Flow and Fluid Behaviour of theMantle - Quantitative Plate Tectonics - page 346

Furthermore,
<

LJ LJ LJ LJ ™D™ 0 D 0

@ §
@r

u ™ .r;™ 0 / D

LJ LJ LJ LJ ™D™ 0 D v 0

(13.90)

1
r

@§
@™

:

u r .r;™ 0 / D

These boundary conditions are compatible
with a solution by separation of variables of the
form:

Fig. 13.10 Corner flows and streamlines at a subduction
zone

§.r;™/ D rf .™/ (13.91)

point. The stream function at this location is given
by: § D ½ Q . The corresponding streamline has
a parabolic shape and is called the stagnation
streamline . It indeed forms the outline of an
object, the Rankine half-body. Streamlines within
the Rankine half-body do not belong to the large-
scale flow surrounding the source, as it is evident
from Fig. 13.6 .
Now we are going to consider another interest-
ing application of the techniques based on stream
function analysis. In Sect. 12.5 , we mentioned
the corner flows that form at subduction zones
(Fig. 13.10 ). Here we shall solve the equations of
motion in the case of a Newtonian incompressible
flow and assuming that the contribution of the
inertial terms is negligible in the Navier-Stokes
equations. In this instance, we have seen that the
equations of motion reduce to the biharmonic Eq.
( 13.71 )for§. In plane polar coordinates, this
equation assumes the form:

Substituting into ( 13.88 )gives:

2 1
r f.™/ C

d 2 f
d™ 2

1
r

4 § Dr

r

r 3 f.™/ C 2 d 2 f d™ 4

d 4 f

1

D

d™ 2 C

D 0

(13.92)

The general solution for f is:

f.™/ D a sin ™ C b cos™ C ™.csin ™ C d cos™/
(13.93)

The constants a , b , c ,and d must be chosen so
that the boundary conditions ( 13.89 )and( 13.90 )
are satisfied. Therefore, we must have:

f.0/ D 0 I f.™ 0 / D 0 (13.94)

Furthermore, by ( 13.91 )wehaveth t
u r D f 0 (™). Consequently,

2 @ 2 §

@™ 2

@ 2 §

1
r

@§
@r C

1
r 2

4 § Dr

r

@r 2 C

D 0

f 0 .0/ D v 0 I f 0 .™ 0 / D v 0 (13.95)

(13.88)

From the condition f (0) D 0 we have immedi-
ately: b D 0. Similarly, from the first of the condi-
tions ( 13.95 ) we obtain: a C d D v 0 . Therefore,
we are left with two linear equations in the
unknown parameters c and d .
The solution for a , c ,and d is then:

Let us assume that the overriding plate of a
subduction system is at rest with respect to the
top transition zone. In this instance, the hinge line
is at rest and we can set up a reference frame as
indicated in Fig. 13.10 . The boundary conditions
of this model are:
8
<

LJ LJ LJ LJ ™D0 D 0

v 0 ™ 0
™ 0 C sin ™ 0 I c D

v 0 .1 C cos™ 0 /
™ 0 C sin ™ 0 I

@§
@r

a D

u ™ .r;0/ D

LJ LJ LJ LJ ™D0 D v 0

(13.89)

1
r

@§
@™

v 0 sin ™ 0
™ 0 C sin ™ 0

:

u r .r;0/ D d D

(13.96)