Vortex Rossby Waves on Smooth Circular Vortices Part II. Idealized Numerical Experiments for Tropical Cyclone and Polar Vortex Interiors Michael T

Dynamics of Atmospheres and Oceans 35 (2002) 179–204

Vortex Rossby waves on smooth circular vortices Part II. Idealized numerical experiments for tropical cyclone and polar vortex interiors Michael T. Montgomery a,∗, Gilbert Brunet b a Colorado State University, Fort Collins, CO 80523, USA b RPN/SMC, Canada, 2121 Route Trans-Canadienne, Dorval, Que., Canada H9P 1J3 Received 18 January 2001; accepted 29 October 2001

Abstract Idealized linear and nonlinear numerical experiments are carried out to test the predictions of the theory developed in Brunet and Montgomery [Vortex Rossby Waves on Smooth Circular Vortices Part I: Theory (pages 153–177, this issue)]. For a monopolar tropical cyclone-like vortex whose strength lies between a tropical depression and tropical storm, linear theory remains uniformly valid in time in the vortex core for all azimuthal wavenumbers. Examples elucidate aspects of the vortex Rossby wave/merger spin up mechanism proposed previously. For the case of the polar vortex, however, wavenumber 1 disturbances in the continuous spectrum are predicted to develop a nonlinear evolution for realistic polar-night stratospheric-jet conﬁgurations and wave breaking is demonstrated to occur within the dynamics of the continuous spectrum. © 2002 Elsevier Science B.V. All rights reserved.

Keywords: Continuous-spectrum; Cylindrical co-ordinates; Vortex Rossby Waves

1. Introduction

In a companion paper by Brunet and Montgomery (2002; hereafter BM02), an exact solution to the linear initial value problem for asymmetric balanced disturbances in the interior of incipient tropical cyclones and the polar vortex was developed in the limit of small and large Rossby deformation radius. When the radial potential vorticity gradient is sufﬁciently strong compared to the radial shear of the tangential velocity, the linear solution remains uniformly valid for small amplitude disturbances. Interior potential vorticity

∗ Corresponding author. Tel.: +1-970-491-8355; fax: +1-970-491-8449. E-mail address: [email protected] (M.T. Montgomery).

0377-0265/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0377-0265(01)00088-4 180 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 perturbations undergo complete axisymmetrization in this case. When the radial shear dom- inates the radial potential vorticity gradient, however, the linear solution no longer remains well ordered. In this case the nonlinear terms become comparable to the linear terms in the potential vorticity equation and wave breaking (overturning vorticity contours in direction opposite the mean shear) is expected. The theory of BM02 appears useful for predicting the onset of Rossby wave breaking in the interior of vortex flows. In this paper we test the theory by examining linear and nonlinear numerical initial value experiments for both tropical cyclone and polar vortex configurations. Unless otherwise stated we will use the same notation as in BM02. The outline of the paper is as follows. Using numerical initial value experiments and the parameter space as defined in BM02 as a guide, Section 2 applies the theory of BM02 to examine particular aspects of vortex Rossby wave dynamics in the interior region of tropical cyclones, and Section 3 examines the dynamics of an upper stratospheric polar-night jet. Section 4 presents the conclusions.

2. Application to TC interiors

2.1. TC intensiﬁcation via convectively generated vorticity anomalies

The usefulness of the theory developed in BM02 can be demonstrated by considering the problem of tropical cyclone development via merger and/or axisymmetrization of convectively generated vorticity anomalies with a dominant (or ‘master’) vortex. Eddy processes such as vortex merger (Melander et al., 1988; Dritschel and Waugh, 1992) and vortex axisymmetrization (Melander et al., 1987a) are becoming more widely recognized as important in both the genesis, development, vertical alignment and evolution of tropical cyclones and hurricanes (Carr and Williams, 1989; Sutyrin, 1989; Guinn and Schubert, 1993; Ritchie and Holland, 1993; Shapiro and Montgomery, 1993; Montgomery and Kallenbach, 1997, hereafter MK; Montgomery and Enagonio, 1998, hereafter ME98; Moller and Montgomery, 1999, hereafter MM99; Moller and Montgomery, 2000, hereafter MM00, Enagonio and Montgomery, 2001, hereafter EM01; Reasor and Montgomery, 2001, hereafter RM01; Davis and Bosart, 2001; Chen and Yau, 2001; Wang, 2001a,b; Prieto et al., 2001). For a barotropic tropical cyclone-like vortex possessing an approximately monopolar vorticity distribution (for tangential velocity data, see, e.g. Willoughby, 1990), we ﬁnd that µ>4(γ + n2 > 4). The small amplitude expansion of BM02 (Section 2) then remains uniformly valid for all time and for all azimuthal wavenumbers in this case. This is the simplest application of the theory and it affords an opportunity to clarify and extend previous work in an idealized setting. A summary of all of the numerical experiments presented here is shown in Fig. 1. The constants γ and n2 deﬁne the parameter space of the model Eq. (2.10) in BM02. Lines of γ +n2 = const. correspond to constant values of G (and µ). As indicated in BM02 (Section 2.2), the region above the solid line corresponds to G ≥ 15/4(γ + n2 > 4), while the region below the dashed line corresponds to G<−1/4 (γ + n2 < 0). The numerical experiments are executed using the semi-spectral numerical model of ME98 (Appendix B) after incorporating the variable Coriolis term needed to simulate the M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 181

