Many scientists have made use of the wavelet method in analyzing time series, often using popular free software. However, at present there are no similar easy to use wavelet packages for analyzing two time series together. We discuss the cross wavelet transform and wavelet coher- ence for examining relationships in time frequency space be- tween two time series. We demonstrate how phase angle statistics can be used to gain confidence in causal relation- ships and test mechanistic models of physical relationships between the time series. As an example of typical data where such analyses have proven useful, we apply the methods to the Arctic Oscillation index and the Baltic maximum sea ice extent record. Monte Carlo methods are used to assess the statistical significance against red noise backgrounds. A software package has been developed that allows users to perform the cross wavelet transform and wavelet coherence (http://www.pol.ac.uk/home/research/waveletcoherence/). As we are interested in extracting low s/n ratio signals in time series we discuss only CWT in this paper. While CWT is a common tool for analyzing localized intermittent os- cillations in a time series, it is very often desirable to ex- amine two time series together that may be expected to be linked in some way. In particular, to examine whether re- gions in time frequency space with large common power have a consistent phase relationship and therefore are sug- gestive of causality between the time series. Many geophys- ical time series are not Normally distributed and we suggest methods of applying the CWT to such time series. From two CWTs we construct the Cross Wavelet Transform (XWT) which will expose their common power and relative phase in time-frequency space. We will further define a measure of Wavelet Coherence (WTC) between two CWT, which can find significant coherence even though the common power is low, and show how confidence levels against red noise back- grounds are calculated. We will present the basic CWT theory before we move on to XWT and WTC. New developments such as quanti- fying the phase relationship and calculating the WTC sig- nificance level will be treated more fully. When using the methods on time series it is important to have solid mecha- nistic foundations on which to base any relationships found, and we caution against using the methods in a "scatter-gun" approach (particularly if the time series probability density functions are modified). To illustrate how the various meth- ods are used we apply them to two data sets from meteo- rology and glaciology. Finally, we will provide links to a MatLab software package.

/pdf/application-of-the-cross-wavelet-transform-and-wavelet-3aorulqrv4.pdf

Application of the cross wavelet transform and wavelet coherence to geophysical time series

The leading modes of variability of the extratropical circulation in both hemispheres are characterized by deep, zonally symmetric or ‘‘annular’’ structures, with geopotential height perturbations of opposing signs in the polar cap region and in the surrounding zonal ring centered near 458 latitude. The structure and dynamics of the Southern Hemisphere (SH) annular mode have been extensively documented, whereas the existence of a Northern Hemisphere (NH) mode, herein referred to as the Arctic Oscillation (AO), has only recently been recognized. Like the SH mode, the AO can be defined as the leading empirical orthogonal function of the sea level pressure field or of the zonally symmetric geopotential height or zonal wind fields. In this paper the structure and seasonality of the NH and SH modes are compared based on data from the National Centers for Environmental Prediction‐National Center for Atmospheric Research reanalysis and supplementary datasets. The structures of the NH and SH annular modes are shown to be remarkably similar, not only in the zonally averaged geopotential height and zonal wind fields, but in the mean meridional circulations as well. Both exist year-round in the troposphere, but they amplify with height upward into the stratosphere during those seasons in which the strength of the zonal flow is conducive to strong planetary wave‐mean flow interaction: midwinter in the NH and late spring in the SH. During these ‘‘active seasons,’’ the annular modes modulate the strength of the Lagrangian mean circulation in the lower stratosphere, total column ozone and tropopause height over mid- and high latitudes, and the strength of the trade winds of their respective hemispheres. The NH mode also contains an embedded planetary wave signature with expressions in surface air temperature, precipitation, total column ozone, and tropopause height. It is argued that the horizontal temperature advection by the perturbed zonal-mean zonal wind field in the lower troposphere is instrumental in forcing this pattern. A companion paper documents the striking resemblance between the structure of the annular modes and observed climate trends over the past few decades.

https://journals.ametsoc.org/downloadpdf/journals/clim/13/5/1520-0442_2000_013_1000_amitec_2.0.co_2.pdf

Annular Modes in the Extratropical Circulation. Part I: Month-to-Month Variability*

