We model the temperature field in the sea ice with a one-dimensional heat diffusion equation. Heating by solar radiation is included as a distributed source term, as calculated in the section on Monte Carlo scattering. The equation describing the conduction of heat in the sea ice is
where the diffusivity is
and the solar driving term is
where temperature is the real part of 
in degrees C,
is a complex-valued
function of depth z and time t, k is thermal conductivity, here
taken to be 2 W.m 
.K ,
is the density of the sea ice, and P is the solar power
absorbed per unit volume (J m 
) of sea ice. A complex-valued
is computationally convenient for analysing oscillations. Depth
z is zero at the upper ice-air interface, and has been rescaled to give
z=1 at the lower ice-water interface, using the factor L, the total
ice thickness, which is taken to be L=2 during spring.
The factor 
accounts for the daily
variation in amplitude as the sun rises and falls, with frequency
radians per day. A has modulus one, which
corresponds to the sun just touching the horizon overnight. The argument
of A sets the zero of time with respect to the sun's periodic
behaviour. 
is the
temperature-dependent heat capacity per unit mass of sea ice (kJ
kg 
C ), and we
use the empirical formula (Schwerdtfeger, 1963; Ono, 1966; Yen, 1981)
where S is the salinity in practical salinity units (psu) of the sea ice, here taken to be 5.5 psu.
The boundary condition used at the
sea-ice interface where z=1 is to fix the temperature at the value
C.
For the boundary condition at the upper ice surface z=0, a Newton-type heat loss is used, with the temperature gradient at the surface proportional to the difference between the ice temperature and the air temperature,
where 
is some positive constant. If 
, the ice is
perfectly insulated from the air. If 
, the
ice temperature at the surface is equal to the air temperature there.
Equation (6) corresponds to the sensible heat
flux, mentioned e.g. in Zeebe and others (1996). Their work suggests
the value 
, at average windspeeds in McMurdo Sound.
This boundary condition may appear crude compared for example with that of Zeebe and others (1996), but incoming and outgoing radiation effects at the upper surface are already accounted for explicitly by the Monte Carlo simulations through the source term P(z), for wavelengths in the range 300-1400 nm, and hence are not needed in the boundary condition. Radiation at wavelengths above 1400 nm is ignored here, because we see very little power in the spectrum of the solar radiation striking the ice surface above 1400 nm. We also ignore latent heat effects at the ice/air interface. For the Weddell sea, latent heat fluxes have been estimated (Eicken, 1992) to be about a quarter of the size of the sensible heat fluxes. Furthermore, we find in the following sections that the oscillatory solar forcing solutions are not very sensitive to the boundary condition used at the upper surface.
