Stochastic weather and climate models

Abstract

Although the partial differential equations that describe the physical climate system are deterministic, there is an important reason why the computational representations of these equations should be stochastic: such representations better respect the scaling symmetries of these underlying differential equations, as described in this Perspective. This Perspective also surveys the ways in which introducing stochasticity into the parameterized representations of subgrid processes in comprehensive weather and climate models has improved the skill of forecasts and has reduced systematic model error, notably in simulating persistent flow anomalies. The pertinence of stochasticity is also discussed in the context of the question of how many bits of useful information are contained in the numerical representations of variables, a question that is critical for the design of next-generation climate models. The accuracy of fluid simulation may be further increased if future-generation supercomputer hardware becomes partially stochastic.

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Fig. 1: Deterministic parameterization is justified only when there is scale separation between resolved and unresolved flow.
Fig. 2: Impact of stochastic parameterization on continuous rank probability skill scores for temperature on the 850 hPa pressure surface in the tropics.
Fig. 3: Power spectra of average sea surface temperature in the Niño-3.4 region in 135-year-long simulations using the National Center for Atmospheric Research Community Atmosphere Model coupled to an ocean model.
Fig. 4: Quasi-stationary circulation regimes over the European–North Atlantic domain and their statistical significance.
Fig. 5: At intermediate noise levels, the typical regime residence times for the time series for the x component of the Lorenz 63 system shift to larger values.

References

  1. 1.

    IPCC Climate Change 2013: The Physical Science Basis (eds Stocker, T. F. et al.) (Cambridge Univ. Press, 2013).

  2. 2.

    IPCC Climate Change 2014: Impacts, Adaptation, and Vulnerability. Part A: Global and Sectoral Aspects (eds Field, C. B. et al.) (Cambridge Univ. Press, 2014).

  3. 3.

    Shepherd, J. G. et al. Geoengineering the Climate: Science, Governance and Uncertainty (The Royal Society Publishing, 2009).

  4. 4.

    Sutton, R. B. et al. Attributing extreme weather to climate change is not a done deal. Nature 561, 177 (2018).

    ADS  Article  Google Scholar 

  5. 5.

    Suckling, E. in Weather and Climate Services for the Energy Industry (ed. Troccoli, A.) 123–137 (Palgrave Macmillan, 2018).

  6. 6.

    Cantelaube, P. & Terres, J.-M. Seasonal weather forecasts for crop yield modelling in Europe. Tellus 57A, 476–487 (2004).

    ADS  Google Scholar 

  7. 7.

    Thomson, M. C. et al. Malaria early warnings based on seasonal climate forecasts from multi-model ensembles. Nature 439, 576–579 (2006).

    ADS  Article  Google Scholar 

  8. 8.

    Palmer, T. N. The prediction of uncertainty in weather and climate forecasting. Rep. Prog. Phys. 63, 71–116 (2000).

    ADS  Article  Google Scholar 

  9. 9.

    Palmer, T. N. The ECMWF ensemble prediction system: looking back (more than) 25 years and projecting forward 25 years. Q. J. R. Meteorol. Soc. https://doi.org/10.1002/qj.3383 (2018).

  10. 10.

    Wilks, D. Statistical Methods in the Atmospheric Sciences (Academic Press, 2011).

  11. 11.

    Weisheimer, A. & Palmer, T. N. On the reliability of seasonal climate forecasts. J. R. Soc. Interface 11, 20131162 (2014).

    Article  Google Scholar 

  12. 12.

    Houghton, J. The Physics of Atmospheres (Cambridge Univ. Press, 2002).

  13. 13.

    Hasselmann, K. Stochastic climate models. Part I. Theory. Tellus 28, 473–485 (1976).

    ADS  Article  Google Scholar 

  14. 14.

    Clement, A. K. et al. The Atlantic Multidecadal Oscillation without a role for ocean circulation. Science 350, 320–324 (2015).

    ADS  Article  Google Scholar 

  15. 15.

    Zhang, R. et al. Comment on “The Atlantic Multidecadal Oscillation without a role for ocean circulation”. Science 352, 1527 (2016).

    ADS  Article  Google Scholar 

  16. 16.

