w 2 x, z
ð Þ= −
S
′
2 x
ð Þ
6
z
3
−
A
′
2 x
ð Þ
2
z
2
− B
′
2 x
ð Þz + C 2 x
ð Þ.
S 1 x
ð Þ ≡
g
μ 1
∂h
∂x
,
S 2 x
ð Þ ≡
ρ 1
ρ 2
μ 1
μ 2
S 1 −
g
μ 2
Δρ
ρ 2
∂ξ
∂x
, Δρ = ρ 2 − ρ 1,
A 1 =
τ
ρ 1 μ 1
,
A 2 x
ð Þ =
ρ 1
ρ 2
μ 1
μ 2
½S 1 ξ + A 1 Š − S 2 ξ,
B 2 x
ð Þ = −
S 2
2
D
2
− A 2 D,
C 1 x
ð Þ =
μ 1
g
S 1 B 1 ,
C 2 x
ð Þ =
S
′
2
6
D
3 +
A
′
2
2
D
2 + B
′
2 D,
where functions S 1 x
ð Þ and B 1 x
ð Þ are solutions of the following equations:
S
0
1 −
rD
3
3
+
rD
2
2
ξ −
rξ
3
6
!
+ S 1 ½ξ + 2D
2 D
′ r −
ξξ
2
2
+
ξ
′ D
2
2
rŠ + F = 0, where
F =
2
3
αξ
3 ξ
′′ + α D
2 −
ξ
2
2
ξ
′
À Á 2 + D
′ r
τ
ρ 1 μ 1
D − ξ
ð
Þ− αξ
′′ ξ
2 D − αξ
′ ξ
2 D
′ −
α
2
ξ
′′ ξD
2
− 3DD
′ αξ
′ ξ −
2
3
αD
3 ξ
′′ − 2D
2 αD
′ ξ
′ −
α
2
ξ
′
À Á 2 ξ
2 − ξ
′ r τ
μ 1 ρ 1
D + ξ
ð
Þ− Dα ξ
′
À Á 2 ξ +
D
2
,
r =
ρ 1
ρ 2
μ 1
μ 2
,
α =
g
μ 2
Δρ
ρ 2
,
and
ξB
′
1 =
μ 1
g
S 1 B 1 −
S
′
1
6
ξ
3
− ξ
′ 1
2
S 1 ξ
2 + A 1 ξ + B 1
&
'
.
An example of the solution is shown in Fig. 9. This particular experiment used
real bathymetry, constant onshore (i.e., downwelling) wind stress at 0.1 N/m
2 , and
Sea of Azov Waters in the Black Sea …
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