.. _rst_Methane Model:
Methane Model
=================
The representation of processes in the methane biogeochemical model
integrated in CLM [CLM4Me; (:ref:`Riley et al. 2011a`)]
is based on several previously published models
(:ref:`Cao et al. 1996`; :ref:`Petrescu et al. 2010`;
:ref:`Tianet al. 2010`; :ref:`Walter et al. 2001`;
:ref:`Wania et al. 2010`; :ref:`Zhang et al. 2002`;
:ref:`Zhuang et al. 2004`). Although the model has similarities
with these precursor models, a number of new process representations and
parameterization have been integrated into CLM.
Mechanistically modeling net surface CH\ :sub:`4` emissions requires
representing a complex and interacting series of processes. We first
(section :numref:`Methane Model Structure and Flow`) describe the overall
model structure and flow of
information in the CH\ :sub:`4` model, then describe the methods
used to represent: CH\ :sub:`4` mass balance; CH\ :sub:`4`
production; ebullition; aerenchyma transport; CH\ :sub:`4`
oxidation; reactive transport solution, including boundary conditions,
numerical solution, water table interface, etc.; seasonal inundation
effects; and impact of seasonal inundation on CH\ :sub:`4`
production.
.. _Methane Model Structure and Flow:
Methane Model Structure and Flow
-------------------------------------
The driver routine for the methane biogeochemistry calculations (ch4, in
ch4Mod.F) controls the initialization of boundary conditions,
inundation, and impact of redox conditions; calls to routines to
calculate CH\ :sub:`4` production, oxidation, transport through
aerenchyma, ebullition, and the overall mass balance (for unsaturated
and saturated soils and, if desired, lakes); resolves changes to
CH\ :sub:`4` calculations associated with a changing inundated
fraction; performs a mass balance check; and calculates the average
gridcell CH\ :sub:`4` production, oxidation, and exchanges with
the atmosphere.
.. _Governing Mass-Balance Relationship:
Governing Mass-Balance Relationship
----------------------------------------
The model (:numref:`Figure Methane Schematic`) accounts for CH\ :sub:`4`
production in the anaerobic fraction of soil (*P*, mol m\ :sup:`-3`
s\ :sup:`-1`), ebullition (*E*, mol m\ :sup:`-3` s\ :sup:`-1`),
aerenchyma transport (*A*, mol m\ :sup:`-3` s\ :sup:`-1`), aqueous and
gaseous diffusion (:math:`{F}_{D}`, mol m\ :sup:`-2` s\ :sup:`-1`), and
oxidation (*O*, mol m\ :sup:`-3` s\ :sup:`-1`) via a transient reaction
diffusion equation:
.. math::
:label: 24.1
\frac{\partial \left(RC\right)}{\partial t} =\frac{\partial F_{D} }{\partial z} +P\left(z,t\right)-E\left(z,t\right)-A\left(z,t\right)-O\left(z,t\right)
Here *z* (m) represents the vertical dimension, *t* (s) is time, and *R*
accounts for gas in both the aqueous and gaseous
phases:\ :math:`R = \epsilon _{a} +K_{H} \epsilon _{w}`, with
:math:`\epsilon _{a}`, :math:`\epsilon _{w}`, and :math:`K_{H}` (-) the air-filled porosity, water-filled
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
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
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
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
solved again for lakes (Chapter 9). For non-inundated areas, the water
table interface is defined at the deepest transition from greater than
95% saturated to less than 95% saturated that occurs above frozen soil
layers. The inundated fraction is allowed to change at each time step,
and the total soil CH\ :sub:`4` quantity is conserved by evolving
CH\ :sub:`4` to the atmosphere when the inundated fraction
decreases, and averaging a portion of the non-inundated concentration
into the inundated concentration when the inundated fraction increases.