Fig. 1. Parameter space for the vortex model Eq. (2.10) of BM02. On the abscissa is n2 and on the ordinate γ . Eight points in parameter space marked by an ‘X’ correspond to 22 numerical experiments (2 for each except for Expt. 1 which entails 4 and Expt. 2 which entails 6): ‘a’ corresponds to the linear run, and ‘b’ corresponds to the nonlinear run. Numerical experiments are performed using the semi-spectral model of ME98 extended to cover the polar vortex as described in Section 2.1.1 in BM02. Points lying above the solid line correspond to G ≥ 15/4, while points lying below the dashed line correspond to G<−1/4. Numerical experiments 0, 1, 2, 4, 6, and 7 are discussed here. See text for details.

polar region as discussed in Section 2.1.1 in BM02. Numerical results are found to be converged at the given resolution. Unless otherwise stated, linear simulations are carried out with zero explicit diffusion (The linear simulations are employed here as an expedient way of representing the complete solution of sheared vortex waves derived in Section 2.3 of BM02). The nonlinear simulations are run with small del-squared diffusion of perturbation vorticity to ensure the integrations remain stable at long times. In this section the Coriolis parameter is taken as constant ( f-plane approximation) and for simplicity we use the nondivergent model (infinite resting depth). Finite depth and finite Rossby number effects are considered briefly in BM02 (Section 3). In tropical cyclone dynamics the nondivergent model has historically served as a useful zeroth-order model for the depth-averaged flow once the primary balance between the mean influx of angular momentum and the loss of angular momentum to the underlying ocean has been removed. Fig. 2 presents the radial distribution of the mean tangential velocity, angular velocity, and relative vorticity of the interior region of an idealized tropical cyclone-like vortex 182 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204

Fig. 2. Basic state tropical cyclone-like vortex employed in Section 2 using the ‘local approximation’ described in Section 2 in BM02. Vortex intensity corresponds to that of a minimal tropical storm (for tangential velocity data, see, e.g. Willoughby, 1990). Only the inner 100 km are shown. The region outside 70 km is not realistic. The v¯ = . × −4 −1 v¯ =− . × −13 −2 −1 parameters defining the vortex are 0 6 0 10 s and 0 4 8 10 m s . The radial distributions of the basic state tangential velocity v(r)¯ , angular rotation rate Ω(r)¯ , and relative vorticity ζ(r)¯ = Ω¯ + dv/¯ dr are shown. Maxima indicated at the top of each plot. Dotted curves represent radial profiles for a Gaussian mean vortex possessing the same circulation inside 50 km radius. M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 183 whose intensity is that of a weak tropical storm. The profile adheres to constraint given by Eq. (4.1) in BM02. The radius of maximum tangential wind (RMW) is 50 km, the maximum tangential velocity is 20 m/s, and the maximum relative vorticity is 1.3 × 10−3 s−1.An ambient Coriolis parameter of f = 5 × 10−5 s−1 is assumed. The radial grid spacing r is 500 m, the time step t is 30 s, and the outer domain is set at 200 km. Beyond 86.5 km radius, the mean tangential velocity reverses in sign and decreases rapidly. This unrealistic feature is an artifact of the local approximation employed in BM02. Since our focus here is on the interior vorticity dynamics (near or within the RMW), the unrealistic behavior of the basic state at large radius does not adversely impact our results. The first initial-value experiment (Expt. 0 in Fig. 1) begins with an azimuthal wavenumber 2 vorticity perturbation whose radial structure is given by r 2 r 2 ¯ |ζ|=0.2ζ(rz) exp 1 − rz rz where rz is the radius at which the perturbation vorticity has maximum amplitude and hat denotes a Fourier-azimuthal transform. For the current example, rz = 40 km. The center of the disturbance then resides just inside the RMW of the basic state vortex. Upon taking the real part, the disturbance vorticity amplitude is 40% of the basic state relative vorticity at r = rz. By traditional measures this disturbance is a small but finite amplitude disturbance. When superposed on the circular basic state vortex, the disturbance produces an ‘elliptical vortex’ in real space. This initial condition is chosen to mimic two weak convective “blow-ups” inside the RMW that generate two regions of positive (cyclonic) relative vorticity in the lower troposphere via vortex tube stretching on the atmospheric mesoscale (e.g. Gentry et al., 1970; ME98; MM99; MM00; EM01). Fig. 3 summarizes the vorticity evolution for Expt. 0. Contours of absolute vorticity (f + ζ¯ + ζ ) are plotted and isopleths less than f have been suppressed. Fig. 3 shows the nonlinear simulation (with an azimuthal truncation of N max = 8) and the linear simulation (with all wave–wave and wave–mean-flow interactions turned off). Despite the relatively large-amplitude perturbation, the similarity between the linear and nonlinear simulations is striking. Vortex axisymmetrization is clearly evident (cf. Fig. 8 in Melander et al., 1987a). The propagation and dispersion of linear sheared vortex Rossby waves captures the essence of the adjustment process here including the filamentation of total vorticity into vorticity bands on the periphery of the cyclonic vorticity region1 . To confirm that these shear waves are indeed sheared vortex Rossby waves, Fig. 4 shows the evolution of the vorticity amplitude for wavenumber 2 in the linear simulation as a function of radius and time. The initial perturbation has its maximum amplitude at r = rz and the figure shows the outward propagation of the primary wave packet and its stagnation near r = 60 km. The center of the wave packet travels approximately 20 km. The outward propagation and stagnation radius are qualitatively in accord with the local WKB theory of MK for sheared vortex Rossby waves. Although the wave packet’s amplitude is governed exactly by the solution to Eq. (2.3) in BM02, for slowly varying basic states, it can be

