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- Author or Editor: Michael E. McIntyre x

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## Abstract

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## Abstract

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## Abstract

After reviewing the background, this article discusses the recently discovered examples of hybrid propagating structures consisting of vortex dipoles and comoving gravity waves undergoing wave capture. It is shown how these examples fall outside the scope of the Lighthill theory of spontaneous imbalance and, concomitantly, outside the scope of shallow-water dynamics. Besides the fact that going from shallow-water to continuous stratification allows disparate vertical scales—small for inertia–gravity waves and large for vortical motion—the key points are 1) that by contrast with cases covered by the Lighthill theory, the wave source feels a substantial radiation reaction when Rossby numbers *R* ≳ 1, so that the source cannot be prescribed in advance; 2) that examples of this sort may supply exceptions to the general rule that spontaneous imbalance is exponentially small in *R*; and 3) that unsteady vortical motion in continuous stratification can stay close to balance thanks to three quite separate mechanisms. These are as follows: first, the near-suppression, by the Lighthill mechanism, of large-scale imbalance (inertia–gravity waves of large horizontal scale), where “large” means large relative to a Rossby deformation length *L _{D}
* characterizing the vortical motion; second, the flaccidity, and hence near-steadiness, of

*L*-wide jets that meander and form loops, Gulf-Stream-like, on streamwise scales ≫

_{D}*L*; and third, the dissipation of small-scale imbalance by wave capture leading to wave breaking, which is generically probable in an environment of random shear and straining. Shallow-water models include the first two mechanisms but exclude the third.

_{D}## Abstract

After reviewing the background, this article discusses the recently discovered examples of hybrid propagating structures consisting of vortex dipoles and comoving gravity waves undergoing wave capture. It is shown how these examples fall outside the scope of the Lighthill theory of spontaneous imbalance and, concomitantly, outside the scope of shallow-water dynamics. Besides the fact that going from shallow-water to continuous stratification allows disparate vertical scales—small for inertia–gravity waves and large for vortical motion—the key points are 1) that by contrast with cases covered by the Lighthill theory, the wave source feels a substantial radiation reaction when Rossby numbers *R* ≳ 1, so that the source cannot be prescribed in advance; 2) that examples of this sort may supply exceptions to the general rule that spontaneous imbalance is exponentially small in *R*; and 3) that unsteady vortical motion in continuous stratification can stay close to balance thanks to three quite separate mechanisms. These are as follows: first, the near-suppression, by the Lighthill mechanism, of large-scale imbalance (inertia–gravity waves of large horizontal scale), where “large” means large relative to a Rossby deformation length *L _{D}
* characterizing the vortical motion; second, the flaccidity, and hence near-steadiness, of

*L*-wide jets that meander and form loops, Gulf-Stream-like, on streamwise scales ≫

_{D}*L*; and third, the dissipation of small-scale imbalance by wave capture leading to wave breaking, which is generically probable in an environment of random shear and straining. Shallow-water models include the first two mechanisms but exclude the third.

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## Abstract

Several different kinds of accurate potential vorticity (PV) inversion operators, and the associated balanced models, are tested for the shallow water equations on a hemisphere in an attempt to approach the ultimate limitations of the balance, inversion, and slow-manifold concepts. The accuracies achieved are far higher than for standard balanced models accurate to one or two orders in Rossby number *R* or Froude number *F* (where *F* = |**u**|/*c*; |**u**| = flow speed; and *c* = gravity wave speed). Numerical inversions, and corresponding balanced-model integrations testing cumulative accuracy, are carried out for cases that include substantial PV anomalies in the Tropics. The balanced models in question are constructed so as to be exactly PV conserving and to have unique velocity fields (implying, incidentally, that they cannot be Hamiltonian). Mean layer depths of 1 and 2 km are tested.

