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## Abstract

It has been shown that the linearized equations for disturbances to a parabolic jet on a β plane, with curvature *U ^{n}_{0}(y)* such that the basic-state absolute vorticity gradient β −

*U*

^{n}_{0}(

*y*) is zero, ultimately become inconsistent in the neighborhood of the jet axis and that nonlinear effects become important. Numerical solutions of the nonlinear long-time asymptotic form of the equations are presented. The numerical results show that the algebraic decay of the disturbances as

*t*

^{−1/2}predicted by the linear equations is inhibited by the nonlinear formation of coherent vortices new the jet axis. These lead to a disturbance amplitude that decays only through the action of weak numerical diffusion but is otherwise as

*t*

^{0}.

The linear theory is extended to the case when the basic-state absolute vorticity gradient is nonzero but weak. When the gradient is weak and negative the decay is modified and is ultimately as *t*
^{−3/2}. When the gradient is weak and positive, on the other hand, a discrete eigenmode is excited and asymptotic decay is inhibited. In both cases linear theory may give a self-consistent description if the amplitude is small enough. Numerical simulation shows that for both signs of the gradient there is a range of amplitudes for which nonlinear effects become directly important. Coherent vortices may form and either inhibit the decay or disrupt the linear mode. The structure of the nonlinear analog of the linear eigenmode is analyzed and shown to have a propagation speed, relative to the jet axis speed, that is a decreasing function of amplitude, tending to zero as the amplitude approaches a finite limiting value.

## Abstract

It has been shown that the linearized equations for disturbances to a parabolic jet on a β plane, with curvature *U ^{n}_{0}(y)* such that the basic-state absolute vorticity gradient β −

*U*

^{n}_{0}(

*y*) is zero, ultimately become inconsistent in the neighborhood of the jet axis and that nonlinear effects become important. Numerical solutions of the nonlinear long-time asymptotic form of the equations are presented. The numerical results show that the algebraic decay of the disturbances as

*t*

^{−1/2}predicted by the linear equations is inhibited by the nonlinear formation of coherent vortices new the jet axis. These lead to a disturbance amplitude that decays only through the action of weak numerical diffusion but is otherwise as

*t*

^{0}.

The linear theory is extended to the case when the basic-state absolute vorticity gradient is nonzero but weak. When the gradient is weak and negative the decay is modified and is ultimately as *t*
^{−3/2}. When the gradient is weak and positive, on the other hand, a discrete eigenmode is excited and asymptotic decay is inhibited. In both cases linear theory may give a self-consistent description if the amplitude is small enough. Numerical simulation shows that for both signs of the gradient there is a range of amplitudes for which nonlinear effects become directly important. Coherent vortices may form and either inhibit the decay or disrupt the linear mode. The structure of the nonlinear analog of the linear eigenmode is analyzed and shown to have a propagation speed, relative to the jet axis speed, that is a decreasing function of amplitude, tending to zero as the amplitude approaches a finite limiting value.

## Abstract

The nonlinear reflection of an isolated Rossby wave train at a low-latitude wave-breaking region is contrasted with the more familiar longitudinally periodic case. General theoretical arguments for nonlinear reflection based on absorptivity bounds do not carry over to the case of an isolated wave train, and detailed investigation is needed to determine the absorption-reflection behavior. Numerical experiments in a single-layer shallow-water model show that wave activity is reflected back into midlatitudes (rather than propagating longitudinally at low latitudes). Finite-amplitudes wave activity diagnostics are used to analyze the nonlinear reflection. Further idealized numerical simulations and simple ideas concerning the propagation of Rossby waves in shear flows are used to give insight into the nonlinear reflection.

