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A diagnostic relationship between finite-amplitude wave activity and the associated adiabatic adjustments to the zonal-mean zonal wind and temperature is developed in the quasigeostrophic (QG) framework and is applied to a 23-yr segment (1979-2001) of the 40-yr ECMWF Re-Analysis (ERA-40) data. Wave activity is defined in terms of an instantaneous areal displacement of QG potential vorticity (PV) from zonal symmetry. Unlike previous forms, the tendency of wave activity equals exactly the negative of the eddy PV flux (Eliassen--Palm flux divergence) in the conservative limit, even at finite amplitude. This allows one to integrate the transformed Eulerian mean (TEM) theory in time and quantify the departure (adiabatic adjustment) of the zonal-mean state from an eddy-free reference state in terms of the observed wave activity. The structure of wave activity identifies synoptic eddies in the extratropics and planetary waves in the high latitudes of winter-to-spring stratosphere. In addition, a thin layer of high wave activity is found at the top of the lowermost stratosphere (~17 km) in the summer extratropics. The reference state is constructed by 'zonalizing' the PV contours conservatively (preserving area) on the isobaric surface and by inverting the resultant PV gradient for the mean flow. The adjustment associated with wave activity depends on the assumed surface boundary condition for the reference state. With a no-slip condition, the observed zonal-mean temperature is on average ~33 (90) K higher than the reference state in the troposphere (stratosphere) of the Arctic winter, while the zonal-mean zonal wind is ~30 m [s.sup.-1] slower in the upper stratosphere. Since the reference state filters out the advective eddy-mean flow interaction, it fluctuates less than the zonal-mean state, potentially improving the signal-to-noise ratio for climate diagnosis. DOI: 10.1175/2010JAS3503.1

An exact diagnostic formalism for finite-amplitude eddy--mean flow interaction is developed for barotropic and quasigeostrophic baroclinic flows on the beta plane. Based on the advection--diffusion--reaction equation for potential vorticity (PV), the formalism quantifies both advective and diffusive contributions to the mean flow modification by eddies, of which the latter is the focus of the present article. The present theory adopts a hybrid Eulerian--Lagrangian-mean description of the flow and defines finite-amplitude wave activity in terms of the areal displacement of PV contours from zonal symmetry. Unlike previous formalisms, wave activity is readily calculable from data and the local Eliassen---Palm relation does not involve cubic or higher-order terms in eddy amplitude. This leads to a natural finite-amplitude extension to the local nonacceleration theorem, as well as the global stability theorems, in the inviscid and unforced limit. The formalism incorporates mixing with effective diffusivity of PV, and the diffusive flux of PV is shown to be a sink of wave activity. The relationship between the advective and diffusive fluxes of PV and its implications for parameterization are discussed in the context of wave activity budget. If all momentum associated with wave activity were returned to the zonal-mean flow, a balanced eddy-free flow would ensue. It is shown that this hypothetical flow [u.sub.REF] is unaffected by the advective PV flux and is driven solely by the diffusive PV flux and forcing. For this reason, [u.sub.REF], rather than the zonal-mean flow, is proposed as a diagnostic for the diffusive mean-flow modification. The formalism is applied to a freely decaying beta-plane turbulence to evaluate the contribution of the diffusive PV flux to the jet formation. DOI: 10.1175/2010JAS3432.1

Formation of multiple jets in forced beta-plane turbulence is studied from the perspective of nonuniform nonconservative arrangement of potential vorticity (PV). Numerical simulations are analyzed to show that mixing and forcing reinforce jets by concentrating PV gradients at the axes of prograde jets. Based on the formalism developed in the companion paper, the nonconservative driving of jets is diagnosed and parameterized through the diffusive flux of PV and the source of wave activity. It is found that the two terms nearly balance on a long time scale, and they are both strongly anticorrelated with the PV gradient, which suggests that PV controls the nonconservative processes and that these processes could be parameterized as functions of the PV gradient. The flux is modeled using the effective diffusivity formula recently obtained by Ferrari and Nikurashin. Consistent with the PV barrier concept, the nonlinear diffusivity is a decreasing function of the squared PV gradient and agrees well with the diffusivity diagnosed from the numerical simulation. The source term is assumed to be inversely proportional to the PV gradient. The parameterization gives rise to a nonlinear partial differential equation (PDE) for the mean flow. A finite-difference model of the PDE predicts formation of a piecewise linear PV (staircase) and piecewise parabolic jets from a near-uniform initial condition when anisotropy and mixing of the flow are sufficiently strong. The origin of the discontinuities is antidiffusive instability of PV gradients, and although nonlinearity allows the discrete model to integrate stably, the solution is sensitive to the initial condition and resolution. The emerging jets in the 1D model have similar characteristics to those in the numerical simulation, but the details of the transient behavior are distinct. Similar discrete models of ill-posed PDEs in which discontinuities form also appear in image processing and granular matter dynamics. DOI: 10.1175/2009JAS3159.1

