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A Quantitative Method of Detecting Transient Rossby Wave Phase Speed: No Evidence of Slowing Down with Global Warming


doi: 10.1007/s00376-022-2164-5

  • Based on the Complex Empirical Orthogonal Functions (CEOFs) of bandpass-filtered daily streamfunction fields, a quantitative method of detecting transient (synoptic) Rossby wave phase speed (RWPhS) is presented. The transient RWPhS can be objectively calculated by the distance between a high (or low) center in the real part of a CEOF mode and its counterpart in the imaginary part of the same CEOF mode divided by the time span between two adjacent peaks (or bottoms) of two principal component curves for the real and imaginary parts of that CEOF mode. The new detection method may partly reveal the spatiotemporal heterogeneity of Rossby wave prorogation. Although the mean westerly jet at 200 hPa doubles the speed of its counterpart at 500 hPa, the estimated RWPhS at both levels are around 1000 km d–1 and quantitatively consistent with the quasigeostrophic-theory-based RWPhS, confirming that the meridional potential vorticity gradient induced by the barotropic and baroclinic shears of mean flow, together with the β effect, play an essential role in Rossby wave propagation. Both observations over the past four decades and a 150-year historical simulation suggest no evidence for slowing wintertime transient Rossby waves in the Northern Hemisphere, but possible regional changes are not excluded. We emphasize that not only the mean flow speed, but also the barotropic and baroclinic shears of the mean flow, and their associated contributions to the meridional potential vorticity (PV) gradient, should be considered in investigating the possible change of Rossby waves with global warming.
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  • Figure 1.  The first mode of complex empirical orthogonal function (CEOF1) and corresponding principal component (PC1) of 2–9-day Lanczos-filtered streamfunction at 200 hPa (a–c) and 500 hPa (d–f) during December, January, and February (DJF) of 2008/09: (a) Real part and (b) imaginary part of CEOF1 (colored contours, m2 s–1), and mean zonal wind speed at 200 hPa in 2008/09 DJF (black contours, m s–1); (c) Real part (black line) and imaginary part (blue dashed line) of PC1, dots mark the data with |PC|≥0.5); (d)–(f): same as (a)–(c), but for 500 hPa. The criterion for the dotted centers in (a) and (b) is the absolute value being larger than 1.5×106 m2 s–1.

    Figure 2.  (a) 200-hPa streamfunction anomaly (original streamfunction on 9 January 2009 minus 2008/09 DJF-averaged streamfunction), (b) 2–9-day Lanczos-filtered streamfunction at 200 hPa on 9 January 2009, (c) reconstructed streamfunction on 9 January 2009 from CEOF1 (i.e., real part of 2008/09 DJF 200-hPa CEOF1 multiplied by real part of PC1 on 9 January 2009 plus imaginary part of CEOF1 multiplied by imaginary part of PC1 on 9 January 2009); (d) reconstructed streamfunction on 9 January 2009 from the first four CEOFs; (f)–(i) are the same as (a)–(d), but for 11 January 2009. Black contours (m s–1) are mean zonal wind speed of 2008/09 DJF at 200 hPa. (e) and (j) are the real and imaginary parts of daily streamfunction at grid point C in Fig. 1 during 2008/09 winter. Black curves are from the original 2–9-day Lanczos-filtered time series [in (e)] and its Hilbert transform [in (j)], while the red curves are reconstructed from the truncated sum of the first 1–4 CEOF modes. Units: 107 m2 s–1. The two days are utilized to illustrate the propagation of Rossby waves during about one-fourth of a wave period.

    Figure A1.  The intraseasonal variability of RWPhS over north Pacific (a) and north Atlantic (b) regions during DJF of 2008/09

    Figure A2.  Box-and-whisker diagrams of zonal Rossby wave phase speed (RWPhS) at 200 hPa between 20°N and 60°N based on (a) annual and (b) decadal statistics separately obtained from the second CEOF modes for the winter of 1979/80 to 2018/2019. (c) and (d) : The same as (a) and (b), but for the third CEOF mode; (e) and (f): The same as (a) and (b), but for the fourth CEOF mode.

    Figure 3.  Box-and-whisker diagrams of zonal Rossby wave phase speed (RWPhS) at 200 hPa between 20°N and 60°N based on (a) annual and (b) decadal statistics from the CEOF1 for the winters of 1979/80 to 2018/19. The short red line in (b) marks the median value. Medians in (a) are connected by a black curve, and the thick red curve is the 5-year Gaussian smoothing of medians. The bottom and top blue lines of the boxes are the 25th- and 75th- percentile values, respectively, while the whiskers extend to the minimum and maximum of the data, except for outliers, which fall outside of 99.3%; (c) and (d) are the same as (a) and (b) but for the zonal RWPhS at 500 hPa.

    Figure 4.  The same as Figs. 3a and 3b, but for RWPhS at 250 hPa during boreal winters from 1850/51 to 1999/2000 in the CESM2 historical simulation.

    Figure 5.  The zonal wind speed (m s−1) at 200 hPa (a) and 500 hPa (b) and the mean $\partial \overline{{\rm{PV}}}/\partial y$ (×10−11 m−1 s−1) at 200 hPa (c) and 500 hPa (d) during boreal winter (DJF) averaged over 1979/80–2018/19.