.. _Figure Methane Schematic:
.. figure:: image1.png
Schematic representation of biological and physical
processes integrated in CLM that affect the net CH\ :sub:`4`
surface flux (:ref:`Riley et al. 2011a`). (left)
Fully inundated portion of a
CLM gridcell and (right) variably saturated portion of a gridcell.
.. _CH4 Production:
CH\ :sub:`4` Production
----------------------------------
Because CLM does not currently specifically represent wetland plant
functional types or soil biogeochemical processes, we used
gridcell-averaged decomposition rates as proxies. Thus, the upland
(default) heterotrophic respiration is used to estimate the wetland
decomposition rate after first dividing off the O\ :sub:`2`
limitation. The O\ :sub:`2` consumption associated with anaerobic
decomposition is then set to the unlimited version so that it will be
reduced appropriately during O\ :sub:`2` competition.
CH\ :sub:`4` production at each soil level in the anaerobic
portion (i.e., below the water table) of the column is related to the
gridcell estimate of heterotrophic respiration from soil and litter
(R\ :sub:`H`; mol C m\ :sup:`-2` s\ :sub:`-1`) corrected for its soil temperature
(:math:`{T}_{s}`) dependence, soil temperature through a
:math:`{A}_{10}` factor (:math:`f_{T}`), pH (:math:`f_{pH}`),
redox potential (:math:`f_{pE}`), and a factor accounting for the
seasonal inundation fraction (*S*, described below):
.. math::
:label: 24.2
P=R_{H} f_{CH_{4} } f_{T} f_{pH} f_{pE} S.
Here, :math:`f_{CH_{4} }` is the baseline ratio between CO\ :sub:`2`
and CH\ :sub:`4` production (all parameters values are given in
:numref:`Table Methane Parameter descriptions`). Currently, :math:`f_{CH_{4} }`
is modified to account for our assumptions that methanogens may have a
higher Q\ :math:`{}_{10}` than aerobic decomposers; are not N limited;
and do not have a low-moisture limitation.
When the single BGC soil level is used in CLM (Chapter :numref:`rst_Decomposition`), the
temperature factor, :math:`f_{T}` , is set to 0 for temperatures equal
to or below freezing, even though CLM allows heterotrophic respiration
below freezing. However, if the vertically resolved BGC soil column is
used, CH\ :sub:`4` production continues below freezing because
liquid water stress limits decomposition. The base temperature for the
:math:`{Q}_{10}` factor, :math:`{T}_{B}`, is 22\ :sup:`o` C and effectively
modified the base :math:`f_{CH_{4}}` value.
For the single-layer BGC version, :math:`{R}_{H}` is distributed
among soil levels by assuming that 50% is associated with the roots
(using the CLM PFT-specific rooting distribution) and the rest is evenly
divided among the top 0.28 m of soil (to be consistent with CLM’s soil
decomposition algorithm). For the vertically resolved BGC version, the
prognosed distribution of :math:`{R}_{H}` is used to estimate CH\ :sub:`4` production.
The factor :math:`f_{pH}` is nominally set to 1, although a static
spatial map of *pH* can be used to determine this factor
(:ref:`Dunfield et al. 1993`) by applying:
.. math::
:label: 24.3
f_{pH} =10^{-0.2235pH^{2} +2.7727pH-8.6} .
The :math:`f_{pE}` factor assumes that alternative electron acceptors
are reduced with an e-folding time of 30 days after inundation. The
default version of the model applies this factor to horizontal changes
in inundated area but not to vertical changes in the water table depth
in the upland fraction of the gridcell. We consider both :math:`f_{pH}`
and :math:`f_{pE}` to be poorly constrained in the model and identify
these controllers as important areas for model improvement.
As a non-default option to account for CH\ :sub:`4` production in
anoxic microsites above the water table, we apply the Arah and Stephen
(1998) estimate of anaerobic fraction:
.. math::
:label: 24.4
\varphi =\frac{1}{1+\eta C_{O_{2} } } .
Here, :math:`\phi` 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
concentration, and :math:`\eta` = 400 mol m\ :sup:`-3`.
