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- Author or Editor: M. E. McIntyre x

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## Abstract

Shear instabilities can result in waves being radiated far from the seat of instability. This phenomenon, and its opposite extreme, the trapping of instabilities, can be described and interdistinguished by linear theory for slowly growing instabilities only. To describe strong instabilities which radiate, laboratory and numerical studies are needed. Linear theory can better describe the somewhat similar phenomenon of “resonant overreflection,” which for a constant incident wave is characterized by linear rather than exponential growth in fine.

The energy budget of sustained radiation, or overreflection, from a shear layer does not necessarily involve net removal of energy from the neighborhood of the shear layer.

## Abstract

Shear instabilities can result in waves being radiated far from the seat of instability. This phenomenon, and its opposite extreme, the trapping of instabilities, can be described and interdistinguished by linear theory for slowly growing instabilities only. To describe strong instabilities which radiate, laboratory and numerical studies are needed. Linear theory can better describe the somewhat similar phenomenon of “resonant overreflection,” which for a constant incident wave is characterized by linear rather than exponential growth in fine.

The energy budget of sustained radiation, or overreflection, from a shear layer does not necessarily involve net removal of energy from the neighborhood of the shear layer.

## Abstract

The one-dimensional propagation of a spectrum of gravity waves through a realistic middle atmosphere is investigated, separating as far as possible the propagation-invariant aspects from the more empirical wave-breaking and other nonlinear aspects. The latter are parameterized by a simple broadband spectral saturation criterion, but the conceptual framework allows for other wave-breaking parameterizations. An upward propagating initial or “launch” spectrum is prescribed in the lower stratosphere. The propagation aspects are handled with careful attention to the mappings and their Jacobians between spectral spaces.

Results for several test cases produce realistic behavior, including cases where some of the waves are back-reflected, as in the summer stratosphere, with much of the spectrum propagating conservatively through substantial altitude ranges. Any launch spectrum can be used in the computational scheme; for definiteness attention is concentrated on the model spectrum of Fritts and VanZandt, but sensitivity tests are also carried out in which the shape and total energy are varied. Other sensitivity tests include varying the steepness of the saturation criterion. The shapes and magnitudes of the computed profiles of wave-induced force, as a function of altitude, are sensitive to some of these changes, especially to the asymptotic shape of the launch spectrum at the smallest values of vertical wavenumber *m*, about which there is little direct observational evidence. However, the maxima and minima of the profiles are located at similar altitudes in each case. Besides pointing, to ways of improving gravity wave parameterization schemes for general circulation models, the results may help to tighten observational constraints on spectra for small *m*.

## Abstract

The one-dimensional propagation of a spectrum of gravity waves through a realistic middle atmosphere is investigated, separating as far as possible the propagation-invariant aspects from the more empirical wave-breaking and other nonlinear aspects. The latter are parameterized by a simple broadband spectral saturation criterion, but the conceptual framework allows for other wave-breaking parameterizations. An upward propagating initial or “launch” spectrum is prescribed in the lower stratosphere. The propagation aspects are handled with careful attention to the mappings and their Jacobians between spectral spaces.

Results for several test cases produce realistic behavior, including cases where some of the waves are back-reflected, as in the summer stratosphere, with much of the spectrum propagating conservatively through substantial altitude ranges. Any launch spectrum can be used in the computational scheme; for definiteness attention is concentrated on the model spectrum of Fritts and VanZandt, but sensitivity tests are also carried out in which the shape and total energy are varied. Other sensitivity tests include varying the steepness of the saturation criterion. The shapes and magnitudes of the computed profiles of wave-induced force, as a function of altitude, are sensitive to some of these changes, especially to the asymptotic shape of the launch spectrum at the smallest values of vertical wavenumber *m*, about which there is little direct observational evidence. However, the maxima and minima of the profiles are located at similar altitudes in each case. Besides pointing, to ways of improving gravity wave parameterization schemes for general circulation models, the results may help to tighten observational constraints on spectra for small *m*.

