Transformed Eulerian Mean Research Articles

A General Three-Dimensional Transformed Eulerian Mean Formulation

A General Three-Dimensional Transformed Eulerian Mean Formulation

Eddy Transport and Mixing in a Wind- and Buoyancy-Driven Jet on the Sphere

Abstract The baroclinically unstable wind- and buoyancy-driven flow in a zonally reentrant pie-shaped sector on a sphere is numerically modeled and then analyzed using the transformed Eulerian-mean (TEM) formalism. Mean fields are obtained by zonal and time averaging performed at fixed height. The very large latitudinal extent of the basin (50.7°S latitude to the equator) allows the latitude variation of the Coriolis parameter to strongly influence the flow. Persistent zonal jets are observed in the statistically steady state. Reynolds stress terms play an important role in redistributing zonal angular momentum: convergence of the lateral momentum flux gives rise to a strong eastward jet, with an adjacent westward jet equatorward and weaker multiple jets poleward. An equally prominent feature of the flow is a strong and persistent eddy that has the structure of a Kelvin cat’s eye and generally occupies the zonal width of the basin at latitudes 15°–10°S. A strongly mixed surface diabatic zone overlies the near-adiabatic interior, within which Ertel potential vorticity (but not thickness) is homogenized along the mean isopycnals everywhere in the basin where eddies have developed (and thus is not homogenized equatorward of the most energetic eastward jet). A region of low potential vorticity (PV) is formed adjacent to the strong baroclinic front associated with that jet and subsequently maintained by strong convective events. The eddy buoyancy flux is dominated by its skew component over large parts of the near-adiabatic interior, with cross-isopycnal components present only in the vicinity of the main jet and in the surface diabatic layer. Close to the main jet, the cross-isopycnal components are dominantly balanced by the triple correlation terms in the buoyancy variance budget, while the advection of buoyancy variance by the mean flow is not a dominant term in the eddy buoyancy variance budget. Along-isopycnal mixing in the near-adiabatic interior is estimated by applying the effective diffusivity diagnostic of Nakamura. The effective diffusivity is large at the flanks of the mean jet and beneath it and small in the jet core. The apparent horizontal diffusivity for buoyancy obtained from the flux–gradient relationship is the same magnitude as the effective diffusivity, but the structures are rather different. The diapycnal diffusivity is strongest in the surface layer and also in a convectively unstable region that extends to depths of hundreds of meters beneath the equatorward flank of the main jet.

The Angular Momentum Budget of the Transformed Eulerian Mean Equations

The axial angular momentum (AAM) budget of zonal atmospheric annuli extending from the surface to a given height and over meridional belts is discussed within the framework of conventional and transformed Eulerian mean (TEM) theory. Conventionally, it is only fluxes of AAM through the boundaries and/or torques at the surface that are able to change the AAM of an annulus. TEM theory introduces new torques in the budget related to the vertically integrated Eliassen–Palm flux divergence and also new AAM fluxes of the residual difference circulation. Some of these torques are displayed for various annuli. In particular, the application of TEM theory generates a large positive torque at tropospheric upper boundaries in the global case. This torque is much larger than the global mountain and friction torques but is cancelled exactly by the new vertical AAM fluxes through the upper boundary. It is concluded that the TEM approach complicates the analysis of AAM budgets but does not provide additional insight. Isentropic pressure torques are believed to be similar to the TEM torques at the upper boundary of an annulus. The isentropic pressure torques are evaluated from data and found to differ in several respects from the TEM torques.

A Model of the Zonally Averaged Stratification and Overturning in the Southern Ocean

