294
4.2 Attractor Basins - Numerical Example
It is possible to work out in some detail an example illustrating the concept of the potential
surface for a two-mode model. In this case the functional F becomes the truncated version
of (22). The function Mis
M(To, T2) = Q l'S S(,l)(To + T2P2(,l))aJ dJL'
+ Q (I S(Il')(To + T2P2(JL') )Ui dJL' - ATo,
ljJ.
where in the two-mode model the ice edge can be expressed in terms of To and T2:
1 (
2(T. - To)) 1/2
11. = V3 1 + T2
(30)
(31)
(32)
Together with the other terms in (22), we have enough information to plot the surface
corresponding to F(To, T2) as a contour diagram in the To, T2 plane. We choose the
parameters such that there is a cusp: Q = 344W 1m2, A = 210W 1m2, B = 1.90W 1m2 rC,
DIB = 0.30, aJ = 0.68, ai = 0.34, S{Il) = 1- 0.5P2(1l), and T. = -lOoC. The operating
curve for this system is shown in the upper left panel of Fig. 2. This choice of parameters
leads to 5 solutions labeled A, B, C, D and E. The potential surface is mapped in the
upper right panel with the steady state points indicated. It can be clearly seen that B
and D are saddle points corresponding to unstable solutions. The lower left panel shows
higher resolution focusing on the right hand basin. It shows that A is a distinct minimum.
The ice-free solution A corresponds to a very shallow minimum and a small but finite
agitat.ion would push it over the hill (B) into the deeper (more stable) basin C. The lower
right panel represents the variation of the functional F(To, T2) shOWing that the variation
vanishes at A, Band C because they are solutions of the energy balance model.
The relative minima in a purely dissipative system such as those we are studying here
are isolated points called attractors. It is easy to map out the att,ractor basins for this
class of problems as illustrated in Fig. 2.
5 Comparing with General Circulation Models
We have conducted some experiments with an atmospheric general circulation model
[GCM] (Lee and North, 1995) and an energy balance model that was tuned (radiation
parameters, diffusion coefficient, etc.) to match the ensemble average solutions of the
GCM. Ensemble averages are necessary in the comparison because of the natural variability in a GCM. In addition, the EBM had a space-time white noise forcing added to its
right hand side. This latter caused the time dependent solutions to exhibit fluctuations
similar to those exhibited in the GCM. The ice cap edge was to follow the instantaneous
-lOoC isotherm. In the GCM solutions we do not find the cusp. The cusp also disappears
in ensemble averages of EBMs forced by noise if the noise level is large enough. As the
level of noise is reduced, the solution trajectory begins to spend more time in the shallow
minimum labeled A in Fig. 2. One could examine the residence times, etc., using the
Fokker-Planck equation. It appears that in the case where the ice cap responds instantaneously, there will be no cusp. We have not yet examined the case where the ice cap
requires a finite time to melt.
4.2 Attractor Basins - Numerical Example
It is possible to work out in some detail an example illustrating the concept of the potential
surface for a two-mode model. In this case the functional F becomes the truncated version
of (22). The function Mis
M(To, T2) = Q l'S S(,l)(To + T2P2(,l))aJ dJL'
+ Q (I S(Il')(To + T2P2(JL') )Ui dJL' - ATo,
ljJ.
where in the two-mode model the ice edge can be expressed in terms of To and T2:
1 (
2(T. - To)) 1/2
11. = V3 1 + T2
(30)
(31)
(32)
Together with the other terms in (22), we have enough information to plot the surface
corresponding to F(To, T2) as a contour diagram in the To, T2 plane. We choose the
parameters such that there is a cusp: Q = 344W 1m2, A = 210W 1m2, B = 1.90W 1m2 rC,
DIB = 0.30, aJ = 0.68, ai = 0.34, S{Il) = 1- 0.5P2(1l), and T. = -lOoC. The operating
curve for this system is shown in the upper left panel of Fig. 2. This choice of parameters
leads to 5 solutions labeled A, B, C, D and E. The potential surface is mapped in the
upper right panel with the steady state points indicated. It can be clearly seen that B
and D are saddle points corresponding to unstable solutions. The lower left panel shows
higher resolution focusing on the right hand basin. It shows that A is a distinct minimum.
The ice-free solution A corresponds to a very shallow minimum and a small but finite
agitat.ion would push it over the hill (B) into the deeper (more stable) basin C. The lower
right panel represents the variation of the functional F(To, T2) shOWing that the variation
vanishes at A, Band C because they are solutions of the energy balance model.
The relative minima in a purely dissipative system such as those we are studying here
are isolated points called attractors. It is easy to map out the att,ractor basins for this
class of problems as illustrated in Fig. 2.
5 Comparing with General Circulation Models
We have conducted some experiments with an atmospheric general circulation model
[GCM] (Lee and North, 1995) and an energy balance model that was tuned (radiation
parameters, diffusion coefficient, etc.) to match the ensemble average solutions of the
GCM. Ensemble averages are necessary in the comparison because of the natural variability in a GCM. In addition, the EBM had a space-time white noise forcing added to its
right hand side. This latter caused the time dependent solutions to exhibit fluctuations
similar to those exhibited in the GCM. The ice cap edge was to follow the instantaneous
-lOoC isotherm. In the GCM solutions we do not find the cusp. The cusp also disappears
in ensemble averages of EBMs forced by noise if the noise level is large enough. As the
level of noise is reduced, the solution trajectory begins to spend more time in the shallow
minimum labeled A in Fig. 2. One could examine the residence times, etc., using the
Fokker-Planck equation. It appears that in the case where the ice cap responds instantaneously, there will be no cusp. We have not yet examined the case where the ice cap
requires a finite time to melt.