## Abstract

The theorems, which exhibit the role of wave dissipation, excitation and transience in the forcing of mean flow changes of second order in wave amplitude by arbitrary, small-amplitude disturbances, are obtained 1) for the primitive equations in pressure coordinates on a sphere, and 2) in a more general form (applicable for instance to nonhydrostatic disturbances in tornadoes or hurricanes) establishing that no approximations beyond axisymmetry of the mean flow are necessary. It is shown how the results reduce to those found by Boyd (1976) for the case of sinusoidal, hydrostatic waves with exponentially growing or decaying amplitude, and it is explained why the approximation used by Boyd in the thermodynamic equation is not needed. The reduction to Boyd's results entails the use of a virial theorem. This theorem amounts to a generalization of the “equipartition” law derived in an earlier paper (Andrews and Mclntyre, 1976). That derivation appeared to rely on an assumption about relative phases of disturbance Fourier components; the present derivation shows that no such assumption is in fact necessary.

## Abstract

The theorems, which exhibit the role of wave dissipation, excitation and transience in the forcing of mean flow changes of second order in wave amplitude by arbitrary, small-amplitude disturbances, are obtained 1) for the primitive equations in pressure coordinates on a sphere, and 2) in a more general form (applicable for instance to nonhydrostatic disturbances in tornadoes or hurricanes) establishing that no approximations beyond axisymmetry of the mean flow are necessary. It is shown how the results reduce to those found by Boyd (1976) for the case of sinusoidal, hydrostatic waves with exponentially growing or decaying amplitude, and it is explained why the approximation used by Boyd in the thermodynamic equation is not needed. The reduction to Boyd's results entails the use of a virial theorem. This theorem amounts to a generalization of the “equipartition” law derived in an earlier paper (Andrews and Mclntyre, 1976). That derivation appeared to rely on an assumption about relative phases of disturbance Fourier components; the present derivation shows that no such assumption is in fact necessary.

## Abstract

If the partial analogy between the behavior of Rossby-Ertel potential vorticity (PV) and the behavior of chemical tracers is to be correctly used in the general case of diabatic, frictional motion, then certain fundamental differences, as well as similarities, between the behavior of PV and that of chemical tracers must be recognized. These differences stem from the well-known kinematical relationship between PV and isentropic circulation (via Stokes' theorem), which has no counterpart for chemical substances.

One way of stating the analogy while recognizing the differences is to say first that PV behaves like the mixing ratio of a peculiar chemical “substance” that has zero source; i.e., is exactly conserved, away from boundaries (conserved not in the material or Lagrangian sense, but in the general sense associated with the idea of an indestructible chemical substance), and second that isentropic surfaces behave exactly as if they were impermeable to this “PV-substance” or “PVS,” even when diabatic heating or cooling, including that associated with turbulent mixing, makes them permeable to mass and chemical substances. In this respect isentropic surfaces can be said to act like semipermeable membranes. The PV itself can of course change, as can the mixing ratio of an exactly conserved chemical substance or decay-corrected radioactive tracer. For instance, all these mixing ratios can change by dilution when cumulonimbus clouds penetrate isentropic surfaces in a tropopause fold.

The net flux or transport of PVS along isentropic surfaces can be either up or down any pre-existing isentropic gradient of PV. For instance the typical effect of the small-scale turbulence due to breaking internal gravity waves is to transport PVS along isentropes in a gradient-independent sense, while transporting chemical substances across isentropes in a downgradient sense. It is the turbulent transport of PVS along isentropes that gives rise to the phenomenon of gravity-wave drag. Such a transport is absent from the formulation given in Danielsen (1990), which supposes that PV always behaves like the mixing ratio of a chemical even in three-dimensionally turbulent flow. The latter supposition is demonstrably incorrect.

