Fix documentation issue #3035

Author: olyson

doc/source/tech_note/Fluxes/CLM50_Tech_Note_Fluxes.rst

@@ -1147,7 +1147,7 @@ where :math:`\overrightarrow{S}_{v}` is the solar radiation absorbed by the vege
 
    \Delta T_{v} =\frac{\overrightarrow{S}_{v} -\overrightarrow{L}_{v} -H_{v} -\lambda E_{v} }{\frac{\partial \overrightarrow{L}_{v} }{\partial T_{v} } +\frac{\partial H_{v} }{\partial T_{v} } +\frac{\partial \lambda E_{v} }{\partial T_{v} } }
 
-where :math:`\Delta T_{v} =T_{v}^{n+1} -T_{v}^{n}` and the subscript "n" indicates the iteration.
+where :math:`\Delta T_{v} =T_{v}^{k+1} -T_{v}^{k}` and the subscript "k" indicates the iteration.
 
 The partial derivatives are
 
@@ -1227,9 +1227,9 @@ The numerical solution for vegetation temperature and the fluxes of momentum, se
 
 #. Latent heat flux from vegetation :math:`\lambda E_{v}` (:eq:`5.101` )
 
-#. If the latent heat flux has changed sign from the latent heat flux computed at the previous iteration (:math:`\lambda E_{v} ^{n+1} \times \lambda E_{v} ^{n} <0`), the latent heat flux is constrained to be 10% of the computed value. The difference between the constrained and computed value (:math:`\Delta _{1} =0.1\lambda E_{v} ^{n+1} -\lambda E_{v} ^{n+1}` ) is added to the sensible heat flux later.
+#. If the latent heat flux has changed sign from the latent heat flux computed at the previous iteration (:math:`\lambda E_{v} ^{k+1} \times \lambda E_{v} ^{k} <0`), the latent heat flux is constrained to be 10% of the computed value. The difference between the constrained and computed value (:math:`\Delta _{1} =0.1\lambda E_{v} ^{k+1} -\lambda E_{v} ^{k+1}` ) is added to the sensible heat flux later.
 
-#. Change in vegetation temperature :math:`\Delta T_{v}` (:eq:`5.129` ) and update the vegetation temperature as :math:`T_{v}^{n+1} =T_{v}^{n} +\Delta T_{v}`. :math:`T_{v}` is constrained to change by no more than 1°K in one iteration. If this limit is exceeded, the energy error is
+#. Change in vegetation temperature :math:`\Delta T_{v}` (:eq:`5.129` ) and update the vegetation temperature as :math:`T_{v}^{k+1} =T_{v}^{k} +\Delta T_{v}`. :math:`T_{v}` is constrained to change by no more than 1°K in one iteration. If this limit is exceeded, the energy error is
 
    .. math::
       :label: 5.138
@@ -1269,7 +1269,7 @@ The error :math:`\lambda \Delta _{3}` is added to the sensible heat flux later.
 
 #. Monin-Obukhov length :math:`L` (:eq:`5.49` )
 
-#. The iteration is stopped after two or more steps if :math:`\tilde{\Delta }T_{v} <0.01` and :math:`\left|\lambda E_{v}^{n+1} -\lambda E_{v}^{n} \right|<0.1` where :math:`\tilde{\Delta }T_{v} =\max \left(\left|T_{v}^{n+1} -T_{v}^{n} \right|,\, \left|T_{v}^{n} -T_{v}^{n-1} \right|\right)`, or after forty iterations have been carried out.
+#. The iteration is stopped after two or more steps if :math:`\tilde{\Delta }T_{v} <0.01` and :math:`\left|\lambda E_{v}^{k+1} -\lambda E_{v}^{k} \right|<0.1` where :math:`\tilde{\Delta }T_{v} =\max \left(\left|T_{v}^{k+1} -T_{v}^{k} \right|,\, \left|T_{v}^{k} -T_{v}^{k-1} \right|\right)`, or after forty iterations have been carried out.
 
 #. Momentum fluxes :math:`\tau _{x}`, :math:`\tau _{y}` (:eq:`5.5`, :eq:`5.6`)
 