The O\ :sub:`2` required to facilitate the vertically resolved
heterotrophic decomposition and root respiration is estimated assuming 1
mol O\ :sub:`2` is required per mol CO\ :sub:`2` produced.
The model also calculates the O\ :sub:`2` required during
nitrification, and the total O\ :sub:`2` demand is used in the
O\ :sub:`2` mass balance solution.
.. _Table Methane Parameter descriptions:
.. table:: Parameter descriptions and sensitivity analysis ranges applied in the methane model
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| Mechanism | Parameter | Baseline Value | Range for Sensitivity Analysis | Units | Description |
+==============+============================+==============================================+==================================================================================================+=============================================+============================================================================================+
| Production | :math:`{Q}_{10}` | 2 | 1.5 – 4 | - | CH\ :sub:`4` production :math:`{Q}_{10}` |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | :math:`f_{pH}` | 1 | On, off | - | Impact of pH on CH\ :sub:`4` production |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | :math:`f_{pE}` | 1 | On, off | - | Impact of redox potential on CH\ :sub:`4` production |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | *S* | Varies | NA | - | Seasonal inundation factor |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | :math:`\beta` | 0.2 | NA | - | Effect of anoxia on decomposition rate (used to calculate *S* only) |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | :math:`f_{CH_{4} }` | 0.2 | NA | - | Ratio between CH\ :sub:`4` and CO\ :sub:`2` production below the water table |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| Ebullition | :math:`{C}_{e,max}` | 0.15 | NA | mol m\ :sup:`-3` | CH\ :sub:`4` concentration to start ebullition |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | :math:`{C}_{e,min}` | 0.15 | NA | - | CH\ :sub:`4` concentration to end ebullition |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| Diffusion | :math:`f_{D_{0} }` | 1 | 1, 10 | m\ :sup:`2` s\ :sup:`-1` | Diffusion coefficient multiplier (Table 24.2) |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| Aerenchyma | *p* | 0.3 | NA | - | Grass aerenchyma porosity |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | *R* | 2.9\ :math:`\times`\ 10\ :sup:`-3` m | NA | m | Aerenchyma radius |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | :math:`{r}_{L}` | 3 | NA | - | Root length to depth ratio |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | :math:`{F}_{a}` | 1 | 0.5 – 1.5 | - | Aerenchyma conductance multiplier |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| Oxidation | :math:`K_{CH_{4} }` | 5 x 10\ :sup:`-3` | 5\ :math:`\times`\ 10\ :math:`{}^{-4}`\ :math:`{}_{ }`- 5\ :math:`\times`\ 10\ :sup:`-2` | mol m\ :sup:`-3` | CH\ :sub:`4` half-saturation oxidation coefficient (wetlands) |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | :math:`K_{O_{2} }` | 2 x 10\ :sup:`-2` | 2\ :math:`\times`\ 10\ :sup:`-3` - 2\ :math:`\times`\ 10\ :sup:`-1` | mol m\ :sup:`-3` | O\ :sub:`2` half-saturation oxidation coefficient |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
| | :math:`R_{o,\max }` | 1.25 x 10\ :math:`{}^{-5}` | 1.25\ :math:`\times`\ 10\ :math:`{}^{-6}` - 1.25\ :math:`\times`\ 10\ :math:`{}^{-4}` | mol m\ :sup:`-3` s\ :sup:`-1` | Maximum oxidation rate (wetlands) |
+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
Ebullition
---------------
Briefly, the simulated aqueous CH\ :sub:`4` concentration in each
soil level is used to estimate the expected equilibrium gaseous partial
pressure (:math:`C_{e}` ), as a function of temperature and depth below
the water table, by first estimating the Henry’s law partitioning
coefficient (:math:`k_{h}^{C}` ) by the method described in
:ref:`Wania et al. (2010)`:
.. math::
:label: 24.5
\log \left(\frac{1}{k_{H} } \right)=\log k_{H}^{s} -\frac{1}{C_{H} } \left(\frac{1}{T} -\frac{1}{T^{s} } \right)
.. math::
:label: 24.6
k_{h}^{C} =Tk_{H} R_{g}
.. math::
:label: 24.7
C_{e} =\frac{C_{w} R_{g} T}{\theta _{s} k_{H}^{C} p}
where :math:`C_{H}` \ is a constant, :math:`R_{g}` is the universal
gas constant, :math:`k_{H}^{s}` is Henry’s law partitioning coefficient
at standard temperature (:math:`T^{s}` ),\ :math:`C_{w}` \ is local
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`
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`)].
