• Baker, N. L., and R. Daley, 2000: Observation and background adjoint sensitivity in the adaptive observation-targeting problem. Quart. J. Roy. Meteor. Soc., 126, 14311454, https://doi.org/10.1002/qj.49712656511.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Barkmeijer, J., F. Bouttier, and M. Van Gijzen, 1998: Singular vectors and estimates of the analysis-error covariance metric. Quart. J. Roy. Meteor. Soc., 124, 16951713, https://doi.org/10.1002/qj.49712454916.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Barkmeijer, J., R. Buizza, T. N. Palmer, K. Puri, and J.-F. Mahfouf, 2001: Tropical singular vectors computed with linearized diabatic physic. Quart. J. Roy. Meteor. Soc., 127, 685708, https://doi.org/10.1002/qj.49712757221.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bejan, A., 2016: Advanced Engineering Thermodynamics. John Wiley & Sons, Inc., 740 pp.

  • Borderies, M., O. Caumont, J. Delanoë, V. Ducrocq, N. Fourrié, and P. Marquet, 2019: Impact of airborne cloud radar reflectivity data assimilation on kilometre-scale numerical weather prediction analyses and forecasts of heavy precipitation events. Nat. Hazards Earth Syst. Sci., 19, 907926, https://doi.org/10.5194/nhess-19-907-2019.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buizza, R., and T. N. Palmer, 1995: The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci., 52, 14341456, https://doi.org/10.1175/1520-0469(1995)052<1434:TSVSOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buizza, R., J. Tribbia, F. Molteni, and T. N. Palmer, 1993: Computation of optimal unstable structures for a numerical weather prediction model. Tellus, 45A, 388407, https://doi.org/10.3402/tellusa.v45i5.14901.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buizza, R., T. N. Palmer, J. Barkmeijer, R. Gelaro, and J.-F. Mahfouf, 1996: Singular vector, norms and large-scale condensation. Preprints, 11th Conf. on Numerical Weather Prediction, Norfolk, VA, Amer. Meteor. Soc., 50–52.

  • Cardinali, C., 2009: Monitoring the observation impact on the short-range forecast. Quart. J. Roy. Meteor. Soc., 135, 239250, https://doi.org/10.1002/qj.366.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chambon, P., L.-F. Meunier, F. Guillaume, J.-M. Piriou, R. Roca, and J.-F. Mahfouf, 2015: Investigating the impact of the water-vapour sounding observations from SAPHIR on board Megha-Tropiques for the ARPEGE global model. Quart. J. Roy. Meteor. Soc., 141, 17691779, https://doi.org/10.1002/qj.2478.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Côté, J., S. Gravel, A. Méthot, A. Patoine, M. Roch, and A. Staniforth, 1998a: The operational CMC-MRB global environmental multiscale (GEM) model. Part I: Design considerations and formulation. Mon. Wea. Rev., 126, 13731395, https://doi.org/10.1175/1520-0493(1998)126<1373:TOCMGE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Côté, J., J.-G. Desmarais, S. Gravel, A. Méthot, A. Patoine, M. Roch, and A. Staniforth, 1998b: The operational CMC-MRB global environmental multiscale (GEM) model. Part II: Results. Mon. Wea. Rev., 126, 13971418, https://doi.org/10.1175/1520-0493(1998)126<1397:TOCMGE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Courtier, P., 1987: Application du contrôle optimal à la prévision numérique en Météorologie (Application of the optimal control to the numerical forecast in meteorology). Ph.D. thesis, Paris-VI University, France, 275 pp.

  • Courtier, P., C. Freydier, J.-F. Geleyn, F. Rabier, and M. Rochas, 1991: The ARPEGE project at Meteo France. Seminar on Numerical Methods in Atmospheric Models, Vol. II, Reading, United Kingdom, ECMWF, 193–232.

  • Courtier, P., J.-N. Thépaut, and A. Hollingsworth, 1994: A strategy for operational implementation of 4D-Var, using an incremental approach. Quart. J. Roy. Meteor. Soc., 120, 13671387, https://doi.org/10.1002/qj.49712051912.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cover, T. M., and J. A. Thomas, 1991: Elements of Information Theory. John Wiley & Sons, Inc., 563 pp.

  • Derber, J., and F. Bouttier, 1999: A reformulation of the background error covariance in the ECMWF global data assimilation system. Tellus, 51A, 195221, https://doi.org/10.3402/tellusa.v51i2.12316.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Descamps, L., D. Ricard, A. Joly, and P. Arbogast, 2007: Is a real cyclogenesis case explained by generalized linear baroclinic instability? J. Atmos. Sci., 64, 42874308, https://doi.org/10.1175/2007JAS2292.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ehrendorfer, M., 2000: The total energy norm in a quasigeostrophic model. J. Atmos. Sci., 57, 34433451, https://doi.org/10.1175/1520-0469(2000)057<3443:NACTEN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ehrendorfer, M., and R. M. Errico, 1995: Mesoscale predictability and the spectrum of optimal perturbations. J. Atmos. Sci., 52, 34753500, https://doi.org/10.1175/1520-0469(1995)052<3475:MPATSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ehrendorfer, M., and J. Tribbia, 1997: Optimal prediction of forecast error covariances through singular vectors. J. Atmos. Sci., 54, 286313, https://doi.org/10.1175/1520-0469(1997)054<0286:OPOFEC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ehrendorfer, M., J. J. Tribbia, and R. M. Errico, 1995: Mesoscale predictability: An assessment through adjoint methods. Seminar on Predictability, Reading, United Kingdom, ECMWF, 157–183.

  • Ehrendorfer, M., R. M. Errico, and K. D. Reader, 1999: Singular-vector perturbation growth in a primitive equation model with moist physics. J. Atmos. Sci., 56, 16271648, https://doi.org/10.1175/1520-0469(1999)056<1627:SVPGIA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eriksson, K.-E., and K. Lindgren, 1987: Structural information in self-organizing systems. Phys. Scr., 35, 388397, https://doi.org/10.1088/0031-8949/35/3/026.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eriksson, K.-E., K. Lindgren, and B. Å. Månsson, 1987: Structure, Context, Complexity, Organization: Physical Aspects of Information and Value. World Scientific Publishing Co., 446 pp.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Errico, R. M., 2000: Interpretations of the total energy and rotational energy norms applied to determination of singular vectors. Quart. J. Roy. Meteor. Soc., 126, 15811599, https://doi.org/10.1256/smsqj.56602.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Errico, R. M., and M. Ehrendorfer, 1995: Moist singular vectors in a primitive-equation regional model. Preprints, 10th Conf. on Atmospheric and Oceanic Waves and Stability, Big Sky, MT, Amer. Meteor. Soc., 235–238.

  • Errico, R. M., K. D. Reader, and M. Ehrendorfer, 2004: Singular vectors for moisture-measuring norms. Quart. J. Roy. Meteor. Soc., 130, 963987, https://doi.org/10.1256/qj.02.227.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gauthier, P., M. Tanguay, S. Laroche, S. Pellerin, and J. Morneau, 2007: Extension of 3DVAR to 4DVAR: Implementation of 4DVAR at the Meteorological Service of Canada. Mon. Wea. Rev., 135, 23392354, https://doi.org/10.1175/MWR3394.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gelaro, R., and Coauthors, 2017: The Modern-Era Retrospective Analysis for Research and Applications, Version 2 (MERRA-2). J. Climate, 30, 54195454, https://doi.org/10.1175/JCLI-D-16-0758.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gelaro, R., R. H. Langland, S. Pellerin, and R. Todling, 2010: The THORPEX observation impact intercomparison experiment. Mon. Wea. Rev., 138, 40094025, https://doi.org/10.1175/2010MWR3393.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holdaway, D., R. Errico, R. Gelaro, and J. G. Kim, 2014: Inclusion of linearized moist physics in NASA’s Goddard earth observing system data assimilation tools. Mon. Wea. Rev., 142, 414433, https://doi.org/10.1175/MWR-D-13-00193.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holdaway, D., R. Errico, R. Gelaro, J. G. Kim, and R. Mahajan, 2015: A linearized prognostic cloud scheme in NASA’s Goddard earth observing system data assimilation tools. Mon. Wea. Rev., 143, 41984219, https://doi.org/10.1175/MWR-D-15-0037.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Honerkamp, J., 1998: Statistical Physics: An Advanced Approach with Applications. Springer-Verlag, 410 pp.

  • Janisková, M., and C. Cardinali, 2017: On the impact of the diabatic component in the Forecast Sensitivity Observation impact diagnostics (ECMWF Tech. Memo. 786). Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications, S. K. Park and L. Xu, Eds., Vol. III., Springer International Publishing, 483–511, https://doi.org/10.1007/978-3-319-43415-5_2.

    • Crossref
    • Export Citation
  • Joly, A., 1995: The stability of steady fronts and the adjoint method: Nonmodal frontal waves. J. Atmos. Sci., 52, 30823108, https://doi.org/10.1175/1520-0469(1995)052<3082:TSOSFA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Joly, A., and A. J. Thorpe, 1991: The stability of time-dependent flows: An application to fronts in developing baroclinic waves. J. Atmos. Sci., 48, 163183, https://doi.org/10.1175/1520-0469(1991)048<0163:TSOTDF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Karbou, F., E. Gérard, and F. Rabier, 2010: Global 4DVAR assimilation and forecast experiments using AMSU observations over land. Part I: Impacts of various land surface emissivity parameterizations. Wea. Forecasting, 25, 519, https://doi.org/10.1175/2009WAF2222243.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Karlsson, S., 1990: Energy, entropy and exergy in the atmosphere. Ph.D. thesis, Institute of Physical Resource Theory, Chalmers University of Technology. Göteborg, Sweden, 121 pp., https://core.ac.uk/download/pdf/70599863.pdf.

  • Kleeman, R., 2002: Measuring dynamical prediction utility using relative entropy. J. Atmos. Sci., 59, 20572072, https://doi.org/10.1175/1520-0469(2002)059<2057:MDPUUR>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kleist, D. T., D. F. Parrish, J. C. Derber, R. Treadon, W.-S. Wu, and S. Lord, 2009: Introduction of the GSI into the NCEP global data assimilation system. Wea. Forecasting, 24, 16911705, https://doi.org/10.1175/2009WAF2222201.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kullback, S., 1959: Information Theory and Statistics. John Wiley & Sons, Inc., 409 pp.

  • Kullback, S., and R. A. Leibler, 1951: On information and sufficiency. Ann. Math. Stat., 22, 7986, https://doi.org/10.1214/aoms/1177729694.

  • Langland, R., and N. Baker, 2004: Estimation of observation impact using the NRL atmospheric variational data assimilation adjoint system. Tellus, 56A, 189201, https://doi.org/10.3402/tellusa.v56i3.14413.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1955: Available potential energy and the maintenance of the general circulation. Tellus, 7A, 157167, https://doi.org/10.3402/tellusa.v7i2.8796.

    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1978: Available energy and the maintenance of a moist circulation. Tellus, 30A, 1531, https://doi.org/10.3402/tellusa.v30i1.10308.

    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1979: Numerical evaluation of moist available energy. Tellus, 31A, 230235, https://doi.org/10.3402/tellusa.v31i3.10429.

    • Search Google Scholar
    • Export Citation
  • Mahfouf, J.-F., and R. Buizza, 1996: On the inclusion of physical processes in linear forward and adjoint models: Impact of large-scale condensation on singular vectors. ECMWF Newsletter Number, No. 72, ECMWF, Reading, United Kingdom, 2–6, https://www.ecmwf.int/node/14652.

