The Lagrangian perspective is widely used in the framework of the atmospheric sciences and has proven to add much to the comprehension of physical and dynamical processes in the atmosphere.

In effect the dynamics and thermodynamics of any mechanical systems, including
fluids and among them the atmosphere, are based on physics' laws (Newton's
and thermodynamics principles) applying to systems that do not exchange matter
with their environment (hereafter *"close"* systems). Thus our basic
representation of atmospheric physics is Lagrangian in nature.

Also in higher level analyses, atmosphere scientists are often interested in
identifying and studying
air-masses that may have a significant influence on the circulation. Among
possibilities, the tracking of air-parcels enables the identification of
"coherent ensembles of trajectories" (Wernli and Davies 1997),
defined as ensembles of air-parcels which experience a common physical history.
More generally the notion of ** coherent air-mass** is of particular
interest.

The most usual technique for the Lagrangian description of an
airflow is the calculation of trajectories, using wind fields extracted from
either analysis systems, global circulation models or limited area models. For
instance, trajectories calculated from analyses have proved to usefully
complement Eulerian diagnostic techniques for the study of flow structure in
the troposphere and stratosphere, when long range transport over several days
is considered (*e.g., *Wernli and Davies 1997).

High-resolution numerical modelling over shorter time ranges has appeared as an efficient tool to capture some features of the dynamics of airflows at the meso-scale. When the question of Lagrangian tracking is considered in this framework, some drawbacks of the trajectory technique can be listed in terms of implementation, treatment, and accuracy.

Most atmospheric models are based on the equations of
fluid mechanics in their Eulerian form. A specific numerical model of
trajectories is then needed in addition to the Eulerian model. The wind fields
generated by the Eulerian model are required at every time-step of the
trajectory model. Thus, as explained by Rössler *et al.*
(1992), trajectories can either be computed in parallel to
the Eulerian model run ("on-line" or "Run-Time" method, hereafter
*RT-method*), or post-processed from the model outputs ("off-line" or
"Post-Mortem" method, hereafter *PM-method*). Compared to the RT-method, the
PM-method has the advantage to allow calculations of forward as well as backward
trajectories.

The number of calculated trajectories rarely exceeds a few hundred, well below the number of model grid-cells. The trajectories hence provide a small part of the available Lagrangian information. Moreover, if parcels are arbitrarily chosen, only a few of the trajectories will actually give physical or dynamical informations which would be relevant to the phenomenon under consideration. Backward trajectories are more suitable to select relevant information, but their use is not possible with the RT-method. Focusing on forward trajectories, either the calculation of a large amount of trajectories or a preselection of parcels are needed. The latter is somewhat arbitrary and requires that the flow is already known, at least from an Eulerian point of view, in order to be relevant. In any case, preselecting trajectories appears to be difficult with the RT-method, thus the PM-method is more usually preferred over the RT-method. However, it is less accurate, due to the time-resolution which is much coarser in the PM- than in the RT-case, since it is hardly ever feasible to store the wind-field data at every time step of the Eulerian model.

Trajectory models in general may have inaccuracies from other causes. In a case of idealised cyclogenesis, Walmsley and Mailhot (1983) estimate the error in parcel location due to various numerical causes. They conclude that the most important error arises from the spatial interpolation of the wind between the main-model grid-points.

Trajectories are not the only possible method for the Lagrangian
description of an airflow. For example, any flow-conservative variable
provides at least part of the information about parcel motions, *e.g., *
potential vorticity or potential temperature in dry airflows, and any passive
tracers in general. Most of the time, motions of passive tracers are simulated
with the aim to study transport of pollutants or chemical species.

Schär and Wernli (1993), however, propose to use passive tracer fields to study the dynamics in adiabatic idealised cyclogenesis with Lagrangian diagnoses. They use three passive tracer fields, initialised with uniform zonal, meridional and vertical gradients respectively. Thus, the initial value of the triplet of tracers for a given grid point can be directly related to the Cartesian coordinates of the grid point. They identify cloudy structures from the vertical displacement field. They remark that surface fronts are more visible in horizontal tracers plots than in or vorticity plots. They also deduce parcel trajectories from the tracer fields.

Considering the drawbacks of the trajectory-based techniques at the meso-scale, and that the method by Schär and Wernli can be easily implemented in a numerical model, the latter method has been retained as Lagrangian analysis tool to be developed for the MesoNH system.

Further than trajectory-based analyses, this method provides a *continuous*
Lagrangian description of flows and enables a wide panel of diagnostics from
which coherent air-masses (that would be otherwise coherent
ensembles of trajectories in a trajectory-based analysis, with possibly some
difficulty to identify them) emerge naturally. Illustrations are given in
Gheusi and Stein (2002).

The present document aims at

- presenting in some details the Lagrangian analysis method based on passive tracers inspired by Schär and Wernli and its specific developments for the MesoNH system;
- guiding the user to carry out Lagrangian analyses of MesoNH simulations.

The document is divided into two main chapters.

The scientific basements of the analysis tools are presented in detail in
chapter 1 *Scientific documentation* - as well as some
specific points of implementation in the MesoNH system.

The practical informations on Lagrangian analyses with MesoNH are
gathered in chapter 2 *User's guide* that is
structured along the three main stages of a MesoNH simulation : MODEL-run;
post-processed computation of DIAGnostics; plots with DIAPROG. As some of the
available Lagrangian diagnostics are not of common use, many examples are
proposed which, hopefully, may inspire the reader.

As in chapter 2 it is often referred to notations or notions introduced in chapter 1, an overview of the latter is recommended. We suggest to read at least §1.2.1 and §1.2.3.

The cited references and an index can be found at the very end of the document.