1 In contrast to the externally perturbed vortex relaxation problem studied by Balmforth et al. (2001), exponential damping and non-linear critical layer processes play no role in the relaxation processes shown here. 184 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 185

Fig. 3. (Continued ). described to first approximation by the conservation of wave action: ∂A + (F ) = ∂t div 0 (2.1) where A is the wave action and F is its two-dimensional flux (Bretherton and Garrett, 1968; J. McWilliams and P. Graves, private communication). Even though the semi-spectral numerical model has been validated against known exact solutions and assorted numerical experiments with idealized tropical cyclone-like vortices using independent pseudo-spectral and finite difference numerical models (MK; ME98; Montgomery et al., 2000), it is instructive to compare the model’s predictions against the theory developed by BM02. One of the more interesting theoretical predictions, derived first by Bassom and Gilbert (1998) and again in the more generalized context of BM02, is the unfamiliar algebraic decay (see Eq. (2.26) in BM02) of perturbation PV and geopo- tential at large times in the inner region r2t ∼ O(1). Fig. 5 verifies this decay in the linear numerical simulation. The solid curve denotes wavenumber 2 perturbation streamfunction amplitude normalized by a constant times t1+µ/4, while the dashed curve denotes 186 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204

Fig. 4. Radius–time contour plot of complex perturbation vorticity amplitude for the linear ‘elliptical vortex’ experiment (Expt. 0a in Fig. 1). Radius in km and time in h. Vorticity amplitude normalized using the scale factor 10−5 s−1. Contour interval is 1–14, incremented by 1.

µ/4 wavenumber 2 perturbation√ vorticity amplitude normalized by a constant times t , where µ = 2 γ + n2 = 4 3. Perturbation vorticity and streamfunction amplitudes from the numerical model were determined at radii given by r2t = 2.8 × 108 m2 s after t = 30 min, beginning at r = 40 km. The curves begin to asymptote to a constant after approximately 5 h, slightly more than one circulation time of the mean vortex at the RMW. The asymptotic behavior of the curves confirms the theoretical prediction and demonstrates that the “large-t” behavior is established rather quickly for near-core wavenumber 2 disturbances. Fig. 6a shows the wave–mean-flow interaction quantified by the change in azimuthal mean tangential velocity (δv¯) for the duration of the 20 h simulation. The bulk of δv¯ occurs well within two circulation times (O(10 h) for this vortex), with the acceleration materializing first followed by the deceleration as the initial wave packet gets absorbed at its stagnation radius. The predicted δv¯ using the linear simulation (so-called ‘quasi-linear approximation’ in ME98 and RM01) and the nonlinear simulation is shown. In the quasi-linear approximation, δv¯ is estimated from the linear solution by temporally averaging the radial divergence of M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 187

Fig. 5. Verifying the temporal decay in the “inner-region” (defined by r2t ∼ O(1)) for the linear ‘elliptical vortex’ experiment (Expt. 0a√ in Fig. 1) with γ = 8 and n = 2. Dashed curve denotes perturbation vorticity amplitude (tv¯ ) 3/|ζˆ | |ζˆ |= (ζˆ (r = r ,t = )) v¯ multiplied by 0 20 , where 20 abs 2 z 0 and 0 denotes the mean√ angular rotation 3+1 ˆ rate at r = 0. Solid curve denotes perturbation streamfunction amplitude multiplied by (tv¯ ) /|φ20|, where ˆ ˆ 0 |φ20|=abs(φ2(r = rz,t = 0)). See text for details. mean eddy-angular-momentum flux, or equivalently, the mean eddy vorticity flux. Using the linearized vorticity equation (Eq. (2.3)) in BM02, an explicit expression for δv¯ is given by

δv(r,t¯ =∞) = 1 A(r,¯ t =∞) − A(r,¯ t = ) r [ 0 ] where r(ζ)2 A(r,¯ t) = (2.2) 2(dζ/¯ dr) is the azimuthally averaged wave activity, or angular pseudo momentum density (Held and Phillips, 1987) which, like the canonical angular momentum, is conserved upon integrat- ing over the ﬂuid. Aside from a multiplicative factor of azimuthal wavenumber, A¯ is the azimuthal mean of wave action deﬁned above. In two-dimensional Euler dynamics, the 188 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204