AbstractThe ocean area south of the Antarctic Circumpolar Current (ACC) frontal system is a region of major watermass modification. Influx of North Atlantic Deep Water (NADW), small-scale mixing, eddy transport and diffusion, as well as the fluxes of momentum and buoyancy at the sea surface combine in a complex array of processes to generate the unique stratification of the Southern Ocean with its southward uprising isopycnals and northward flux of Antarctic Intermediate Water (AAIW) and Antarctic Bottom Water. Comprehensive analytical models of this scenario are rare. The authors develop and apply a model based on zonally and temporally averaged theory to explain the conversion of NADW into AAIW with all of the aforementioned processes contained in an extremely simplified way. Eddies appear via a transformed Eulerian mean (TEM) approach with a conventional downgradient parameterization of the meridional density flux. The structure of the eddy coefficient is estimated from hydrographic and wind stress data by a simple inverse approach. Mixing is limited to a near-surface layer and is treated in a most simple entrainment form. The model determines the zonal mean density stratification in the Southern Ocean and the baroclinic transport of the ACC from the applied wind stress and the surface density flux and unravels the role and importance of the different processes responsible for shaping the stratification (Ekman and eddy-induced advection and pumping, mixing, surface buoyancy flux, and eddy-induced diffusion). All of these processes must be present to yield an agreement between the simulated stratification and the observed one, but details of their parameterization might not be too critical. The ACC transport is shown to have a contribution forced by the local wind stress as well as another contribution relating to the nonlocal forcing by wind stress and density flux over the entire Antarctic zone.

Diagnosis of Meridional Ozone Transport Based on Mass-Weighted Isentropic Zonal Means

Abstract A transport equation based on mass-weighted isentropic zonal means applies to the diagnosis for the meridional ozone transport in the troposphere and stratosphere. The mean and eddy ozone fluxes are estimated from the global distributions of the temperature, wind, and ozone. In comparison with the conventional Eulerian mean and transformed Eulerian mean (TEM), the present diagnosis has advantages for the expression of eddy transport terms. The adiabatic eddy flux is separated from the diabatic eddy flux, which is parallel to the isentropic surface. The analysis shows that the eddy flux is almost adiabatic except that it is significantly affected by diabatic effects near the lower troposphere. Another advantage lies in the mean meridional transport. Although it is almost similar to the TEM, significant differences can be found near the Antarctic polar vortex due to nongeostrophic effects. Furthermore, the isentropic diagnosis expresses a strong equatorward flux near the lower boundary, while the TEM hardly does this because of inadequate treatment of the lower-boundary conditions. The life cycle of ozone can be understood through the exact estimation of the transport terms. Although the stratospheric meridional transport is mainly performed by the Brewer–Dobson circulation, the strong poleward eddy ozone flux is caused by planetary wave breaking, especially in the winter hemisphere. In the extratropics, the ozone subsides from the stratosphere to the troposphere by mean downward motions, mainly diffused to the lower latitudes probably due to strong baroclinic waves and effectively lost through chemical processes in the lower troposphere.

Transformed Eulerian-Mean Theory. Part I: Nonquasigeostrophic Theory for Eddies on a Zonal-Mean Flow

Abstract A theoretical formalism for nongeostrophic eddy transport in zonal-mean flows, using a transformed Eulerian-mean (TEM) approach in z coordinates, is discussed. By using Andrews and McIntyre’s coordinate-independent definition of the “quasi-Stokes streamfunction,” it is argued that the surface boundary condition can be dealt with more readily than when the widely used quasigeostrophic definition is adopted. Along with the “residual mean circulation,” the concept of “residual eddy flux” arises naturally within the TEM framework, and it is argued that it is this residual eddy flux, and not the “raw” eddy flux, that might reasonably be expected to be downgradient. This distinction is shown to be especially important for Ertel potential vorticity (PV). The authors show how a closed set of transformed mean equations can be generated, and how the eddy forcing appears in the TEM momentum equations. Under adiabatic conditions, the “eddy drag” is just proportional to the residual eddy flux of PV along the mean isopycnals; in the diabatic layer close to the surface, it is more complicated, but becomes very simple for small Rossby number (without any assumption of small isopycnal slope).

