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<!--
########################################
## @author Benjamin Thomas Schwertfeger (July 2021)
## copyright by Benjamin Thomas Schwertfeger (July 2021)
############
https://paleodyn.uni-bremen.de/study/MES/ebm/index.html
-->
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta http-quiv="X-UA-Compatible" content="ie-edge">
<!-- <link rel="icon" type="image/x-icon"
href="https://www.awi.de/typo3conf/ext/sms_boilerplate/Resources/Public/Images/AWI/awi_logo.svg"> -->
<title>Enegie Balance Models</title>
<link rel="stylesheet" href="format.css">
<!-- CALCULATION -->
<script type="module" src="main_bundle.js"></script>
<!-- CHART JS -->
<script src='https://cdn.jsdelivr.net/npm/chart.js@3.3.1/dist/chart.min.js'> </script>
<!-- PLOTLY -->
<script src="https://cdn.plot.ly/plotly-2.0.0-rc.3.min.js"></script>
<!-- to display mathematical equations-->
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<!-- <script src='//cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML'></script> -->
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
</head>
<body>
<!-- <img src="https://upload.wikimedia.org/wikipedia/commons/f/f7/AWI_Logo_2017.svg"
style="position:absolute; right: 10px; top: 10px; width: 15%;" \> -->
<div id="Headline" style="margin:auto; padding:10px; text-align:center;">
<h1>Energy balance models</h1>
</div>
<div>
<p class="p_information">
"Previous studies identified instabilities for a shrinking
ice
cover in two types of idealized climate models: (i)
annual-mean
latitudinally varying diffusive energy balance models (EBMs)
and
(ii) seasonally varying single-column models (SCMs). The
instabilities in these low-order models stand in contrast
with
results from compre- hensive global climate models (GCMs),
which
typically do not simulate any such instability. To help
bridge
the gap between low-order models and GCMs, an idealized
model is
developed that includes both latitudinal and seasonal
variations. The following EBM is a idealized representation
of
sea ice and climate with seasonal and latitudinal variations
in
a global domain." <i>(Extraced from: Till J.W Wagner and Ian
Eisenman, "How Climate Model Complexity Influences Sea
Ice
Stability" (2015) p.2 Abstract)</i>
For simplicity, I will call the paper "paper" so that I can refer to it more easily on this page.
<br><br>
</p>
<p class="p_information">
  ☞ <a target="_blank"
href="https://www.researchgate.net/publication/276276426_How_Climate_Model_Complexity_Influences_Sea_Ice_Stability">Paper
and Documentation</a>
</p>
<p class="p_information">
  ☞ Python source code: <a target="_blank"
href="http://eisenman.ucsd.edu/code/python/EBM_simple_WE15.py">simple EBM</a>,
<a target="_blank" href="http://eisenman.ucsd.edu/code/python/sea_ice_EBM_WE15.py">complex EBM</a>
</p>
</div>
<div class="placeholder" style="width:100%;height:67px;"></div>
<!-- BEGIN SIMPLE EBM -->
<div class="tebm-all" style="width:95%; margin: 0 auto;">
<div id="tebm_headline" style="padding:10px; text-align:center;">
<h2>Temperature EBM</h2>
</div>
<div class="tebm-description" style="padding:30px;">
<table style="margin: 0 auto; width: 95%;">
<tr>
<td>Diffusy for heat transport: </td>
<td>\(D^*=Wm^{-2}K^{-1}\)</td>
<td>   </td>
<td>OLR when \(T=T_m\):</td>
<td>\(A=Wm^{-2}\)</td>
<td>   </td>
<td>OLR temperature dependence:</td>
<td>\(B=Wm^{-2}K^{-1}\)</td>
