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## Abstract

Effective diffusivity calculated from a scalar field that obeys the advection–diffusion equation has proved useful for estimating the permeability of unsteady boundaries of air masses such as the edge of the stratospheric polar vortex and the extratropical tropopause. However, the method does not discriminate the direction of transport—whereas some material crosses the boundary from one side to the other, some material does so in the other direction—yet the extant method concerns only the net transport.

In this paper, the diagnostic is extended to allow partitioning of fluxes of mass and tracer into opposing directions. This is accomplished by discriminating the regions of “inward” and “outward” wave breaking with the local curvature of the tracer field. The utility of the new method is demonstrated for nonlinear Kelvin– Helmholtz instability and Rossby wave breaking in the stratosphere using a numerically generated tracer. The method successfully quantifies two-way transport and hence the direction of wave breaking—the predominantly equatorward breaking of Rossby waves in the extratropical middle stratosphere, for example. Isolated episodes of mixing are identified well, particularly by the mass flux that primarily arises from the tracer filaments.

Comparison of different transport schemes suggests that the results are reasonably robust under a varying subgrid representation of the model.

## Abstract

Effective diffusivity calculated from a scalar field that obeys the advection–diffusion equation has proved useful for estimating the permeability of unsteady boundaries of air masses such as the edge of the stratospheric polar vortex and the extratropical tropopause. However, the method does not discriminate the direction of transport—whereas some material crosses the boundary from one side to the other, some material does so in the other direction—yet the extant method concerns only the net transport.

In this paper, the diagnostic is extended to allow partitioning of fluxes of mass and tracer into opposing directions. This is accomplished by discriminating the regions of “inward” and “outward” wave breaking with the local curvature of the tracer field. The utility of the new method is demonstrated for nonlinear Kelvin– Helmholtz instability and Rossby wave breaking in the stratosphere using a numerically generated tracer. The method successfully quantifies two-way transport and hence the direction of wave breaking—the predominantly equatorward breaking of Rossby waves in the extratropical middle stratosphere, for example. Isolated episodes of mixing are identified well, particularly by the mass flux that primarily arises from the tracer filaments.

Comparison of different transport schemes suggests that the results are reasonably robust under a varying subgrid representation of the model.

## Abstract

An exact, two-dimensional (height–latitude type) diagnostic model of tracer transport in the stratosphere is formulated. The model is a generalization of the area analysis method devised by Butchart and Remsberg but uses the mass element enclosed by the contours of a quasi-conservative tracer on a given isentropic surface as a “meridional” coordinate. This modified Lagrangian-mean coordinate is transparent to the advective effects of the winds, and thus unambiguously extracts the nonconservative effects on the tracer distribution. The derived transport equation takes an equivalent advective form; that is, the tracer contours are “advected” by the mean nonconservative mass flow while them are no “eddy flux” contributions. Hence, the mass flux is implied in the motion of the tracer contours. Not only is this model conceptually simple, it is also computationally economical for analyzing large, high-resolution datasets since time averaging can be omitted to define a robust mean field of the tracer.

The model is used to diagnose the N_{2}O mixing ratio and potential vorticity simulated in the high-resolution, Geophysical Fluid Dynamics Laboratory SKYHI GCM. The analysis not only identifies the boundary of the polar vortex better than the Eulerian zonal-mean models but highlights how the nonconservative processes (diabatic heating and friction) contribute to the formation of the vortex edge. It also reveals the different kinematics for potential vorticity and chemical tracers.

## Abstract

An exact, two-dimensional (height–latitude type) diagnostic model of tracer transport in the stratosphere is formulated. The model is a generalization of the area analysis method devised by Butchart and Remsberg but uses the mass element enclosed by the contours of a quasi-conservative tracer on a given isentropic surface as a “meridional” coordinate. This modified Lagrangian-mean coordinate is transparent to the advective effects of the winds, and thus unambiguously extracts the nonconservative effects on the tracer distribution. The derived transport equation takes an equivalent advective form; that is, the tracer contours are “advected” by the mean nonconservative mass flow while them are no “eddy flux” contributions. Hence, the mass flux is implied in the motion of the tracer contours. Not only is this model conceptually simple, it is also computationally economical for analyzing large, high-resolution datasets since time averaging can be omitted to define a robust mean field of the tracer.

