1. Overview
One important mechanism of gravity wave production in the atmosphere is latent heat release by moist updrafts. Another, connected to mountain waves, is topographic forcing. A third, more subtle mechanism, is unforced radiation from jets and vortices, that is, spontaneous radiation. For example, it has been shown that inertia–gravity (IG) waves can emerge from a jet as a by-product of baroclinic instability (e.g., O’Sullivan and Dunkerton 1995). It has also been shown that IG waves can emanate from the vortical motions of ageostrophic turbulence (e.g., Métais and Herring 1989; Polvani et al. 1994). In this paper, we will revisit the paradigm of spontaneous IG wave radiation from a single vortex.
Recent articles have speculated that IG wave radiation can play an important role in the life cycle of mesoscale atmospheric vortices over land or sea. Chimonas and Hauser (1997) proposed that it can weaken supercell mesocyclones, under proper conditions, and thereby reduce the chance that they spawn tornadoes. Similarly, Chow and Chan (2003) argued that IG wave radiation can provide an important sink of angular momentum for hurricanes. In the outer region of a hurricane, it might also trigger the formation of spiral rainbands (e.g., Kurihara 1976; Willoughby et al. 1984; Chow et al. 2002).
In section 2, we will identify the basic parameter regime in which spontaneous IG wave radiation can have a relatively large amplitude and appreciably affect the vortex circulation. We will argue that intense tropical cyclones (TCs), supercell mesocyclones, polar lows, and island wake vortices penetrate this regime. Nonetheless, these vortices may not dramatically radiate. The main purpose of this paper is to find conditions that inhibit the spontaneous radiation of IG waves from an otherwise capable vortex.
We will limit our study to pure spontaneous radiation (PSR) (see Ford 1994a, b; Plougonven and Zeitlin 2002; Schecter and Montgomery 2004). By pure, we mean spontaneous radiation that sustains its own growth, as opposed to that which is passively amplified by, say, baroclinic or Kelvin–Helmholtz instabilities. An ideal vortex that is immune to such breakdowns, and symmetric instability, is a monotonic shallow-water cyclone (e.g., Montgomery and Shapiro 1995). We will base our theoretical discussion on this model because of its analytical and computational simplicity.
Linear perturbation theory provides the following analysis of PSR: A generic deformation of the cyclone, perhaps a defect of its creation, projects onto a set of discrete azimuthally propagating Rossby waves. Each vortex Rossby (VR) wave emits a frequency-matched spiral IG wave, of proportional strength, into the environment. The radiation has positive feedback on the VR wave, causing both to grow. However, as it propagates, the VR wave resonantly disturbs potential vorticity (PV) in a critical layer centered at the radius r*. The critical layer disturbance can have negative feedback on the VR wave and cause it to decay. That is, critical layer damping can oppose radiative pumping and thereby inhibit PSR.
In sections 3 and 4, we will illustrate and closely examine the competition between radiative pumping and critical layer damping in the context of linear theory (cf. Schecter and Montgomery 2004). From an equation that expresses conservation of wave activity, we will derive a formula [Eq. (63)] for the growth rate γ of a discrete VR wave and its coupled radiation field. The formula states that γ is the sum of a positive radiation term and a negative critical layer term. The magnitude of the critical layer term increases with the cyclone’s radial PV gradient at r*. If the gradient exceeds a threshold, the negative critical layer term prevails. As a result, the VR wave and its radiation field decay with time.
Damping will persist if linear theory remains valid forever. On the other hand, nonlinear stirring of the critical layer can destabilize the VR wave by removing the mean PV gradient that opposes radiative pumping (cf. Briggs et al. 1970; Pillai and Gould 1994; Montgomery and Enagonio 1998; Schecter et al. 2000; Balmforth et al. 2001). This nonlinear radiative instability occurs if the initial wave amplitude exceeds a critical value, which we will estimate in section 5 [see Eq. (70)]. In section 6, we will verify the estimated critical value with numerical simulations.
