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Results for “Brewer–Dobson circulation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1999

Climate Change and the Brewer-Dobson Circulation

Using GCM (General Circulation Model) simulations of the troposphere-stratosphere circulation in different climate regimes, the sensitivity of the Brewer-Dobson Circulation to climate change will be discussed. Included will be simulations of the stratosphere with increasing/doubled CO2 as done with different versions of the GISS (Goddard Institute for Space Studies) Global Climate Middle Atmosphere Model, with and without ozone response. To put the results in a broader perspective, stratospheric circulation changes from several paleoclimate simulations will also be presented, including the Last Glacial Maximum and the Paleocene, each with several different (plausible) latitudinal sea surface temperature gradients. Hence the climate changes investigated will include strong variations in baroclinic and topographic tropospheric planetary wave and gravity wave forcing, as well as global mean temperature and atmospheric CO2 levels. Results will be analyzed in terms of wave driving via E-P flux convergences and gravity wave effects, and will be shown to be strongly dependent on wave propagation characteristics.

Rind, David↗
NASA NTRS1999

Climate Change and the Brewer-Dobson Circulation

Using GCM simulations of the troposphere-stratosphere circulation in different climate regimes, the sensitivity of the Brewer-Dobson Circulation to climate change will be discussed. Included will be simulations of the stratosphere with increasing/doubled C02 as done with different versions of the GISS Global Climate Middle Atmosphere Model, with and without ozone response. To put the results in a broader perspective, stratospheric circulation changes from several paleoclimate simulations will also be presented, including the Last Glacial Maximum and the Paleocene, each with several different (plausible) latitudinal sea surface temperature gradients. Hence the climate changes investigated will include strong variations in baroclinic and topographic tropospheric planetary wave and gravity wave forcing, as well as global mean temperature and atmospheric C02 levels. Results will be analyzed in terms of wave driving via E-P flux convergences and gravity wave effects, and will be shown to be strongly dependent on wave propagation characteristics.

Rind, David↗
NASA NTRS2008

Stratospheric Ozone in the Post-CFC Era

Vertical and latitudinal changes in the stratospheric ozone in the post-chlorofluorocarbon (CFC) era are investigated using simulations of the recent past and the 21st century with a coupled chemistry-climate model. Model results reveal that, in the 2060s when the stratospheric halogen loading is projected to return to its 1980 values, the extratropical column ozone is significantly higher than that in 1975-1984, but the tropical column ozone does not recover to 1980 values. Upper and lower stratospheric ozone changes in the post- CFC era have very different patterns. Above 15 hPa ozone increases almost latitudinally uniformly by 6 Dobson Unit (DU), whereas below 15 hPa ozone decreases in the tropics by 8 DU and increases in the extratropics by up to 16 DU. The upper stratospheric ozone increase is a photochemical response to greenhouse gas induced strong cooling, and the lower stratospheric ozone changes are consistent with enhanced mean advective transport due to a stronger Brewer-Dobson circulation. The model results suggest that the strengthening of the Brewer-Dobson circulation plays a crucial role in ozone recovery and ozone distributions in the post-CFC era.

Li, Feng↗
NASA NTRS2009

Stratospheric Ozone in the Post-CFC Era

Vertical and latitudinal changes in the stratospheric ozone in the post-chlorofluorocarbon (CFC) era are investigated using simulations of the recent past and the 21st century with a coupled chemistry-climate model. Model results reveal that, in the 2060s when the stratospheric halogen loading is projected to return to its 1980 values, the extratropical column ozone is significantly higher than that in 1975-1984, but the tropical column ozone does not recover to 1980 values. Upper and lower stratospheric ozone changes in the post-CFC era have very different patterns. Above 15 hPa ozone increases almost latitudinally uniformly by 6 Dobson Unit (DU), whereas below 15 hPa ozone decreases in the tropics by 8 DU and increases in the extratropics by up to 16 DU. The upper stratospheric ozone increase is a photochemical response to greenhouse gas induced strong cooling, and the lower stratospheric ozone changes are consistent with enhanced mean advective transport due to a stronger Brewer-Dobson circulation. The model results suggest that the strengthening of the Brewer-Dobson circulation plays a crucial role in ozone recovery and ozone distributions in the post-CFC era.