Transformed Eulerian-Mean Theory. Part II: Potential Vorticity Homogenization and the Equilibrium of a Wind- and Buoyancy-Driven Zonal Flow

Abstract The equilibrium of a modeled wind- and buoyancy-driven, baroclinically unstable, flow is analyzed using the transformed Eulerian-mean (TEM) approach described in Part I. Within the near-adiabatic interior of the flow, Ertel potential vorticity is homogenized along mean isopycnals—a finding readily explained using TEM theory, given the geometry of the domain. The equilibrium, zonal-mean buoyancy structure at the surface is determined entirely by a balance between imposed surface fluxes and residual mean and eddy buoyancy transport within a “surface diabatic layer.” Balance between these same processes and the wind stress determines the stratification, and hence potential vorticity, immediately below this layer. Ertel potential vorticity homogenization below then determines the mean buoyancy structure everywhere. Accordingly, the equilibrium structure of this flow can be described—and quantitatively reproduced—from knowledge of the eddy mixing rates within the surface diabatic zone and the depth of this zone, together with potential vorticity homogenization beneath. These results emphasize the need to include near-surface buoyancy transport, as well as interior PV transport, in eddy parameterization schemes. They also imply that, in more realistic models, the surface buoyancy balances may be impacted by processes in remote locations that allow diapycnal flow.

Isopycnal Averaging at Constant Height. Part II: Relating to the Residual Streamfunction in Eulerian Space

Abstract In Part I, the “vertical” transport streamfunction was defined as resulting from isopycnic averaging at constant height in the same way that the meridional streamfunction results from averaging at constant latitude. Part II here discusses the relationship between these two isopycnic streamfunctions and the Eulerian residual streamfunction that arises from the transformed Eulerian mean (TEM). It is known that the meridional isopycnic streamfunction can be approximated by a Taylor expansion to give an Eulerian residual streamfunction involving the horizontal eddy flux. This Taylor expansion approximation works well in the interior, removing the spurious mixing associated with the simple Eulerian-averaged streamfunction. However, it fails near the surface where isopycnals outcrop to the surface. It can be shown in a similar way that the vertical isopycnic streamfunction can formally be approximated by a residual streamfunction involving the vertical eddy flux. However, if horizontal isopycnal displacements are large, this approximation fails even in the ocean interior. Inspired by the two different residual streamfunctions, a more general form of TEM formulation is explored. It is shown that the different TEM residual streamfunctions arise from decomposing the eddy flux into a component along isopycnals, which leads to advective flow, and a remaining diffusive component, which is oriented either vertically or horizontally. In theory the diffusive flux can be oriented in any direction, although in practice the orientation should be such that neither the advective flow nor the diffusive flux cross any boundary (surface, sidewalls, and bottom). However, it is not clear how to merge the continuously changing orientation in a physically meaningful way. A variety of approaches are discussed.

Eliassen–Palm Flux Diagnosis Based on Isentropic Representation

Abstract This paper demonstrates an isentropic representation of the Eliassen–Palm flux (EP flux) as a tool for diagnosis of the atmospheric zonal mean states. This is a full extension of the transformed Eulerian mean (TEM) to express lower boundary conditions and finite-amplitude effects of waves. The isentropic EP flux is diagnosed from the NCEP–NCAR reanalysis for 12 yr and compared with the TEM. In the free atmosphere, the two diagnoses show similarities in the mass streamfunction and in the EP flux except that the EP flux divergence is somewhat different near the polar vortex. The similarities reflect the equivalency between the two schemes under the quasigeostrophic assumptions. However, the lower boundary values are very different between the two schemes. In the TEM, the vertical EP flux inaccurately suggests a much larger momentum flux coming out from the surface. In the isentropic diagnosis, the lower boundary value of vertical EP flux, which expresses the momentum exchange with the earth's surfa...

On advection and diffusion in the mesosphere and lower thermosphere: The role of rotational fluxes

A formalism to describe the advective and diffusive eddy transport in terms of the mean tracer is presented. It is based on Eulerian averaging, the flux‐gradient relation, and the decomposition of the eddy flux of a tracer into advective, diffusive, and rotational components. The rotational (nondivergent) flux arises because the conservation equation for the mean tracer contains only a divergence of the eddy flux of the tracer. To provide a closure, a modification to the flux separation technique based on the eddy variance equation is introduced. The “eddy‐induced” advective velocity is represented as the sum of two velocities v1 and v2. Velocity v1 is similar to that in the Transformed Eulerian Mean (TEM) formulation but is generalized to account for both horizontal and vertical eddy fluxes and mean gradients. The velocity v2 depends on the flux of eddy variance of the tracer. The diffusion coefficient is represented as a sum of K1, which may serve as a diagnostic of an irreversible mixing, and K2, which describes up‐ or down‐gradient eddy fluxes of the tracer due to local transformations of the eddy variance. Both v2 and K2 arise from taking account of the rotational fluxes. The scheme is applied to output from a global circulation model of the middle atmosphere. It is shown that in the meridional plane the correction v2 to the TEM velocity is small in the mesosphere and lower thermosphere. For the diffusion coefficient, however, the correction K2 must be accounted for above approximately 110 km.

