Velocity is assumed linear between surface and bed.
where a and b are given by:
and
Here the convention is that a positive surface accumulation rate corresponds to a negative vertical velocity (down is negative), and a positive basal accretion rate corresponds to a positive vertical velocity (up is positive).
The temperature equation is:
Writing the total derivative in terms of partials and advection terms, with the partial with respect to time and the heat generation term both equal to zero we get:
where
Substituting 
and 
the
equation becomes:
Separating variables:
and integrating:
or:
where 
and 
.
Integrating this from S to z:
becomes:
The constant A is evaluated by substituting B for z in
the above and requiring that 
. From this the
constant is given by:
where the meaning of the shorthand 
is obvious.
The final expression for the temperature as a function of
depth within the ice shelf for a vertical velocity field
defined as a linear variation from the surface accumulation
rate to the basal accretion rate is given by:
Alternatively the basal (geothermal) heat flux, 
, might be
defined, rather than the basal temperature. In this case A is defined
by equation (
) evaluated at B.
and 
is still given by equation ().
The integrals indicated schematically in the above expression
are similar to error functions (
)
and can be evaluated numerically.