Li, F.↗
DOE OSTI2025

Recent Warming of the Southern Hemisphere Subtropical Lower Stratosphere and Antarctic Ozone Healing

Observed temperature changes from 2002 to 2022 reveal a pronounced warming of the Southern Hemisphere (SH) subtropical lower stratosphere, and a cooling of the Antarctic lower stratosphere. In contrast, model simulations of 21st-century stratospheric temperature changes show widespread cooling driven by increasing greenhouse gases, with local warming in the Antarctic lower stratosphere due to ozone healing. We provide evidence that these discrepancies between observed and simulated stratospheric temperature changes are linked to a slowdown of the Brewer-Dobson Circulation, particularly in the SH. These changes in the stratospheric circulation are strongest from October through December. This altered circulation warms the SH subtropical lower stratosphere while cooling the Antarctic lower stratosphere, canceling and even reversing the Antarctic ozone recovery that would have occurred in its absence during this period. When circulation changes are accounted for, the SH subtropical lower-stratospheric warming is removed, and Antarctic lower-stratospheric warming is revealed with enhanced ozone healing, highlighting the crucial role of the stratospheric circulation in shaping temperature and ozone changes.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1999

Properties of QBO and SAO Generated by Gravity Waves

We present an extension for the 2D (zonal mean) version of our Numerical Spectral Mode (NSM) that incorporates Hines' Doppler spread parameterization (DSP) for small scale gravity waves (GW). This model is applied to describe the seasonal variations and the semi-annual and quasi-biennial oscillations (SAO and QBO). Our earlier model reproduced the salient features of the mean zonal circulation in the middle atmosphere, including the QBO extension into the upper mesosphere inferred from UARS measurements. In the present model we incorporate also tropospheric heating to reproduce the upwelling at equatorial latitudes associated with the Brewer-Dobson circulation that affects significantly the dynamics of the stratosphere as Dunkerton had pointed out. Upward vertical winds increase the period of the QBO observed from the ground. To compensate for that, one needs to increase the eddy diffusivity and the GW momentum flux, bringing the latter closer to values recommended in the DSP. The QBO period in the model is 30 months (mo), which is conducive to synchronize this oscillation with the seasonal cycle of solar forcing. Multi-year interannual oscillations are generated through wave filtering by the solar driven annual oscillation in the zonal circulation. Quadratic non-linearities generate interseasonal variations to produce a complicated pattern of variability associated with the QBO. The computed temperature amplitudes for the SAO and QBO are in substantial agreement with observations at equatorial and extratropical latitudes. At high latitudes, however, the observed QBO amplitudes are significantly larger, which may be a signature of propagating planetary waves not included in the present model. The assumption of hydrostatic equilibrium not being imposed, we find that the effects from the vertical Coriolis force associated with the equatorial oscillations are large for the vertical winds and significant for the temperature variations even outside the tropics but are relatively small for the zonal winds.

Mayr, H. G.↗
DOE OSTI2026

Experimental protocol for phase 1 of the APARC QUOCA (QUasibiennial oscillation and Ozone Chemistry interactions in the Atmosphere) working group

The quasi-biennial oscillation (QBO) is the main mode of variability in the tropical stratosphere, influencing the predictability of other regions in the atmosphere through its teleconnections to the stratospheric polar vortices and coupling to surface tropical and extratropical variability. However, climate and forecasting models consistently underestimate QBO amplitudes in the lower stratosphere, likely contributing to their failure to simulate these teleconnections. One underexplored contributor to model biases is missing representation of ozone-radiative feedbacks, which enhance temperature variability in the lower stratosphere, particularly at periods at and greater than the QBO (>28 months). While previous studies suggest that ozone-radiative feedbacks can impact QBO periods, amplitudes and the associated secondary circulation in the lower stratosphere, the reported impacts differ widely among models and are hard to interpret due to differences in methodology. To this end, here we propose a coordinated experimental protocol – held joint between the Atmospheric Processes and their Role in Climate (APARC) Quasi-Biennial Oscillation Initiative (QBOi) and Chemistry Climate Modeling Initiative (CCMI) activities – which is aimed at assessing the coupling between stratospheric ozone, temperature and the circulation. We use the proposed experiments to define the ozone feedback on the QBO in both present-day and idealized (abrupt quadrupling of carbon dioxide) climates. While primary focus is on the QBO, the proposed protocol also enables analysis of other aspects of ozone-radiative-dynamical coupling in the atmosphere, including impacts on the Brewer-Dobson Circulation and tropospheric eddy-driven jet responses to future climate change. Here we document the scientific rationale and design of the QUOCA Phase 1 experiments, summarize the data request, and give a brief overview of participating models. Preliminary results using the NASA Goddard Institute for Space Studies E2-2 climate model are used to illustrate sensitivities to certain methodological choices.

