Energy balance models

"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 comprehensive 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." (Extracted from: Till J.W Wagner and Ian Eisenman, "How Climate Model Complexity Influences Sea Ice Stability" (2015) p.2 Abstract) For simplicity, I will call the paper "paper" so that I can refer to it more easily on this page.

  ☞ Paper and Documentation

  ☞ Python source code: simple EBM, complex EBM

Temperature EBM

Diffusy for heat transport: \(D^*=Wm^{-2}K^{-1}\)     OLR when \(T=T_m\): \(A=Wm^{-2}\)     OLR temperature dependence: \(B=Wm^{-2}K^{-1}\)
Ocean mixed layer heat capacity: \(cw=W\cdot yr\cdot m^{-2}K^{-1}\)     Insolation at equator: \(S_0=Wm^{-2}\)     Insolation spatial dependence: \(S_2=Wm^{-2}\)
Ice-free coalbedo at equator: \(A_0\)     Ice-free coalbedo spatial dependence: \(A_2\)     Coalbedo when there is sea ice: \(\alpha_i\)
Radiative forcing: \(F=Wm^{-2}\)     Gamma: \(\gamma\)
Latitude(s):
\(D^*\)
0.6
\(A\)
193
\(B\)
2.1
\(cw\)
9.8
\(S_0\)
420
\(S_2\)
240
\(A_0\)
0.7
\(A_2\)
0.1
\(\alpha_i\)
0.4
\(F\)
0.0
\(\gamma\)
1

Aqua planet EBM with seasonal cycle

"This model is an idealized representation of sea ice and climate with seasonal and latitudinal variations in a global domain. The surface is an aqua planet with an ocean mixed layer that includes sea ice when conditions are sufficiently cold." (paper p.3; Chapter 2)

"Horizontal diffusion occurs in a ghost layer 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." (paper p.7; Chapter 2.e)

Diffusy for heat transport: \(D^*=Wm^{-2}K^{-1}\)     OLR when \(T=T_m\): \(A=Wm^{-2}\)     Coalbedo when there is sea ice: \(\alpha_i\)
Insolation at equator: \(S_0=Wm^{-2}\)     OLR temperature dependence: \(B=Wm^{-2}K^{-1}\)     Ghost layer heat capacity: \(cg=W\cdot yr\cdot m^{-2}K^{-1}\)
Insolation seasonal dependence: \(S^*_1=Wm^{-2}\)     Ocean mixed layer heat capacity: \(cw=W\cdot yr\cdot m^{-2}K^{-1}\)     Ghost layer coupling timescale: \(\tau_g\)
Insolation spatial dependence: \(S_2=Wm^{-2}\)     Heat flux from ocean below: \(F_b=Wm^{-2}\)     Duration in years: \(yr\)
Ice-free coalbedo at equator: \(A_0\)     Sea ice latent heat fusion: \(L_f=W\cdot yr\cdot m^{-3}\)     time of coldest \(T\): \(T_{cold}\)
Ice-free coalbedo spatial dependence: \(A_2\)     Sea ice thermal conductivity: \(K=Wm^{-1}K^{-1}\)     Time of warmest \(T\): \(T_{hot}\)
Radiative forcing: \(F=Wm^{-2}\)     Gamma: \(\gamma\)
\(D^*\):
\(A\):
\(\alpha_i\):
\(S_0\):
\(B\):
\(c_g\):
\(S^{*}_{1}\):
\(c_w\):
\(\tau_g\):
\(S_2\):
\(F_b\):
\(yr\):
\(A_0\):
\(L_f\):
\(A_2\):
\(K\):
\(T_{hot}\):
\(T_{cold}\):

With default parameters the seasonal cycle of the equilibrated climate is plotted.

(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.

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 (\(1-\alpha\)).

\(T_m\) - melting temperature
OLR - outgoing longwave radiation
\(S_0\) - annual mean insolation at the equator
\(S^{*}_1\) - determines the amplitude of seasonal insolation variations (annual frequency: \(\omega = 2\pi yr^{-1}\) )
\(S^{*}_1\) - determines the equator-to-pole insolation gradient


NOTE: Cloud cover and water vapor are not included in this idealized sea ice model.