[1] Atmospheric gravity waves have been a subject of intense research activity in recent years because of their myriad effects and their major contributions to atmospheric circulation, structure, and variability. Apart from occasionally strong lower-atmospheric effects, the major wave influences occur in the middle atmosphere, between ∼ 10 and 110 km altitudes because of decreasing density and increasing wave amplitudes with altitude. Theoretical, numerical, and observational studies have advanced our understanding of gravity waves on many fronts since the review by Fritts [1984a]; the present review will focus on these more recent contributions. Progress includes a better appreciation of gravity wave sources and characteristics, the evolution of the gravity wave spectrum with altitude and with variations of wind and stability, the character and implications of observed climatologies, and the wave interaction and instability processes that constrain wave amplitudes and spectral shape. Recent studies have also expanded dramatically our understanding of gravity wave influences on the large-scale circulation and the thermal and constituent structures of the middle atmosphere. These advances have led to a number of parameterizations of gravity wave effects which are enabling ever more realistic descriptions of gravity wave forcing in large-scale models. There remain, nevertheless, a number of areas in which further progress is needed in refining our understanding of and our ability to describe and predict gravity wave influences in the middle atmosphere. Our view of these unknowns and needs is also offered.

/pdf/gravity-wave-dynamics-and-effects-in-the-middle-atmosphere-4huexaq7y8.pdf

Gravity wave dynamics and effects in the middle atmosphere

[i] An inventory of air pollutant emissions in Asia in the year 2000 is developed to support atmospheric modeling and analysis of observations taken during the TRACE-P experiment funded by the National Aeronautics and Space Administration (NASA) and the ACE-Asia experiment funded by the National Science Foundation (NSF) and the National Oceanic and Atmospheric Administration (NOAA). Emissions are estimated for all major anthropogenic sources, including biomass burning, in 64 regions of Asia. We estimate total Asian emissions as follows: 34.3 Tg SO 2 , 26.8 Tg NO x , 9870 Tg CO 2 , 279 Tg CO, 107 Tg CH 4 , 52.2 Tg NMVOC, 2.54 Tg black carbon (BC), 10.4 Tg organic carbon (OC), and 27.5 Tg NH 3 . In addition, NMVOC are speciated into 19 subcategories according to functional groups and reactivity. Thus we are able to identify the major source regions and types for many of the significant gaseous and particle emissions that influence pollutant concentrations in the vicinity of the TRACE-P and ACE-Asia field measurements. Emissions in China dominate the signature of pollutant concentrations in this region, so special emphasis has been placed on the development of emission estimates for China. China's emissions are determined to be as follows: 20.4 Tg SO 2 , 11.4 Tg NO x , 3820 Tg CO 2 , 116 Tg CO, 38.4 Tg CH 4 , 17.4 Tg NMVOC, 1.05 Tg BC, 3.4 Tg OC, and 13.6 Tg NH 3 . Emissions are gridded at a variety of spatial resolutions from 1° × 1° to 30 s x 30 s, using the exact locations of large point sources and surrogate GIS distributions of urban and rural population, road networks, landcover, ship lanes, etc. The gridded emission estimates have been used as inputs to atmospheric simulation models and have proven to be generally robust in comparison with field observations, though there is reason to think that emissions of CO and possibly BC may be underestimated. Monthly emission estimates for China are developed for each species to aid TRACE-P and ACE-Asia data interpretation. During the observation period of March/ April, emissions are roughly at their average values (one twelfth of annual). Uncertainties in the emission estimates, measured as 95% confidence intervals, range from a low of ±16% for SO 2 to a high of ±450% for OC.

