Chapter 2: SEA SURFACE MICROLAYER
2
0
1
0
1/ 2
, ;
exp
exp
4
'
4
'
' ,
t
i
T
T
z
I z t i
t t
t t
d dt
K
D K
N
N S
K
f
­
½
°
°
u
®
¾
°
°
¯
¿
ª
º
¬
¼
³ ³
(2.81)
2
0
2
0
1/ 2
, ;
exp
exp
4
'
4
'
' ,
t
i
T
T
z
I z t i
t t
t t
d dt
K
D K
N
N S
K
f
­
½
°
°
u
®
¾
°
°
¯
¿
ª
º
¬
¼
³ ³
(2.82)
and
0
R
q
is the scaled solar irradiance just below the sea surface
mathematically defined as
1
1
0
0
1
R
p
R
z
p
q
c
I z
c
A I
U
U
o
6
.
Change
of
variables
in
(2.81)
and
(2.82)
'
t t
T
,
1/ 2
1/ 2
4 T
i
T
u z K N T
D N T
, and
1/ 2
'
4 T
i
T
u
z K N T
D N T
leads to:
9
2
0
1/ 2
1
0
1/ 2
1/ 2
2
1/ 2
,
e x p
1
4
exp
1
.
4
t
R
R
i i
T i
i
i
T
i
T
T i
i
i
T
T
z
T z t q
a
z
erf
z
z
erf
d
D
N DT D
N T
D N T
N D T D
D N T
T
N T
­
ª
§
°
«
¨
'
®
¨
«
°
©
¬
¯
½
ª
º
§
· °
º
«
»
¨
¸
¾
» ¼
¨
¸
«
» °
©
¹
¬
¼ ¿
¦ ³
(2.83)
Equation (2.83) is integrated to obtain:
^
`
9
1
2
0
1
2
1/ 2
2
1
,
e x p
1
2
exp
1
2exp
4
e x p
R
R
T
ii
i
i
i
i
i
i
i
i
i
T z t
q
a
z
erf
z
e r f
z
erf
N
D
D
G
[ G
D
G
[ G
D
G S
[
[ [
[
ª
º
'
¬
¼
ª
º
¬
¼
ª
º
¬
¼
¦
(2.84)
where
1/ 2
4 T
z
t
[
N
and
1/ 2
i
i
T t
G D N
are non-dimensional depth and
time, respectively.
According to (2.76) and (2.84), the near-surface temperature difference
between the renewal events evolves in the following way:
109