Difference between revisions of "Upwelling Diffusion Entrainment Implementation"

Latest revision as of 17:12, 17 June 2013

model description
- carbon cycle
- non-CO2 concentrations
- radiative forcing routines
- climate model
- upwelling-diffusion-entrainment implementation

Implementation of upwelling-diffusion-entrainment equations

This section details how the equations governing the upwelling-diffusion-entrainment (UDE) ocean (Eqs. A62, A63) are implemented and modified by entrainment terms and depth-dependent ocean area (see Fig. A2). These equations represent the core of the UDE model and build on the initial work by (Hoffert et al. (1980).

The entrainment is here modeled so that the upwelling velocity in the main column is the same in each layer. Thus, the three area correction factors, <math>\theta_z^{\rm top}</math>, <math>\theta_z^{b}</math> and <math>\theta_z^{\rm dif}</math>, applied below are:

<math>\theta_z^{\rm top} = \frac{A_z}{(A_{z+1}+A_z)/2}</math>

<math>\theta_z^{b} = \frac{A_{z+1}}{(A_{z+1}+A_z)/2} </math>

<math>\theta_z^{\rm dif} = \frac{A_{z+1}-A_{z}}{(A_{z+1}+A_z)/2}\label{eq_areacorrection_thetatop}</math>

(A67)

where <math>A_z</math> is the area at the top of layer z or bottom of layer z-1 and the denominator is thus an approximation for the mean area of each ocean layer.

For the mixed layer, all terms in Eq. (A62) involving <math>\Delta T^{t+1}_{\rm NO,1}</math> are collected on the left hand side in variable <math>A(1)</math>. All terms involving <math>\Delta T^{t+1}_{\rm NO,2}</math> are collected in variable <math>B(1)</math> on the left hand side. All other terms are held in variable <math>D(1)</math> on the right hand side, so that the equation reads:

<math>\Delta T_{\rm NO,1}^{t+1} = -\frac{B(1)}{A(1)}\Delta T_{\rm NO,2}^{t+1} + \frac{D(1)}{A(1)} \label{eq_udebm_coding_ALL1}</math>		(A68)
with
<math>A(1) = 1.0+\theta_1^{\rm top}\Delta t\frac{ \lambda_O\alpha}{\zeta_o}</math>	:feedback over ocean	(A69)
<math>+\theta_1^{b}\Delta t\frac{ K_z}{0.5h_m h_d}</math>	:diffusion to layer 2
<math>+\theta_1^{b}\Delta t\frac{ w^t \beta}{h_m}</math>	:downwelling
<math>+\theta_1^{\rm top}\Delta t\frac{ k_{\rm LO}\lambda_L\mu\alpha }{\zeta_o f_{\rm NO} (\frac{k_{\rm LO}}{f_{\rm NL}} + \lambda_L)}</math>	:feedback over land
<math>B(1) = -\theta_1^{b}\Delta t\frac{ K_z}{0.5h_m h_d}</math>	:diffusion from layer 2	(A70)
<math>-\theta_1^{b}\Delta t\frac{ w^t}{h_m}</math>	:upwelling from layer 2
<math>D(1) = \Delta T_{\rm NO,1}^{t}</math>	:previous temp	(A71)
<math>+ \theta_1^{\rm top}\Delta t\frac{1}{\zeta_o}DeltaQ_{NO}</math>	: forcing ocean
<math>+ \theta_1^{\rm top}\Delta t\frac{\alpha k_{NS}}{\zeta_o f_{NO}}(\Delta T^t_{\rm SO,1}-\Delta T^t_{NO,1})</math>	:inter-hemis. exch.
<math>+ \theta_1^{\rm top}\Delta t\frac{ k_{LO}\Delta Q_{NL}}{\zeta_o f_{NO} (\frac{k_{LO}}{f_{NL}} + \lambda_L)} </math>	: land forcing
<math>+ \theta_1^{b}\Delta t\frac{\Delta w^t}{h_m}(T^0_{\rm NO,2}-T^0_{NO,sink})</math>	: variable upwelling

For the interior layers (2<math>{\leq}</math>z<math>{\leq}</math><math>n</math>), i.e., all layers except the top mixed layer and the bottom layer, the terms are re-ordered, so that <math>A(z)</math> comprises the terms for <math>\Delta T^{t+1}_{\rm NO,z-1}</math>, <math>B(z)</math> the terms for <math>\Delta T^{t+1}_{\rm NO,z}</math>, <math>C(z)</math> the terms for <math>\Delta T^{t+1}_{\rm NO,z+1}</math> and <math>D(z)</math> the remaining terms, according to:

