On the dominance of changes in planetary angular momentum in large scale extra-tropical flows



[1] Considerations of the temporal changes in angular momentum are employed to diagnose air trajectories over large scale distances calculated from the publicly available global data base of NCEP/NCAR. It is shown that outside the Tropics both the total angular momentum and its Lagrangian (material) time-derivative are dominated by the planetary term, regardless of the direction of the transport, whereas the contribution of the non-linear term is always negligible. In the Tropics the planetary term and the acceleration terms alternately dominated the angular momentum evolution. These Lagrangian results are reinforced by Eulerian calculations of instantaneous maps of the relative contributions of the various terms in the evolution equation of the angular momentum. The simple equation of angular momentum evolution (compared to the more complex zonal momentum equation) is accurately satisfied by NCEP/NCAR re-analyzed fields even over Africa, where the data for the reanalysis is scant. The only exceptions to the general dominance of the planetary term over the angular momentum evolution in the extra-Tropics are along trough-/ridge-lines (where the flow changes direction sharply) and at the entrance/exit regions of straight jets, where the flow accelerates/decelerates.

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

equation image

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 = Mp + Mr where Mp = Ω R2 cos2ϕ is the planetary angular momentum and Mr = R cos ϕ · u is the relative angular momentum. Since ΩR = 465 m/s it is clear that for realistic velocities the planetary angular momentum dominates since MpMr.

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

equation image

[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), equation imagev sin ϕ (2Ω + equation image) = −equation imageequation image [Holton, 1992, equation 2.19], in equation (2) yields:

equation image

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 2ΩRcosϕ (>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:

equation image

so that when the nonlinear term is ignored one obtains

equation image

[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., fva = equation image. 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.

2. Data and Methods

[9] The atmospheric data used in this study were taken from the NCEP/NCAR Reanalysis database [Kalnay et al., 1996; Kistler et al., 2001], with 2.5° × 2.5° and 6-h resolution, at 12 pressure levels in the 0°–50°N, 20°W–50°E domain, which covers most of Africa. This is the domain where Rubin et al. [2007] identified examples of large scale transport with strong zonal forces after 1988. Following Ziv [2001] and Knippertz and Martin [2005] isentropic coordinates were incorporated in order to construct simulations of air trajectories.

[10] Trajectories are constructed using the two-dimensional NCEP/NCAR wind field data in isentropic coordinates. The calculation of the atmospheric fields and the air parcel position, P(t + Δt) is done in two steps, similar to HYSPLIT model method [Draxler and Hess, 1997]. In the first step, an intermediate estimate of the parcel's position P′(t + Δt) is obtained by a first-order approximation, using the wind velocity V(P, t) and position P(t) at the starting time:

equation image

In the second step, a second-order approximation is achieved by using the average of the wind velocity at the initial position V(P, t) and that at the intermediate position V(P′, t + Δt):

equation image

The physical quantities used throughout this study were derived by linearly interpolating the gridded atmospheric fields. Forward differencing was used for evaluating the spatial derivatives in Eulerian maps and for evaluating the changes of angular momentum and its components along the trajectory i.e., the Lagrangian time-derivative. The application of central differencing in the Lagrangian time-derivative would have eliminated a time step and spot comparisons between the two calculations showed no significant difference. The local time derivatives and equation image(where the use of isentropic coordinates mandates that this pressure gradient term is given by the gradient of the corresponding Montgomery streamfunction) along the trajectory were both obtained by central differencing, which yielded intermediate values between forward and backward differencing.

[11] Isentropic coordinates are used for trajectory production in order to track the actual motion of air parcels, since in the absence of diabatic processes an air parcel remains on the same isentropic surface. Our trajectory calculation was done using our own MATLAB written program that followed the 3D HYSPLIT model algorithm explained earlier. Since the angular momentum conservation law applies to Lagrangian formalism, isentropic coordinates are also used for the ‘snapshot’ presentation as it represents actual air motion closer than isobaric charts. This is particularly significant for finite-differencing in the north-south direction where the slopes of the two surfaces are opposite to one another. The isentropic analysis was done on the 315°K level, chosen since it lies within the NCEP/NCAR standard pressure levels everywhere in our domain.