</tr>
<tr>
<td>Ocean mixed layer heat capacity:</td>
<td>\(cw=W\cdot yr\cdot m^{-2}K^{-1}\)</td>
<td>   </td>
<td>Insolation at equator:</td>
<td>\(S_0=Wm^{-2}\)</td>
<td>   </td>
<td>Insolation spatial dependence:</td>
<td>\(S_2=Wm^{-2}\)</td>
</tr>
<tr>
<td>Ice-free coalbedo at equator:</td>
<td>\(A_0\)</td>
<td>   </td>
<td>Ice-free coalbedo spatial dependence:</td>
<td>\(A_2\)</td>
<td>   </td>
<td>Coalbedo when there is sea ice:</td>
<td>\(\alpha_i\)</td>
</tr>
<tr>
<td>Radiative forcing:</td>
<td>\(F=Wm^{-2}\)</td>
<td>   </td>
<td>Gamma:</td>
<td>\(\gamma\)</td>
</tr>
</table>
</div>
<div id="tebm_content_wrapper" class="tebm-content-wrapper">
<div id="tebm_content" class="tebm-content">
<div class="tebm-plot-container">
<div id="tebm_all_graphs_container" class="ebm-all-graphs-container">
<canvas id="tempEBMchart_all"></canvas>
</div>
<div id="graph-container" class="tebm-graph-container">
<canvas id="tempEBMchart"></canvas>
</div>
</div>
<div class="tebm-input-section">
<div class="input-wrapper">
<div>Latitude(s):
<select name="xlatitudes" id="xlatitudes-container" class="selection"
onchange="window.updateTYPlot(this.value)"></select>
</div>
</div>
<div class="input-wrapper">
<div>\(D^*\)</div>
<div id="D_sliderAmount" name="D_sliderAmount" class="slide-value">0.6</div>
<input id="D_slide" type="range" class="slide" name="tebm_slider" min="0" max="1" step="0.01"
value="0.6">
</div>
<div class="input-wrapper">
<div>\(A\)</div>
<div id="A_sliderAmount" name="A_sliderAmount" class="slide-value">193</div>
<input id="A_slide" type="range" class="slide" name="tebm_slider" min="0" max="500" step="1"
value="193">
</div>
<div class="input-wrapper">
<div>\(B\)</div>
<div id="B_sliderAmount" name="B_sliderAmount" class="slide-value">2.1</div>
<input id="B_slide" type="range" class="slide" name="tebm_slider" min="0" max="10" step="0.1"
value="2.1">
</div>
<div class="input-wrapper">
<div>\(cw\)</div>
<div id="cw_sliderAmount" name="cw_sliderAmount" class="slide-value">9.8</div>
<input id="cw_slide" type="range" class="slide" name="tebm_slider" min="0.1" max="50" step="0.1"
value="9.8">
</div>
<div class="input-wrapper">
<div>\(S_0\)</div>
<div id="S0_sliderAmount" name="S0_sliderAmount" class="slide-value">420</div>
<input id="S0_slide" type="range" class="slide" name="tebm_slider" min="0" max="1000" step="1"
value="420">
</div>
<div class="input-wrapper">
<div>\(S_2\)</div>
<div id="S2_sliderAmount" name="S2_sliderAmount" class="slide-value">240</div>
<input id="S2_slide" type="range" class="slide" name="tebm_slider" min="0" max="1000" step="1"
value="240">
</div>
<div class="input-wrapper">
<div>\(A_0\)</div>
<div id="a0_sliderAmount" name="a0_sliderAmount" class="slide-value">0.7</div>
<input id="a0_slide" type="range" class="slide" name="tebm_slider" min="0" max="1" step="0.001"
value="0.7">
</div>
<div class="input-wrapper">
<div>\(A_2\)</div>
<div id="a2_sliderAmount" name="a2_sliderAmount" class="slide-value">0.1</div>
<input id="a2_slide" type="range" class="slide" name="tebm_slider" min="0" max="1" step="0.001"
value="0.1">
</div>
<div class="input-wrapper">
<div>\(\alpha_i\)</div>
<div id="ai_sliderAmount" name="ai_sliderAmount" class="slide-value">0.4</div>
<input id="ai_slide" type="range" class="slide" name="tebm_slider" min="0" max="1" step="0.001"
value="0.4">
</div>
<div class="input-wrapper">
<div>\(F\) </div>