Fig. 6. Change in mean tangential velocity δv¯ as a function of radius for both quasi-linear and nonlinear numerical model simulations of Expts. 0 and 1 in Fig. 1 using tropical cyclone-like vortex in Fig. 2. (a) Double ‘blow up’ (i.e. elliptical vortex) experiment, Expt. 0. (b) Single ‘blow up’ experiment, Expt. 1. In both (a) and (b), maximum perturbation vorticity is 40% of the local mean relative vorticity. change in the mean flow due to vortex Rossby waves is thus one over radius times the change in wave activity. Fig. 6a shows that the acceleration is greater than the deceleration (angular momentum conservation), and the zero in δv¯ is nearly coincident with the final stagnation radius of the primary wave packet (cf. Fig. 4 in MK). (Although the latter is valid for radially localized vorticity perturbations that propagate small radial distances, the wave activity expression for δv¯ is valid for perturbations having arbitrary radial structure.) The fact that sheared vorticity disturbances yield a nontrivial momentum flux divergence with a corresponding acceleration/deceleration dipole is not new (e.g. ‘negative viscosity’, Starr, 1968). Only recently, however, it has been emphasized that sheared disturbances in M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 189 vortices behave as sheared Rossby waves propagating azimuthally and radially on the radial PV gradient waveguide of the mean vortex, and produce an irreversible up-gradient transport of like-sign eddy vorticity and irreversible down-gradient transport of like- and opposite-sign eddy vorticity (MK). Up-gradient eddy vorticity flux is another way of describing ‘vortex merger’ or ‘vortex coalescence’. The weakly nonlinear formulation thus appears useful for obtaining insight into ‘complete merger’ events as defined by Dritschel and Waugh (1992), or the asymptotic limit of initially separated vortices which undergo merger as in Melander et al. (1988) or Ritchie and Holland (1993). The vortex merger property of sheared vortex Rossby waves is implicit in Fig. 6a. By Stokes’ theorem, the maximum δv¯ observed in Fig. 6a must be associated with an inward transport of like-sign eddy vorticity, and the minimum must be associated with an outward transport of both like and opposite-sign eddy vorticity (cf. Fig. 8 in MK). The same con- clusion follows using wave activity ideas. As Rossby wave activity diverges outward from the excitation region, the mean tangential flow is accelerated and like-sign eddy vorticity is transported inward (ζ u < 0). As wave activity converges outside the excitation region, the mean flow is decelerated and like-sign eddy vorticity is transported outward (ζ u > 0). The inward and outward transport of like-sign eddy vorticity by linear vortex Rossby waves is evident in the linear simulation of Fig. 3. The Lagrangian trajectory calculations of RM01 provide additional confirmation. These Rossby wave–mean-flow predictions have been shown to remain approximately valid for finite amplitude near-core vorticity anomalies on vortex monopoles in 2D Euler, shallow water, quasi-geostrophic, and AB vortex dynamics (MM00; EM01; RM01). The uniform validity of the linear solution in the interior region of vortex monopoles provides mathematical support for why this is so. Further insight into why linear theory remains a useful approximation for vortex monopoles subject to finite amplitude disturbances can be gleaned from an analogy with the ‘beta Rossby number’. The beta Rossby number has been used successfully to estimate the ratio of the nonlinear terms in the vorticity equation to the linear Rossby restoring term in studies of Rossby wave turbulence (Rhines, 1975) and vortex motion on the beta plane (McWilliams and Flierl, 1979; Reznik and Dewar, 1994). The ‘vortex beta Rossby number’ R␤, defined as the ratio of the nonlinear advection terms in the (potential) vorticity equation to the corresponding linear vortex Rossby wave restoring term, can be used to estimate the range of applicability of linear vortex Rossby wave theory for finite amplitude (potential) vorticity disturbances on an initially circular monopolar vortex (MM00; EM01; cf. Held and Phillips, 1987, Section 3, Eqs. (8) and (9)). The monotonicity condition ensures that exponentially unstable or algebraically unstable Rossby wave modes do not dominate the dynamics (e.g. Schubert et al., 1999). When R␤ 1, the large ‘Rossby elasticity’ inhibits the ‘roll up’ of vorticity contours forced by the non-linear advection terms. Farther out in radius, however, where the mean radial vorticity gradient decreases rapidly, the weakly nonlinear and fully nonlinear models predict drastically different results with strong vorticity anomalies in the nonlinear case tending to co-rotate about the vorticity centroid with little or no shearing (Melander et al., 1988). To illustrate some of these points, two additional numerical experiments are performed keeping the same mean vortex in Fig. 2. The first initializes with a wavenumber 1 vorticity perturbation (Expt. 1 in Fig. 1) having identical radial structure as in the wavenumber 2 experiment. The azimuthal truncation for this nonlinear simulation is N = 4. Fig. 6b shows 190 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204