  • Mahfouf, J.-F., and B. Bilodeau, 2007: Adjoint sensitivity of surface precipitation to initial conditions. Mon. Wea. Rev., 135, 28792896, https://doi.org/10.1175/MWR3439.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mahfouf, J.-F., R. Buizza, and R. M. Errico, 1996: Strategy for including physical processes in the ECMWF variational data assimilation system. Workshop on Non-linear Aspects of Data Assimilation, Reading, United Kingdom, ECMWF, 595–632, https://www.ecmwf.int/node/10924.

  • Majda, A. J., R. Kleeman, and D. Cai, 2002: A mathematical framework for quantifying predictability through relative entropy. Methods Appl. Anal., 9, 425444, https://doi.org/10.4310/MAA.2002.V9.N3.A8.

    • Search Google Scholar
    • Export Citation
  • Margules, M., 1910: The mechanical equivalent of any given distribution of atmospheric pressure, and the maintenance of a given difference in pressure (Translation by C. Abbe of a lecture read at the meeting of the imperial academy of science, Vienna, July, 11, 1901). Smithson. Misc. Collect., 51, 501532, https://www3.nd.edu/~powers/ame.20231/gibbs1873b.pdf.

    • Search Google Scholar
    • Export Citation
  • Marquet, P., 1991: On the concept of exergy and available enthalpy: Application to atmospheric energetics. Quart. J. Roy. Meteor. Soc., 117, 449475, https://doi.org/10.1002/qj.49711749903.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marquet, P., 1993: Exergy in meteorology: Definition and properties of moist available enthalpy. Quart. J. Roy. Meteor. Soc., 119, 567590, https://doi.org/10.1002/qj.49711951112.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marquet, P., 2003: The available-enthalpy cycle. I: Introduction and basic equations. Quart. J. Roy. Meteor. Soc., 129, 24452466, https://doi.org/10.1256/qj.01.62.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marquet, P., and T. Dauhut, 2018: Reply to “Comments on ‘A third-law isentropic analysis of a simulated hurricane.”’ J. Atmos. Sci., 75, 37353747, https://doi.org/10.1175/JAS-D-18-0126.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Palmer, T. N., R. Gelaro, J. Barkmeijer, and R. Buizza, 1998: Singular-vectors, metrics, and adaptative observations. J. Atmos. Sci., 55, 633653, https://doi.org/10.1175/1520-0469(1998)055<0633:SVMAAO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pearce, R. P., 1978: On the concept of available potential energy. Quart. J. Roy. Meteor. Soc., 104, 737755, https://doi.org/10.1002/qj.49710444115.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Procaccia, I., and R. D. Levine, 1976: Potential work: A statistical-mechanical approach for systems in disequilibrium. J. Chem. Phys., 65, 33573364, https://doi.org/10.1063/1.433482.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Putman, W., 2007: Development of the finite-volume dynamical core on the cubed-sphere. Ph.D. thesis, Florida State University, 91 pp., https://diginole.lib.fsu.edu/islandora/object/fsu%3A168667.

  • Rabier, F., E. Klinker, P. Courtier, and A. Hollingsworth, 1996: Sensitivity of forecast errors to initial conditions. Quart. J. Roy. Meteor. Soc., 122, 121150, https://doi.org/10.1002/qj.49712252906.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rivière, O., G. Lapeyre, and O. Talagrand, 2009: A novel technique for nonlinear sensitivity analysis: Application to moist predictability. Quart. J. Roy. Meteor. Soc., 135, 15201537, https://doi.org/10.1002/qj.460.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Shannon, C. E., 1948: A mathematical theory of communication. Bell Syst. Tech. J., 27, 379423, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Szargut, J., and T. Styrylska, 1969: Die exergetische Analyse von Prozessen der feuchten Luft (An exergetic analysis of processes for damp air). Heiz.-Lüft.-Haustechn., 20 (5), 173178.

    • Search Google Scholar
    • Export Citation
  • Talagrand, O., 1981: A study of the dynamics of four-dimensional data assimilation. Tellus, 33A, 4360, https://doi.org/10.3402/tellusa.v33i1.10693.

    • Search Google Scholar
    • Export Citation
  • Thépaut, J.-N., and P. Courtier, 1991: Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model. Quart. J. Roy. Meteor. Soc., 117, 12251254, https://doi.org/10.1002/qj.49711750206.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thomson, W., 1853: On the restoration of mechanical energy from an unequally heated space. Philos. Mag., 5 (30), 102105, https://doi.org/10.1080/14786445308562743.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Trémolet, Y., 2008: Computation of observation sensitivity and observation impact in incremental variational data assimilation. Tellus, 60A, 964978, https://doi.org/10.1111/j.1600-0870.2008.00349.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wu, W.-S., R. J. Purser, and D. F. Parrish, 2002: Three-dimensional variational analysis with spatially inhomogeneous covariances. Mon. Wea. Rev., 130, 29052916, https://doi.org/10.1175/1520-0493(2002)130<2905:TDVAWS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Xu, Q., 2007: Measuring information content from observations for data assimilation: Relative entropy versus Shannon entropy difference. Tellus, 59A, 198209, https://doi.org/10.1111/j.1600-0870.2006.00222.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zadra, A., M. Buenner, S. Laroche, and J.-F. Mahfouf, 2004: Impact of the GEM model simplified physics on extratropical singular vectors. Quart. J. Roy. Meteor. Soc., 130, 25412569, https://doi.org/10.1256/qj.03.208.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • View in gallery

    The seasonal averages of the RMS of analysis increments for water Sq (g kg−1) are computed for ARPEGE outputs every 6 h and plotted in latitude–pressure sections for (a) winter (DJF) and (c) summer (JJA). The corresponding seasonal averages of the exergy SqV term Vq given by (18) are plotted for (b) winter (DJF) and (d) summer (JJA).

  • View in gallery

    As in Figs. 1b and 1d, but for the DJF and JJA seasonal average of the MB07 water term Vq2 given by (9).

  • View in gallery

    Vertical profiles of horizontal mean of seasonal averages computed from ARPEGE outputs every 6 h and for three latitude domains: (a),(d) southern extratropical midlatitudes from −60° to −30°; (b),(e) tropical latitudes from −30° to +30°; (c),(f) northern extratropical midlatitudes from +30° to +60°. The vertical profiles of the DJF means are plotted in (a)–(c); those for the JJA means in (d)–(f). The E99 water terms Vq1 (purple dotted lines) are given by (8) with wq = 1.0 and wq = 0.3. The exergy water term Vq (red dashed lines) is given by (18). The MB07 water term Vq2 (black dashed lines) is given by (10). The RMS of analysis increments in water vapor is Sq (blue solid lines).

  • View in gallery

    As in Figs. 1a and 1b, but for temperature (K).

  • View in gallery

    The same ARPEGE seasonal mean (DJF) as in Fig. 3a but for temperature (K) and for the RMS of analysis increments ST (solid blue), the E99 term VT10.77 (dotted purple), and the exergy term VT (red dashed).

  • View in gallery

    Latitude–pressure sections and vertical profiles of horizontal averages for the water term for 26 Dec 2002: (a),(c),(e) GEM-CMC; (b),(d),(f) GEOS-MERRA-2. (a),(b) Sections of the RMS of analysis increments Sq (g kg−1). (c),(d) Sections of exergy norms Vq (g kg−1). (e),(f) Vertical profiles of horizontal averages of E99 (dotted purple), MB07 (dashed black), and exergy (dashed red) norms and the analysis increments Sq (solid blue).

  • View in gallery

    The dimensionless exergy weighting factor wq(qυ) given by (19) plotted with qυ in ordinates.

  • View in gallery

    The dimensionless exergy weighting factor wq(z) given by (19) for the vertical profile of average values qυ¯(p) of the GEM-CMC dataset used to plot the Fig. 6c.

  • View in gallery

    The 24-h forecast observation impacts per analysis for each observation system. Comparisons of (i) the dry norm (white); (ii) the moist norm E99 with wq = 0.3 (gray), namely the same as Fig. 9 in H14; and (iii) the moist exergy norm (dark).

  • View in gallery

    The two functions F(X) = X − ln(1 + X) and X2/2 plotted for −1 > X > +2.5.

  • View in gallery

    The two functions H(X) = (1 + X) ln(1 + X) − X and X2/2 plotted for −1 < X < +2.5.

  • View in gallery

    The separation of the flow into an uneven basic state (x¯, solid lines) plus the eddies (dashed lines), defined by xxx¯. The x term stands for the meteorological variables T, p, Zυ, or rυ, also u and υ.

All Time Past Year Past 30 Days
Abstract Views 16 0 0
Full Text Views 375 273 24
PDF Downloads 196 109 8

Definition of the Moist-Air Exergy Norm: A Comparison with Existing “Moist Energy Norms”

View More View Less
  • 1 Météo-France, CNRM-CNRS UMR-3589, Toulouse, France
  • | 2 Global Modeling and Assimilation Office, NASA Goddard Space Flight Center, Greenbelt, Maryland, and University Corporation for Atmospheric Research, Boulder, Colorado
© Get Permissions
Free access

Abstract

This study presents a new formulation for the norms and scalar products used in tangent linear or adjoint models to determine forecast errors and sensitivity to observations and to calculate singular vectors. The new norm is derived from the concept of moist-air available enthalpy, which is one of the availability functions referred to as exergy in general thermodynamics. It is shown that the sum of the kinetic energy and the moist-air available enthalpy can be used to define a new moist-air squared norm that is quadratic in 1) wind components, 2) temperature, 3) surface pressure, and 4) water vapor content. Preliminary numerical applications are performed to show that the new weighting factors for temperature and water vapor are significantly different from those used in observation impact studies, and are in better agreement with observed analysis increments. These numerical applications confirm that the weighting factors for water vapor and temperature exhibit a large increase with height (by several orders of magnitude) and a minimum in the midtroposphere, respectively.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Pascal Marquet, pascal.marquet@meteo.fr

Abstract

This study presents a new formulation for the norms and scalar products used in tangent linear or adjoint models to determine forecast errors and sensitivity to observations and to calculate singular vectors. The new norm is derived from the concept of moist-air available enthalpy, which is one of the availability functions referred to as exergy in general thermodynamics. It is shown that the sum of the kinetic energy and the moist-air available enthalpy can be used to define a new moist-air squared norm that is quadratic in 1) wind components, 2) temperature, 3) surface pressure, and 4) water vapor content. Preliminary numerical applications are performed to show that the new weighting factors for temperature and water vapor are significantly different from those used in observation impact studies, and are in better agreement with observed analysis increments. These numerical applications confirm that the weighting factors for water vapor and temperature exhibit a large increase with height (by several orders of magnitude) and a minimum in the midtroposphere, respectively.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Pascal Marquet, pascal.marquet@meteo.fr

1. Introduction

Several inner products, based on “energy” squared norms, have been used in four-dimensional variational assimilation tools to minimize cost functions (Talagrand 1981; Courtier 1987; Thépaut and Courtier 1991). It was supposed that the energy corresponding to observational errors could be distributed equally among these different basic prognostic fields. Inner products based on these energy squared norms are used to define dry semi-implicit operators and dry normal modes of GCMs or NWP models, as long as they are invariant by the linear set of primitive equations (Thépaut and Courtier 1991).

Here, the term energy means that the sum of quadratic terms is considered for perturbations of the wind components (u′)2 + (υ′)2, temperature (T′)2, and surface pressure (ps)2 or [ln(ps)]2 (see appendix A for the list of symbols). Moist-air generalizations of the energy squared norm have been suggested by Courtier (1987, hereafter C87), Ehrendorfer et al. (1999, hereafter E99), or Mahfouf and Bilodeau (2007, hereafter MB07), among others, by including the water vapor content via an additional quadratic term (qυ)2.