<div id="F_sliderAmount" name="F_sliderAmount" class="slide-value">0.0</div>
<input id="F_slide" type="range" class="slide" name="tebm_slider" min="0" max="4" step="0.05"
value="0.0">
</div>
<div class="input-wrapper">
<div>\(\gamma\)</div>
<div id="gamma_sliderAmount" name="gamma_sliderAmount" class="slide-value">1</div>
<input id="gamma_slide" type="range" class="slide" name="tebm_slider" min="0" max="2"
step="0.01" value="1">
</div>
<div class="input-wrapper" style="margin: 0px; vertical-align: top;">
<button type="button" id="add_tebm_graph" class="someBtn">Add</button>
<button type="button" id="tebm_resetbtn" class="someBtn">Reset</button>
</div>
</div>
</div>
</div>
</div> <!-- END SIMPLE TEBM -->
<div class="between-placeholder" style="height: 100px;"></div>
<!-- BEGIN COMPLEX EBM -->
<div class="complex-ebm-all" style="width:95%; margin: 0 auto;">
<div id="complex_ebm_headline" style="text-align:center;">
<h2>Aquaplanet EBM with seasonal cycle</h2>
</div>
<div id="complex_ebm_introduction">
<p class="p_information">
"This model is an idealized representation of sea ice and climate with seasonal and latitudinal
variations in a global domain. The surface is an aquaplanet with an ocean mixed layer that includes sea
ice when conditions are sufficiently cold." <i>(paper p.3; Chapter 2)</i><br>
<br>
"Horizontal diffusion occurs in a <i>ghost layer</i> with heat capacity \(cg\), all other processes
occur in the main layer, and the temperature of the ghost layer is relaxed toward the temperature of the
main layer with time scale \(tg\). [...] the ghost layer does not represent a separate physical layer
such as the atmosphere, which would add physical complexity to the mode." <i>(paper p.7; Chapter
2.e)</i>
</p>
</div>
<div class="complex-ebm-description" style="width:95%; margin:0 auto; padding:30px;">
<table style="margin: 0 auto; width: 100%;">
<tr>
<td>Diffusy for heat transport: </td>
<td>\(D^*=Wm^{-2}K^{-1}\)</td>
<td>   </td>
<td>OLR when \(T=T_m\):</td>
<td>\(A=Wm^{-2}\)</td>
<td>   </td>
<td>Coalbedo when there is sea ice:</td>
<td>\(\alpha_i\)</td>
</tr>
<tr>
<td>Insolation at equator:</td>
<td>\(S_0=Wm^{-2}\)</td>
<td>   </td>
<td>OLR temperature dependence:</td>
<td>\(B=Wm^{-2}K^{-1}\)</td>
<td>   </td>
<td>Ghost layer heat capacity:</td>
<td>\(cg=W\cdot yr\cdot m^{-2}K^{-1}\)</td>
</tr>
<tr>
<td>Insolation seasonal dependence:</td>
<td>\(S^*_1=Wm^{-2}\)</td>
<td>   </td>
<td>Ocean mixed layer heat capacity:</td>
<td>\(cw=W\cdot yr\cdot m^{-2}K^{-1}\)</td>
<td>   </td>
<td>Ghost layer coupling timescale:</td>
<td>\(\tau_g\)</td>
</tr>
<tr>
<td>Insolation spatial dependence:</td>
<td>\(S_2=Wm^{-2}\)</td>
<td>   </td>
<td>Heat flux from ocean below:</td>
<td>\(F_b=Wm^{-2}\)</td>
<td>   </td>
<td>Duration in years:</td>
<td>\(yr\)</td>
</tr>
<tr>
<td>Ice-free coalbedo at equator:</td>
<td>\(A_0\)</td>
<td>   </td>
<td>Sea ice latent heat fusion:</td>
<td>\(L_f=W\cdot yr\cdot m^{-3}\)</td>
<td>   </td>
<td>time of coldest \(T\):</td>
<td>\(T_{cold}\)</td>
</tr>
<tr>
<td>Ice-free coalbedo spatial dependence:</td>
<td>\(A_2\)</td>
<td>   </td>
<td>Sea ice thermal conductivity:</td>
<td>\(K=Wm^{-1}K^{-1}\)</td>
<td>   </td>
<td>Time of warmest \(T\):</td>
<td>\(T_{hot}\)</td>
</tr>
</tr>
<tr>
<td>Radiative forcing:</td>
<td>\(F=Wm^{-2}\)</td>
<td>   </td>
<td>Gamma:</td>
<td>\(\gamma\)</td>
</tr>
</table>
</div>