<math>\Delta T_{\rm NO,z-1}^{t+1} = -\frac{B(z)}{A(z)}\Delta T_{\rm NO,z}^{t+1} - \frac{C(z)}{A(z)}\Delta T_{\rm NO,z+1}^{t+1} + \frac{D(z)}{A(z)}</math>		(A72)
with
<math>A(z) = - \theta_z^{top}\Delta t\frac{K_z}{0.5(h_d+h_d')h_d}</math>	: diffusion from layer above	(A73)
<math>B(z) = 1.0 + \theta_z^{b}\Delta t\frac{K_z}{h_d^2}</math>	:diffusion to layer below
<math>+\theta_z^{top}\Delta t\frac{K_z}{0.5(h_d+h_d')h_d}</math>	:diffusion to layer above
<math>+\theta_z^{top}\Delta t\frac{ w^t}{h_d}</math>	:upwelling to layer above	(A74)
<math>C(z) = - \theta_z^{b}\Delta t\frac{K_z}{h_d^2}</math>	:diffusion from layer below
<math>-\theta_z^{b}\Delta t\frac{ w^t}{h_d}</math>	:upwelling from layer below	(A75)
<math>D(z) = \Delta T_{\rm NO,z}^{t}</math>	:previous temp
<math>+\Delta t\frac{\Delta w^t}{h_d} (\theta_z^{b}T^{0}_{\rm NO,z+1}-\theta_z^{top}T^{0}_{\rm NO,z})</math>	:variable upwelling
<math>+\theta_z^{\rm dif}\Delta t\frac{ w^t}{h_d}\beta\Delta T_{\rm NO,1}^{t-1}</math>	:entrainment
<math>+\theta_z^{\rm dif}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{\rm NO,sink}</math>	:variable entrainment	(A76)

where <math>h_d'</math> is zero for the layer below the mixed layer and <math>h_d</math> otherwise. For the bottom layer, the respective sum factor <math>A(n)</math> for <math>\Delta T^{t+1}_{\rm NO,n-1}</math>, <math>B(n)</math> for <math>\Delta T^{t+1}_{\rm NO,n}</math> and <math>D(n)</math> for the remaining terms is:

<math>\Delta T_{{\rm NO},n-1}^{t+1} = -\frac{B(n)}{A(n)}\Delta T_{{\rm NO},n}^{t+1} + \frac{D(n)}{A(n)}</math>		(A77)
with
<math>A(n) = - \theta_{n}^{\rm top}\Delta t\frac{K_z}{h_d^2}</math>	:diffusion from layer n-1	(A78)
<math>B(n) = 1.0 + \theta_{n}^{\rm top}\Delta t\frac{K_z}{h_d^2}</math>	:diffusion to layer n-1	(A79)
<math>+\theta_{n}^{\rm top}\Delta t\frac{ w^t}{h_d}</math>	:upwelling to layer n-1
<math>D(n) = \Delta T_{{\rm NO},n}^{t}</math>	:previous temp	(A80)
<math>+\theta_{n}^{\rm top}\Delta t\frac{w^t}{h_d} \beta\Delta T^{t-1}_{\rm NO,1}</math>	:downwelling from top layer
<math>-\theta_{n}^{\rm top}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{{\rm NO},n}</math>	:variable upwelling
<math>+\theta_{n}^{\rm top}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{\rm NO,sink}</math>	:variable downweilling

With these Eqs. (A68-A80), the ocean temperatures can be solved consecutively from the bottom to the top layer at each time step.