## Abstract

The nonlinear reflection of an isolated Rossby wave train at a low-latitude wave-breaking region is contrasted with the more familiar longitudinally periodic case. General theoretical arguments for nonlinear reflection based on absorptivity bounds do not carry over to the case of an isolated wave train, and detailed investigation is needed to determine the absorption-reflection behavior. Numerical experiments in a single-layer shallow-water model show that wave activity is reflected back into midlatitudes (rather than propagating longitudinally at low latitudes). Finite-amplitudes wave activity diagnostics are used to analyze the nonlinear reflection. Further idealized numerical simulations and simple ideas concerning the propagation of Rossby waves in shear flows are used to give insight into the nonlinear reflection.

## Abstract

The combined effects of advection and diffusion on the equilibrium spatial structure of a tracer whose spatial variation is maintained by a large-scale forcing are considered. Motivated by the lower stratosphere, the flow is taken to be large-scale, time-dependent, and purely horizontal but varying in the vertical, with the vertical shear much larger than horizontal velocity gradients. As a result, the ratio *α* between horizontal and vertical tracer scales is large. (For the lower stratospheric surf zone *α* has been shown to be about 250.) The diffusion parameterizes the mixing effects of small-scale processes.

The three space dimensions and the large range between the forcing scale and the diffusive scale mean that direct numerical simulation would be prohibitively expensive for this problem. Instead, an ensemble approach is used that takes advantage of the separation between the large scale of the flow and the small scale of the tracer distribution. This approach, which has previously been used in theoretical investigations of two-dimensional flows, provides an efficient technique to derive statistical properties of the tracer distributions such as horizontal-wavenumber spectrum.

First, the authors consider random-strain models in which the velocity gradient experienced by a fluid parcel is modeled by a random process. The results show the expected *k*
^{−1} Batchelor spectrum at large scales, with a deviation from this form at a scale that is larger by a factor *α* than the diffusive scale found in the absence of vertical shear. This effect may be crudely captured by replacing the diffusivity *κ* by an &ldquo=uivalent diffusivity” *α*
^{2}
*κ,* but the diffusive dissipation is then substantially overestimated, and the spectrum at large *k* is too steep. This may be attributed to the failure of the equivalent diffusivity to capture the variability of the vertical shear.

The technique is then applied to lower-stratospheric velocity fields. For realistic values of the diffusivity *κ,* the spectrum is found to be affected by diffusion at surprisingly large scales. For the value *κ* ∼ 10^{−2} m^{2} s^{−1} suggested in several recent papers, the diffusion is sufficiently strong that there is no clear *k*
^{−1} regime, consistent with observations. The spectrum is then relatively well approximated by the observed *k*
^{−2} power law in the range 20–200 km, but significantly steeper at smaller scales. For the molecular value *κ* = 10^{−4} m^{2} s^{−1}, in contrast, an unrealistic *k*
^{−1} regime appears.

## Abstract

The combined effects of advection and diffusion on the equilibrium spatial structure of a tracer whose spatial variation is maintained by a large-scale forcing are considered. Motivated by the lower stratosphere, the flow is taken to be large-scale, time-dependent, and purely horizontal but varying in the vertical, with the vertical shear much larger than horizontal velocity gradients. As a result, the ratio *α* between horizontal and vertical tracer scales is large. (For the lower stratospheric surf zone *α* has been shown to be about 250.) The diffusion parameterizes the mixing effects of small-scale processes.

The three space dimensions and the large range between the forcing scale and the diffusive scale mean that direct numerical simulation would be prohibitively expensive for this problem. Instead, an ensemble approach is used that takes advantage of the separation between the large scale of the flow and the small scale of the tracer distribution. This approach, which has previously been used in theoretical investigations of two-dimensional flows, provides an efficient technique to derive statistical properties of the tracer distributions such as horizontal-wavenumber spectrum.