The same inner products and norms are currently used for computing dry or moist singular vectors and for determining forecast errors or sensitivity to observations based on tangent linear and adjoint models (Buizza and Palmer 1995; Palmer et al. 1998; Mahfouf and Bilodeau 2007; Janisková and Cardinali 2017).

However, all these norms suffer from a lack of consistency with physical relationships in thermodynamics, because (i) these energy squared norms are not based on the standard definition of energy as expressed in general thermodynamics; (ii) the use of the squared norm for water including the quadratic term (qυ)2 is poorly justified; (iii) these definitions are not unique, with for instance an arbitrary tuning parameters, which is often left undetermined for the water component.

Ideally, all these quadratic terms should be derived from some general laws of physics. This is true for the average of the kinetic energy (u2¯+υ2¯)/2, which is the sum of the terms (u¯)2/2+(υ¯)2/2 computed with the mean state of wind plus the terms (u)2¯/2+(υ)2¯/2 computed with the perturbations of wind. This result is true if u=uu¯ and υ=υυ¯, leading to (u)¯=(υ)¯=0. The squared norm for the wind components is computed in assimilation, singular vector and sensitivity studies with (u)2¯/2+(υ)2¯/2, where for instance u′ and υ′ are the unbiased departures between analyses and short-range forecasts.

In contrast, the usual temperature component of the squared norm (T2)¯/2=(T¯)2/2+(T)2¯/2 cannot be derived from the general definition of the energy and the first law of thermodynamics. Indeed, the dry-air internal energy or enthalpy varies linearly with temperature, with hcpdT for the enthalpy up to constant reference values. Consequently, the true energy and enthalpy cannot generate quadratic terms, due to h¯=cpdT¯=0.

To derive quadratic squared norms in both wind components and temperature, a relevant method might be based on the study of the sum of the kinetic energy and “a form of the Available Potential Energy” (APE) of Lorenz (1955). This method is chosen in Talagrand (1981), the old ARPEGE-IFS documentation (1989, unpublished), Joly and Thorpe (1991), Joly (1995), Ehrendorfer and Errico (1995), Errico and Ehrendorfer (1995), E99, Ehrendorfer (2000), Errico (2000), and Descamps et al. (2007).

In these studies, the specific value of the approximate APE is written as (T)2¯/(2Γ¯), where both the perturbation of temperature T=TT¯ and the stability parameter Γ¯ depend on T¯, where
Γ¯=T¯cpdpRdT¯p.
The calculations of Γ¯ are explicitly performed in Talagrand (1981) and Descamps et al. (2007) by using a standard atmosphere for defining a reference profile T¯(p) that varies with height.
On the other hand, the stability parameter is often computed by using a constant reference value for T¯, which is denoted by Tr or an equivalent. This leads to T¯/p=0 in (1) and to Γ¯=Tr/cpd. This is an explanation for the quadratic term:
(T)2¯2Γ¯=cpd(T)2¯2Tr,
which is retained in almost all present formulations of the temperature component of norms. A constant value Tr is used in Courtier (1987), Thépaut and Courtier (1991), Buizza et al. (1993), Ehrendorfer and Errico (1995), Buizza et al. (1996), Mahfouf and Buizza (1996), E99, Errico (2000), Barkmeijer et al. (2001), Zadra et al. (2004), Errico et al. (2004), Mahfouf and Bilodeau (2007), Rivière et al. (2009), Holdaway et al. (2014), Janisková and Cardinali (2017), among others.

However, it is worth noting that the use of a constant value Tr for T¯(p) in (2) is not compatible with the stability term (1) that appears in the formulation of APE expressed with pressure coordinate, where T¯ must be defined as the isobaric average of T according to Lorenz (1955). No other definition is allowed, and the use of a constant temperature Tr makes the theory incompatible with that of Lorenz and weakens the theoretical basis for present formulations of the norm for temperature.

All temperature, pressure and water vapor components of existing squared norms correspond to the quadratic terms (T)2, (ps)2 or [ln(ps)]2 and (qυ)2. It is thus tempting to consider these components as forming a “total energy” squared norm. However, it is explained in Errico (2000) that these squared norms are not based on clear thermodynamic definitions nor on any obvious energy norm of pressure or moisture (“Although it is called a measure of the energy, it has not been demonstrated that it is indeed such in the contexts to which it has been applied. The fact that it has units of energy per unit mass does not by itself qualify it as a measure of energy”). Moreover, the moist-air generalization of the APE by Lorenz (1978, 1979) does not lead to any easy-to-use analytical formulation that could replace (T)2¯/(2Γ¯) with a moist-air version for Γ¯. This means that the APE approach cannot be easily generalized to moist air.

Therefore, other ideas had to be tested in order to solve the problems described so far. Since the temperature component (2) is presently derived from an approximate version of the APE of Lorenz, which was improved by Pearce (1978) and Marquet (1991) for the dry air, and then by Marquet (1993, hereafter M93) for the moist air, this article examines the possibility of deriving the quadratic terms in temperature, pressure and water content from a general principle based on the concept of “moist available enthalpy” defined in M93.

The available enthalpy is one form of what is known as “exergy” in general thermodynamics. This new exergy norm is used in Borderies et al. (2019) to measure the relative impact of the assimilation of observations on the analysis and short-term forecasts for the French AROME model, with a large impact of the new water-content quadratic term. Indeed, the weighting factors of the exergy norm are significantly different from those used up to now in dry and moist squared norms, in particular by several orders of magnitude for the water content.

To achieve some numerical validation of the theoretical formulations for the exergy norm, the same comparisons of the squared norms with inverse analysis increment estimates are made as in MB07.

The motivations for these comparisons can be found in Errico et al. (2004), where a moist norm was used with weights “proportional to estimates of the variances of analysis uncertainty”. It was also explained in Barkmeijer et al. (2001) that “in the case of forecast-error covariance prediction, a norm at initial time based on the analysis-error covariance matrix is the more appropriate” (Ehrendorfer and Tribbia 1997; Palmer et al. 1998; Barkmeijer et al. 1998). At that time, “the analysis-error covariance metric became the reciprocal of the total-energy metric currently used at ECMWF to compute singular vectors for the EPS” (Barkmeijer et al. 1998). And “a specific-humidity norm based on error variances” was experimented by Derber and Bouttier (1999) at ECMWF, leading to a specific-humidity norm defined in Barkmeijer et al. (2001) from the ECMWF “averaged error variances for qυ,” with a strong decrease of this norm above 500 hPa, a property that has remained unexplained until now.

This paper is organized as follows. Existing moist-air squared norms are recalled in section 2a. Section 2b presents some theoretical motivations for the use of exergy functions based on the concepts of relative entropy and Kullback–Leibler divergence. The derivations of the moist-air available-enthalpy are conducted in appendixes BG and the corresponding quadratic approximate squared norm components are shown in section 2c for temperature, pressure, and water. The datasets from the Canadian Meteorological Centre (CMC), the NASA Goddard Earth Observing System (GEOS), and the French ARPEGE models are described in section 3. These datasets are used to compare the norm components for water and temperature with the root-mean-square (RMS) of analysis increments, with cross sections and vertical profiles shown in sections 4ac for the three models, leading to an explanation of the decrease with height of the water vapor exergy terms described in section 4d. Forecast observation impacts are described in section 4e for the GEOS model. Conclusions are drawn in section 5.

2. Theoretical considerations

a. Existing moist-air energy norms

A moist squared norm is defined in E99 by
NE99=(u)2+(υ)22dmΣ+RdTrgpr(ps)22dΣΣ+cpdTr(T)22dmΣ+wq(z)(Lυ)2cpdTr(qυ)22dmΣ.
The state vector is represented by the local departure from mean values of basic quantities, denoted by u′, υ′, T′, ps, and qυ. The differential mass dm = ρdτ is equal to dpdΣ/g, where Σ is the horizontal surface area. The volume integrals over dm/Σ and the surface integral over dΣ/Σ represent energies per unit of horizontal area, all expressed in units of J m−2. The pressure component is expressed in E99 as a volume integral of RdTr(ps)2/(2pr2), but the expressions are equivalent providing that ps¯pr.
The surface pressure contribution of the squared norm is often expressed differently, in terms of the logarithm of surface pressure, leading to
RdTrprg[{ln(ps)}]22dΣΣ.
This formalism is retained in C87, Thépaut and Courtier (1991), Buizza et al. (1993), Buizza and Palmer (1995), Rabier et al. (1996), Palmer et al. (1998), and Errico (2000).

The two formalisms using the surface pressure or its logarithm are nearly equivalent, providing that ps¯pr. Indeed, the departure term must be computed as {ln(ps)}=ln(ps)ln(ps)¯ in (4) and the perturbation of pressure is equal to ps=psps¯ in (3), leading to {ln(ps)}=ln(1+ps/ps¯)ln(1+ps/ps¯)¯ps/ps¯ up to small higher-order terms.

The justification for the last integral of (3) depending on the variance of water vapor content can be found in Ehrendorfer et al. (1995), Buizza et al. (1996), Mahfouf et al. (1996) and E99. The water contribution of the squared norm is derived from the temperature contribution cpd(T′)2/(2Tr) with the additional hypothesis that changes of temperature and moisture are related by cpdTLυqυ, namely by assuming a conservation of the moist static energy cpdT+Lυqυ+ϕ at constant height for all moist (condensation) process. A similar quadratic term was suggested in C87, where two scale factors for height (Hr) and water content (Qr) were defined, leading to the equivalent formulation gHr(qυ)2/(Qr)2.

The question addressed in E99 is the relevance of that special formulation for the water contribution. Due to the uncertainty in the assumption cpdT+Lυqυ0 (particularly in frequently undersaturated moist areas without condensation processes), an additional relative weight wq(z) (also denoted by w2 or ϵ, depending on papers) is added in the last integral of (3). The effects of making this relative weight larger or smaller than the standard value 1 are discussed in E99 and Barkmeijer et al. (2001), where wq(z) may increase with height in the upper troposphere and in the stratosphere.

An alternative definition of the water contribution of the squared norm is proposed in MB07 by replacing the assumption of conservation of perturbed moist static energy by a conservation of relative humidity approximated by qυ/qsw. This assumption is expected to be realistic in cloudy areas where relative humidity reaches 100%, however, it may not be realistic in frequently undersaturated moist areas. The constraint of zero departure (at constant pressure) in the quantity qυ/qsw/(T, p) corresponds to qυ=(Γ¯q)T, where Γ¯q=qυ¯ln(qsw¯)/T. The alternative contribution proposed in MB07 can be written as
cpdTr1(Γ¯q)2(qυ)22dmΣ.
MB07 found that this revised formulation for the water component of the norm better match the RMS of the analysis increments than the E99 norm. Indeed, the MB07 formulation better reflects the typical size of perturbations produced by data assimilation systems and (5) accounts for the exponential decrease of specific humidity with altitude, leading to much smaller absolute errors than with the original constant contribution in the last integral of (3). This result agrees with the increase of wq(z) with altitude considered in Zadra et al. (2004) in moist singular vector computations. The aim was to suppress the impact of humidity perturbations in the stratosphere according to the results of Buizza et al. (1996) and E99, who showed that for increasing wq the contribution of the dry fields dominates initially, whereas the contribution of moisture dominates at the final time (and vice versa when wq is smaller).
According to Errico et al. (2004) and MB07, the gridpoint discretization of either (3) or (5) can be written as the inverse variance weighted squared norm:
ijk[(uijk)2Vu+(υijk)2Vυ+(Tijk)2(VT1)jk]ωijΔσk+ij[(ps)ijk2(Vp1)jk]ωij+ijk[(qυ)ijk2(Vq1)jk]ωijΔσk,
where Δσk is the thickness of the layer k in the σ vertical coordinate and ωij is the fractional coverage of the model grid box defined by the zonal (i) and meridional (j) indices.