It is well known that the shallow-water equations are isomorphic with the 2D compressible gas equations, with ratio of specific heats cp/cυ = 2. As a result, our study parallels that of acoustic radiation from a vortex in a compressible fluid (e.g., Lighthill 1952; Broadbent and Moore 1979; Kop’ev and Leont’ev 1983, 1985, 1988; Kambe 1986; Sozou 1987; Zeitlin 1991; Chan et al. 1993; Howe 2001). Although numerical investigations have distinguished vortices that spontaneously emit sound from those that are quiet (e.g., Chan et al. 1993), to our knowledge, the results were not explained in terms of VR wave dynamics. In particular, the results were not explained as the outcome of a competition between critical layer damping and radiative pumping, as they are here.
This paper also overlaps astrophysical studies of modal stability in compressible, differentially rotating discs, with gravitational centers of attraction (e.g., Papaloizou and Pringle 1987; Shukhman 1991; Papaloizou and Lin 1995; Li et al. 2001). The usual example is a stellar accretion disc. The astrophysics literature explicitly considers the effects of both critical layer stirring and spiral radiation on the growth of linear (e.g., Papaloizou and Pringle 1987) and weakly nonlinear (e.g., Shukhman 1991) modes. In addition, the numerical simulations of Li et al. (2001) provide vivid illustrations of spiral radiation that is produced by Rossby-like waves and mesovortices within compressible discs.
More generally, our discussion of nonlinear PSR connects to the broad study of waves with nonlinear critical layers (for an early review, see Maslowe 1986). Such waves include planetary Rossby waves (e.g., Killworth and McIntyre 1985), shallow-water waves in sheared channel flow (e.g., Balmforth 1999), waves in stellar accretion disks (see above), and collisionless plasma waves (e.g., O’Neil 1965). The list also includes nonradiative VR waves in a 2D Euler vortex (Briggs et al. 1970; Pillai and Gould 1994; Schecter et al. 2000; Balmforth et al. 2001).
The remainder of this paper is organized as follows: section 2 identifies the parameter regime in which an atmospheric vortex can spontaneously radiate IG waves with notable intensity. Sections 3 and 4 outline the linear theory of PSR from a monotonic cyclone in the context of the shallow-water model. Section 5 addresses the limitations of linear theory. Section 6 presents a computational study of nonlinear PSR. Section 7 recapitulates our conclusions.
2. The potential to spontaneously radiate
The diagram indicates that, due to small values of Ro, one should not expect vortex-scale vibrations to generate IG wave radiation from oceanic eddies, the circumpolar stratospheric vortex, synoptic-scale cyclones, or mesoscale convective vortices (MCVs). In addition, due to large values of ε, typical oscillations of tornadoes and dust devils should not produce IG wave radiation. We must note, however, that tornadoes can have Mach numbers as large as one-half and emit detectable levels of analogous acoustic radiation (e.g., Abdullah 1966; Bedard 2005).
On the other hand, our estimates suggest that TCs, supercell mesocyclones, polar lows, and island wake vortices can penetrate the parameter regime in which spontaneous IG wave radiation is dynamically relevant. That is, we assert that such vortices can satisfy conditions (3) and (4).
3. Pure spontaneous radiation
In this section, we will briefly analyze the exponentially growing eigenmodes that account for PSR from a monotonic shallow-water cyclone. We will show that each such eigenmode consists of a core VR wave coupled to an outer spiral IG wave. In addition, we will show that smoothing the PV profile of the cyclone can negate the radiative instability.
a. Shallow-water model on the f plane
In what follows, we will use a polar coordinate system, in which r and φ denote the radius and azimuth. In addition, we will let u and υ denote the radial and azimuthal components of u. Overbars and primes will mark equilibrium and perturbation fields, respectively.
b. Monotonic cyclones
c. Eigenmodes
Equations (24) and (25) provide mixed boundary conditions, that is, relations between Φ and dΦ/dr at points near the origin and in the radiation zone. The values of ω that permit regular solutions to Eq. (22) with these boundary conditions are obtained numerically by a standard shooting technique. Appendix B provides formulas for relating the eigenfunction Φ to the velocity and PV wavefunctions.
d. Radiative instability
Figure 3 illustrates the responses of two hyperbolic cyclones to slight elliptical deformations. We first consider the top row, which is from a numerical simulation of the nonlinear shallow-water equations (see section 6a).
e. Quasimodes
Figure 4 shows that increasing the smoothness parameter Δ of the hyperbolic cyclone decreases the growth rate γ of its n = 2 radiative eigenmode. Moreover, increasing Δ above a threshold quenches the radiative instability. Before explaining the physics of this transition (see section 4d), there is an important technical issue to address.