The model is used to diagnose the N_{2}O mixing ratio and potential vorticity simulated in the high-resolution, Geophysical Fluid Dynamics Laboratory SKYHI GCM. The analysis not only identifies the boundary of the polar vortex better than the Eulerian zonal-mean models but highlights how the nonconservative processes (diabatic heating and friction) contribute to the formation of the vortex edge. It also reveals the different kinematics for potential vorticity and chemical tracers.

## Abstract

Effects of isolated transport barriers on the global mixing and fluxes of a tracer are investigated, where a barrier is defined as a local minimum in effective diffusivity. An idealized 1D model with a prescribed diffusivity profile, with or without forcing, is used to show that the structure, flux, and decay rates of the tracer are all very sensitive to the barrier geometry, particularly when it is deep and narrow. Although the tracer gradients and the variance dissipation are concentrated to the barrier region, the flux shows a more global response to the barrier, decreasing everywhere. The harmonic mean of effective diffusivity is proposed as a useful first-order predictor of the global transport. This 1D model is used to diagnose the isentropic transport in the upper troposphere and lower stratosphere with offline transport calculations driven by the Met Office winds. The global tracer variance in these calculations decays approximately exponentially, and the time-mean decay rate and tracer structure are well captured by the gravest 1D eigenmode with the time-averaged effective diffusivity. However, the decay rate and the flux of the full solution are 15%–20% smaller than those of the eigenmode because of a negative temporal correlation between the effective diffusivity and the gradient. The vertical and decadal variations of the decay rates are consistent with the corresponding variations in the harmonic mean effective diffusivity. To the extent that the global mixing is sensitive to the local barrier properties, and to the extent that the latter are sensitive to the errors in advecting winds and model numerics, modeling of global atmospheric transport remains a challenge. This may explain, at least partially, the disparate model estimates reported in the literature.

## Abstract

Effects of isolated transport barriers on the global mixing and fluxes of a tracer are investigated, where a barrier is defined as a local minimum in effective diffusivity. An idealized 1D model with a prescribed diffusivity profile, with or without forcing, is used to show that the structure, flux, and decay rates of the tracer are all very sensitive to the barrier geometry, particularly when it is deep and narrow. Although the tracer gradients and the variance dissipation are concentrated to the barrier region, the flux shows a more global response to the barrier, decreasing everywhere. The harmonic mean of effective diffusivity is proposed as a useful first-order predictor of the global transport. This 1D model is used to diagnose the isentropic transport in the upper troposphere and lower stratosphere with offline transport calculations driven by the Met Office winds. The global tracer variance in these calculations decays approximately exponentially, and the time-mean decay rate and tracer structure are well captured by the gravest 1D eigenmode with the time-averaged effective diffusivity. However, the decay rate and the flux of the full solution are 15%–20% smaller than those of the eigenmode because of a negative temporal correlation between the effective diffusivity and the gradient. The vertical and decadal variations of the decay rates are consistent with the corresponding variations in the harmonic mean effective diffusivity. To the extent that the global mixing is sensitive to the local barrier properties, and to the extent that the latter are sensitive to the errors in advecting winds and model numerics, modeling of global atmospheric transport remains a challenge. This may explain, at least partially, the disparate model estimates reported in the literature.

## Abstract

Equilibration of two-dimensional Eady waves is numerically investigated using the geostrophic momentum equations incorporating heat and momentum diffusion. Extended solutions are obtained beyond what would be the collapse of surface fronts in the inviscid theory, and are found to accurately reproduce the equilibration of baroclinic waves simulated with the primitive equations. Potential vorticity anomalies produced at the surface fronts are essential in the initial amplitude saturation and in the asymptotic behavior of the equilibrated flow. A supergeostrophic shear spun up nonlinearly in the zonal flow also plays an important role in causing the reversal of the tilt.