doc/source/tech_note/Radiative_Fluxes/CLM50_Tech_Note_Radiative_Fluxes.rst

@@ -141,9 +141,9 @@ where
 .. math::
    :label: 4.14
 
-   \begin{array}{l} {L_{vg} \, \uparrow =\left(1-\varepsilon _{g} \right)\left(1-\varepsilon _{v} \right)\left(1-\varepsilon _{v} \right)L_{atm} \, \downarrow } \\ {\qquad \qquad +\varepsilon _{v} \left[1+\left(1-\varepsilon _{g} \right)\left(1-\varepsilon _{v} \right)\right]\sigma \left(T_{v}^{n} \right)^{3} \left[T_{v}^{n} +4\left(T_{v}^{n+1} -T_{v}^{n} \right)\right]} \\ {\qquad \qquad +\varepsilon _{g} \left(1-\varepsilon _{v} \right)\sigma \left(T_{g}^{n} \right)^{4} } \\ {\qquad =\left(1-\varepsilon _{g} \right)\left(1-\varepsilon _{v} \right)\left(1-\varepsilon _{v} \right)L_{atm} \, \downarrow } \\ {\qquad \qquad +\varepsilon _{v} \sigma \left(T_{v}^{n} \right)^{4} } \\ {\qquad \qquad +\varepsilon _{v} \left(1-\varepsilon _{g} \right)\left(1-\varepsilon _{v} \right)\sigma \left(T_{v}^{n} \right)^{4} } \\ {\qquad \qquad +4\varepsilon _{v} \sigma \left(T_{v}^{n} \right)^{3} \left(T_{v}^{n+1} -T_{v}^{n} \right)} \\ {\qquad \qquad +4\varepsilon _{v} \left(1-\varepsilon _{g} \right)\left(1-\varepsilon _{v} \right)\sigma \left(T_{v}^{n} \right)^{3} \left(T_{v}^{n+1} -T_{v}^{n} \right)} \\ {\qquad \qquad +\varepsilon _{g} \left(1-\varepsilon _{v} \right)\sigma \left(T_{g}^{n} \right)^{4} } \end{array}
+   \begin{array}{l} {L_{vg} \, \uparrow =\left(1-\varepsilon _{g} \right)\left(1-\varepsilon _{v} \right)\left(1-\varepsilon _{v} \right)L_{atm} \, \downarrow } \\ {\qquad \qquad +\varepsilon _{v} \left[1+\left(1-\varepsilon _{g} \right)\left(1-\varepsilon _{v} \right)\right]\sigma \left(T_{v}^{n+1,k} \right)^{3} \left[T_{v}^{n+1,k} +4\left(T_{v}^{n+1,k+1} -T_{v}^{n+1} \right)\right]} \\ {\qquad \qquad +\varepsilon _{g} \left(1-\varepsilon _{v} \right)\sigma \left(T_{g}^{n} \right)^{4} } \\ {\qquad =\left(1-\varepsilon _{g} \right)\left(1-\varepsilon _{v} \right)\left(1-\varepsilon _{v} \right)L_{atm} \, \downarrow } \\ {\qquad \qquad +\varepsilon _{v} \sigma \left(T_{v}^{n+1,k} \right)^{4} } \\ {\qquad \qquad +\varepsilon _{v} \left(1-\varepsilon _{g} \right)\left(1-\varepsilon _{v} \right)\sigma \left(T_{v}^{n+1,k} \right)^{4} } \\ {\qquad \qquad +4\varepsilon _{v} \sigma \left(T_{v}^{n+1,k} \right)^{3} \left(T_{v}^{n+1,k+1} -T_{v}^{n+1} \right)} \\ {\qquad \qquad +4\varepsilon _{v} \left(1-\varepsilon _{g} \right)\left(1-\varepsilon _{v} \right)\sigma \left(T_{v}^{n+1,k} \right)^{3} \left(T_{v}^{n+1,k+1} -T_{v}^{n+1} \right)} \\ {\qquad \qquad +\varepsilon _{g} \left(1-\varepsilon _{v} \right)\sigma \left(T_{g}^{n} \right)^{4} } \end{array}
 
-where :math:`\varepsilon _{v}` is the vegetation emissivity and :math:`T_{v}^{n+1}` and :math:`T_{v}^{n}` are the vegetation temperatures at the current and previous time steps, respectively (:ref:`rst_Momentum, Sensible Heat, and Latent Heat Fluxes`). The first term in the equation above is the atmospheric longwave radiation that is transmitted through the canopy, reflected by the ground, and transmitted through the canopy to the atmosphere. The second term is the longwave radiation emitted by the canopy directly to the atmosphere. The third term is the longwave radiation emitted downward from the canopy, reflected by the ground, and transmitted through the canopy to the atmosphere. The fourth term is the increase (decrease) in longwave radiation due to an increase (decrease) in canopy temperature that is emitted by the canopy directly to the atmosphere. The fifth term is the increase (decrease) in longwave radiation due to an increase (decrease) in canopy temperature that is emitted downward from the canopy, reflected from the ground, and transmitted through the canopy to the atmosphere. The last term is the longwave radiation emitted by the ground and transmitted through the canopy to the atmosphere.
+where :math:`\varepsilon _{v}` is the vegetation emissivity, :math:`T_{v}^{n+1}` and :math:`T_{v}^{n}` are the vegetation temperatures at the current and previous time steps, respectively, and :math:`k` is the iteration level (:ref:`rst_Momentum, Sensible Heat, and Latent Heat Fluxes`). The first term in the equation above is the atmospheric longwave radiation that is transmitted through the canopy, reflected by the ground, and transmitted through the canopy to the atmosphere. The second term is the longwave radiation emitted by the canopy directly to the atmosphere. The third term is the longwave radiation emitted downward from the canopy, reflected by the ground, and transmitted through the canopy to the atmosphere. The fourth term is the increase (decrease) in longwave radiation due to an increase (decrease) in canopy temperature that is emitted by the canopy directly to the atmosphere. The fifth term is the increase (decrease) in longwave radiation due to an increase (decrease) in canopy temperature that is emitted downward from the canopy, reflected from the ground, and transmitted through the canopy to the atmosphere. The last term is the longwave radiation emitted by the ground and transmitted through the canopy to the atmosphere.
 
 The upward longwave radiation from the ground is
 
@@ -157,7 +157,7 @@ where :math:`L_{v} \, \downarrow` is the downward longwave radiation below the v
 .. math::
    :label: 4.16
 
-   L_{v} \, \downarrow =\left(1-\varepsilon _{v} \right)L_{atm} \, \downarrow +\varepsilon _{v} \sigma \left(T_{v}^{n} \right)^{4} +4\varepsilon _{v} \sigma \left(T_{v}^{n} \right)^{3} \left(T_{v}^{n+1} -T_{v}^{n} \right).
+   L_{v} \, \downarrow =\left(1-\varepsilon _{v} \right)L_{atm} \, \downarrow +\varepsilon _{v} \sigma \left(T_{v}^{n+1,k} \right)^{4} +4\varepsilon _{v} \sigma \left(T_{v}^{n+1,k} \right)^{3} \left(T_{v}^{n+1,k+1} -T_{v}^{n+1} \right).
 
 The net longwave radiation flux for the ground is (positive toward the atmosphere)