    Palmer, T. N. A nonlinear dynamical perspective on model error: A proposal for non-local stochastic-dynamic parameterization in weather and climate prediction. Q. J. R. Meteorol. Soc. 127, 279–304 (2001).

    ADS  Google Scholar 

  17. 17.

    Palmer, T. N., Doering, A. & Seregin, G. The real butterfly effect. Nonlinearity 27, R123–R141 (2014).

    ADS  Article  Google Scholar 

  18. 18.

    Arakawa, A. The cumulus parameterization problem: past, present, and future. J. Clim. 17, 2493–2525 (2004).

    ADS  Article  Google Scholar 

  19. 19.

    Palmer, T. N., Shutts, G. J. & Swinbank, R. Alleviation of a systematic westerly bias in general circulation and numerical weather prediction models through an orographic gravity wave drag parameterization. Q. J. R. Meteorol. Soc. 112, 1001–1031 (1986).

    ADS  Article  Google Scholar 

  20. 20.

    Gent, P. R. & McWilliams, J. C. Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr. 20, 150–155 (1990).

    ADS  Article  Google Scholar 

  21. 21.

    Neelin, J. D., Peters, O., Lin, J. W.-B., Hales, K. & Holloway, C. E. Rethinking convective quasi-equilibrium: observational constraints for stochastic convective schemes in climate models. Phil. Trans. R. Soc. A 366, 2579–2602 (2008).

    ADS  Article  Google Scholar 

  22. 22.

    Majda, A. J. & Bertozzi, A. L. Vorticity and Incompressible Flow (Cambridge Univ. Press, 2002).

  23. 23.

    Nastrom, G. D. & Gage, K. S. A climatology of atmospheric wavenumber spectra observed by commercial aircraft. J. Atmos. Sci. 42, 950–960 (1985).

    ADS  Article  Google Scholar 

  24. 24.

    Lovejoy, S. & Schertzer, D. The Weather and Climate (Cambridge Univ. Press, 2013).

  25. 25.

    Zhang, X., Liu, H. & Zhang, M. Double ITCZ in coupled ocean–atmosphere models: from CMIP3 to CMIP5. Geophys. Res. Lett. 42, 8651–8659 (2015).

    ADS  Article  Google Scholar 

  26. 26.

    Masato, G., Hoskins, B. J. & Woollings, T. Winter and summer Northern Hemisphere blocking in CMIP5 models. J. Clim. 26, 7044–7059 (2013).

    ADS  Article  Google Scholar 

  27. 27.

    Palmer, T. N. A personal perspective on modelling the climate system. Proc. R. Soc. A 472, 20150772 (2016).

    ADS  Article  Google Scholar 

  28. 28.

    Buizza, R., Miller, M. & Palmer, T. N. Stochastic representation of model uncertainties in the ECMWF ensemble prediction system. Q. J. R. Meteorol. Soc. 125, 2887–2908 (1999).

    ADS  Article  Google Scholar 

  29. 29.

    Palmer, T. N. et al. Stochastic Parametrization and Model Uncertainty ECMWF Technical Memoranda (ECMWF, 2009); http://www.ecmwf.int/sites/default/files/elibrary/2009/11577-stochastic-parametrization-and-model-uncertainty.pdf

  30. 30.

    Majda, A. J., Timofeyev, I. & Vanden Eijnden, E. A mathematical framework for stochastic climate models. Commun. Pure Appl. Math. 54, 891–974 (2001).

    MathSciNet  Article  Google Scholar 

  31. 31.

    Palmer, T. N. Towards the probabilistic Earth-system simulator: a vision for the future of climate and weather prediction. Q. J. R. Meteorol. Soc. 138, 841–861 (2012).

    ADS  Article  Google Scholar 

  32. 32.

    Holm, D. D. Variational principles for stochastic fluid dynamics. Proc. R. Soc. A 471, 20140963 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  33. 33.

    Plant, R. S. & Craig, G. C. A stochastic parameterization for deep convection based on equilibrium statistics. J. Atmos. Sci. 65, 87–105 (2008).

    ADS  Article  Google Scholar 

  34. 34.