The weighting factors Vu, Vυ, (VT1)jk, (Vp1)j, and (Vq1)jk will hereafter be referred to as “V terms.” They are interpreted as variances of analysis errors in Errico et al. (2004) and MB07. The indices j and k mean that temperature, surface pressure and water variances can a priori depend on latitude (j) and/or altitude (k).

From (3) and (6), the V terms in E99 can be written as
Vu=Vυ=2=V0,VT1=V0Trcpd=V0(Tr)2cpdTr,
Vp1=V0(pr)2RdTr,(Vq1)k=V0wq(z)cpdTr(LυQr)2(Qr)2.
The four terms Vu, Vυ, VT1, and Vp1 are all constant, whereas (Vq1)k may depend on altitude for water, via the arbitrary weight wq(z).

All terms in parentheses in (6) are dimensionless in Errico et al. (2004) and MB07, where the dimensions of the square root of (VT1)jk, (Vp1)j, and (Vq1)jk are K, hPa, and kg kg−1, respectively. The square root of these V terms will be called “SqV terms” hereafter. The dimensionless characteristic of (6) can be explained by first multiplying all terms of (3) by the dimensionless value 2, and then by dividing all terms by the same energy term V0 = 2 J kg−1. Therefore, the dimensions of cpdTr, RdTr, and LυQr are same as the one of Vu = Vυ = V0, namely in units of m2 s−2 or J kg−1. The value of the dummy specific content Qr has no impact in (8); it is introduced to highlight the relevant dimension of kg2 kg−2 for (Vq1)jk.

The definition (5) proposed by MB07 corresponds to
(Vq2)jk=V0Trcpd(T¯)2(T¯qsw¯qsw¯T)2(qυ¯)2,
(Vq2)jkV0Trcpd[Lυqυ¯Rυ(T¯)2]2.
From (9), (Vq2)jk is expressed in kg2 kg−2, because cpd(T¯)2/Tr has the same dimension as V0. This means that the dimension of the square root of (Vq2)jk is the same as the specific content qυ¯, which is expressed in kg kg−1 and, from (10), varies with altitude via the ratio of the average terms qυ¯ and (T¯)2.

b. Relative entropy, exergy, and available enthalpy

Due to the uncertainty and plurality in VT1, Vq1, or Vq2 defined in E99 or MB07, and due to the arbitrary values for wq(z), it is necessary to find a more general and comprehensive “measure,” “norm” or “distance” between a perturbed thermodynamic state defined by (T2, qυ2, ps2) and a reference one defined by (T1, qυ1, ps1).

It is explained in section 3 of Marquet and Dauhut (2018) that this distance can be measured by the quantity referred to as “relative entropy” by Shannon (1948) and then defined in Kullback and Leibler (1951) and Kullback (1959) by
K(xy)=j=1nxjlog(xj/yj),
where the xjs represent a real state (x) and the yjs represent a reference state (y) of the system (see Cover and Thomas 1991).1

This Kullback–Leibler divergence K is usually interpreted as being a nonsymmetric measure of how much the xjs deviate from the yjs. It also represents the “gain in information” of the state characterized by the distribution (xj) with respect to the equilibrium distribution (yj). Therefore, it is unclear whether K corresponds to the measure or the distance between the two thermodynamic states (T2, qυ2, ps2) and (T1, qυ1, ps1).

The main difficulty lies in determining the xjs and the yjs that correspond to these two thermodynamic states. Moreover, the relative entropy K is clearly different from the entropy s(x)=j=1nxjlog(xj) of Shannon (1948), with a change of sign and another reference state yj included in (11). However, it is possible to show that the macroscopic value of K roughly corresponds to the free energy function eiTrs, which is different from the entropy s because it depends on the internal energy ei and a reference temperature Tr. More precisely, it is shown for instance in Procaccia and Levine (1976), Eriksson and Lindgren (1987), and Karlsson (1990) that the exergy of moist air can be computed by the “available energy” function ae = kBTr K, with K(x||y) given by (11). This function ae can be written in terms of the local atmospheric variables (p, T, qn), leading to
ae=(eieir)+pr(ααr)Tr(ssr)nμrn(qnqrn),
where the subscript “r” denotes a reference state and where the sum over “n” represents the dry air, water vapor, liquid water and ice species. The specific volume is α = 1/ρ and the specific contents qn are multiplied by the reference Gibbs functions μrn = hrnTr srn. The quantity ae given by (12) is called “maximum available work from a nonflow system” by Bejan (2016, Eq. 5.12) for system at rest reaching a pressure equilibrium with the environment (the laboratory). The last sum over n in (12) is called “chemical exergy” by Bejan, while the other terms form the “nonflow exergy.”
The sum of the terms (eieir) and −pr(ααr) in (12) must be replaced by the difference in specific enthalpy (hhr) to form the “thermomechanical and chemical flow exergy” defined in Bejan (2016, Eq. 5.25). It is the same available enthalpy function as that studied in Marquet (1991) and M93 and corresponding to (B1), with all other terms of (12) remaining the same, leading to
am=(hhr)Tr(ssr)nμrn(qnqrn).
The use of the specific enthalpy h to replace the internal energy is motivated by the natural application of h to the flowing moist-air atmosphere. No hypothesis is made from this point of view, since the use of enthalpy does not impose movements that would be made “at constant pressure.” The change in the variable h = ei + p/ρ is simply mathematical, with no underlying physical assumptions. One of the interests of the introduction of the enthalpy h is the existence of the Bernoulli function h + gz + (u2 + υ2)/2, which is constant during stationary, adiabatic and frictionless motions, with a similar Bernoulli’s law derived in M93 for am + gz + (u2 + υ2)/2.

The flow exergy am given by (13) ensures the definition of the aforementioned general distance between a perturbed atmospheric state and a reference one. Indeed, since the available enthalpy is the maximum work (or energy) that a system can deliver when passing from a reference state to the real state, this work is produced by transformations from different forms of energy to other forms of energy.

In particular, it is shown in M93 that a Bernoulli equation exists and that the sum am(T, p, qυ, ql, qi) + (u2 + υ2)/2 + ϕ is conserved along any streamline of an adiabatic frictionless and reversible steady flow of a closed parcel of moist air. This means that the conversions between the potential energy, the kinetic energy and the temperature, pressure and water components of am(T, p, qυ, ql, qi) given by (13) can be evaluated with the weighting factors VT, Vp, and Vq, ensuring relevant thermodynamic transformations of energy from one form to another.

c. The new moist-air available-enthalpy norm

The three components of the squared norm based on the M93 exergy function given by (13) are derived in appendixes BG. They can be written in terms of the square of the perturbations of temperature (G4), surface pressure (G11)(G13), and water vapor (G18), leading to
NT=[cpdTr(T¯)2](T)22dmΣ,
Np=RdTrgps¯(ps)2¯2=[RdTr(ps¯)2](ps)22dmΣ,
Nυ=[RdTr(r0rυ¯)](rυ)22dmΣ.
The new V terms corresponding to (7)(10) for temperature, pressure, and water content can be written as
(VT)jk=V0Trcpd(T¯Tr)2,(Vp)j=V0pr2RdTr(ps¯pr)2,
(Vq)jk=V0r0rυ¯RdTr=prererV0rrrυ¯RdTr=V0rυ¯RυTr.
From the first formulation in (18), (Vq)jk is independent of rr. The last formulation in (18) is obtained with Rυ = Rd/r0 and r0 = rr (prer)/er ≈ 622 g kg−1, where r0 is proportional to the reference mixing ratio rr. This shows that the dimensions of (Vq)jk and of rrrυ¯ are both kg2 kg−2, since V0 = 2 m2 s−2 and RdTr have the same dimension. Therefore, the square root of (Vq)jk has the dimension of a mixing ratio, as expected.

From (8) and (17) the pressure V terms Vp1 and (Vp)j may be close to each other if prps¯1000hPa, with (Vp)j only depending on ps¯ and being independent on pr.

Differently, the temperature and water V terms can differ significantly because T¯ and rυ¯ vary with height. This is especially true for (Vq)jk since rυ¯ may vary by three orders of magnitude from the surface to the stratosphere.

The comparison of (18) with (8) allows a computation of the unknown dimensionless weighting factor wq(z) in E99, leading to
wq(z)=cpdRυ(Tr)2(Lυ)21rυ¯(z),
wq(z)=(cpdTr)(RdTr)(Lυrr)2(erprer)rrrυ¯(z),
where Rυ=(Rder)/[(prer)rr] is used to derive the formulation (20), which better shows the dimensionless feature due to the compensation of the terms cpdTr and RdTr with Lυrr, also of er with prer and of rr with rυ.
The exergy weighting factor (20) explains the expected behavior for wq(z), which increases with height for decreasing values of rυ¯(z). A similar decrease holds with the MB07 value derived from the comparison of the constant relative humidity V term in (10) with the constant MSE V term in (8), leading to
wq2(z)(cpdTr)2(RdTr)2(Lυrr)4(erprer)2[rrqυ¯(z)]2.
A comparison of (21) with (20) shows that wq2 ≈ (wq)2 because rυ¯qυ¯. Therefore, the MB07 value is approximately the square of the available enthalpy value, leading to an enhanced variation of wq2(z) with height in MB07.

Although the reference value of water content has no impact on the water term (Vq)jk given by (18), it is possible to compute, for the sake of internal consistency and realism, both er and rr for several of the values of Tr and pr that, from Table 1, are typically used in atmospheric research (semi-implicit algorithms, computation of singular vectors and studies of sensitivity to observations or forecast errors). The result is shown in Table 2 for saturating pressures er = esw(Tr) or esi(Tr) with respect to the more stable state (liquid water or ice), depending on the temperature Tr. The zero Celsius and 280 K temperatures are added to show the rapid increase of both er and rr with Tr for an increase of a few degrees between 270 and 280 K. The higher temperature Tr = 350 K leads to unrealistically large values of rr, which are even undefined (negative) for 367.8 hPa. The explanation for these impossible values for some couple (Tr, pr) comes from the fact that er is defined as the saturation pressure at the temperature Tr. We therefore assume that pr > er, which is not verified for example for Tr = 350 K for which er = 411 hPa is greater than 367.8 hPa in Table 2. But this assumption pr > er does not limit the validity of the theory, in the same way that the assumption p > e for humid air does not limit the two state equations for dry air and water vapor. Therefore the available enthalpy function and the exergy norm are well-defined for values Tr < 300 K for which the ratios |Xυ/Yυ| are greater than 10 in Table E1, regardless of the pressure pr.

Table 1.

The reference temperatures Tr (K) and pressures pr (hPa) used (from the left to the right) in Pearce (1978) and M93, Buizza et al. (1996) and Mahfouf and Buizza (1996), E99 and Holdaway et al. (2014), Errico et al. (2004) and MB07, and Janisková and Cardinali (2017).

Table 1.
Table 2.

The reference mixing ratio rr(Tr, pr) defined as r0 er(Tr)/[prer(Tr)] in g kg−1 and the saturated pressure er(Tr) in hPa computed for several reference temperatures Tr in K and pressures pr in hPa.

Table 2.

3. The datasets

The RMS of analysis increments Sq and the SqV terms are computed for three systems using 3DVAR or 4DVAR algorithms. The periods correspond to either individual days, month or seasonal periods. The aim is to show that the temperature and water components of the exergy norm lead to robust results (i.e., that are valid for a wide range of durations and for different systems).

