649 lines
40 KiB
ReStructuredText
649 lines
40 KiB
ReStructuredText
.. _rst_Methane Model:
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Methane Model
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=================
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The representation of processes in the methane biogeochemical model
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integrated in CLM [CLM4Me; (:ref:`Riley et al. 2011a<Rileyetal2011a>`)]
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is based on several previously published models
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(:ref:`Cao et al. 1996<Caoetal1996>`; :ref:`Petrescu et al. 2010<Petrescuetal2010>`;
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:ref:`Tianet al. 2010<Tianetal2010>`; :ref:`Walter et al. 2001<Walteretal2001>`;
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:ref:`Wania et al. 2010<Waniaetal2010>`; :ref:`Zhang et al. 2002<Zhangetal2002>`;
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:ref:`Zhuang et al. 2004<Zhuangetal2004>`). Although the model has similarities
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with these precursor models, a number of new process representations and
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parameterization have been integrated into CLM.
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Mechanistically modeling net surface CH\ :sub:`4` emissions requires
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representing a complex and interacting series of processes. We first
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(section :numref:`Methane Model Structure and Flow`) describe the overall
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model structure and flow of
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information in the CH\ :sub:`4` model, then describe the methods
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used to represent: CH\ :sub:`4` mass balance; CH\ :sub:`4`
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production; ebullition; aerenchyma transport; CH\ :sub:`4`
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oxidation; reactive transport solution, including boundary conditions,
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numerical solution, water table interface, etc.; seasonal inundation
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effects; and impact of seasonal inundation on CH\ :sub:`4`
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production.
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.. _Methane Model Structure and Flow:
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Methane Model Structure and Flow
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-------------------------------------
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The driver routine for the methane biogeochemistry calculations (ch4, in
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ch4Mod.F) controls the initialization of boundary conditions,
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inundation, and impact of redox conditions; calls to routines to
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calculate CH\ :sub:`4` production, oxidation, transport through
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aerenchyma, ebullition, and the overall mass balance (for unsaturated
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and saturated soils and, if desired, lakes); resolves changes to
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CH\ :sub:`4` calculations associated with a changing inundated
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fraction; performs a mass balance check; and calculates the average
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gridcell CH\ :sub:`4` production, oxidation, and exchanges with
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the atmosphere.
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.. _Governing Mass-Balance Relationship:
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Governing Mass-Balance Relationship
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----------------------------------------
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The model (:numref:`Figure Methane Schematic`) accounts for CH\ :sub:`4`
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production in the anaerobic fraction of soil (*P*, mol m\ :sup:`-3`
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s\ :sup:`-1`), ebullition (*E*, mol m\ :sup:`-3` s\ :sup:`-1`),
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aerenchyma transport (*A*, mol m\ :sup:`-3` s\ :sup:`-1`), aqueous and
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gaseous diffusion (:math:`{F}_{D}`, mol m\ :sup:`-2` s\ :sup:`-1`), and
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oxidation (*O*, mol m\ :sup:`-3` s\ :sup:`-1`) via a transient reaction
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diffusion equation:
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.. math::
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:label: 24.1
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\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)
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Here *z* (m) represents the vertical dimension, *t* (s) is time, and *R*
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accounts for gas in both the aqueous and gaseous
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phases:\ :math:`R = \epsilon _{a} +K_{H} \epsilon _{w}`, with
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:math:`\epsilon _{a}`, :math:`\epsilon _{w}`, and :math:`K_{H}` (-) the air-filled porosity, water-filled
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porosity, and partitioning coefficient for the species of interest,
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respectively, and :math:`C` represents CH\ :sub:`4` or O\ :sub:`2` concentration with respect to water volume (mol m\ :sup:`-3`).
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An analogous version of equation is concurrently solved for
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O\ :sub:`2`, but with the following differences relative to
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CH\ :sub:`4`: *P* = *E* = 0 (i.e., no production or ebullition),
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and the oxidation sink includes the O\ :sub:`2` demanded by
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methanotrophs, heterotroph decomposers, nitrifiers, and autotrophic root
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respiration.