The results show that, in the cases studied, the dynamical information contained in PV distributions is remarkably close to being complete even though *R* = ∞ at the equator and even though local maximum Froude numbers, *F*
_{max}, approach unity in some cases. For example, in a 10-day integration of the balanced model corresponding to one of the most accurate inversion operators, “third-order normal mode inversion,” the mean depth was 1 km, the minimum depth less than 0.5 km, and *F*
_{max} ≃ 0.7, hardly small in comparison with unity. At the end of 10 days of integration, the cumulative rms error in the layer depth was less than 15 m, that is, less than 5% of the typical rms spatial variation of 310 m. At the end of the first day of integration the rms error was 5 m, that is, less than 2%. Here “error” refers to a comparison between the results of a balanced integration and those of a corresponding primitive equation integration initialized to have low gravity wave activity on day 0. Contour maps of the PV distributions remained almost indistinguishable by eye over the 10-day period. This remarkable cumulative accuracy, far beyond anything that could have been expected from standard scale analysis, is probably related to the weakness of the spontaneous-adjustment emission or “Lighthill radiation” studied in the companion paper by

## Abstract

Several different kinds of accurate potential vorticity (PV) inversion operators, and the associated balanced models, are tested for the shallow water equations on a hemisphere in an attempt to approach the ultimate limitations of the balance, inversion, and slow-manifold concepts. The accuracies achieved are far higher than for standard balanced models accurate to one or two orders in Rossby number *R* or Froude number *F* (where *F* = |**u**|/*c*; |**u**| = flow speed; and *c* = gravity wave speed). Numerical inversions, and corresponding balanced-model integrations testing cumulative accuracy, are carried out for cases that include substantial PV anomalies in the Tropics. The balanced models in question are constructed so as to be exactly PV conserving and to have unique velocity fields (implying, incidentally, that they cannot be Hamiltonian). Mean layer depths of 1 and 2 km are tested.

The results show that, in the cases studied, the dynamical information contained in PV distributions is remarkably close to being complete even though *R* = ∞ at the equator and even though local maximum Froude numbers, *F*
_{max}, approach unity in some cases. For example, in a 10-day integration of the balanced model corresponding to one of the most accurate inversion operators, “third-order normal mode inversion,” the mean depth was 1 km, the minimum depth less than 0.5 km, and *F*
_{max} ≃ 0.7, hardly small in comparison with unity. At the end of 10 days of integration, the cumulative rms error in the layer depth was less than 15 m, that is, less than 5% of the typical rms spatial variation of 310 m. At the end of the first day of integration the rms error was 5 m, that is, less than 2%. Here “error” refers to a comparison between the results of a balanced integration and those of a corresponding primitive equation integration initialized to have low gravity wave activity on day 0. Contour maps of the PV distributions remained almost indistinguishable by eye over the 10-day period. This remarkable cumulative accuracy, far beyond anything that could have been expected from standard scale analysis, is probably related to the weakness of the spontaneous-adjustment emission or “Lighthill radiation” studied in the companion paper by

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## Abstract

The propagation of Kelvin–Helmholtz (KH) shear-generated gravity waves through the summer stratosphere is investigated using ray tracing, taking into account back-reflection as well as wave dissipation due to precritical-layer breaking, radiative damping, and viscous diffusion. It is found that the transmission rate of upward–eastward waves to the mesosphere is surprisingly good, provided that the horizontal scale of the waves is large enough to prevent back-reflection. A rough upper bound on the net momentum flux into the mesosphere due to a large ensemble of mutually incoherent KH-generated clear-air turbulence events is then estimated, using reasonable-looking assumptions about the statistics of such events. It is found that on this basis the wave source cannot safely be neglected in the global angular momentum budget.

## Abstract

The propagation of Kelvin–Helmholtz (KH) shear-generated gravity waves through the summer stratosphere is investigated using ray tracing, taking into account back-reflection as well as wave dissipation due to precritical-layer breaking, radiative damping, and viscous diffusion. It is found that the transmission rate of upward–eastward waves to the mesosphere is surprisingly good, provided that the horizontal scale of the waves is large enough to prevent back-reflection. A rough upper bound on the net momentum flux into the mesosphere due to a large ensemble of mutually incoherent KH-generated clear-air turbulence events is then estimated, using reasonable-looking assumptions about the statistics of such events. It is found that on this basis the wave source cannot safely be neglected in the global angular momentum budget.

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## Abstract

This paper considers stratified and shallow water non-Hamiltonian potential-vorticity-based balanced models (PBMs). These are constructed using the exact (Rossby or Rossby–Ertel) potential vorticity (PV). The most accurate known PBMs are those studied by McIntyre and Norton and by Mohebalhojeh and Dritschel. It is proved that, despite their astonishing accuracy, these PBMs all fail to conserve mass locally. Specifically, they exhibit velocity splitting in the sense of having two velocity fields, **v** and **v**
*
_{m}
*, the first to advect PV and the second to advect mass. The difference

**v**−

**v**

*is nonzero in general, even if tiny. Unlike the different velocity splitting found in all Hamiltonian balanced models, the present splitting can be healed. The result is a previously unknown class of balanced models, here called “hyperbalance equations,” whose formal orders of accuracy can be made as high as those of any other PBM. The hyperbalance equations use a single velocity field*

_{m}**v**to advect mass as well as to advect and evaluate the exact PV.

