312
T13 = (1- f(x)(( - z),
(2.18)
and Glen's law is
(2.19)
If A = 1 is constant, then two integrations of (2.19) give the ice flux Q = f~ ndz as
(2.20)
where H = ( - h is the depth, and nb is the sliding velocity. Integration of the mass conservation equation, together with an appropriate kinematic surface boundary condition,
then leads to the integral conservation law,
aH aQ
-+-=a
at ax
'
(2.21)
where a is the dimensionless accumulation rate. (2.21) is an equation of convective diffusion type, with the diffusive term being that proportional to f.
Note that if transverse variations were to be included, we should solve
aA aQ
-+-=a
at ax
'
(2.22)
where A is the cross sectional area, and Q would be given by Q = fA ndS, where 1L solves
(2.17) in A, together with appropriate boundary conditions.
Ice sheets
A model for ice sheets can be derived in much the same way - typical aspect ratios
are 10- 3 .~ but there is no 'downslope' gravity term pg sin ex (effectively ex = 0), and
the appropriate balance determines the driving shear stress at the base in terms of the
surface slope. Effectively, the advection term is lost and f = 1. Another difference is that
x '" y '" I ('" 3000 km) while z '" 3 km is the only small position variable. An isothermal
version of (2.21) is then (with V = (a/ax, a/ay))
[{ IV(ln-l Hn+2}
]
H t = V.
n + 2 V( + HUb + a,
(2.23)
and is a nonlinear diffusion type equation for H, since ( = H + h. The sliding velocity Ub
is apparently a convective term, but in fact the sliding law usually has Ub in the direction
of shear stress, whence Ub ex - V (, and this term also is diffusive.