As the growth rate of the radiative mode becomes negative, it transforms from an eigenmode into a quasi- mode. Quasimodes are not exact solutions to the eigenmode problem. For example, the PV disturbance of a quasimode decays with time over the bulk of the cyclone, but grows with time in a thin critical layer (e.g., Schecter et al. 2000, 2002). In contrast, an eigenmode would decay everywhere.2
A formal Laplace transform solution to the initial value problem leads to a simple procedure for computing the complex frequency ω of a quasimode. The procedure has been derived for the quasimodes of 2D Euler vortices (Briggs et al. 1970; Corngold 1995; Spencer and Rasband 1997) and for the quasimodes of 3D barotropic cyclones in density stratified fluids (Schecter et al. 2002; Schecter and Montgomery 2004). Below, we describe the analogous procedure for a shallow-water cyclone, without derivation.
The bottom row of Fig. 3 illustrates the behavior of a quasimode in a hyperbolic cyclone, with Δ = 0.3. In this case, the data were obtained from a numerical integration of the linearized shallow-water equations (appendix A). A sponge ring was applied at large radii to absorb outward-propagating waves. The initial conditions were given by the n = 2 quasimode wavefunction, computed along ρ(r). The pressure anomaly closely resembles a pure eigenmode, but its time dependence deviates slightly from exponential decay. The deviation is an artifact of the PV blemish seeded in the critical layer of the wavefunction.
4. Radiative pumping versus critical layer damping
In this section, we will explain why a VR wave is compelled to grow upon emitting a frequency matched IG wave into the environment. In addition, we will explain why resonant stirring of PV in the critical layer (at r*) counters such growth. Importantly, we will find that critical layer damping prevails over radiative pumping if the magnitude of d
a. Angular pseudomomentum
b. Conservation of wave activity
One may view the growth of a radiative mode (e.g., Fig. 3, top row) to be a consequence of conservation of wave activity. Figure 6 is a snapshot of the wave activity densities, Λq and Λυϕ, of the n = 2 radiative mode of a hyperbolic cyclone. Over the bulk of the cyclone the positive PV component Λq dominates; that is, the VR wave has positive wave activity. In contrast, the negative Λυϕ component dominates the outer IG wave activity. This suggests that IG wave radiation involves the ejection of negative wave activity into the environment. To conserve total wave activity, it would seem that the bulk VR wave must grow.4 On the other hand, the following subsection will show that the growth is slowed, and possibly reversed, by transfer of positive wave activity from the bulk VR wave into the critical layer.
c. The growth rate of a radiative VR wave: Linear theory
d. Interpretation and verification of the growth rate formula
To interpret the analytical growth rate formula [Eq. (63)], we first observe that the M integral is positive if rυ is not too large. Figure 6 validates our assertion for the n = 2 radiative mode of a hyperbolic cyclone. Simply note that the integrand of M is directly proportional to the net-positive bulk wave activity density in this figure.
According to Eq. (63), the growth rate γ of a VR wave is the sum of two terms. The first term, ϵrad/M, accounts for the feedback of radiation. It is positive if the VR wave excites an outward-propagating spiral IG wave at rυ. The second term, −ϵcl/M, gives the feedback of PV stirring in the critical layer. It is negative for monotonic cyclones. Moreover, if the magnitude of d
We now return to Fig. 4, which shows the growth rate of a radiative VR wave of a hyperbolic cyclone, with variable smoothness. The × markers correspond to Eq. (63). To evaluate the right-hand side of this equation, we used the wavefunctions (U, V, Φ, Q) and critical layer parameters (r*, lγ) of the computed eigenmodes or quasimodes. Evidently, Eq. (63) accurately describes the competition between radiative pumping and critical layer damping, near marginal stability.