An inventory of gaseous and primary aerosol emissions in Asia in the year 2000 : NASA global tropospheric experiment transport and chemical evolution over the pacific (TRACE-P): Measurements and analysis (TRACEP1)

The quasi-biennial oscillation (QBO) dominates the variability of the equatorial stratosphere (∼16–50 km) and is easily seen as downward propagating easterly and westerly wind regimes, with a variable period averaging approximately 28 months. From a fluid dynamical perspective, the QBO is a fascinating example of a coherent, oscillating mean flow that is driven by propagating waves with periods unrelated to that of the resulting oscillation. Although the QBO is a tropical phenomenon, it affects the stratospheric flow from pole to pole by modulating the effects of extratropical waves. Indeed, study of the QBO is inseparable from the study of atmospheric wave motions that drive it and are modulated by it. The QBO affects variability in the mesosphere near 85 km by selectively filtering waves that propagate upward through the equatorial stratosphere, and may also affect the strength of Atlantic hurricanes. The effects of the QBO are not confined to atmospheric dynamics. Chemical constituents, such as ozone, water vapor, and methane, are affected by circulation changes induced by the QBO. There are also substantial QBO signals in many of the shorter-lived chemical constituents. Through modulation of extratropical wave propagation, the QBO has an effect on the breakdown of the wintertime stratospheric polar vortices and the severity of high-latitude ozone depletion. The polar vortex in the stratosphere affects surface weather patterns, providing a mechanism for the QBO to have an effect at the Earth's surface. As more data sources (e.g., wind and temperature measurements from both ground-based systems and satellites) become available, the effects of the QBO can be more precisely assessed. This review covers the current state of knowledge of the tropical QBO, its extratropical dynamical effects, chemical constituent transport, and effects of the QBO in the troposphere (∼0–16 km) and mesosphere (∼50–100 km). It is intended to provide a broad overview of the QBO and its effects to researchers outside the field, as well as a source of information and references for specialists. The history of research on the QBO is discussed only briefly, and the reader is referred to several historical review papers. The basic theory of the QBO is summarized, and tutorial references are provided.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999RG000073

The quasi-biennial oscillation

An analysis is made of equatorial wave disturbances in the upper stratosphere and mesosphere by the use of meteorological rocket and satellite observations, to clarify their structure and behavior in relation to the semiannual oscillation of the mean zonal wind. From a power spectral analysis of wind and temperature over Ascension Island during the four years from 1969 to 1972, it is found that in the height region between 25 and 60 km there exists a wave disturbance with a characteristic vertical scale of 15–20 km; the wave is more active in the easterlies than in the westerlies, showing marked semiannual variation. Compared with the well-known characteristics of equatorial waves in the lower stratosphere, this wave is likely to he identified as a Kelvin wave with a period of about 10 days. It is suggested that this wave plays an essential role in producing the semiannual reversal of the mean zonal wind by supplying the westerly momentum to the equatorial mesospheric levels. Further discussions ...

Equatorial Waves in the Upper Stratosphere and Mesosphere in Relation to the Semiannual Oscillation of the Zonal Wind

Fluctuations in vertical profiles of atmospheric temperature and horizontal wind in the 20-60 km altitude range have been isolated from meteorological rocket measurements during 1977-87 at 15 widely separated sites. The seasonal, geographical, and vertical variability of the variance of horizontal velocities, u' 2 + υ' 2 , and relative-temperature perturbations, T' 2 *, were studied. The bulk of the variance of both quantities in the 2- 10 km and 2-20 km vertical-wavelength bands was associated with gravity-wave motions, although in-depth study of the wave polarization shows that planetary-scale equatorial wave modes contribute to the variance at equatorial sites. Annual mean variances varied widely among the 15 stations, suggesting appreciable geographical variability in stratospheric wave activity

Gravity wave and equatorial wave morphology of the stratosphere derived from long‐term rocket soundings

Climatology of gravity waves in the middle atmosphere

Observational studies on the semiannual oscillation in the tropical stratosphere and mesosphere are reviewed. Results of many statistics based on rocket and satellite observations reveal that the long-term behavior of the mean zonal wind exhibits two semiannual cycles which have their maximum amplitudes centered at the stratopause level and the mesopause level, each one being associated with the semiannual temperature variations predominating at levels about 10 km lower.

Isamu Hirota

Papers

The quasi-biennial oscillation

Equatorial Waves in the Upper Stratosphere and Mesosphere in Relation to the Semiannual Oscillation of the Zonal Wind

Gravity wave and equatorial wave morphology of the stratosphere derived from long‐term rocket soundings

Climatology of gravity waves in the middle atmosphere

Observational evidence of the semiannual oscillation in the tropical middle atmosphere—A review