What Is the Trajectory of Arctic Sea Ice? - Arctic Sea Ice Decline: Observations, Projections, Mechanisms, and Implications - page 178

figure 9 shows the approach to equilibrium. The black
dots are different September initial conditions from through-
out the domain. Black and gray paths trace subsequent Sep-
tembers obtained by integrating equations (4) with α = -3
for June-July and α = -6 for July-August. Black trajectories
lead to the equilibrium September value for perennial ice;
gray trajectories lead to the equilibrium September value
for seasonal ice. notice how the September trajectories all
rapidly approach a curving “attractor” and then continue
along the attractor to the equilibrium point (gray circle).
The triangles are the 48 Septembers of the PIOMAS model,
clustered loosely around the perennial ice equilibrium point
and along the attractor. Although the black and gray paths
are derived from equations (4) with coefficients fit to the
PIOMAS model output, we also show the 150 Septembers
of the CCSM3 model (squares and pluses). The CCSM3
Septembers for 1950-2005 (squares) do not match those of
PIOMAS because of the different thin ice/thick ice cutoffs
(Tables 1 and 2) and because of differences between the two
models. However, after about 2025 the CCSM3 Septembers
follow a path close to the curving attractor toward the sea-
sonal ice equilibrium near (0,0).

Figure 9. Approach to equilibrium. The black dots, with spacing
0.07, are initial conditions in September. The black and gray paths
are the September trajectories obtained by integrating equations
(4). Black trajectories lead to the equilibrium September value at
(0.42, 0.36) (gray circle). Gray trajectories lead to the equilibrium
September value at (0.04, 0.00) (gray circle). Triangles are the
48 Septembers of the PIOMAS model (1958-2005). Squares and
pluses are the 150 Septembers of the CCSM3 model (1950-2099).

4. DISCUSSIOn

We reiterate that the equations (2) and (4) do not explic-
itly model physical processes. The annually periodic coeffi-
cients, derived empirically, are meant to simulate the annual
cycles of the mean external forcing fields, with no interan-
nual variability. The linear equations (2) are perhaps the sim-
plest possible nontrivial evolution equations for thin ice and
thick ice, and the quadratic term in equation (4) is a simple
nonlinearity. nevertheless, these empirical models illustrate
that (1) it is not difficult to formulate a system with multiple
stable states and (2) the inclusion of a quadratic nonlinearity
that mimics the ice-albedo feedback leads to the emergence
of a stable annual cycle with ice-free summers (August-
September). Serreze et al. [2007] raised the question of how
a seasonally ice-free Arctic Ocean might be realized: through
gradual decline or through a rapid transition once the ice
thins to a more vulnerable state? Our empirical model would
allow the second scenario to exist if there were a mechanism
to nudge the system across the “tipping point” from one sta-
ble state to another.
The concept of a tipping point has recently gained pop-
ularity in the press [ Walker , 2006]. A tipping point is an

Figure 8. Perennial ice equilibrium cycle (circles) and seasonal ice
equilibrium cycle (squares) obtained by integrating equations (4).
Initial conditions for September within the gray triangle (bottom
left) lead to the seasonal ice cycle. Initial conditions for September
outside the gray triangle lead to the perennial ice cycle. The 48-year
PIOMAS trajectory is shown in gray.