  • Barnes, E. A., 2013: Revisiting the evidence linking Arctic amplification to extreme weather in midlatitudes. Geophys. Res. Lett., 40, 4734−4739, https://doi.org/10.1002/grl.50880.
    Barnett, T. P., 1983: Interaction of the monsoon and Pacific trade wind system at interannual time scales part I: The equatorial zone. Mon. Wea. Rev., 111, 756−773, https://doi.org/10.1175/1520-0493(1983)111<0756:IOTMAP>2.0.CO;2.
    Blackport, R., and J. A. Screen, 2020: Insignificant effect of Arctic amplification on the amplitude of midlatitude atmospheric waves. Science Advances, 6, eaay2880, https://doi.org/10.1126/sciadv.aay2880.
    Chen, X. D., D. H. Luo, Y. T. Wu, E. Dunn-Sigouin, and J. Lu, 2021: Nonlinear response of atmospheric blocking to early winter Barents–Kara seas warming: An idealized model study. J. Climate, 34, 2367−2383, https://doi.org/10.1175/JCLI-D-19-0720.1.
    Danabasoglu, G., 2019: NCAR CESM2-FV2 Model Output Prepared for CMIP6 CMIP Historical. Version 20191120. Earth System Grid Federation. Available from
    Dee, D. P., and Coauthors, 2011: The ERA-interim reanalysis: Configuration and performance of the data assimilation system. Quart. J. Roy. Meteor. Soc., 137, 553−597, https://doi.org/10.1002/qj.828.
    Duchon, C. E., 1979: Lanczos filtering in one and two dimensions. J. Appl. Meteorol. Climatol., 18, 1016−1022, https://doi.org/10.1175/1520-0450(1979)018<1016:LFIOAT>2.0.CO;2.
    Feldstein, S. B., and I. M. Held, 1989: Barotropic decay of baroclinic waves in a two-layer beta-plane model. J. Atmos. Sci., 46, 3416−3430, https://doi.org/10.1175/1520-0469(1989)046<3416:BDOBWI>2.0.CO;2.
    Fragkoulidis, G., and V. Wirth, 2020: Local Rossby wave packet amplitude, phase speed, and group velocity: Seasonal variability and their role in temperature extremes. J. Climate, 33, 8767−8787, https://doi.org/10.1175/JCLI-D-19-0377.1.
    Francis, J. A., and S. J. Vavrus, 2012: Evidence linking Arctic amplification to extreme weather in mid-latitudes. Geophys. Res. Lett., 39, L06801, https://doi.org/10.1029/2012GL051000.
    Hayashi, Y., 1979: A generalized method of resolving transient disturbances into standing and traveling waves by space-time spectral analysis. J. Atmos. Sci., 36, 1017−1029, https://doi.org/10.1175/1520-0469(1979)036<1017:AGMORT>2.0.CO;2.
    Held, I. M., 1999: The macroturbulence of the troposphere. Tellus B: Chemical and Physical Meteorology, 51, 59−70, https://doi.org/10.3402/tellusb.v51i1.16260.
    Hoskins, B., and T. Woollings, 2015: Persistent extratropical regimes and climate extremes. Current Climate Change Reports, 1, 115−124, https://doi.org/10.1007/s40641-015-0020-8.
    Hoskins, B. J., I. N. James, and G. H. White, 1983: The shape, propagation and mean-flow interaction of large-scale weather systems. J. Atmos. Sci., 40, 1595−1612, https://doi.org/10.1175/1520-0469(1983)040<1595:TSPAMF>2.0.CO;2.
    Lu, J. H., and M. Cai, 2010: Quantifying contributions to polar warming amplification in an idealized coupled general circulation model. Climate Dyn., 34, 669−687, https://doi.org/10.1007/s00382-009-0673-x.
    Luo, D. H., and W. Q. Zhang, 2020: A nonlinear multiscale theory of atmospheric blocking: Dynamical and thermodynamic effects of meridional potential vorticity gradient. J. Atmos. Sci., 77, 2471−2500, https://doi.org/10.1175/JAS-D-20-0004.1.
    Luo, D. H., X. D. Chen, J. Overland, I. Simmonds, Y. T. Wu, and P. F. Zhang, 2019: Weakened potential vorticity barrier linked to recent winter Arctic sea ice loss and midlatitude cold extremes. J. Climate, 32, 4235−4261, https://doi.org/10.1175/JCLI-D-18-0449.1.
    McIntyre, M. E., 2015: DYNAMICAL METEOROLOGY | Potential vorticity. Encyclopedia of Atmospheric Sciences, 2nd ed., G. R. North et al., Eds., Academic Press, 375−383,
    Orlanski, I., and B. Gross, 2000: The life cycle of baroclinic eddies in a storm track environment. J. Atmos. Sci., 57, 3498−3513, https://doi.org/10.1175/1520-0469(2000)057<3498:TLCOBE>2.0.CO;2.
    Phillips, N. A., 1990: Dispersion processes in large-scale weather prediction. WMO-No.700, 126 pp.
    Pratt, R. W., and J. M. Wallace, 1976: Zonal propagation characteristics of large-scale fluctuations in the mid-latitude troposphere. J. Atmos. Sci., 33, 1184−1194, https://doi.org/10.1175/1520-0469(1976)033<1184:ZPCOLF>2.0.CO;2.
    Randel, W. J., and I. M. Held, 1991: Phase speed spectra of transient eddy fluxes and critical layer absorption. J. Atmos. Sci., 48, 688−697, https://doi.org/10.1175/1520-0469(1991)048<0688:PSSOTE>2.0.CO;2.
    Riboldi, J., F. Lott, F. D'Andrea, and G. Rivière, 2020: On the linkage between Rossby wave phase speed, atmospheric blocking, and Arctic amplification. Geophys. Res. Lett., 47, e2020GL087796, https://doi.org/10.1029/2020GL087796.
    Smith, B. T., J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler, 1974: Matrix Eigensystem Routines - EISPACK Guide. Springer, 389 pp,
    Staten, P. W., J. Lu, K. M. Grise, S. M. Davis, and T. Birner, 2018: Re-examining tropical expansion. Nature Climate Change, 8, 768−775, https://doi.org/10.1038/s41558-018-0246-2.
    Sun, W. B., and Coauthors, 2021: The assessment of global surface temperature change from 1850s: The C-LSAT2.0 ensemble and the CMST-Interim datasets. Adv. Atmos. Sci., 38, 875−888, https://doi.org/10.1007/s00376-021-1012-3.
    Thorncroft, C. D., B. J. Hoskins, and M. E. McIntyre, 1993: Two paradigms of baroclinic-wave life-cycle behaviour. Quart. J. Roy. Meteor. Soc., 119, 17−55, https://doi.org/10.1002/qj.49711950903.
    von Storch, H., and F. W. Zwiers, 2003: Statistical Analysis in Climate Research. Cambridge University Press, 484 pp.
