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The signature of the tropospheric gravity wave background in observed mesoscale motion

How convection couples to mesoscale vertical motion and what determines these motions is poorly understood. This study diagnoses profiles of area-averaged mesoscale divergence from measurements of horizontal winds collected by an extensive upper-air sounding network of a recent campaign over the western tropical North Atlantic, the Elucidating the Role of Clouds-Circulation Coupling in Climate (EUREC4A) campaign. Observed area-averaged divergence amplitudes scale approximately inversely with area-equivalent radius. This functional dependence is also confirmed in reanalysis data and a global, freely evolving simulation run at 2.5 km horizontal resolution. Based on the numerical data it is demonstrated that the energy spectra of inertia gravity waves can explain the scaling of divergence amplitudes with area. At individual times, however, few waves can dominate the region. Nearly monochromatic tropospheric waves are diagnosed in the soundings by means of an optimized hodograph analysis. For one day, results suggest that an individual wave directly modulated the satellite-observed cloud pattern. However, because such immediate wave impacts are rare, the systematic modulation of vertical motion due to inertia–gravity waves may be more relevant as a convection-modulating factor. The analytic relationship between energy spectra and divergence amplitudes proposed in this article, if confirmed by future studies, could be used to design better external forcing methods for regional models.
Full reference: Stephan, C.C. and Mariaccia, A., 2021. The signature of the tropospheric gravity wave background in observed mesoscale motion. Weather and Climate Dynamics, 2(2), pp.359-372. https://doi.org/10.5194/wcd-2-359-2021.

An assessment of scale-dependent variability and bias in global prediction models

The paper presents a method for the scale-dependent validation of the spatio-temporal variability in global weather or climate models and for their bias quantification in relation to dynamics. The method provides a relationship between the bias and simulated spatial and temporal variance by a model in comparison with verifying reanalysis data. For the low resolution (T30L8) subset of ERA-20C data, it was found that 80–90% (depending on season) of the global interannual variance is at planetary scales (zonal wavenumbers k = 0−3), and only about 1% of the variance is at scales with k>7. The reanalysis is used to validate a T30L8 GCM in two configurations, one with the prescribed sea-surface temperature (SST) and another using a slab ocean model. Although the model with the prescribed SST represents the average properties of surface fields well, the interannual variability is underestimated at all scales. Similar to variability, model bias is strongly scale dependent. Biases found in the experiment with the prescribed SST are largely increased in the experiment using a slab ocean, especially in k=0, in scales with missing variability and in seasons with poorly simulated energy distribution. The perfect model scenario (a comparison between the GCM coupled to a slab ocean vs. the same model with prescribed SSTs) shows that the representation of the ocean is not critical for synoptic to subsynoptic variability, but essential for capturing the planetary scales.
Full reference: Žagar, N., Kosovelj, K., Manzini, E., Horvat, M. and Castanheira, J., 2020. An assessment of scale-dependent variability and bias in global prediction models. Climate Dynamics, 54(1), pp.287-306. https://doi.org/10.1007/s00382-019-05001-x

The Predictability Limit of Midlatitude Weather

Comments on “What Is the predictability limit of midlatitude weather?”
Full reference: Žagar, N. and I. Szunyogh, 2020: Comments on “What Is the Predictability Limit of Midlatitude Weather?”. J. Atmos. Sci., 77(2). 781-785, https://doi.org/10.1175/JAS-D-19-0166.1

A reduced-order representation of the Madden–Julian oscillation based on reanalyzed normal mode coherences

The Madden–Julian oscillation (MJO) is presented as a series of interacting Rossby and inertial gravity waves of varying vertical scales and meridional extents. These components are isolated by decomposing reanalysis fields into a set of normal mode functions (NMF), which are orthogonal eigenvectors of the linearized primitive equations on a sphere. The NMFs that demonstrate spatial properties compatible with the MJO are inertial gravity waves of zonal wavenumber k = 1 and the lowest meridional index n = 0, and Rossby waves with (k, n) = (1, 1). For these horizontal scales, there are multiple small vertical-scale baroclinic modes that have temporal properties indicative of the MJO. On the basis of one such eastward-propagating inertial gravity wave (i.e., a Kelvin wave), composite averages of the Japanese 55-year Reanalysis demonstrate an eastward propagation of the velocity potential, and oscillation of outgoing longwave radiation and precipitation fields over the Maritime Continent, with an MJO-appropriate temporal period. A cross-spectral analysis indicates that only the MJO time scale is coherent between this Kelvin wave and the more energetic modes. Four mode clusters are identified: Kelvin waves of correct phase period and direction, Rossby waves of correct phase period, energetic Kelvin waves of larger vertical scales and meridional extents extending into the extratropics, and energetic Rossby waves of spatial scales similar to that of the energetic Kelvin waves. We propose that within this normal mode framework, nonlinear interactions between the aforementioned mode groups are required to produce an energetic MJO propagating eastward with an intraseasonal phase period. By virtue of the selected mode groups, this theory encompasses both multiscale and tropical–extratropical interactions.
Full reference: Kitsios, V., O’Kane, T.J. and Žagar, N., 2019. A reduced-order representation of the Madden–Julian oscillation based on reanalyzed normal mode coherences. Journal of the Atmospheric Sciences, 76(8), pp.2463-2480. https://doi.org/10.1175/JAS-D-18-0197.1.