First, the authors consider random-strain models in which the velocity gradient experienced by a fluid parcel is modeled by a random process. The results show the expected *k*
^{−1} Batchelor spectrum at large scales, with a deviation from this form at a scale that is larger by a factor *α* than the diffusive scale found in the absence of vertical shear. This effect may be crudely captured by replacing the diffusivity *κ* by an &ldquo=uivalent diffusivity” *α*
^{2}
*κ,* but the diffusive dissipation is then substantially overestimated, and the spectrum at large *k* is too steep. This may be attributed to the failure of the equivalent diffusivity to capture the variability of the vertical shear.

The technique is then applied to lower-stratospheric velocity fields. For realistic values of the diffusivity *κ,* the spectrum is found to be affected by diffusion at surprisingly large scales. For the value *κ* ∼ 10^{−2} m^{2} s^{−1} suggested in several recent papers, the diffusion is sufficiently strong that there is no clear *k*
^{−1} regime, consistent with observations. The spectrum is then relatively well approximated by the observed *k*
^{−2} power law in the range 20–200 km, but significantly steeper at smaller scales. For the molecular value *κ* = 10^{−4} m^{2} s^{−1}, in contrast, an unrealistic *k*
^{−1} regime appears.

## Abstract

The effect of vertical differencing on equatorial inertial instability is studied and explicit results obtained for growth rates as a function of the vertical resolution. It is found that for a basic state independent of height, the form of the growing modes is the same as that without vertical discretization except that the vertical wavenumber is replaced by an *effective* vertical wavenumber in the differential equation for the horizontal structure. This effective vertical wavenumber is bounded above by a value that depends on the spacing of the model levels, which implies that growing modes only occur when the shear exceeds a certain value.

The upper bound is crucially dependent on the form of the difference scheme. For a scheme in which horizontal velocities and geopotential are evaluated on full levels and temperature and vertical velocity are evaluated on half levels (the Charney–Phillips scheme) the upper bound on the effective vertical wavenumber is 2/*δ* in the Boussinesq limit, where *δ* is the spacing between the model levels. For a scheme in which the horizontal velocity, geopotential, and temperature are evaluated on full levels, and only the vertical velocity on half levels (the Lorenz scheme), there is no upper bound on the effective vertical wavenumber in the Boussinesq limit so that growing modes occur for any nonzero value of the shear. This is contrary to the expectation that there is a minimum critical shear for instability because the vertical resolution limits the vertical wavenumber.

The effect of Newtonian cooling is also considered and an expression for the growth rate as a function of the cooling coefficient and the effective vertical wavenumber is found. It is found that provided the shear at the equator is nonzero, there are growing modes for all vertical wavenumbers, unlike the case without Newtonian cooling, where a mode grows only if its vertical wavenumber exceeds a critical value that depends on the shear. The consequences for numerical models with finite vertical resolution are discussed.

## Abstract

The effect of vertical differencing on equatorial inertial instability is studied and explicit results obtained for growth rates as a function of the vertical resolution. It is found that for a basic state independent of height, the form of the growing modes is the same as that without vertical discretization except that the vertical wavenumber is replaced by an *effective* vertical wavenumber in the differential equation for the horizontal structure. This effective vertical wavenumber is bounded above by a value that depends on the spacing of the model levels, which implies that growing modes only occur when the shear exceeds a certain value.

The upper bound is crucially dependent on the form of the difference scheme. For a scheme in which horizontal velocities and geopotential are evaluated on full levels and temperature and vertical velocity are evaluated on half levels (the Charney–Phillips scheme) the upper bound on the effective vertical wavenumber is 2/*δ* in the Boussinesq limit, where *δ* is the spacing between the model levels. For a scheme in which the horizontal velocity, geopotential, and temperature are evaluated on full levels, and only the vertical velocity on half levels (the Lorenz scheme), there is no upper bound on the effective vertical wavenumber in the Boussinesq limit so that growing modes occur for any nonzero value of the shear. This is contrary to the expectation that there is a minimum critical shear for instability because the vertical resolution limits the vertical wavenumber.