    White, R. H., K. Kornhuber, O. Martius, and V. Wirth, 2022: From atmospheric waves to heatwaves: A waveguide perspective for understanding and predicting concurrent, persistent, and extreme extratropical weather. Bull. Amer. Meteor. Soc., 103, E923−E935, https://doi.org/10.1175/BAMS-D-21-0170.1.
    Wirth, V., M. Riemer, E. K. M. Chang, and O. Martius, 2018: Rossby wave packets on the midlatitude waveguide—a review. Mon. Wea. Rev., 146, 1965−2001, https://doi.org/10.1175/MWR-D-16-0483.1.
    Yao, Y., D. H. Luo, A. G. Dai, and I. Simmonds, 2017: Increased quasi stationarity and persistence of winter Ural blocking and Eurasian extreme cold events in response to arctic warming. Part I: Insights from observational analyses. J. Climate, 30, 3549−3568, https://doi.org/10.1175/JCLI-D-16-0261.1.
    Yeh, T.-C, 1949: On energy dispersion in the atmosphere. J. Atmos. Sci., 6, 1−16, https://doi.org/10.1175/1520-0469(1949)006<0001:OEDITA>2.0.CO;2.
    Yeh, T.-C., and P.-C. Chu, 1958: Some Fundamental Problems of the General Circulation of the Atmosphere. Science Press, 159 pp. (in Chinese)
    Yeh, T.-C., and Y.-S. Chen, 1963: The vertical structure and its relation to the speed of movement and development of the long waves. Acta Meteorologica Sinica, 21, 25−36, https://doi.org/10.11676/qxxb1963.003. (in Chinese with English abstract
    Zeng, Q. C, 1983: The evolution of a Rossby-wave packet in a three-dimensional baroclinic atmosphere. J. Atmos. Sci., 40, 73−84, https://doi.org/10.1175/1520-0469(1983)040<0073:TEOARW>2.0.CO;2.
  • [1] Yaokun LI, Jiping CHAO, Yanyan KANG, 2022: Variations in Amplitudes and Wave Energy along the Energy Dispersion Paths for Rossby Waves in the Quasigeostrophic Barotropic Model, ADVANCES IN ATMOSPHERIC SCIENCES, 39, 876-888.  doi: 10.1007/s00376-021-1244-2
    [2] Brian HOSKINS, 2015: Potential Vorticity and the PV Perspective, ADVANCES IN ATMOSPHERIC SCIENCES, 32, 2-9.  doi: 10.1007/s00376-014-0007-8
    [3] Luo Dehai, 1999: Nonlinear Three-Wave Interaction among Barotropic Rossby Waves in a Large-scale Forced Barotropic Flow, ADVANCES IN ATMOSPHERIC SCIENCES, 16, 451-466.  doi: 10.1007/s00376-999-0023-2
    [4] MENG Xiangfeng, WU Dexing, LIN Xiaopei, LAN Jian, 2006: A Further Investigation of the Decadal Variation of ENSO Characteristics with Instability Analysis, ADVANCES IN ATMOSPHERIC SCIENCES, 23, 156-164.  doi: 10.1007/BF02656936
    [5] H.L. Kuo, 1995: Three-dimensional Global Scale Permanent-wave Solutions of the Nonlinear Quasigeostrophic Potential Vorticity Equation and Energy Dispersion, ADVANCES IN ATMOSPHERIC SCIENCES, 12, 387-404.  doi: 10.1007/BF02657001
    [6] Chen Zhongming, Liu Fuming, Li Xiaoping, Tao Jie, 1994: Oscillatory Rossby Solitary Waves in the Atmosphere, ADVANCES IN ATMOSPHERIC SCIENCES, 11, 65-73.  doi: 10.1007/BF02656995
    [7] Tianju WANG, Zhong ZHONG, Ju WANG, 2018: Vortex Rossby Waves in Asymmetric Basic Flow of Typhoons, ADVANCES IN ATMOSPHERIC SCIENCES, 35, 531-539.  doi: 10.1007/s00376-017-7126-y
    [8] Jiang Guorong, 1996: CISK-related Rossby Waves in the Tropical Atmosphere, ADVANCES IN ATMOSPHERIC SCIENCES, 13, 115-123.  doi: 10.1007/BF02657032
    [9] Zhao Ping, 1991: The Effects of Zonal Flow on Nonlinear Rossby Waves, ADVANCES IN ATMOSPHERIC SCIENCES, 8, 299-306.  doi: 10.1007/BF02919612
    [10] ZHONG Wei, LU Han-Cheng, Da-Lin ZHANG, 2010: Mesoscale Barotropic Instability of Vortex Rossby Waves in Tropical Cyclones, ADVANCES IN ATMOSPHERIC SCIENCES, 27, 243-252.  doi: 10.1007/s00376-009-8183-7
    [11] Y. L. McHall, 1993: Large Scale Perturbations in Extratropical Atmosphere-Part I: On Rossby Waves, ADVANCES IN ATMOSPHERIC SCIENCES, 10, 169-180.  doi: 10.1007/BF02919139
    [12] Yi Zengxin, T. Warn, 1987: A NUMERICAL METHOD FOR SOLVING THE EVOLUTION EQUATION OF SOLITARY ROSSBY WAVES ON A WEAK SHEAR, ADVANCES IN ATMOSPHERIC SCIENCES, 4, 43-54.  doi: 10.1007/BF02656660
    [13] Zuohao CAO, Da-Lin ZHANG, 2004: Tracking Surface Cyclones with Moist Potential Vorticity, ADVANCES IN ATMOSPHERIC SCIENCES, 21, 830-835.  doi: 10.1007/BF02916379
    [14] Chanh Q. KIEU, Da-Lin ZHANG, 2012: Is the Isentropic Surface Always Impermeable to the Potential Vorticity Substance?, ADVANCES IN ATMOSPHERIC SCIENCES, 29, 29-35.  doi: 10.1007/s00376-011-0227-0
    [15] Li Liming, Huang Feng, Chi Dongyan, Liu Shikuo, Wang Zhanggui, 2002: Thermal Effects of the Tibetan Plateau on Rossby Waves from the Diabatic Quasi-Geostrophic Equations of Motion, ADVANCES IN ATMOSPHERIC SCIENCES, 19, 901-913.  doi: 10.1007/s00376-002-0054-4
    [16] Yaokun LI, Jiping CHAO, Yanyan KANG, 2021: Variations in Wave Energy and Amplitudes along the Energy Dispersion Paths of Nonstationary Barotropic Rossby Waves, ADVANCES IN ATMOSPHERIC SCIENCES, 38, 49-64.  doi: 10.1007/s00376-020-0084-9
    [17] Shuguang WANG, Juan FANG, Xiaodong TANG, Zhe-Min TAN, 2022: A Survey of Statistical Relationships between Tropical Cyclone Genesis and Convectively Coupled Equatorial Rossby Waves, ADVANCES IN ATMOSPHERIC SCIENCES, 39, 747-762.  doi: 10.1007/s00376-021-1089-8
    [18] Zuohao CAO, Da-Lin ZHANG, 2005: Sensitivity of Cyclone Tracks to the Initial Moisture Distribution: A Moist Potential Vorticity Perspective, ADVANCES IN ATMOSPHERIC SCIENCES, 22, 807-820.  doi: 10.1007/BF02918681
    [19] Olivia MARTIUS, Cornelia SCHWIERZ, Michael SPRENGER, 2008: Dynamical Tropopause Variability and Potential Vorticity Streamers in the Northern Hemisphere ---A Climatological Analysis, ADVANCES IN ATMOSPHERIC SCIENCES, 25, 367-380.  doi: 10.1007/s00376-008-0367-z
    [20] Zuohao Cao, G.W.K. Moore, 1998: A Diagnostic Study of Moist Potential Vorticity Generation in an Extratropical Cyclone, ADVANCES IN ATMOSPHERIC SCIENCES, 15, 152-166.  doi: 10.1007/s00376-998-0036-2