<!-- INPUT SECTION -->
<div class="complex-ebm-input-section" style="width:95%; margin: 0px auto; text-align: center;">
<div class="complex-input-wrapper">
\(D^*\):
<input id="complex_input_D" type="text" class="complex_ebm_input_field" value="0.6" />
</div>
<div class="complex-input-wrapper">
\(A\):
<input id="complex_input_A" type="text" class="complex_ebm_input_field" value="193"></td>
</div>
<div class="complex-input-wrapper">
\(\alpha_i\):
<input id="complex_input_ai" type="text" class="complex_ebm_input_field" value="0.4"></td>
</div>
<div class="complex-input-wrapper">
\(S_0\):
<input id="complex_input_S0" type="text" class="complex_ebm_input_field" value="420"></td>
</div>
<div class="complex-input-wrapper">
\(B\):
<input id="complex_input_B" type="text" class="complex_ebm_input_field" value="2.1"></td>
</div>
<div class="complex-input-wrapper">
\(c_g\):
<input id="complex_input_cg" type="text" class="complex_ebm_input_field" value="0.01"></td>
</div>
<div class="complex-input-wrapper">
\(S^{*}_{1}\):
<input id="complex_input_S1" type="text" class="complex_ebm_input_field" value="338"></td>
</div>
<div class="complex-input-wrapper">
\(c_w\):
<input id="complex_input_cw" type="text" class="complex_ebm_input_field" value="9.8"></td>
</div>
<div class="complex-input-wrapper">
\(\tau_g\):
<input id="complex_input_tau" type="text" class="complex_ebm_input_field" value="0.00001"></td>
</div>
<div class="complex-input-wrapper">
\(S_2\):
<input id="complex_input_S2" type="text" class="complex_ebm_input_field" value="240"></td>
</div>
<div class="complex-input-wrapper">
\(F_b\):
<input id="complex_input_Fb" type="text" class="complex_ebm_input_field" value="4"></td>
</div>
<div class="complex-input-wrapper">
\(yr\):
<input id="complex_input_years" type="text" class="complex_ebm_input_field" value="30"></td>
</div>
<div class="complex-input-wrapper">
\(A_0\):
<input id="complex_input_a0" type="text" class="complex_ebm_input_field" value="0.7"></td>
</div>
<div class="complex-input-wrapper">
\(L_f\):
<input id="complex_input_Lf" type="text" class="complex_ebm_input_field" value="9.5"></td>
</div>
<div class="complex-input-wrapper">
\(A_2\):
<input id="complex_input_a2" type="text" class="complex_ebm_input_field" value="0.1"></td>
</div>
<div class="complex-input-wrapper">
\(K\):
<input id="complex_input_k" type="text" class="complex_ebm_input_field" value="2"></td>
</div>
<div class="complex-input-wrapper">
\(T_{hot}\):
<input id="complex_input_summer" type="text" class="complex_ebm_input_field" value="76"></td>
</div>
<div class="complex-input-wrapper">
\(T_{cold}\):
<input id="complex_input_winter" type="text" class="complex_ebm_input_field" value="26"></td>
</div>
<div>
<button type="button" id="complex_ebm_submitbtn" class="someBtn" style="z-index:1">Go</button>
<button type="button" id="complex_ebm_resetbtn" class="someBtn" style="z-index:1">Reset</button>
</div>
<div id="complex_is_Running" style="color:red; display:none; margin-top:20px">please wait...<br>(this may
take a few moments)</div>
</div>
<div class="placeholder" style="height:55px;"></div>
<!-- PLOTTING AREA -->
<div id="complex-ebm-plot-container" class="complex-ebm-plot-container" style="display:none;">
<div class="complex-contour-plots-wrapper">
<div class="complex-contour-plots-wrapper-row">
<div id="complex_ebm_seas_enthalpy_plot" class="complex-contour-plot-container"></div>