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As currently implemented, each gridcell contains an inundated and a
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non-inundated fraction. Therefore, equation is solved four times for
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each gridcell and time step: in the inundated and non-inundated
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fractions, and for CH\ :sub:`4` and O\ :sub:`2`. If desired,
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the CH\ :sub:`4` and O\ :sub:`2` mass balance equation is
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solved again for lakes (Chapter 9). For non-inundated areas, the water
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table interface is defined at the deepest transition from greater than
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95% saturated to less than 95% saturated that occurs above frozen soil
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layers. The inundated fraction is allowed to change at each time step,
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and the total soil CH\ :sub:`4` quantity is conserved by evolving
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CH\ :sub:`4` to the atmosphere when the inundated fraction
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decreases, and averaging a portion of the non-inundated concentration
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into the inundated concentration when the inundated fraction increases.
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.. _Figure Methane Schematic:
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.. figure:: image1.png
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Schematic representation of biological and physical
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processes integrated in CLM that affect the net CH\ :sub:`4`
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surface flux (:ref:`Riley et al. 2011a<Rileyetal2011a>`). (left)
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Fully inundated portion of a
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CLM gridcell and (right) variably saturated portion of a gridcell.
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.. _CH4 Production:
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CH\ :sub:`4` Production
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----------------------------------
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Because CLM does not currently specifically represent wetland plant
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functional types or soil biogeochemical processes, we used
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gridcell-averaged decomposition rates as proxies. Thus, the upland
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(default) heterotrophic respiration is used to estimate the wetland
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decomposition rate after first dividing off the O\ :sub:`2`
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limitation. The O\ :sub:`2` consumption associated with anaerobic
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decomposition is then set to the unlimited version so that it will be
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reduced appropriately during O\ :sub:`2` competition.
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CH\ :sub:`4` production at each soil level in the anaerobic
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portion (i.e., below the water table) of the column is related to the
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gridcell estimate of heterotrophic respiration from soil and litter
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(R\ :sub:`H`; mol C m\ :sup:`-2` s\ :sub:`-1`) corrected for its soil temperature
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(:math:`{T}_{s}`) dependence, soil temperature through a
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:math:`{A}_{10}` factor (:math:`f_{T}`), pH (:math:`f_{pH}`),
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redox potential (:math:`f_{pE}`), and a factor accounting for the
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seasonal inundation fraction (*S*, described below):
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.. math::
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:label: 24.2
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P=R_{H} f_{CH_{4} } f_{T} f_{pH} f_{pE} S.
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Here, :math:`f_{CH_{4} }` is the baseline ratio between CO\ :sub:`2`
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and CH\ :sub:`4` production (all parameters values are given in
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:numref:`Table Methane Parameter descriptions`). Currently, :math:`f_{CH_{4} }`
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is modified to account for our assumptions that methanogens may have a
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higher Q\ :math:`{}_{10}` than aerobic decomposers; are not N limited;
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and do not have a low-moisture limitation.
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When the single BGC soil level is used in CLM (Chapter :numref:`rst_Decomposition`), the
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temperature factor, :math:`f_{T}` , is set to 0 for temperatures equal
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to or below freezing, even though CLM allows heterotrophic respiration
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below freezing. However, if the vertically resolved BGC soil column is
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used, CH\ :sub:`4` production continues below freezing because
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liquid water stress limits decomposition. The base temperature for the
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:math:`{Q}_{10}` factor, :math:`{T}_{B}`, is 22\ :sup:`o` C and effectively
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modified the base :math:`f_{CH_{4}}` value.
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For the single-layer BGC version, :math:`{R}_{H}` is distributed
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among soil levels by assuming that 50% is associated with the roots
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(using the CLM PFT-specific rooting distribution) and the rest is evenly
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divided among the top 0.28 m of soil (to be consistent with CLM’s soil
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decomposition algorithm). For the vertically resolved BGC version, the
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prognosed distribution of :math:`{R}_{H}` is used to estimate CH\ :sub:`4` production.
