where
Q 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
g i + 0.5 h ̄
i + 0.5
f i + 0.5
;
ð10Þ
here g i , f i are the acceleration due to gravity and the Coriolis parameter in the
middle of the space between the i-th and i + 1-st stations, h ̄
i + 0.5 is the depth
averaged over the mentioned space. Variable Q t represents the total deviation of the
transport in time along the section in the middle of the spaces between the stations
and it is the component of the distortion of the barotropic flow that can be taken into
account and estimated using the altimetry data.
If we subtract the transport across the section from instantaneous barotropic
transport Q adcp t
ð Þ, we obtain an anomaly of the barotropic transport approximating
the ADCP measurements at the stations:
ΔQ adcp t
ð Þ = ∑
i = 1, n
Δq adcp x i , t
ð Þ≡ Δζ x n , t
ð
Þ− Δζ x n − 0.5 , t
ð
Þ
ð
Þ
g n h ̄
n
fn +
+ Δζ x 1.5 , t
ð
Þ− Δζ x 1 , t
ð
Þ
ð
Þ
g1h ̄
1
f1 + ∑
i = 2, n − 1
Δζ x i + 0.5 , t
ð
Þ− Δζ x i − 0.5 , t
ð
Þ
ð
Þ
gih ̄
i
fi
,
ð11Þ
where Δζ x i±0.5 , t
ð
Þ= ζ x i±0.5 , t
ð
Þ− ζ x i±0.5 , t i
ð
Þ. It is important that the value of
ΔQ adcp t
ð Þ does not depend on the mean dynamic topography of the ocean and like
the deviation in time Q t over spaces between stations can be calculated on the basis
of the SLA satellite data. To proceed to the discrete distribution of the barotropic
transport in time it is necessary to replace t by t j in formula (11).
Two-dimensional distribution Δq x i , t j
À
Á
per unit length (top panel) and distribution ΔQ adcp t j
À Á
(bottom panel) for the section carried out in the Drake Passage in
October 2010 are shown in Fig. 3 as an example; time counts t j correspond to the
moments of the stations. The points on curve ΔQ adcp t j
À Á
are the sum of elements
Δq x i , t j
À
Á
for fixed t j . Diagonal elements Δq x i , t i
ð
Þ (i.e., when i = j) are zero. The
range of values ΔQ adcp t j
À Á
determines the variability of the barotropic transport
approximating the ADCP measurements during the time of the section. An example
in Fig. 3 also shows that the barotropic transport over the section can appear beyond
the variability of the instantaneous barotropic transport during the time of the section.
Instantaneous barotropic currents calculated from the level values at stations can
be written as:
Q
′
alt t
ð Þ = ∑
i = 1, n − 1
q alt x i + 0.5 , t
ð
Þ≡ ∑
i = 1, n − 1
ζ x i + 1 , t
ð
Þ− ζ x i , t
ð Þ
ð
Þ
g i + 0.5 h ̄
i + 0.5
f i + 0.5
. ð12Þ
Influence of the Current Field Non-stationarity …
373
Q 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
g i + 0.5 h ̄
i + 0.5
f i + 0.5
;
ð10Þ
here g i , f i are the acceleration due to gravity and the Coriolis parameter in the
middle of the space between the i-th and i + 1-st stations, h ̄
i + 0.5 is the depth
averaged over the mentioned space. Variable Q t represents the total deviation of the
transport in time along the section in the middle of the spaces between the stations
and it is the component of the distortion of the barotropic flow that can be taken into
account and estimated using the altimetry data.
If we subtract the transport across the section from instantaneous barotropic
transport Q adcp t
ð Þ, we obtain an anomaly of the barotropic transport approximating
the ADCP measurements at the stations:
ΔQ adcp t
ð Þ = ∑
i = 1, n
Δq adcp x i , t
ð Þ≡ Δζ x n , t
ð
Þ− Δζ x n − 0.5 , t
ð
Þ
ð
Þ
g n h ̄
n
fn +
+ Δζ x 1.5 , t
ð
Þ− Δζ x 1 , t
ð
Þ
ð
Þ
g1h ̄
1
f1 + ∑
i = 2, n − 1
Δζ x i + 0.5 , t
ð
Þ− Δζ x i − 0.5 , t
ð
Þ
ð
Þ
gih ̄
i
fi
,
ð11Þ
where Δζ x i±0.5 , t
ð
Þ= ζ x i±0.5 , t
ð
Þ− ζ x i±0.5 , t i
ð
Þ. It is important that the value of
ΔQ adcp t
ð Þ does not depend on the mean dynamic topography of the ocean and like
the deviation in time Q t over spaces between stations can be calculated on the basis
of the SLA satellite data. To proceed to the discrete distribution of the barotropic
transport in time it is necessary to replace t by t j in formula (11).
Two-dimensional distribution Δq x i , t j
À
Á
per unit length (top panel) and distribution ΔQ adcp t j
À Á
(bottom panel) for the section carried out in the Drake Passage in
October 2010 are shown in Fig. 3 as an example; time counts t j correspond to the
moments of the stations. The points on curve ΔQ adcp t j
À Á
are the sum of elements
Δq x i , t j
À
Á
for fixed t j . Diagonal elements Δq x i , t i
ð
Þ (i.e., when i = j) are zero. The
range of values ΔQ adcp t j
À Á
determines the variability of the barotropic transport
approximating the ADCP measurements during the time of the section. An example
in Fig. 3 also shows that the barotropic transport over the section can appear beyond
the variability of the instantaneous barotropic transport during the time of the section.
Instantaneous barotropic currents calculated from the level values at stations can
be written as:
Q
′
alt t
ð Þ = ∑
i = 1, n − 1
q alt x i + 0.5 , t
ð
Þ≡ ∑
i = 1, n − 1
ζ x i + 1 , t
ð
Þ− ζ x i , t
ð Þ
ð
Þ
g i + 0.5 h ̄
i + 0.5
f i + 0.5
. ð12Þ
Influence of the Current Field Non-stationarity …
373