7 TRACMASS—A Lagrangian Trajectory Model
227
Blanke and Raynaud (1997). The present code is written in Fortran 95 and can be
driven by velocity fields from most GCMs based on finite differences. The TRACMASS code is continuously upgraded and adapted. The code can be downloaded
from http://tracmass.org/. The user must be familiar, in order to be able to use the
TRACMASS code, with (1) the equations of motion for the ocean-atmosphere circulation (described in Chaps. 2, 4 and 6), (2) the finite differences of these equations
(Chap. 3), (3) the TRACMASS theory (the introduction to which is presented in this
chapter), (4) Unix and (5) Fortran.
The Lagrangian trajectory approach has many similarities with the Eulerian
tracer approach but at the same time many differences. The two approaches are
often confused due to their similarities. They are both advected passively by the
velocity fields of the GCM, which makes it possible to trace water/air masses or
substances such as pollutants as they are carried with the ocean currents or winds.
The tracer equation generally needs to be integrated ‘on-line’ with the GCM while
the Lagrangian trajectories can be both ‘on-line’ and ‘off-line’. The ‘off-line’ calculation of Lagrangian trajectories is by far the most rapid way since one only
needs to read the already simulated velocity fields in order to calculate the trajectories.
The tracer equation includes explicitly a diffusion term, which represents a parameterization of the unresolved subgrid scales. There is also a numerical reason
to include this since GCMs generally need some diffusion and viscosity to remain
numerically stable in order to dissipate energy or to eliminate numerical noise due
to the truncation errors in the numerical schemes. The passive tracers also have
a numerical diffusion due to the finite difference approximation error, which by
itself often would be enough as diffusion. The tracer approach is therefore often
too diffusive but has been improved with better numerical advection schemes during the last decade. The Lagrangian trajectories are passively advected with the
currents or winds and the subgrid parameterization is included in the sense that
the GCM has been integrated with viscosity and diffusion. An extra diffusion can,
however, if desired, be added to the trajectories. Another advantage of the trajectories is that it is possible to follow particles from their release points to the end
both forward and backwards, which is impossible with passive tracers that cannot be integrated backward in time due to the numerical and parameterized diffusion.
The present chapter will describe the basic theory for the TRACMASS trajectory
calculations and is organized as follows. In Sect. 7.2 we present the basic equations
for a rectangular grid, which is then extended in Sect. 7.3 to the more general case
with non-rectangular grids and for atmospheric GCMs in Sect. 7.4. The TRACMASS analytical time dependent scheme based on de Vries and Döös (2001) is
presented in Sect. 7.5 followed by the presentation of two simple sub-grid parameterizations in Sect. 7.6 and how the mass conservation in TRACMASS enables
analysis of the water/air mass transports in Sect. 7.7. In Sect. 7.8, we summarize
and discuss the TRACMASS approach and its possible improvements in the future.
227
Blanke and Raynaud (1997). The present code is written in Fortran 95 and can be
driven by velocity fields from most GCMs based on finite differences. The TRACMASS code is continuously upgraded and adapted. The code can be downloaded
from http://tracmass.org/. The user must be familiar, in order to be able to use the
TRACMASS code, with (1) the equations of motion for the ocean-atmosphere circulation (described in Chaps. 2, 4 and 6), (2) the finite differences of these equations
(Chap. 3), (3) the TRACMASS theory (the introduction to which is presented in this
chapter), (4) Unix and (5) Fortran.
The Lagrangian trajectory approach has many similarities with the Eulerian
tracer approach but at the same time many differences. The two approaches are
often confused due to their similarities. They are both advected passively by the
velocity fields of the GCM, which makes it possible to trace water/air masses or
substances such as pollutants as they are carried with the ocean currents or winds.
The tracer equation generally needs to be integrated ‘on-line’ with the GCM while
the Lagrangian trajectories can be both ‘on-line’ and ‘off-line’. The ‘off-line’ calculation of Lagrangian trajectories is by far the most rapid way since one only
needs to read the already simulated velocity fields in order to calculate the trajectories.
The tracer equation includes explicitly a diffusion term, which represents a parameterization of the unresolved subgrid scales. There is also a numerical reason
to include this since GCMs generally need some diffusion and viscosity to remain
numerically stable in order to dissipate energy or to eliminate numerical noise due
to the truncation errors in the numerical schemes. The passive tracers also have
a numerical diffusion due to the finite difference approximation error, which by
itself often would be enough as diffusion. The tracer approach is therefore often
too diffusive but has been improved with better numerical advection schemes during the last decade. The Lagrangian trajectories are passively advected with the
currents or winds and the subgrid parameterization is included in the sense that
the GCM has been integrated with viscosity and diffusion. An extra diffusion can,
however, if desired, be added to the trajectories. Another advantage of the trajectories is that it is possible to follow particles from their release points to the end
both forward and backwards, which is impossible with passive tracers that cannot be integrated backward in time due to the numerical and parameterized diffusion.
The present chapter will describe the basic theory for the TRACMASS trajectory
calculations and is organized as follows. In Sect. 7.2 we present the basic equations
for a rectangular grid, which is then extended in Sect. 7.3 to the more general case
with non-rectangular grids and for atmospheric GCMs in Sect. 7.4. The TRACMASS analytical time dependent scheme based on de Vries and Döös (2001) is
presented in Sect. 7.5 followed by the presentation of two simple sub-grid parameterizations in Sect. 7.6 and how the mass conservation in TRACMASS enables
analysis of the water/air mass transports in Sect. 7.7. In Sect. 7.8, we summarize
and discuss the TRACMASS approach and its possible improvements in the future.