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.

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.

Area equivalent latitude based on potential vorticity (PV) is a widely used diagnostic for isentropic transport in the stratosphere and upper troposphere. Here, an alternate method for calculating equivalent latitude is explored, namely, a numerical synthesis of a PV-like tracer from a long-term integration of the advection-diffusion equation on isentropic surfaces. It is found that the tracer equivalent latitude (TrEL) behaves much like the traditional PV equivalent latitude (PVEL) despite the simplified governing physics; this is evidenced by examining the kinematics of the Arctic lower stratospheric vortex. Yet in some cases TrEL performs markedly better as a coordinate for long-lived trace species such as ozone. These instances include analysis of lower tratospheric ozone during the Stratospheric Aerosol and Gas Experiment (SAGE) III Ozone Loss and Validation Experiment (SOLVE) campaign and three-dimensional reconstruction of total column ozone during November-December 1999 from fitted ozone-equivalent latitude relationship. It is argued that the improvement is due to the tracer being free from the diagnostic errors and certain diabatic processes that affect PV. The sensitivity of TrEL to spatial and temporal resolution, advection scheme, and driving winds is also examined.

It is shown that the Eulerian-mean advection-diffusion equation for a passive tracer, q, can be cast, uniquely and exactly into [delta]/[delta]t[bar]q + [gradient] * ([bar][v.sup.*][bar]q) = [gradient] * ([K.sup.*] [gradient] [bar]q), where the bar denotes an arbitrary Eulerian average (spatial, temporal, or both), [[bar]v.sup.*] is a nondivergent effective transport velocity, and [K.sup.*] is a scalar that varies in space and time. Furthermore, [K.sup.*] = [K.sub.k] + [K.sub.m], where [K.sub.k] represents (possibly nonlocal) along-gradient dispersion of tracer isosurface, whereas [K.sub.m] = D[bar]|[gradient]q[|.sup.2]/|[gradient][bar]q[|.sup.2] [greater than or equal to] D represents local mixing due to diffusion at unresolved scales (D) amplified by resolved (but averaged) fluid dynamical stirring. The amplifying factor, [bar]|[gradient]q[|.sup.2]/|[gradient][bar]q[|.sup.2] measures local roughness of tracer field and determines the spatiotemporal structure of [K.sub.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.

An efficient solution procedure is developed for a nonseparable semigeostrophic Eady problem on a semi-infinite f plane. A model is designed to study the effects of meridionally isolated jets and tropopause morphology on baroclinic instability. Potential vorticity (PV) is assumed to be piecewise constant and discontinuous at the tropopause, whose height is allowed to vary with latitude. The computational domain is discretized in a stretched coordinate to maintain adequate resolution in the vicinity of the tropopause and axis of the jet. The basic state is numerically inverted from the two PV values with the specification of only the meridional profiles of potential temperature (PT) at the surface and (unspecified) tropopause height. The nonseparable eigenvalue problem about this basic state is solved for the discrete (zero-PV) normal modes. Since continuum modes are not sought, the size of the problem is reduced greatly, keeping the storage and CPU requirements moderate even at relatively high spatial resolutions. This model is used to investigate changes in the zonal-mean state and the stability thereof, in response to arrangements of the zonal-mean PT at the surface and tropopause. In particular, (a) partial mixing of surface PT and (b) appearance of a local minimum of PT on the tropopause are considered as models of baroclinic adjustment and tropopause folds, respectively. The former renders the mean flow more barotropic and shifts up the zonal scale of baroclinic instability. The latter gives rise to a markedly dipped tropopause that is barotropically unstable at various ranges of wavenumbers. The results' implications on the life cycle simulations and roll-up of stratospheric intrusions are discussed.

