What Is the Trajectory of Arctic Sea Ice? - Arctic Sea Ice Decline: Observations, Projections, Mechanisms, and Implications - page 175

included. Physical models often have to be “tuned” to re-
produce observed behavior. Alternatively, empirical models
reproduce observed behavior by design.

concentration is less than 0.002, with a standard deviation
of 0.008. Apparently the annual cycle has only one degree
of freedom per year, an amplitude that determines the Sep-
tember minimum. The success of this one-dimensional fit
motivates the following two-dimensional case.

3.3. Empirical Models

3.3.2. Two-dimensional linear model. Let x equal thin ice
concentration and y equal thick ice concentration. We wish
to formulate simple equations for x and y that reproduce, in
an average sense, the annual trajectories of figure 3. Based
on the one-dimensional case, we choose linear equations
with annually periodic coefficients:

While Merryfield et al. [this volume] constructed a simple
physical model to mimic the essential behavior of the sea ice
component of CCSM3, we adopt an empirical approach to-
ward “modeling the model” in which we postulate a simple
form for the evolution equations of thin ice and thick ice that
includes undetermined coefficients, use the output of PIO-
MAS or CCSM3 to find the coefficients that best reproduce
the thin/thick trajectory, and then study the stability proper-
ties of the resulting equations.

d x / d t = a ( t )( 1 − x − y ) + b ( t ) x + c ( t ) y
d y / d t = d ( t )( 1 − x − y ) + e ( t ) x + f ( t ) y .

(2)

3.3.1. One-dimensional linear model. We start with a one-
dimensional formulation. Let z equal the total sea ice con-
centration in the Arctic. We postulate d z /d t = p( t )(1 - z ),
where p( t ) is an annually periodic function of the form p( t )
= A cos(ω( t - τ)). With t measured in months, the frequency
ω is 2π/12. The constant τ = 10.3 months is a phase shift to
align the minima of z ( t ) with the end of summer. The mo-
tivation for the form of this equation is that it is linear in z ,
the solutions are periodic, and it gives the correct shape of
the annual cycle (as shown presently). If A is a constant then
the amplitude of each annual cycle is the same. If we allow a
different value of A each year, we can reproduce interannual
variability. figure 4 shows the solution of this one-dimen-
sional equation (solid line), together with the monthly Arc-
tic sea ice concentration from PIOMAS (black dots). The
annual values of A used in this 10-year interval are [0.98,
0.96, 0.97, 0.99, 0.98, 0.83, 0.81, 0.90, 0.84, 1.19]. The fit
is remarkably good: the bias between z and the PIOMAS ice

The term 1 - x - y is the open water concentration. The six
functions a , b , c , d , e , and f are determined from the 48-
year time series of PIOMAS thin ice and thick ice concen-
trations using a least squares method. Let time be measured
in months, and replace the continuous d x /d t with Δ x , which
represents the change in thin ice concentration from one
month to the next. Consider the change from, say, january to
february. We write

(3)

D x k = a ( 1 − x k − y k ) + bx k + cy k + error k ,

where x k and y k are the concentrations of thin ice and thick
ice in january of year k and Δ x k is the change in thin ice
concentration from january to february of year k . This is 48
equations (48 years, k = 1 to 48) with three unknowns: a , b ,
and c . We find a , b , and c by using a standard least squares
procedure that minimizes the variance of the error term. The
process is repeated for the change from february to March,
March to April, etc. Thus a ( t ) consists of 12 values, one for
each month, and similarly for b ( t ) and c ( t ). finally, the same
procedure is applied to the thick ice equation (for Δ y k ) to ob-
tain the 12 values of d ( t ), e ( t ), and f ( t ). The equations (2) can
then be integrated in time, starting from some initial state
( x 0 , y 0 ). figure 5 shows the resulting equilibrium trajectory
that is obtained, no matter what the initial state. The full 48-
year thin ice/thick ice trajectory from PIOMAS is shown in
gray. The equilibrium trajectory is clearly an average of the
48 years, by construction. The coefficients a ( t ), b ( t ), c ( t ),
d ( t ), e ( t ), and f ( t ) parameterize the annual cycle of forcing
fields such as air temperature (i.e., cold in winter and warm
in summer). If the coefficients were constants instead of pe-
riodic functions, the equilibrium would be a single point.

Figure 4. Solid curve is the solution of d z /d t = p( t )(1 - z ), where
p( t ) is a cosine function with a period of 1 year and a different
amplitude for each year shown by solid curve. Black dots indicate
monthly Arctic sea ice concentration from PIOMAS.