Bubbles are immediately added to the surface flux for saturated columns
and are placed immediately above the water table interface in
unsaturated columns.
.. _Aerenchyma Transport:
Aerenchyma Transport
-------------------------
Aerenchyma transport is modeled in CLM as gaseous diffusion driven by a
concentration gradient between the specific soil layer and the
atmosphere and, if specified, by vertical advection with the
transpiration stream. There is evidence that pressure driven flow can
also occur, but we did not include that mechanism in the current model.
The diffusive transport through aerenchyma (*A*, mol m\ :sup:`-2` s\ :sup:`-1`) from each soil layer is represented in the model as:
.. math::
:label: 24.8
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
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
aerodynamic resistance between the surface and the atmospheric reference
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 .
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
tree and shrub PFTs:
.. math::
:label: 24.9
T=\frac{4 f_{N} N_{a}}{0.22} \pi R^{2} .
Here :math:`N_{a}` is annual net primary production (NPP, mol
m\ :sup:`-2` s\ :sup:`-1`); *R* is the aerenchyma radius
(2.9 :math:`\times`\ 10\ :sup:`-3` m); :math:`{f}_{N}` is the
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
oxygen.
CLM also simulates the direct emission of CH\ :sub:`4` from leaves
to the atmosphere via transpiration of dissolved methane. We calculate
this flux (:math:`F_{CH_{4} -T}` ; mol m\ :math:`{}^{-}`\ :sup:`2`
s\ :sup:`-1`) using the simulated soil water methane concentration
(:math:`C_{CH_{4} ,j}` (mol m\ :sup:`-3`)) in each soil layer *j*
and the CLM predicted transpiration (:math:`F_{T}` ) for each PFT,
assuming that no methane was oxidized inside the plant tissue:
.. math::
:label: 24.10
F_{CH_{4} -T} =\sum _{j}\rho _{r,j} F_{T} C_{CH_{4} ,j} .
.. _CH4 Oxidation:
CH\ :sub:`4` Oxidation
---------------------------------
CLM represents CH\ :sub:`4` oxidation with double Michaelis-Menten
kinetics (:ref:`Arah and Stephen 1998`; :ref:`Segers 1998`),
dependent on both the gaseous CH\ :sub:`4` and O\ :sub:`2` concentrations:
.. math::
:label: 24.11
R_{oxic} =R_{o,\max } \left[\frac{C_{CH_{4} } }{K_{CH_{4} } +C_{CH_{4} } } \right]\left[\frac{C_{O_{2} } }{K_{O_{2} } +C_{O_{2} } } \right]Q_{10} F_{\vartheta }
where :math:`K_{CH_{4} }` and :math:`K_{O_{2} }` \ are the half
saturation coefficients (mol m\ :sup:`-3`) with respect to
CH\ :sub:`4` and O\ :sub:`2` concentrations, respectively;
:math:`R_{o,\max }` is the maximum oxidation rate (mol
m\ :sup:`-3` s\ :sup:`-1`); and :math:`{Q}_{10}`
specifies the temperature dependence of the reaction with a base
temperature set to 12 :sup:`o` C. The soil moisture limitation
factor :math:`F_{\theta }` is applied above the water table to
represent water stress for methanotrophs. Based on the data in
:ref:`Schnell and King (1996)`, we take
:math:`F_{\theta } = {e}^{-P/{P}_{c}}`, where *P* is the soil moisture
potential and :math:`{P}_{c} = -2.4 \times {10}^{5}` mm.