The effect of Newtonian cooling is also considered and an expression for the growth rate as a function of the cooling coefficient and the effective vertical wavenumber is found. It is found that provided the shear at the equator is nonzero, there are growing modes for all vertical wavenumbers, unlike the case without Newtonian cooling, where a mode grows only if its vertical wavenumber exceeds a critical value that depends on the shear. The consequences for numerical models with finite vertical resolution are discussed.

## Abstract

A quasigeostrophic, two-layer, *β*-plane channel model is used to investigate the dynamics of baroclinic wave packets. A series of experiments are performed in which an unstable flow is maintained by lower-level Ekman friction and radiative relaxation toward a temperature profile that corresponds to a broad parabolic upper-level jet. The final statistically steady state achieved in each experiment is found to depend on the magnitude of the hyperdiffusivity *ν*
_{0} and the supercriticality, which is controlled by *β.* The most important qualitative difference in such states between experiments is found to be the degree to which a waveguide in the upper level is found to develop. The mechanism for this upper-level waveguide development is the mixing effect of the eddies at the flanks of the jet, which leads to a strong potential vorticity gradient at the center of the channel, with well-mixed regions to the north and south.

Two distinct regimes with different qualitative behavior are observed and illustrated by two particular experiments. In the first regime strong hyperdiffusivity inhibits the development of the waveguide. Steady wave packets are shown to stabilize the background flow upstream by increasing the meridional shear of the jet. This upstream stabilization is argued to be a mechanism for packet maintenance in this regime. In the second regime the diffusivity is lower, and a well-developed upper-level waveguide results. The wave packets in this regime are unsteady and are shown to stabilize the background flow at, and slightly upstream of, their maxima. Wave activity diagnostics suggest that the most important mechanism in maintaining these packets is the zonal convergence of wave activity, indicating that the wave packets are undergoing a form of nonlinear self-focusing, analogous to that identified in weakly nonlinear models.

Finally, results are presented from a 10-level primitive equation model with parameter values relevant to the real atmosphere. In this experiment the nonlinear response of the background flow to the wave packets is shown to be qualitatively very similar to that observed in the low-diffusivity two-layer model experiment.

## Abstract

A quasigeostrophic, two-layer, *β*-plane channel model is used to investigate the dynamics of baroclinic wave packets. A series of experiments are performed in which an unstable flow is maintained by lower-level Ekman friction and radiative relaxation toward a temperature profile that corresponds to a broad parabolic upper-level jet. The final statistically steady state achieved in each experiment is found to depend on the magnitude of the hyperdiffusivity *ν*
_{0} and the supercriticality, which is controlled by *β.* The most important qualitative difference in such states between experiments is found to be the degree to which a waveguide in the upper level is found to develop. The mechanism for this upper-level waveguide development is the mixing effect of the eddies at the flanks of the jet, which leads to a strong potential vorticity gradient at the center of the channel, with well-mixed regions to the north and south.

Two distinct regimes with different qualitative behavior are observed and illustrated by two particular experiments. In the first regime strong hyperdiffusivity inhibits the development of the waveguide. Steady wave packets are shown to stabilize the background flow upstream by increasing the meridional shear of the jet. This upstream stabilization is argued to be a mechanism for packet maintenance in this regime. In the second regime the diffusivity is lower, and a well-developed upper-level waveguide results. The wave packets in this regime are unsteady and are shown to stabilize the background flow at, and slightly upstream of, their maxima. Wave activity diagnostics suggest that the most important mechanism in maintaining these packets is the zonal convergence of wave activity, indicating that the wave packets are undergoing a form of nonlinear self-focusing, analogous to that identified in weakly nonlinear models.

Finally, results are presented from a 10-level primitive equation model with parameter values relevant to the real atmosphere. In this experiment the nonlinear response of the background flow to the wave packets is shown to be qualitatively very similar to that observed in the low-diffusivity two-layer model experiment.