Figure 4 also shows how the radiative pumping rate (ϵrad/M) and critical layer damping rate (ϵcl/M) vary with the smoothness parameter Δ. For these curves, ϵrad and M were evaluated with5 rυ = 12ro. As Δ increases from 0.05 to 0.3, the radiative pumping rate increases slightly. However, the PV gradient at r* explosively grows, and the critical layer damping rate increases by orders of magnitude. Consequently, γ becomes negative.
Note that we did not plot the analytical growth rates for strongly damped waves. If γ is too large, the critical layer, |r − r*| < c lγ, covers a sizable fraction of the vortex. This contradicts a key assumption of our analysis.
5. Breakdown of linear theory
The above linear theory suggests that a sufficiently large PV gradient in the critical layer inhibits radiation from a VR wave, regardless of the wave amplitude. In reality, the wave amplitude must fall below a finite threshold. In what follows, we will estimate this threshold’s upper bound. We will do so with a basic model of critical layer flow that is useful for introducing concepts and making “back of the envelope” calculations. It is not our intention to present a rigorous theory for the nonlinear interaction of a VR wave and its critical layer (cf. Balmforth et al. 2001; Shukhman 1991). Our simplified analysis seems adequate to explain the principal simulation results of section 6.
a. Basic model of critical layer flow
Suppose that a single exponentially damped VR wave propagates around a monotonic cyclone. Let us consider the flow near the critical radius r*. In this region, a tracer (Lagrangian parcel) approximately obeys Hamiltonian dynamics.
b. The effect of VR wave amplitude on the nature of critical layer flow
Figure 7a shows three snapshots of the critical layer flow as it evolves with time. The streamlines correspond to contours of constant H. A separatrix (dashed curve) instantaneously divides the flow into regions of trapped and untrapped fluid. The maximum radial half-width of the trapping region is ltrapeγτ/2Ωb. Since γ is negative, the trapping region thins exponentially with time. Near the center of the trapping region, a tracer has a quasi-elliptical orbit with instantaneous frequency Ωbeγτ/2Ωb.
Figures 7b and 7c show the radial motions of two tracers, for increasing values of the frequency ratio, Ωb/|γ|. Both tracers are placed near the center of the trapping region at t = 0. Specifically, the initial coordinates and momenta are (μ, p) = (−π/2, 0.1) and (−π/2, 0.001). Both plots are nearly identical.
If the VR wave is very weak, that is, if Ωb/|γ| ≪ 1, the tracers complete only a small fraction of their orbits before the separatrix shrinks beneath them. As a result, the tracers become untrapped and maintain near constant radial positions. If the VR wave is very strong, that is, if Ωb/|γ| ≫ 1, the tracers complete many orbital cycles.
Since PV follows tracer trajectories, its distribution in the critical layer would overturn in the case of a strong VR wave. Because the rate of overturning varies between the center of the trapping region and the separatrix,
6. Nonlinear simulations
The main purpose of this section is to verify, by numerical experiments, that condition (70) suffices to cause PSR that does not occur in linear theory. In addition, we will analyze in detail the flow of wave activity during the nonlinear evolution of a radiative VR wave. At the end of this section, we will demonstrate that sustained radiation can lead to vortex fission if unimpeded over many vortex rotation periods. Discussion of the latter topic is kept brief because of its close relation to previous numerical studies of acoustic radiation from a compressible vortex (e.g., Chan et al. 1993).
a. The numerical model
Our numerical integration of the shallow-water equations (8) and (9) is based on the enstrophy-conserving staggered grid model of Sadourny (1975). The simulation domain is a square box with doubly periodic boundary conditions. At large radii, a circular sponge ring absorbs outward-propagating IG waves and keeps the fluid near rest. In addition, ∇4 hyperdiffusion dissipates grid-scale fluctuations to ensure numerical stability.
As shown in Fig. 3, a vortex-scale deformation of a shallow-water cyclone tends to generate longwave radiation in the environment. In our simulations, lrad is typically 10 times greater than the vortex scale. In contrast, PV stirring in the critical layer generates finescale filaments. Over the course of our simulations, these filaments can thin to a small fraction of the vortex scale. Because the dynamical length scales differ considerably in the cyclone and in the environment, we introduced nested inner and outer grids. The spatial resolution of the inner grid is 6 times greater than that of the outer grid.
