Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

4.3. The surface inteyral approach to the modelling problem
In the surface integral approach, within the anolr[alous slab the
equation (4.4b) , i. e.
cur-2~ + k 2 ' ~ = - k 2 ~
-a .-a a-n
is solved by finite differences or an equivalent method. The re-
quired field values one grid point width above the upper horizontal
boundary at z = z and below the lower boundary at z = zL are
1
expressed as surface integrals in terms of the tangential component
of ga at z = z and z2, respectively.
.
1
~
Le-t V and V2 be the half-spaces z < z and z > z respectively,
1 1
and let S m = 1.2 be the planes z = z . Let C!mi (r r , r E Vm.
mr in -1 -0 - -0
- r E vm U S be a solution of
m
(i=1,2,3; m=1,2) satisfying for - r E S the boundary condition
m
In Vl and V2, Ea is a solution of .
cur12i3 + k2 E = 0
-a n -a
Multiply (4.34) by G!~), (4.32) by E , integrate the difference with
-1 -a
respect to - r over Vm, and obtain on using (4.61, (4.33) and ga + 0
-
for r -t -:
' r E Vm, or in tensor notation
-0'
,
where
E (r) = (-I)~J curl
-a --o
'm
3 ,-.
x.
(m)
curlG. .
-1 -1
A
(r Irl;{z x E (r)}d~
-0- - -a