The zonal mean potential temperature profile in the equilibrium state is similar to the prediction by the adjustment hypothesis of Gutowski when only horizontal diffusion is present. However, it is closer to an Eady's basic state with enhanced static stability when vertical diffusion is also present.

The difference in the ageostrophic streamfunctions between the geostrophic momentum and primitive equations reveals rich features of inertia-gravity waves radiating from the surface fronts in the latter model. The role of the gravity waves in the equilibration process, however, is found to be minor.

## Abstract

Equilibration of two-dimensional Eady waves is numerically investigated using the geostrophic momentum equations incorporating heat and momentum diffusion. Extended solutions are obtained beyond what would be the collapse of surface fronts in the inviscid theory, and are found to accurately reproduce the equilibration of baroclinic waves simulated with the primitive equations. Potential vorticity anomalies produced at the surface fronts are essential in the initial amplitude saturation and in the asymptotic behavior of the equilibrated flow. A supergeostrophic shear spun up nonlinearly in the zonal flow also plays an important role in causing the reversal of the tilt.

The zonal mean potential temperature profile in the equilibrium state is similar to the prediction by the adjustment hypothesis of Gutowski when only horizontal diffusion is present. However, it is closer to an Eady's basic state with enhanced static stability when vertical diffusion is also present.

The difference in the ageostrophic streamfunctions between the geostrophic momentum and primitive equations reveals rich features of inertia-gravity waves radiating from the surface fronts in the latter model. The role of the gravity waves in the equilibration process, however, is found to be minor.

## Abstract

Baroclinic adjustment hypothesis fails to account for the enhancement of the barotropic jet observed in idealized baroclinic-wave life-cycle simulations. In this paper, an adjustment theory more consistent with the numerical results is developed through a careful examination of life-cycle experiments and nonseparable eigenvalue problems using the two-layer model.

In all cases examined, nonlinear eddies emanate from an unstable normal mode of meridionally concentrated gradients of the zonal-mean potential vorticity (PV). The flows are neither forced nor damped, except that moderate second-order horizontal diffusion is used to achieve an eddy-free state in a finite computational time. The final zonal-mean states are typically characterized by a well-defined barotropic jet that is not sufficiently stable in the sense of Charney and Stern but stable for all zonal wavenumbers allowed by the geometry of the channel. It is shown that vertical asymmetry in the meridional arrangement of PV leads to (a) production of barotropically sheared jet and (b) shift in the zonal scale of baroclinic instability and subsequent neutralization. It is argued that the extent of the meridional arrangement necessary to suppress the most momentum-transporting baroclinic wave determines the width of the adjusted flow. This width is roughly proportional to the initial zonal scale of the mode on the *f* plane but constrained by a beta-related critical mixing length on the beta plane.

Relationship to other theories (e.g., barotropic governor and geostrophic turbulence) is discussed, along with the relevance of the theory to the earth’s midlatitude troposphere.

## Abstract

Baroclinic adjustment hypothesis fails to account for the enhancement of the barotropic jet observed in idealized baroclinic-wave life-cycle simulations. In this paper, an adjustment theory more consistent with the numerical results is developed through a careful examination of life-cycle experiments and nonseparable eigenvalue problems using the two-layer model.

In all cases examined, nonlinear eddies emanate from an unstable normal mode of meridionally concentrated gradients of the zonal-mean potential vorticity (PV). The flows are neither forced nor damped, except that moderate second-order horizontal diffusion is used to achieve an eddy-free state in a finite computational time. The final zonal-mean states are typically characterized by a well-defined barotropic jet that is not sufficiently stable in the sense of Charney and Stern but stable for all zonal wavenumbers allowed by the geometry of the channel. It is shown that vertical asymmetry in the meridional arrangement of PV leads to (a) production of barotropically sheared jet and (b) shift in the zonal scale of baroclinic instability and subsequent neutralization. It is argued that the extent of the meridional arrangement necessary to suppress the most momentum-transporting baroclinic wave determines the width of the adjusted flow. This width is roughly proportional to the initial zonal scale of the mode on the *f* plane but constrained by a beta-related critical mixing length on the beta plane.