ARPEGE is the NWP model used at the French weather service at Météo-France (Courtier et al. 1991). The horizontal Gauss grid is based on a Schmidt projection with a spectral truncation T1198 and a stretching factor of 2.2 (i.e., with a varying resolution from 7 km over France to 33 km over the South Pacific). The vertical grid has 105 hybrid levels extending from 10 m to 0.1 hPa. The data assimilation is based on a 6-hourly incremental 4DVAR (Courtier et al. 1994), with increments computed at the truncations T149c1 (135 km) and T399c1 (50 km).

The Global Environment Multiscale (GEM) model (Côté et al. 1998a,b) studied in MB07 is used at the Canadian Meteorological Centre (CMC). The global horizontal grid has a uniform resolution of 1.5° in longitude and latitude. The resolution is variable in the vertical, with 28 σ levels extending from the surface up to 10 hPa. The analysis increments are diagnosed by the CMC 3DVAR system (Gauthier et al. 2007).

The Goddard Earth Observing System version 5 (GEOS-5) is an atmospheric global circulation model developed by the National Aeronautics and Space Administration’s (NASA) Global Modeling and Assimilation Office (GMAO). The model is based on the finite volume cubed-sphere (FV3) dynamical core (Putman 2007). The Modern-Era Retrospective Analysis for Research and Applications (MERRA-2), version 2 (Gelaro et al. 2017), is a global reanalysis produced by GMAO using the GEOS forecast model and gridpoint statistical analysis data assimilation system (Wu et al. 2002; Kleist et al. 2009). The 3D-Var system MERRA-2 produces an analysis every 6 h from 1980 to the present day. The horizontal resolution of the data assimilation and model is around 50 km, or 0.5°. In the vertical, 72 hybrid sigma-pressure levels are used, reaching from the surface to 0.01 hPa. The linearized version of GEOS includes the FV3 dynamical core and a linearization of the relaxed Arakawa–Schubert convection scheme (Holdaway et al. 2014, hereafter H14), single moment cloud scheme (Holdaway et al. 2015) and a simplified boundary layer scheme.

4. The results

a. Seasonal means of ARPEGE: The water norms

The ARPEGE seasonal averages of RMS of analysis increments Sq and exergy SqV term Vq are shown in Fig. 1. The winter and summer averages are computed with data four times per day (0000, 0600, 1200, 1800 UTC).

Fig. 1.
Fig. 1.

The seasonal averages of the RMS of analysis increments for water Sq (g kg−1) are computed for ARPEGE outputs every 6 h and plotted in latitude–pressure sections for (a) winter (DJF) and (c) summer (JJA). The corresponding seasonal averages of the exergy SqV term Vq given by (18) are plotted for (b) winter (DJF) and (d) summer (JJA).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The general patterns for Sq and Vq are roughly similar, with a large vertical decrease with height (from 0.5 to less than 0.005 g kg−1) and seasonal latitude oscillations following the regions of maximum surface temperatures (from −15° in DJF to +15° in JJA).

Values close to the ground are of the same order of magnitude for the analysis increments (≈0.7 g kg−1), the exergy term (≈0.4 g kg−1) and the E99 term (Vq10.31 g kg−1 or 0.57 g kg−1 computed with Tr = 300 K and wq = 1.0 or wq = 0.3).

The JJA and DJF seasonal means of the “constant RH” value Vq2 derived in MB07 are shown in Fig. 2. The seasonal latitude oscillation is similar to that of Sq and Vq in Fig. 1. The decrease with height of Vq2 is larger than for the exergy norm, due to the property wq2 ≈ (wq)2 derived from (20)(21) leading to values of Vq2 smaller than 0.0005 g kg−1 in the stratosphere (purple color). These values of Vq2 are close to those for the RMS of analysis increments above the level 200 hPa.

Fig. 2.
Fig. 2.

As in Figs. 1b and 1d, but for the DJF and JJA seasonal average of the MB07 water term Vq2 given by (9).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Vertical profiles are plotted in Fig. 3 for the horizontal means of the RMS of analysis increment Sq and for the V terms Vq (exergy), Vq1 (E99) and Vq2 (MB07).

Fig. 3.
Fig. 3.

Vertical profiles of horizontal mean of seasonal averages computed from ARPEGE outputs every 6 h and for three latitude domains: (a),(d) southern extratropical midlatitudes from −60° to −30°; (b),(e) tropical latitudes from −30° to +30°; (c),(f) northern extratropical midlatitudes from +30° to +60°. The vertical profiles of the DJF means are plotted in (a)–(c); those for the JJA means in (d)–(f). The E99 water terms Vq1 (purple dotted lines) are given by (8) with wq = 1.0 and wq = 0.3. The exergy water term Vq (red dashed lines) is given by (18). The MB07 water term Vq2 (black dashed lines) is given by (10). The RMS of analysis increments in water vapor is Sq (blue solid lines).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Almost the same features are observed for the two seasons and for the three latitude domains. The large decrease with height by at least three orders of magnitude for the analysis increments Sq cannot be represented by the E99 constant values Vq10.31 or 0.57 g kg−1 with wq = 1.0 or wq = 0.3, nor for any other constant value for wq.

The differences between the vertical profiles of the RMS of analysis increments, those for the exergy terms and those for the MB07 term remain small from the surface up to about 200 hPa (less than one order of magnitude). The exergy term Vq is almost similar to the RMS of analysis increments for the layer 500–250 hPa in the tropical and summer extratropical regions, with the blue and red lines intersecting each other. For levels above 100 hPa, the MB07 term is closer to the RMS of analysis increments than the exergy term, with a rapid decrease with height that the exergy term cannot reproduce.

For these reasons, the RMS of the analysis increments, the exergy norm and the MB07 norm are thus similar to each other, while the values for E99 are more different from the other three. The aim was not to perfectly simulate the RMS of the analysis increments, but to approach them qualitatively, both for their vertical variation and for their order of magnitude.

The lack of a contribution from condensed water species to the moist-air exergy norm, together with the absence of any latent heat terms Lυ or Ls, may seem surprising. However, the condensed water contents ql and qi do exist in (B1) for the moist-air exergy function am, which forms the starting point for deriving the moist exergy squared norm.

It is this theory that ultimately allows ql and qi to be neglected in the squared norm components NT, Np and Nυ, as small correction terms. Moreover, the seasonal averages plotted in Fig. 1 for ARPEGE confirm that there is no need to add independent norms related to the condensates ql or qi, because the comparisons between the latitude-section of Sq and Vq do not reveal missing structures related to the convective regions where ql and qi are large (tropical cumulus and extratropical frontal regions).

b. Seasonal means of ARPEGE: The temperature norms

The exergy norm seemed able to induce new results, especially for the moisture term Vq due to the term rυ¯(p) in (18), a result confirmed in the previous section. Similarly, since the ratio T¯(p)/Tr in (17) varies with pressure, and therefore with height, one may wonder whether this variation predicted by the theory is realistic or not.

For this purpose, ARPEGE winter averages of the RMS of analysis increments for temperature ST and of the temperature exergy term VT are shown in Fig. 4. The corresponding vertical profiles of horizontal mean values are plotted in Fig. 5. The summer averages exhibit similar results (not shown).

Fig. 4.
Fig. 4.

As in Figs. 1a and 1b, but for temperature (K).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Fig. 5.
Fig. 5.

The same ARPEGE seasonal mean (DJF) as in Fig. 3a but for temperature (K) and for the RMS of analysis increments ST (solid blue), the E99 term VT10.77 (dotted purple), and the exergy term VT (red dashed).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Although the comparisons of norms for each latitude and pressure are less relevant for the temperature components than for the water components (especially within the tropics), the general appearance for ST and VT is approximately similar, with a maximum near the surface (between 1000 and 800 hPa), and a minimum in the tropical troposphere for medium and high levels (between 600 and 100 hPa).

The variations with height of VT are similar to those for ST, while the constant value deduced from the E99 temperature component of the norm (VT10.77 for Tr = 300 K) is further from the ST profile.

Therefore, although variations with height of ST and VT are smaller than those for Sq and Vq, the similarity between the vertical profiles of the seasonal averages of ST and VT confirms the possible crude interpretation of the RMS of analysis increments with the temperature term computed from the squared exergy norm, and with a realistic impact for the ratio [T¯(p)/Tr]2 in (17).

c. A specific day for CMC and GEOS systems

The results presented in the previous sections regarding ARPEGE seasonal averages are encouraging, but the need for daily applications of the exergy norm would require similar variations with height and latitude for a given situation for both the analysis increments and the norms. In addition, the encouraging results obtained with the 4D-Var incremental assimilation of the ARPEGE variable mesh model must be confirmed with different models and/or assimilation schemes.

To do this, the results obtained for the humidity variable are shown in Fig. 6 for one single analysis (0000 UTC 26 December 2002). Outputs from the GEM-CMC system are on the left in Figs. 6a, 6c, and 6e and those from the GEOS-MERRA-2 system are on the right in Figs. 6b, 6d, and 6f. The latitude–pressure sections for the RMS of analysis increments Sq in Figs. 6a and 6b are similar to those in Figs. 1a and 1b. The vertical profiles of the exergy term Vq, the MB07 term Vq2 and the E99 terms Vq1 computed with Tr = 300 K and wq = 1.0 or wq = 0.3 are similar to those in Fig. 3a.

Fig. 6.
Fig. 6.

Latitude–pressure sections and vertical profiles of horizontal averages for the water term for 26 Dec 2002: (a),(c),(e) GEM-CMC; (b),(d),(f) GEOS-MERRA-2. (a),(b) Sections of the RMS of analysis increments Sq (g kg−1). (c),(d) Sections of exergy norms Vq (g kg−1). (e),(f) Vertical profiles of horizontal averages of E99 (dotted purple), MB07 (dashed black), and exergy (dashed red) norms and the analysis increments Sq (solid blue).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

While the RMS of analysis increments are noisier for those GEM-CMC and GEOS-MERRA-2 daily outputs than for the ARPEGE seasonal averages, the same decay with height and relative maxima in the lower layers in the tropics is observed for this particular day. The differences between the three ARPEGE, GEM-CMC, and GEOS-MERRA-2 systems are more pronounced above 200 hPa in the upper troposphere and in the stratosphere, where GEM-CMC exhibits larger analysis increments than ARPEGE, while those for GEOS-MERRA-2 are smaller than ARPEGE.

The latitude–pressure sections plotted for the water component of the exergy norm in Figs. 6c and 6d for GEM-CMC and GEOS-MERRA-2 are similar to those for ARPEGE in Figs. 1a and 1b.

The water exergy SqV term Vq is relatively smooth and not noisy because it depends on the averaged value of the water vapor qυ¯ computed on a circle of latitude, which is less variable in space than the daily RMS of analysis increments Sq.

The results presented in this section for a specific day and for two different systems are therefore broadly comparable to those shown for the ARPEGE seasonal averages. We can therefore be confident that the results derived in this paper from the exergy norm will be robust for other systems with similar patterns of analysis fields.

d. The decrease with height of wq

The advantage of the exergy approach is that it provides an analytic formulation for the weighting factor wq given by (19). As an example, values of wq(qυ) are plotted in Fig. 7 for 0.1 < qυ < 25 g kg−1.

Fig. 7.
Fig. 7.

The dimensionless exergy weighting factor wq(qυ) given by (19) plotted with qυ in ordinates.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The weighting factor wq(rυ) is smaller than unity for moist low levels where qυ > 6.7 g kg−1 for Tr = 300 K, and it is equal to 0.33 for qυ ≈ 20 g kg−1. Conversely, it is much larger than unity for small values of qυ, reaching wq ≈ 67 for qυ ≈ 0.1 g kg−1 in the upper troposphere and wq ≈ 6700 for qυ ≈ 0.001 g kg−1 in the stratosphere.

