5.4 Diffusive effects
5.4.3 Calculation of displacements
Here we describe the calculation of the displacements for each water parcel. The dis-
placements in each of the three spatial directions are distributed by the 1DGaussian (5.39),
therefore each displacement is calculated separately. There are a large number of good algo-
rithms available (ex. Press et al 1992) for generating uniform random numbers on the interval
[0,1]
y∈[R]10. (5.44)
The uniform distribution defines the probability of generating a number betweenyandy+dy
by
P(y)dy = dy for y∈[, ] (5.45)
Using a transformation of variables these uniform deviates can be used to generate random
numbers following other distributions. The multi-dimensional transformation law of proba-
bilities (eg. Feller 1950) is given by
P(x1, x2, . . . , xn)dx1dx2. . . dxn=P(y1, y2, . . . , yn) ∂(y1, y2, . . . , yn) ∂(x1, x2, . . . , xn) dx1 dx2. . . dxn. (5.47)
For the uniform distribution on the interval [0,1] we have P(y) = 1, therefore in 1Dwe need
to find a function y(x) that satisfies the differential equation
dy dx =P(x, t). (5.48)
ForP(x, t) as in (5.39) the solution is given by
y(x) = Z x −∞ P(x0, t)dx0 = 1 2erf x √ Dt + , (5.49)
where erf(x) is the error function as given in (5.30). This mapping is illustrated in figure 5.2.
The random deviates following distributionP(x, t) are given by the inverse of (5.49)
x(y) =√4Dt erf−1 (y−
). (5.50)
This expression can be used to generate Gaussian deviatesxfrom uniform deviatesy, although
the inverse of the error function (erf−1) must be approximated numerically.
However in 2D a transformation has been introduced by Box and Muller (1958), which
gives an expression for two Gaussian deviates that can be calculated analytically. A 2D
Gaussian distribution for two independent deviatesx1 andx2 distributed according to (5.39)
is given by P(x1, x2, t) = 1 √ 4πDt exp " −x21 4Dt # ·√ 1 4πDt exp " −x22 4Dt # . (5.51)
Figure 5.2: Mapping (5.49) for generating 1D Gaussian deviates x from uniform deviatesy.
Given two uniform deviatesy1, y2 ∈[R]10 we can use the transformation law (5.47) to generate
two Gaussian deviatesx1 and x2. In order for this we need transformations
y1 =f1(x1, x2), (5.52) y2 =f2(x1, x2), (5.53) such that ∂(y1, y2) ∂(x1, x2) =P(x1, x2, t). (5.54)
This condition is satisfied by the following transformations, which are similar to those intro-
duced by Box and Muller (1958)
f1(x1, x2) = 1−exp " −x 2 1+x22 4Dt # (5.55) f2(x1, x2) = 1 2π arctan x 1 x2 . (5.56)
These transformations are invertible and give the following expressions forx1 and x2
x1=f1−1(y1, y2) =
q
4Dtlog [1−y1] cos (2πy2) (5.57)
x2 =f2−1(y1, y2) =
q
Expressions (5.57) and (5.58) represent the x- andy-components of a point in the 2D plane
with a distance ofp
4Dtlog [1−y1] from the origin, and an angle of 2πy2 with respect to the
x-axis.
We use the transformations (5.57) and (5.58) once for the two horizontal displacements
using two uniform deviates y1 and y2 and the horizontal eddy viscosity D =AH. Then we
repeat this process using two new uniform deviates y3 andy4 and the vertical eddy viscosity
D=AZ, and use only one of the resulting deviates. This leads to the following displacements
xd =
q
4AHtlog [1−y1] cos (2πy2), (5.59)
yd =
q
4AHtlog [1−y1] sin (2πy2), (5.60)
zd =
q
4AZtlog [1−y3] cos (2πy4), (5.61)
which are added to to the position of the particle after each time-step of lengtht.
Previously a random walk model has been introduced in models for oil spill trajectories by
Al-Rabeh and Gunay (1992) and Evans and Noye (1995). In these models the two horizontal
dimensions are combined, and random jumps (xd, yd) are generated by choosing a point in the
2D plane. The distance from the starting point rd, and the angle θ are chosen as uniformly
distributed random numbers. The standard deviation of the horizontal displacement is chosen
to be equal to the standard deviation of a 2D random walk
σ2D =
p
σ2+σ2 =√4Dt, (5.62)
whereσ is the standard deviation found for the 1D random walk in (5.41).
For the vertical dimension Al-Rabeh and Gunay (1992) apply a 1D equivalent of this
[−1,1] is obtained using the following transformation
z= 2y−1, y∈[R]10. (5.63)
The mean and standard deviation ofz are given by
µ(z) = 2µ(y)−1 = 2 Z 1 0 yP(y)dy−1 = 0, (5.64) σ(z) = 2σ(y) = 2 s Z 1 0 (y−µ(z))2P(y)dy= √1 3. (5.65)
The displacement zd is then chosen such that its standard deviation is equal to (5.41), as
found for the 1D random walk. This gives the displacement
zd =
√
3σz=√6Dt(2y−1), y ∈[R]10, (5.66)
which is distributed uniformly within the rangeh−√3σ,√3σi.