.. _Reactive Transport Solution:
Reactive Transport Solution
--------------------------------
The solution to equation 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
table or into the atmosphere; calculate the overall CH\ :sub:`4`
or O\ :sub:`2` source term based on production, aerenchyma
transport, ebullition, and oxidation; establish boundary conditions,
including surface conductance to account for snow, ponding, and
turbulent conductances and bottom flux condition
(section :numref:`CH4 and O2 Source Terms`); calculate diffusivity
(section :numref:`Aqueous and Gaseous Diffusion`); and solve the resulting
mass balance using a tridiagonal solver (section
:numref:`Crank-Nicholson Solution Methane`).
.. _Competition for CH4and O2:
Competition for CH\ :sub:`4` and O\ :sub:`2`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
For each time step, the unlimited CH\ :sub:`4` and
O\ :sub:`2` demands in each model depth interval are computed. If
the total demand over a time step for one of the species exceeds the
amount available in a particular control volume, the demand from each
process associated with the sink is scaled by the fraction required to
ensure non-negative concentrations. Since the methanotrophs are limited
by both CH\ :sub:`4` and O\ :sub:`2`, the stricter
limitation is applied to methanotroph oxidation, and then the
limitations are scaled back for the other processes. The competition is
designed so that the sinks must not exceed the available concentration
over the time step, and if any limitation exists, the sinks must sum to
this value. Because the sinks are calculated explicitly while the
transport is semi-implicit, negative concentrations can occur after the
tridiagonal solution. When this condition occurs for O\ :sub:`2`,
the concentrations are reset to zero; if it occurs for
CH\ :sub:`4`, the surface flux is adjusted and the concentration
is set to zero if the adjustment is not too large.
.. _CH4 and O2 Source Terms:
CH\ :sub:`4` and O\ :sub:`2` Source Terms
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The overall CH\ :sub:`4` net source term consists of production,
oxidation at the base of aerenchyma, transport through aerenchyma,
methanotrophic oxidation, and ebullition (either to the control volume
above the water table if unsaturated or directly to the atmosphere if
saturated). For O\ :sub:`2` below the top control volume, the net
source term consists of O\ :sub:`2` losses from methanotrophy, SOM
decomposition, and autotrophic respiration, and an O\ :sub:`2`
source through aerenchyma.
.. _Aqueous and Gaseous Diffusion:
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
:ref:`Lerman (1979) ` and applied by
:ref:`Wania et al. (2010)`
(:numref:`Table Temperature dependence of aqueous and gaseous diffusion`).
.. _Table Temperature dependence of aqueous and gaseous diffusion:
.. 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` |
+==========================================================+==========================================================+========================================================+
| Aqueous | 0.9798 + 0.02986\ *T* + 0.0004381\ *T*\ :sup:`2` | 1.172+ 0.03443\ *T* + 0.0005048\ *T*\ :sup:`2` |
+----------------------------------------------------------+----------------------------------------------------------+--------------------------------------------------------+
| Gaseous | 0.1875 + 0.0013\ *T* | 0.1759 + 0.0011\ *T* |
+----------------------------------------------------------+----------------------------------------------------------+--------------------------------------------------------+
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
properties can be represented as:
.. math::
:label: 24.12
D_{e} =D_{0} \theta _{a}^{2} \left(\frac{\theta _{a} }{\theta _{s} } \right)^{{\raise0.7ex\hbox{$ 3 $}\!\mathord{\left/ {\vphantom {3 b}} \right. \kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$ b $}} } ,
where :math:`\theta _{a}` and :math:`\theta _{s}` are the air-filled
and total (saturated water-filled) porosities (-), respectively, and *b*
is the slope of the water retention curve (-). However, :ref:`Iiyama and
Hasegawa (2005)` have shown that the original Millington-Quirk
(:ref:`Millington and Quirk 1961`) relationship matched
measurements more closely in unsaturated peat soils:
.. math::
:label: 24.13
D_{e} =D_{0} \frac{\theta _{a} ^{{\raise0.7ex\hbox{$ 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
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
(:ref:`Moldrup et al. (2003)`):
.. math::
:label: 24.14
D_{e} =D_{0} \theta _{s} ^{2} .