<div id="complex_ebm_seas_T_plot" class="complex-contour-plot-container"></div>
<div id="complex_ebm_seas_SeaIce_plot" class="complex-contour-plot-container"></div>
</div>
</div>
<div class="placeholder" style="height:15px;"></div>
<div id="complex_ebm_graphs" class="complex-ebm-4graphs-container">
<div class="complex-graph-row">
<div class="col-left">
<div id="complex_tsurf_graph_container" class="complex-graph-container">
<canvas id="complex_tsurf_graph"></canvas>
</div>
</div>
<div class="col-right">
<div id="complex_iceThickness_graph_container" class="complex-graph-container">
<canvas id="complex_iceThickness_graph"></canvas>
</div>
</div>
</div>
<div class="complex-graph-row">
<div class="col-left">
<div id="complex_seasCycleIceThickness_graph_container" class="complex-graph-container">
<canvas id="complex_seasCycleIceThickness_graph"></canvas>
</div>
</div>
<div class="col-right">
<div id="complex_seasCycleIceEdge_graph_container" class="complex-graph-container">
<canvas id="complex_seasCycleIceEdge_graph"></canvas>
</div>
</div>
</div>
</div>
</div>
<div id="complex_ebm_placeholder" style="height:50px;"></div>
<!-- SOME INFORMATION ABOUT THIS EBM-->
<div id="complex-ebm-information">
<p class="p_information">
With default parameters the seasonal cycle of the equilibrated climate is plotted.<br><br>
(a) shows the seasonal cycle of \(E(t, x)\), which fully represents the model state since \(E\) is the
only prognostic variable and the forcing varies seasonally.
The associated surface temperature (b) and ice thickness (c) are roughly consistent with present- day
climate observations in the Northern Hemisphere. <br><br>
The red curve in Fig. (a) - (c) indicates the ice edge. The blue line in Fig. \(c\)
indicates the time of coldest (winter) and the red line the time of warmest (summer). \(x = 0\)
represents the latitude at the equator and \(x = 1\) at the North Pole. Coalbedo is the fraction of
incident solar radiation \(S\) that is absorbed and not reflected to space (\(1-\alpha\)).
<br>
<br>
</p>
<table style="width: 75%; margin: 0 auto;">
<tr>
<td>\(T_m\) -</td>
<td>melting temperature</td>
</tr>
<tr>
<td>OLR -</td>
<td>outgoing longwave radiation</td>
</tr>
<tr>
<td>\(S_0\) -</td>
<td>annual mean insolation at the equator</td>
</tr>
<tr>
<td>\(S^{*}_1\) -</td>
<td>determines the amplitude of seasonal insolation variations (annual frequency: \(\omega = 2\pi
yr^{-1}\) )</td>
</tr>
<tr>
<td>\(S^{*}_1\) -</td>
<td>determines the equator-to-pole insolation gradient </td>
</tr>
</table>
<p class="p_information">
<br>
NOTE: Cloud cover and water vapor are not included in this idealized sea ice model.
</p>
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<p>
Mathematical basics described in <a
href="https://www.researchgate.net/publication/276276426_How_Climate_Model_Complexity_Influences_Sea_Ice_Stability"
target="_blank"><i>"How
Climate Model Complexity Influences Sea Ice Stability" (2015)</i></a> by Till J. W. Wagner and Ian
Eisenman <br><br>
JavaScript implementation with additional variations, and creation of this page by <a
href="https://www.b-schwertfeger.de" target="_blank">Benjamin T. Schwertfeger</a> (2021) <br>
<a href="https://www.awi.de" target="_blank">Alfred Wegener Institute, Helmholtz Centre for Polar and Marine
Research, Bremerhaven, Germany</a>
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