z}{exp}$}\\ K_0e^{-f\subtext{z}{exp}} & \text{if $z \ge \subtext{z}{exp}$}. \end{cases} -``` +$$ It is also possible to provide a $\subtext{K}{sat}$ value per soil layer by specifying `ksat_profile` "layered", these $\subtext{K}{sat}$ values are used directly to compute the vertical @@ -517,10 +519,12 @@ A S-curve (see plot below) is used to make a smooth transition (a c-factor ($c$) used): $$ - b = \frac{1.0}{1.0 - \subtext{\mathrm{cf}}{soil}}\\~\\ - \mathrm{soilinfredu} = \frac{1.0}{b + \exp(-c (T_s - a))} + \subtext{\mathrm{cf}}{soil}\\~\\ +\begin{gathered} + b = \frac{1.0}{1.0 - \subtext{\mathrm{cf}}{soil}}\\ + \mathrm{soilinfredu} = \frac{1.0}{b + \exp(-c (T_s - a))} + \subtext{\mathrm{cf}}{soil}\\ a = 0.0\\ c = 8.0 +\end{gathered} $$ ![Infiltration correction factor as a function of soil temperature](../../images/soil_frozeninfilt.png) @@ -658,7 +662,7 @@ apply an irrigation rate higher than the soil infiltration capacity. To account irrigation efficiency the net irrigation demand is divided by the irrigation efficiency for non-paddy crops (`irrigation_efficiency` $\SIb{}{-}$, default is $1.0$), resulting in gross irrigation demand $\SIb{}{mm t^{-1}}$. Finally, the gross irrigation demand is limited by the maximum -irrigation rate (`maximum_irrigation_rate` $\SIb{}{mm t^{-1}}$, default is $\SI{25}^{mm\;day-1}$). If +irrigation rate (`maximum_irrigation_rate` $\SIb{}{mm t^{-1}}$, default is $\SI{25}{mm\;day-1}$). If the maximum irrigation rate is applied, irrigation continues at subsequent time steps until field capacity is reached. Irrigation is added to the `SBM` variable `avail_forinfilt` $\SIb{}{mm t^{-1}}$, the amount of water available for infiltration. diff --git a/docs/model_docs/vertical/sediment.qmd b/docs/model_docs/vertical/sediment.qmd index b1873d3d5..db1ab7095 100644 --- a/docs/model_docs/vertical/sediment.qmd +++ b/docs/model_docs/vertical/sediment.qmd @@ -57,7 +57,7 @@ use. Kinetic energies from both direct throughfall and leaf drainage are then multiplied by the respective depths of direct throughfall and leaf drainage (mm) and added to get the total -rainfall kinetic energy ``\mathrm{KE}``. The soil detached by rainfall ``\SIb{D_R}{g m^{-2}}`` is +rainfall kinetic energy $\mathrm{KE}$. The soil detached by rainfall $\SIb{D_R}{g m^{-2}}$ is then: $$ D_R = k\,\mathrm{KE}\,e^{-\varphi h} @@ -110,9 +110,11 @@ The other methods to estimate the USLE $K$ factor are to use either topsoil comp topsoil geometric mean diameter. $K$ estimation from topsoil composition is estimated with the equation developed in the EPIC model (Williams et al, 1983): $$ +\begin{gathered} \subtext{K}{USLE} = \left[ 0.2 + 0.3\exp\left(-0.0256\;\mathrm{SAN}\frac{(1-\mathrm{SIL})}{100}\right) \right] - \left(\frac{\mathrm{SIL}}{\mathrm{CLA}+\mathrm{SIL}}\right)^{0.3} \\~\\ + \left(\frac{\mathrm{SIL}}{\mathrm{CLA}+\mathrm{SIL}}\right)^{0.3} \\ \left(1-\frac{0.25\;\mathrm{OC}}{\mathrm{OC}+e^{3.72-2.95\;\mathrm{OC}}}\right)\left(1-\frac{0.75\;\mathrm{SN}}{\mathrm{SN}+e^{-5.51+22.9\;\mathrm{SN}}}\right) +\end{gathered} $$ where $\SIb{\mathrm{CLA}}{\%}$, $\SIb{\mathrm{SIL}}{\%}$, $\SIb{\mathrm{SAN}}{\%}$ are respectively the clay, silt and sand fractions of the topsoil, $\SIb{OC}{\%}$ is the topsoil organic carbon content and $\mathrm{SN} = 1-\mathrm{SAN}/100$. diff --git a/docs/model_docs/vertical/shared_processes.qmd b/docs/model_docs/vertical/shared_processes.qmd index 8c154ae2a..33740ecc2 100644 --- a/docs/model_docs/vertical/shared_processes.qmd +++ b/docs/model_docs/vertical/shared_processes.qmd @@ -19,7 +19,7 @@ The respective rates of snow melt and refreezing are: $$ \begin{align*} - Q_m &=& \subtext{\mathrm{cf}}{max}(T_a−\mathrm{tt})\, &&T_a > \mathrm{tt} \\~\\ + Q_m &=& \subtext{\mathrm{cf}}{max}(T_a−\mathrm{tt})\, &&T_a > \mathrm{tt} \\ Q_r &=& \subtext{\mathrm{cf}}{max} \, \mathrm{cf}_r(\mathrm{tt}−T_a) &&T_a < \mathrm{tt} \end{align*} $$ @@ -161,11 +161,9 @@ can be determined through a lookup table with land cover based on literature (Pi Lui 1998). Next the `cmax` (leaves) is determined using: $$ -cmax(leaves) = sl \, LAI + \mathrm{cmax}(\mathrm{leaves}) = \mathrm{sl} \cdot \mathrm{LAI} $$ - \mathrm{cmax}(\mathrm{leaves}) = \mathrm{sl} \cdot \mathrm{LAI} -``` To get to total storage (`cmax`) the woody part of the vegetation also needs to be added. As for `sl`, the storage of the woody part `swood` can also be related to land cover (lookup table). @@ -173,9 +171,9 @@ table). The canopy gap fraction is determined using the extinction coefficient `kext` (van Dijk and Bruijnzeel 2001): -```math +$$ \mathrm{canopygapfraction} = \exp(-\subtext{k}{ext} \cdot \mathrm{LAI}) -``` +$$ The extinction coefficient `kext` can be related to land cover.