A New Theory for the Generation of the Equatorial Subsurface Countercurrents

Abstract An idealized numerical simulation of the tropical Atlantic Ocean is used to study the dynamics of an Atlantic subsurface countercurrent, the South Equatorial Undercurrent (SEUC). The particular structure of the SEUC between 28° and 10°W allows for a reformulation of the transformed Eulerian mean (TEM) equations with which the momentum balance of the SEUC can be explored. With this modified TEM framework, it is shown that between 28° and 10°W the SEUC is maintained against dissipation by the convergence of the Eliassen–Palm flux. The source of this Eliassen–Palm flux is the tropical instability waves that are generated along the shear between the Equatorial Undercurrent and the South Equatorial Current.

Zonal‐mean transport characteristics of ECMWF re‐analysis data

AbstractTransport characteristics for a dataset from a global data assimilation system (European Centre for Medium‐Range Weather Forecasts re‐analyses for 1993) are analysed by constructing an advection–diffusion operator with tracer‐transport properties as close as possible to the zonally and seasonally averaged tracer advection. This is achieved by computing a best‐fit flux–gradient relation for a large number of artificial tracers. The meridional circulation of the advection–diffusion operator (the transport velocity) is thermally direct almost everywhere. It is similar to the Transformed Eulerian Mean (TEM) residual circulation in most of the stratosphere, but is much weaker than this circulation in the extratropical troposphere. The diffusivity tensor has a complicated structure where transport barriers and ‘surf zones’ can be identified. The structure is strongly connected to the zonal wind, with distinct minima in the jet‐stream areas. The zonal acceleration due to effects other than the transport circulation is studied. It is shown that this acceleration has a tendency to be strong when the mixing is strong and vice versa. Copyright © 2003 Royal Meteorological Society.

Planetary-Wave-Induced Transport in the Stratosphere

In the stratosphere, chemical tracers are drawn systematically from the equator to the pole. This observed Brewer–Dobson circulation is driven by wave drag, which in the stratosphere arises mainly from the breaking and dissipation of planetary-scale Rossby waves. While the overall sense of the circulation follows from fundamental physical principles, a quantitative theoretical understanding of the connection between wave drag and Lagrangian transport is limited to linear, small-amplitude waves. However, planetary waves in the stratosphere generally grow to a large amplitude and break in a strongly nonlinear fashion. This paper addresses the connection between stratospheric wave drag and Lagrangian transport in the presence of strong nonlinearity, using a mechanistic three-dimensional primitive equations model together with offline particle advection. Attention is deliberately focused on a weak forcing regime, such that sudden warmings do not occur and a quasi-steady state is reached, in order to examine this question in the cleanest possible context. Wave drag is directly linked to the transformed Eulerian mean (TEM) circulation, which is often used as a surrogate for mean Lagrangian motion. The results show that the correspondence between the TEM and mean Lagrangian velocities is quantitatively excellent in regions of linear, nonbreaking waves (i.e., outside the surf zone), where streamlines are not closed. Within the surf zone, where streamlines are closed and meridional particle displacements are large, the agreement between the vertical components of the two velocity fields is still remarkably good, especially wherever particle paths are coherent so that diabatic dispersion is minimized. However, in this region the meridional mean Lagrangian velocity bears little relation to the meridional TEM velocity, and reflects more the kinematics of mixing within and across the edges of the surf zone. The results from the mechanistic model are compared with those from the Canadian Middle Atmosphere Model to test the robustness of the conclusions.