## Abstract

Numerical simulations in multilevel baroclinic turbulence in a *β*-plane channel model are discussed, focusing on the transport and mixing behavior. The temperature field in the model is relaxed toward a field consistent with a broad zonal jet with vertical shear that is a Gaussian function of the cross-channel coordinate. The resulting statistical equilibrium flow includes an active baroclinic eddy field. The transport and mixing properties are analyzed by considering the fields of potential vorticity and a passive tracer (from which effective diffusivities/equivalent lengths are calculated). The upper part of the flow organizes itself in such a way that there is a transport barrier in the center of the channel, with eddy mixing regions on either side. In the lower part of the flow the eddy mixing occurs across a single broad region, with no central transport barrier. The transition between these two regimes takes place abruptly at a height *z _{T}*. A large set of simulations is used to map out the variation of

*z*as a function of external parameters including

_{T}*β*, the thermal relaxation rate

*κ*, and the (lower boundary) frictional relaxation rate

_{T}*κ*(applied in the lowest model layer only). The transition height

_{M}*z*is argued to be relevant to sharp vertical transitions in transport and mixing observed in atmospheric and oceanic flows.

_{T}## Abstract

Numerical simulations in multilevel baroclinic turbulence in a *β*-plane channel model are discussed, focusing on the transport and mixing behavior. The temperature field in the model is relaxed toward a field consistent with a broad zonal jet with vertical shear that is a Gaussian function of the cross-channel coordinate. The resulting statistical equilibrium flow includes an active baroclinic eddy field. The transport and mixing properties are analyzed by considering the fields of potential vorticity and a passive tracer (from which effective diffusivities/equivalent lengths are calculated). The upper part of the flow organizes itself in such a way that there is a transport barrier in the center of the channel, with eddy mixing regions on either side. In the lower part of the flow the eddy mixing occurs across a single broad region, with no central transport barrier. The transition between these two regimes takes place abruptly at a height *z _{T}*. A large set of simulations is used to map out the variation of

*z*as a function of external parameters including

_{T}*β*, the thermal relaxation rate

*κ*, and the (lower boundary) frictional relaxation rate

_{T}*κ*(applied in the lowest model layer only). The transition height

_{M}*z*is argued to be relevant to sharp vertical transitions in transport and mixing observed in atmospheric and oceanic flows.

_{T}## Abstract

A simple model of the tropospheric circulation, based on a 10-level primitive equation model, is forced by linearly relaxing the potential temperature toward an idealized, zonally symmetric equilibrium field. The model equations are integrated in time until a statistically steady state is obtained. The local relationship between the state of the background flow, the direction of wave propagation, and subsequent wave breaking at the tropopause level is then investigated. Maps of potential vorticity (PV) on isentropic surfaces are analyzed and all four different types of wave breaking described recently by Peters and Waugh are shown to occur. It is found that cyclonic wave breaking events are usually initiated by poleward fluxes of wave activity, and anticyclonic events by equatorward fluxes. Composites are then used to show that equatorward fluxes are associated with a jet that is locally broad and weak, with relatively strong isentropic PV gradients to its equatorward flank. By contrast, poleward fluxes are associated with a narrow, strong jet, with very weak or even negative PV gradients on its equatorward side. It is argued that this result is consistent with nonlinear critical-layer theory, as under certain conditions an isolated region of homogenized potential vorticity must remain a perfect reflector of wave activity for all time.

The variability exhibited by the zonal flow field is then investigated using a cross-sectional EOF method. The first EOF is found to have similar structure in the latitude–height plane to the baroclinic waves themselves, and describes much of the variability associated with them. The second EOF has structure that corresponds to a sharp, narrow jet in its positive phase and a weak, broad jet in its negative phase. Its phase is shown to be well correlated with the wave activity flux index, with the maximum occurring at a space and time lag, with the phase of the EOF preceding the index. Most of the variability associated with this EOF occurs on the scale of zonal wavenumbers 2–4, suggesting that the direction of meridional propagation of the baroclinic waves is determined locally. Strikingly, the phase of the second EOF propagates in a wavelike manner, with wavenumber and period (≈11–14 days) quite distinct from those of the baroclinic waves. Individual phase maxima of these long waves can persist for up to ≈20–25 days, as they do not decay rapidly due to downstream radiation.