δv¯ for both linear and nonlinear simulations. The quasi-linear and nonlinear predictions give a slightly stronger maximum acceleration than the corresponding predictions for the wavenumber 2 perturbation. This is to be expected based on the local WKB predictions of MK together with wave activity ideas. For perturbations possessing identical radial structure, wavenumber 1 perturbations propagate farther radially than wavenumber 2 perturbations, hence, the difference in wave activity (and δv¯) near the excitation radius will be larger for wavenumber 1 than for wavenumber 2. Although the difference in δv¯ between these cases is not large, this is a general property: Low wavenumber disturbances that axisymmetrize are more effective in spinning up the mean vortex than high wavenumber disturbances. The second experiment begins with the same set up as Expt. 1, but with double the anomaly amplitude (perturbation vorticity amplitude now 80% of the local mean relative vorticity). Linear and nonlinear numerical integrations are presented in Fig. 7. The corresponding δv¯ prediction after 20 h integration time is shown in Fig. 8 (the bulk of the acceleration ma- terializes sooner than this). The difference between the linear and nonlinear experiments is more noticeable but not qualitatively different. The nonlinear simulation exhibits radial dispersion of vortex Rossby waves later in the symmetrization process. Considering that R␤ increases with increasing amplitude, the discrepancy is not surprising and is in accord with the general tendency of nonlinearity to limit wave dispersion, as in solitary waves (e.g. Whitham, 1974). As expected, the maximum δv¯ in the quasi-linear calculation is exactly four times its half-amplitude counterpart, and the nonlinear simulation gives a somewhat reduced acceleration maximum. Note that the nonlinear run is more effective at spinning up the central region (implying a more efficient up-gradient eddy vorticity transport, by Stokes’ theorem). Angular momentum conservation then implies a slightly stronger deceleration outside the RMW. Nonlinear (wave–mean-flow and wave–wave interactions) processes tend to enhance the irreversible eddy vorticity transport predicted by linear theory. Looking more closely at Figs. 3 and 7 one observes that the area bounded by interior vorticity contours in the linear run decreases somewhat with time. Recall that in the strictly linear approximation, the sum of basic state plus perturbation vorticity is conserved following fluid particles and the basic state is not allowed to change. Fluid particles with initially like-sign perturbation vorticity that get transported up the basic state vorticity gradient must consequently experience a decrease in their perturbation vorticity in order to compensate for the increase in basic state vorticity. This property of the linear approximation accounts for the reduction in area enclosed by the vorticity isopleths near the vortex center. It may be of interest to note that axisymmetric vorticity perturbations located at the vortex center yield the greatest spin up without any Rossby waves whatsoever (ME98). It should be remembered, though, that during the early stage of the tropical cyclone lifecycle observations suggest that convection tends to be weakly coupled to the vortex. Convective blow-ups are therefore not expected at the center of the mean vortex. The present calculations support the idea that vortex intensification via convective PV anomalies and sheared vortex Rossby waves can be reasonably efficient provided the blow ups are not too far from the circulation center (ME98; MM00). When a strong blow up occurs, or if a blow up occurs sufficiently far away (so R␤ is large), the new vorticity anomaly may become the master vortex (Melander et al., 1987b; EM01). M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 191

Fig. 7. Single ‘blow up’ (or ‘lopsided vortex’) experiment (Expt. 1 in Fig. 1) with maximum perturbation vorticity now 80% of local mean relative vorticity. The total vorticity contours f + ζ at t = 0, 5, and 10 h are shown. Left column summarizes nonlinear simulation, and right column summarizes linear simulation. Contour intervals are deﬁned by 5 × 10−5 s−1 + j × 1.2 × 10−4 s−1,j = 0, 1,... ,14. Negative vorticity on periphery not shown.

2.2. The limiting case of n = 1 in the local approximation

The discussion of Section 2.4 in BM02 for a large (but ﬁnite) radius of deformation when applied to the TC case holds in general, except for the case n = 1. It is worthwhile to make brief comment here, since it is the only one where the condition√ that insures an increase of the wave-activity in the matching region is not satisﬁed, i.e. 8 + n2 − n = 0, 2, 4,....In 192 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204

Fig. 8. The δv¯ as a function of radius for the single ‘blow up’ experiment of Fig. 7. Quasi-linear and nonlinear predictions are indicated. fact, this particular case gives a simple special solution for the vorticity and streamfunction in the nondivergent limit: 2 ζ(r,t)ˆ = 4r r2 − exp(−r2(1 + it)) (1 + it) and r φ(r,t)ˆ = exp(−r2(1 + it)) (2.3) (1 + it)2 The vorticity in Eq. (2.3) shows only one extremum of wave activity as expected for n = 1. We expect an extra maximum in the matching region when n ≥ 2. A second interior maximum of wave-activity for n ≥ 2 was also observed in numerical simulations by MK for more realistic tangential velocity proﬁles with a single extremum.

3. Application to the polar vortex

3.1. Some preliminary discussions

In BM02 (Section 4), it was pointed out that the large radial tropospheric gradient in potential vorticity γ indicates that the continuous spectrum evolves quite slowly and that the shearing process is quite marginal in the upper tropospheric polar jet. In contrast, the negative or near-zero values of γ in the wintertime stratosphere suggest that shearing processes become important. The discussion in BM02 (Section 2.5) shows generally that for all γ the inner region is a region of wave-activity maximum. The special case γ = 0 for all n is an exception to the latter statement. This case does not satisfy the condition that guarantees an increase of M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 193 wave activity in the inner region. In this case the Rossby restoring term vanishes identically and hence we expect no radial wave propagation. In fact, the asymptotic solution Eq. (2.28) in BM02 is easily shown to be an exact solution of Eq. (2.10) in BM02 for all n in the nondivergent limit, the vorticity being n ζ(r,t)ˆ = r exp(−r2(1 + it)) (3.1) and the streamfunction, using Abramowitz and Stegun (1970, Eqs. (13.1.27) and (13.2.2)), rn 1 φ(r,t)ˆ = exp − r2(1 + it) 4n 2n(1 + it) 2 1 1 n− × exp − r2(1 + it)s (1 + s) 1 ds (3.2) −1 2 The corresponding streamfunction for n = 1 can be integrated exactly. The result is 1 φ(r,t)ˆ = exp(−r2(1 + it) − 1) (3.3) 4r(1 + it)2