Relationship to other theories (e.g., barotropic governor and geostrophic turbulence) is discussed, along with the relevance of the theory to the earth’s midlatitude troposphere.

## Abstract

A simple two-dimensional linear stability analysis is presented to study the characteristics of the scale selection and structure of baroclinic cyclogenesis. Effects of a realistic vertical thermal structure of the environment and the related nongeostrophic wind are investigated in detail. Both short (mesoscale) waves trapped in the lower layer with a small static stability and long waves extending throughout the column are unstable, which supports Blumen's results. The integral constraints imply that the vertical structure of the stratification can provide an “internal lid” to confine the wave into the mixed layer. Nongeostrophic effects become important as Richardson number decreases as pointed out by Stone; the scale of the most unstable Eady's mode becomes sensitive to the change of shear, unlike the geostrophic case. The measure of the horizontal scale of the disturbance is the modified Rossby radius of deformationwhere Λ is the vertical shear and provides O(Ri^{−½}) correction to the conventional definition. However this shear dependence is drastically different between the full nongeostrophic system and the one with geostrophic momentum approximation. Another nongeostrophic mode—the weakly unstable, small-scale waves found by Stone—is identified as an instability due to the inertia critical layer which is sustained by the resonance between one of the boundary modes and inertia–gravity modes. It is analytically shown that this mode may exist even without one of the boundaries, unlike Eady's mode. The growth rates and the shallow structure of the most unstable short Eady waves are generally in agreement with observed early stages of winter storms (coastal cyclogenesis, polar lows, and comma clouds), though some observed cases have a considerably deep structure.

## Abstract

A simple two-dimensional linear stability analysis is presented to study the characteristics of the scale selection and structure of baroclinic cyclogenesis. Effects of a realistic vertical thermal structure of the environment and the related nongeostrophic wind are investigated in detail. Both short (mesoscale) waves trapped in the lower layer with a small static stability and long waves extending throughout the column are unstable, which supports Blumen's results. The integral constraints imply that the vertical structure of the stratification can provide an “internal lid” to confine the wave into the mixed layer. Nongeostrophic effects become important as Richardson number decreases as pointed out by Stone; the scale of the most unstable Eady's mode becomes sensitive to the change of shear, unlike the geostrophic case. The measure of the horizontal scale of the disturbance is the modified Rossby radius of deformationwhere Λ is the vertical shear and provides O(Ri^{−½}) correction to the conventional definition. However this shear dependence is drastically different between the full nongeostrophic system and the one with geostrophic momentum approximation. Another nongeostrophic mode—the weakly unstable, small-scale waves found by Stone—is identified as an instability due to the inertia critical layer which is sustained by the resonance between one of the boundary modes and inertia–gravity modes. It is analytically shown that this mode may exist even without one of the boundaries, unlike Eady's mode. The growth rates and the shallow structure of the most unstable short Eady waves are generally in agreement with observed early stages of winter storms (coastal cyclogenesis, polar lows, and comma clouds), though some observed cases have a considerably deep structure.

## Abstract

A number of idealized life-cycle simulations of baroclinically unstable waves are systematically analyzed to study the effects of eddy momentum flux and of zonal mean horizontal shear on the finite-amplitude evolution of the waves. Twenty-level quasigeostrophic and primitive equation models with channel geometry are numerically integrated with the most unstable linear normal mode as an initial condition. The flows are inviscid except for weak second-order horizontal diffusion.