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The factor :math:`f_{pH}` is nominally set to 1, although a static
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spatial map of *pH* can be used to determine this factor
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(:ref:`Dunfield et al. 1993<Dunfieldetal1993>`) by applying:
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.. math::
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:label: 24.3
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f_{pH} =10^{-0.2235pH^{2} +2.7727pH-8.6} .
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The :math:`f_{pE}` factor assumes that alternative electron acceptors
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are reduced with an e-folding time of 30 days after inundation. The
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default version of the model applies this factor to horizontal changes
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in inundated area but not to vertical changes in the water table depth
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in the upland fraction of the gridcell. We consider both :math:`f_{pH}`
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and :math:`f_{pE}` to be poorly constrained in the model and identify
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these controllers as important areas for model improvement.
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As a non-default option to account for CH\ :sub:`4` production in
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anoxic microsites above the water table, we apply the Arah and Stephen
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(1998) estimate of anaerobic fraction:
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.. math::
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:label: 24.4
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\varphi =\frac{1}{1+\eta C_{O_{2} } } .
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Here, :math:`\phi` is the factor by which production is inhibited
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above the water table (compared to production as calculated in equation
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, :math:`C_{O_{2}}` (mol m\ :sup:`-3`) is the bulk soil oxygen
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concentration, and :math:`\eta` = 400 mol m\ :sup:`-3`.
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The O\ :sub:`2` required to facilitate the vertically resolved
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heterotrophic decomposition and root respiration is estimated assuming 1
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mol O\ :sub:`2` is required per mol CO\ :sub:`2` produced.
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The model also calculates the O\ :sub:`2` required during
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nitrification, and the total O\ :sub:`2` demand is used in the
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O\ :sub:`2` mass balance solution.
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.. _Table Methane Parameter descriptions:
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.. table:: Parameter descriptions and sensitivity analysis ranges applied in the methane model
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| Mechanism | Parameter | Baseline Value | Range for Sensitivity Analysis | Units | Description |
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+==============+============================+==============================================+==================================================================================================+=============================================+============================================================================================+
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| Production | :math:`{Q}_{10}` | 2 | 1.5 – 4 | - | CH\ :sub:`4` production :math:`{Q}_{10}` |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | :math:`f_{pH}` | 1 | On, off | - | Impact of pH on CH\ :sub:`4` production |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | :math:`f_{pE}` | 1 | On, off | - | Impact of redox potential on CH\ :sub:`4` production |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | *S* | Varies | NA | - | Seasonal inundation factor |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | :math:`\beta` | 0.2 | NA | - | Effect of anoxia on decomposition rate (used to calculate *S* only) |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | :math:`f_{CH_{4} }` | 0.2 | NA | - | Ratio between CH\ :sub:`4` and CO\ :sub:`2` production below the water table |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| Ebullition | :math:`{C}_{e,max}` | 0.15 | NA | mol m\ :sup:`-3` | CH\ :sub:`4` concentration to start ebullition |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | :math:`{C}_{e,min}` | 0.15 | NA | - | CH\ :sub:`4` concentration to end ebullition |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| Diffusion | :math:`f_{D_{0} }` | 1 | 1, 10 | m\ :sup:`2` s\ :sup:`-1` | Diffusion coefficient multiplier (Table 24.2) |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| Aerenchyma | *p* | 0.3 | NA | - | Grass aerenchyma porosity |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | *R* | 2.9\ :math:`\times`\ 10\ :sup:`-3` m | NA | m | Aerenchyma radius |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | :math:`{r}_{L}` | 3 | NA | - | Root length to depth ratio |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | :math:`{F}_{a}` | 1 | 0.5 – 1.5 | - | Aerenchyma conductance multiplier |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| 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) |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | :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 |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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| | :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) |
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+--------------+----------------------------+----------------------------------------------+--------------------------------------------------------------------------------------------------+---------------------------------------------+--------------------------------------------------------------------------------------------+
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Ebullition
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---------------
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Briefly, the simulated aqueous CH\ :sub:`4` concentration in each
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soil level is used to estimate the expected equilibrium gaseous partial
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pressure (:math:`C_{e}` ), as a function of temperature and depth below
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the water table, by first estimating the Henry’s law partitioning
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coefficient (:math:`k_{h}^{C}` ) by the method described in
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:ref:`Wania et al. (2010)<Waniaetal2010>`:
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.. math::
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:label: 24.5
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\log \left(\frac{1}{k_{H} } \right)=\log k_{H}^{s} -\frac{1}{C_{H} } \left(\frac{1}{T} -\frac{1}{T^{s} } \right)
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.. math::
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:label: 24.6
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k_{h}^{C} =Tk_{H} R_{g}
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.. math::
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:label: 24.7
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C_{e} =\frac{C_{w} R_{g} T}{\theta _{s} k_{H}^{C} p}
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where :math:`C_{H}` \ is a constant, :math:`R_{g}` is the universal
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gas constant, :math:`k_{H}^{s}` is Henry’s law partitioning coefficient
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at standard temperature (:math:`T^{s}` ),\ :math:`C_{w}` \ is local
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aqueous CH\ :sub:`4` concentration, and *p* is pressure.