3. Results

[12] The left column of Figure 1 shows five trajectories that typify different flow patterns and which were analyzed for their angular momentum evolution. The contours of the Montgomery streamfunction field 36 hours after the initial time are also indicated on these plots. All five trajectories were calculated with NCEP/NCAR reanalyzed fields on the 315°K isentropic surface over Africa as described in section 2. The right column shows the corresponding temporal changes of the total angular momentum (M, thick solid line), the planetary term (Mp, thin solid line) and the relative term (Mr, dashed line) along the trajectory.

Figure 1.

(left) Forward isentropic air trajectories (thick line) along the 315°K surface starting at different times. Starting point is marked by an asterisk and 6-hour time increments are marked by plus signs. Contours (thin lines) represent Montgomery streamfunction (1000 m2s−2) 36 hours after the starting time of the trajectory. Starting time is (a) 12 UTC 31 December 1993, (b) 00 UTC 30 December 1993, (c) 18 UTC 19 January 2003, (d) 00 UTC 30 December 1993, and (e) 18 UTC 31 December 1993. (right) Corresponding Lagrangian time-derivatives of M (thick solid line), Mp (thin solid line), and Mr (dashed line) along these trajectories.

[13] The first row in Figure 1 (a and a′) shows the case of purely meridional movement from north to south. The second row (panels b and b′) represents a meridional flow from south to north ahead of a Tropical Plume that extends from north Africa to the Middle East. The third row (panels c and c′) represents a purely zonal flow associated with a Subtropical Jet while the fourth row (panels d and d′) shows the case of a highly curved flow encircling an upper level trough. The bottom row (panels e and e′) shows the case of an easterly flow on the equator. Except for the equatorial case (panels e and e′) and during the first time increment in the case of a pure zonal flow (panels c and c′) the temporal changes in the total angular momentum followed closely those of the planetary term. In equatorial trajectories no consistent behavior was observed: in the case shown in panels e and e′ of Figure 1 the changes in Mr dominated the changes in M while in other cases examined (not shown) the changes in M were dominated either by the changes in Mp or by similar contributions of Mp and Mr.

[14] The results of the Lagrangian calculations along the trajectories shown in Figure 1 are also evident in Eulerian calculations of the time derivatives of the various terms in equation (2). Figure 2 shows various related quantities for 18 UTC 31 December 1993, which is included in the examples shown in Figures 1a, 1b, 1d, and 1e. These are the angular momentum (Figure 2a, black contours), its material time-derivative (Figure 2a, color shading), the negative of zonal gradient of the associated Montgomery streamfunction (Figure 2b). Also shown in Figure 2 are the material time-derivative of the Planetary term (i.e., the 1st term on RHS of equation 2, shown in Figure 2c) and the ratio of the absolute value of the acceleration term (i.e., the third term in equation 2) to the sum of absolute values of the three terms on RHS of equation (2), which is shown in Figure 2d.

Figure 2.

Various quantities plotted on 315°K isentropic surface at 18 UTC 31 December 1993. (a) Angular momentum (109 m2s−1, black contour) and its material time-derivative (1000 m2s−2, color shaded contour). (b) Negative of zonal gradient of the associated Montgomery streamfunction (1000 m2s−2). (c) Material time-derivative of the planetary term, i.e., the 1st term on RHS of equation (2). (d) The ratio of the absolute value of the acceleration term (see equation (2)) to the sum of absolute values of the three terms on the RHS of equation (2).

[15] The contours of total angular momentum (Figure 2a) clearly demonstrate that its spatial variation is mainly meridional and even the presence of a pronounced trough (see the Montgomery streamfunction contours in Figure 1b, which are 6 hours before this snapshot) does not significantly alter the primary meridional variation. The material temporal changes in total angular momentum (Figure 2a, shading) and the zonal gradient of the Montgomery streamfunction (Figure 2b) show a high degree of similarity between their spatial patterns. This confirms the applicability of NCEP/NCAR fields to the balance of equation (3). Similar relation was observed also in an examination of these quantities along the trajectories of Figure 1 (not shown). The main two features in the zonal gradient field of Montgomery streamfunction are two dipoles (couple of centers with opposite signs), one over southwestern Europe (upper left corner of Figure 2b) and another over the eastern part of North Africa and the Mediterranean. The former dipole is a manifestation of a mid-latitude upper level trough while the latter reflects a subtropical upper level trough associated with a Tropical Plume [Rubin et al., 2007]. The material temporal change in the planetary term, Mp (Figure 2c), also shows similar patterns to those of the temporal changes of the total angular momentum, Figure 2a. This agreement between the spatial pattern of the temporal changes in M and in Mp reconfirms our Lagrangian findings regarding the dominance of the temporal changes of Mp over those of M presented in the right column of Figure 1 as discussed earlier.