It is also possible to plot the vertical profiles of wq in terms of the horizontal mean value qυ¯(p) computed from the GEM-CMC simulation, shown in Fig. 8. The large increase of wq with height, with a factor varying nonlinearly from 1 to 40 for the pressure varying from 1000 to 300 hPa, is similar to the one proposed empirically in previous studies; for instance, a weight of wq(rυ) ≈ 5 was evaluated for the lower part of the atmosphere in Barkmeijer et al. (2001) from the ECMWF averaged error variances for qυ, with wq(rυ) strongly increasing above 500 hPa. This description is consistent with the exergy weight displayed in Fig. 8.

Fig. 8.
Fig. 8.

The dimensionless exergy weighting factor wq(z) given by (19) for the vertical profile of average values qυ¯(p) of the GEM-CMC dataset used to plot the Fig. 6c.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The same relation used to plot these diagrams “wq in terms of qυ” is used to plot the exergy norm in the pressure (p) and latitude (φ) sections shown in Figs. 1b, 1d, 6c, and 6d, where the zonal averages qυ¯(φ,p) varies with both pressure and latitude.

e. FSOI

The forecast sensitivity to observation impact (FSOI) method can be used to assess and compare the capacity of various observing systems to reduce a given short-range forecast error produced by a NWP model (e.g., Baker and Daley 2000; Langland and Baker 2004; Cardinali 2009; Gelaro et al. 2010). Typically, fields from a 24 h forecast are compared against a verifying analysis, in terms of u, υ, T, ps, and qυ using an inner product based on the E99 energy norm with different values of wq in the moist term. The adjoint of the forecast model is used to propagate a sensitivity backward from verifying time (24 h) to obtain a sensitivity at analysis time (0 h). The adjoint model can include both dry physical processes (turbulent diffusion, radiation, gravity wave drag) and moist processes (large-scale condensation, moist convection).

Impacts shown in the present paper are examined in averages per observation system and for the global domain with the E99 norms (7)(8) where Tr = 270 K, pr = 1000 hPa, and wq = 0.3. The value of 0.3 is chosen empirically in H14 to produce approximately equal weighting between the temperature and specific humidity components of the norm.

The metrics monitored at GMAO are the following: impact per analysis, impact per observation, fraction of beneficial observations, and observation count per analysis. The observation impacts are computed as reductions in the final 24 h forecast errors due to any given extra set of observations included in the initial analysis. The adjoint model can be used to propagate the final energy norm gradient backward 24 h in order to obtain sensitivities of these forecast errors at the initial time (Trémolet 2008). These sensitivities are then passed through the adjoint of the data assimilation system to convert them into observation space and to provide the impacts.

Figure 9 compares the 24 h forecast error reductions produced by various observing systems included in the MERRA-2 data assimilation system with three different inner products for the estimation of the global forecast error: the E99 “dry energy squared norm” with wq = 0.0, the E99 “moist energy squared norm” with wq = 0.3, and the “exergy squared norm” NT + Np + Nυ introduced in Eqs. (14)(16) of section 2c.

Fig. 9.
Fig. 9.

The 24-h forecast observation impacts per analysis for each observation system. Comparisons of (i) the dry norm (white); (ii) the moist norm E99 with wq = 0.3 (gray), namely the same as Fig. 9 in H14; and (iii) the moist exergy norm (dark).

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The impacts of the dry energy E99 squared norms are those computed and studied in Fig. 9 of H14 for the month 17 March–17 April 2012. The impacts for the two moist squared norms (E99 with wq = 0.3 and exergy formulations) are computed for another month (1–30 September 2015). For convenience, the impacts of the three dry and moist squared norms are compared on the same plot despite having been computed over those two distinct periods. In all experiments, the adjoint model includes a comprehensive set of physical processes with moist processes as described in H14.

As expected from the definition of the moist energy norm, impacts are larger when they include the moist term, as already shown in H14. It is interesting to note that the increase in observation impacts not only holds for observations sensitive to atmospheric water vapor, such as radiosoundings, but also for observation systems where only a small subset of the observations directly measure moisture, such as the Infrared Atmosphere Sounding Interferometer (IASI) radiances, Advanced Microwave Sounding Unit (AMSU-A) radiances that are sensitive to atmospheric temperature, and atmospheric motion vectors (AMVs) that are directly sensitive to horizontal wind components. These results show that a reduction of forecast error in the moisture field is possible through observations of temperature and wind. This could occur through dynamical balance, for example.

The ranking, in terms of contributions of the various observing systems to the forecast error reduction, is unchanged when moving from E99/wq = 0.0 to E99/wq = 0.3. Similarly, when examining the impact with the exergy norm instead, it is clear that the overall observation impact is larger, but that the ranking of the observation systems relative to each other is almost the same. Larger values come from the difference in the weighting factor wq applied to the moisture at upper levels, which does not depend on height for the E99 norm and increases with height for the exergy norm according to Fig. 8.

The most striking feature, when using the exergy norm, is the very large increase by a factor of three (or > +200%, see Table 3) of the only observing system highly sensitive to atmospheric water vapor: the Microwave Humidity Sounder (MHS). According to Fig. 9, radiosonde observations (RAOBs) ranks first for the exergy norm, which may have important implications given that the operational radiosonde observing network is expensive to operate. These results suggest that radiosonde humidity sensors play an important role in the 24 h forecast accuracy, even more than MHS.

Table 3.

The increase in observation impacts (in percentage) corresponding to Fig. 9 for the change of the dry norms to the moist E99 with wq = 0.3 (first line), and then to the moist exergy (second line).

Table 3.

5. Conclusions

The main objective of this paper is to provide a general and more satisfactory method for combining thermodynamic variables of the atmosphere into a norm. There are several formulations for these norms currently in use for a wide variety of important applications, yet until now all have been derived using heuristic methods and approximations.

It is argued in this paper that such approximations can be avoided by instead considering the principles of fundamental physics more carefully. Specifically, the approach is to start with some general exergy functions, which are constructed by combining the first (enthalpy) and second (entropy) law of thermodynamics, leading to the available enthalpy function am derived in M93. This kind of exergy function is also based on the concept of relative entropy or Kullback distance, two equivalent concepts that are already used in many papers dealing with assimilation techniques.

The choice of the exergy (available enthalpy) squared norms provides not only the quadratic terms (T′)2, (ps)2, and (qυ)2, but also values for the weighting factors which multiply these quadratic terms. It is shown in the present paper that the weighting factors for T and qυ vary with height in the same way as the RMS of analysis increments. This ensures an even weighting of all variables and all levels when computing the global norm. Such results are valid for both seasonal average periods and for a particular day.

The fact that the weights for the exergy norm for T and qυ are close to the RMS of analysis increments is not straightforward. Indeed, if the observation system is radically changed, the increments could be very different, while the exergy-norm weights would not be modified. To better understand the complex links that can exist between fields as different as thermodynamics, information theory and data assimilation, it is possible to refer to papers cited in section 3 of Marquet and Dauhut (2018).

Inspired by previous studies by Kleeman (2002) and Majda et al. (2002), the paper of Xu (2007) examined the use of the relative entropy or Kullback-Leibler distance K(x||y) given by (11)to measure the information content of the pdf produced by an optimal analysis of observations (or compressed super-observations) with respect to a prior background pdf used by the analysis (…) where the background pdf can be always considered as an approximation of the analysis pdf.” Xu showed that the integral form of the relative entropy K(x||y) “is a quadratic form of the analysis increment vector weighted by B−1”, and “yields an explicit formulation in which the signal part is given by the inner-product of the analysis increment vector weighted by the inverse of the background covariance matrix” (B−1).

Since Xu (2007) demonstrated a close relationship between K(x||y) and the weighting factors Vu, Vυ, Vυ, Vp, VT, and Vq, the next step is to use the close relationship shown by Procaccia and Levine (1976), Eriksson and Lindgren (1987), Eriksson et al. (1987), Karlsson (1990) and Honerkamp (1998) between K(x||y) and exergy functions, to foresee a direct link between the moist-air exergy defined in thermodynamics and the weighting factors used in data assimilation.

The new exergy (available enthalpy) squared norm may solve the main disadvantage of using the constant E99 moist V term stated in Rivière et al. (2009), namely that the weight for water is no longer proportional to the weight for temperature with the exergy formulation, leading to new results with the use of the Vq term.

A first usage of the exergy norm in the context of FSOI experiments has shown that it increases observation impact in a way similar to what has previously been described when going from a dry energy norm to a moist energy norm (e.g., H14). However, the enhancement of the impact is larger, since the exergy norm accounts more evenly for moisture forecast errors between the various atmospheric layers, whereas the moist energy norm penalizes the upper-tropospheric levels. The results are very similar among the various observing systems, however, with a noticeable difference for the MHS and RAOBs, for which the contributions are particularly enhanced with the exergy norm. This is in agreement with the known impact of microwave humidity sounders from direct observing system experiments (Karbou et al. 2010; Chambon et al. 2015). In consequence, it is expected that the various observing systems would be more fairly ranked through more balanced contributions between wind, temperature and water vapor forecast errors through the use of the exergy norm in FSOI experiments.

Another usage of the exergy norm has been shown by Borderies et al. (2019) to demonstrate the impact of airborne cloud radar reflectivity data assimilation.

The important point is that the analytical formulation of the exergy norm is not complicated. It is comparable in complexity to existing formulations (E99; MB07) and can be easily coded and used in operational systems, for moist singular vector and FSOI calculations as well as forecast verifications. The only new aspect is the need to take into account horizontal averages, or averages on each latitude circle, for the mean temperature and vapor content variables T¯ and rυ¯ that appear in (14)(18) to define NT, Nυ, VT, and Vq.

Acknowledgments

The definitions of the squared norm components NT, Np, and Nυ were obtained during the Pan-GCSS meeting in Athens, Greece, in May 2005. The results presented in this paper are thanks to Philippe Courtier’s initial encouragements, with numerous preliminary tests carried out between 2005 and 2018. The authors wish to thank the editor and the three reviewers for their comments, which helped to improve the manuscript.