For those interested, appendix C tabulates all of the numerical parameters for the simulations that appear in this paper.
b. Balanced initial conditions
c. Quasimode revival
Figure 8 plots wave intensity (Ω2b) versus time for excited VR waves on both Gaussian and hyperbolic cyclones. Table 1 fully describes the cyclone and wave parameters. Notably, each wave is a quasimode; that is, its linear growth rate γ is negative.
Early on, the simulated VR waves decay exponentially with time, in accord with linear theory. Two distinct behaviors occur at later times depending of the value of Ωb(0)/|γ|. If this ratio exceeds unity, the VR wave intensifies after a bounce period, 2π/Ωb(0). If this ratio is less than unity, as in curves G3 and G4, the VR wave continues to decay.7 This dichotomy was anticipated from the basic considerations of section 5.
d. Nonlinear critical layer stirring
The two columns on the right-hand side of Fig. 8 show the evolutions of deformed Gaussian cyclones. Different colors represent different levels of PV. In one case, G1, the deformation excites a strong VR wave; that is, Ωb(0)/|γ| > 1. In the other case, G3, the deformation excites a weak VR wave; that is, Ωb(0)/|γ| < 1. According to these simulations, PV coils in the critical layer of a strong VR wave, but not in the critical layer of a weak VR wave. Figure 9 provides a closer look at the critical layer flows and leads us to the same conclusion.
Figure 10 shows the initial and late-time profiles of azimuthally averaged PV 〈q〉φ in the critical layers of a strong VR wave (G1) and of two weaker VR waves (G3 and G4). The strong wave levels the PV profile, whereas the weaker waves do not. Apparently, the weakest wave (G4) has a negligible effect. In section 5, we argued that leveling the critical layer is what enables radiative pumping to act unopposed. The growth of strong waves but not of weak waves, in Fig. 8, adds merit to this claim.
e. Wave activity analysis
The top dotted curve is the gain of bulk VR wave activity by IG wave radiation, that is, the time integral of S(rυ, t). The bottom dotted curve is the loss of bulk VR wave activity by critical layer absorption, that is, minus the time integral of the right-hand side of Eq. (49). Initially, critical layer damping is more intense than radiative pumping, and the bulk wave activity decays. However, as the cat’s eyes coil PV, the critical layer periodically returns some positive wave activity. This reduces the efficacy of critical layer damping. On the other hand, radiative pumping persists and causes the bulk wave activity to grow.
Note that, overall, the nonlinear critical layer continues to gain wave activity at the expense of the bulk VR wave. There are at least two plausible explanations. To begin with, as the cat’s eyes grow, they entrain and stir exterior PV. This seems likely to produce a net gain of critical layer wave activity. In addition, the inner region of our fixed annular critical layer happens to include part of the core PV distribution. So, as the core becomes increasingly elliptical, it should add some wave activity to the critical layer.
f. Vortex fission
To complete our discussion, we briefly examine the long-term effect of nonlinear PSR. Figure 12 illustrates the evolution of an elliptically deformed hyperbolic cyclone that continuously radiates IG waves. The simulation is identical to H1, but with half the grid resolution. Initially, the ellipticity of the core PV distribution decays. From Fig. 8, we know that the decay rate agrees with linear theory. Shortly, the PV turns over in the critical layer, and the core ellipticity starts to grow. Simultaneously, the cat’s eyes expand radially, and siphon potential vorticity from the core. Ultimately, the cyclone splits in two. The splitting is consistent with a simplified theory by Ford (1994b) and is analogous simulations of a 2D compressible vortex by Chan et al. (1993).
7. Concluding remarks
In this paper, we examined the pure spontaneous radiation (PSR) of IG waves from a monotonic shallow-water cyclone, with Ro greater than unity. Specifically, we considered radiation that is generated by a discrete VR wave in the vortex core. We verified that the radiation has positive feedback and can split the vortex if left unchecked. However, stirring of PV in the critical layer reduces the growth rate of the instability, and can even quench it. Quenching occurs only if
(i) the initial magnitude of the negative PV gradient, d
q /dr, is sufficiently large at r* so that linear critical layer damping prevails over linear radiative pumping [see Eq. (63)], and(ii) the nonlinear mixing rate Ωb of PV in the critical layer is less than the linear decay rate |γ| of the VR wave.
