The next logical step is to allow spatial resolution into our model. We will do this by breaking the northern hemisphere into 9 latitude regions of 10^{o} each. The general geometry is shown in the following diagram:
In the model we will consider, the solar flux and albedo
both have a latitude dependence (S_{i} and
α
_{i}, where i runs from 1 to 9 indicating the latitude band).
Also, if a latitudinal band is
colder or warmer than the average global temperature, heat will flow
into/out-of the region. We will assume that this heat flow depends linearly
on the temperature difference between the region and the global temperature,
in other words it is: F×(T_{i}-T_{Ave}).
Putting this additional term into
Equation 3
from the Global Climate Modeling Project
results in an energy balance
equation of:
S_{i} | Solar Flux in Latitude Band i |
This is the product of S/4 (the average global solar constant) times the insolation s_{i} | |
s_{i} | Solar Insolation |
The fraction of solar flux incident on each latitudinal band. | |
α _{i} | Albedo in Latitude Band i |
The albedo of ice is much larger than the albedo of land/water. We can do a crude model of the temperature dependence of the albedo by using α _{i}=0.3 for T_{i} > T_{c} or α _{i}=0.6 for T_{i} <= T_{c} | |
T_{c} | Below this Temperature, we assume a Permanent Ice Pack (T_{c} = -10^{o}C) |
F | Heat Transport Coefficient (F=3.80 W m^{-2} ^{o}C^{-1}) |
T_{Ave} | Global Weighted Average Temperature |
This temperature is the weighted average of the temperature in all of the latitude zones on the previous iteration. The weighting factors f_{i} are just the relative fraction of the surface area of the sphere in each latitude zone. | |
A & B | Coefficients expressing Infrared Radiation Loss (A=204 W m^{-2} and B=2.17 W m^{-2} ^{o}C^{-1}) |
C_{E} | Heat Capacity (C_{E} = 2.08 × 10^{8} J/m^{2} ^{o}C) |
The Rough Flow of the
Matlab script ebm2.m
(which
iterates Eqn. 5) is:
keyboard
command to allow you to
easily change any variables
MaxTempDiff=1E6
- we will be checking
for a small value in the next step
MaxTempDiff
)
is smaller than the tolerance specified
by TolTempDiff
EXERCISE 4:
Run the Matlab script using the parameters given in the program
and this handout.
Try adjusting the factor SX
(which is multiplied by the Solar Constant) to find out
how sensitive the model is to variations in the solar constant.
By what factor does the solar
constant need to decrease before the earth is completely
glaciated? Before all the permanent ice pack melts?
Modify your program so instead of finding a steady-state solution
it will find a time dependent solution (using Eqn. 4).
Zone | Annual Mean Temp (^{o}C)^{a} T_{i} | Solar Insolation (Fraction of Solar Constant) ^{b} s_{i} |
Albedo^{c}
α _{i} |
Heat Transfer Into/Out of Zone (W/m^{2})^{c} |
80-90^{o} N | -16.9 | 0.500 | 0.589 | -103 |
70-80 | -12.3 | 0.531 | 0.544 | -94 |
60-70 | -5.1 | 0.624 | 0.452 | -72 |
50-60 | 2.2 | 0.770 | 0.407 | -47 |
40-50 | 8.8 | 0.892 | 0.357 | -21 |
30-40 | 16.2 | 1.021 | 0.309 | 1 |
20-30 | 22.9 | 1.120 | 0.272 | 18 |
10-20 | 26.1 | 1.189 | 0.248 | 46 |
0-10^{o} N | 26.4 | 1.219 | 0.254 | 59 |
0-10^{o} S | 26.1 | 1.219 | 0.241 | 56 |
10-20 | 24.6 | 1.189 | 0.236 | 41 |
20-30 | 21.4 | 1.120 | 0.251 | 22 |
30-40 | 16.5 | 1.021 | 0.296 | 0 |
40-50 | 9.9 | 0.892 | 0.358 | -27 |
50-60 | 2.9 | 0.770 | 0.426 | -57 |
60-70 | -6.9 | 0.624 | 0.513 | -86 |
70-80 | -29.5 | 0.531 | 0.602 | -90 |
80-90^{o} S | -42.3 | 0.500 | 0.617 | -88 |
http://physics.gac.edu/~huber/envision/instruct/ebm2doc.html