## Abstract

If the partial analogy between the behavior of Rossby-Ertel potential vorticity (PV) and the behavior of chemical tracers is to be correctly used in the general case of diabatic, frictional motion, then certain fundamental differences, as well as similarities, between the behavior of PV and that of chemical tracers must be recognized. These differences stem from the well-known kinematical relationship between PV and isentropic circulation (via Stokes' theorem), which has no counterpart for chemical substances.

One way of stating the analogy while recognizing the differences is to say first that PV behaves like the mixing ratio of a peculiar chemical “substance” that has zero source; i.e., is exactly conserved, away from boundaries (conserved not in the material or Lagrangian sense, but in the general sense associated with the idea of an indestructible chemical substance), and second that isentropic surfaces behave exactly as if they were impermeable to this “PV-substance” or “PVS,” even when diabatic heating or cooling, including that associated with turbulent mixing, makes them permeable to mass and chemical substances. In this respect isentropic surfaces can be said to act like semipermeable membranes. The PV itself can of course change, as can the mixing ratio of an exactly conserved chemical substance or decay-corrected radioactive tracer. For instance, all these mixing ratios can change by dilution when cumulonimbus clouds penetrate isentropic surfaces in a tropopause fold.

The net flux or transport of PVS along isentropic surfaces can be either up or down any pre-existing isentropic gradient of PV. For instance the typical effect of the small-scale turbulence due to breaking internal gravity waves is to transport PVS along isentropes in a gradient-independent sense, while transporting chemical substances across isentropes in a downgradient sense. It is the turbulent transport of PVS along isentropes that gives rise to the phenomenon of gravity-wave drag. Such a transport is absent from the formulation given in Danielsen (1990), which supposes that PV always behaves like the mixing ratio of a chemical even in three-dimensionally turbulent flow. The latter supposition is demonstrably incorrect.

## Abstract

Using a new generalization of the Eliassen-Palm relations, we discuss the zonal-mean-flow tendency ∂*ū*/∂*t* due to waves in a stratified, rotating atmosphere, with particular attention to equatorially trapped modes. Wave transience, forcing and dissipation are taken into account in a very general way. The theory makes it possible to discuss the latitudinal (*y*) and vertical (*z*) dependence of ∂*ū*/∂*t* qualitatively and calculate it directly from an approximate knowledge of the wave structure. For equatorial modes it reveals that the *y* profile of ∂*ū*/∂*t* is strongly dependent on the nature of the forcing or dissipation mechanism. A by-product of the theory is a far-reaching generalization of the theorems of Charney-Drazin, Dickinson and Holton on the forcing of ∂*ū*/∂*t* by conservative linear waves.

Implications for the quasi-biennial oscillation in the equatorial stratosphere are discussed. Graphs of *y* profiles of ∂*ū*/∂*t* are given for the equatorial waves considered in the recent analysis of observational data by Lindzen and Tsay (1975). The *y* profile of ∂*ū*/∂*t* for Rossby-gravity and inertio-gravity modes, in Lindzen and Tsay's parameter ranges, prove extremely sensitive to whether or not small amounts of mechanical dissipation are present alongside the radiative-photochemical dissipation of the waves.

The probable importance of low-frequency Rossby waves for the momentum budget of the descending easterlies is suggested.

## Abstract

Using a new generalization of the Eliassen-Palm relations, we discuss the zonal-mean-flow tendency ∂*ū*/∂*t* due to waves in a stratified, rotating atmosphere, with particular attention to equatorially trapped modes. Wave transience, forcing and dissipation are taken into account in a very general way. The theory makes it possible to discuss the latitudinal (*y*) and vertical (*z*) dependence of ∂*ū*/∂*t* qualitatively and calculate it directly from an approximate knowledge of the wave structure. For equatorial modes it reveals that the *y* profile of ∂*ū*/∂*t* is strongly dependent on the nature of the forcing or dissipation mechanism. A by-product of the theory is a far-reaching generalization of the theorems of Charney-Drazin, Dickinson and Holton on the forcing of ∂*ū*/∂*t* by conservative linear waves.

