7 TRACMASS—A Lagrangian Trajectory Model
239
In the present study we will repeat one of the Fabbroni (2009) tests for the two
TRACMASS schemes and evaluate them by comparing them with the exact analytical inertial oscillation solution. The test consists of using the analytical solution
of damped inertial oscillations, which are carried away with a mean geostrophic
current so that the equations of motion are
∂u
∂t
− f v = −γ u,
∂v
∂t
+ f u = −γ v + f u g ,
(7.34)
which describe particle circles with a drift to the east due to a geostrophic velocity
u g and with a decreasing oscillation radius depending on the linear friction coefficient γ . The solutions for the velocities are
u = u g e
−γ g t
+ (u 0 − u g )e
−γ t cos f t,
v = −(u 0 − u g )e
−γ t sin f t
(7.35)
and for the particle trajectories
x = x 0 +
u g
γ g
1 − e
−γ g t
+
(u 0 − u g )f
f 2 + γ 2
γ
f
+ e
−γ t
sin f t −
γ
f
cos f t
,
y = y 0 −
(u 0 − u g )f
f 2 + γ 2
1 − e
−γ t
cos f t +
γ
f
sin f t
.
(7.36)
We used the same coefficients as Fabbroni (2009) with u 0 = 0.3 m/s, u g = 0.04 m/s
and a damping time of t d = 1/γ = 2.89 days and t g = 1/γ g = 28.9 days. The latitude was set to be 45 ◦ N. The velocities are read into TRACMASS every hour
(t G = 1 hour) to mimic a GCM that stores the data once an hour. This in contrast
to Fabbroni (2009) who read in the velocities as often as every 3 minutes, which is
unrealistically high to be run off-line with.
TRACMASS was then integrated forward in time using the time-stepping
method with intermediate time steps so the velocities were updated with linear interpolation between two such mimicked ‘GCM’ velocities. The results are shown
in Fig. 7.8. Only the red curve that corresponds to the case with no intermediate
time steps deviates clearly from the true analytical solution. The blue curve calculated with 10 intermediate time steps has a slight difference. The green (for which
1000 intermediate time steps have been used) lies almost exactly under the purple
curve (that reflects the analytical time integration scheme). The small differences
between the results from the truly analytical solution and the analytical time integration scheme and time-stepping scheme with 1000 intermediate time steps are
likely due to that the velocities are only read into TRACMASS every hour on the
model grid and not continuously in both time and space since it is suppose to mimic
the reading of GCM fields, which are stored only every hour.
We do not know why we obtain clearly different and better results using TRACMASS here compared to what (Fabbroni 2009) got with Ariane, since both codes,
239
In the present study we will repeat one of the Fabbroni (2009) tests for the two
TRACMASS schemes and evaluate them by comparing them with the exact analytical inertial oscillation solution. The test consists of using the analytical solution
of damped inertial oscillations, which are carried away with a mean geostrophic
current so that the equations of motion are
∂u
∂t
− f v = −γ u,
∂v
∂t
+ f u = −γ v + f u g ,
(7.34)
which describe particle circles with a drift to the east due to a geostrophic velocity
u g and with a decreasing oscillation radius depending on the linear friction coefficient γ . The solutions for the velocities are
u = u g e
−γ g t
+ (u 0 − u g )e
−γ t cos f t,
v = −(u 0 − u g )e
−γ t sin f t
(7.35)
and for the particle trajectories
x = x 0 +
u g
γ g
1 − e
−γ g t
+
(u 0 − u g )f
f 2 + γ 2
γ
f
+ e
−γ t
sin f t −
γ
f
cos f t
,
y = y 0 −
(u 0 − u g )f
f 2 + γ 2
1 − e
−γ t
cos f t +
γ
f
sin f t
.
(7.36)
We used the same coefficients as Fabbroni (2009) with u 0 = 0.3 m/s, u g = 0.04 m/s
and a damping time of t d = 1/γ = 2.89 days and t g = 1/γ g = 28.9 days. The latitude was set to be 45 ◦ N. The velocities are read into TRACMASS every hour
(t G = 1 hour) to mimic a GCM that stores the data once an hour. This in contrast
to Fabbroni (2009) who read in the velocities as often as every 3 minutes, which is
unrealistically high to be run off-line with.
TRACMASS was then integrated forward in time using the time-stepping
method with intermediate time steps so the velocities were updated with linear interpolation between two such mimicked ‘GCM’ velocities. The results are shown
in Fig. 7.8. Only the red curve that corresponds to the case with no intermediate
time steps deviates clearly from the true analytical solution. The blue curve calculated with 10 intermediate time steps has a slight difference. The green (for which
1000 intermediate time steps have been used) lies almost exactly under the purple
curve (that reflects the analytical time integration scheme). The small differences
between the results from the truly analytical solution and the analytical time integration scheme and time-stepping scheme with 1000 intermediate time steps are
likely due to that the velocities are only read into TRACMASS every hour on the
model grid and not continuously in both time and space since it is suppose to mimic
the reading of GCM fields, which are stored only every hour.
We do not know why we obtain clearly different and better results using TRACMASS here compared to what (Fabbroni 2009) got with Ariane, since both codes,