## Abstract

A simple model of the tropospheric circulation, based on a 10-level primitive equation model, is forced by linearly relaxing the potential temperature toward an idealized, zonally symmetric equilibrium field. The model equations are integrated in time until a statistically steady state is obtained. The local relationship between the state of the background flow, the direction of wave propagation, and subsequent wave breaking at the tropopause level is then investigated. Maps of potential vorticity (PV) on isentropic surfaces are analyzed and all four different types of wave breaking described recently by Peters and Waugh are shown to occur. It is found that cyclonic wave breaking events are usually initiated by poleward fluxes of wave activity, and anticyclonic events by equatorward fluxes. Composites are then used to show that equatorward fluxes are associated with a jet that is locally broad and weak, with relatively strong isentropic PV gradients to its equatorward flank. By contrast, poleward fluxes are associated with a narrow, strong jet, with very weak or even negative PV gradients on its equatorward side. It is argued that this result is consistent with nonlinear critical-layer theory, as under certain conditions an isolated region of homogenized potential vorticity must remain a perfect reflector of wave activity for all time.

The variability exhibited by the zonal flow field is then investigated using a cross-sectional EOF method. The first EOF is found to have similar structure in the latitude–height plane to the baroclinic waves themselves, and describes much of the variability associated with them. The second EOF has structure that corresponds to a sharp, narrow jet in its positive phase and a weak, broad jet in its negative phase. Its phase is shown to be well correlated with the wave activity flux index, with the maximum occurring at a space and time lag, with the phase of the EOF preceding the index. Most of the variability associated with this EOF occurs on the scale of zonal wavenumbers 2–4, suggesting that the direction of meridional propagation of the baroclinic waves is determined locally. Strikingly, the phase of the second EOF propagates in a wavelike manner, with wavenumber and period (≈11–14 days) quite distinct from those of the baroclinic waves. Individual phase maxima of these long waves can persist for up to ≈20–25 days, as they do not decay rapidly due to downstream radiation.

## 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

The forcing of planetary wave variability in the stratosphere by synoptic-scale baroclinic eddies in the troposphere is considered. Simple forced–dissipative numerical experiments are performed in a primitive equation model using a deep hemispheric model domain. The flow is thermally relaxed toward zonally symmetric notional wintertime conditions. No zonally asymmetric thermal or topographic forcing is applied. All planetary-scale zonal asymmetry arises solely through the nonlinear wave–wave interaction of the baroclinic eddies in the troposphere. The numerical experiments indicate that realistic stratospheric planetary wave amplitudes and variability, comparable to those observed in the Southern Hemisphere, can be forced through this mechanism. No evidence is found in these simulations for planetary-scale disturbances arising through in situ instability in the stratosphere.

The nonlinear tropospheric forcing mechanism in the numerical simulations is further investigated by reproducing the stratospheric planetary wave response with a linear model that is forced by the nonlinear eddy forcing that acted in the troposphere of the nonlinear simulation. The forced linear model experiments indicate that (i) as anticipated, both the eddy vorticity forcing and the eddy temperature forcing are required to account for the planetary wave response, (ii) only the low-frequency component of the nonlinear forcing is important, (iii) the vertical structure of the eddy forcing is equivalent to a compact source near tropopause level, and (iv) the variability of the planetary wave response in the stratosphere arises primarily from the variability of the nonlinear eddy forcing in the troposphere, rather than from the variability of the wave propagation characteristics associated with the basic-state zonally averaged flow.