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Manuscript received: 14 June 2022
Manuscript revised: 03 July 2022
Manuscript accepted: 06 July 2022
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A Quantitative Method of Detecting Transient Rossby Wave Phase Speed: No Evidence of Slowing Down with Global Warming

    Corresponding author: Jianhua LU, lvjianhua@mail.sysu.edu.cn
  • 1. School of Atmospheric Sciences, Sun Yat-Sen University and Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai 519082, China
  • 2. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing100029, China
  • 3. Guangdong Province Key Laboratory for Climate Change and Natural Disaster Studies, Sun Yat-sen University, Guangzhou 510275, China

Abstract: Based on the Complex Empirical Orthogonal Functions (CEOFs) of bandpass-filtered daily streamfunction fields, a quantitative method of detecting transient (synoptic) Rossby wave phase speed (RWPhS) is presented. The transient RWPhS can be objectively calculated by the distance between a high (or low) center in the real part of a CEOF mode and its counterpart in the imaginary part of the same CEOF mode divided by the time span between two adjacent peaks (or bottoms) of two principal component curves for the real and imaginary parts of that CEOF mode. The new detection method may partly reveal the spatiotemporal heterogeneity of Rossby wave prorogation. Although the mean westerly jet at 200 hPa doubles the speed of its counterpart at 500 hPa, the estimated RWPhS at both levels are around 1000 km d–1 and quantitatively consistent with the quasigeostrophic-theory-based RWPhS, confirming that the meridional potential vorticity gradient induced by the barotropic and baroclinic shears of mean flow, together with the β effect, play an essential role in Rossby wave propagation. Both observations over the past four decades and a 150-year historical simulation suggest no evidence for slowing wintertime transient Rossby waves in the Northern Hemisphere, but possible regional changes are not excluded. We emphasize that not only the mean flow speed, but also the barotropic and baroclinic shears of the mean flow, and their associated contributions to the meridional potential vorticity (PV) gradient, should be considered in investigating the possible change of Rossby waves with global warming.

    • Rossby waves never simply propagate. Because of baroclinic or barotropic instabilities and nonlinear advection of potential vorticity (PV), they may grow, saturate, break, and decay, leading to the macroturbulence of the troposphere (Held, 1999). Rossby waves, also called large-scale eddies, play a key role in maintaining the general circulation of the atmosphere via meridional transport of heat and momentum (Yeh and Chu, 1958), i.e., via the stirring and mixing of PV and via wave–mean flow interactions (e.g., McIntyre, 2015). Even in terms of their propagation, Rossby waves do not propagate like single waves with constant wave numbers and frequencies, but more like wave packets (e.g., Zeng, 1983; Wirth et al., 2018). Due to their dispersive nature, Rossby waves originating in one region (such as over Asia) may stir up new Rossby waves far downstream (say, over North America) with a speed (group speed) much faster than their phase speed (Yeh, 1949; see also a thorough review in Phillips, 1990). The propagation of Rossby waves may also be regulated by seasonally varying tropical–polar heat contrast, land–sea contrast, and topographic features, hence forming regionally enhanced or weakened storm tracks (Hoskins, et al., 1983).

      Given the intricacies of Rossby wave dynamics, caution must be taken in inferring possible changes of Rossby waves with climate change from over-simplified arguments. Indeed, the unsettled debates on Francis and Vavrus (2012, FV12 hereafter) largely reflect the complexity and multifaceted nature of the possible response of Rossby waves to global and regional climate change (Barnes, 2013; Yao et al., 2017; Blackport and Screen, 2020; White et al., 2022).

      FV12 suggested that Rossby wave phase speed (RWPhS) may slow down with climate change because of reduced zonal wind associated with polar warming amplification and reduced tropical–polar temperature gradient. However, Barnes (2013), Blackport and Screen (2020), and Riboldi et al. (2020) found that there is no evidence of slowing RWPhS and no evidence of ridge elongation. Indeed, in the lower troposphere the meridional temperature gradient decreases due to amplified Arctic warming, but it increases in the upper troposphere due to amplified tropical warming (Lu and Cai, 2010; Hoskins and Woollings, 2015).

      Even if the abovementioned studies had provided evidence against the FV12 hypothesis, uncertainty would remain largely due to the methodology of diagnosing RWPhS, which has been based on space–time spectral analysis following Hayashi (1979) and Randel and Held (1991). Confusion is also caused by not distinguishing between Rossby waves with different scales: synoptic or planetary, transient or (quasi) stationary. Luo et al. (2019) pointed out that the abovementioned linear propagation arguments of RWPhS are not suitable to explain the movement of quasi-stationary, persistent blockings, and Chen et al. (2021) further proved by idealized simulations that nonlinear RWPhS may well explain the response of Ural blocking to Barents–Kara-Sea warming.

      Therefore, we focus on transient (synoptic) Rossby waves in this study and provide a new method to calculate their propagation speed. This method is based on the complex empirical orthogonal function (CEOF) analysis and a detailed automatic detection procedure from the CEOFs. The results show that, overall, there is no significant linear trend in transient RWPhS during boreal winter over the last four decades, from 1979 to 2019, with a global surface air temperature (SAT) warming of about 0.6°C (Fig. 9 in Sun et al., 2021), though we do not exclude possible regional changes in RWPhS with global warming.

