Preventive Methods for Coastal Protection pp 225-249 | Cite as

# TRACMASS—A Lagrangian Trajectory Model

## Abstract

A detailed description of the Lagrangian trajectory model TRACMASS is presented. The theory behind the original scheme for steady state velocities is derived for rectangular and curvilinear grids with different vertical coordinates for the oceanic and atmospheric circulation models. Two different ways to integrate the trajectories in time in TRACMASS are presented. These different time schemes are compared by simulating inertial oscillations, which show that both schemes are sufficiently accurate not to deviate from the analytical solution.

The TRACMASS are exact solutions to differential equations and can hence be integrated both forward and backward with unique solutions. Two low-order trajectory subgrid parameterizations, which are available in TRACMASS, are explained. They both enable an increase of the Lagrangian dispersion, but are, however, too simple to simulate some of the Lagrangian properties that are desirable. The mass conservation properties of TRACMASS are shown to make it possible to follow the water or air masses both forward and backward in time, which also opens up for all sorts of calculations of water/air mass exchanges as well as Lagrangian stream functions.

## Keywords

General Circulation Model Volume Transport Inertial Oscillation Surface Drifter Curvilinear Grid## Notes

### Acknowledgements

The authors wish to thank Tarmo Soomere and Ewald Quak for constructive comments. This work was originally motivated by the BONUS+ project *BalticWay* that was supported by the funding from the European Community’s Seventh Framework Programme (FP7 2007–2013) under grant agreement No. 217246 made with the joint Baltic Sea research and development programme BONUS+ and by the Swedish Research Council for Environment, Agricultural Sciences and Spatial Planning (Formas, Ref. No. 2008–1900).