On the meridional circulation and balance of momentum in the Southern Ocean of POP

The circulation of the Southern Ocean is studied in the eddy-resolving model POP (Parallel Ocean Program) by an analysis of zonally integrated balances. The TEM formalism (Transformed Eulerian Mean) is extended to include topography and continental boundaries, thus deviations from a zonally integrated state involve transient and standing eddies. The meridional circulation is presented in terms of the Eulerian, eddy-induced, and residual streamfunctions. It is shown that the splitting of the meridional circulation into Ekman and geostrophic transports and the component induced by subgrid and Reynolds stresses is identical to a particular form of the zonally integrated balance of zonal momentum. In this balance, the eddy-induced streamfunctions represent the interfacial form stresses by transient and standing eddies and the residual streamfunction represents the acceleration of the zonal current by density fluxes in a zonally integrated frame. The latter acceleration term is directly related to the surface flux of density and interior fluxes due to the resolved and unresolved eddies. The eddy-induced circulation is extremely vigorous in POP. In the upper ocean a shallow circulation, reversed in comparison to the Deacon cell and mainly due to standing eddies, appears to the north of Drake Passage latitudes, and in the Drake Passage belt of latitudes a deep-reaching cell is induced by transient eddies. In the resulting residual circulation the Deacon cell is largely cancelled and the residual advection of the zonal mean potential density is balanced by diapycnal eddy and subgrid fluxes which are strong in the upper few hundred meters but small in the ocean interior. The balance of zonal momentum is consistent with other eddy-resolving models; a new aspect is the clear identification of density effects in the zonally integrated balance. We show that the wind stress and the stress induced by the residual circulation drive the eastward current, whereas both eddy species result in a braking. Finally, we extend the Johnson–Bryden model of zonal transport to incorporate all relevant terms from the zonal momentum balance. It is shown that wind stress and induction by the residual circulation carry an eastward transport while bottom form stress and the stress induced by standing eddies yield westward components of transport.

QBO and Pinatubo signals in the mass flux at 100hPa during the period from 1986 to 1995

The residual mean meridional circulation is calculated by the transformed Eulerian-mean (TEM) momentum and continuity equations for 10 years from December 1985 to November 1995. The tropical upward mass flux at 100 hPa is estimated from the extratropical downward mass fluxes in the southern and northern hemispheres. The multiple regression statistical model is used to obtain quantitatively the signal of the quasi-biennial oscillation (QBO) and the trend. The results show that the tropical upward mass flux is enhanced during QBO easterly phase and reduced during QBO westerly phase. The signal of Mount Pinatubo volcanic eruption in June 1991 is detected. An abrupt increase of the mass flux is shown in the latter half of 1991, and the increase ceases at the end of 1993, about 2.5 years after the eruption.

Lagrangian transport experiments in the MLT region

In order to evaluate material transports in the upper mesosphere and lower thermosphere (MLT) region, Lagrangian transport experiments are performed using wind fields simulated by the Kyushu University middle atmosphere general circulation model (Miyahara and Miyoshi, 1997; Miyahara et al., 1993). The Lagrangian mean meridional circulation is different from the residual mean circulation defined by the Transformed Eulerian Mean equation system, which is thought to be approximately equal to the Lagrangian mean meridional circulation in the middle atmosphere in case of weak dissipation and transience (Andrews and McIntyre, 1978; Andrews et al., 1987). It is considered that the diffusivity caused by the transience and dissipation of various wave motions in the MLT region has significant effects on the material transport in this region, and makes them different from each other.

成層圏・上部対流圏の残差平均子午面循環

The climatological variation of the residual mean meridional circulation in the stratosphere, and the upper troposphere is investigated for 10 years from December 1985 to November 1995. The residual circulation is calculated from the transformed Eulerian-mean (TEM) zonal momentum and continuity equations. The method takes account of the Rayleigh friction and the mean zonal flow change. The tropical upward mass flux across the 100 hPa surface is maximum in November and minimum in June, and the annual mean is 101.8 × 10 8 kg/s, with a turnover time of 1.6 years for the atmosphere above 100 hPa. The cause of differences in magnitude and seasonal variation of the Mass flux between previous studies, and the present study, is discussed. It is shown that the present results is more reasonable than previous studies. In the stratospheric mass budget, it is found that the spring and summer circulations are stronger in the southern hemisphere as compared with the same seasons in the north, and the reverse is true during fall and winter. The annual mean circulation is stronger in the northern hemisphere than in the southern hemisphere. The poleward flow is weak in the winter middle stratosphere. In addition, it is suggested that the mass exchange at 100 hPa surface is mainly controlled by wave forcings in the lower stratosphere.