@@ Line 11: / Line 11: @@
 This section details how the equations governing the upwelling-diffusion-entrainment (UDE) ocean (Eqs. [[#eq_62|A62]],  [[#eq_63|A63]]) are implemented and modified by entrainment terms and depth-dependent ocean area (see Fig. [[#Fig-A2a|A2]]). These equations represent the core of the UDE model and build on the initial work by ([[References#Hoffert_1980_Role_DeapSea, Harvey_Schneider_1985_PartII, Harvey_Schneider_1985_PartI|Hoffert et al. (1980)]].
-The entrainment is here modeled so that the upwelling velocity in the main column is the same in each layer. Thus, the three area correction factors, <m>\theta_z^{\rm top}</m>, <m>\theta_z^{b}</m> and
+The entrainment is here modeled so that the upwelling velocity in the main column is the same in each layer. Thus, the three area correction factors, <math>\theta_z^{\rm top}</math>, <math>\theta_z^{b}</math> and
-<m>\theta_z^{\rm dif}</m>, applied below are:
+<math>\theta_z^{\rm dif}</math>, applied below are:
-<m>\theta_z^{\rm top} = \frac{A_z}{(A_{z+1}+A_z)/2}</m>
+<math>\theta_z^{\rm top} = \frac{A_z}{(A_{z+1}+A_z)/2}</math>
-<m>\theta_z^{b} = \frac{A_{z+1}}{(A_{z+1}+A_z)/2} </m>
+<math>\theta_z^{b} = \frac{A_{z+1}}{(A_{z+1}+A_z)/2} </math>
-<m>\theta_z^{\rm dif} = \frac{A_{z+1}-A_{z}}{(A_{z+1}+A_z)/2}\label{eq_areacorrection_thetatop}</m><span id="eq_A67"></span><div style="float: right; clear: right;">('''A67''')</div>
+<math>\theta_z^{\rm dif} = \frac{A_{z+1}-A_{z}}{(A_{z+1}+A_z)/2}\label{eq_areacorrection_thetatop}</math><span id="eq_A67"></span><div style="float: right; clear: right;">('''A67''')</div>
-where <m>A_z</m> is the area at the top of layer z or bottom of layer z-1 and the denominator is thus an approximation for the mean area of each ocean layer.
+where <math>A_z</math> is the area at the top of layer z or bottom of layer z-1 and the denominator is thus an approximation for the mean area of each ocean layer.
-For the mixed layer, all terms in Eq. ([[#eq_62|A62]]) involving <m>\Delta T^{t+1}_{\rm NO,1}</m> are collected on the left hand side in variable <m>A(1)</m>. All terms involving <m>\Delta T^{t+1}_{\rm NO,2}</m> are collected in variable <m>B(1)</m> on the left hand side. All other terms are held in variable <m>D(1)</m> on the right hand side, so that the
+For the mixed layer, all terms in Eq. ([[#eq_62|A62]]) involving <math>\Delta T^{t+1}_{\rm NO,1}</math> are collected on the left hand side in variable <math>A(1)</math>. All terms involving <math>\Delta T^{t+1}_{\rm NO,2}</math> are collected in variable <math>B(1)</math> on the left hand side. All other terms are held in variable <math>D(1)</math> on the right hand side, so that the
 equation reads:
 {|
-| <m>\Delta T_{\rm NO,1}^{t+1} = -\frac{B(1)}{A(1)}\Delta T_{\rm NO,2}^{t+1} + \frac{D(1)}{A(1)} \label{eq_udebm_coding_ALL1}</m> ||  || <span id="eq_A68"></span><div style="float: right; clear: right;">('''A68''')</div>
+| <math>\Delta T_{\rm NO,1}^{t+1} = -\frac{B(1)}{A(1)}\Delta T_{\rm NO,2}^{t+1} + \frac{D(1)}{A(1)} \label{eq_udebm_coding_ALL1}</math> ||  || <span id="eq_A68"></span><div style="float: right; clear: right;">('''A68''')</div>
 |-
 | with ||  ||
 |-
-| <m>A(1) = 1.0+\theta_1^{\rm top}\Delta t\frac{ \lambda_O\alpha}{\zeta_o}</m> || :feedback over ocean || <span id="eq_A69"></span><div style="float: right; clear: right;">('''A69''')</div>