APPENDIX A

List of Symbols and Acronyms

Bp

A dummy notation for a pressure norm

APE

The global available potential energy (Lorenz)

α

The specific mass of moist air (the density 1/ρ)

ae

The moist specific available energy

ah, am

The dry and moist specific available enthalpies

aT, ap

Temperature and pressure components of ah and am

aυ

The water component of am

cpd

Specific heat of dry air (1004.7 J K−1 kg−1)

cpv

Specific heat of water vapor (1846.1 J K−1 kg−1)

cl

Specific heat of liquid water (4218 J K−1 kg−1)

ci

Specific heat of ice (2106 J K−1 kg−1)

cp

The specific heat at constant pressure for moist air, = qdcpd + qυcpv + qlcl + qici

δ

= Rυ/Rd − 1 ≈ 0.608

e

The water vapor partial pressure

ei

The specific internal energy

er

The water vapor reference partial pressure, with er = esw(Tr) ≈ 6.11 hPa

F, H

Dimensionless functions of X or Y

g

Magnitude of Earth’s gravity (9.8065 m s−2)

Γ¯

The Lorenz stability parameter

Γ¯q

A weight in the water component of MB07 norm

GCM

General circulation model

h, H

Specific and global enthalpies

Hr

A dummy-scale height (C87)

kB

The Boltzmann constant

K

Kullback function, contrast, relative entropy

Lf

= hlhi: latent heat of melting

Lυ

= hυhl: latent heat of vaporization

Ls

= hυhi: latent heat of sublimation

Lf(Tr)

= 0.334 × 106 J kg−1

Lυ(Tr)

= 2.501 × 106 J kg−1

Ls(Tr)

= 2.835 × 106 J kg−1

m

A mass of moist air

dm

The element of mass (= ρdτ)

N

The global available enthalpy squared norms

NWP

Numerical weather prediction

ωij

The fractional coverage of the model grid box

p

The pressure (p = pd + e)

ps

The surface pressure

q

The specific content (e.g., qυ = ρυ/ρ)

Qr

A dummy specific water content (C87)

r

The mixing ratio (e.g., rυ = ρυ/ρd)

r0

= Rd/Rυ ≈ 0.622 = 1/1.608

ρ

Specific mass of moist air (= ρd + ρυ + ρl + ρi)

Rd

Dry-air gas constant (287.06 J K−1 kg−1)

Rυ

Water vapor gas constant (461.52 J K−1 kg−1)

R

Gas constant for moist air (= qdRd + qυRυ)

s

The specific entropy

σ

The vertical coordinate of the model grid box

Σ, dΣ

Global and element of horizontal surface of Earth

T

The absolute temperature

Tr

The reference zero Celsius temperature (273.15 K)

U

The horizontal wind and its components (u, υ)

U

The horizontal wind speed u2+υ2

μ

The specific Gibbs’ function (hTs)

ϕ

The gravitational potential energy (gz + Cste)

V

The variances of analysis errors

V0

A special variance of 2 J kg−1

wq

A relative weight in water components of norms

xj, yj

The micro states that define the function K

Z

A dimensionless water vapor variable

a. Lower indices (for h, s, p, μ, ρ, q, r, V, X, Y, Z):

r

Reference value (entropy, available enthalpy)

d, υ

Dry-air and water vapor gases phases

l, i

Liquid water and ice condensed phases

sw, si

Saturating value (over liquid or ice)

t

Total water value (vapor plus liquid plus ice)

T, p, υ

Temperature, pressure, and water components

T1, p1

Notations for pressure components (V)

q, q2

Notations for water components (V)

1, 2

Notations in separating laws

i, j, k

Indices for longitude, latitude and altitude

b. Upper indices/operator:

()

Departure terms from average values

()¯

Average values

APPENDIX B

The Specific Moist-Air Available Enthalpy

The specific moist available enthalpy am is an exergy function defined in M93 [see Eq. (17), p. 574] as a sum of four partial moist available enthalpies for dry air (am)d, water vapor (am)υ, liquid water (am)l, and ice (am)i, leading to
am=qd(am)d+qυ(am)υ+ql(am)l+qi(am)i,
(am)d=[hd(hd)r]Tr[sd(sd)r],
(am)υ=[hυ(hυ)r]Tr[sυ(sυ)r],
(am)l=[hl(hl)r]Tr[sl(sl)r],
(am)i=[hi(hi)r]Tr[si(si)r],
where Tr is a constant reference pressure.
Differences in enthalpy and in entropy can be computed for dry air, water vapor and condensed species by assuming that the specific heat at constant pressure (cpd, cpv, cl, ci) and gas constants (Rd, Rυ) are all constant for the atmospheric range of temperature (from 180 to 320 K), leading to
hd(hd)r=cpd(TTr),hυ(hυ)r=cpv(TTr),
hl(hl)r=cl(TTr),hi(hi)r=ci(TTr),
and
sd(sd)r=cpdln(T/Tr)Rdln[pd/(pd)r],
sυ(sυ)r=cpvln(T/Tr)Rυln[e/er],
sl(sl)r=clln(T/Tr),si(si)r=ciln(T/Tr).
The reference partial pressure er is equal to the ice-vapor value esi(Tr) for Tr < 0°C or to the liquid-vapor value esw(Tr) for Tr > zero Celsius. The moist available enthalpy (B1) is computed by including (B6)(B10) in (B2)(B5), yielding
am=cp[TTrTrln(TTr)]+Tr{qdRdln[pd(pd)r]+qυRυln(eer)}.
Here ql and qi are not neglected, but appear in the moist values of cp and qd = 1 − qυqlqi, anywhere else.

APPENDIX C

The Temperature Component of am

The first term on the rhs of (B11) is the Motivity defined by Lord Kelvin (Thomson 1853). It corresponds to the moist temperature component aT of the available enthalpy defined in Marquet (1991, hereafter M91) and M93 in terms of the function F(X) according to
aT(T)=cpTrF(XT),XT=T/Tr1>1,
F(X)=Xln(1+X).
The difference from the dry case studied in M91 is that cp is equal to qdcpd + qυcpv + qlcl + qici and is not a constant, since it depends on varying specific contents of dry air and water species.

F(X) is positive and asymmetric with respect to X = 0, see Fig. C1. It is a quadratic-like function because F(X) ≈ X2/2 for |X| < 0.3. This terminology “quadratic-like” corresponds to functions with Taylor series of the form: X2/2 + aX3 + bX4 + …, where the quadratic term X2/2 is the first-order approximation and where the other higher-order terms can be discarded. This approximation is typically valid for 210 K < T < 390 K if Tr = 300 K. F(X) = 0 only for X = 0, namely for T = Tr.

Fig. C1.
Fig. C1.

The two functions F(X) = X − ln(1 + X) and X2/2 plotted for −1 > X > +2.5.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

APPENDIX D

The Pressure Components of am

Terms in the second line of (B11) can be rearranged in order to compute the separate quadratic contributions due to total pressure p = pd + e on the one hand, and to water species contents (qυ, ql, or qi) on the other hand.

The three state functions for moist air, dry air, and water vapor can be written as p = ρRT, pd = qdρRdT and e = qυρRυT, respectively, leading to
TrqdRd=pdTr/(ρT)=RTrpd/p,
TrqυRυ=eTr/(ρT)=RTre/p,
where the moist gas constant R = qd Rd + qυ Rυ is not a constant since it varies with qd and qυ.
The terms qd Rd and qυ Rυ given by (D1) and (D2) can be inserted into (B11), yielding
am=aT+RTr{pdpln[pd(pd)r]+eerln(eer)}.
The next step is to isolate the pressure component ap defined by (D4), leading to the separation of am into
am=aT+ap+aυ,ap=RTrln(ppr),
aυ=RTr{pdpln[pdppr(pd)r]+epln(epprer)},
where the remaining terms grouped in (D5) form the water components aυ.
It is not possible to define directly a squared norm starting from the term ln(p/pr), since it is negative for p < pr. This apparent drawback was already mentioned in M91 and M93. However, it is possible to integrate by parts ap in (D4) with respect to p, leading to
ap=RTrprp[pprln(ppr)(pprC)].
A new quadratic-like function H(X) can be introduced by choosing the constant of integration C = 1, yielding
ap=RTrp[prH(Xp)],Xp=ppr1,
H(X)=(1+X)ln(1+X)X,
where Xp is the dimensionless pressure control variable.

It is shown in Fig. D1 that H(X) is positive and asymmetric with respect to X = 0. It is a quadratic-like function because H(X) ≈ X2/2 up to higher-order terms. H(X) = 0 only for X = 0, namely for p = pr.

Fig. D1.
Fig. D1.

The two functions H(X) = (1 + X) ln(1 + X) − X and X2/2 plotted for −1 < X < +2.5.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

The constant reference pressure pr can enter the derivative in (D7) and the term pr H(Xp) is equal to the function prp + p ln(p/pr) ≈ (ppr)2/(2pr) called “store of work for any layer under isothermal conditions” in Margules (1910) and studied in Eq. (Ia)′ page 505, the bottom of page 506 and the top of page 507 of this old paper.

APPENDIX E

The Water Components of am

The aim of this section is to show that aυ given by (D5), which depends on the six pressures p, pr, pd, (pd)r, e, and er, can be expressed in terms of the sole water mixing ratios rυ = qυ/qd and rr. In this way, aυ will be interpreted as the water vapor component of am.

Division of (D2) by (D1) leads to e/pd = rυ/r0, where r0Rd/Rυ = 0.622. The same result is valid for the reference state, leading to er/(pd)r = rr/r0 and to a reference value for the mixing ratio given by
rr=r0er/(pd)r.
This reference mixing ratio is fully determined if Tr and pr are known, because pr = (pd)r + er and er(Tr) is the saturation pressure of water at Tr and (pd)r = prer(Tr) can then be computed.
The four pressure terms involved in (D5) are computed by using e/pd = rυ/r0 and er/(pd)r = rr/r0, leading to
pdp=pdpd+e=r0rυ+r0,pr(pd)r=rr+r0r0,
ep=epd+e=rυrυ+r0,prer=rr+r0rr.
The component aυ given by (D5) can then be written as
aυ=RTr[(rυrυ+r0)ln(rυrr)ln(rυ+r0rr+r0)].
This formulation of aυ has already been derived in the exergetic analysis of moist-air processes described (in German) in Szargut and Styrylska [1969, Eq. (10)] and recalled in Bejan [2016, Eq. (5.48), p. 207], though with different notations.

The bracketed terms in (E4) only depend on rυ and on the two known reference values Tr and pr, since the reference mixing ratio is rr = r0 er(Tr)/[prer(Tr)]. Therefore, aυ will be called the water vapor component of am.

The impacts of ql and qi are not neglected up to this point, because the condensed water contents impact the gas constant R, which depends on qυ and qd = 1 − qυqlqi. Conversely, the bracketed terms in (E4), which generates the quadratic-like part of aυ, do not depend on ql or qi. These results could not be expected and are just imposed by the exact computations.

Let us introduce the water variables
Zυ=ep=rυrυ+r0,Zr=erpr=rrrr+r0.
which are computed with (E3). The water component aυ given by (E4) can be transformed into the sum of the two terms depending on the function H, leading to
aυ=RTrZrH(Xυ)+RTr(1Zr)H(Yυ),
where
Xυ=ZυZr1=r0rr(rυrrrυ+r0),
Yυ=(1Zυ)(1Zr)1=(rυrrrυ+r0).
The equality of (E6) with (E4) can be checked by using basic algebra. This result has been obtained via a lengthy trial and error process, with the aim of introducing any of the quadratic-like functions F or H of the variable (rυrr)/rr.

The ratio |Xυ/Yυ| = r0/rr = [prer(Tr)]/er(Tr) shown in Table E1 is computed for the set of reference values Tr and pr used in Tables 1 and 2. The ratio is larger than 20 for Tr ≤ 300 K and pr = 800 or 1000 hPa. This result justifies the name “large” and “small” given to Xυ and Yυ, respectively.

Table E1.

The ratio |Xυ/Yυ| = [prer(Tr)]/er(Tr) computed for several reference temperatures Tr and pressures pr. See the Table 2 for values of er(Tr).

Table E1.

The higher temperature Tr = 350 K leads to small values of |Xυ/Yυ|, which are close to unity, with an undefined (negative) ratio for 367.8 hPa. Values of Tr > 300 K are thus beyond the scope of the next definition for the water component of the exergy norm, where both rυ and rr are much lower than r0 ≈ 622 g kg−1 only for Tr ≤ 300 K, leading to Xυ ≈ (rυrr)/rr and Yυ ≈ −(rυrr)/r0. The best candidate for a water dimensionless variable similar to XT = (TTr)/Tr is thus the large component Xυ.

APPENDIX F

Separating Properties of F and H

Previous results cannot be used as such by replacing the terms (TTr)2, (pspr)2, and (rυrr)2 by the departure terms (T′)2, (ps)2, and (rυ)2, respectively. This issue is motivated by the usual applications where the perturbation terms T′, ps, and rυ may need to get zero average values, whereas TTr, pspr, and rυrr cannot cancel for all vertical levels and for constant values of Tr, pr, and rr.

It is thus important to introduce the mean values T¯, ps¯, and rυ¯, which denote averages of T, ps, and rυ computed for a given circle of latitudes, or for a given pressure level, or for any other kind of average like those considered in Fig. F1. The eddy departure terms will then be defined in the usual way by T=TT¯, ps=psps¯, and rυ=rυrυ¯.