Modal decomposition of the global response to tropical heating perturbations resembling MJO

The paper presents four ensembles of numerical experiments that compare the response to monopole and dipole heating perturbations resembling different phases of the Madden–Julian oscillation (MJO). The results quantify the Rossby and inertio-gravity (IG) wave response using the normal-mode function decomposition. The day 3 response is characterized by about 60% variance in the IG modes, with about 85% of it belonging to the Kelvin waves. On day 14, only 10% of the response variance is due to the Kelvin waves. Although the n = 1 Rossby mode is the main contributor to the Rossby variance at all time scales, the n > 1 Rossby modes contribute over 50% of the balanced response to the MJO heating. In the short range, dipole perturbations produce a response with the maximal variance in zonal wavenumbers k = 2–3 whereas in the medium range the response maximizes at k = 1 in all experiments. Furthermore, the medium-range response to the heating perturbation mimicking MJO phase 6 is found also over Europe.
Full reference: Kosovelj, K., F. Kucharski, F. Molteni and N. Žagar, 2019: Modal decomposition of the global response to tropical heating perturbations resembling MJO. J. Atmos. Sci., 76, 1457-1469, https://doi.org/10.1175/JAS-D-18-0203.1.

Systematic decomposition of the MJO and its Northern Hemispheric extratropical response into Rossby and inertio-gravity components

The Madden–Julian Oscillation (MJO) is the dominant form of intraseasonal variability in the Tropics. The MJO is a complex convectively coupled phenomenon, which is still poorly represented in the current generation of climate models, and our understanding of its essential dynamics and its influence on the midlatitude circulation is still incomplete. Here, we use a normal-mode decomposition method to decompose the MJO systematically into Kelvin, inertio-gravity (IG), and Rossby-wave components in the ERA-Interim reanalysis data for the period 1980–2015 to provide a climatology of the eight MJO phases for the Kelvin, IG, and Rossby-wave components.
Our analysis shows that the Rossby modes provide a larger contribution to the magnitude of the MJO in terms of geopotential height and winds than the Kelvin wave and IG modes. Moreover, the kinetic energy associated with the Rossby modes of the MJO accounts for about 93% of the kinetic energy. Our decomposition also shows that the Kelvin wave is the dominant mode in the unbalanced wave part, which is flanked by Rossby waves on both sides of the Equator, consistent with previous studies. The extratropical response to the MJO also consists of both IG and Rossby-wave components in the Northern Hemisphere (NH). The midlatitude MJO response is also linked to well-known teleconnection patterns like the North Atlantic Oscillation and the Pacific–North American pattern. The transient NH atmospheric response is fast, of the order of 5–7 days. While the extratropical response is dominated by Rossby waves, IG waves also show a prominent response in the NH.
Full reference: Franzke, C.L., Jelic, D., Lee, S. and Feldstein, S.B., 2019. Systematic decomposition of the MJO and its Northern Hemispheric extratropical response into Rossby and inertio‐gravity components. Quarterly Journal of the Royal Meteorological Society, 145(720), pp.1147-1164.

Estimating subseasonal variability and trends in global atmosphere using reanalysis data

A new measure of subseasonal variability is introduced that provides a scale-dependent estimation of vertically and meridionally integrated atmospheric variability in terms of the normal modes of linearized primitive equations. Applied to the ERA-Interim data, the new measure shows that subseasonal variability decreases for larger zonal wave numbers. Most of variability is due to balanced (Rossby mode) dynamics but the portion associated with the inertio-gravity (IG) modes increases as the scale reduces. Time series of globally integrated variability anomalies in ERA-Interim show an increase in variability after year 2000. In recent years the anomalies have been about 2% above the 1981–2010 average. The relative increase in variability projecting on the IG modes is larger and more persistent than for the Rossby modes. Although the IG part is a small component of the subseasonal variability, it is an important effect likely reflecting the observed increase in the tropical precipitation variability.
Full reference: Žagar, N., D. Jelić, M.J. Alexander and E. Manzini, 2018: Estimating subseasonal variability and trends in global atmosphere using reanalysis data. Geophys. Res. Lett., 45, 12999-13007, https://doi.org/10.1029/2018GL080051.