Environmental sciences↗
NASA NTRS1992

Buffering of stratospheric circulation by changing amounts of tropical ozone - A Pinatubo case study

Stratospheric aerosol from Mount Pinatubo heated the tropical lower stratosphere by about 0.3 K/day mainly due to absorption of terrestrial infrared radiation. This heating was dissipated by: (1) an observed increase in stratospheric temperatures, which enhanced the radiation cooling; (2) additional mean upward motion, observed for the aerosol cloud, which led to adiabatic cooling; and (3) reductions in ozone concentrations resulting from enhanced upward motions. Each of these processes operated on a different time scale: maximum temperatures were observed after about 90 days; maximum ozone losses of about -1.5 ppm occurred after 140 days when the enhanced vertical velocities effectively lifted the ozone profile by about 2 km. We believe this shows that ozone plays an important role in buffering vertical motion in the tropical lower stratosphere, and hence the residual Brewer Dobson circulation of the whole stratosphere.

Kinne, S.↗
DOE OSTI2026

Oceanic fronts shape hemispheric contrasts in polar stratospheric extremes

Sudden Stratospheric Warmings (SSWs) and Polar Stratospheric Clouds (PSCs) exhibit striking inter-hemispheric asymmetries: SSWs are frequent in the Arctic but rare in the Antarctic, while PSCs are more persistent in the Antarctic. Although land–sea thermal contrast and orography (LSCO) have been traditionally invoked to explain these asymmetries, here we show using semi-idealized model experiments that even though LSCO strongly impacts the mean state of SSWs and PSCs, it alone cannot fully account for the observed differences. Using model experiments, we reveal that midlatitude oceanic sea surface temperature (SST) fronts represent a crucial supporting additional driver of the hemispheric stratospheric differences. Like LSCO, SST fronts enhance stratospheric convergence of resolved waves, strengthening the Brewer–Dobson Circulation and inducing high-latitude adiabatic warming. This warming significantly enhances Arctic SSW frequency and strongly suppresses PSC formation. Sub-grid-scale non–resolved wave forcing modulates the stratospheric resolved waves effect. The oceanic impact is dominated by North Pacific SST fronts. Our results highlight the indispensable role of SST fronts in shaping Arctic–Antarctic asymmetries in stratospheric dynamics and associated extremes.

Omrani, Nour-Eddine [Univ. of Bergen (Norway)] (OR↗
NASA NTRS1998

Observational and Modeling Studies of Radiative, Chemical, and Dynamical Interactions in the Earth''s Atmosphere

A 3-dimensional model was developed to support mechanistic studies. The model solves the global primitive equations in isentropic coordinates, which directly characterize diabatic processes forcing the Brewer-Dobson circulation of the middle atmosphere. It's numerical formulation is based on Hough harmonics, which partition horizontal motion into its rotational and divergent components. These computational features, along with others, enable 3D integrations to be performed practically on RISC computer architecture, on which they can be iterated to support mechanistic studies. The model conserves potential vorticity quite accurately under adiabatic conditions. Forced by observed tropospheric structure, in which integrations are anchored, the model generates a diabatic circulation that is consistent with satellite observations of tracer behavior and diabatic cooling rates. The model includes a basic but fairly complete treatment of gas-phase photochemistry that represents some 20 chemical species and 50 governing reactions with diurnally-varying shortwave absorption. The model thus provides a reliable framework to study transport and underlying diabatic processes, which can then be compared against chemical and dynamical structure observed and in GCM integrations. Integrations with the Langley GCM were performed to diagnose feedback between simulated convection and the tropical circulation. These were studied in relation to tropospheric properties controlling moisture convergence and environmental conditions supporting deep convection, for comparison against mechanistic integrations of wave CISK that successfully reproduce the Madden-Julian Oscillation (MJO) of the tropical circulation. These comparisons were aimed at identifying and ultimately improving aspects of the convective simulation, with the objective of recovering a successful simulation of the MJO in the Langley GCM, behavior that should be important to budgets of upper-tropospheric water vapor and chemical species.