## Abstract

This paper considers stratified and shallow water non-Hamiltonian potential-vorticity-based balanced models (PBMs). These are constructed using the exact (Rossby or Rossby–Ertel) potential vorticity (PV). The most accurate known PBMs are those studied by McIntyre and Norton and by Mohebalhojeh and Dritschel. It is proved that, despite their astonishing accuracy, these PBMs all fail to conserve mass locally. Specifically, they exhibit velocity splitting in the sense of having two velocity fields, **v** and **v**
*
_{m}
*, the first to advect PV and the second to advect mass. The difference

**v**−

**v**

*is nonzero in general, even if tiny. Unlike the different velocity splitting found in all Hamiltonian balanced models, the present splitting can be healed. The result is a previously unknown class of balanced models, here called “hyperbalance equations,” whose formal orders of accuracy can be made as high as those of any other PBM. The hyperbalance equations use a single velocity field*

_{m}**v**to advect mass as well as to advect and evaluate the exact PV.

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## Abstract

The effects of enforcing local mass conservation on the accuracy of non-Hamiltonian potential-vorticity- based balanced models (PBMs) are examined numerically for a set of chaotic shallow-water *f*-plane vortical flows in a doubly periodic square domain. The flows are spawned by an unstable jet and all have domain-maximum Froude and Rossby numbers Fr ∼0.5 and Ro ∼1, far from the usual asymptotic limits Ro → 0, Fr → 0, with Fr defined in the standard way as flow speed over gravity wave speed. The PBMs considered are the plain and hyperbalance PBMs defined in Part I. More precisely, they are the plain-*δδ*, plain-*γγ*, and plain-*δγ* PBMs and the corresponding hyperbalance PBMs, of various orders, where “order” is related to the number of time derivatives of the divergence equation used in defining balance and potential-vorticity inversion. For brevity the corresponding hyperbalance PBMs are called the hyper-*δδ*, hyper-*γγ*, and hyper-*δγ* PBMs, respectively. As proved in Part I, except for the leading-order plain-*γγ* each plain PBM violates local mass conservation. Each hyperbalance PBM results from enforcing local mass conservation on the corresponding plain PBM. The process of thus deriving a hyperbalance PBM from a plain PBM is referred to for brevity as plain-to-hyper conversion. The question is whether such conversion degrades the accuracy, as conjectured by McIntyre and Norton.

Cumulative accuracy is tested by running each PBM alongside a suitably initialized primitive equation (PE) model for up to 30 days, corresponding to many vortex rotations. The accuracy is sensitively measured by the smallness of the ratio *ϵ* = ||*Q*
_{PBM} − *Q*
_{PE}||_{2}/||*Q*
_{PE}||_{2}, where *Q*
_{PBM} and *Q*
_{PE} denote the potential vorticity fields of the PBM and the PEs, respectively, and || ||_{2} is the *L*
_{2} norm. At 30 days the most accurate PBMs have *ϵ* ≈ 10^{−2} with PV fields hardly distinguishable visually from those of the PEs, even down to tiny details. Most accurate is defined by minimizing *ϵ* over all orders and truncation types *δδ*, *γγ*, and *δγ*. Contrary to McIntyre and Norton’s conjecture, the minimal *ϵ* values did not differ systematically or significantly between plain and hyperbalance PBMs. The smallness of *ϵ* suggests that the slow manifolds defined by the balance relations of the most accurate PBMs, both plain and hyperbalance, are astonishingly close to being invariant manifolds of the PEs, at least throughout those parts of phase space for which Ro ≲ 1 and Fr ≲ 0.5.

As another way of quantifying the departures from such invariance, that is, of quantifying the fuzziness of the PEs’ slow quasimanifold, initialization experiments starting at days 1, 2, . . . 10 were carried out in which attention was focused on the amplitudes of inertia–gravity waves representing the imbalance arising in 1-day PE runs. With balance defined by the most accurate PBMs, and imbalance by departures therefrom, the results of the initialization experiments suggest a negative correlation between early imbalance and late cumulative error *ϵ*. In such near-optimal conditions the imbalance seems to be acting like weak background noise producing an effect analogous to so-called stochastic resonance, in that a slight increase in noise level brings PE behavior closer to the balanced behavior defined by the most accurate PBMs when measured cumulatively over 30 days.