Implications for the quasi-biennial oscillation in the equatorial stratosphere are discussed. Graphs of *y* profiles of ∂*ū*/∂*t* are given for the equatorial waves considered in the recent analysis of observational data by Lindzen and Tsay (1975). The *y* profile of ∂*ū*/∂*t* for Rossby-gravity and inertio-gravity modes, in Lindzen and Tsay's parameter ranges, prove extremely sensitive to whether or not small amounts of mechanical dissipation are present alongside the radiative-photochemical dissipation of the waves.

The probable importance of low-frequency Rossby waves for the momentum budget of the descending easterlies is suggested.

## Abstract

A detailed comparison is made between the Stewartson-Warn-Warn analytical solution for a fully nonlinear, nondiffusive Rossby-wave critical layer and a new analytical solution for the corresponding zonally truncated, “wave-mean” or “quasi-linear” problem in which the motion is represented by the zonal mean and a single zonal harmonic only. The effect of adding harmonics one by one is also considered. The results illustrate the extent to which zonally truncated models, which inevitably miss certain aspects of the fluid behavior, nevertheless contrive to mimic some important dynamical features of the evolution of the nonlinear critical layer, particularly the absorption-reflection behavior. The zonally truncated and fully nonlinear models predict almost the same reflection coefficient up to the time *T*
_{r} when a state of perfect reflection is first reached, and the predicted values of *T*
_{r} itself differ by only 5%. The agreement deteriorates only subsequently, during the first overreflecting stage. The reasons for this behavior are clarified by visualizing the way in which the truncated model misrepresents the (potential) vorticity field. Solutions, analytical and numerical, are also presented for zonally truncated models of nonlinear critical layers in which Rayleigh friction, or viscous diffusion, play an important role in the vorticity balance, as they may do in some numerical models. These solutions are compared with the corresponding fully nonlinear solutions. Implications for the numerical modeling of dynamical and tracer-transport processes involving planetary or Rossby wave “breaking” are discussed.

## Abstract

A detailed comparison is made between the Stewartson-Warn-Warn analytical solution for a fully nonlinear, nondiffusive Rossby-wave critical layer and a new analytical solution for the corresponding zonally truncated, “wave-mean” or “quasi-linear” problem in which the motion is represented by the zonal mean and a single zonal harmonic only. The effect of adding harmonics one by one is also considered. The results illustrate the extent to which zonally truncated models, which inevitably miss certain aspects of the fluid behavior, nevertheless contrive to mimic some important dynamical features of the evolution of the nonlinear critical layer, particularly the absorption-reflection behavior. The zonally truncated and fully nonlinear models predict almost the same reflection coefficient up to the time *T*
_{r} when a state of perfect reflection is first reached, and the predicted values of *T*
_{r} itself differ by only 5%. The agreement deteriorates only subsequently, during the first overreflecting stage. The reasons for this behavior are clarified by visualizing the way in which the truncated model misrepresents the (potential) vorticity field. Solutions, analytical and numerical, are also presented for zonally truncated models of nonlinear critical layers in which Rayleigh friction, or viscous diffusion, play an important role in the vorticity balance, as they may do in some numerical models. These solutions are compared with the corresponding fully nonlinear solutions. Implications for the numerical modeling of dynamical and tracer-transport processes involving planetary or Rossby wave “breaking” are discussed.

## Abstract

This paper describes a new computationally efficient, ultrasimple nonorographic spectral gravity wave parameterization model. Its predictions compare favorably, though not perfectly, with a model of gravity wave propagation and breaking that computes the evolution with altitude of a full, frequency- and wavenumber-dependent gravity wave spectrum. The ultrasimple model depends on making the midfrequency (hydrostatic, nonrotating) approximation to the dispersion relation, as in Hines’ parameterization. This allows the full frequency–wavenumber spectrum of pseudomomentum flux to be integrated with respect to frequency, and thus reduced to a spectrum that depends on vertical wavenumber *m* and azimuthal direction *ϕ* only. The ultrasimple model treats the *m* dependence as consisting of up to three analytically integrable segments, or “parts.” This allows the total pseudomomentum flux to be evaluated by using analytical expressions for the areas under the parts rather than by performing numerical quadratures. The result is a much greater computational efficiency.

