diff --git a/doc/source/tech_note/Methane/CLM50_Tech_Note_Methane.rst b/doc/source/tech_note/Methane/CLM50_Tech_Note_Methane.rst index 7c89f857e3..875689558b 100644 --- a/doc/source/tech_note/Methane/CLM50_Tech_Note_Methane.rst +++ b/doc/source/tech_note/Methane/CLM50_Tech_Note_Methane.rst @@ -66,7 +66,7 @@ phases:\ :math:`R = \epsilon _{a} +K_{H} \epsilon _{w}`, with porosity, and partitioning coefficient for the species of interest, respectively, and :math:`C` represents CH\ :sub:`4` or O\ :sub:`2` concentration with respect to water volume (mol m\ :sup:`-3`). -An analogous version of equation is concurrently solved for +An analogous version of equation :eq:`24.1` is concurrently solved for O\ :sub:`2`, but with the following differences relative to CH\ :sub:`4`: *P* = *E* = 0 (i.e., no production or ebullition), and the oxidation sink includes the O\ :sub:`2` demanded by @@ -74,7 +74,7 @@ methanotrophs, heterotroph decomposers, nitrifiers, and autotrophic root respiration. As currently implemented, each gridcell contains an inundated and a -non-inundated fraction. Therefore, equation is solved four times for +non-inundated fraction. Therefore, equation :eq:`24.1` is solved four times for each gridcell and time step: in the inundated and non-inundated fractions, and for CH\ :sub:`4` and O\ :sub:`2`. If desired, the CH\ :sub:`4` and O\ :sub:`2` mass balance equation is @@ -173,9 +173,9 @@ anoxic microsites above the water table, we apply the Arah and Stephen \varphi =\frac{1}{1+\eta C_{O_{2} } } . -Here, :math:`\phi` is the factor by which production is inhibited +Here, :math:`\varphi` is the factor by which production is inhibited above the water table (compared to production as calculated in equation -, :math:`C_{O_{2}}` (mol m\ :sup:`-3`) is the bulk soil oxygen +:eq:`24.2`, :math:`C_{O_{2}}` (mol m\ :sup:`-3`) is the bulk soil oxygen concentration, and :math:`\eta` = 400 mol m\ :sup:`-3`. The O\ :sub:`2` required to facilitate the vertically resolved @@ -259,8 +259,8 @@ aqueous CH\ :sub:`4` concentration, and *p* is pressure. The local pressure is calculated as the sum of the ambient pressure, water pressure down to the local depth, and pressure from surface ponding (if applicable). When the CH\ :sub:`4` partial pressure -exceeds 15% of the local pressure (Baird et al. 2004; Strack et al. -2006; Wania et al. 2010), bubbling occurs to remove CH\ :sub:`4` +exceeds 15% of the local pressure (:ref:`Baird et al. 2004`; :ref:`Strack et al. +2006`; :ref:`Wania et al. 2010`), bubbling occurs to remove CH\ :sub:`4` to below this value, modified by the fraction of CH\ :sub:`4` in the bubbles [taken as 57%; (:ref:`Kellner et al. 2006`; :ref:`Wania et al. 2010`)]. @@ -286,14 +286,14 @@ The diffusive transport through aerenchyma (*A*, mol m\ :sup:`-2` s\ :sup:`-1`) A=\frac{C\left(z\right)-C_{a} }{{\raise0.7ex\hbox{$ r_{L} z $}\!\mathord{\left/ {\vphantom {r_{L} z D}} \right. \kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$ D $}} +r_{a} } pT\rho _{r} , -where *D* is the free-air gas diffusion coefficient (m:sup:`2` s\ :sup:`-1`); *C(z)* (mol m\ :sup:`-3`) is the gaseous +where *D* is the free-air gas diffusion coefficient (m\ :sup:`2` s\ :sup:`-1`); *C(z)* (mol m\ :sup:`-3`) is the gaseous concentration at depth *z* (m); :math:`r_{L}` is the ratio of root length to depth; *p* is the porosity (-); *T* is specific aerenchyma -area (m:sup:`2` m\ :sup:`-2`); :math:`{r}_{a}` is the +area (m\ :sup:`2` m\ :sup:`-2`); :math:`{r}_{a}` is the aerodynamic resistance between the surface and the atmospheric reference -height (s m:sup:`-1`); and :math:`\rho _{r}` is the rooting +height (s m\ :sup:`-1`); and :math:`\rho _{r}` is the rooting density as a function of depth (-). The gaseous concentration is -calculated with Henry’s law as described in equation . +calculated with Henry’s law as described in equation :eq:`24.7`. Based on the ranges reported in :ref:`Colmer (2003)`, we have chosen baseline aerenchyma porosity values of 0.3 for grass and crop PFTs and 0.1 for @@ -310,7 +310,7 @@ m\ :sup:`-2` s\ :sup:`-1`); *R* is the aerenchyma radius belowground fraction of annual NPP; and the 0.22 factor represents the amount of C per tiller. O\ :sub:`2` can also diffuse in from the atmosphere to the soil layer via the reverse of the same pathway, with -the same representation as Equation but with the gas diffusivity of +the same representation as Equation :eq:`24.8` but with the gas diffusivity of oxygen. CLM also simulates the direct emission of CH\ :sub:`4` from leaves @@ -358,7 +358,7 @@ potential and :math:`{P}_{c} = -2.4 \times {10}^{5}` mm. Reactive Transport Solution -------------------------------- -The solution to equation is solved in several sequential steps: resolve +The solution to equation :eq:`24.11` is solved in several sequential steps: resolve competition for CH\ :sub:`4` and O\ :sub:`2` (section :numref:`Competition for CH4and O2`); add the ebullition flux into the layer directly above the water @@ -416,7 +416,7 @@ Aqueous and Gaseous Diffusion For gaseous diffusion, we adopted the temperature dependence of molecular free-air diffusion coefficients (:math:`{D}_{0}` -(m:sup:`2` s\ :sup:`-1`)) as described by +(m\ :sup:`2` s\ :sup:`-1`)) as described by :ref:`Lerman (1979) ` and applied by :ref:`Wania et al. (2010)` (:numref:`Table Temperature dependence of aqueous and gaseous diffusion`). @@ -426,7 +426,7 @@ molecular free-air diffusion coefficients (:math:`{D}_{0}` .. table:: Temperature dependence of aqueous and gaseous diffusion coefficients for CH\ :sub:`4` and O\ :sub:`2` +----------------------------------------------------------+----------------------------------------------------------+--------------------------------------------------------+ - | :math:`{D}_{0}` (m\ :sup:`2` s\ :sup:`-1`) | CH\ :sub:`4` | O\ :sub:`2` | + | :math:`{D}_{0}` (cm\ :sup:`2` s\ :sup:`-1`) | CH\ :sub:`4` | O\ :sub:`2` | +==========================================================+==========================================================+========================================================+ | Aqueous | 0.9798 + 0.02986\ *T* + 0.0004381\ *T*\ :sup:`2` | 1.172+ 0.03443\ *T* + 0.0005048\ *T*\ :sup:`2` | +----------------------------------------------------------+----------------------------------------------------------+--------------------------------------------------------+ @@ -437,7 +437,7 @@ Gaseous diffusivity in soils also depends on the molecular diffusivity, soil structure, porosity, and organic matter content. :ref:`Moldrup et al. (2003)`, using observations across a range of unsaturated mineral soils, showed that the relationship between -effective diffusivity (:math:`D_{e}` (m:sup:`2` s\ :sup:`-1`)) and soil +effective diffusivity (:math:`D_{e}` (m\ :sup:`2` s\ :sup:`-1`)) and soil properties can be represented as: .. math:: @@ -457,8 +457,8 @@ measurements more closely in unsaturated peat soils: D_{e} =D_{0} \frac{\theta _{a} ^{{\raise0.7ex\hbox{$ 10 $}\!