## Abstract

The effects of enforcing local mass conservation on the accuracy of non-Hamiltonian potential-vorticity- based balanced models (PBMs) are examined numerically for a set of chaotic shallow-water *f*-plane vortical flows in a doubly periodic square domain. The flows are spawned by an unstable jet and all have domain-maximum Froude and Rossby numbers Fr ∼0.5 and Ro ∼1, far from the usual asymptotic limits Ro → 0, Fr → 0, with Fr defined in the standard way as flow speed over gravity wave speed. The PBMs considered are the plain and hyperbalance PBMs defined in Part I. More precisely, they are the plain-*δδ*, plain-*γγ*, and plain-*δγ* PBMs and the corresponding hyperbalance PBMs, of various orders, where “order” is related to the number of time derivatives of the divergence equation used in defining balance and potential-vorticity inversion. For brevity the corresponding hyperbalance PBMs are called the hyper-*δδ*, hyper-*γγ*, and hyper-*δγ* PBMs, respectively. As proved in Part I, except for the leading-order plain-*γγ* each plain PBM violates local mass conservation. Each hyperbalance PBM results from enforcing local mass conservation on the corresponding plain PBM. The process of thus deriving a hyperbalance PBM from a plain PBM is referred to for brevity as plain-to-hyper conversion. The question is whether such conversion degrades the accuracy, as conjectured by McIntyre and Norton.

Cumulative accuracy is tested by running each PBM alongside a suitably initialized primitive equation (PE) model for up to 30 days, corresponding to many vortex rotations. The accuracy is sensitively measured by the smallness of the ratio *ϵ* = ||*Q*
_{PBM} − *Q*
_{PE}||_{2}/||*Q*
_{PE}||_{2}, where *Q*
_{PBM} and *Q*
_{PE} denote the potential vorticity fields of the PBM and the PEs, respectively, and || ||_{2} is the *L*
_{2} norm. At 30 days the most accurate PBMs have *ϵ* ≈ 10^{−2} with PV fields hardly distinguishable visually from those of the PEs, even down to tiny details. Most accurate is defined by minimizing *ϵ* over all orders and truncation types *δδ*, *γγ*, and *δγ*. Contrary to McIntyre and Norton’s conjecture, the minimal *ϵ* values did not differ systematically or significantly between plain and hyperbalance PBMs. The smallness of *ϵ* suggests that the slow manifolds defined by the balance relations of the most accurate PBMs, both plain and hyperbalance, are astonishingly close to being invariant manifolds of the PEs, at least throughout those parts of phase space for which Ro ≲ 1 and Fr ≲ 0.5.

As another way of quantifying the departures from such invariance, that is, of quantifying the fuzziness of the PEs’ slow quasimanifold, initialization experiments starting at days 1, 2, . . . 10 were carried out in which attention was focused on the amplitudes of inertia–gravity waves representing the imbalance arising in 1-day PE runs. With balance defined by the most accurate PBMs, and imbalance by departures therefrom, the results of the initialization experiments suggest a negative correlation between early imbalance and late cumulative error *ϵ*. In such near-optimal conditions the imbalance seems to be acting like weak background noise producing an effect analogous to so-called stochastic resonance, in that a slight increase in noise level brings PE behavior closer to the balanced behavior defined by the most accurate PBMs when measured cumulatively over 30 days.