+| <math>A(1) = 1.0+\theta_1^{\rm top}\Delta t\frac{ \lambda_O\alpha}{\zeta_o}</math> || :feedback over ocean || <span id="eq_A69"></span><div style="float: right; clear: right;">('''A69''')</div>
 |-
-| <m>+\theta_1^{b}\Delta t\frac{ K_z}{0.5h_m h_d}</m> || :diffusion to layer 2 ||
+| <math>+\theta_1^{b}\Delta t\frac{ K_z}{0.5h_m h_d}</math> || :diffusion to layer 2 ||
 |-
-| <m>+\theta_1^{b}\Delta t\frac{ w^t \beta}{h_m}</m> || :downwelling ||
+| <math>+\theta_1^{b}\Delta t\frac{ w^t \beta}{h_m}</math> || :downwelling ||
 |-
-| <m>+\theta_1^{\rm top}\Delta t\frac{ k_{\rm LO}\lambda_L\mu\alpha }{\zeta_o f_{\rm NO} (\frac{k_{\rm LO}}{f_{\rm NL}} + \lambda_L)}</m> || :feedback over land ||
+| <math>+\theta_1^{\rm top}\Delta t\frac{ k_{\rm LO}\lambda_L\mu\alpha }{\zeta_o f_{\rm NO} (\frac{k_{\rm LO}}{f_{\rm NL}} + \lambda_L)}</math> || :feedback over land ||
 |-
-| <m>B(1) = -\theta_1^{b}\Delta t\frac{ K_z}{0.5h_m h_d}</m> || :diffusion from layer 2 || <span id="eq_A70"></span><div style="float: right; clear: right;">('''A70''')</div>
+| <math>B(1) = -\theta_1^{b}\Delta t\frac{ K_z}{0.5h_m h_d}</math> || :diffusion from layer 2 || <span id="eq_A70"></span><div style="float: right; clear: right;">('''A70''')</div>
 |-
-| <m>-\theta_1^{b}\Delta t\frac{ w^t}{h_m}</m> || :upwelling from layer 2 ||
+| <math>-\theta_1^{b}\Delta t\frac{ w^t}{h_m}</math> || :upwelling from layer 2 ||
 |-
-| <m>D(1) = \Delta T_{\rm NO,1}^{t}</m> || :previous temp || <span id="eq_A71"></span><div style="float: right; clear: right;">('''A71''')</div>
+| <math>D(1) = \Delta T_{\rm NO,1}^{t}</math> || :previous temp || <span id="eq_A71"></span><div style="float: right; clear: right;">('''A71''')</div>
 |-
-| <m>+ \theta_1^{\rm top}\Delta t\frac{1}{\zeta_o}DeltaQ_{NO}</m> || : forcing ocean ||
+| <math>+ \theta_1^{\rm top}\Delta t\frac{1}{\zeta_o}DeltaQ_{NO}</math> || : forcing ocean ||
 |-
-| <m>+ \theta_1^{\rm top}\Delta t\frac{\alpha k_{NS}}{\zeta_o f_{NO}}(\Delta T^t_{\rm SO,1}-\Delta T^t_{NO,1})</m>  || :inter-hemis. exch. ||
+| <math>+ \theta_1^{\rm top}\Delta t\frac{\alpha k_{NS}}{\zeta_o f_{NO}}(\Delta T^t_{\rm SO,1}-\Delta T^t_{NO,1})</math>  || :inter-hemis. exch. ||
 |-
-| <m>+ \theta_1^{\rm top}\Delta t\frac{ k_{LO}\Delta Q_{NL}}{\zeta_o f_{NO} (\frac{k_{LO}}{f_{NL}} + \lambda_L)} </m> || : land forcing ||
+| <math>+ \theta_1^{\rm top}\Delta t\frac{ k_{LO}\Delta Q_{NL}}{\zeta_o f_{NO} (\frac{k_{LO}}{f_{NL}} + \lambda_L)} </math> || : land forcing ||
 |-
-| <m>+ \theta_1^{b}\Delta t\frac{\Delta w^t}{h_m}(T^0_{\rm NO,2}-T^0_{NO,sink})</m>  || : variable upwelling ||
+| <math>+ \theta_1^{b}\Delta t\frac{\Delta w^t}{h_m}(T^0_{\rm NO,2}-T^0_{NO,sink})</math>  || : variable upwelling ||
 |}
-For the interior layers (2<m>{\leq}</m>z<m>{\leq}</m><m>n</m>), i.e., all layers except the top mixed layer and the bottom layer, the terms are re-ordered, so that <m>A(z)</m> comprises the terms for <m>\Delta T^{t+1}_{\rm NO,z-1}</m>, <m>B(z)</m> the terms for <m>\Delta T^{t+1}_{\rm NO,z}</m>, <m>C(z)</m> the terms for <m>\Delta T^{t+1}_{\rm NO,z+1}</m> and <m>D(z)</m> the remaining terms, according to:
+For the interior layers (2<math>{\leq}</math>z<math>{\leq}</math><math>n</math>), i.e., all layers except the top mixed layer and the bottom layer, the terms are re-ordered, so that <math>A(z)</math> comprises the terms for <math>\Delta T^{t+1}_{\rm NO,z-1}</math>, <math>B(z)</math> the terms for <math>\Delta T^{t+1}_{\rm NO,z}</math>, <math>C(z)</math> the terms for <math>\Delta T^{t+1}_{\rm NO,z+1}</math> and <math>D(z)</math> the remaining terms, according to:
