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K. Döös et al.
flow. This can be illustrated by a cyclist, who passes an immobile traffic jam. In this
case the static car driver sees the moving cyclist from an Eulerian perspective, while
the moving cyclist observes the static traffic jam from a Lagrangian perspective. The
zigzagging path of the cyclist between the cars constitutes a Lagrangian trajectory.
Most analytical and numerical models in fluid dynamics are made in the Eulerian
framework, since it is then straightforward to describe the motion as a function of
position and time. This is why in nearly all ocean general circulation models the
equations of motion are discretized with finite differences on a fixed grid so that the
motion of the water and its tracers such as salinity and temperature are described
from the Eulerian perspective with different values in each grid box, even if the
vertical discretization often has a time dependent component related to the motion
of the fluid. Lagrangian trajectories are, however, still possible to calculate from the
model simulated Eulerian velocity fields on the model grid.
The present chapter will present the TRACMASS Lagrangian trajectory model,
which uses the Eulerian velocity fields, which have been simulated by ocean or
atmosphere general circulation models (GCM). The trajectories are calculated offline, i.e., after the GCM has been integrated and the velocity fields have been stored.
This makes it possible to calculate many more trajectories than would be possible
on-line, i.e., simultaneously with the GCM run. TRACMASS has been applied to
many different general circulation models, both for the ocean and for the atmosphere.
The original feature of the method is that it solves the trajectory path through
each grid cell with an analytical solution of a differential equation which depends
on the velocities on the walls of the grid box. The scheme was originally developed in Döös (1995), Blanke and Raynaud (1997) for stationary velocity fields and
hereafter further developed in de Vries and Döös (2001) for time-dependent fields
by solving a linear interpolation of the velocity field both in time and in space over
each grid box. This is in contrast to the time schemes such as simple Euler forward or
more advanced fourth order Runge–Kutta methods (Butcher 2008; Fabbroni 2009)
where the trajectories are integrated forward in time with as short time steps as possible.
A consequence of solving the trajectory paths analytically over a certain time is
that the solutions are unique and can be integrated forward in time and then backward in time and arriving exactly at the same position, which is not possible with
the other trajectory methods. This makes it possible to trace origins of water or air
masses as long as the subgrid parameterization is not activated.
The TRACMASS code has been further developed over the years and used in
many studies of the global ocean (Döös and Coward 1997; Drijfhout et al. 2003;
Döös et al. 2008) and regional ones for the Mediterranean and Baltic Seas (Döös
et al. 2004; Jönsson et al. 2004; Engqvist et al. 2006; Soomere et al. 2011) as well
as the large scale atmospheric circulation (Kjellsson and Döös 2012).
The code was originally written in Fortran 77 for the FRAM ocean model at
the Institute of Oceanographic Sciences, Deacon Laboratory (IOSDL) in Wormley,
UK in the early 1990s. The name TRACMASS comes from the EU project with
the same name, where it served together with the similar trajectory code Ariane by
K. Döös et al.
flow. This can be illustrated by a cyclist, who passes an immobile traffic jam. In this
case the static car driver sees the moving cyclist from an Eulerian perspective, while
the moving cyclist observes the static traffic jam from a Lagrangian perspective. The
zigzagging path of the cyclist between the cars constitutes a Lagrangian trajectory.
Most analytical and numerical models in fluid dynamics are made in the Eulerian
framework, since it is then straightforward to describe the motion as a function of
position and time. This is why in nearly all ocean general circulation models the
equations of motion are discretized with finite differences on a fixed grid so that the
motion of the water and its tracers such as salinity and temperature are described
from the Eulerian perspective with different values in each grid box, even if the
vertical discretization often has a time dependent component related to the motion
of the fluid. Lagrangian trajectories are, however, still possible to calculate from the
model simulated Eulerian velocity fields on the model grid.
The present chapter will present the TRACMASS Lagrangian trajectory model,
which uses the Eulerian velocity fields, which have been simulated by ocean or
atmosphere general circulation models (GCM). The trajectories are calculated offline, i.e., after the GCM has been integrated and the velocity fields have been stored.
This makes it possible to calculate many more trajectories than would be possible
on-line, i.e., simultaneously with the GCM run. TRACMASS has been applied to
many different general circulation models, both for the ocean and for the atmosphere.
The original feature of the method is that it solves the trajectory path through
each grid cell with an analytical solution of a differential equation which depends
on the velocities on the walls of the grid box. The scheme was originally developed in Döös (1995), Blanke and Raynaud (1997) for stationary velocity fields and
hereafter further developed in de Vries and Döös (2001) for time-dependent fields
by solving a linear interpolation of the velocity field both in time and in space over
each grid box. This is in contrast to the time schemes such as simple Euler forward or
more advanced fourth order Runge–Kutta methods (Butcher 2008; Fabbroni 2009)
where the trajectories are integrated forward in time with as short time steps as possible.
A consequence of solving the trajectory paths analytically over a certain time is
that the solutions are unique and can be integrated forward in time and then backward in time and arriving exactly at the same position, which is not possible with
the other trajectory methods. This makes it possible to trace origins of water or air
masses as long as the subgrid parameterization is not activated.
The TRACMASS code has been further developed over the years and used in
many studies of the global ocean (Döös and Coward 1997; Drijfhout et al. 2003;
Döös et al. 2008) and regional ones for the Mediterranean and Baltic Seas (Döös
et al. 2004; Jönsson et al. 2004; Engqvist et al. 2006; Soomere et al. 2011) as well
as the large scale atmospheric circulation (Kjellsson and Döös 2012).
The code was originally written in Fortran 77 for the FRAM ocean model at
the Institute of Oceanographic Sciences, Deacon Laboratory (IOSDL) in Wormley,
UK in the early 1990s. The name TRACMASS comes from the EU project with
the same name, where it served together with the similar trajectory code Ariane by