Conditions (i) and (ii) were derived in sections 4 and 5, respectively. Condition (ii) was verified by numerical experiments in section 6.
In addition to the above results, we analyzed the flow of conserved wave activity during nonlinear PSR, from a cyclone that is stable in linear theory. Figure 14 is a diagrammatic summary. On the one hand, the resonant disturbance of PV at r* tends to transfer positive wave activity from the VR wave into its critical layer. The result is damping of the VR wave. On the other hand, IG wave radiation involves the ejection of negative wave activity into the environment, compelling the VR wave to grow. As PV coils inside, the critical layer becomes less absorbent and periodically returns some of its wave activity. This enables radiative pumping to dominate.
Details of the cyclonic equilibrium and its perturbation determine whether or not conditions (i) and (ii) are satisfied. A general survey is beyond the scope of this paper, but some qualitative hypotheses are appropriate. Figure 4 indicates that discrete VR waves in smooth cyclones (those closer to Gaussian than square) are likely to satisfy condition (i). Past computational studies further suggest that deforming a smooth cyclone hardly excites discrete VR waves if the intrinsic Rossby deformation radius LR [Eq. (7)] is much less than the radius of maximum wind (e.g., Reasor et al. 2004). In this case, only a very intense impulse might suffice to excite a wave that violates condition (ii).
Section 2 proposed that tropical cyclones, supercell mesocyclones, polar lows, and island wake vortices can spontaneously radiate large-scale spiral IG waves. Such radiation may or may not prove significant in future studies. The contents of this paper may help explain why, but cannot provide the full answer. Atmospheric vortices, of course, are far more complex than inviscid monotonic shallow-water cyclones. They are generally baroclinic and have secondary (poloidal) circulations; in addition, they are often nonmonotonic. Furthermore, their perturbations are influenced by eddy viscosity, boundary layer drag, and moist diabatic processes. One should merely view this paper as another stepping-stone toward a more comprehensive theory of spontaneous radiation from vortical flows in the atmosphere.
Acknowledgments
This work was supported by NSF Grants ATM-0132006, ATM-0349980, ATM-0101781, and ONR Grant N00014-02-1-0474.
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APPENDIX A
Linearized Shallow-Water Model
APPENDIX B
Polarization Equations
APPENDIX C
Numerical Parameters
Table C1 lists the numerical parameters, defined below, that were used for all simulations (G1–H4) in this paper. Simulation H1-lr corresponds to the low-resolution vortex fission experiment. It has the same physical parameters as H1 (see Table 1).
Discretization parameters
The numerical model integrates the shallow-water equations on two concentric square grids.8 The inner fine grid has spacing δx and the outer coarse grid has spacing ΔX. Each side of the fine grid has length Lfg, and each side of the coarse grid has length Lcg. The flow is evolved forward in time using a fourth-order Runge–Kutta scheme, in increments of size δt.
Artificial damping parameters
APPENDIX D
Initialization
The above transformation reduces initialization to the simultaneous solution of two coupled Poisson equations (D2) and (D3), with periodic boundary conditions, and an integral equation [(D4)]. We solve a discretized version of this system for ψ and ϕ by iterative relaxation. The initial velocity field is then obtained from Eq. (D1).
APPENDIX E
Data Filtering
Two curves in Fig. 8, G3 and G4, are filtered signals of radiative VR waves. Filtering was necessary at late times when the VR wave amplitudes fell beneath the low-frequency noise threshold. The following explains how the filtered signals were obtained from the raw data.
To begin with, Fig. E1 contrasts the filtered and raw signals for both simulations. The top and bottom plots show the amplitude and phase of u2(rp, t) ∝ Ω2beiθu. Let us first consider the raw data. In both cases the exponential decay of the perturbation amplitude halts near t = 80. Simultaneously, the oscillation frequency, −dθu/dt, decreases by orders of magnitude. The dashed curves in all plots were obtained from a numerical integration of the linearized shallow-water equations. They accurately describe the dominant component of the raw signal; that is, the abrupt change at t = 80 is a linear effect.