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The local pressure is calculated as the sum of the ambient pressure,
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water pressure down to the local depth, and pressure from surface
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ponding (if applicable). When the CH\ :sub:`4` partial pressure
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exceeds 15% of the local pressure (Baird et al. 2004; Strack et al.
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2006; Wania et al. 2010), bubbling occurs to remove CH\ :sub:`4`
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to below this value, modified by the fraction of CH\ :sub:`4` in
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the bubbles [taken as 57%; (:ref:`Kellner et al. 2006<Kellneretal2006>`;
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:ref:`Wania et al. 2010<Waniaetal2010>`)].
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Bubbles are immediately added to the surface flux for saturated columns
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and are placed immediately above the water table interface in
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unsaturated columns.
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.. _Aerenchyma Transport:
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Aerenchyma Transport
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-------------------------
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Aerenchyma transport is modeled in CLM as gaseous diffusion driven by a
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concentration gradient between the specific soil layer and the
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atmosphere and, if specified, by vertical advection with the
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transpiration stream. There is evidence that pressure driven flow can
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also occur, but we did not include that mechanism in the current model.
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The diffusive transport through aerenchyma (*A*, mol m\ :sup:`-2` s\ :sup:`-1`) from each soil layer is represented in the model as:
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.. math::
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:label: 24.8
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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} ,
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where *D* is the free-air gas diffusion coefficient (m:sup:`2` s\ :sup:`-1`); *C(z)* (mol m\ :sup:`-3`) is the gaseous
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concentration at depth *z* (m); :math:`r_{L}` is the ratio of root
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length to depth; *p* is the porosity (-); *T* is specific aerenchyma
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area (m:sup:`2` m\ :sup:`-2`); :math:`{r}_{a}` is the
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aerodynamic resistance between the surface and the atmospheric reference
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height (s m:sup:`-1`); and :math:`\rho _{r}` is the rooting
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density as a function of depth (-). The gaseous concentration is
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calculated with Henry’s law as described in equation .
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Based on the ranges reported in :ref:`Colmer (2003)<Colmer2003>`, we have chosen
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baseline aerenchyma porosity values of 0.3 for grass and crop PFTs and 0.1 for
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tree and shrub PFTs:
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.. math::
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:label: 24.9
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T=\frac{4 f_{N} N_{a}}{0.22} \pi R^{2} .
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Here :math:`N_{a}` is annual net primary production (NPP, mol
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m\ :sup:`-2` s\ :sup:`-1`); *R* is the aerenchyma radius
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(2.9 :math:`\times`\ 10\ :sup:`-3` m); :math:`{f}_{N}` is the
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belowground fraction of annual NPP; and the 0.22 factor represents the
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amount of C per tiller. O\ :sub:`2` can also diffuse in from the
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atmosphere to the soil layer via the reverse of the same pathway, with
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the same representation as Equation but with the gas diffusivity of
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oxygen.
