ζ x i + 1 , t i + 1
ð
Þ− ζ x i , t i
ð
Þ≡
≡ ζ x i + 1 , t i + 1
ð
Þ− ζ x i + 0.5 , t i + 1
ð
Þ
ð
Þ + ζ x i + 0.5 , t i + 1
ð
Þ − ζ x i + 0.5 , t i
ð
Þ
ð
Þ + ζ x i + 0.5 , t i
ð
Þ− ζ x i , t i
ð
Þ
ð
Þ .
ð5Þ
Summing over all the spaces between the stations (from the 1-st to the n-th), we
get:
ζ x n , t n
ð
Þ− ζ x 1 , t 1
ð
Þ≡ ∑
i = 1, n − 1
ζ x i + 1 , t i + 1
ð
Þ− ζ x i , t i
ð
Þ
ð
Þ ≡ ∑ Δζ = ∑ Δζ x + ∑ Δζ t ,
ð6Þ
where
∑ Δζ x = ζ x 1.5 , t 1
ð
Þ− ζ x 1 , t 1
ð
Þ
ð
Þ + ∑
i = 2, n − 1
ζ x i + 0.5 , t i
ð
Þ− ζ x i − 0.5 , t i
ð
Þ
ð
Þ + ζ x n , t n
ð
Þ− ζ x n − 0.5 , t n
ð
Þ
ð
Þ ,
ð6aÞ
∑ Δζ t = ∑
i = 1, n − 1
ζ x i + 0.5 , t i + 1
ð
Þ− ζ x i + 0.5 , t i
ð
Þ
ð
Þ .
ð6bÞ
Here, ∑ Δζ x is the sum of the level differences at the intervals between adjacent
interstation midpoints at fixed time moments, ∑ Δζ t is the sum of the level deviations in time in the middle of the spaces between stations, and n is the number of
stations. In the sketch shown in Fig. 2 explaining formula (6), the bold line shows
the ocean level curve consisting of Δζ x and Δζ t , the dashed line is the curve formed
of Δζ. In the sum over the entire section, these two curves give the same level
difference.
The Δζ x value at the i-th interval of the section approximates the barotropic
component of the geostrophic current at fixed time corresponding to the moment of
the i-th station. In the physical sense, this component is absolutely similar to the
barotropic component of the geostrophic current measured using ADCP. Thus, the
level deviation in time in the middle of the spaces between stations determines the
physical difference between the calculations of the barotropic component of
the absolute geostrophic current from the satellite altimetry and ADCP data.
In practice, it is not possible to determine the barotropic component of the
current directly from the ADCP velocity measurements at one station, since the
measured baroclinic profile contains unknown non-geostrophic components (e.g.,
purely drift current, inertial oscillations). In addition, the measurements in the upper
several tens of meters are usually obtained with a very large error or these data are
not available in the case of shipboard ADCP. Therefore, the barotropic component
based on the ADCP measurements is determined for the baroclinic geostrophic
profile calculated from the CTD data from two neighboring adjacent stations, which
usually consists of the following procedure [1, 6, 11, 13]. Initially, the measured
velocity profiles of two neighboring stations are averaged at each level up to the
greatest common depth of the CTD and ADCP measurements at these stations.
Then, the depth averaged values of the velocity for this averaged profile ⟨u adcp ⟩ and
Influence of the Current Field Non-stationarity …
369
ð
Þ− ζ x i , t i
ð
Þ≡
≡ ζ x i + 1 , t i + 1
ð
Þ− ζ x i + 0.5 , t i + 1
ð
Þ
ð
Þ + ζ x i + 0.5 , t i + 1
ð
Þ − ζ x i + 0.5 , t i
ð
Þ
ð
Þ + ζ x i + 0.5 , t i
ð
Þ− ζ x i , t i
ð
Þ
ð
Þ .
ð5Þ
Summing over all the spaces between the stations (from the 1-st to the n-th), we
get:
ζ x n , t n
ð
Þ− ζ x 1 , t 1
ð
Þ≡ ∑
i = 1, n − 1
ζ x i + 1 , t i + 1
ð
Þ− ζ x i , t i
ð
Þ
ð
Þ ≡ ∑ Δζ = ∑ Δζ x + ∑ Δζ t ,
ð6Þ
where
∑ Δζ x = ζ x 1.5 , t 1
ð
Þ− ζ x 1 , t 1
ð
Þ
ð
Þ + ∑
i = 2, n − 1
ζ x i + 0.5 , t i
ð
Þ− ζ x i − 0.5 , t i
ð
Þ
ð
Þ + ζ x n , t n
ð
Þ− ζ x n − 0.5 , t n
ð
Þ
ð
Þ ,
ð6aÞ
∑ Δζ t = ∑
i = 1, n − 1
ζ x i + 0.5 , t i + 1
ð
Þ− ζ x i + 0.5 , t i
ð
Þ
ð
Þ .
ð6bÞ
Here, ∑ Δζ x is the sum of the level differences at the intervals between adjacent
interstation midpoints at fixed time moments, ∑ Δζ t is the sum of the level deviations in time in the middle of the spaces between stations, and n is the number of
stations. In the sketch shown in Fig. 2 explaining formula (6), the bold line shows
the ocean level curve consisting of Δζ x and Δζ t , the dashed line is the curve formed
of Δζ. In the sum over the entire section, these two curves give the same level
difference.
The Δζ x value at the i-th interval of the section approximates the barotropic
component of the geostrophic current at fixed time corresponding to the moment of
the i-th station. In the physical sense, this component is absolutely similar to the
barotropic component of the geostrophic current measured using ADCP. Thus, the
level deviation in time in the middle of the spaces between stations determines the
physical difference between the calculations of the barotropic component of
the absolute geostrophic current from the satellite altimetry and ADCP data.
In practice, it is not possible to determine the barotropic component of the
current directly from the ADCP velocity measurements at one station, since the
measured baroclinic profile contains unknown non-geostrophic components (e.g.,
purely drift current, inertial oscillations). In addition, the measurements in the upper
several tens of meters are usually obtained with a very large error or these data are
not available in the case of shipboard ADCP. Therefore, the barotropic component
based on the ADCP measurements is determined for the baroclinic geostrophic
profile calculated from the CTD data from two neighboring adjacent stations, which
usually consists of the following procedure [1, 6, 11, 13]. Initially, the measured
velocity profiles of two neighboring stations are averaged at each level up to the
greatest common depth of the CTD and ADCP measurements at these stations.
Then, the depth averaged values of the velocity for this averaged profile ⟨u adcp ⟩ and
Influence of the Current Field Non-stationarity …
369