To simplify the solution, we assumed that gaseous diffusion dominates
above the water table interface and aqueous diffusion below the water
table interface. Descriptions, baseline values, and dimensions for
parameters specific to the CH\ :sub:`4` model are given in
:numref:`Table Methane Parameter descriptions`. For freezing or frozen
soils below the water table, diffusion is limited to the remaining
liquid (CLM allows for some freezing point depression), and the diffusion
coefficients are scaled by the
volume-fraction of liquid. For unsaturated soils, Henry’s law
equilibrium is assumed at the interface with the water table.
.. _Boundary Conditions:
Boundary Conditions
^^^^^^^^^^^^^^^^^^^^^^^^^^
We assume the CH\ :sub:`4` and O\ :sub:`2` surface fluxes
can be calculated from an effective conductance and a gaseous
concentration gradient between the atmospheric concentration and either
the gaseous concentration in the first soil layer (unsaturated soils) or
in equilibrium with the water (saturated
soil\ :math:`w\left(C_{1}^{n} -C_{a} \right)` and
:math:`w\left(C_{1}^{n+1} -C_{a} \right)` for the fully explicit and
fully implicit cases, respectively (however, see
:ref:`Tang and Riley (2013)`
for a more complete representation of this process). Here, *w* is the
surface boundary layer conductance as calculated in the existing CLM
surface latent heat calculations. If the top layer is not fully
saturated, the :math:`\frac{D_{m1} }{\Delta x_{m1} }` term is replaced
with a series combination:
:math:`\left[\frac{1}{w} +\frac{\Delta x_{1} }{D_{1} } \right]^{-1}` ,
and if the top layer is saturated, this term is replaced with
:math:`\left[\frac{K_{H} }{w} +\frac{\frac{1}{2} \Delta x_{1} }{D_{1} } \right]^{-1}` ,
where :math:`{K}_{H}` is the Henry’s law equilibrium constant.
When snow is present, a resistance is added to account for diffusion
through the snow based on the Millington-Quirk expression :eq:`24.13`
and CLM’s prediction of the liquid water, ice, and air fractions of each
snow layer. When the soil is ponded, the diffusivity is assumed to be
that of methane in pure water, and the resistance as the ratio of the
ponding depth to diffusivity. The overall conductance is taken as the
series combination of surface, snow, and ponding resistances. We assume
a zero flux gradient at the bottom of the soil column.
.. _Crank-Nicholson Solution Methane:
Crank-Nicholson Solution
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Equation 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
.. math::
:label: 24.15
\frac{R_{j}^{n+1} C_{j}^{n+1} -R_{j}^{n} C_{j}^{n} }{\Delta t} =\frac{1}{\Delta x_{j} } \left[\frac{D_{p1}^{n} }{\Delta x_{p1}^{} } \left(C_{j+1}^{n} -C_{j}^{n} \right)-\frac{D_{m1}^{n} }{\Delta x_{m1}^{} } \left(C_{j}^{n} -C_{j-1}^{n} \right)\right]+S_{j}^{n} ,
where *j* refers to the cell in the vertically discretized soil column
(increasing downward), *n* refers to the current time step,
:math:`\Delta`\ *t* is the time step (s), *p1* is *j+½*, *m1* is *j-½*,
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
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 .
