Numerical Modelling of Jet-Streak Dynamics

Philip Cunningham and Daniel Keyser

State University of New York at Albany
Albany, New York, USA

1. INTRODUCTION

Jet streaks, defined as localised wind speed maxima situated along the axis of a jet stream at the level of maximum wind (Palmén and Newton 1969, p. 199, pp. 206-212), are ubiquitous in extratropical flow regimes and, because of their importance as upper-level precursors to cyclogenesis and their association with severe weather, they have received a significant amount of attention from the synoptic community (e.g., Carlson 1991, pp. 404-419; Bluestein 1993, pp. 394-407).

Previous investigations of jet streaks typically have focused upon the transverse vertical circulations associated with these disturbances, and it is apparent that the kinematic fields accompanying upper-tropospheric jet streaks are well understood for a wide variety of flow configurations (e.g., Keyser and Shapiro 1986; Moore and VanKnowe 1992). Nevertheless, little attention has been devoted to the fundamental issues of jet-streak dynamics, either through analytical or numerical studies. Carlson (1991, p. 409) and Bluestein (1993, p. 395) link jet-streak motion and formation, respectively, directly to vertical circulations, and conceptual models describing characteristic patterns of vertical motion in relation to jet streaks are ingrained in the synoptic literature. However, previous diagnostic (e.g., Krishnamurti 1968; Keyser et al. 1989, 1992; Loughe et al. 1995) and numerical modelling (Moore and VanKnowe 1992) studies suggest that the cross-contour ageostrophic flow in jet-streak entrance and exit regions is predominantly rotational. This finding offers the possibility that nondivergent barotropic dynamics may be of primary importance for jet streaks, and that divergent circulations and vertical motion play only a secondary, restoring role.

A common feature of jet streaks [illustrated schematically in Fig. 3c of Uccellini and Kocin (1987)] is the positive-negative couplet of relative vorticity associated with the wind speed maximum. Furthermore, Bluestein (1993, p. 237) has proposed that a jet streak may be thought of as a positive-negative couplet of perturbation potential vorticity (PV) on an isentropic surface. In support of this representation, Fig. 1 shows fields associated with a jet streak that was located over the North Central United States at 1200 UTC 3 November 1995, obtained from initialised analyses of the Eta model of the National Centers for Environmental Prediction (NCEP). The jet streak clearly is accompanied by a couplet, or dipole, of relative vorticity, ζ, on the 300 hPa surface (Fig. 1a) that is asymmetric in the sense that its cyclonic member is significantly larger in magnitude than its anticyclonic member. Moreover, a large part of the total ageostrophic wind (Vag, Fig. 1a) is provided by the rotational component (Vagr, Fig. 1b), which is significantly larger in magnitude than the divergent component (Vagd, not shown). The streamfunction associated with the rotational ageostrophic wind (ψagr, Fig. 1b) displays a four-cell pattern that is cyclonic in the entrance and exit regions and anticyclonic on the flanks of the jet streak.

Figure 1. Initialised analyses from NCEP's Eta model, valid at 1200 UTC 3 November 1995, of the following: (a) 300 hPa Vag and ζ (contour interval 3 x 10-5 s-1, negative values dashed); (b) 300 hPa Vagr and ψagr (contour interval 16 x 105 m2 s-1, negative values dashed). In both panels total wind speeds greater than 50 m s-1 are shaded as indicated, and arrow length scale is given at bottom left.

The foregoing discussion raises issues that need to be addressed regarding the dynamics of jet streaks, such as their representation, motion, and life cycles (i.e., origin and evolution) in relation to the large-scale flow environment in which they invariably are embedded. Consideration of these issues prompts the question of how these features may be examined using a body of idealised conceptual and numerical models. This presentation will address a number of these issues: in particular, whether there exist coherent dynamical structures that may be used to represent jet streaks and account for their observed signatures, such as PV distributions and ageostrophic circulations. We also shall examine whether these structures are able to explain jet-streak motion under idealised conditions. Of considerable interest is whether a nondivergent barotropic framework is sufficient to explain the fundamental dynamics of jet streaks, or if divergent vertical circulations are indeed required for a realistic representation of their structure, motion, and evolution. These issues will be examined using idealised f- and β-plane dynamical models (nondivergent barotropic and shallow-water) solved both analytically and numerically. In particular, we shall use numerical channel models to examine the structure, motion, and evolution of arbitrarily imposed initial conditions that are representative of jet streaks and of their synoptic-scale environments. Finally, in an attempt to identify coherent structures that may help explain jet-streak dynamics, we shall examine some analytical solutions that may be applied to these models.

2. JET-STREAK REPRESENTATION

A major objective of this study is to determine the simplest possible representation of jet streaks that still retains the essential dynamical signatures. In this regard, it is possible that the dipole of relative vorticity often associated with jet streaks may provide some insight into their dynamics.

2.1 Numerical Representation

As an initial investigation into the possibility of using dipolar vortices to represent jet streaks, we shall examine the structure, motion, and evolution of a pair of counter-rotating elliptical vortices of similar strength and size. This feature will be called a ``dipolar elliptical vortex'' (DEV). It is stressed that the DEV representation is somewhat arbitrary and does not constitute a steady solution to the governing equations.