    Bengtsson, L., Steinheimer, M., Bechtold, P. & Geleyn, J.-F. A stochastic parametrization for deep convection using cellular automata. Q. J. R. Meteorol. Soc. 139, 1533–1543 (2013).

    ADS  Article  Google Scholar 

  35. 35.

    Gottwald, G. A., Peters, K. & Davies, L. A data-driven method for the stochastic parametrisation of subgrid-scale tropical convective area fraction. Q. J. R. Meteorol. Soc. 142, 349–359 (2016).

    ADS  Article  Google Scholar 

  36. 36.

    Khouider, B., Biello, J. & Majda, A. J. A stochastic multicloud model for tropical convection. Commun. Math. Sci. 8, 187–216 (2010).

    MathSciNet  Article  Google Scholar 

  37. 37.

    Tompkins, A. M. & Berner, J. A stochastic convective approach to account for model uncertainty due to unresolved humidity variability. J. Geophys. Res. 113, D18101 (2008).

    ADS  Article  Google Scholar 

  38. 38.

    Berner, J. et al. Stochastic parameterization: toward new view of weather and climate models. Bull. Am. Meteorol. Soc. 98, 565–587 (2017).

    ADS  Article  Google Scholar 

  39. 39.

    Mason, P. J. & Thomson, D. J. Stochastic backscatter in large-eddy simulations of boundary layers. J. Fluid. Mech. 242, 51–78 (1992).

    ADS  Article  Google Scholar 

  40. 40.

    Shutts, G. A kinetic energy backscatter algorithm for use in ensemble prediction systems. Q. J. R. Meteorol. Soc. 131, 3079–3102 (2005).

    ADS  Article  Google Scholar 

  41. 41.

    Berner, J., Shutts, G. J., Leutbecher, M. & Palmer, T. N. A spectral stochastic kinetic energy backscatter scheme and its impact on flow-dependent predictability in the ECMWF ensemble prediction system. J. Atmos. Sci. 66, 603–626 (2009).

    ADS  Article  Google Scholar 

  42. 42.

    Palmer, T. N. & Williams, P. Stochastic Physics and Climate Modelling (Cambridge Univ. Press, 2017).

  43. 43.

    Wouters, J. & Lucarini, V. Disentangling multi-level systems: averaging, correlations and memory. J. Stat. Mech. 3, P03003 (2012).

  44. 44.

    Wouters, J. & Lucarini, V. Multi-level dynamical systems: connecting the Ruelle response theory and the Mori–Zwanzig approach. J. Stat. Phys. 151, 850–860 (2013).

    ADS  MathSciNet  Article  Google Scholar 

  45. 45.

    Franzke, C. L., O’Kane, T. J., Berner, J., Williams, P. D. & Lucarini, V. Stochastic climate theory and modeling. WIREs Clim. Change 6, 63–78 (2015).

    Article  Google Scholar 

  46. 46.

    Vissio, G. & Lucarini, V. A proof of concept for scale-adaptive parametrizations: the case of the Lorenz ‘96 model. Q. J. R. Meteorol. Soc. 144, 63–75 (2018).

    ADS  Article  Google Scholar 

  47. 47.

    Bengtsson, L.et al. A model framework for stochastic representation of uncertainties associated with physical processes in NOAA’s Next Generation Global Prediction System (NGGPS). Mon. Weather Rev., https://doi.org/10.1175/MWR-D-18-0238.1 (2019).

    ADS  Article  Google Scholar 

  48. 48.

    Shutts, G. J. & Palmer, T. N. Convective forcing fluctuations in a cloud-resolving model: relevance to the stochastic parameterization problem. J. Clim. 20, 187–202 (2007).

    ADS  Article  Google Scholar 

  49. 49.

    Shutts, G. J. & Callado Pallarès, A. Assessing parameterization uncertainty associated with horizontal resolution in numerical weather prediction models. Phil. Trans. R. Soc. Lond. A 372, 20130284 (2014).

    ADS  Article  Google Scholar 

  50. 50.

    Christensen, H. M. Constraining stochastic parametrisation schemes using high-resolution simulations. Preprint at arXiv https://arxiv.org/abs/1904.04503 (2019).

  51. 51.

