Seasonal Energy Balance Climate Model

This handout accompanies the second lab class. The seasonal energy balance model is that adpated from that originally used by North and Coakley (1979). It solves two coupled one-dimensional energy balance equations:

$$C_L {d T_L \over d t} = Q S(x,t) (1-\alpha_L) - (A + B T_L) +... ...{d}{dx} D (1-x^2) \frac{dT_L}{dx} - {\nu \over f_L(x)}(T_L-T_W)$$

and

$$C_W {d T_W \over d t} = Q S(x,t) (1-\alpha_W) - (A + B T_W) +... ...{d}{dx} D (1-x^2) \frac{dT_W}{dx} - {\nu \over f_W(x)}(T_W-T_L)$$

The subscripts L and W refer to land and water respectively. The model represents a fraction of land at each latitude ($f_L(x)$), with the remainder being ocean (i.e. $f_W = 1 - f_L$). $\nu$ is the coupling coefficient and changes how strongly the land and ocean temperatures are tied together.

The standard set of parameters and functions for the model are the following:

$$Q=338.5 \Wm2$$

$$A=203.0 \Wm2$$

$$B=2.09 \Wm2 \hbox{$^\circ {\rm C}$}^{-1}$$

$$D = 0.44 \Wm2 \hbox{$^\circ {\rm C}$}^{-1}$$

$$C_L=0.45 \Wm2 \hbox{$^\circ {\rm C}$}^{-1} yr$$

$$C_W=9.8 \Wm2 \hbox{$^\circ {\rm C}$}^{-1} yr$$

$$\nu = 3.0 \Wm2 \hbox{$^\circ {\rm C}$}^{-1}$$

Note that $C_{L,W}/B$ gives the adjustment time scale for land/ocean in $yrs/\hbox{$^\circ {\rm C}$}$. In this model the albedo has been parameterized a little differently from the previous model, in order to account for the differing albedos over land and ocean:

$$\alpha_L=\cases{0.363+0.08(3x.^2-1)/2; & snow free: $T > -2... ...0.6; & snow covered: $T \le -2 \hbox{$^\circ {\rm C}$}$.\cr }$$

$$\alpha_W=\cases{0.263+0.08(3x.^2-1)/2; & ice free: $T > -2 ... ... 0.6; & ice covered: $T \le -2 \hbox{$^\circ {\rm C}$}$.\cr }$$

We have also included the orbital parameters as quantities you can vary. The values for the present day are:

Eccentricity, $e$, = 0.01672
Obliquity, $\epsilon$ = 23.44$^o$
Perihelion, $\omega$ = 102.07$^o$

The eccentricity is the square root of one minus the square of the ratio of the major and minor axes. The obliquity is the tilt of the rotational axis relative to the orbital plane. The perihelion sets the phase of the seasons relative to the perihelion (or point of closest approach to the sun).

You should play around with all of these parameters to get a feel for what they do.

Option to include explicit sea ice

This sea ice model is a basic slab model with no leads, to match the simplicity of the EBM. We ignore the salinity of the sea ice in this model and assume all parameters are those of freshwater ice.

The net flux into the top ice surface is

$$F_{\rm net} = Q S(x,t) (1-\alpha_W) - (A + B T_W) + \frac{d}{dx} D (1-x^2) \frac{dT_W}{dx} - {\nu \over f_W(x)}(T_W-T_L)$$

The first step is to see if

$$F_{\rm net}(T_W) +k{-2-T_W \over h}=0$$

yields a physical value for $T_w$ (i.e., $T_W\le-2\hbox{$^\circ {\rm C}$}$). If not, we solve for $F_{\rm net}$ subject to the constraint: $T_W=-2\hbox{$^\circ {\rm C}$}$. Regardless of $T_w$, we compute the ablation/accretion from

$$F_{\rm net} - F_w = L_f {dh \over dt}$$

where the density is absorbed into $L_f$. The ocean-ice flux $F_w$ is crudely parameterized here so that it is 4 $\rm W~m^{-2}$ if the ice area is only one gridbox wide and it decreases linearly to 0 as the ice covers the globe. A separate value is computed for each hemisphere. The heat that is supplied to the ice via $F_w$ is subsequently taken away from the ice-free ocean grid cells in an energy conserving manner. This is a rather ad hoc attempt to approximate heat transport in the ocean that affects sea ice from below.

Finally we must make a special calculation anytime the ocean temperature drops below -2 $^\circ {\rm C}$, where we grow just enough ice to bring the temperature back to -2.