      It should be noted that although the reduction of RWPhS under weaker mean westerly flow (as FV12 suggested) seems consistent with Rossby’s long-wave formula (c=Uβ/k2), we find the transient RWPhSs are more consistent with the quasigeostrophic Rossby wave speed (see Eq. 7 in section 4) in which the mean potential vorticity gradient ($\partial \overline{{\rm{PV}}}/\partial y$), including both the β effect and the (−UyyUzz) term induced by barotropic and baroclinic wind shears (Uy and Uz), determines the RWPhS together with the mean zonal flow. Indeed, the reduction of the mean flow and associated reduction in PV gradient have opposite effects on phase propagation of Rossby waves, and hence, RWPhS may remain unchanged although the mean zonal flow (U) becomes weaker with global warming.

    2.   Data and methods
    • The daily zonal and meridional velocity (u and v) data from the ERA-Interim reanalysis of the European Centre for Medium-Range Weather Forecast (ECMWF) at 200 hPa and 500 hPa (Dee et al., 2011) with a resolution of 2.5°×2.5° are utilized. The time period covers 1979–2019, and we focus on boreal winter from 1 November to 31 March. The daily u and v data during the period 1850–2000 from one of the historical simulations of NCAR’s CESM2-FV2 from the sixth phase of the Coupled Model Intercomparison Project (CMIP6) are also used (Danabasoglu, 2019).

      Indeed, this analysis method can be applied on many other variables, such as wind, temperature, and geopotential height fields, but here, we only use the streamfunction field calculated from the u and v fields. We perform the CEOF analysis separately for each year. The procedure of detecting RWPhS is as follows:

      (1) The daily streamfunction fields from 1 November to 31 March of each year are first bandpass-filtered to retain only the transient (2–9 days) part of the variance by applying the standard Lanczos filter (Duchon, 1979). Note that after being filtered, the data for the first and last 10 days (i.e., 1–10 November and 22–31 March) are cut off and not used in further calculations.

      (2) Denoting the filtered time series at a grid point as $ x\left(t\right) $, we obtain its Hilbert transform $\hat{x}\left(t\right)$, which lags $ x\left(t\right) $ by π/2 phase, based on:

      where $ L=20 $, which provides an adequate amplitude response in the frequency domain (Barnett, 1983). After the Hilbert transform, the data for 11–30 November and 2–21 March are not available because of tapering, and hence, we only use the data of the 90 days since 1 December, i.e., December, January, and February (DJF), in the following analysis.

      (3) The complex time series $ X\left(t\right)=x\left(t\right)+i\hat{x}\left(t\right) $ at all grid points over the Northern Hemisphere forms a complex space–time field $ {\boldsymbol{X}}_{n\times m} $ (here n = 90, the length of $ X\left(t\right) $ at any grid point in one winter; m = 144 × 37, the number of grid points in the Northern Hemisphere) on which the decomposition of CEOF is performed. The decomposition procedure is as follows, and readers may refer to the appendix part of Barnett et al. (1983) and chapters 13 and 16 of von Storch and Zwiers (2003) for a thorough mathematical background.

      By obtaining the complex covariance matrix ${\boldsymbol{S}}_{n\times n}={\boldsymbol{X}}_{n\times m}{\tilde{\boldsymbol{X}}}_{m\times n}/n$, where $\tilde{\boldsymbol{X}}$ is the complex conjugate transposition of $ \boldsymbol{X} $, we may calculate its eigenvalues and eigenvectors. As $ {\boldsymbol{S}}_{n\times n} $ is a Hermitian matrix, there exists a unitary matrix $ {\boldsymbol{U}}_{n\times n} $ satisfying

      where ${\boldsymbol{\varLambda }}_{n\times n}$ is a real diagonal matrix with eigenvalues $ {\lambda }_{1} > {\lambda }_{2}\dots > {\lambda }_{n} $ as its diagonal elements, the j-th row of $ {\boldsymbol{U}}_{n\times n} $ is the complex eigenvector corresponding to the eigenvalue $ {\lambda }_{j} $, and $ {\tilde{\boldsymbol{U}}}_{n\times n} $ is the complex conjugate transposition of $ {\boldsymbol{U}}_{n\times n} $, which can be easily obtained by utilizing the subroutines in standard math libraries such as EISPACK (Smith, et al., 1974).

      Then, the complex EOF matrix $ {\boldsymbol{V}}_{m\times n} $ is obtained by

      and the complex principal component (PC) matrix $ {\boldsymbol{T}}_{n\times n} $ is

      As such, the CEOF decomposition is achieved by

      where $ {\boldsymbol{T}}^{\text{T}} $ is the transposition of $ \boldsymbol{T} $ and $\tilde{\boldsymbol{V}}$ is the complex conjugate transposition of $ \boldsymbol{V} $. The j-th mode of CEOFs (CEOFj) is the j-th column of $ {\boldsymbol{V}}_{m\times n} $, as shown in Figs. 1a and 1b for the real and imaginary parts of CEOF1. The j-th row of $ {\boldsymbol{T}}_{n\times n} $ represents the real and imaginary parts of the j-th complex PC, as shown in Figs. 1c and 1d for the first PC.

      Figure 1.  The first mode of complex empirical orthogonal function (CEOF1) and corresponding principal component (PC1) of 2–9-day Lanczos-filtered streamfunction at 200 hPa (a–c) and 500 hPa (d–f) during December, January, and February (DJF) of 2008/09: (a) Real part and (b) imaginary part of CEOF1 (colored contours, m2 s–1), and mean zonal wind speed at 200 hPa in 2008/09 DJF (black contours, m s–1); (c) Real part (black line) and imaginary part (blue dashed line) of PC1, dots mark the data with |PC|≥0.5); (d)–(f): same as (a)–(c), but for 500 hPa. The criterion for the dotted centers in (a) and (b) is the absolute value being larger than 1.5×106 m2 s–1.