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## Abstract

A longstanding mystery about Jupiter has been the straightness and steadiness of its weather-layer jets, quite unlike terrestrial strong jets with their characteristic unsteadiness and long-wavelength meandering. The problem is addressed in two steps. The first is to take seriously the classic Dowling–Ingersoll (DI) 1½-layer scenario and its supporting observational evidence, pointing toward deep, massive, zonally symmetric zonal jets in the underlying dry-convective layer. The second is to improve the realism of the model stochastic forcing used to represent the effects of Jupiter’s moist convection, as far as possible within the 1½-layer dynamics of the DI scenario. The real moist convection should be strongest in the belts where the interface to the deep flow is highest and coldest and should generate cyclones as well as anticyclones, with the anticyclones systematically stronger. The new model forcing reflects these insights. Also, it acts quasi frictionally on large scales to produce statistically steady turbulent weather-layer regimes without any need for explicit large-scale dissipation, and with weather-layer jets that are approximately straight thanks to the influence of the deep jets, allowing shear stability despite nonmonotonic potential vorticity gradients when the Rossby deformation length scale is not too large. Moderately strong forcing produces chaotic vortex dynamics and realistic belt–zone contrasts in the model’s convective activity, through an eddy-induced sharpening and strengthening of the weather-layer jets relative to the deep jets, tilting the interface between them. Weak forcing, for which the only jet-sharpening mechanism is the passive, Kelvin shearing of vortices (as in the zonostrophic instability mechanism), produces unrealistic belt–zone contrasts.

## Abstract

A longstanding mystery about Jupiter has been the straightness and steadiness of its weather-layer jets, quite unlike terrestrial strong jets with their characteristic unsteadiness and long-wavelength meandering. The problem is addressed in two steps. The first is to take seriously the classic Dowling–Ingersoll (DI) 1½-layer scenario and its supporting observational evidence, pointing toward deep, massive, zonally symmetric zonal jets in the underlying dry-convective layer. The second is to improve the realism of the model stochastic forcing used to represent the effects of Jupiter’s moist convection, as far as possible within the 1½-layer dynamics of the DI scenario. The real moist convection should be strongest in the belts where the interface to the deep flow is highest and coldest and should generate cyclones as well as anticyclones, with the anticyclones systematically stronger. The new model forcing reflects these insights. Also, it acts quasi frictionally on large scales to produce statistically steady turbulent weather-layer regimes without any need for explicit large-scale dissipation, and with weather-layer jets that are approximately straight thanks to the influence of the deep jets, allowing shear stability despite nonmonotonic potential vorticity gradients when the Rossby deformation length scale is not too large. Moderately strong forcing produces chaotic vortex dynamics and realistic belt–zone contrasts in the model’s convective activity, through an eddy-induced sharpening and strengthening of the weather-layer jets relative to the deep jets, tilting the interface between them. Weak forcing, for which the only jet-sharpening mechanism is the passive, Kelvin shearing of vortices (as in the zonostrophic instability mechanism), produces unrealistic belt–zone contrasts.

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## Abstract

An initial zonally symmetric quasigeostrophic potential vorticity (PV) distribution *q _{i}
*(

*y*) is subjected to complete or partial mixing within some finite zone |

*y*| <

*L*, where

*y*is latitude. The change in

*M*, the total absolute angular momentum, between the initial and any later time is considered. For standard quasigeostrophic shallow-water beta-channel dynamics it is proved that, for any

*q*(

_{i}*y*) such that

*dq*/

_{i}*dy*> 0 throughout |

*y*| <

*L*, the change in

*M*is always negative. This theorem holds even when “mixing” is understood in the most general possible sense. Arbitrary stirring or advective rearrangement is included, combined to an arbitrary extent with spatially inhomogeneous diffusion. The theorem holds whether or not the PV distribution is zonally symmetric at the later time. The same theorem governs Boussinesq potential-energy changes due to buoyancy mixing in the vertical. For the standard quasigeostrophic beta-channel dynamics to be valid the Rossby deformation length

*L*≫

_{D}*ϵL*where

*ϵ*is the Rossby number; when

*L*= ∞ the theorem applies not only to the beta channel but also to a single barotropic layer on the full sphere, as considered in the recent work of Dunkerton and Scott on “PV staircases.” It follows that the

_{D}*M*-conserving PV reconfigurations studied by those authors must involve processes describable as PV unmixing, or antidiffusion, in the sense of time-reversed diffusion. Ordinary jet self-sharpening and jet-core acceleration do not, by contrast, require unmixing, as is shown here by detailed analysis. Mixing in the jet flanks suffices. The theorem extends to multiple layers and continuous stratification. A least upper bound and greatest lower bound for the change in

*M*is obtained for cases in which

*q*is neither monotonic nor zonally symmetric. A corollary is a new nonlinear stability theorem for shear flows.