Salby, Murry↗
NASA NTRS1999

Dynamical and Chemical Behavior of the Lower Stratosphere and Interactions with the Troposphere

Equivalent-barotropic calculations, in tandem with Lagrangian analyses, reveal how changes of total ozone follow from vertical and horizontal transport by planetary waves. Those calculations also throw light on how diabatic motions comprising the Brewer-Dobson circulation develop from quasi-horizontal advection by planetary waves. Potential temperature along a material surface indicates organized subsidence inside the polar-night vortex, resembling tracer observations from UARS. Lagrangian histories illustrate that this sinking motion follows in large part from parcels being driven out of thermodynamic equilibrium by planetary waves, especially at high latitudes. Irreversible heat transfer then produces a net drift of air across isentropic surfaces as parcels orbit about the displaced vortex. By driving mean-meridional overturning in the stratosphere, this downward drift is ultimately responsible for transferring ozone from the tropics to the extratropical lower stratosphere. It also introduces horizontal structure into the distribution of total ozone, which surfaces clearly in ozone trends. High-resolution global cloud imagery constructed from 6 satellites simultaneously observing the Earth was used to investigate the spectrum of equatorial waves generated by tropical convection and propagating vertically into the stratosphere. The results indicate that temperature variability is dominated by planetary-scale equatorial waves like the Kelvin mode, which agrees with satellite observations of the tropical stratosphere. However, the Kelvin mode accounts for only about 30 - 50% of the eastward momentum flux radiating into the stratosphere, the remainder coming from gravity waves. An algorithm was developed to determine 3-dimensional atmospheric motion from satellite tracer measurements. Based on Lagrangian constraints, the algorithm circumvents limitations of the traditional scheme for inferring motion from temperature measurements and determines the circulation in the tropics as reliably as elsewhere. A study of deep convection revealed that the highest towers (those penetrating into stratospheric air and controlling tropopause height and composition through convective mixing) occur in close association with the diurnal cycle of convection. Clouds colder than 220 K develop almost entirely in association with the diurnal cycle of convection over tropical landmasses and substantially in association with it even over maritime regions.

Salby, Murry L.↗
NASA NTRS1999

Synoptic Mapping of Chemical Composition, Thermal Structure, and Air Motion from UARS Observations

An operational mapping algorithm was developed to process measurements of individual species observed by different satellite instruments on board UARS. Based on Fast Fourier Synoptic Mapping (FFSM), the algorithm accounts for the precessing orbit of UARS, the error of individual instruments, and gaps associated with instrument duty cycle and the satellite yaw maneuver. It provides synoptic structure and evolution on periods as short as 1 day, derived collectively from all observations of an individual species. The algorithm was applied to synoptically map temperature, thickness, and several chemical constituents observed by the instruments: MLS, CLAES, ISAMS, and HALOE. Each field variable observed by these instruments was mapped twice-daily in continuous global time series several months long. Mapped behavior produced via FFSM was compared against standard archived products generated via Kalman filtering. The standard map products reveal similar behavior, but are limited to features of larger scale. Tracer structure that develops through flow deformation and attending transport is therefore represented in those products more coarsely, eventually being sheared down to scales that are no longer represented properly. The synoptic time series also reveal a diurnal cycle for several of the constituents, one that emerges clearly in their space-time spectra. The zonal and meridional structure of diurnal variability was mapped by filtering the space-time spectrum, an intermediate product of FFSM, to those scales resolved by the UARS sampling. Geographical variations of the diurnal cycle have also been evaluated in its seasonal-mean structure by compositing distributions at individual local times. Both were compared to diurnal variations in chemical models and in dynamical models that account for transport by the diurnal tide. Diurnal variations were found to introduce spurious behavior into the archived products generated via Kalman filtering, behavior that is misrepresented in those synoptic time series. Aliasing by under-sampled diurnal variations introduces a bias into the time-mean structure of archived products. That systematic error is manifested as anomalous wave structure in the time-mean fields. By contrast, synoptic time series generated via FFSM faithfully represent such behavior as varying diurnally, so it does not corrupt time-mean structure. Tracer structure mapped via the above algorithm was being applied to determine the Brewer-Dobson circulation of the stratosphere through Lagrangian considerations. Preliminary results are encouraging: They reveal diabatic ascent in the tropics, balanced by diabatic descent over the winter hemisphere. This Lagrangian description of transport derived from UARS tracer measurements would be potentially valuable for diagnosing interannual changes of transport, specifically, in relation to interannual changes of total ozone. Regrettably, support was reduced to 50% of the approved funding level, leaving this component incomplete.