Multivariate analysis of Kelvin wave seasonal variability in ECMWF L91 analyses

This paper presents a multivariate analysis of the linear Kelvin waves (KWs) represented by the operational 91-level ECMWF analyses in the 2007–2013 period, with a focus on seasonal variability. The applied method simultaneously filters KW wind and temperature perturbations in the continuously stratified atmosphere on the spherical Earth. The spatial filtering of the three-dimensional KW structure in the upper troposphere and lower stratosphere is based on the Hough harmonics using several tens of linearized shallow-water equation systems on the spherical Earth with equivalent depths ranging from 10 km to a few metres.
Results provide the global KW energy spectrum. It shows a clear seasonal cycle with the KW activity predominantly in zonal wavenumbers 1–2, where up to 50 % more energy is observed during the solstice seasons in comparison with boreal spring and autumn.
Seasonal variability in KWs in the upper troposphere and lower stratosphere is examined in relation to the background wind and stability. A spectral bandpass filtering is used to decompose variability into three period ranges: seasonal, intra-seasonal and intra-monthly variability components. Results reveal a slow seasonal KW component with a robust dipole structure in the upper troposphere with its position determined by the location of the dominant convective outflow throughout the seasons. Its maximal strength occurs during boreal summer when easterlies in the eastern hemisphere are strongest. The two other components represent vertically propagating KWs and are observed throughout the year with seasonal variability mostly found in the wave amplitudes being dependent on the seasonality of the background easterly winds and static stability.
Full reference: Blaauw, M. and N. Žagar, 2018: Multivariate analysis of Kelvin wave seasonal variability in ECMWF L91 analyses. Atmos. Chem. Phys., 18, 8313-8330, https://doi.org/10.5194/acp-18-8313-2018

Gravity waves excited during a minor sudden stratospheric warming

An exceptionally deep upper-air sounding launched from Kiruna airport (67.82∘ N, 20.33∘ E) on 30 January 2016 stimulated the current investigation of internal gravity waves excited during a minor sudden stratospheric warming (SSW) in the Arctic winter 2015/16. The analysis of the radiosonde profile revealed large kinetic and potential energies in the upper stratosphere without any simultaneous enhancement of upper tropospheric and lower stratospheric values. Upward-propagating inertia-gravity waves in the upper stratosphere and downward-propagating modes in the lower stratosphere indicated a region of gravity wave generation in the stratosphere. Two-dimensional wavelet analysis was applied to vertical time series of temperature fluctuations in order to determine the vertical propagation direction of the stratospheric gravity waves in 1-hourly high-resolution meteorological analyses and short-term forecasts. The separation of upward- and downward-propagating waves provided further evidence for a stratospheric source of gravity waves. The scale-dependent decomposition of the flow into a balanced component and inertia-gravity waves showed that coherent wave packets preferentially occurred at the inner edge of the Arctic polar vortex where a sub-vortex formed during the minor SSW.
Full reference: Doernbrack, A., S. Gisinger, N. Keifler, T. Portele, M. Bamberger, M. Rapp, M. Gerding, J. Soder, N. Žagar and D. Jelic, 2018: Gravity waves excited during a minor sudden stratospheric warming. Atmos. Chem. Phys.,18, 12915-12931, https://doi.org/10.5194/acp-18-12915-2018

Normal modes of atmospheric variability in observations, numerical weather prediction and climate models

Full reference: Žagar, N., J. Boyd, A. Kasahara, J. Tribbia, E. Källén, H. Tanaka and J.-I. Yano, 2016: Normal modes of atmospheric variability in observations, numerical weather prediction and climate models. Bull. Amer. Meteor. Soc., https://doi.org/10.1175/BAMS-D-15-00325.1

Application of Normal-Mode Function Decomposition in Weather and Climate Research. in: Modal VIew of Atmospheric Variability