3.2. Some numerical experiments for a model polar vortex

To illustrate the polar vortex dynamics captured by the model Eq. (2.10) in BM02 (and its nonlinear extension), numerical simulations are now presented. We assume that the polar vortex has been perturbed by upward propagating planetary Rossby waves from the troposphere. Rather than model the forcing as a continuous or intermittent source of either topographic or baroclinic Rossby waves, we adhere to our initial-value theme and model the excitation via an initial condition in perturbation vorticity. Extension of the upcoming results examining continuous forcing, ﬁnite depth effects, ﬁnite Rossby number effects, and wave-induced mixing of chemical constituents within the vortex are certainly topics worthy of further study.

3.2.1. An example of wave breaking in the continuous spectrum As a check of our implementation of the numerical semi-spectral model for the polar vortex case, the exact solution given in Section 3.1 was veriﬁed for n = 1 (Expt. 2a in Fig. 1, results not shown). Following the convention of Section 2, the linear experiment was designated Expt. 2a and the nonlinear experiment Expt. 2b. The radial grid spacing was δr = 8 km, the time step was δt = 30 min, and the outer domain was rmax = 4800 km. From the estimate ¯ tmax ≤ π/[nδr|dΩ/dr|max] (Appendix A in Smith and Montgomery, 1995) the linear inviscid integration is valid for times of O(1000 days). According to Fig. 1, for this choice of γ and n the nonlinear terms can attain similar magnitudes to the linear terms and wave breaking (i.e. overturning vorticity contours in the direction opposite the mean shear) becomes possible. The mean vortex for γ = 0 is shown in Fig. 9, which displays the tangential wind, angular velocity, and relative vorticity proﬁle to an outer radius of rmax = 4800 km. The initial perturbation is an azimuthal wavenumber 1 perturbation whose radial structure is given by r r 2 ˆ ¯ 1 |ζ1(r)|=0.22ζ(rz) exp 1 − rz 2 rz 194 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204

γ = v = v¯ = . × −18 −2 −1 Fig. 9. Basic state polar vortex for 0 associated with a value of 0 0 and 0 2 695 10 m s . The radial distribution of mean tangential velocity v(r)¯ , angular rotation rate Ω(r)¯ , and relative vorticity ζ(r)¯ = Ω¯ + dv/¯ dr out to the limiting radius of 4800 km in the polar tangent-plane approximation discussed in BM02 is shown. Maxima are indicated at the top of each plot. M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 195

Fig. 10. Radius–time (Hovmoller) contour plot of perturbation vorticity amplitude for the linear polar vortex experiment with γ = 0 and n = 1 (Expt. 2a in Fig. 1). Radius in km and time in days. Vorticity amplitudes have been normalized using the scale factor 10−7 s−1. Contour interval is 1–8, incremented by 1. See text for details.

√ where rz = 2×1000 km. The maximum perturbation vorticity is 44% of the basic state relative vorticity at rz. The azimuthal truncation is N = 8 for the nonlinear simulation and N = 1 for the linear simulation. The results of these experiments are summarized in Figs. 10–13. A contour plot of the Fourier vorticity amplitude as a function of radius and time (Hovmoller plot) from the linear numerical model is shown in Fig. 10. The absence of radial wave propagation is clear, as expected with γ = 0. Fig. 11 veriﬁes the temporal decay of perturbation streamfunction (solid curve) and perturbation vorticity (solid curve) for the inner region r2t ∼ O(1) as simulated by the linear numerical model. The modi- ﬁcation of this decay by nonlinear effects is superposed on the plots (dashed curves) as one means of estimating when the nonlinear dynamics begin to deviate from linear theory. The data was processed in a similar manner to Fig. 5. The perturbation streamfunction +µ/ amplitude was normalized by a constant multiplied by t1 4. The perturbation vorticity amplitude was normalized by a constant times tµ/4, where µ = 2 γ + n2 = 2. Perturba- tion vorticity and streamfunction from the numerical model were evaluated at radii given by r2t = 4 × 1017 m2 s after t = 4 days, beginning at r = 1070 km. The curves from the linear model begin to asymptote to a constant after a time of O(30 days). The relatively 196 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204

