THE NEAR-SURFACE LAYER OF THE OCEAN
3
3
0
0
exp
z
r z r
ar
§
·
¨
¸
©
¹
or 0
0
exp
3
z
r z r
ar
§
·
¨
¸
©
¹
.
(1.71)
The drop-size distribution at depth z changes according to
0
0
z
r
N
N
n r
r
r r
w
w
w
w
w w
,
(1.72)
which, with (1.71) and (1.65), leads to
0
0
0
0
exp 2
/ 1
3
3
z
z
z
n r n
r
ar
ar
§
·§
·
/
¨
¸¨
¸
©
¹©
¹
.
(1.73)
The freshwater volume at depth z is given by integral
3
4
3
c
u
z
r
V z V
r n r dr
S
f
³
.
(1.74)
Substituting (1.71) and
0
0
0
exp
1
3
3
z
z
dr
dr
ar
ar
§
·§
·
¨
¸¨
¸
©
¹©
¹
(1.75)
into (1.74) results in the following integral:
4
3
0
0
0
0
0
8
exp
2
3
c
r
z
V z
V r
r dr
ar
f
§
·
/
/
¨
¸
©
¹
³
,
(1.76)
which is equivalent to
4
3
3
0
0
0
0
0
0
0
0
0
0
0
8
exp
2
exp
2
3
c
r
z
z
V z
V r
r dr
r
r dr
ar
ar
f
ª
º
§
·
§
·
/
/
/
«
»
¨
¸
¨
¸
«
»
©
¹
©
¹
¬
¼
³
³
. (1.77)
With the first integral on the right side of (1.77) expressed via the
MacDonald function (or modified Bessel function) 4
K (see Gradshtein and
Ryzhik, 2000), the source function for penetrating rain in the near-surface
38