## 1. Introduction

[2] Reconstruction of air trajectories is commonly used for tracing thermodynamic characteristics or Potential Vorticity and their temporal evolution. In the present work we reconstruct air trajectories for tracing the evolution of angular momentum of air parcels (unit volumes) that undergo large displacements over Africa. The Lagrangian analysis is supplemented by Eulerian analysis of the various terms that contribute to the changes in angular momentum. Both analyses employ the publicly available NCEP/NCAR re-analyzed fields.

[3] The vector of angular momentum per unit mass of air on the rotating Earth is **r** × **U**, where **U** is the total velocity measured in an inertial frame of reference and **r** is the position vector in this reference frame. The angular momentum component parallel to the axis of rotation (regarded hereafter as the angular momentum, *M*) does not vanish over long times and is expressed as

where *R* is Earth radius, Ω is the angular frequency of the rotation of Earth, *ϕ* is latitude and *u* is the eastward component of the velocity of an air parcel relative to Earth's surface. Angular momentum was successfully applied as a diagnostic variable in the analysis of equatorial crossing by Monsoon systems [*Dvorkin and Paldor*, 1999] and for estimating the torques exerted by mountain ridges on the moving air [*Egger and Hoinka*, 2005].

[4] Using the designations of *Straub* [1993] the angular momentum is divided into two components: *M* = *M*_{p} + *M*_{r} where *M*_{p} = Ω *R*^{2} cos^{2}*ϕ* is the planetary angular momentum and *M*_{r} = *R* cos *ϕ* · *u* is the relative angular momentum. Since Ω*R* = 465 m/s it is clear that for realistic velocities the planetary angular momentum dominates since *M*_{p} ≫ *M*_{r}.

[5] The material time-derivative in Eulerian formulation (i.e. the counterpart of the ordinary time derivative in Lagrangian form), , is given by = + + , and its application to *M* defined in equation (1) yields:

[6] The designation of each of the terms is self-explanatory and is used below. Substituting the x-momentum equation (while neglecting both the vertical velocity and frictional forces), − *v* sin *ϕ* (2Ω + ) = − [*Holton*, 1992, equation 2.19], in equation (2) yields:

where Φ is the geopotential height in isobaric coordinates (and Montgomery streamfunction in isentropic coordinates, see e.g., *Holton* [1992]). Thus, in the absence of zonal pressure gradient forces, *M* is conserved along a trajectory, which is the usual application of equation (3) [*Paldor and Sigalov*, 2001; *Rom-Kedar et al.*, 1997].

[7] In realistic circumstances the contribution of the non-linear term (i.e., the second term in equation 2) is always negligible since the zonal velocity, *u*, is much smaller than 2Ω*R*cos*ϕ* (>700 ms^{−1} for *ϕ* ≤ 40°). It is clear from equations (2) and (3) that in meridional geostrophic flows the planetary term is the primary contributor to the temporal changes in angular momentum, since geostrophy implies:

so that when the nonlinear term is ignored one obtains

[8] Equations (2) and (5) along with the fact (explained above) that the nonlinear term is small imply that the acceleration term determines the ageostrophic component of the meridional wind i.e., *fv*_{a} = . In contrast, in zonal flows and in cases when the meridional flow is not geostrophic it is not evident from these equations which of the terms on the Right Hand Side (RHS) of equation (2) dominates the temporal changes in angular momentum. In the present study we examine the angular momentum evolution in various cases including strong zonal pressure gradient force (when the geostrophic velocity is primarily meridional) and cases of meridional non-geostrophic flows, zonal flows and equatorial flows. Our results show that the planetary term dominates the change of *M* in nearly all cases except for flows on the equator.