Kelvin waves and the quasi‐biennial oscillation: An observational analysis

A method is derived for estimating the Kelvin wave contribution to the vertical flux of westerly momentum in the equatorial stratosphere, which is based on temperature and geopotential perturbations. This method is used to estimate the momentum transfer due to Kelvin wave activity as derived from the cryogenic limb array etalon spectrophotometer (CLAES) temperature data set, for the onset of the westerly phase of the QBO during 1992 and the first few months of 1993, that is, during the first part of the Upper Atmosphere Research Satellite (UARS) mission. The results are compared with the zonal winds as observed by the high‐resolution Doppler imager (HRDI) also flown on board UARS, and the United Kingdom Meteorological Office (UKMO) data assimilation model product, a correlative data set to the UARS mission. The analysis shows that although the Kelvin wave momentum flux convergence is occasionally sufficient to account for the observed QBO westerly acceleration, the observed flux is sporadic in nature and virtually disappears during the second half of the sample, when the westerly vertical shear zone approaches the 100 hPa level. An estimate of the total westerly momentum flux necessary to produce the observed descent of the westerly phase of the QBO is made using the transformed Eulerian mean (TEM) formalism. The results suggest that Kelvin waves are not sufficient to force the descent of the westerly phase of the QBO. There appears to be a need for enhanced westerly forcing throughout the descent of the westerly phase of the QBO. This enhanced forcing is most likely provided by gravity waves that are unresolved by the satellite observations. These results are in agreement with the results derived from general circulation models.

QBO and Pinatubo signals in the mass flux at 100 hPa and stratospheric circulation

The residual mean meridional circulation is calculated by the transformed Eulerian‐mean (TEM) momentum and continuity equations for the period from December 1988 to November 1995. Our method does not rely on radiative calculations. The tropical upward mass flux at 100 hPa is estimated from the extratropical downward mass fluxes in the southern and northern hemispheres. The multiple regression statistical model is used to obtain quantitatively the signal of the quasi‐biennial oscillation (QBO) and the trend. The results show that the tropical upward mass flux is enhanced during QBO easterly phase and reduced during QBO westerly phase. As a residual, the signal of Mount Pinatubo volcanic eruption in June 1991 is detected. This reflects the aerosol effects to the residual circulation in the lower stratosphere after the eruption. An abrupt increase of the mass flux is shown in the latter half of 1991, and the increase ceases at the end of 1993, about 2.5 years after the eruption.

On anomalous meridional circulations and Eliassen-Palm flux divergences in an idealized model of dissipating, non-breaking Rossby waves

Even in idealized models of steady, dissipating, non-breaking Rossby waves at small wave amplitude, and even in the absence of barotropic and baroclinic shear instabilities, there can be an anomalous Eliassen-Palm flux divergence in the sense that the divergence is positive when the background potential-vorticity gradient is also positive, implying upgradient eddy potential-vorticity transport. The phenomenon is illustrated in the simplest possible case of dissipation by Rayleigh friction and Newtonian cooling, and is shown by a more general argument not to be restricted to that case. The physical reason is that infrared radiative damping can act anti-dissipatively on potential-vorticity anomalies whenever the vertical disturbance structure is diffractive or evanescent, as with most real stratospheric synoptic and sub-synoptic-scale disturbances forced from below. Associated with this phenomenon are anomalous (eastward) phase tilts with height, and equatorward transformed Eulerian-mean (TEM) meridional velocities. It is pointed out that the latter is a clearcut example of a TEM circulation whose sense is opposite to that of the generalized Lagrangian-mean circulation (and the effective transport circulation in the sense of Plumb and Mahlman) induced by the same steady, small-amplitude disturbance.