## References

- Berloff P, McWilliams J (2002) Material transport in oceanic gyres. Part II: Hierarchy of stochastic models. J Phys Oceanogr 32:797–830 CrossRefGoogle Scholar
- Berloff P, McWilliams J, Bracco A (2002) Material transport in oceanic gyres. Part I: Phenomenology. J Phys Oceanogr 32:764–796 CrossRefGoogle Scholar
- Berrisford P, Kållberg P, Kobayashi S, Dee D, Uppala S, Simmons AJ, Poli P, Sato H (2011) Atmospheric conservation properties in ERA-Interim. Q J R Meteorol Soc 137:1381–1399 CrossRefGoogle Scholar
- Blanke B, Raynaud S (1997) Kinematics of the Pacific Equatorial Undercurrent: a Eulerian and Lagrangian approach from GCM results. J Phys Oceanogr 27:1038–1053 CrossRefGoogle Scholar
- Blanke B, Arhan M, Madec G, Roche S (1999) Warm water paths in the equatorial Atlantic as diagnosed with a general circulation model. J Phys Oceanogr 29:2753–2768 CrossRefGoogle Scholar
- Bracco A, LaCasce J, Provenzale A (2000a) Velocity probability density functions for oceanic floats. J Phys Oceanogr 30:461–474 CrossRefGoogle Scholar
- Bracco A, LaCasce J, Pasquero C, Provenzale A (2000b) The velocity distribution in barotropic turbulence. Phys Fluids 12:2478–2488 CrossRefGoogle Scholar
- Bracco A, Chassignet EP, Garraffo Z, Provenzale A (2003) Lagrangian velocity distributions in a high resolution numerical simulation of the North Atlantic. J Atmos Ocean Technol 8:1212–1220 CrossRefGoogle Scholar
- Butcher JC (2008) Numerical methods for ordinary differential equations, 2nd edn. Wiley, New York zbMATHCrossRefGoogle Scholar
- de Vries P, Döös K (2001) Calculating Lagrangian trajectories using time-dependent velocity fields. J Atmos Ocean Technol 18:1092–1101 CrossRefGoogle Scholar
- Döös K (1995) Inter-ocean exchange of water masses. J Geophys Res—Oceans 100:13,499–13,514 Google Scholar
- Döös K, Coward AC (1997) The Southern Ocean as the major upwelling zone of the North Atlantic Deep Water, pp 3–17. WOCE Newsletter, No 27, July 1997 Google Scholar
- Döös K, Engqvist A (2007) Assessment of water exchange between a discharge region and the open sea—a comparison of different methodological concepts. Estuar Coast Shelf Sci 74:585–597 CrossRefGoogle Scholar
- Döös K, Meier HEM, Döscher R (2004) The Baltic haline conveyor belt or the overturning circulation and mixing in the Baltic. Ambio 33:261–266 Google Scholar
- Döös K, Nycander J, Coward AC (2008) Lagrangian decomposition of the Deacon Cell. J Geophys Res—Oceans 113:C07028 CrossRefGoogle Scholar
- Döös K, Rupolo V, Brodeau L (2011) Dispersion of surface drifters and model-simulated trajectories. Ocean Model 39:301–310 CrossRefGoogle Scholar
- Drijfhout S, de Vries P, Döös K, Coward A (2003) Impact of eddy-induced transport of the Lagrangian structure of the upper branch of the thermohaline circulation. J Phys Oceanogr 33:2141–2155 CrossRefGoogle Scholar
- Engqvist A, Döös K, Andrejev O (2006) Modeling water exchange and contaminant transport through a Baltic coastal region. Ambio 35:435–447 CrossRefGoogle Scholar
- Fabbroni N (2009) Numerical simulations of passive tracers dispersion in the sea. Alma Mater Studiorum—Universita di Bologna, PhD Thesis, 164 pp Google Scholar
- Griffa A (1996) Applications of stochastic particle models to oceanographic problems. In: Adler RJ, Müller P, Rozovoskii BL (eds) Stochastic modelling in physical oceanography. Birkhäuser, Basel, pp 114–140 Google Scholar
- Jönsson B, Lundberg P, Döös K (2004) Baltic sub-basin turnover times examined using the Rossby Centre Ocean model. Ambio 33:2257–2260 Google Scholar
- Kjellsson J, Döös K (2012) Lagrangian decomposition of the Hadley and Ferrel Cells. Geophys Res Lett 39:L15807 CrossRefGoogle Scholar
- Koch-Larrouy A, Madec G, Blanke B, Molcard R (2008) Water mass transformation along the Indonesian throughflow in an OGCM. Ocean Dyn 58:289–309 CrossRefGoogle Scholar
- Levine RC (2005) Changes in shelf waters due to air-sea fluxes and their influence on the Arctic Ocean circulation as simulated in the OCCAM global ocean model. University of Southampton, Faculty of Engineering Science and Mathematics, School of Ocean and Earth Science, PhD Thesis, 225 pp Google Scholar
- Marsh R, Megann AP (2002) Tracing water masses with particle trajectories in an isopycnic-coordinate model of the global ocean. Ocean Model 4:27–53 CrossRefGoogle Scholar
- Mesinger F, Arakawa A (1976) Numerical methods used in atmospheric models. GARP publications series 17, vol I. WMO, Geneva, 64 pp Google Scholar
- Pasquero C, Provenzale A, Babiano A (2001) Parameterization of dispersion in two-dimensional turbulence. J Fluid Mech 439:279–303 zbMATHCrossRefGoogle Scholar
- Rupolo V (2007) Observing turbulence regimes and Lagrangian dispersal properties in the ocean. In: Griffa A, Kirwan AD, Mariano AJ, Özgökmen TM, Rossby T (eds) Lagrangian analysis and prediction of coastal and ocean dynamics (LAPCOD). Cambridge University Press, Cambridge, pp 423–479 Google Scholar
- Simmons AJ, Burridge DM (1981) An energy and angular-momentum conserving vertical finite-difference scheme and hybrid vertical coordinates. Mon Weather Rev 109:758–766 CrossRefGoogle Scholar
- Soomere T, Delpeche N, Viikmäe B, Quak E, Meier HEM, Döös K (2011) Patterns of current-induced transport in the surface layer of the Gulf of Finland. Boreal Environ Res 16(Suppl A):49–63 Google Scholar