The model equation (2) for temperature is nonlinear, and as it stands it is difficult to extract information about solution responses to penetrating solar radiation. However, large temperature variations on time scales of the order of a week or more (slow time), and smaller daily temperature oscillations (fast time), are visible in the temperature data presented in figures 1 and 2. Hence we will make progress by proceeding to analyse the daily small-scale changes as perturbations on the longer-term trends.
We seek a temperature expansion in the form
where (following, e.g., Kevorkian and Cole, 1980) we are using a
multiple time-scales analysis, with t a fast time, 
a
small parameter, and 
a slow time scale. Note that
we are explicitly taking the leading behaviour 
to be slowly
varying in time. If we choose Q to also be of size , we
find the background temperature solves the steady-state problem
which has a solution of the form
Substituting this into the boundary conditions then gives
The temperature at the top of the sea ice (z=0) is
Note that as 
, 
, as expected since large 
corresponds to good thermal contact between ice and air. The leading
temperature behaviour is linear in z, taking the values
at z=1 and 
at z=0.
The 
problem is
where 
and 
.
This is a linear partial differential equation for v. The right-hand side consists of oscillatory and non-oscillatory forcing terms, and we are only interested in the oscillatory behaviour of v. Hence we let
and we want to determine V(z). |V| gives the
amplitude and 
gives the phase shift of the oscillations in v. Substitution
into equation (12) and equating coefficients of
gives
which is to be solved for V subject to the boundary conditions V(1) = 0 and
This is a boundary-value problem for a linear inhomogeneous second-order ordinary differential equation with variable coefficients, which cannot be solved by elementary methods. However, numerical solutions will be discussed in the following sections, and asymptotic solutions are the subject of a paper in preparation.