    Christensen, H. M., Lock, S. J., Moroz, I. M. & Palmer, T. N. Introducing independent patterns into the stochastically perturbed parametrization tendencies (SPPT) scheme. Q. J. R. Meteorol. Soc. 143, 2168–2181 (2017).

    ADS  Article  Google Scholar 

  52. 52.

    Chevallier, F., Chéruy, F., Scott, N. A. & Chédin, A. A neural network approach for a fast and accurate computation of longwave radiative budget. J. Appl. Meteorol. 37, 1385–1397 (1998).

    ADS  Article  Google Scholar 

  53. 53.

    Krasnopolsky, V. M. The Aapplication of Neural Networks in the Earth-system Sciences Atmospheric and Oceanographic Sciences Library, Vol. 46 (Springer, 2013).

  54. 54.

    Andrejczuk, M. et al. Oceanic stochastic parameterizations in a seasonal forecast system. Mon. Weather Rev. 144, 1867–1875 (2016).

    ADS  Article  Google Scholar 

  55. 55.

    Porta Mana, P. G. L. & Zanna, L. Toward a stochastic parameterization of ocean mesoscale eddies. Ocean Model. 79, 1–20 (2014).

    ADS  Article  Google Scholar 

  56. 56.

    MacLeod, D. A., Cloke, H. L., Pappenberger, F. & Weisheimer, A. Improved seasonal prediction of the hot summer of 2003 over Europe through better representation of uncertainty in the land surface. Q. J. R. Meteorol. Soc. 142, 79–90 (2016).

    ADS  Article  Google Scholar 

  57. 57.

    Juricke, S. & Jung, T. Influence of stochastic sea ice parameterization on climate and the role of atmosphere–sea ice–ocean interaction. Phil. Trans. R. Soc. Lond. A 372, 20130283 (2014).

  58. 58.

    Williams, P. D. Climatic impacts of stochastic fluctuations in air–sea fluxes. Geophys. Res. Lett. 39, L10705 (2012).

    ADS  Google Scholar 

  59. 59.

    Murphy, A. H. A note on the utility of probabilistic predictions and the probability score in the cost-loss ratio decision situation. J. Appl. Meteorol. 5, 534–537 (1966).

    ADS  Article  Google Scholar 

  60. 60.

    Palmer, T. N. & Richardson, D. Decisions, decisions…! In ECMWF Newsletter 12–14 (ECMWF, 2014); https://www.ecmwf.int/sites/default/files/elibrary/2014/14584-newsletter-no141-autumn-2014.pdf

  61. 61.

    Weisheimer, A., Corti, S., Palmer, T. & Vitart, F. Addressing model error through atmospheric stochastic physical parametrizations: impact on the coupled ECMWF seasonal forecasting system. Phil. Trans. R. Soc. A 372, 20130290 (2014).

  62. 62.

    Subramanian, A., Weisheimer, A., Palmer, T., Vitart, F. & Bechtold, P. Impact of stochastic physics on tropical precipitation in the coupled ECMWF model. Q. J. R. Meteorol. Soc. 143, 852–865 (2017).

    ADS  Article  Google Scholar 

  63. 63.

    Christensen, H. M., Berner, J., Coleman, D. & Palmer, T. N. Stochastic parameterization and the El Niño–Southern Oscillation. J. Clim. 30, 17–38 (2017).

    ADS  Article  Google Scholar 

  64. 64.

    Strommen, K., Christensen, H. M., Berner, J. & Palmer, T. N. The impact of stochastic parametrisations on the representation of the Asian summer monsoon. Clim. Dyn. 50, 2269–2282 (2018).

    Article  Google Scholar 

  65. 65.

    Mo, K. & Ghil, M. Cluster analysis of multiple planetary flow regimes. J. Geophys. Res. 93, 10927–10952 (1988).

    ADS  Article  Google Scholar 

  66. 66.

    Weaver, M. Summer 2018 was UK’s joint hottest on record, Met Office says. The Guardian https://www.theguardian.com/uk-news/2018/sep/03/summer-2018-uk-joint-hottest-on-record-met-office-says (2018).

  67. 67.