Fig. 11. Verification of temporal decay in the “inner-region” defined by r2t ∼ O(1) for the polar vortex experiment with γ = 0 and n = 1 (Expt. 2 in Fig. 1). (a) Perturbation streamfunction amplitude multiplied by 3 × ((tΩ )3/2/|φˆ |) |φˆ |= (φˆ (r = r ,t = )) Ω 10 e 10 , where 10 abs 1 z 0 √ and e denotes the rotation rate of the Earth. 3/2 ˆ ˆ ˆ (b) Perturbation vorticity amplitude multiplied by 10 × ((tΩe) /ζ10), where ζ10 = abs(ζ1(r = rz,t = 0)). In both (a) and (b), the solid curve portrays the linear simulation (Expt. 2a), and the dashed curve portrays the nonlinear simulation (Expt. 2b). See text for details. weak radial shear of the mean angular velocity near 1000 km radius, compared to the shear farther out, accounts for the appreciable time needed to achieve the predicted decay. Based on streamfunction decay, the nonlinear dynamics begin to deviate from linear theory after a time of O(20 days). The nature of this deviation is described further below. Fig. 12 shows three snapshots of the total relative vorticity ζ(r,t)¯ + ζ (r,λ,t). The planetary vorticity distribution given by f(r)is not shown. As in Section 3, the nonlinear results comprise the left column and the linear results comprise the right column. By t = 44 days, the vorticity contours near 1500 km radius have already undergone significant folding opposite the mean shear. Small-scale vortical structures are evident at t = 68 days in the region beyond 1000 km. The difference between the linear and fully nonlinear simulation is striking and quite unlike the examples of Section 2. Another illustration of the difference between linear and nonlinear dynamics is presented in Fig. 13 which shows perturbation vorticity ζ (r,λ,t)at t = 0, 20, and 44 days (a close-up view of the evolution leading to t = 44 days, shown in Fig. 12). Fig. 13 shows the emergence of a coherent vortex in the nonlinear case, and illustrates a dramatic change in the topology of the vorticity contours. The perturbation vorticity field as simulated with the nonlinear model portrays characteristics similar to the instability of sheared disturbances (Haynes, 1987; Nauta and Toth, 1998) as well as the nonlinear Rossby wave critical layer (Haynes, 1989; Brunet and Haynes, 1995). M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 197

Fig. 12. Map-plots of total relative vorticity ζ(r,t)¯ + ζ (r,λ,t)for the polar vortex experiment with γ = 0 and n = 1 (Expt. 2 in Fig. 1—an interior displacement of the polar vortex [a lopsided polar vortex, r.e.]). Contribution 2 2 from planetary vorticity f(r) = Ωe(2 − (r /a )) not included in plot. The nonlinear simulation is shown in the left column, while the linear simulation is shown in the right column. Results at t = 0, 44, and 68 days. Contour intervals are 1 × 10−10 s−1 + j × 0.5 × 10−6 s−1,j = 0, 1,... ,20.

To verify that the breaking behavior observed in Figs. 12 and 13 is consistent with the theory of Section 2 in BM02, auxiliary experiments have been performed. According to the theory, wave breaking at γ = 0 should occur only with n = 1 (see Fig. 1). Disturbances of higher wavenumber should not break, at least for small but ﬁnite amplitude disturbances. We have tested this prediction by running the two numerical experiments, designated 198 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204

Fig. 13. Map-plots of perturbation relative vorticity ζ (r,λ,t)for the polar vortex Expt. 2 in Fig. 12. The nonlinear simulation is shown in the left column, and linear simulation is shown in the right column. Results at t = 0, 20, and 44 days. Positive contour intervals are 2 × 10−8 s−1 + j × 3.2 × 10−7 s−1,j = 0, 1,... ,5; and negative contour intervals are −16.2 × 10−7 s−1 + j × 3.2 × 10−7 s−1,j = 0, 1,... ,5.

Expts. 6 and 7 in Fig. 1, for azimuthal wavenumbers 2 and 3, respectively. The initial perturbations had the same maximum amplitude and radius of maximum amplitude as Expt. 2, but possessed a radial structure varying as r2 exp(−r2) and r3 exp(−r2) for n = 2 and 3, respectively, consistent with the leading-order time-asymptotic analytical solution presented in Eq. (2.25) in BM02. The linear and nonlinear experiments were found to remain in close agreement throughout the 100 days numerical experiment and no wave breaking was M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 199

Fig. 14. Map-plots of perturbation relative vorticity ζ (r,λ,t)for the polar vortex Expt. 6 in Fig. 1. The parameters (n2,γ) for this case fall precisely on the dividing line between breaking and no breaking. The fully nonlinear simulation is shown in the left column, while the linear simulation is shown in the right column. Results at t = 0, 40, and 80 days. Positive contour intervals are 2 × 10−8 s−1 + j × 3.2 × 10−7 s−1,j = 0, 1,... ,5 and negative contour intervals are −16.2 × 10−7 s−1 + j × 3.2 × 10−7 s−1,j = 0, 1,... ,5.