The model performs significantly better than an earlier model that treated the *m* dependence as consisting of up to two parts. Numerical experiments show that similar models with more than three parts using the midfrequency approximation yield little further improvement. The limiting factor is the midfrequency approximation and not the number of parts.

## Abstract

This paper describes a new computationally efficient, ultrasimple nonorographic spectral gravity wave parameterization model. Its predictions compare favorably, though not perfectly, with a model of gravity wave propagation and breaking that computes the evolution with altitude of a full, frequency- and wavenumber-dependent gravity wave spectrum. The ultrasimple model depends on making the midfrequency (hydrostatic, nonrotating) approximation to the dispersion relation, as in Hines’ parameterization. This allows the full frequency–wavenumber spectrum of pseudomomentum flux to be integrated with respect to frequency, and thus reduced to a spectrum that depends on vertical wavenumber *m* and azimuthal direction *ϕ* only. The ultrasimple model treats the *m* dependence as consisting of up to three analytically integrable segments, or “parts.” This allows the total pseudomomentum flux to be evaluated by using analytical expressions for the areas under the parts rather than by performing numerical quadratures. The result is a much greater computational efficiency.

The model performs significantly better than an earlier model that treated the *m* dependence as consisting of up to two parts. Numerical experiments show that similar models with more than three parts using the midfrequency approximation yield little further improvement. The limiting factor is the midfrequency approximation and not the number of parts.

## Abstract

Some consequences of regarding potential vorticity as a tracer are considered. It is shown that neither diabatic heating, nor frictional forces, nor external forces such as might be used to model gravity-wave drag, can bring about any net transport or Rossby-Ertel potential vorticity (PV) across an isotropic surface—notwithstanding the diabatic, cross-isentropic transport of mass and chemical tracers. Nor can PV be created or destroyed within a layer bounded by two isentropic surface. It can only be transported along the layer. and diluted or concentrated by cross-isentropic mass inflow or outflow. This constitutes a systematic difference between the behavior of PV and that of other tracers, recognition of which simplifies thinking about PV budgets and gives insight into the relationships between dynamical processes, departures from radiatively determined temperatures, and chemical tracer transport including stratosphere-troposphere exchange.

The results just stated are true by virtue of the way in which the PV is constructed mathematically, and are therefore true not only of the exact PV constructed from the exact wind and potential-temperature fields, but true also, for example, of any “coarse-gain PV” constructed from observed or avenged fields.

Some related results on vorticity and on generalizations of the potential vorticity concept are noted, together with their implications for vorticity and potential-vorticity budgets (in the tropics and elsewhere) and for the cumulonimbus parameterization problem.

## Abstract

Some consequences of regarding potential vorticity as a tracer are considered. It is shown that neither diabatic heating, nor frictional forces, nor external forces such as might be used to model gravity-wave drag, can bring about any net transport or Rossby-Ertel potential vorticity (PV) across an isotropic surface—notwithstanding the diabatic, cross-isentropic transport of mass and chemical tracers. Nor can PV be created or destroyed within a layer bounded by two isentropic surface. It can only be transported along the layer. and diluted or concentrated by cross-isentropic mass inflow or outflow. This constitutes a systematic difference between the behavior of PV and that of other tracers, recognition of which simplifies thinking about PV budgets and gives insight into the relationships between dynamical processes, departures from radiatively determined temperatures, and chemical tracer transport including stratosphere-troposphere exchange.

The results just stated are true by virtue of the way in which the PV is constructed mathematically, and are therefore true not only of the exact PV constructed from the exact wind and potential-temperature fields, but true also, for example, of any “coarse-gain PV” constructed from observed or avenged fields.