The eddy vorticity and eddy temperature forcing fields are combined into a single expression by introducing a transformation of the equations that govern the Fourier decomposition of deviations away from the zonally averaged flow, referred to as the transformed Fourier decomposition (TFD). The TFD transformation is essentially a generalization of that used in the transformed Eulerian mean formalism. The spatial and temporal characteristics of the total eddy forcing are then analyzed.

The baroclinic eddies in the troposphere of the full simulation show strong organization into wave packets with a dominant wave-2 structure in amplitude. There is a strong, high-frequency, nonlinear wave-2 forcing associated with these packets. However, the propagation characteristics of the background flow in the simulation do not allow upward propagation of wave-2 disturbances with the corresponding frequency and there is little associated signal in the stratosphere. Experiments with a linear model, applying the same nonlinear forcing, show that there are background zonal flows, with plausibly realistic velocity fields, that allow upward propagation of such disturbances. It is therefore suggested that baroclinic wave packets may be an important mechanism for forcing higher-frequency wave-2 disturbances observed in the real Southern Hemisphere stratosphere. The low-frequency stratospheric disturbances obtained in the nonlinear simulations appear to be associated with more subtle aspects of the baroclinic wave packets such as their spatial and temporal variability.

## Abstract

The forcing of planetary wave variability in the stratosphere by synoptic-scale baroclinic eddies in the troposphere is considered. Simple forced–dissipative numerical experiments are performed in a primitive equation model using a deep hemispheric model domain. The flow is thermally relaxed toward zonally symmetric notional wintertime conditions. No zonally asymmetric thermal or topographic forcing is applied. All planetary-scale zonal asymmetry arises solely through the nonlinear wave–wave interaction of the baroclinic eddies in the troposphere. The numerical experiments indicate that realistic stratospheric planetary wave amplitudes and variability, comparable to those observed in the Southern Hemisphere, can be forced through this mechanism. No evidence is found in these simulations for planetary-scale disturbances arising through in situ instability in the stratosphere.

The nonlinear tropospheric forcing mechanism in the numerical simulations is further investigated by reproducing the stratospheric planetary wave response with a linear model that is forced by the nonlinear eddy forcing that acted in the troposphere of the nonlinear simulation. The forced linear model experiments indicate that (i) as anticipated, both the eddy vorticity forcing and the eddy temperature forcing are required to account for the planetary wave response, (ii) only the low-frequency component of the nonlinear forcing is important, (iii) the vertical structure of the eddy forcing is equivalent to a compact source near tropopause level, and (iv) the variability of the planetary wave response in the stratosphere arises primarily from the variability of the nonlinear eddy forcing in the troposphere, rather than from the variability of the wave propagation characteristics associated with the basic-state zonally averaged flow.

The eddy vorticity and eddy temperature forcing fields are combined into a single expression by introducing a transformation of the equations that govern the Fourier decomposition of deviations away from the zonally averaged flow, referred to as the transformed Fourier decomposition (TFD). The TFD transformation is essentially a generalization of that used in the transformed Eulerian mean formalism. The spatial and temporal characteristics of the total eddy forcing are then analyzed.

The baroclinic eddies in the troposphere of the full simulation show strong organization into wave packets with a dominant wave-2 structure in amplitude. There is a strong, high-frequency, nonlinear wave-2 forcing associated with these packets. However, the propagation characteristics of the background flow in the simulation do not allow upward propagation of wave-2 disturbances with the corresponding frequency and there is little associated signal in the stratosphere. Experiments with a linear model, applying the same nonlinear forcing, show that there are background zonal flows, with plausibly realistic velocity fields, that allow upward propagation of such disturbances. It is therefore suggested that baroclinic wave packets may be an important mechanism for forcing higher-frequency wave-2 disturbances observed in the real Southern Hemisphere stratosphere. The low-frequency stratospheric disturbances obtained in the nonlinear simulations appear to be associated with more subtle aspects of the baroclinic wave packets such as their spatial and temporal variability.

## 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.