      Note that the original anomaly relative to the seasonal mean (Figs. 2a and f) includes not only the 2–9-day transient eddies, but also the longer-period transient eddies, quasi-stationary eddies, and stationary eddies. But the fields reconstructed simply from CEOF1 (Figs. 2c and h) have spatial patterns very similar to those of the 2–9-day Lanczos-filtered fields (Figs. 2b and g) with the wave amplitude of the reconstructed field being about 40% of the latter, corresponding to the variance of CEOF1 being about 16%−20% of the total. If we utilize the first four CEOFs to perform the reconstruction, then both the spatial patterns (Figs. 2d and i) and the temporal evolution (Figs. 2e and j for point C in Fig. 1) of the reconstructed fields are close to the original Lanczos-filtered fields. The results illustrate well that CEOFs may be used to analyze the dominant travelling transient Rossby waves.

      Figure 2.  (a) 200-hPa streamfunction anomaly (original streamfunction on 9 January 2009 minus 2008/09 DJF-averaged streamfunction), (b) 2–9-day Lanczos-filtered streamfunction at 200 hPa on 9 January 2009, (c) reconstructed streamfunction on 9 January 2009 from CEOF1 (i.e., real part of 2008/09 DJF 200-hPa CEOF1 multiplied by real part of PC1 on 9 January 2009 plus imaginary part of CEOF1 multiplied by imaginary part of PC1 on 9 January 2009); (d) reconstructed streamfunction on 9 January 2009 from the first four CEOFs; (f)–(i) are the same as (a)–(d), but for 11 January 2009. Black contours (m s–1) are mean zonal wind speed of 2008/09 DJF at 200 hPa. (e) and (j) are the real and imaginary parts of daily streamfunction at grid point C in Fig. 1 during 2008/09 winter. Black curves are from the original 2–9-day Lanczos-filtered time series [in (e)] and its Hilbert transform [in (j)], while the red curves are reconstructed from the truncated sum of the first 1–4 CEOF modes. Units: 107 m2 s–1. The two days are utilized to illustrate the propagation of Rossby waves during about one-fourth of a wave period.

      (4) For each mode of the CEOFs, most high/low centers in the real part can find their counterparts in the imaginary part, which are downstream of the former with their phase difference being $ \pi /2 $ for a traveling wave (see grid points A, B, and C in Fig. 1a and A', B', and C' in Fig. 1b for examples). Then the RWPhS can be calculated as the distance between each pair of high/low centers (the wave propagation distance, i.e., the distance of AA', BB', and CC' in Fig. 1b) divided by the time intervals between the peaks (bottoms) in the real-part principal component and their adjacent counterparts in the imaginary part (i.e., T1T1', T2T2', etc. in Fig. 1c).

      Therefore, for each red or blue dot (including those labeled A, B, and C, for a total of about 12 points) in Fig. 1a, we may obtain about 26 RWPhSs. The total number of RWPhSs obtained from CEOF1 during 2008/09 DJF is 312, but the locations and number (between 300 and 400) of RWPhSs differ in other years. Although the calculated RWPhSs are spatiotemporally discrete, we may obtain regional statistics of these RWPhSs for each season. This allows the spatiotemporal heterogeneity of Rossby wave propagation to be partly represented in this method, as shown in Fig. A1 in the Appendix which shows the RWPhS every three days during DJF of 2008/09 over the north Pacific and North Atlantic. This result implies that the RWPhS over the north Pacific is generally larger than that over the North Atlantic, but the two oceans’ intra-seasonal variations are somewhat similar.

      Figure A1.  The intraseasonal variability of RWPhS over north Pacific (a) and north Atlantic (b) regions during DJF of 2008/09

    3.   No evidence of slowing transient RWPhS
    • First, we examine the variation of RWPhS during the winters of 1979–2019, obtained from the ERA-Interim reanalysis data. Taken as an example, the real and imaginary parts of CEOF1 for the 200-hPa streamfuntion during the 2008/09 winter (DJF) (Figs. 1a and 1b) show that transient Rossby waves travel along the subtropical westerly waveguide and the waves are particularly strong along the storm tracks over the Pacific and the North Atlantic. It is also clear that the waves may grow at the exit of jet streaks over East Asia and North America. By definition, the imaginary PC (the blue curve in Fig. 1c) lags π/2 phase compared to its real counterpart (the black curve in Fig. 1c), which is about 2 days in Fig. 1c, corresponding well to the typical period of synoptic/transient Rossby waves (6–8 days). The real and imaginary PCs also show subseasonal oscillations in their amplitudes, indicating the active and inactive periods of the corresponding mode. Note that the CEOF1 for 500-hPa streamfunction and the corresponding real and imaginary PCs (Figs. 1d, 1e, and 1f) are similar to those shown in Figs. 1ac. Indeed, there also exist interannual and interdecadal variations in the spatial structure of CEOFs and their temporal evolution (PCs) if we carefully check the CEOFs for each year from 1979 to 2019. By following the detection method laid out above, we examine the variation of RWPhS over the four decades from 1979/80 to 2018/19 (Fig. 3). Here, we show the zonal RWPhS only, but the results hold also for the total phase speed — note that Rossby waves also propagate meridionally. The box-and-whisker diagrams in Fig. 3, which is based on the statistics of the abovementioned individual, spatiotemporally discrete RWPhS, show the yearly-based (with about 300–400 samples of RWPhSs in each DJF) and decade-based statistics (with all samples in ten DJFs) of RWPhS at 200 hPa (Figs. 3a and b) and 500 hPa (Figs. 3c and d) over the midlatitudes between 20°N and 60°N. Indeed, while salient interannual variations exist, the trends of RWPhS at both 200 hPa and 500 hPa are close to zero, as can easily be seen during boreal winter (Fig. 3). The decade-based statistics (Figs. 3b and d) further confirm (since the statistical uncertainty is largely reduced with a tenfold increase in the number of samples) that there is no apparent trend in the zonal RWPhS during the last four decades. Indeed, the results hold for the RWPhS of other CEOF modes, as shown in Fig. A2 for the 2nd, 3rd, and 4th CEOF modes. The overall RWPhS trend obtained from the four CEOF modes over 1979–2019 is about 70 km d–1 per degree of global warming, being negligible compared to the absolute value of RWPhS and being smaller than the intrinsic error (about ±100 km d–1) caused by the spatial and temporal resolutions of the reanalysis data.

      Figure A2.  Box-and-whisker diagrams of zonal Rossby wave phase speed (RWPhS) at 200 hPa between 20°N and 60°N based on (a) annual and (b) decadal statistics separately obtained from the second CEOF modes for the winter of 1979/80 to 2018/2019. (c) and (d) : The same as (a) and (b), but for the third CEOF mode; (e) and (f): The same as (a) and (b), but for the fourth CEOF mode.

