Regions where S > 0 and S < 0 are located respectively to the left and to the right
from the bold line in Fig. 8. They are related to ice drifts to the “north” and to the
“south” respectively. Figure 8 show the existence of the roots of Eq. (19) in both
regions S > 0 and S < 0 when ϕ < π ̸ 2. The angle of wind direction ϕ is specified in
Fig. 8a.
It follows from Eqs. (17) and (18) that:
v y =
δρ i hf
2
ffiffiffiffiffi
3γ
p ρ w C w
ffiffiffiffiffiffiffiffiffiffiffi
1 + Δ
p
− δ
, Δ =
4
ffiffiffiffiffi
3γ
p ρ w C w ρ a C a V a ð
ffiffiffiffiffi
3γ
p jV ay j − V ax Þ + ρ i ghη x
À
Á
ρ i hf
ð
Þ
2
.
ð20Þ
Simple estimates show that Δ > 60 when h ≤ 1 m, V a > 5 m ̸ s, ϕ ∈ ð0.5, π ̸ 2Þ
and jη x j < 2 × 10
− 6 . In this case,
ffiffiffiffiffiffiffiffiffiffiffi
1 + Δ
p
≈ Δ, and the drift velocity and the ES
width are approximated with the formulas
v y =
δρ i hf
ffiffiffi ffi
Δ
p
2
ffiffiffiffiffi
3γ
p ρ w C w
, Δx =
2hP
*
δh 2 P * ̸ L 2 + 3ρ a C a V a V ax − 3ρ i ghη x
.
ð21Þ
It follows from Eq. (1) that inside the PS the ice thickness changes according to
the equation
2P
*
3
dh
dx
= ρ i hfv y + ρ a C a V a V ax − ρ i ghη x ,
ð22Þ
where v y is determined by formula (20).
Further, the analysis of the ice drift to the “north” (δ = 1) is performed with more
details. Blue and pink lines in Fig. 10 are described respectively by the equations
Δ = 0, Δx = 0.
ð23Þ
The regions, in which conditions Δ > 0, Δx > 0 are satisfied, are located between
blue and pink lines shown on the plane ðϕ, η x Þ in Fig. 9. The lines are constructed
with different values of the wind speed V a shown in the figures. The effect of ice
thickness is visible from the comparison of Fig. 9a (h = 0.5 m) and Fig. 9b
(h = 1 m). Vertical distances between blue and pink lines constructed with the same
values of V a and ϕ decrease proportionally to the ice thickness. The distances reach
maximum when ϕ ≈ π ̸ 2, i.e. when the wind is parallel to the y-axis, and tend to
zero when ϕ → π and ϕ → 0. In the case of the onshore winds with ϕ ∈ ð0.3, π ̸ 2Þ
conditions Δ > 0, Δx > 0 are satisfied with the positive and negative sea surface tilts.
In the case when onshore wind is almost perpendicular to the shoreline and
ϕ ∈ ð0, 0.3Þ these conditions are satisfied only when the sea surface tilt is positive.
In the case of offshore winds with ϕ ∈ ðπ ̸ 2, πÞ the conditions Δ > 0, Δx > 0 are
satisfied only when the sea surface tilt is negative.
550
A. Marchenko
from the bold line in Fig. 8. They are related to ice drifts to the “north” and to the
“south” respectively. Figure 8 show the existence of the roots of Eq. (19) in both
regions S > 0 and S < 0 when ϕ < π ̸ 2. The angle of wind direction ϕ is specified in
Fig. 8a.
It follows from Eqs. (17) and (18) that:
v y =
δρ i hf
2
ffiffiffiffiffi
3γ
p ρ w C w
ffiffiffiffiffiffiffiffiffiffiffi
1 + Δ
p
− δ
, Δ =
4
ffiffiffiffiffi
3γ
p ρ w C w ρ a C a V a ð
ffiffiffiffiffi
3γ
p jV ay j − V ax Þ + ρ i ghη x
À
Á
ρ i hf
ð
Þ
2
.
ð20Þ
Simple estimates show that Δ > 60 when h ≤ 1 m, V a > 5 m ̸ s, ϕ ∈ ð0.5, π ̸ 2Þ
and jη x j < 2 × 10
− 6 . In this case,
ffiffiffiffiffiffiffiffiffiffiffi
1 + Δ
p
≈ Δ, and the drift velocity and the ES
width are approximated with the formulas
v y =
δρ i hf
ffiffiffi ffi
Δ
p
2
ffiffiffiffiffi
3γ
p ρ w C w
, Δx =
2hP
*
δh 2 P * ̸ L 2 + 3ρ a C a V a V ax − 3ρ i ghη x
.
ð21Þ
It follows from Eq. (1) that inside the PS the ice thickness changes according to
the equation
2P
*
3
dh
dx
= ρ i hfv y + ρ a C a V a V ax − ρ i ghη x ,
ð22Þ
where v y is determined by formula (20).
Further, the analysis of the ice drift to the “north” (δ = 1) is performed with more
details. Blue and pink lines in Fig. 10 are described respectively by the equations
Δ = 0, Δx = 0.
ð23Þ
The regions, in which conditions Δ > 0, Δx > 0 are satisfied, are located between
blue and pink lines shown on the plane ðϕ, η x Þ in Fig. 9. The lines are constructed
with different values of the wind speed V a shown in the figures. The effect of ice
thickness is visible from the comparison of Fig. 9a (h = 0.5 m) and Fig. 9b
(h = 1 m). Vertical distances between blue and pink lines constructed with the same
values of V a and ϕ decrease proportionally to the ice thickness. The distances reach
maximum when ϕ ≈ π ̸ 2, i.e. when the wind is parallel to the y-axis, and tend to
zero when ϕ → π and ϕ → 0. In the case of the onshore winds with ϕ ∈ ð0.3, π ̸ 2Þ
conditions Δ > 0, Δx > 0 are satisfied with the positive and negative sea surface tilts.
In the case when onshore wind is almost perpendicular to the shoreline and
ϕ ∈ ð0, 0.3Þ these conditions are satisfied only when the sea surface tilt is positive.
In the case of offshore winds with ϕ ∈ ðπ ̸ 2, πÞ the conditions Δ > 0, Δx > 0 are
satisfied only when the sea surface tilt is negative.
550
A. Marchenko