    King, A. & Henley, B. It’s a savage summer in the Northern Hemisphere — and climate change is slashing the odds of more heatwaves. The Conversation https://theconversation.com/its-a-savage-summer-in-the-northern-hemisphere-and-climate-change-is-slashing-the-odds-of-more-heatwaves-100582 (2018).

  68. 68.

    Heatwave in northern Europe, summer 2018. World Weather Attribution https://www.worldweatherattribution.org/attribution-of-the-2018-heat-in-northern-europe/ (2018).

  69. 69.

    de la Hamaide, S., Devitt, P. & Hogan, M. Heatwave ravages European fields, sending wheat prices soaring. Reuters https://www.reuters.com/article/us-europe-wheat-harvest/heatwave-ravages-european-fields-sending-wheat-prices-soaring-idUSKBN1KN0L9 (2018).

  70. 70.

    Vaughan, A. UK summer ‘wind drought’ puts green revolution into reverse. The Guardian https://www.theguardian.com/environment/2018/aug/27/uk-summer-wind-drought-puts-green-revolution-into-reverse (2018).

  71. 71.

    Woollings, T. et al. Blocking and its response to climate change. Curr. Clim. Change Rep. 4, 287–300 (2018).

    Article  Google Scholar 

  72. 72.

    Schiemann, R., Demory, M.-E., Shaffrey, L. C., Strachan, J. & Vidale, P.-L. The resolution sensitivity of Northern Hemisphere blocking in four 25-km atmospheric global circulation models. J. Clim. 30, 337–358 (2017).

    ADS  Article  Google Scholar 

  73. 73.

    Green, J. S. A. The weather during July 1976: some dynamical considerations of the drought. Weather 32, 120–126 (1977).

    ADS  Article  Google Scholar 

  74. 74.

    Dawson, A. & Palmer, T. N. Simulating weather regimes: impact of model resolution and stochastic parameterization. Clim. Dyn. 44, 2177–2193 (2015).

    Article  Google Scholar 

  75. 75.

    Lorenz, E. N. Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963).

    ADS  Article  Google Scholar 

  76. 76.

    Kwasniok, F. Enhanced regime predictability in atmospheric low-order models due to stochastic forcing. Phil. Trans. R. Soc. A 372, 20130286 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  77. 77.

    Charney, J. G. & DeVore, J. G. Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci. 36, 1205–1216 (1979).

    ADS  Article  Google Scholar 

  78. 78.

    Kondrashov, D., Ide, K. & Ghil, M. Weather regimes and preferred transition paths in a three-level quasigeostrophic model. J. Atmos. Sci. 61, 568–587 (2004).

    ADS  MathSciNet  Article  Google Scholar 

  79. 79.

    Lorenz, E. N. in Predictability of Weather and Climate (eds Palmer, T. N. & Hagedorn, R.) 40–58 (Cambridge Univ. Press, 1996).

  80. 80.

    Christensen, H. M., Moroz, I. M. & Palmer, T. N. Simulating weather regimes: impact of stochastic and perturbed parameter schemes in a simple atmospheric model. Clim. Dyn. 44, 2195–2214 (2015).

    Article  Google Scholar 

  81. 81.

    Strommen, K. & Palmer, T. N. Signal and noise in regime systems: a hypothesis on the predictability of the North Atlantic Oscillation. Q. J. R. Meteorol. Soc. 145, 147–163 (2019).

    ADS  Article  Google Scholar 

  82. 82.

    Eade, R. et al. Do seasonal-to-decadal climate predictions underestimate the predictability of the real world?. Geophys. Res. Lett. 41, 5620–5628 (2014).

    ADS  Article  Google Scholar 

  83. 83.

    Palmer, T. N. Build high-resolution global climate models. Nature 515, 338–339 (2014).

    ADS  Article  Google Scholar 

  84. 84.

    Palmer, T. N. More reliable forecasts with less precise computations: a fast-track route to cloud-resolved weather and climate simulators. Phil. Trans. R. Soc. A 372, 20130391 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  85. 85.

    Váňa, F. et al. Single precision in weather forecasting models: an evaluation with the IFS. Mon. Weather Rev. 145, 495–502 (2017).

    ADS  Article  Google Scholar 

  86. 86.