observed. Note that γ = 0, and n = 2 falls precisely on the boundary between breaking and no-breaking. Fig. 14 summarizes the outcome of Expt. 6. The linear and nonlinear simulations evolve nearly identically with no overturning vorticity contours. As additional conﬁrmation that the breaking behavior shown in Figs. 12–14 is consistent with the theory, numerical experiments at double, half, and quarter the initial wave 200 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 amplitude have been executed for n = 1 using the same n = 1 structure as above (results not shown). At double the initial wave amplitude the discrepancy between linear and nonlinear dynamics manifests itself at earlier times O(10 days) with the appearance of coherent eddy structures near 2000 km radius. At half the initial wave amplitude, the discrepancy between linear and nonlinear dynamics becomes apparent after O(70 days), with a roll up in the vorticity contours in the inner region near 1200 km radius. This behavior is consistent with the theory which predicts that the breaking time goes as O(ε−2). At one-quarter the initial wave amplitude, only a slight tendency for the perturbation vorticity contours to resist the mean shear near the center is observed, but there is no wave breaking throughout the vortex. The linear and nonlinear results are virtually identical in all respects in this case. The fact that breaking does not occur for the weak amplitude case does not contradict the theory, however. Recall that the breaking criterion given by γ + n2 < 4 is only a necessary condition and supposes that the nonlinear terms can catch up to the linear terms. As the initial wave amplitude goes to zero, the matching region where this becomes possible shrinks in radius and breaking becomes unlikely in this case as this region shrinks to zero. The present examples show that perturbation velocity amplitudes need not be unreal- ζ ≥ ( × −6 −1) istically large ( max O 1 10 s ) to initiate wave breaking and stirring in the interior region of the polar vortex when γ is less than or equal to 0. Note that breaking occurs here in the absence of exponential shear instability and occurs entirely within the dynamics of the continuous spectrum. Our results suggest that the dividing line between linear and nonlinear dynamics is captured by the simple model Eq. (2.10) in BM02 and that wavenumber 1 is the preferred breaking and stirring mode since it should break soonest. The linear theory of Section 2 in BM02 cannot, of course, predict the coherent structures that emerge. The fully nonlinear equations of motion are required for this purpose.

3.2.2. An example of a discrete Rossby wave Before closing this section we mention some initial-value experiments that have been carried out to determine if the necessary condition derived for the existence of discrete vortex Rossby waves (G<−1/4) is also sufﬁcient for the simple model given by Eq. (2.10) in BM02. From the analysis of Leblond (1964), a discrete set of polar-basin Rossby waves is expected in the limit of a resting basic state. We can then interpret the condition G<−1/4 as a parameter boundary that enables these waves to persist in the presence of shear. A complete study of the discrete spectrum in such circumstances is not carried out. Here, we present just one example of such a mode captured by both linear and nonlinear models. Fig. 15 summarizes the linear and nonlinear simulation using γ =−3 and n = 1 for an initial perturbation with radial structure given by r r 2 ˆ ¯ 1 |ζ(r)|=0.1ζ(rz) exp 1 − rz 2 rz where rz = 1000 km is the radius at which the perturbation vorticity has maximum amplitude. The disturbance vorticity amplitude is 20% of the basic state relative vorticity at rz. This choice of γ and n2 places us well below the dashed curve of Fig. 1. If the criterion M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204 201

Fig. 15. Map-plots of perturbation relative vorticity ζ (r,λ,t)for the polar vortex experiment with γ =−3 and n = 1 (Expt. 4 in Fig. 1). The left column shows the nonlinear simulation, while the right column shows the linear simulation at t = 0, 44, and 92 days. Contour interval is 8 × 10−8 s−1. were sufficient, we would expect a discrete wave mode to emerge in time and retrograde in clockwise sense around the pole maintaining a fixed radial structure and amplitude. Fig. 15 confirms this prediction. The linear and nonlinear models predict a wavenumber 1 feature rotating anticyclonically around the pole, and other plots between days 80 and 116 (not shown) indicate an approximately constant amplitude in both linear and nonlinear experiments. At this small but finite amplitude, the azimuthal phase speed observed in the linear model is slightly greater than the phase speed observed in the nonlinear model. 202 M.T. Montgomery, G. Brunet / Dynamics of Atmospheres and Oceans 35 (2002) 179–204

4. Conclusions

Several numerical experiments have been considered here in order to demonstrate the usefulness of the vortex wave theory developed by BM02. 1. The wave breaking/no-breaking prediction has been verified with numerical experiments for polar vortex and tropical cyclone configurations. 2. An example of a discrete Rossby wave in the presence of weak shear was demonstrated using both linear and nonlinear numerical models for the polar vortex configuration. The theory may prove useful in the diagnosis of full physics hurricane models and in interpreting the observed breakdown of the stratospheric vortex and the Lagrangian stirring of chemical constituents within it. Given the ability of the barotropic model to discriminate between breaking and nonbreaking events for small but finite amplitude disturbances, the theory may also prove useful in examining vertical Rossby wave propagation and the at- tendant wave breaking in the interior of the polar vortex without the constraints implicit in contour dynamical models possessing a small number of PV contours (BG98, p. 134, 3rd para; Polvani and Saravanan, 2000). These problems are left for future study.

Acknowledgements

We thank T. Warn, M. McIntyre and A. Zadra for helpful discussions, and P. Reasor and T. Cram for assistance with some of the ﬁgures. P. Reasor is thanked for helping MTM set up the numerical model in the winter of 1997. This work was supported in part by the Isaac Newton Institute through a 1996 Program of the Mathematics of Atmosphere and Ocean Dynamics; NSF ATM-9732678and ATM-0101781.A ﬁrst draft of this work was completed during the Newton program. MTM wishes to thank Recherche Prévision Numérique of the Meteorological Service of Canada and McGill University for their hospitality that enabled the completion of this work, and LKP and MPM for their support.

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