Some related results on vorticity and on generalizations of the potential vorticity concept are noted, together with their implications for vorticity and potential-vorticity budgets (in the tropics and elsewhere) and for the cumulonimbus parameterization problem.

## Abstract

A review is given that focuses on why the sideways mixing of potential vorticity (PV) across its background gradient tends to be inhomogeneous, arguably a reason why persistent jets are commonplace in planetary atmospheres and oceans, and why such jets tend to sharpen themselves when disturbed. PV mixing often produces a sideways layering or banding of the PV distribution and therefore a corresponding number of jets, as dictated by PV inversion. There is a positive feedback in which mixing weakens the “Rossby wave elasticity” associated with the sideways PV gradients, facilitating further mixing. A partial analogy is drawn with the Phillips effect, the spontaneous layering of a stably stratified fluid, in which vertically *homogeneous* stirring produces vertically *inhomogeneous* mixing of the background buoyancy gradient. The Phillips effect has been extensively studied and has been clearly demonstrated in laboratory experiments. However, the “eddy-transport barriers” and sharp jets characteristic of extreme PV inhomogeneity, associated with strong PV mixing and strong sideways layering into Jupiter-like “PV staircases,” with sharp PV contrasts Δ*q*
_{barrier}, say, involve two additional factors besides the Rossby wave elasticity concentrated at the barriers. The first is shear straining by the colocated eastward jets. PV inversion implies that the jets are an essential, not an incidental, part of the barrier structure. The shear straining increases the barriers’ resilience and amplifies the positive feedback. The second is the role of the accompanying radiation-stress field, which mediates the angular-momentum changes associated with PV mixing and points to a new paradigm for Jupiter, in which the radiation stress is excited not by baroclinic instability but by internal convective eddies nudging the Taylor–Proudman roots of the jets.

Some examples of the shear-straining effects for strongly nonlinear disturbances are presented, helping to explain the observed resilience of eddy-transport barriers in the Jovian and terrestrial atmospheres. The main focus is on the important case where the nonlinear disturbances are vortices with core sizes ∼*L _{D}*, the Rossby (deformation) length. Then a nonlinear shear-straining mechanism that seems significant for barrier resilience is the shear-induced disruption of vortex pairs. A sufficiently strong vortex pair, with PV anomalies ±Δ

*q*

_{vortex}, such that Δ

*q*

_{vortex}≫ Δ

*q*

_{barrier}, can of course punch through the barrier. There is a threshold for substantial penetration through the barrier, related to thresholds for vortex merging. Substantial penetration requires Δ

*q*

_{vortex}≳ Δ

*q*

_{barrier}, with an accuracy or fuzziness of order 10% when core size ∼

*L*, in a shallow-water quasigeostrophic model. It is speculated that, radiation stress permitting, the barrier-penetration threshold regulates jet spacing in a staircase situation. For instance, if a staircase is already established by stirring and if the stirring is increased to produce Δ

_{D}*q*

_{vortex}values well above threshold, then the staircase steps will be widened (for given background PV gradient

*β*) until the barriers hold firm again, with Δ

*q*

_{barrier}increased to match the new threshold. With the strongest-vortex core size ∼

*L*this argument predicts a jet spacing 2

_{D}*b*= Δ

*q*

_{barrier}/

*β*∼

*L*

^{2}

_{Rh}(

*U*

_{vortex})/

*L*in order of magnitude, where

_{D}*L*

_{Rh}(

*U*

_{vortex}) = (

*U*

_{vortex}/

*β*)

^{1/2}, the Rhines scale based on the peak vortex velocity

*U*

_{vortex}, when 2

*b*≳

*L*

_{D}. The resulting jet speeds

*U*

_{jet}are of the same order as

*U*

_{vortex}; thus also 2

*b*∼

*L*

^{2}

_{Rh}(

*U*

_{jet})/

*L*. Weakly inhomogeneous turbulence theory is inapplicable here because there is no scale separation between jets and vortices, both having scales ∼

_{D}*L*in this situation.