It is found that the finite-amplitude baroclinic waves are sensitively influenced by the vertically integrated eddy momentum flux of the normal mode via the large barotropic shear it spins up in the mean flow. This “barotropic governor” mechanism prevents the eddy from attaining all the available potential energy stored in the domain, leading to irreversible barotropic decay. Only in the purely baroclinic, *f*-plane, quasigeostrophic problem, where the vertically integrated eddy momentum flux identically vanishes due to symmetry, is the growth of baroclinic waves unaffected by the barotropic governor and bounded solely by the total available potential energy. Barotropic shear in the basic flow, the earth's spherical geometry, and nonquasigeostrophic motion all introduce spatial asymmetry into the normal mode, whose nonlinear evolution therefore rapidly departs from the purely baroclinic solution. The details of the departure depend sensitively on the shape of the initial asymmetry, however.

The results suggest the natural tendency of baroclinic waves toward barotropic decay in nearly inviscid atmospheres.

## Abstract

A number of idealized life-cycle simulations of baroclinically unstable waves are systematically analyzed to study the effects of eddy momentum flux and of zonal mean horizontal shear on the finite-amplitude evolution of the waves. Twenty-level quasigeostrophic and primitive equation models with channel geometry are numerically integrated with the most unstable linear normal mode as an initial condition. The flows are inviscid except for weak second-order horizontal diffusion.

It is found that the finite-amplitude baroclinic waves are sensitively influenced by the vertically integrated eddy momentum flux of the normal mode via the large barotropic shear it spins up in the mean flow. This “barotropic governor” mechanism prevents the eddy from attaining all the available potential energy stored in the domain, leading to irreversible barotropic decay. Only in the purely baroclinic, *f*-plane, quasigeostrophic problem, where the vertically integrated eddy momentum flux identically vanishes due to symmetry, is the growth of baroclinic waves unaffected by the barotropic governor and bounded solely by the total available potential energy. Barotropic shear in the basic flow, the earth's spherical geometry, and nonquasigeostrophic motion all introduce spatial asymmetry into the normal mode, whose nonlinear evolution therefore rapidly departs from the purely baroclinic solution. The details of the departure depend sensitively on the shape of the initial asymmetry, however.

The results suggest the natural tendency of baroclinic waves toward barotropic decay in nearly inviscid atmospheres.

## Abstract

Two-dimensional mixing of a tracer is diagnosed using *A*, the area between the tracer contour and a reference contour, as the horizontal coordinate. In the absence of sources or sinks, the tracer distribution in the area coordinate is governed bywhereis the square of the equivalent length of the tracer contour and *k* is the constant microscale diffusion coefficient. While diffusion is necessary to invoke transport, the large-scale kinematics regulate the evolution of *q* through the stretching and redistribution of *L _{e}*

^{2}. It is argued that

*L*

_{e}^{2}is a useful, easy to compute diagnostic for irreversible transport, especially for identifying a barrier. Typical behaviors of

*L*

_{e}^{2}in various flow regimes are illustrated using the numerically simulated Kelvin-Helmholtz billow. Signature of mixing is found in the irreversible growth of

*L*

_{e}^{2}, but the precise time dependence is complex due to interplay between advection and diffusion.

Formation of the edges (concentrated gradients) and their permeability to mass are addressed using a kinematic model with prescribed *L _{e}*

^{2}. Relevance of this model to the stratospheric polar vortex, midlatitude tropopause, and oceanic thermocline is discussed.

## Abstract

Two-dimensional mixing of a tracer is diagnosed using *A*, the area between the tracer contour and a reference contour, as the horizontal coordinate. In the absence of sources or sinks, the tracer distribution in the area coordinate is governed bywhereis the square of the equivalent length of the tracer contour and *k* is the constant microscale diffusion coefficient. While diffusion is necessary to invoke transport, the large-scale kinematics regulate the evolution of *q* through the stretching and redistribution of *L _{e}*

^{2}. It is argued that

*L*

_{e}^{2}is a useful, easy to compute diagnostic for irreversible transport, especially for identifying a barrier. Typical behaviors of

*L*

_{e}^{2}in various flow regimes are illustrated using the numerically simulated Kelvin-Helmholtz billow. Signature of mixing is found in the irreversible growth of

*L*

_{e}^{2}, but the precise time dependence is complex due to interplay between advection and diffusion.