    Dueben, P. D. & Palmer, T. N. Benchmark tests for numerical weather forecasts on inexact hardware. Mon. Weather Rev. 142, 3809–3829 (2014).

    ADS  Article  Google Scholar 

  87. 87.

    Hatfield, S., Subramanian, A., Palmer, T. & Düben, P. Improving weather forecast skill through reduced-precision data assimilation. Mon. Weather Rev. 146, 49–62 (2018).

    ADS  Article  Google Scholar 

  88. 88.

    Dawson, A., Düben, P. D., MacLeod, D. A. & Palmer, T. N. Reliable low precision simulations in land surface models. Clim. Dyn. 51, 2658–2666 (2018).

    Article  Google Scholar 

  89. 89.

    Jeffress, S., Düben, P. & Palmer, T. Bitwise efficiency in chaotic models. Proc. R. Soc. A 473, 20170144 (2017).

    ADS  Article  Google Scholar 

  90. 90.

    Thornes, T., Düben, P. & Palmer, T. A power law for reduced precision at small spatial scales: experiments with an SQG model. Q. J. R. Meteorol. Soc. 144, 1179–1188 (2018).

    ADS  Article  Google Scholar 

  91. 91.

    Chantry, M., Thornes, T. & Palmer, T. N. Scale-selective precision for weather and climate forecasting. Mon. Weather Rev. 147, 645–655 (2019).

    ADS  Article  Google Scholar 

  92. 92.

    Dueben, P. D., Russell, F. P., Niu, X., Luk, W. & Palmer, T. N. On the use of programmable hardware and reduced numerical precision in earth-system modeling. J. Adv. Model. Earth Syst. 7, 1393–1408 (2015).

    ADS  Article  Google Scholar 

  93. 93.

    Subramanian, A., Juricke, S., Dueben, P. & Palmer, T. N. A stochastic representation of sub-grid uncertainty for dynamical core development. Bull. Am. Meteorol. Soc. https://doi.org/10.1175/BAMS-D-17-0040.1 (2019).

    Article  Google Scholar 

  94. 94.

    Palem, K. V. Energy aware computing through probabilistic switching: a study of limits. IEEE Trans. Comput. 54, 1123–1137 (2005).

    Article  Google Scholar 

  95. 95.

    Palem, K. V. Inexactness and a future of computing. Phil. Trans. R. Soc. A 372, 20130281 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  96. 96.

    Palmer, T. N. Modelling: build imprecise supercomputers. Nature 526, 32–33 (2015).

    ADS  Article  Google Scholar 

  97. 97.

    Palmer, T. N. & O’Shea, M. Solving difficult problems creatively: a role for energy optimised deterministic/stochastic hybrid computing. Front. Comput. Neurosci. 9, 124 (2015).

    Article  Google Scholar 

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Acknowledgements

The author thanks H. Christensen for helpful input on and improvements to this paper. This work was supported by the ERC Advanced Grant ITHACA grant number DCR00620.

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Glossary

Albedo

A non-dimensional measure of the diffuse reflection of sunlight from a surface. More specifically, the ratio of radiosity to irradiance.

Biosphere

The regions of the Earth system occupied by living organisms.

Cryosphere

Those portions of Earth’s surface where water is in solid form.

Madden–Julian oscillation

A planetary-scale eastward-propagating oscillation in surface pressure (and related variables), primarily located in the tropics, with a periodicity of about 30–60 days.

Orographic

Related to the topography of mountains.

Outset

In dynamical systems theory, a saddle-point instability has both a stable and unstable manifold in state space, corresponding to the saddle point’s attractor and repeller, respectively. The outset corresponds to the unstable manifold.

Rossby waves

Planetary-scale wavelike disturbances in surface pressure (and related variables) whose existence and properties are dependent on the rotation of the Earth.

Skill

A measure of the practical value of weather forecasts. For probabilistic forecasts, the skill combines the reliability of the forecast probabilities and the sharpness of the forecast probability distributions.

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Palmer, T.N. Stochastic weather and climate models. Nat Rev Phys 1, 463–471 (2019). https://doi.org/10.1038/s42254-019-0062-2

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