Using an average of the fully implicit and fully explicit relationships
gives:
.. math::
:label: 24.16
\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.:
.. math::
:label: 24.17
aC_{j-1}^{n+1} +bC_{j}^{n+1} +cC_{j+1}^{n+1} =r,
with coefficients specified in equation .
Two methane balance checks are performed at each timestep to insure that
the diffusion solution and the time-varying aggregation over inundated
and non-inundated areas strictly conserves methane molecules (except for
production minus consumption) and carbon atoms.
.. _Interface between water table and unsaturated zone:
Interface between water table and unsaturated zone
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
We assume Henry’s Law equilibrium at the interface between the saturated
and unsaturated zone and constant flux from the soil element below the
interface to the center of the soil element above the interface. In this
case, the coefficients are the same as described above, except for the
soil element above the interface:
.. math:: \frac{D_{p1} }{\Delta x_{p1} } =\left[K_{H} \frac{\Delta x_{j} }{2D_{j} } +\frac{\Delta x_{j+1} }{2D_{j+1} } \right]^{-1}
.. math:: b=\left[\frac{R_{j}^{n+1} }{\Delta t} +\frac{1}{2\Delta x_{j} } \left(K_{H} \frac{D_{p1}^{} }{\Delta x_{p1} } +\frac{D_{m1}^{} }{\Delta x_{m1} } \right)\right]
.. math::
:label: 24.18
r=\frac{R_{j}^{n} }{\Delta t} C_{j}^{n} +\frac{1}{2\Delta x_{j} } \left[\frac{D_{p1}^{} }{\Delta x_{p1} } \left(C_{j+1}^{n} -K_{H} 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]
and the soil element below the interface:
.. math:: \frac{D_{m1} }{\Delta x_{m1} } =\left[K_{H} \frac{\Delta x_{j-1} }{2D_{j-1} } +\frac{\Delta x_{j} }{2D_{j} } \right]^{-1}
.. math:: a=-K_{H} \frac{1}{2\Delta x_{j} } \frac{D_{m1}^{} }{\Delta x_{m1} }
.. math::
:label: 24.19
r=\frac{R_{j}^{n} }{\Delta t} +C_{j}^{n} +\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} -K_{H} C_{j-1}^{n} \right)\right]+\frac{1}{2} \left[S_{j}^{n} +S_{j}^{n+1} \right]
.. _Inundated Fraction Prediction:
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
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)
from 1993 - 2000. They suggested that the IGBP estimate for inundation
could be used as a measure of sensitivity of their detection approach at
low inundation. We therefore used the sum of their satellite-derived
:math:`{F}_{i}` and the constant IGBP estimate when it was less than
10% to perform a simple inversion for the inundated fraction for methane
production (:math:`{f}_{s}`). The method optimized two parameters
(:math:`{fws}_{slope}` and :math:`{fws}_{intercept}`) for each
grid cell in a simple model based on simulated total water storage
(:math:`{TWS}`):
.. math::
:label: 24.20
f_{s} =fws_{slope} TWS + fws_{intercept} .
These parameters were evaluated at the
0.5\ :sup:`o` resolution, and aggregated for
coarser simulations. Ongoing work in the hydrology
submodel of CLM may alleviate the need for this crude simplification of
inundated fraction in future model versions.
.. _Seasonal Inundation:
Seasonal Inundation
------------------------
A simple scaling factor is used to mimic the impact of
seasonal inundation on CH\ :sub:`4` production (see appendix B in
:ref:`Riley et al. (2011a)` for a discussion of this
simplified expression):
.. math::
:label: 24.21
S=\frac{\beta \left(f-\bar{f}\right)+\bar{f}}{f} ,S\le 1.
Here, *f* is the instantaneous inundated fraction, :math:`\bar{f}` is
the annual average inundated fraction (evaluated for the previous
calendar year) weighted by heterotrophic respiration, and
:math:`\beta` is the anoxia factor that relates the fully anoxic
decomposition rate to the fully oxygen-unlimited decomposition rate, all
other conditions being equal.