_{D}## Abstract

A review is given that focuses on why the sideways mixing of potential vorticity (PV) across its background gradient tends to be inhomogeneous, arguably a reason why persistent jets are commonplace in planetary atmospheres and oceans, and why such jets tend to sharpen themselves when disturbed. PV mixing often produces a sideways layering or banding of the PV distribution and therefore a corresponding number of jets, as dictated by PV inversion. There is a positive feedback in which mixing weakens the “Rossby wave elasticity” associated with the sideways PV gradients, facilitating further mixing. A partial analogy is drawn with the Phillips effect, the spontaneous layering of a stably stratified fluid, in which vertically *homogeneous* stirring produces vertically *inhomogeneous* mixing of the background buoyancy gradient. The Phillips effect has been extensively studied and has been clearly demonstrated in laboratory experiments. However, the “eddy-transport barriers” and sharp jets characteristic of extreme PV inhomogeneity, associated with strong PV mixing and strong sideways layering into Jupiter-like “PV staircases,” with sharp PV contrasts Δ*q*
_{barrier}, say, involve two additional factors besides the Rossby wave elasticity concentrated at the barriers. The first is shear straining by the colocated eastward jets. PV inversion implies that the jets are an essential, not an incidental, part of the barrier structure. The shear straining increases the barriers’ resilience and amplifies the positive feedback. The second is the role of the accompanying radiation-stress field, which mediates the angular-momentum changes associated with PV mixing and points to a new paradigm for Jupiter, in which the radiation stress is excited not by baroclinic instability but by internal convective eddies nudging the Taylor–Proudman roots of the jets.

Some examples of the shear-straining effects for strongly nonlinear disturbances are presented, helping to explain the observed resilience of eddy-transport barriers in the Jovian and terrestrial atmospheres. The main focus is on the important case where the nonlinear disturbances are vortices with core sizes ∼*L _{D}*, the Rossby (deformation) length. Then a nonlinear shear-straining mechanism that seems significant for barrier resilience is the shear-induced disruption of vortex pairs. A sufficiently strong vortex pair, with PV anomalies ±Δ

*q*

_{vortex}, such that Δ

*q*

_{vortex}≫ Δ

*q*

_{barrier}, can of course punch through the barrier. There is a threshold for substantial penetration through the barrier, related to thresholds for vortex merging. Substantial penetration requires Δ

*q*

_{vortex}≳ Δ

*q*

_{barrier}, with an accuracy or fuzziness of order 10% when core size ∼

*L*, in a shallow-water quasigeostrophic model. It is speculated that, radiation stress permitting, the barrier-penetration threshold regulates jet spacing in a staircase situation. For instance, if a staircase is already established by stirring and if the stirring is increased to produce Δ

_{D}*q*

_{vortex}values well above threshold, then the staircase steps will be widened (for given background PV gradient

*β*) until the barriers hold firm again, with Δ

*q*

_{barrier}increased to match the new threshold. With the strongest-vortex core size ∼

*L*this argument predicts a jet spacing 2

_{D}*b*= Δ

*q*

_{barrier}/

*β*∼

*L*

^{2}

_{Rh}(

*U*

_{vortex})/

*L*in order of magnitude, where

_{D}*L*

_{Rh}(

*U*

_{vortex}) = (

*U*

_{vortex}/

*β*)

^{1/2}, the Rhines scale based on the peak vortex velocity

*U*

_{vortex}, when 2

*b*≳

*L*

_{D}. The resulting jet speeds

*U*

_{jet}are of the same order as

*U*

_{vortex}; thus also 2

*b*∼

*L*

^{2}

_{Rh}(

*U*

_{jet})/

*L*. Weakly inhomogeneous turbulence theory is inapplicable here because there is no scale separation between jets and vortices, both having scales ∼

_{D}*L*in this situation.

_{D}