_{i}## Abstract

An initial zonally symmetric quasigeostrophic potential vorticity (PV) distribution *q _{i}
*(

*y*) is subjected to complete or partial mixing within some finite zone |

*y*| <

*L*, where

*y*is latitude. The change in

*M*, the total absolute angular momentum, between the initial and any later time is considered. For standard quasigeostrophic shallow-water beta-channel dynamics it is proved that, for any

*q*(

_{i}*y*) such that

*dq*/

_{i}*dy*> 0 throughout |

*y*| <

*L*, the change in

*M*is always negative. This theorem holds even when “mixing” is understood in the most general possible sense. Arbitrary stirring or advective rearrangement is included, combined to an arbitrary extent with spatially inhomogeneous diffusion. The theorem holds whether or not the PV distribution is zonally symmetric at the later time. The same theorem governs Boussinesq potential-energy changes due to buoyancy mixing in the vertical. For the standard quasigeostrophic beta-channel dynamics to be valid the Rossby deformation length

*L*≫

_{D}*ϵL*where

*ϵ*is the Rossby number; when

*L*= ∞ the theorem applies not only to the beta channel but also to a single barotropic layer on the full sphere, as considered in the recent work of Dunkerton and Scott on “PV staircases.” It follows that the

_{D}*M*-conserving PV reconfigurations studied by those authors must involve processes describable as PV unmixing, or antidiffusion, in the sense of time-reversed diffusion. Ordinary jet self-sharpening and jet-core acceleration do not, by contrast, require unmixing, as is shown here by detailed analysis. Mixing in the jet flanks suffices. The theorem extends to multiple layers and continuous stratification. A least upper bound and greatest lower bound for the change in

*M*is obtained for cases in which

*q*is neither monotonic nor zonally symmetric. A corollary is a new nonlinear stability theorem for shear flows.

_{i}^{ }

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## Abstract

The ultimate limitations of the balance, slow-manifold, and potential vorticity inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneous-adjustment emission, or Lighthill radiation, of inertia–gravity waves by unsteady, two-dimensional or layerwise-two-dimensional vortical flow (the wave emission mechanism sometimes being called “geostrophic” adjustment even though it need not take the flow toward geostrophic balance). Spontaneous-adjustment emission is studied in detail for the case of unbounded *f*-plane shallow-water flow, in which the potential vorticity anomalies are confined to a finite-sized region, but whose distribution within the region is otherwise completely general. The approach assumes that the Froude number **F**
**R**
**F**
**R**
*O*(**F**
^{4}), which is a high enough order to describe not only the weakly emitted waves but also, explicitly, the correspondingly weak radiation reaction upon the vortical flow, accounting for the loss of vortical energy. Exact evolution on a slow manifold, in its usual strict sense, would be incompatible with the arrow of time introduced by this radiation reaction and energy loss. The magnitude *O*(**F**
^{4}) of the radiation reaction may thus be taken to measure the degree of “fuzziness” of the entity that must exist in place of the strict slow manifold. That entity must, presumably, be not a simple invariant manifold, but rather an *O*(**F**
^{4})-thin, multileaved, fractal “stochastic layer” like those known for analogous but low-order coupled oscillator systems. It could more appropriately be called the “slow quasimanifold.”

## Abstract

The ultimate limitations of the balance, slow-manifold, and potential vorticity inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneous-adjustment emission, or Lighthill radiation, of inertia–gravity waves by unsteady, two-dimensional or layerwise-two-dimensional vortical flow (the wave emission mechanism sometimes being called “geostrophic” adjustment even though it need not take the flow toward geostrophic balance). Spontaneous-adjustment emission is studied in detail for the case of unbounded *f*-plane shallow-water flow, in which the potential vorticity anomalies are confined to a finite-sized region, but whose distribution within the region is otherwise completely general. The approach assumes that the Froude number **F**
**R**
**F**
**R**
*O*(**F**
^{4}), which is a high enough order to describe not only the weakly emitted waves but also, explicitly, the correspondingly weak radiation reaction upon the vortical flow, accounting for the loss of vortical energy. Exact evolution on a slow manifold, in its usual strict sense, would be incompatible with the arrow of time introduced by this radiation reaction and energy loss. The magnitude *O*(**F**
^{4}) of the radiation reaction may thus be taken to measure the degree of “fuzziness” of the entity that must exist in place of the strict slow manifold. That entity must, presumably, be not a simple invariant manifold, but rather an *O*(**F**
^{4})-thin, multileaved, fractal “stochastic layer” like those known for analogous but low-order coupled oscillator systems. It could more appropriately be called the “slow quasimanifold.”