Conservation equations are all based on the consideration of the flux of some state variable flowing into and out of some region of the domain. In general the sources and sinks within this region are also considered in arriving at a conservation equation. These fluxes generally depend on position, but they may also depend on the state variable itself, or they may represent fluxes of state variable carried into the region by moving material.
As an example consider a 1-D problem with flux flowing from right to
left through some region, and with some distributed sources that are a
function of position. We will have a flux in the right-hand side of the
region of 
, balanced by a flux out of the left hand
side of 
. We will also have a flux due to the distributed
source which will be equal to 
, where dx is the width of the
differential area. If we are considering a time-dependent problem, the
balance of fluxes will also include a term that represents
``retention'' of flux. This will manifest as a rate of change of the
state variable with time, 
. Putting
all these pieces together provides the following expression.
This says that neglecting time-dependent effects, the gradient of the
flux at some point in the domain is just equal to the source at that
point. An imbalance between these two terms will result in a change in
the local state variable, manifest in the time-dependent term. If a
portion of the distributed source is proportional to the state variable
we may include a term in the right-hand side 
. If material
moving at a velocity 
is carrying flux into the region we may
also add a term 
to the right-hand
side.
Equation (
) is not a differential equation, because it is
still expressed in terms of the flux, 
, and the state variable,
u. To complete the differential equation we need a constitutive
relation that relates the flux of state variable to the gradient of the
state variable. A simple example of such a constitutive relation is
Fourier's Law for heat flow problems. This simply expresses the fact
that heat flows down a temperature gradient. The flux of heat flowing
from a warm region to a cold region is proportional to the temperature
gradient between these two regions, where the constant of
proportionality is the usual conductivity of the medium. A generic
constitutive relation has the form given in the following expression.
Combining equations () and () we obtain a
second-order differential equation.
For 2- and 3-D domains we will have an analogous equation
where k, u, c, b and f are functions of position within their
respective domains, and the 
and 
operators are
either 2-dimensional or 3-dimensional gradient and divergence
operators.
Thus far we have been dealing with a generic conservation equation. We
can apply equation () or () to the primary
conservation laws of glaciology by making the identifications in Table
1. For example, the generic state variable, u, corresponds to the
height of the ice surface for the mass conservation equation, to the
velocity vector, 
, for the momentum conservation equation, and
to the temperature, T, for the energy equation.
Each conservation equation is transformed into a differential equation
in terms of its own state variable by the use of a particular
constitutive equation corresponding to the generic equation
().
For mass conservation this will take the form of the column-averaged flow law which can be expressed for pure flow by the following expression.
From this expression we can identify the constant of proportionality,
k, as itself being dependent on the surface gradient, 
, and
the thickness, H.
This nonlinear problem must then be solved by an iterative process, whereby an initial uniform distribution of k is assumed, a solution for h and H is obtained, a new nonuniform k is obtained from this solution, and the process is repeated until it has converged to a solution. This type of iterative linearized solution is a common technique in dealing with such nonlinear constitutive equations.
For the momentum conservation equation we have a generalized flow law relating stress components to stain-rate components, given by the following expression.
Here 
is the usual strain invariant which depends on
all the components of the strain-rate tensor, A is the ice hardness
parameter, and the term in the square brackets represents an effective
viscosity. Strain rates are related to the gradients of velocity
through the following relationship.
Again, because of the nonlinear dependence of the flux-like variable on the gradient of the state variable in this constitutive equation, the solution will require a linearization constant, with an iterative procedure to arrive at a self-consistent solution for the velocity vector.
Finally, energy conservation uses Fourier's Law relating the heat flux to the temperature gradient given by the following expression
where k is the thermal conductivity. Because this conductivity can itself be a function of temperature, this equation must also be solved by an iterative process.