 {|
-| <m>\Delta T_{\rm NO,z-1}^{t+1} = -\frac{B(z)}{A(z)}\Delta T_{\rm NO,z}^{t+1} - \frac{C(z)}{A(z)}\Delta T_{\rm NO,z+1}^{t+1} + \frac{D(z)}{A(z)}</m> ||  || <span id="eq_A72"></span><div style="float: right; clear: right;">('''A72''')</div>
+| <math>\Delta T_{\rm NO,z-1}^{t+1} = -\frac{B(z)}{A(z)}\Delta T_{\rm NO,z}^{t+1} - \frac{C(z)}{A(z)}\Delta T_{\rm NO,z+1}^{t+1} + \frac{D(z)}{A(z)}</math> ||  || <span id="eq_A72"></span><div style="float: right; clear: right;">('''A72''')</div>
 |-
 | with ||  ||
 |-
-| <m>A(z) = - \theta_z^{top}\Delta t\frac{K_z}{0.5(h_d+h_d')h_d}</m>|| : diffusion from layer above || <span id="eq_A73"></span><div style="float: right; clear: right;">('''A73''')</div>
+| <math>A(z) = - \theta_z^{top}\Delta t\frac{K_z}{0.5(h_d+h_d')h_d}</math>|| : diffusion from layer above || <span id="eq_A73"></span><div style="float: right; clear: right;">('''A73''')</div>
 |-
-| <m>B(z) = 1.0 + \theta_z^{b}\Delta t\frac{K_z}{h_d^2}</m> || :diffusion to layer below ||
+| <math>B(z) = 1.0 + \theta_z^{b}\Delta t\frac{K_z}{h_d^2}</math> || :diffusion to layer below ||
 |-
-| <m>+\theta_z^{top}\Delta t\frac{K_z}{0.5(h_d+h_d')h_d}</m> || :diffusion to layer above ||
+| <math>+\theta_z^{top}\Delta t\frac{K_z}{0.5(h_d+h_d')h_d}</math> || :diffusion to layer above ||
 |-
-| <m>+\theta_z^{top}\Delta t\frac{ w^t}{h_d}</m> || :upwelling to layer above || <span id="eq_A74"></span><div style="float: right; clear: right;">('''A74''')</div>
+| <math>+\theta_z^{top}\Delta t\frac{ w^t}{h_d}</math> || :upwelling to layer above || <span id="eq_A74"></span><div style="float: right; clear: right;">('''A74''')</div>
 |-
-| <m>C(z) = - \theta_z^{b}\Delta t\frac{K_z}{h_d^2}</m> || :diffusion from layer below ||
+| <math>C(z) = - \theta_z^{b}\Delta t\frac{K_z}{h_d^2}</math> || :diffusion from layer below ||
 |-
-| <m>-\theta_z^{b}\Delta t\frac{ w^t}{h_d}</m> || :upwelling from layer below || <span id="eq_A75"></span><div style="float: right; clear: right;">('''A75''')</div>
+| <math>-\theta_z^{b}\Delta t\frac{ w^t}{h_d}</math> || :upwelling from layer below || <span id="eq_A75"></span><div style="float: right; clear: right;">('''A75''')</div>
 |-
-| <m>D(z) = \Delta T_{\rm NO,z}^{t}</m>  || :previous temp ||
+| <math>D(z) = \Delta T_{\rm NO,z}^{t}</math>  || :previous temp ||
 |-
-| <m>+\Delta t\frac{\Delta w^t}{h_d} (\theta_z^{b}T^{0}_{\rm NO,z+1}-\theta_z^{top}T^{0}_{\rm NO,z})</m> || :variable upwelling ||
+| <math>+\Delta t\frac{\Delta w^t}{h_d} (\theta_z^{b}T^{0}_{\rm NO,z+1}-\theta_z^{top}T^{0}_{\rm NO,z})</math> || :variable upwelling ||
 |-
-| <m>+\theta_z^{\rm dif}\Delta t\frac{ w^t}{h_d}\beta\Delta T_{\rm NO,1}^{t-1}</m> || :entrainment ||
+| <math>+\theta_z^{\rm dif}\Delta t\frac{ w^t}{h_d}\beta\Delta T_{\rm NO,1}^{t-1}</math> || :entrainment ||
 |-
-| <m>+\theta_z^{\rm dif}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{\rm NO,sink}</m> || :variable entrainment || <span id="eq_A76"></span><div style="float: right; clear: right;">('''A76''')</div>
+| <math>+\theta_z^{\rm dif}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{\rm NO,sink}</math> || :variable entrainment || <span id="eq_A76"></span><div style="float: right; clear: right;">('''A76''')</div>
 |}