Evidently, a slow discrete mode appears to overtake the signal of the radiative VR wave. This mode exists because the cyclone is only quasi monotonic: recall from section 3b that the PV jumps slightly at the distant radius r = rb. The slow mode is peaked at rb, but is nonzero in the vortex core, which contains the probe radius rp. It does not produce IG wave radiation because its frequency is merely one-tenth of the Coriolis parameter f.
The slow mode is of little interest to our discussion. Presumably, it has negligible influence on the much faster radiative mode. That is, it merely adds linearly to the signal. To remove the slow mode from the raw signal we follow a simple procedure. First, we fit the late-time measurements of u2(rp, t) to a function of the form aslowe−iωslowt. Here aslow and ωslow are complex parameters. We then subtract the fit function from the original u2(rp, t). We refer to the difference as the filtered signal.
For simulation G3, in which Ωb(0)/|γ| = 0.79, the filtering leaves a nearly monochromatic signal, corresponding to the radiative VR wave. For simulation G4, in which Ωb(0)/|γ| = 0.35, the filtered signal is less clean, especially past t = 160. Beyond this time, we can no longer assert that it represents the original VR wave. Moreover, we can no longer connect its amplitude to the mixing rate of the critical layer. A combination of physical waves and discretization noise likely accounts for the broadened frequency spectrum.
Estimated parameter regimes for a spectrum of atmospheric vortices and mesoscale oceanic eddies. Balanced shallow-water cyclones reside to the left of the dashed curve. The table on the right provides the characteristic scales that were used to evaluate the ranges of Ro, Fr, and ε for each type of vortex.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Balanced (a) Gaussian and (b) hyperbolic cyclones. All plots are in natural units [see Eq. (17)].
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Simulations of (top) sustained and (bottom) damped spontaneous radiation from hyperbolic cyclones with Δ = 0.1 and Δ = 0.3, respectively. The contour plots show the geopotential perturbation ϕ′. In both cases, a core VR wave excites an outward- propagating spiral IG wave in the environment. Section 4 explains why the radiation intensifies in the top case but decays in the bottom case. The graphs to the right show the VR wave amplitude and oscillation frequency vs time. In both cases, the evolution agrees with linear eigenmode theory (dashed curves), generalized to include quasimodes (see section 3e). The wave measurements were obtained from the n = 2 complex Fourier coefficient of the geopotential perturbation, ϕ2(r, t) ≡ |ϕ2|eiθϕ, evaluated at (top) r = 1.03ro and (bottom) r = ro. Times and lengths are in natural units; ϕmax is the initial maximum value of the geopotential (approximately the ambient geopotential).
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Growth rate γ of the n = 2 radiative VR wave vs the smoothness parameter Δ of a hyperbolic cyclone. Numerical solutions to the eigenmode/quasimode problem (○) are compared to analytical growth rates (×), where applicable. The dashed curves show the radiative pumping and critical layer damping contributions, ϵrad/M and −ϵcl/M, to the analytical growth rates [see Eq. (63)]. A growth rate of 0.1 in natural units corresponds to an e-folding time of 1.59 vortex rotation periods.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
The ϒ function that appears in the definition of angular pseudomomentum
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Wave activity densities for the n = 2 radiative mode of a hyperbolic cyclone, with Δ = 0.1. The solid curve is the quadratic approximation of Λq, whereas the dashed curve is Λυϕ. Both are scaled by the same arbitrary factor. Their sum, the total wave activity density, is mostly positive for the inner VR wave but is negative for the outer IG wave (see inset). The horizontal bar on the cross-hair at the base of the plot indicates the region that the M integral [Eq. (53)] is supposed to exclude (|r − r*| < 10lγ). Radius r is in natural units.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Basic critical layer dynamics. (a) Three snapshots of the instantaneous flow in the critical layer of a damped VR wave. The reference frame corotates with the wave. (b), (c) Radial motions of two fluid parcels placed near the central fixed point of the trapping region. Different curves correspond to different values of the initial wave amplitude (Ωb/|γ|).