Salby, Murry L.↗
NASA NTRS1999

Statistical Perspectives on Stratospheric Transport

Long-lived tropospheric source gases, such as nitrous oxide, enter the stratosphere through the tropical tropopause, are transported throughout the stratosphere by the Brewer-Dobson circulation, and are photochemically destroyed in the upper stratosphere. These chemical constituents, or "tracers" can be used to track mixing and transport by the stratospheric winds. Much of our understanding about the stratospheric circulation is based on large scale gradients and other spatial features in tracer fields constructed from satellite measurements. The point of view presented in this paper is different, but complementary, in that transport is described in terms of tracer probability distribution functions (PDFs). The PDF is computed from the measurements, and is proportional to the area occupied by tracer values in a given range. The flavor of this paper is tutorial, and the ideas are illustrated with several examples of transport-related phenomena, annotated with remarks that summarize the main point or suggest new directions. One example shows how the multimodal shape of the PDF gives information about the different branches of the circulation. Another example shows how the statistics of fluctuations from the most probable tracer value give insight into mixing between different regions of the atmosphere. Also included is an analysis of the time-dependence of the PDF during the onset and decline of the winter circulation, and a study of how "bursts" in the circulation are reflected in transient periods of rapid evolution of the PDF. The dependence of the statistics on location and time are also shown to be important for practical problems related to statistical robustness and satellite sampling. The examples illustrate how physically-based statistical analysis can shed some light on aspects of stratospheric transport that may not be obvious or quantifiable with other types of analyses. An important motivation for the work presented here is the need for synthesis of the large and growing database of observations of the atmosphere and the vast quantities of output generated by atmospheric models.

Sparling, L. C.↗
NASA NTRS1998

EOS Interdisciplinary Investigation: Observational and Modeling Studies of Radiative, Chemical, and Dynamical Interactions in the Earth's Atmosphere

A 3-dimensional model of atmospheric dynamics and photochemistry has been developed from the primitive equations in isentropic coordinates. The model extends from the upper troposphere through the middle atmosphere and is driven by observed dynamical and chemical structure in the troposphere. The model's formulation is entirely spectral: Horizontal structure is represented in terms of vector Hough functions, which explicitly partition the motion into its rotational and divergent components. This formulation enables the model's computational performance to be increased dramatically by filtering high-frequency gravity waves, which do not affect PV conservation. Vertical structure is represented in terms of eigenfunctions that follow from the primitive equations in isentropic coordinates. The model's fully spectral formulation enables scale-selective dissipation, necessary for numerical stability, to be applied at 6th order-in all 3 coordinate directions. This feature leaves all but the shortest vertical scales undamped and, consequently, allows potential vorticity and chemical tracers to be conserved quite accurately, distinctly better than in the model's layered counterpart. These dynamical features are complemented by a basic but fairly complete treatment of gas phase photochemistry, which accounts for some 48 chemical species, diurnally-varying SW absorption by O2 and O3, and LW cooling calculated via a detailed band calculation of radiative transfer. Driven by observed tropospheric structure, the 3D model reproduces observed structure in the middle atmosphere, inclusive of transport by the Brewer-Dobson circulation and accompanying chemical variations. Calculated residual motion is consistent with diabatic cooling rates, poleward transport, and descent in the polar night derived from satellite measurements of chemical tracers made from LIMS, SAGE, and UARS. The model also reproduces the observed structure of chemical species like O3, HNO3, and ClO. At the same time, it provides a respectable 3D simulation of the wintertime increase of midlatitude total ozone, as well as its geographical structure observed by Nimbus-7 TOMS. The 3D model has been used to explore the origin of diabatic upwelling in the tropical lower stratosphere, which has been related to planetary wave absorption in the winter- time stratosphere, as well as to influences from tropospheric convection at the tropical tropopause.