      Figure 3.  Box-and-whisker diagrams of zonal Rossby wave phase speed (RWPhS) at 200 hPa between 20°N and 60°N based on (a) annual and (b) decadal statistics from the CEOF1 for the winters of 1979/80 to 2018/19. The short red line in (b) marks the median value. Medians in (a) are connected by a black curve, and the thick red curve is the 5-year Gaussian smoothing of medians. The bottom and top blue lines of the boxes are the 25th- and 75th- percentile values, respectively, while the whiskers extend to the minimum and maximum of the data, except for outliers, which fall outside of 99.3%; (c) and (d) are the same as (a) and (b) but for the zonal RWPhS at 500 hPa.

      However, there exists an apparent difference in the inter-annual variability of RWPhS between the upper- and mid-troposphere, and this will be discussed in the next section.

      Additionally, we calculated the RWPhS by applying the same method to the Lanczos-filtered daily streamfunction at 250 hPa during the 150 boreal winters from 1850/51 to 1999/2000, with an observed global SAT warming of about 0.6°C (Fig. 9 in Sun et al., 2021), using the historical simulation from CESM2. Even with a much longer time span, the annual and decadal box-and-whisker diagrams of RWPhS (Fig. 4) exhibit a near-zero [5 km d–1 (100) yr-1] trend, although there do exist interannual and interdecadal variations.

      Figure 4.  The same as Figs. 3a and 3b, but for RWPhS at 250 hPa during boreal winters from 1850/51 to 1999/2000 in the CESM2 historical simulation.

      Therefore, based on our method, there is so far no evidence of slowing transient Rossby wave propagation, but we note that we do not include quasi-stationary Rossby waves in our analysis.

    4.   Why is the transient RWPhS at 200 hPa so close to the RWPhS at 500 hPa?
    • We can see from Fig. 3 that during 1979–2018 the range of mean zonal wave speed at 200 hPa is mainly between 900 km d–1 and 1200 km d–1, with the median being about 1000 km d–1. The range and average of zonal RWPhS at 500 hPa are not much different from those at 200 hPa, with the median being about 950 km d–1, slightly less than that at 200 hPa, although the mean zonal wind speed at 200 hPa is twice as large as that at 500 hPa. First, we should mention that, physically, the similar RWPhSs in the upper- and mid-troposphere are not surprising from the viewpoint of vertical coupling associated with the life cycle of baroclinic waves (e.g., Feldstein and Held, 1989; Thorncroft et al., 1993; Orlanski and Gross, 2000). Furthermore, because the synoptic/transient Rossby waves are baroclinic in nature, their propagations are determined not only by the mean wind speeds and β [the meridional gradient of the Coriolis parameter (f)], as indicated by Rossby’s long-wave formula (c=Uβ/k2), but also by the other parts of the meridional gradient of potential vorticity (PV), i.e., the meridional and vertical second-order derivatives of the time-mean zonal flow, as shown in the well-known formula [e.g., Eq. (4.40) in Phillips,1990]:

      where c is Rossby-wave phase speed, $ \bar{U} $ is the mean zonal wind speed, $ \beta =\partial f/\partial y $, $S=H^2N^2/f^2 $, N is the Brunt–Väisalä frequency, $ H $ is the scale height of the atmosphere, $Z=-\log (p/ p_0)$, and k, l, and m are zonal, meridional, and vertical wave numbers, respectively.

      The 40-winter-averaged zonal flow ($\bar{U}$) at 200 hPa (Fig. 5a) and 500 hPa (Fig. 5b) shows the strongest westerly jet stream located mainly between 20°N and 40°N, particularly over western Pacific and North Atlantic. The zonally averaged $ \bar{U} $ at 200 hPa along the westerly jet (36 m s–1) is much larger than its counterpart at 500 hPa (16 m s–1). Accordingly, the $ \bar{U} $-related part of the meridional gradient of PV ($\partial \overline{{\rm{PV}}}/\partial y$), i.e., $ -{\bar{U}}_{yy}-\frac{1}{p}\frac{\partial }{\partial Z}\left(\frac{p}{S}\frac{\partial \bar{U}}{\partial Z}\right) $ in Eq. (7), at 200 hPa (6.04×1011 m–1 s–1 between 20°N and 40°N) is also much larger than that at 500 hPa (3.22×1011 m–1 s–1 between 20°N and 40°N), leading to a larger total $\partial \overline{{\rm{PV}}}/\partial y$ at 200 hPa (Fig. 5c) than at 500 hPa (Fig. 5d).

      Figure 5.  The zonal wind speed (m s−1) at 200 hPa (a) and 500 hPa (b) and the mean $\partial \overline{{\rm{PV}}}/\partial y$ (×10−11 m−1 s−1) at 200 hPa (c) and 500 hPa (d) during boreal winter (DJF) averaged over 1979/80–2018/19.

      If we take from Fig. 1 the following typical values in Eq. (7):

      k=7/(a cosφ),i.e., zonal 7 waves; l=2π/(aπ/3)=6/a;

      m~2π /3H, derived from the phase difference (about 0.3 cycles) between 1000 hPa and 200 hPa of geopotential height in Fig. 6 of Pratt and Wallace (1976);

      H~10 km, and other parameters calculated from the reanalysis data,we obtain the mean zonal RWPhS between 20°N and 40°N as 11.6 m s–1 (1002 km d–1) at 200 hPa and 10.5 m s–1 ( 907 km d–1) at 500 hPa, very close to our estimate obtained from the CEOF-based detection method. Note the above estimate may differ depending on the three-dimensional scale (i.e., the wavenumbers k,l,m) of Rossby waves, and hence the consistency between CEOF-based RWPhS and the RWPhS obtained from quasigeostrophic theory should be considered as qualitative, rather than quantitatively precise.

      In short, we should consider not only the increase or decrease of the mean flow speed, but also the associated changes in barotropic and baroclinic shears and in the meridional PV gradient. Such a more thorough consideration is essential not only to correctly understand the changes of Rossby wave propagation, but also to understand the possible changes of barotropic and baroclinic instabilities with climate change. For example, except by changing the location and strength of mean westerly flow, the expansion of the upper-tropospheric tropical area with global warming (e.g., Staten et al., 2018) may well modulate RWPhS by enhancing PV gradient.