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Nonlinear evolution of linearly damped, radiative VR waves. (left) Wave amplitude, measured by the ratio |Ωb/γ|2, vs time. The dashed curves indicate the early exponential decay that linear theory predicts. The cross-hair beneath each simulation curve indicates the initial bounce period, tb = 2π/Ωb(0), for that simulation. The asterisks indicate filtered data (see appendix E). (right) Potential vorticity vs time for simulations G1 and G3. See Tables 1 and C1 for simulation details. Time t is in natural units.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Potential vorticity stirring in the critical layer. (a) Simulation G1, Ωb(0)/|γ| > 1. (b) Simulation G3, Ωb(0)/|γ| < 1. Times and lengths are in natural units.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Nonlinear changes to the azimuthally averaged potential vorticity in the critical layer, centered at r* = 2.36. (a) Simulation G1, Ωb(0)/|γ| = 1.58. (b) Simulation G3, Ωb(0)/|γ| = 0.79. (c) Simulation G4, Ωb(0)/|γ| = 0.35. The solid curves represent the initial conditions, whereas the dashed curves correspond to late-time averages in the intervals (a) 138.7 < t < 427.5, (b) 138.7 < t < 345.0, and (c) 138.7 < t < 420.9.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
A closer look at the nonlinear evolution of a supercritical VR wave (simulation H3). (a) The undulatory growth of bulk wave activity (solid curve). The theoretical growth (dashed curve) is the gain by radiative pumping plus the loss by critical layer damping (dotted curves). (b) The growth of critical layer wave activity. By the end of the simulation, the measured value of δAcl (solid curve) is 30% less than the influx of wave activity into the critical layer (dashed curve). Hyperdiffusion is the likely cause. The wave activities in (a) and (b) are divided by the initial (positive) value of Ab. Time t is in natural units.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Fission of a cyclone, induced by IG wave radiation. Time t is in natural units, so t = 2π corresponds to one vortex rotation period.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Temporal decay of vortex energy due to radiation. The solid curves are total, kinetic, and available potential energy integrals, inside the vortex radius rυ. The dashed curve is the initial total energy minus the energy carried away by IG waves. All energies are normalized to the initial value of total energy.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Flow of wave activity during nonlinear PSR from a monotonic cyclone. Overall, the bulk VR wave ejects negative IG wave activity into the environment and deposits positive wave activity into the critical layer. At later times, the critical layer periodically sends some positive wave activity back to the VR wave. Such transfers are labeled secondary nonlinear exchanges (SNLE) in the diagram.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Fig. E1. Comparison of raw and filtered data for numerical experiments (a) G3 and (b) G4. The solid/dashed curves were obtained from nonlinear/linear simulations. Only nonlinear data were filtered. The data were sampled every 1.7 natural time units. The horizontal bars in the phase plots indicate the linear oscillation period of the n = 2 radiative VR wave.
Citation: Journal of the Atmospheric Sciences 63, 2; 10.1175/JAS3641.1
Physical simulation parameters: G1–H3 correspond to the simulations in Figs. 8 –13 and 15, and H4 corresponds to the top simulation in Fig. 3. All dimensional parameters are in natural units. Ωb and ltrap are evaluated at t = 0.
Table C1. Numerical simulation parameters, in natural units.
In addition, we generalized the method to find
Note also that the asymptotic wavefunction [Eq. (25)] of a quasimode ultimately increases with r, since γ < 0. We assert that such behavior is physical, since the outer radiation field travels at a finite radial velocity, and is generated by a central source (the VR wave) that decays exponentially with time.
In case rF does not vanish at infinity, we redefine dAen/dt to be the rate at which wave activity enters the environment only from the vortex [Eq. (48)]. This would differ from the time derivative of the improper integral in Eq. (43).
Similarly, Kop’ev and Leont’ev (1983, 1985, 1988) used an energy argument to explain spontaneous acoustic radiation. From their perspective, a negative energy VR wave grows by the ejection of a positive energy sound wave.
We have verified that the dashed curves in Fig. 4 are qualitatively the same for any vortex radius in the range r* + 10lγ < rυ < 48ro. We have not examined larger values of rυ.
Equations (69) are analogous to those of a nonlinear pendulum with a time-dependent gravity (or length) parameter.
Curves G3 and G4 were filtered at late times to remove low-frequency noise. Appendix E describes the filtering technique.
With staggering, there are actually eight total grids.