-where <m>h_d'</m> is zero for the layer below the mixed layer and <m>h_d</m> otherwise. For the bottom layer, the respective sum factor <m>A(n)</m> for <m>\Delta T^{t+1}_{\rm NO,n-1}</m>, <m>B(n)</m> for <m>\Delta T^{t+1}_{\rm NO,n}</m> and <m>D(n)</m> for the remaining terms is:
+where <math>h_d'</math> is zero for the layer below the mixed layer and <math>h_d</math> otherwise. For the bottom layer, the respective sum factor <math>A(n)</math> for <math>\Delta T^{t+1}_{\rm NO,n-1}</math>, <math>B(n)</math> for <math>\Delta T^{t+1}_{\rm NO,n}</math> and <math>D(n)</math> for the remaining terms is:
 {|
-| <m>\Delta T_{{\rm NO},n-1}^{t+1} = -\frac{B(n)}{A(n)}\Delta T_{{\rm NO},n}^{t+1} + \frac{D(n)}{A(n)}</m> ||  || <span id="eq_A77"></span><div style="float: right; clear: right;">('''A77''')</div>
+| <math>\Delta T_{{\rm NO},n-1}^{t+1} = -\frac{B(n)}{A(n)}\Delta T_{{\rm NO},n}^{t+1} + \frac{D(n)}{A(n)}</math> ||  || <span id="eq_A77"></span><div style="float: right; clear: right;">('''A77''')</div>
 |-
 | with ||  ||
 |-
-| <m>A(n) = - \theta_{n}^{\rm top}\Delta t\frac{K_z}{h_d^2}</m> || :diffusion from layer n-1 || <span id="eq_A78"></span><div style="float: right; clear: right;">('''A78''')</div>
+| <math>A(n) = - \theta_{n}^{\rm top}\Delta t\frac{K_z}{h_d^2}</math> || :diffusion from layer n-1 || <span id="eq_A78"></span><div style="float: right; clear: right;">('''A78''')</div>
 |-
-| <m>B(n) = 1.0 + \theta_{n}^{\rm top}\Delta t\frac{K_z}{h_d^2}</m> || :diffusion to layer n-1 || <span id="eq_A79"></span><div style="float: right; clear: right;">('''A79''')</div>
+| <math>B(n) = 1.0 + \theta_{n}^{\rm top}\Delta t\frac{K_z}{h_d^2}</math> || :diffusion to layer n-1 || <span id="eq_A79"></span><div style="float: right; clear: right;">('''A79''')</div>
 |-
-| <m>+\theta_{n}^{\rm top}\Delta t\frac{ w^t}{h_d}</m> || :upwelling to layer n-1 ||
+| <math>+\theta_{n}^{\rm top}\Delta t\frac{ w^t}{h_d}</math> || :upwelling to layer n-1 ||
 |-
-| <m>D(n) = \Delta T_{{\rm NO},n}^{t}</m> || :previous temp || <span id="eq_A80"></span><div style="float: right; clear: right;">('''A80''')</div>
+| <math>D(n) = \Delta T_{{\rm NO},n}^{t}</math> || :previous temp || <span id="eq_A80"></span><div style="float: right; clear: right;">('''A80''')</div>
 |-
-| <m>+\theta_{n}^{\rm top}\Delta t\frac{w^t}{h_d} \beta\Delta T^{t-1}_{\rm NO,1}</m>  || :downwelling from top layer ||
+| <math>+\theta_{n}^{\rm top}\Delta t\frac{w^t}{h_d} \beta\Delta T^{t-1}_{\rm NO,1}</math>  || :downwelling from top layer ||
 |-
-| <m>-\theta_{n}^{\rm top}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{{\rm NO},n}</m> || :variable upwelling ||
+| <math>-\theta_{n}^{\rm top}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{{\rm NO},n}</math> || :variable upwelling ||
 |-
-| <m>+\theta_{n}^{\rm top}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{\rm NO,sink}</m> || :variable downweilling ||
+| <math>+\theta_{n}^{\rm top}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{\rm NO,sink}</math> || :variable downweilling ||
 |}
 With these Eqs. ([[#eq_68|A68]]-[[#eq_80|A80]]), the ocean temperatures can be solved consecutively from the bottom to the top layer at each time step.

Difference between revisions of "Upwelling Diffusion Entrainment Implementation"

Latest revision as of 17:12, 17 June 2013

Implementation of upwelling-diffusion-entrainment equations

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