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CLM also simulates the direct emission of CH\ :sub:`4` from leaves
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to the atmosphere via transpiration of dissolved methane. We calculate
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this flux (:math:`F_{CH_{4} -T}` ; mol m\ :math:`{}^{-}`\ :sup:`2`
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s\ :sup:`-1`) using the simulated soil water methane concentration
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(:math:`C_{CH_{4} ,j}` (mol m\ :sup:`-3`)) in each soil layer *j*
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and the CLM predicted transpiration (:math:`F_{T}` ) for each PFT,
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assuming that no methane was oxidized inside the plant tissue:
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.. math::
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:label: 24.10
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F_{CH_{4} -T} =\sum _{j}\rho _{r,j} F_{T} C_{CH_{4} ,j} .
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.. _CH4 Oxidation:
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CH\ :sub:`4` Oxidation
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---------------------------------
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CLM represents CH\ :sub:`4` oxidation with double Michaelis-Menten
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kinetics (:ref:`Arah and Stephen 1998<ArahStephen1998>`; :ref:`Segers 1998<Segers1998>`),
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dependent on both the gaseous CH\ :sub:`4` and O\ :sub:`2` concentrations:
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.. math::
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:label: 24.11
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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 }
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where :math:`K_{CH_{4} }` and :math:`K_{O_{2} }` \ are the half
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saturation coefficients (mol m\ :sup:`-3`) with respect to
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CH\ :sub:`4` and O\ :sub:`2` concentrations, respectively;
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:math:`R_{o,\max }` is the maximum oxidation rate (mol
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m\ :sup:`-3` s\ :sup:`-1`); and :math:`{Q}_{10}`
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specifies the temperature dependence of the reaction with a base
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temperature set to 12 :sup:`o` C. The soil moisture limitation
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factor :math:`F_{\theta }` is applied above the water table to
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represent water stress for methanotrophs. Based on the data in
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:ref:`Schnell and King (1996)<SchnellKing1996>`, we take
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:math:`F_{\theta } = {e}^{-P/{P}_{c}}`, where *P* is the soil moisture
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potential and :math:`{P}_{c} = -2.4 \times {10}^{5}` mm.
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.. _Reactive Transport Solution:
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Reactive Transport Solution
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--------------------------------
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The solution to equation is solved in several sequential steps: resolve
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competition for CH\ :sub:`4` and O\ :sub:`2` (section
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:numref:`Competition for CH4and O2`); add the ebullition flux into the
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layer directly above the water
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table or into the atmosphere; calculate the overall CH\ :sub:`4`
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or O\ :sub:`2` source term based on production, aerenchyma
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transport, ebullition, and oxidation; establish boundary conditions,
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including surface conductance to account for snow, ponding, and
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turbulent conductances and bottom flux condition
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(section :numref:`CH4 and O2 Source Terms`); calculate diffusivity
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(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) <Lerman1979>` and applied by
|
||
:ref:`Wania et al. (2010)<Waniaetal2010>`
|
||
(: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)<Moldrupetal2003>`, 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 $}} } ,
|
||
|
||
where :math:`\theta _{a}` and :math:`\theta _{s}` are the air-filled
|
||
and total (saturated water-filled) porosities (-), respectively, and *b*
|
||
is the slope of the water retention curve (-). However, :ref:`Iiyama and
|
||
Hasegawa (2005)<IiyamaHasegawa2005>` have shown that the original Millington-Quirk
|
||
(:ref:`Millington and Quirk 1961<MillingtonQuirk1961>`) relationship matched
|
||
measurements more closely in unsaturated peat soils:
|
||
|
||
.. math::
|
||
:label: 24.13
|
||
|
||
D_{e} =D_{0} \frac{\theta _{a} ^{{\raise0.7ex\hbox{$ 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)<Moldrupetal2003>`):
|
||
|
||
.. 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)<TangRiley2013>`
|
||
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<Pressetal1992>`),
|
||
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)<Prigentetal2007>` 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)<Rileyetal2011a>` 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.
|
||
|