Jökull - 01.12.1983, Blaðsíða 61

I^XXU| [ðyyU|«|ðzzUj (7)
|8xxV| = |ðyyV|«|8ZZV|
Let u(S) = (u,v) now be the horizontal velocity
vector and — V| the 2-dimensional Laplacian. The
viscous term on the right of (1) then takes the form
fl = (g/2v) (z-f) (z+f—2h) V2h (13)
Equation (3) can now be used to eliminate the
2-dimensional velocity u(S). We integrate (3) with
respect to z from z = f to z = h and apply the
boundary conditions
V2fl = (ðzzu, 8zzv). (8)
(d) As a further consequence of the thin layer
creep, the vertical viscous forces can be neglected as
compared with the gravitational forces. We will
thus assume for the fluid pressure in the layer the
hvdrostatic relation
p=ge (h - z) (9)
The pressure gradient thus takes the form
Vp = (g(?ðxh, gpðyh, -gp). (10)
As a consequence of the above approximations, the
horizontal component of equation (1) reads now
0 = -ggV2h + pðzzú , (11)
where V2 is the horizontal gradient. The vertical
component simply reduces to the equation for the
hvdrostatic pressure in the layer.
Since the íirst term of the right of (11) is indepen-
dent of z, the equation is easily integrated with
respect to z. The boundary conditions require that
the fluid velocity vanish at the bottom of the layer
and we will assume that there is no solid cover and
therfore no viscous force at the top. Hence, the
conditions are
z = f, ú = 0
z = h, 8zú = 0
(12)
and the integration of (11) thus results in
z = f w = 0
z = h w = 8th
(14)
Hence, interchanging the vertical integration and
the horizontal diíferentiation, we obtain
h
V2-[f údz] + ð.h = Q (15)
^ f
where Q is the specific vertically integrated source
volume output that is being introduced to complete
equation (15). Infact, this quantity is different from
zero only at the volcanic vents. The integral is
obtained from (13) resultingin
fldz = — (g/3v) (h — f)3 V2h .
(16)
The final equation is obtained by inserting (16)
into (15), viz.,
ðth - (g/3v)V2-[(h - f)3V2h] = Q (17)
This is the result of our approximations and repres-
ents the equation of a creeping thin Newtonean
fluid layer where the altitude of the surface is the
only remaining unknown.
To solve this equation, we need data on the
kinematic viscosity v, the bottom form f(S) the
source density Q(S) and the front boundary cond-
ition that has yet to be formulated. As already stat-
ed the front condition is enigmatic, but to obtain
some semi-quantitative results, we propose the
following procedure.
A possible approach, although rather proble-
matic, consists in regarding the front velocity vQ
measured in the direction of the front normal as a
purely empirical function of the front thickness
(hQ-f), indicated in Fig. 1, and the lava viscosity.
JÖKULL 33. ÁR 59