Antonius Golly at 15:12, 17 June 2013

2013-06-17T15:12:04Z

Antonius Golly: Created page with "*model description carbon cycle non-CO2 concentrations **[[Radiative Forcing | radiative forcing r..."

2013-06-17T11:31:38Z

Created page with "*model description ** carbon cycle ** non-CO2 concentrations **[[Radiative Forcing | radiative forcing r..."

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*[[Model Description|model description]]
**[[The Carbon Cycle| carbon cycle]]
**[[Non-CO2 Concentrations | non-CO2 concentrations]]
**[[Radiative Forcing | radiative forcing routines]]
**[[Upwelling_diffusion_climate_model | climate model]]
**[[Upwelling Diffusion Entrainment Implementation | upwelling-diffusion-entrainment implementation]]

==Implementation of upwelling-diffusion-entrainment equations==

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
<m>\theta_z^{\rm dif}</m>, applied below are:

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

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

<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>

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.

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
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>
|-
| 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>
|-
| <m>+\theta_1^{b}\Delta t\frac{ K_z}{0.5h_m h_d}</m> || :diffusion to layer 2 ||
|-
| <m>+\theta_1^{b}\Delta t\frac{ w^t \beta}{h_m}</m> || :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 ||
|-
| <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>
|-
| <m>-\theta_1^{b}\Delta t\frac{ w^t}{h_m}</m> || :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>
|-
| <m>+ \theta_1^{\rm top}\Delta t\frac{1}{\zeta_o}DeltaQ_{NO}</m> || : 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. ||
|-
| <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 ||
|-
| <m>+ \theta_1^{b}\Delta t\frac{\Delta w^t}{h_m}(T^0_{\rm NO,2}-T^0_{NO,sink})</m> || : 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:

{|
| <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>
|-
| 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>
|-
| <m>B(z) = 1.0 + \theta_z^{b}\Delta t\frac{K_z}{h_d^2}</m> || :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 ||
|-
| <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>
|-
| <m>C(z) = - \theta_z^{b}\Delta t\frac{K_z}{h_d^2}</m> || :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>
|-
| <m>D(z) = \Delta T_{\rm NO,z}^{t}</m> || :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 ||
|-
| <m>+\theta_z^{\rm dif}\Delta t\frac{ w^t}{h_d}\beta\Delta T_{\rm NO,1}^{t-1}</m> || :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>
|}

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:

{|
| <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>
|-
| 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>
|-
| <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>
|-
| <m>+\theta_{n}^{\rm top}\Delta t\frac{ w^t}{h_d}</m> || :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>
|-
| <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 ||
|-
| <m>-\theta_{n}^{\rm top}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{{\rm NO},n}</m> || :variable upwelling ||
|-
| <m>+\theta_{n}^{\rm top}\Delta t\frac{\Delta w^t}{h_d} T^{0}_{\rm NO,sink}</m> || :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.

Upwelling Diffusion Entrainment Implementation - Revision history

Antonius Golly at 15:12, 17 June 2013

Antonius Golly: Created page with "*model description ** carbon cycle ** non-CO2 concentrations **[[Radiative Forcing | radiative forcing r..."

Antonius Golly: Created page with "*model description carbon cycle non-CO2 concentrations **[[Radiative Forcing | radiative forcing r..."