258
J. Kjellsson et al.
with the indices having the same meaning as above. Similarly, the acceleration was
calculated using a similar finite difference scheme applied to the particular velocity
component:
a i,m (t n ) ≡
du i,m (t n )
dt
≈
u i,m (t n ) − u i,m (t n−1 )
t n − t n−1
.
(8.7)
Note that velocity is not defined at the first time instant (or position), and acceleration is not defined at the first and second time instant.
The Lagrangian velocity autocorrelation describes the correlation of the velocity
at one time with that of previous times:
R(τ ) =
σ 2 (τ )
σ 2 (τ = 0)
≈ R q =
σ 2
q
σ 2
0
,
(8.8)
where σ 2 (τ ) and σ 2 (τ = 0) are the Lagrangian velocity autocovariances for time
lag τ and no lag (τ = 0), respectively. Here q is the discrete time step and R q is the
autocorrelation at time step q. The quantity σ 2 (τ ) is defined as
σ
2 (τ ) = lim
T →∞
1
T
T
0
u
(t + τ ) · u
(t) dt ≈ σ
2
q ≡
2
i=1
1
N − q − 1
N −q−1
n=1
u
i,n u
i,n+q ,
(8.9)
where u
i,n = u i,n − u i and u i is a time average of the segment. Note that the total
velocity autocovariance is the sum of its zonal and meridional components: σ 2 =
σ 2
i=1 + σ 2
i=2 .
Using the autocorrelation R(τ ), the Lagrangian integral time scale T L is defined
as
T L =
t 1
0
R(τ ) dτ.
(8.10)
This is a measure of the memory of a trajectory, that is, the time lag during which the
Lagrangian velocity is correlated. When computing this integral, the upper bound,
t 1 , is the point where R(τ ) = 0 occurs for the first time. This truncation is perhaps
the most commonly used one, due to the often noisy character of the autocorrelation
function R(τ ) for large τ . Lumpkin et al. (2002) compared this choice with several
other approximations, and found that all approaches produced essentially the same
results. Thus it is natural to assume that the approximation used here is a robust one.
8.5 Results
8.5.1 The Surface Drifters
The 12 drifter trajectories were of different length because of variable drifter lifetime. The mean drifter lifetime was ∼80 days (Table 8.1). For this reason, each
J. Kjellsson et al.
with the indices having the same meaning as above. Similarly, the acceleration was
calculated using a similar finite difference scheme applied to the particular velocity
component:
a i,m (t n ) ≡
du i,m (t n )
dt
≈
u i,m (t n ) − u i,m (t n−1 )
t n − t n−1
.
(8.7)
Note that velocity is not defined at the first time instant (or position), and acceleration is not defined at the first and second time instant.
The Lagrangian velocity autocorrelation describes the correlation of the velocity
at one time with that of previous times:
R(τ ) =
σ 2 (τ )
σ 2 (τ = 0)
≈ R q =
σ 2
q
σ 2
0
,
(8.8)
where σ 2 (τ ) and σ 2 (τ = 0) are the Lagrangian velocity autocovariances for time
lag τ and no lag (τ = 0), respectively. Here q is the discrete time step and R q is the
autocorrelation at time step q. The quantity σ 2 (τ ) is defined as
σ
2 (τ ) = lim
T →∞
1
T
T
0
u
(t + τ ) · u
(t) dt ≈ σ
2
q ≡
2
i=1
1
N − q − 1
N −q−1
n=1
u
i,n u
i,n+q ,
(8.9)
where u
i,n = u i,n − u i and u i is a time average of the segment. Note that the total
velocity autocovariance is the sum of its zonal and meridional components: σ 2 =
σ 2
i=1 + σ 2
i=2 .
Using the autocorrelation R(τ ), the Lagrangian integral time scale T L is defined
as
T L =
t 1
0
R(τ ) dτ.
(8.10)
This is a measure of the memory of a trajectory, that is, the time lag during which the
Lagrangian velocity is correlated. When computing this integral, the upper bound,
t 1 , is the point where R(τ ) = 0 occurs for the first time. This truncation is perhaps
the most commonly used one, due to the often noisy character of the autocorrelation
function R(τ ) for large τ . Lumpkin et al. (2002) compared this choice with several
other approximations, and found that all approaches produced essentially the same
results. Thus it is natural to assume that the approximation used here is a robust one.
8.5 Results
8.5.1 The Surface Drifters
The 12 drifter trajectories were of different length because of variable drifter lifetime. The mean drifter lifetime was ∼80 days (Table 8.1). For this reason, each