A DEV initialised in the shallow-water model is shown in Fig. 2 for t = 0 h. Free parameters in the DEV representation are chosen to replicate length scales and relative vorticity magnitudes characteristic of jet streaks. There is no background flow and f-plane geometry is adopted for simplicity. It is evident that the dipole of relative vorticity (Fig. 2a) is associated with a maximum in fluid speed (Fig. 2b; henceforth referred to as a ``jet''). Although the divergence (Fig. 2c) is configured in the four-cell pattern commonly associated with straight jet streaks, it is at least an order of magnitude smaller than the ageostrophic relative vorticity (Fig. 2d). The latter displays a four-cell pattern, similar to that seen in Fig. 1b, that is cyclonic in the entrance and exit regions of the jet and anticyclonic on its flanks. The dominance of the rotational ageostrophic wind suggests that the structure and subsequent motion and evolution of the DEV may be well-described by nondivergent dynamics. Indeed the time evolution of the DEV in the nondivergent barotropic model (not shown) is extremely similar to that in the shallow-water model, and the divergence in the latter appears to be of secondary importance throughout this evolution.

Figure 2. Charts at t = 0 h for the DEV in the shallow-water model of: (a) ζ (contour interval 1 x 10-5 s-1, negative values dashed), (b) Vag (vectors plotted every 10 grid points, vector length scale in m s-1 shown at bottom right) and |V| (contour interval 2 m s-1), (c) ∇ ⋅ V (contour interval 4 x 10-8 s-1, negative values dashed), and (d) ζag (contour interval 2 x 10-6 s-1, negative values dashed).

Examination of the DEV in both models shows its structure to be qualitatively similar to observations. However, evolution of the DEV shows this feature to become significantly less elliptic with time, and after a period of transience its structure bears a strong resemblance to that found in analytical solutions of dipolar vortices. A range of numerical experiments with differing parameters in the DEV representation (not shown) indicates that an arbitrary DEV on an f-plane tends to evolve towards an analytical dipolar vortex structure [the numerical simulations of Couder and Basdevant (1986) also support this apparent tendency]. Hence it is suggested that analytical solutions of dipolar vortices may provide a prototypical representation of jet streaks.

2.2 Analytical Representation

Analytical solutions of steadily translating dipolar vortex structures may be found for a variety of dynamical frameworks; however, for simplicity, we shall choose the solution due to Lamb (1932, Art. 165; henceforth referred to as the ``Lamb dipole'') that is valid in the f-plane nondivergent barotropic model. Properties of this solution are illustrated in Fig. 3. Free parameters again are chosen to replicate length scales and relative vorticity magnitudes characteristic of jet streaks; for the Lamb dipole these parameters are the translation speed, chosen to be 4.5 m s-1 towards the east, and the dipole radius, chosen to be 750 km, in which case the maximum relative vorticity is approximately 6.6 x 10-5 s-1. It is apparent that the dipole in relative vorticity (Fig. 3a) is associated with a jet (Fig. 3b) and that the ageostrophic circulation (Figs. 3c,d), which is entirely rotational, is of the four-cell pattern documented previously.

Figure 3. The Lamb dipole: (a) ζ, (b) |V|, (c) Vag, (d) ζag. Contour intervals and vector representation are as in Fig. 2.

This analytical approach may be generalised to beta plane geometry by using the modon solution due to Larichev and Reznik (1976), valid in the nondivergent barotropic model. Similar extensions are possible to the shallow-water model by using the modon solution to the equivalent barotropic [shallow-water quasi-geostrophic (QG)] system due to Flierl et al. (1980). This latter solution is not steady in the shallow-water model; however, after dynamic initialisation an approximate solution exists in which advection by the ageostrophic velocity is important, as is the case for the Lamb dipole. In addition to a rotational ageostrophic flow that essentially is identical to that of the Lamb dipole, this approximate solution displays a four-cell pattern of divergence.

3. JET-STREAK MOTION AND EVOLUTION

To examine the dynamics of jet-streak motion and evolution, we employ piecewise PV inversion, in which the vortex dipole under consideration is divided spatially into its cyclonic and anticyclonic members. This inversion allows quantification of the contributions of these respective members to the motion and evolution of the dipole as a whole, and evaluation of the extent to which they advect and deform themselves and their opposing dipole member.

For the Lamb dipole, the effect of each component vortex is predominantly to advect its dipole partner to the east. There also is a small westward motion due to self-advection of the vortices. Note that since motion of a vortex in the absence of β is given by the velocity vector at its centre, the speed of motion of the dipole as a whole is likely to be significantly slower than the speed in the jet core, in accordance with observations of jet streaks (e.g., Palmén and Newton 1969, pp. 206-207).

4. CONCLUSIONS AND FURTHER WORK

It is suggested that vortex dipoles provide a reasonably realistic representation of jet-streak structure and motion, and in particular the Lamb dipole appears to be the simplest feature than contains the salient dynamical characteristics observed in many jet streaks (e.g., Fig. 1). However, the vortex dipoles described above are unable to explain the asymmetry in the relative vorticity field seen in Fig. 1a, and, moreover, they do not include the effects of synoptic-scale background flows.

Future work involves examining the unsteady evolution of dipolar vortices in the nondivergent barotropic and shallow-water models in the presence of more general background flows, such as synoptic-scale jet streams and waves. These investigations may provide insight into the life cycles of jet streaks in the atmosphere, and also may help explain characteristics that are not accounted for by the dipoles in isolation. We also shall examine the possibility of applying dipolar vortex solutions valid in two-layer (Flierl et al. 1980) and continuously stratified (e.g., Berestov 1979; Kizner 1984) QG models to the dynamics of jet streaks; consideration of vertical structure will allow further investigation into the importance of divergent circulations for these features. It is hoped that our analytical and numerical investigations may shed some light on the fundamental dynamics associated with jet streaks in the atmosphere.

5. ACKNOWLEDGMENTS

This research was funded by the National Science Foundation through Grant ATM-9421678 awarded to the State University of New York at Albany.

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Phil Cunningham
cunning@atmos.albany.edu