The real ocean is not flat and the walls are not vertical. In this case, the
instantaneous barotropic transport across the section line changes with time due to
the displacement of the level isolines (stream lines of the absolute geostrophic
velocity at the ocean surface) relative to the isobaths. When calculating in finite
differences, the transport also depends on the particular scheme of the section
splitting into segments. Instantaneous barotropic transport calculated from the level
values in the middle of the spaces between stations, i.e. the transport approximating
ADCP measurements, can be written as:
Q adcp t
ð Þ = ∑
i = 1, n
q adcp x i , t
ð Þ≡ ζ x n , t
ð
Þ− ζ x n − 0.5 , t
ð
Þ
ð
Þ
g n h ̄
n
fn + ζ x 1.5 , t
ð
Þ− ζ x 1 , t
ð
Þ
ð
Þ
g 1 h ̄
1
f1 +
+ ∑
i = 2, n − 1
ζ x i + 0.5 , t
ð
Þ− ζ x i − 0.5 , t
ð
Þ
ð
Þ
gih ̄
i
f i
,
ð7Þ
where g i , f i are the acceleration due to gravity and the Coriolis parameter at the
point of the i-th station, h ̄
i is the depth averaged over the interval of the section
corresponding to the i-th station (over the i-th interval between the interstation
midpoints of the spaces), q adcp x i , t
ð Þ is the barotropic transport in this interval of the
section. The sign of the transport summation by stations is determined in (7)
conventionally. For example, for quasi-zonal sections, it is convenient to choose the
positive direction to the left of the direction of the section if it runs from west to
east. Then, the positive transport values correspond to the flow to the north, the
negative transport values correspond to the flow to the south. For example, in the
case of quasi-meridional sections it is convenient to select the positive direction to
the left of the ship motion along the section from west to east. Then, the positive
values correspond to the easterly flow, the negative values correspond to the
westerly flow.
The barotropic transport across a particular section calculated according to the
same scheme consists of elements q adcp x i , t i
ð
Þ corresponding to the moments of the
execution time of the stations:
Q adcp = ∑
i = 1, n
q adcp x i , t i
ð
Þ≡ ζ x n , t n
ð
Þ− ζ x n − 0.5 , t n
ð
Þ
ð
Þ
g n h ̄
n
fn + ζ x 1.5 , t 1
ð
Þ− ζ x 1 , t 1
ð
Þ
ð
Þ
g 1 h ̄
1
f1 +
+ ∑
i = 2, n − 1
ζ x i + 0.5 , t i
ð
Þ− ζ x i − 0.5 , t i
ð
Þ
ð
Þ
gih ̄
i
fi
.
ð8Þ
The total barotropic transport (i.e. determined by the total differential of the
ocean level) is represented in the following form:
Q alt = Q adcp + Q t ,
ð9Þ
372
R. Yu. Tarakanov
instantaneous barotropic transport across the section line changes with time due to
the displacement of the level isolines (stream lines of the absolute geostrophic
velocity at the ocean surface) relative to the isobaths. When calculating in finite
differences, the transport also depends on the particular scheme of the section
splitting into segments. Instantaneous barotropic transport calculated from the level
values in the middle of the spaces between stations, i.e. the transport approximating
ADCP measurements, can be written as:
Q adcp t
ð Þ = ∑
i = 1, n
q adcp x i , t
ð Þ≡ ζ x n , t
ð
Þ− ζ x n − 0.5 , t
ð
Þ
ð
Þ
g n h ̄
n
fn + ζ x 1.5 , t
ð
Þ− ζ x 1 , t
ð
Þ
ð
Þ
g 1 h ̄
1
f1 +
+ ∑
i = 2, n − 1
ζ x i + 0.5 , t
ð
Þ− ζ x i − 0.5 , t
ð
Þ
ð
Þ
gih ̄
i
f i
,
ð7Þ
where g i , f i are the acceleration due to gravity and the Coriolis parameter at the
point of the i-th station, h ̄
i is the depth averaged over the interval of the section
corresponding to the i-th station (over the i-th interval between the interstation
midpoints of the spaces), q adcp x i , t
ð Þ is the barotropic transport in this interval of the
section. The sign of the transport summation by stations is determined in (7)
conventionally. For example, for quasi-zonal sections, it is convenient to choose the
positive direction to the left of the direction of the section if it runs from west to
east. Then, the positive transport values correspond to the flow to the north, the
negative transport values correspond to the flow to the south. For example, in the
case of quasi-meridional sections it is convenient to select the positive direction to
the left of the ship motion along the section from west to east. Then, the positive
values correspond to the easterly flow, the negative values correspond to the
westerly flow.
The barotropic transport across a particular section calculated according to the
same scheme consists of elements q adcp x i , t i
ð
Þ corresponding to the moments of the
execution time of the stations:
Q adcp = ∑
i = 1, n
q adcp x i , t i
ð
Þ≡ ζ x n , t n
ð
Þ− ζ x n − 0.5 , t n
ð
Þ
ð
Þ
g n h ̄
n
fn + ζ x 1.5 , t 1
ð
Þ− ζ x 1 , t 1
ð
Þ
ð
Þ
g 1 h ̄
1
f1 +
+ ∑
i = 2, n − 1
ζ x i + 0.5 , t i
ð
Þ− ζ x i − 0.5 , t i
ð
Þ
ð
Þ
gih ̄
i
fi
.
ð8Þ
The total barotropic transport (i.e. determined by the total differential of the
ocean level) is represented in the following form:
Q alt = Q adcp + Q t ,
ð9Þ
372
R. Yu. Tarakanov