The difference between our 1D displacements (5.59)-(5.61) and Al-Rabeh and Gunay’s
(1992) 1D displacement (5.66) is illustrated in figure 5.3. The former is a Gaussian bell
curve which matches the Laplacian viscosity terms in the momentum equations, while the
latter is represented by a rectangle. While the Gaussian distribution has 68%/95% of the
displacements contained in the range [−σ, σ]/[−2σ,2σ], the rectangle distribution has 100%
of the displacements contained in the rangeh−√3σ,√3σi.
We can approximate the magnitude of the diffusive jumps using an estimate of the time-
steptneeded to cross a grid-box. The dimension ∆xof a horizontal grid box is approximately
14km. A particle in a weak flow of 1cm/s will cross this grid box in approximately 106s.
Therefore the standard deviation of the horizontal jumps is given by
σ=p2AHt= 20km, (5.67)
where AH is equal to the horizontal eddy viscosity of 2·106cm2/s. Therefore in this case
Figure 5.3: Comparison of 1D probability distributions for particle displacements. Black:
Distribution (5.66) as used by Al Rabeh and Gunay (1992). Blue: Distribution (5.39) with
σ=√2Dt.
horizontal grid-box size and gives us an upper estimate for the size of the horizontal random
jumps.
The dimension ∆z of a vertical grid box in the upper ocean is approximately 100m. A
particle in a weak flow of 10−4cm/swill cross this grid box in approximately 108s. Therefore the standard deviation of the vertical jumps is given by
σ=p2AZt≈141m, (5.68)
whereAZ is equal to the vertical eddy viscosity of 1cm2/s. Therefore in this case 68% of the
jumps are within the range [−141m,141m], which is of the same order as the vertical grid-box
size. This is a severe upper limit as the horizontal flow must be very weak for a particle to
the range of vertical displacements is approximately [−14m,14m], which is smaller than the
dimension of most vertical grid-boxes. Therefore any oscillations in the particle trajectories
that are larger than the size of a horizontal or vertical grid-box are not likely to be caused by
Particle tracking - application
6.1
Introduction
Here we apply the method described in the previous chapter to calculate pathways of shelf
waters through the Arctic Ocean and assess their influence on the circulation. We start by
comparing the time-independent, time-dependent, and diffusive time-dependent methods for
pathways of Barents Sea Water. We then calculate pathways for the other inflows of Atlantic
Water and Pacific Water. Also we assess time-scales for signals to propagate from the Barents
Sea and Chukchi Sea shelves. These are two of the most important shelf seas as they directly
receive and modify water from the Atlantic and Pacific Oceans.
Various observations on the Canadian side of the Lomonosov Ridge have shown the pres-
ence of AW and BSW (Schauer et al 1997, McLaughlin et al 2002). However the reach of the
water masses could depend significantly on the phase of the Arctic Oscillation, as shown by
Maslowski et al (2000). The OCCAM model simulation was forced by constant atmospheric
conditions from 1986-1988, therefore the particle trajectories are free from atmospheric vari-
ability. Nevertheless they will give an insight into what processes determine the circumpolar
boundary current pathway.
Using the method we can follow changes in water properties along pathways of the bound-
ary current, and we also use the method to calculate backward trajectories using the inverse
velocity field to determine the origin of BSW that is involved in the high pressure region at
the St Anna Trough.
For each calculation we start particles flowing through a fixed vertical section. The sections
for the calculations in this chapter are shown in figure 6.1. The number of particles starting
from each grid-box on the section is based upon the strength of the transport such that
strong, important, flows have a larger number of particles than weak flows. The process of
selecting the particles is described in appendix A. Each particle is injected instantaneously,
and therefore initially represents an infinitesimally small part of the transport through the
starting section. As the flow is flux-conserving at each moment in time, and a current is
completely determined by the particles that compose it, the flux associated with each particle
(particle flux) is conserved during the integration.
Although all N particles approximately represent the same transport, there are small
differences in particle fluxes as described in appendix A. Therefore statistics based on the
number of particles are not accurate. In order to visualize the trajectories we use statistics
that are weighted by the particle fluxes. The statistic P(i, j) is defined as the percentage
of the initial flux (FT) that has passed through the horizontal grid-box (i, j) throughout the
whole length of the integration, counting each particle only once for each grid-box
P(i, j) = X p∈I Fp FT ·100%, (6.1)
wherepis an index for each particle,Irepresents the set of particles that have passed through
We also visualize the mean depths and passing times of the particles for each horizontal
grid-box, which are weighted by the particle fluxes
D(i, j) = X p∈I FpDp(i, j) X p∈I Fp , (6.2) T(i, j) = X p∈I FpTp(i, j) X p∈I Fp , (6.3)
whereDp(i, j) andTp(i, j) represent the depth and time respectively at which particleppasses
through horizontal grid-box (i, j).
In the calculations we use one year’s worth of 10-daily instantaneous velocity and property
fields for the time-dependent trajectories. At the end of each year we loop back to the first set
of fields. We choose the starting day as the time of the maximum fluxes through the starting
section.
The instantaneous datasets may contain signs of inertial oscillations, for which averaging
is needed to reduce the net effect (Jayne and Tokmakian 1997). However we only have one
year of data, and to reduce the inertial oscillations by a significant amount would require
averaging over several datasets. This would lead to a severely reduced time resolution, and
we therefore choose not to correct for this effect in this study. We also want the effect of the
random motions associated with time-dependence in the trajectories, which would be severely
reduced by averaging over multiple time samples. Another feature that is not corrected for
is the jump from the last dataset to the first dataset at the end of each year. This can