Formation of the edges (concentrated gradients) and their permeability to mass are addressed using a kinematic model with prescribed *L _{e}*

^{2}. Relevance of this model to the stratospheric polar vortex, midlatitude tropopause, and oceanic thermocline is discussed.

## Abstract

*q*, can be cast, uniquely and exactly into

*K** is a scalar that varies in space and time. Furthermore,

*K** =

*K*

_{k}+

*K*

_{m}, where

*K*

_{k}represents (possibly nonlocal) along-gradient dispersion of tracer isosurface, whereas

*D*) amplified by resolved (but averaged) fluid dynamical stirring. The amplifying factor,

*q*|

^{2}

*q*

^{2}, measures local roughness of tracer field and determines the spatiotemporal structure of

*K*

_{m}. This factor is called “mixing efficiency” and it is proposed as a diagnostic of mixing. The mixing efficiency diagnostic is demonstrated using potential vorticity and nitrous oxide from the outputs of the Geophysical Fluid Dynamics Laboratory (GFDL) SKYHI general circulation model (GCM) analyzed on the 320-K isentropic surface for the month of March. It is found that mixing is severely suppressed along the jet axes at the midlatitude tropopause with marked zonal localization in the Northern Hemisphere.

The formalism's relationship to the previously derived modified Lagrangian-mean theory is also discussed.

## Abstract

*q*, can be cast, uniquely and exactly into

*K** is a scalar that varies in space and time. Furthermore,

*K** =

*K*

_{k}+

*K*

_{m}, where

*K*

_{k}represents (possibly nonlocal) along-gradient dispersion of tracer isosurface, whereas

*D*) amplified by resolved (but averaged) fluid dynamical stirring. The amplifying factor,

*q*|

^{2}

*q*

^{2}, measures local roughness of tracer field and determines the spatiotemporal structure of

*K*

_{m}. This factor is called “mixing efficiency” and it is proposed as a diagnostic of mixing. The mixing efficiency diagnostic is demonstrated using potential vorticity and nitrous oxide from the outputs of the Geophysical Fluid Dynamics Laboratory (GFDL) SKYHI general circulation model (GCM) analyzed on the 320-K isentropic surface for the month of March. It is found that mixing is severely suppressed along the jet axes at the midlatitude tropopause with marked zonal localization in the Northern Hemisphere.

The formalism's relationship to the previously derived modified Lagrangian-mean theory is also discussed.

## Abstract

An analytic model is formulated to study the characteristics of shear instabilities in meridionally and vertically sheared flows. The model is based on the quasigeostrophic equations in two layers. The layers are divided into sections of piecewise uniform potential vorticity. An algebraic dispersion relation is obtained for the complex phase speed *c*. The magnitude and the sign of the potential vorticity jumps, their meridional separation, the barotropic shear, and the wavenumber of the modes determine the stability of the system. Solutions describe not only pure baroclinic and barotropic instabilities, but also mixtures of these instabilities. The influences of linearly sheared barotropic flows on baroclinic instability are studied in detail, with an emphasis on the direction of vertically integrated momentum flux. The model's implications for the nonlinear life cycle of baroclinic waves are also discussed.

## Abstract

An analytic model is formulated to study the characteristics of shear instabilities in meridionally and vertically sheared flows. The model is based on the quasigeostrophic equations in two layers. The layers are divided into sections of piecewise uniform potential vorticity. An algebraic dispersion relation is obtained for the complex phase speed *c*. The magnitude and the sign of the potential vorticity jumps, their meridional separation, the barotropic shear, and the wavenumber of the modes determine the stability of the system. Solutions describe not only pure baroclinic and barotropic instabilities, but also mixtures of these instabilities. The influences of linearly sheared barotropic flows on baroclinic instability are studied in detail, with an emphasis on the direction of vertically integrated momentum flux. The model's implications for the nonlinear life cycle of baroclinic waves are also discussed.