[16] The fact that these findings are not obvious is bolstered in the colored shading of Figure 2d, where the ratio of the absolute value of the acceleration term (see equation 2) to the sum of absolute values of the three terms in equation (2) is shown. There are several “islands” (located mostly, but not exclusively, in the lower latitudes, equatorward of 15°N) where the temporal changes in M are dominated (contours exceeding 50%) by those of the acceleration term.

[17] The non-linear term in equation (2) is the smallest of the three terms and its contribution to the temporal changes in M is two-orders of magnitude smaller than that of the other two terms so it can be always ignored in all practical applications.

4. Summary and Discussion

[18] Eulerian and Lagrangian analyses of temporal material changes of angular momentum over North Africa and the Mediterranean Basin show that to a high degree of accuracy it equals the zonal pressure gradient force. This demonstrates the applicability of NCEP/NCAR reanalysis wind data over eastern North Africa, where radiosondes observations are scarce. The results presented in this study show that although not used explicitly in the reanalysis of NCEP/NCAR fields, the angular momentum is accurately estimated by the wind field. The reliability of the NCEP/NCAR wind data and the dominance of Mp beyond its expected realm of applicability, suggest the use of angular momentum as a diagnostic tool in future large scale and synoptic analyses in extra-tropical latitudes.

[19] As was explained in the Introduction in geostrophic meridional flows the dominating term in the change in M (equation 2) should be the planetary term, Mp. Since the magnitude of Mp depends on latitude (ϕ) and on v one can expect that it will not dominate the changes in M near the equator and when the wind has a strong ageostrophic v-component (i.e., large zonal acceleration). In cases when the planetary term does not dominate the changes in M, the other contributor to the changes in M (besides Mp) is the zonal acceleration term.

[20] Our calculations using real data demonstrate that the changes in Mp are 1 – 2 orders of magnitude larger than those in the other two terms in many cases, including those where the velocity is not meridional and geostrophic. This dominance of the changes in Mp prevails even in the extreme case shown in Figure 2c, where the meridional wind component (be it geostrophic or not) is negligible compared to the total wind vector. Yet, the meridional velocity component is crucial for determining the angular momentum changes along the flow (via Mp).

[21] The cases where Mp does not dominate the changes in M are the zonally accelerating flow (the 1st time step in the trajectory of Figure 1c) and the equatorial flow (Figure 1e, e′). In the first case the changes in Mp and those in Mr cancelled one another while in equatorial flows neither of the two terms dominated the changes in M consistently and there were cases where Mr dominated (such as that shown in Figure 1e) and cases in which Mp dominated (not shown). Figure 2d also reaffirms the finding that changes in Mp are not dominant near the equator. As explained above, the acceleration term is the only alternate contributor (other than the planetary one) to the changes in M. Thus, this term is expected to dominate the changes in M near trough- or ridge-lines (when the flow changes its direction sharply) or at the entrance/exit of straight jets where the flow accelerates/decelerates. Indeed, our results confirm these expectations and Figure 2d provides such an example near a trough-line (located in the transition between the positive and negative centers of ∂Φ/∂λ in Figure 2b along 15°E between 20° and 30°N latitudes) which coincides with the strip of high ratio in Figure 2d signaling that the acceleration term contributes significantly to the changes in M.

[22] It should be noted that although the velocity component that determines M is u, the temporal changes in M are determined by either vg or vaDu/Dt.


[23] This work was supported by the Israel Science Foundation grant 579/05 to HU and by the Research Authority of the Open University of Israel.