      Despite the similarities of RWPhSs and of their long-term trends at the upper- and mid-troposphere, their interannual variations (Figs. 3a and 3c) are not necessarily consistent. Both physical mechanism and the statistical method may contribute to the inconsistency. While there exists vertical coupling of baroclinic Rossby waves, the propagation and growth/decay during their life cycle could be quite different (Thorncroft et al., 1993; Orlanski and Gross, 2000). Indeed, the difference in the propagation speed of baroclinic Rossby waves in the upper, middle, and lower troposphere associated with different vertical structures (baroclinicty) of mean flow was pointed out long ago by Yeh and Chen (1963) and Zeng (1983). Accordingly, these differences may be reflected in the CEOFs and corresponding PCs. Second, these differences may be retained in the yearly-based statistics, which are derived from only about 300–400 samples for each winter, but can be largely removed in the decade-based statistics by the smoothing effect of using a large sample size. Understandably, these interannual variances can be considered as internal noise, and hence they do not contribute to the externally forced long-term trend of RWPhS.

    5.   Summary and discussion
    • In this study, we propose a quantitative CEOF-based method for detecting synoptic/transient Rossby wave phase speed (RWPhS) at the midlatitudes between 20°N and 60°N and analyze its variation over the 40 winters from 1979/80 to 2018/19 in the ERA-Interim reanalysis and the 150 winters from 1850/51 to 1999/2000 in a CMIP6 historical simulation using CESM2-FV2. The merits of the method are that it may account for the dominant modes of Rossby waves for each season and account partly for the spatiotemporal heterogeneity of Rossby wave propagation. No evidence of transient Rossby-wave slowing during the boreal winter is found in either the reanalysis or the historical simulation. Our conclusion supports the conclusions based on spectral analysis, such as those by Barnes (2013) and Riboldi et al. (2020). Note, however, because our statistics cover all of the northern midlatitudes, we did not exclude possible regional changes of RWPhS with global warming, and regionally dependent results based on different methods will be reported in following publications.

      Despite the twofold difference in the mean zonal flow between at 200 hPa and 500 hPa, the CEOF-based estimates of the RWPhS at the two levels are very close, with both being about 1000 km d–1. This result is consistent with the estimate based on the quasigeostrophic Rossby wave dispersion relation, in which the barotropic and baroclinc shears of the mean zonal flow can affect the Rossby wave propagation via the influence on the mean meridional PV gradient ($\partial \overline{{\rm{PV}}}/\partial y$), reflecting the vertical coupling of Rossby wave propagation. However, there is no reason to expect a close correlation in terms of their interannual variations due to the different developmental, propagating, and decaying features of transient Rossby waves, as was pointed out long ago by Yeh and Chen (1963), Zeng (1983), and Thorncroft et al. (1993) among others. Given the limited number of wave samples for each season, these different vertical features lead to an interannual randomness in RWPhS, but the externally forced long-term trend may robustly emerge from such randomness by including a much larger number of wave samples. Our results suggest that the effects of barotropic and baroclinic shears, and more generally the PV gradient, should be included in the consideration of Rossby wave response to global warming. Indeed, the reduction of the mean flow and associated reduction in PV gradient have opposite effects on phase propagation of Rossby waves, and hence, RWPhS may remain unchanged although the mean zonal flow (U) becomes weaker with global warming.

      There is no reason to assume that Rossby waves should remain unchanged with global climate change given the robust changes in the tropical–polar temperature gradient and mean circulation. It is probable that regional positive or negative trends of RWPhS do exist (Fragkoulidis and Wirth, 2020) although the hemispheric-averaged trend is weak. Indeed, our CEOF-based method is spatiotemporally discrete but may reflect the dominant modes of transient Rossby waves, while the local wave speed method of Fragkoulidis and Wirth (2020) is spatiotemporally continuous but the calculation may be sensitive to the determination of local wave numbers and frequencies. In this regard, it will be helpful to compare our metric with that of Fragkoulidis and Wirth (2020), and even to use the two metrics together.

      Note that in this study, we focus on the propagation of transient Rossby waves only. It is necessary to discriminate them from slow-moving Rossby waves with a longer period (say, above 10 days), including blockings and other quasi-stationary waves, due to their totally different underlying mechanisms. Indeed, not distinguishing between different Rossby waves does cause confusion in the ongoing debate over their possible responses to global warming. There is an intrinsic limitation on the sole consideration of linear propagation of slow-moving quasi-stationary Rossby waves because they are in nature nonlinear and associated with multiscale interaction and the change in wave amplitude (e.g., Luo and Zhang, 2020). But even for these slow-moving quasi-stationary Rossby waves, the PV gradient perspective is still useful in understanding both their dynamics and their responses to climate change (Luo, et al., 2019). We also stress that our metric may be useful in investigating the interaction between transient waves and quasi-stationary waves.

      Acknowledgements. The authors appreciate the support from National Natural Science Foundation of China (Grant Nos. 42175070 and 41875130) and Guangdong Province Key Laboratory for Climate Change and Natural Disaster Studies (Grant No. 2020B1212060025). The authors are grateful to State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics for hosting their visit during the summer of 2021. We acknowledge the World Climate Research Programme, which, through its Working Group on Coupled Modelling, coordinated and promoted CMIP6. We thank NCAR’s CESM modelling team for producing and making available their model output, the Earth System Grid Federation (ESGF) for archiving the data and providing access, and the multiple funding agencies who support CMIP6 and ESGF.

      Author Contributions. J. L. developed the original CEOF code, initiated the idea of detecting RWPhS based on CEOF, and conceptualized the theoretical explanation. Y. W. developed the RWPhS detection procedure and performed the diagnostics. Both authors contributed to the writing and reviewing of the manuscript.

      Data Availability ERA-Interim reanalysis datasets used in this study have been obtained from ECMWF (https://www.ecmwf.int/en/forecasts/datasets/reanalysis-datasets/era-interim). The NCAR CESM2-FV2 model output prepared for CMIP6 is available at http://esgf-node.llnl.gov/search/cmip6/?mip_era=CMIP6&activity_id=CMIP&institution_id=NCAR&source_id=CESM2-FV2&experiment_id=historical.

      APPENDIX

Reference

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