\mathord{\left/ {\vphantom {10 3}} \right. \kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$ 3 $}} } }{\theta _{s} ^{2} } -In CLM, we applied equation for soils with zero organic matter content -and equation for soils with more than 130 kg m\ :sup:`-3` organic +In CLM, we applied equation :eq:`24.12` for soils with zero organic matter content +and equation :eq:`24.13` for soils with more than 130 kg m\ :sup:`-3` organic matter content. A linear interpolation between these two limits is applied for soils with SOM content below 130 kg m\ :sup:`-3`. For aqueous diffusion in the saturated part of the soil column, we applied @@ -518,10 +518,10 @@ a zero flux gradient at the bottom of the soil column. Crank-Nicholson Solution ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -Equation is solved using a Crank-Nicholson solution -(:ref:`Press et al. 1992`), +Equation :eq:`24.1` is solved using a Crank-Nicholson solution +(:ref:`Press et al., 1992`), which combines fully explicit and implicit representations of the mass -balance. The fully explicit decomposition of equation can be written as +balance. The fully explicit decomposition of equation :eq:`24.1` can be written as .. math:: :label: 24.15 @@ -535,11 +535,11 @@ and :math:`S_{j}^{n}` is the net source at time step *n* and position *j*, i.e., :math:`S_{j}^{n} =P\left(j,n\right)-E\left(j,n\right)-A\left(j,n\right)-O\left(j,n\right)`. The diffusivity coefficients are calculated as harmonic means of values -from the adjacent cells. Equation is solved for gaseous and aqueous +from the adjacent cells. Equation :eq:`24.15` is solved for gaseous and aqueous concentrations above and below the water table, respectively. The *R* term ensure the total mass balance in both phases is properly accounted for. An analogous relationship can be generated for the fully implicit -case by replacing *n* by *n+1* on the *C* and *S* terms of equation . +case by replacing *n* by *n+1* on the *C* and *S* terms of equation :eq:`24.15`. Using an average of the fully implicit and fully explicit relationships gives: @@ -548,14 +548,14 @@ gives: \begin{array}{l} {-\frac{1}{2\Delta x_{j} } \frac{D_{m1}^{} }{\Delta x_{m1}^{} } C_{j-1}^{n+1} +\left[\frac{R_{j}^{n+1} }{\Delta t} +\frac{1}{2\Delta x_{j} } \left(\frac{D_{p1}^{} }{\Delta x_{p1}^{} } +\frac{D_{m1}^{} }{\Delta x_{m1}^{} } \right)\right]C_{j}^{n+1} -\frac{1}{2\Delta x_{j} } \frac{D_{p1}^{} }{\Delta x_{p1}^{} } C_{j+1}^{n+1} =} \\ {\frac{R_{j}^{n} }{\Delta t} +\frac{1}{2\Delta x_{j} } \left[\frac{D_{p1}^{} }{\Delta x_{p1}^{} } \left(C_{j+1}^{n} -C_{j}^{n} \right)-\frac{D_{m1}^{} }{\Delta x_{m1}^{} } \left(C_{j}^{n} -C_{j-1}^{n} \right)\right]+\frac{1}{2} \left[S_{j}^{n} +S_{j}^{n+1} \right]} \end{array}, -Equation is solved with a standard tridiagonal solver, i.e.: +Equation :eq:`24.16` is solved with a standard tridiagonal solver, i.e.: .. math:: :label: 24.17 aC_{j-1}^{n+1} +bC_{j}^{n+1} +cC_{j+1}^{n+1} =r, -with coefficients specified in equation . +with coefficients specified in equation :eq:`24.16`. Two methane balance checks are performed at each timestep to insure that the diffusion solution and the time-varying aggregation over inundated @@ -599,7 +599,7 @@ Inundated Fraction Prediction ---------------------------------- A simplified dynamic representation of spatial inundation -based on recent work by :ref:`Prigent et al. (2007)` is used. Prigent et al. (2007) described a +based on recent work by :ref:`Prigent et al. (2007)` is used. :ref:`Prigent et al. (2007)` described a multi-satellite approach to estimate the global monthly inundated fraction (:math:`{F}_{i}`) over an equal area grid (0.25 :math:`\circ` \ :math:`\times`\ 0.25\ :math:`\circ` at the equator)