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 by [Mathematical Expression Omitted], where [Mathematical Expression Omitted] is 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 regulates the evolution of q through the stretching and redistribution of ??. It is argued that ??, is a useful, easy to compute diagnostic for irreversible transport, especially for identifying a barrier. Typical behaviors of ?? in various flow regimes are illustrated using the numerically simulated Kelvin-Helmholtz billow. Signature of mixing is found in the irreversible growth of ??, 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 ??. Relevance of this model to the stratospheric polar vortex, midlatitude tropopause, and oceanic thermocline is discussed.

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 there 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.sub.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.

Equilibration of two-dimensional Eady waves is numerically investigated using the geostrophic moment equations incorporating heat and momentum diffusion. Extended solutions are obtained beyond what wou the collapse of surface fronts in the inviscid theory, and are found to accurately reproduce the equ baroclinic waves simulated with the primitive equations. Potential vorticity anomalies produced at t fronts are essential in the initial amplitude saturation and in the asymptotic behavior of the equil supergeostrophic shear spun up nonlinearly in the zonal flow also plays an important role in causing of the tilt. The zonal mean potential temperature profile in the equilibrium state is similar to the prediction b adjustment hypothesis of Gutowski when only horizontal diffusion is present. However, it is closer t basic state with enhanced static stability when vertical diffusion is also present. The difference in the ageostrophic streamfunctions between the geostrophic momentum and primitive eq reveals rich features of inertia-gravity waves radiating from the surface fronts in the latter model role of the gravity waves in the equilibration process, however, is found to be minor.

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.

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.

The equilibration of two-dimensional baroclinic waves differs fundamentally from equilibration in three dimensions because two-dimensional eddies cannot develop meridional temperature or velocity structure. It was shown in an earlier paper that frontogenesis together with diffusive mixing in a two-dimensional Eady wave brings positive potential vorticity (PV) anomalies deep into the atmosphere from both boundaries and allows the disturbance to settle into a steady state without meridional gradients. Here we depart from the earlier explanation of this equilibration and associate the PV intrusions with essentially the same kind of vortex 'roll-up' that characterizes the evolution of barotropic shear layers. To avoid subgrid turbulence parameterizations and computational diffusion, the analogy is developed using Eady's generalized baroclinic instability problem. Eady's generalized model has two semi-infinite regions of large PV surrounding a layer of relatively small PV. Without boundaries, frontal collapse, or strong diffusion the model still produces equilibrated states, with structure similar to the vortex streets that emerge from unstable barotropic shear layers. The similarity is greatest when the baroclinic development is viewed in isentropic coordinates. The contrast between the present equilibrated solutions, which exhibit no vertical tilt, and Blumen's diffusive frontogenesis model, which allows the wave to retain its phase tilt, is briefly discussed.

Finite-amplitude Rossby wave activity (FAWA) proposed by Nakamura and Zhu measures the waviness of quasigeostrophic potential vorticity (PV) contours and the associated modification of the zonal-mean zonal circulation, but it does not distinguish longitudinally localized weather anomalies, such as atmospheric blocking. In this article, FAWA is generalized to local wave activity (LWA) to diagnose eddy-mean flow interaction on the regional scale. LWA quantifies longitude-by-longitude contributions to FAWA following the meridional displacement of PV from the circle of equivalent latitude. The zonal average of LWA recovers FAWA. The budget of LWA is governed by the zonal advection of LWA and the radiation stress of Rossby waves. The utility of the diagnostic is tested with a barotropic vorticity equation on a sphere and meteorological reanalysis data. Compared with the previously derived Eulerian impulse-Casimir wave activity, LWA tends to be less filamentary and emphasizes large isolated vortices involving reversals of meridional gradient of potential vorticity. A pronounced Northern Hemisphere blocking episode in late October 2012 is well captured by a high-amplitude, near-stationary LWA. These analyses reveal that the nonacceleration relation holds approximately over regional scales: the growth of phase-averaged LWA and the deceleration of local zonal wind are highly correlated. However, marked departure from the exact nonacceleration relation is also observed during the analyzed blocking event, suggesting that the contributions from nonadiabatic processes to the blocking development are significant. [ABSTRACT FROM AUTHOR]

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.