Salby, Murry L.↗
NASA NTRS2001

Large-Scale Stratospheric Transport Processes

The paper discusses the following: 1. The Brewer-Dobson circulation: tropical upwelling. 2. Mixing into polar vortices. 3. The latitudinal structure of "age" in the stratosphere. 4. The subtropical "tracer edges". 5. Transport in the lower troposphere. 6. Tracer modeling during SOLVE. 7. 3D modeling of "mean age". 8. Models and measurements II.

Plumb, R. Alan↗
NASA NTRS2003

Overview on Role of the Troposphere

The troposphere is the lower boundary for the stratosphere. As such, it provides a radiative boundary that is a large source of IR flux into for the stratosphere and scattered UV-Vis energy. Planetary-scale waves that are forced in the troposphere can propagate into the stratosphere, providing a large additional source of energy for the stratosphere. This wave energy is the principal energy source for the Brewer-Dobson circulation. Medium-scale wave can also propagate into the lower stratosphere, but have a smaller effect. The final source of energy for the stratosphere comes from smaller scale disturbances such as gravity waves, mixed Rossby-gravity modes, and Kelvin waves. These smaller scale waves provide an important source of energy for the stratosphere. In this presentation we will cover these tropospheric sources of energy that effect the stratosphere. In particular, we will focus attention on planetary scale waves and their forcings.

Newman, Paul A.↗
NASA NTRS2008

Detection and Attribution of the Recovery of Polar Ozone

The Antarctic ozone hole develops each year and culminates by early spring (late September - early October). The severity of the hole has been assessed from satellites using the minimum total ozone value from the October monthly mean (depth of the hole), calculating the average area coverage during this September-October period, and by estimating ozone mass deficit. Profile information shows that ozone is completely destroyed in the 14-2 1 km layer by early October. Ozone is mainly destroyed by halogen (chlorine and bromine) catalytic cycles, and these losses are modulated by temperature variations. Because atmospheric halogen levels are responding to international agreements that limit or phase out production, the amount of halogens in the stratosphere should decrease over the next few decades. Both models and projections of ozone depleting substances (ODSs) into the 21St century reveal that polar ozone levels should recover in the 2060- 2070 period. In this talk, we will review current projections of polar ozone recovery. Using models and ODs projections, we explore both the past, near future (2008-2025), and far future (> 2025) levels of polar ozone. Finally, we will discuss various factors that complicate recovery such as greenhouse gas changes (e.g., cooling in the upper stratosphere) and the acceleration of the Brewer-Dobson circulation.

Newman, Paul A.↗
NASA NTRS2008

The Quasi-biennial Oscillation and Annual Variations in Tropical Ozone from SHADOZ and HALOE

We examine the tropical ozone mixing ratio perturbation fields generated from a monthly ozone climatology using 1998 to 2006 ozonesonde data from the Southern Hemisphere Additional Ozonesondes (SHADOZ) network and the 13-year satellite record from 1993 to 2005 obtained from the Halogen Occultation Experiment (HALOE). The long time series and high vertical resolution of the ozone and temperature profiles from the SHADOZ sondes coupled with good tropical coverage north and south of the equator gives a detailed picture of the ozone structure in the lowermost stratosphere down through the tropopause where the picture obtained from HALOE measurements is blurred by coarse vertical resolution. Ozone perturbations respond to annual variations in the Brewer-Dobson Circulation (BDC) in the region just above the cold-point tropopause to around 20 km. Annual cycles in ozone and temperature are well correlated. Above 20 km, ozone and temperature perturbations are dominated by the Quasi-biennial Oscillation (QBO). Both satellite and sonde records show good agreement between positive and negative ozone mixing ratio anomalies and alternating QBO westerly and easterly wind shears from the Singapore rawinsondes with a mean periodicity of 26 months for SHADOZ and 25 months for HALOE. There is a temporal offset of one to three months with the QBO wind shear ahead of the ozone anomaly field. The meridional length scales for the annual cycle and the QBO, obtained using the temperature anomalies and wind shears in the thermal wind equation, compare well with theoretical calculations.

Witte, J. C.↗