The book introduces the theory and computation of the normal-mode functions, reviews the applications of the normal modes in data assimilation and initialization of numerical weather prediction models, and for predictability research, and offers an up-to-date overview of research of atmospheric variability and energy transfers in terms of the Rossby and inertia-gravity waves across scales.
Full reference: Application of Normal-Mode Function Decomposition in Weather and Climate Research. in: Modal VIew of Atmospheric Variability, Eds.: N. Žagar, and J. Tribbia. Mathematics of Planet Earth (8), Springer (2020), ISBN 978-3-030-60963-4, https://doi.org/10.1007/978-3-030-60963-4

Atmospheric Subseasonal Variability and Circulation Regimes: Spectra, Trends, and Uncertainties

The globally integrated subseasonal variability associated with the two main atmospheric circulation regimes, the balanced (or Rossby) and unbalanced (or inertia–gravity) regimes, is evaluated for the four reanalysis datasets: ERA-Interim, JRA-55, MERRA, and ERA5. The results quantify amplitudes and trends in midlatitude traveling and quasi-stationary Rossby wave patterns as well as in the equatorial wave activity across scales. A statistically significant reduction of subseasonal variability is found in Rossby waves with zonal wavenumber k = 6 along with an increase in variability in wavenumbers k = 3–5 in the summer seasons of both hemispheres. The four reanalyses also agree regarding increased variability in the large-scale Kelvin waves, mixed Rossby–gravity waves, and westward-propagating inertio-gravity waves with the lowest meridional mode. The amplitude and sign of trends in inertia–gravity modes with smaller zonal scales and greater meridional modes differ between the ERA-Interim and JRA-55 datasets on the one hand and the ERA5 and MERRA data on the other. An increased variability in the ERA-Interim and JRA-55 accounts for positive trends in their total subseasonal variability.
Full reference: Žagar, N., Ž. Zaplotnik, and K. Karami, 2020: Atmospheric Subseasonal Variability and Circulation Regimes: Spectra, Trends, and Uncertainties. J. Climate, 33 (21), 9375–9390, https://doi.org/10.1175/JCLI-D-20-0225.1

Energy spectra in inertio-gravity waves

MODES has provided a novel approach for the spatial filtering of the inertia-gravity (IG) waves in global analysis data.
The method is applied to the ECMWF interim reanalysis and the operational 2014–16 analysis fields. The derived spectrum of IG wave energy is divided into three regimes: a part associated with the large-scale unbalanced circulations that has a slope close to −1 for zonal wavenumbers up to k=6, a synoptic-scale range between 3000 and around 500 km that is characterized by a nearly −5/3 slope, and a mesoscale range below 500 km where the slope of the IG energy spectrum in the 2015/16 analyses is steeper.
In contrast, the energy spectrum of the Rossby waves has a −3 slope for all zonal wavenumbers greater than k=6.
Presented results suggest that energy associated with the IG modes exceeds the level of energy associated with the Rossby waves around zonal wavenumber 35. The exact wavenumber depends on the season and considered atmospheric depth and it is suggested as a cutoff scale for studies of gravity waves.
Full reference: Žagar, N., Jelić, D., Blaauw, M. and Bechtold, P., 2017. Energy spectra and inertia–gravity waves in global analyses. Journal of the Atmospheric Sciences, 74(8), pp.2447-2466. https://doi.org/10.1175/JAS-D-16-0341.1.

A global view of the limits of prediction skill of NWP models

The scale-dependent growth of the global forecast uncertainties simulated by the operational ensemble prediction system of the European Centre for Medium-Range Weather Forecasts. It is shown that the initial uncertainties are largest in the tropics and have biggest amplitudes at the large scales. The growth of forecast uncertainties (ensemble spread) takes place at all scales from the beginning of forecasts. The growth is nearly uniform in the zonal wavenumbers 1–5 and strongly scale-dependent in the larger wavenumbers. Moreover, the growth from initial uncertainties at large scales appears dominant over the impact of errors cascading up from small scales.
The growth of uncertainties is found to be faster in the balanced than in the unbalanced modes and after 0.5–1 day of forecasts the balanced errors become dominant except at the subsynoptic scales.
Full reference: Žagar, N., 2017. A global perspective of the limits of prediction skill of NWP models. Tellus A: Dynamic Meteorology and Oceanography, 69(1), p.1317573. https://doi.org/10.1080/16000870.2017.1317573.

Information Content of Observations in the Global EnKF Data Assimilation

Modal approach has been developed to estimate the efficiency of data assimilation to reduce the prior uncertainties in the global data assimilation systems.
The approach has been applied to the ensemble Kalman filter data assimilation system DART/CAM. Observing ssytem simulation experiments employed the perfect-model framework and a globally homogeneous network of wind and temperature profiles.
The scale-dependent representation of variance reduction of the prior ensemble by the data assimilation shows that the peak efficiency of data assimilation is on the synoptic scales in the midlatitudes that are associated with quasigeostrophic dynamics. In contrast, the variance associated with the inertia–gravity modes is less successfully reduced on all scales. A smaller information content of observations on planetary scales with respect to the synoptic scales is discussed in relation to the large-scale tropical uncertainties that current data assimilation methodologies do not address successfully.
It is shown that a smaller reduction of the large-scale uncertainties in the prior state for NWP in the tropics than in the midlatitudes is associated with the applied radius for the covariance localization.
Full reference: Žagar, N., Anderson, J., Collins, N., Hoar, T., Raeder, K., Lei, L. and Tribbia, J., 2016. Scale-dependent representation of the information content of observations in the global ensemble Kalman filter data assimilation. Monthly Weather Review, 144(8), pp.2927-2945. https://doi.org/10.1175/MWR-D-15-0401.1

Scale-dependent estimates of the growth of global forecast uncertainties

MODES has been applied for a scale-dependent estimates of the growth of forecast uncertainties in a global prediction system.
A new parametric model for the representation of the forecast error growth is formulated and applied independently to every zonal wavenumber. In contrast to the standard fitting method, the new fitting function involves no time derivatives and provides the asymptotic values of the forecast errors as a function of the fitting parameters.
The new model is easily transformed to the widely used model of Dalcher and Kalnay (1987) to discuss the scale-dependent growth as a sum of two terms, the so-called alpha and beta terms. Their comparison shows that at planetary scales their contributions to the growth in the first two days are similar whereas at small scales the term describes most of a rapid exponential growth of errors towards saturation. Full reference: Žagar, N., Horvat, M., Zaplotnik, Ž. and Magnusson, L., 2017. Scale-dependent estimates of the growth of forecast uncertainties in a global prediction system. Tellus A: Dynamic Meteorology and Oceanography, 69(1), p.1287492. https://doi.org/10.1080/16000870.2017.1287492.

Short-term forecast errors and balance issues

Some properties of the short-term forecast errors are modelled in the background-error covariance matrix for data assimilation. The three-dimensional decomposition of forecast errors derived from the ensemble of analyses and forecasts of the ECMWF system suggests that a significant part of the short-term forecast errors (as modelled by the ensemble spread) is associated with the inertio-gravity mods.
Full reference: Žagar, N. , L. Isaksen, D. Tan and J. Tribbia, 2013: Balance properties of the short-range forecast errors in the ECMWF 4D-Var ensemble. Q. J. R. Meteorol. Soc., 139, 1229-1238. DOI: 10.1002/qj.2033. https://doi.org/10.1002/qj.2033

Modal view of the global predictability with application to ECMWF ensemble prediction system

A new methodology for the analysis of atmospheric predictability has been developed and applied to the ECMWF operational ensemble prediction system. It is based on the representation of atmospheric dynamical variables in terms of normal mode functions and the computation of the ensemble spread in modal space.
Full reference: N. Žagar, R. Buizza and J. Tribbia, 2015: A Three-Dimensional Multivariate Modal Analysis of Atmospheric Predictability with Application to the ECMWF Ensemble. J. Atmos. Sci., 72, 4423–4444. https://doi.org/10.1175/JAS-D-15-0061.1

Description of the MODES software

Paper on the theory and technical development of the MODES software has been published in Geoscientific Model Development
Full reference: N. Žagar, A. Kasahara, K. Terasaki, J. Tribbia and H. Tanaka, 2015: Normal-mode function representation of global 3-D data sets: open-access software for the atmospheric research community. https://doi.org/10.5194/gmd-8-1169-2015.
MODES is compiled by using gfortran although other options have been succesfully tested. The application requires the use of the netcdf and (optionally) grib-api libraries.

Representation of MJO in terms of 3D normal-mode functions

MODES has been applied for a three-dimensional multivariate decomposition of the Madden-Julian Oscillation into balanced and inertio-gravity components. The decomposition provides a quantitative comparison between the roles of the Kelvin mode and the equatorial Rossby mode as well as the MJO teleconnections.
Full reference: N. Žagar and C. Franzke, 2015: Systematic decomposition of the Madden-Julian Oscillation into balanced and inertio-gravity components, Geophys. Res. Lett., 42, 6829–6835. https://doi.org/10.1002/2015GL065130.