Fig. F1.
Fig. F1.

The separation of the flow into an uneven basic state (x¯, solid lines) plus the eddies (dashed lines), defined by xxx¯. The x term stands for the meteorological variables T, p, Zυ, or rυ, also u and υ.

Citation: Monthly Weather Review 148, 3; 10.1175/MWR-D-19-0081.1

Therefore, the aim is to express the available-enthalpy functions aT, ap and aυ depending on TTr, pspr, and rυrr in terms of the “energies of the mean state,” which depend on (T¯Tr)2/2, (ps¯pr)2/2, and (rυ¯rr)2/2 plus the “energies of the eddies,” which depend on (T)2¯/2, (ps)2¯/2, and (rυ)2¯/2.

For pure quadratic quantities, such as the kinetic energy, the basic separating property is given by the binomial law:
(X1+X2)2=(X1)2+(X2)2+2X1X2.
If the flow X is separated into a mean part X1 for which X1¯X1, plus an eddy part X2 for which X2¯0, the separating property writes as
(X1+X2)2¯=(X1)2¯+(X2)2¯.
A similar exact separating property is derived for F(X) in Marquet (1991, 2003), and the one valid for H(X) is shown in this appendix. For any variable written as X = X1 + X2 + X1X2, the two properties
F(X)=F(X1)+F(X2)+X1X2,
H(X)=(1+X2)H(X1)+(1+X1)H(X2)+X1X2,
are valid for X1 > −1 and X2 > −1, which means X1 + X2 + X1X2 = (1 + X1)(1 + X2) − 1 > −1. The flow X is then separated into the same mean and eddy parts used to derived (F2) and with X1¯X1 and X2¯0, leading to
F(X)¯=F(X1)¯+F(X2)¯,
H(X)¯=H(X1)¯+(1+X1¯)H(X2)¯.

The physical consequence of (F2), (F5), and (F6) is the appearance of exact self-similarity properties verified by the total, mean and eddy parts of the flow: quadratic F or H functions generate quadratic F or H functions for the mean and the eddy parts of the flow. More precisely, the quadratic approximation of (F5) will allow computations of (TTr)2/2 in terms of (T¯Tr)2/2 and (T′)2/2, with similar results derived from (F6) and valid for surface pressure and water vapor mixing ratio.

APPENDIX G

Mean and Eddy Components of aT, ap, aυ

Mean and eddy components of aT given by (C1) can be computed by replacing XT = T/Tr − 1 in (C2) by
XT=(TT¯)+(T¯Tr1)+(TT¯)(T¯Tr1),
where T¯/Tr1 and T/T¯1=T/T¯ correspond to X1 and X2 in (F5), respectively. It is then assumed that cpcpd and F(X) ≈ X2/2, leading to
aT¯cpdTrF(T¯Tr1)+cpdTrF(TT¯)¯,
aT¯cpd(T¯Tr)22Tr+cpd(TrT¯)2(T)22Tr¯.
The three-dimensional integral of the first quadratic term in the rhs of (G3) represents the “unavailable enthalpy” of the mean state T¯ with respect to the isothermal reference state Tr. The integral of the second quadratic term represents the “available enthalpy” of the perturbations T′ of the actual state T with respect to the mean state T¯, and it forms the temperature contribution of the squared norm that can be written as
NTcpdTr(T¯)2(T)22dmΣ.
This squared norm is studied in sections 2c and 4b.
The integral of ap given by (D7) is computed by assuming that RRd, leading to
ApRdTrpr[0psH(Xp)pdpg]dΣΣ,
ApRdTr[H(Xps)1]prgdΣΣ,
where Xps=ps/pr1. The term −1 is due to Xp = −1 for p = 0 and H(−1) = 1, leading to a constant value RdTr pr /g that will not enter the definition of the squared norm component for pressure. The aim is thus to compute mean and eddy components of
Bp=Ap+RdTrprgRdTrprgH(Xps)¯,
by replacing Xps by
Xps=(psps¯)+(ps¯pr1)+(psps¯)(ps¯pr1),
where ps¯/pr1 and ps/ps¯1=ps/ps¯ correspond to X1 and X2 in (F6), respectively. The separating property (F6) can then be applied to (G7), leading to
BpRdTrprg[H(ps¯pr1)+(ps¯pr)H(psps¯)¯],
BpRdTrgpr(ps¯pr)22+RdTrgps¯(ps)2¯2,
where it is assumed that H(X) ≈ X2/2.
The first quadratic term of Bp in the rhs of (G10) represents the unavailable enthalpy of the mean state ps¯ with respect to the constant reference pressure pr. The second quadratic term represents the available enthalpy of the perturbations ps of the actual state ps with respect to the mean state ps¯. This pressure contribution of the squared norm can be transformed back into a three-dimensional integral, leading to
NpRdTrgps¯(ps)2¯2=RdTr(ps¯)2psg(ps)2¯2dΣΣ,
Np=RdTr(ps¯)2(0psdpg)(ps)2¯2dΣΣ,
Np=RdTr(ps¯)2(ps)2¯2dmΣ.
This squared norm is studied in section 2c.
It is shown in appendix E that the first term in the rhs of (E6) is much larger than the second term, due to |Xυ| ≫ |Yυ|. This result is used together with the assumptions RRd, r0rυ, and r0rr, leading to Zυrυ/r0, Zrrr/r0, and Xυrυ/rr − 1, to approximate the (isobaric, horizontal or uneven) surface mean value of aυ by
aυ¯RυTrrrH(rυrr1)¯,
where Rυ = Rd/r0 has been used.
The separating property (F6) can then be applied to (G14) and with the exact property
(rυrr1)=(rυrυ¯)+(rυ¯rr1)+(rυrυ¯)(rυ¯rr1),
where rυ=rυrυ¯. The terms rυ¯/rr1 and rυ/rυ¯1=rυ/rυ¯ correspond to X1 and X2 in (F6), respectively, with the property rυ¯=0 leading to
aυ¯RυTrrr[H(rυ¯rr1)+(rυ¯rr)H(rυrυ¯)¯].
It is finally assumed that H(X) ≈ X2/2, leading to
aυ¯RυTrrr(rυ¯rr)22+RυTrrυ¯(rυ)2¯2.
The integral of the first quadratic term in the rhs of (G17) represents the unavailable enthalpy of the mean state rυ¯ with respect to the constant reference pressure rr. The integral of the second quadratic term represents the available enthalpy of the perturbations rυ of the actual state rυ with respect to the mean state rυ¯, and it forms the water contribution of the squared norm, which can be written as
NυRυTrrυ¯(rυ)22dmΣ.
This squared norm is studied in sections 2c, 4a, 4c, and 4e.

If the exact moist value R = (1 − qt) Rd + qυ Rυ was not approximated by Rd in (E6), leading to Rd/r0 = Rυ in (G14)(G17), then a factor (1+2δrυ¯) would exist (computations not shown) in the factor of Rυ in (G18), but leading to small terms in comparison with the definition (G18) for Nυ.

REFERENCES

  • Baker, N. L., and R. Daley, 2000: Observation and background adjoint sensitivity in the adaptive observation-targeting problem. Quart. J. Roy. Meteor. Soc., 126, 14311454, https://doi.org/10.1002/qj.49712656511.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Barkmeijer, J., F. Bouttier, and M. Van Gijzen, 1998: Singular vectors and estimates of the analysis-error covariance metric. Quart. J. Roy. Meteor. Soc., 124, 16951713, https://doi.org/10.1002/qj.49712454916.

    • Search Google Scholar
    • Export Citation
  • Barkmeijer, J., R. Buizza, T. N. Palmer, K. Puri, and J.-F. Mahfouf, 2001: Tropical singular vectors computed with linearized diabatic physic. Quart. J. Roy. Meteor. Soc., 127, 685708, https://doi.org/10.1002/qj.49712757221.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bejan, A., 2016: Advanced Engineering Thermodynamics. John Wiley & Sons, Inc., 740 pp.

  • Borderies, M., O. Caumont, J. Delanoë, V. Ducrocq, N. Fourrié, and P. Marquet, 2019: Impact of airborne cloud radar reflectivity data assimilation on kilometre-scale numerical weather prediction analyses and forecasts of heavy precipitation events. Nat. Hazards Earth Syst. Sci., 19, 907926, https://doi.org/10.5194/nhess-19-907-2019.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buizza, R., and T. N. Palmer, 1995: The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci., 52, 14341456, https://doi.org/10.1175/1520-0469(1995)052<1434:TSVSOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buizza, R., J. Tribbia, F. Molteni, and T. N. Palmer, 1993: Computation of optimal unstable structures for a numerical weather prediction model. Tellus, 45A, 388407, https://doi.org/10.3402/tellusa.v45i5.14901.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buizza, R., T. N. Palmer, J. Barkmeijer, R. Gelaro, and J.-F. Mahfouf, 1996: Singular vector, norms and large-scale condensation. Preprints, 11th Conf. on Numerical Weather Prediction, Norfolk, VA, Amer. Meteor. Soc., 50–52.

  • Cardinali, C., 2009: Monitoring the observation impact on the short-range forecast. Quart. J. Roy. Meteor. Soc., 135, 239250, https://doi.org/10.1002/qj.366.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chambon, P., L.-F. Meunier, F. Guillaume, J.-M. Piriou, R. Roca, and J.-F. Mahfouf, 2015: Investigating the impact of the water-vapour sounding observations from SAPHIR on board Megha-Tropiques for the ARPEGE global model. Quart. J. Roy. Meteor. Soc., 141, 17691779, https://doi.org/10.1002/qj.2478.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Côté, J., S. Gravel, A. Méthot, A. Patoine, M. Roch, and A. Staniforth, 1998a: The operational CMC-MRB global environmental multiscale (GEM) model. Part I: Design considerations and formulation. Mon. Wea. Rev., 126, 13731395, https://doi.org/10.1175/1520-0493(1998)126<1373:TOCMGE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Côté, J., J.-G. Desmarais, S. Gravel, A. Méthot, A. Patoine, M. Roch, and A. Staniforth, 1998b: The operational CMC-MRB global environmental multiscale (GEM) model. Part II: Results. Mon. Wea. Rev., 126, 13971418, https://doi.org/10.1175/1520-0493(1998)126<1397:TOCMGE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Courtier, P., 1987: Application du contrôle optimal à la prévision numérique en Météorologie (Application of the optimal control to the numerical forecast in meteorology). Ph.D. thesis, Paris-VI University, France, 275 pp.

  • Courtier, P., C. Freydier, J.-F. Geleyn, F. Rabier, and M. Rochas, 1991: The ARPEGE project at Meteo France. Seminar on Numerical Methods in Atmospheric Models, Vol. II, Reading, United Kingdom, ECMWF, 193–232.

  • Courtier, P., J.-N. Thépaut, and A. Hollingsworth, 1994: A strategy for operational implementation of 4D-Var, using an incremental approach. Quart. J. Roy. Meteor. Soc., 120, 13671387, https://doi.org/10.1002/qj.49712051912.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cover, T. M., and J. A. Thomas, 1991: Elements of Information Theory. John Wiley & Sons, Inc., 563 pp.

  • Derber, J., and F. Bouttier, 1999: A reformulation of the background error covariance in the ECMWF global data assimilation system. Tellus, 51A, 195221, https://doi.org/10.3402/tellusa.v51i2.12316.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Descamps, L., D. Ricard, A. Joly, and P. Arbogast, 2007: Is a real cyclogenesis case explained by generalized linear baroclinic instability? J. Atmos. Sci., 64, 42874308, https://doi.org/10.1175/2007JAS2292.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ehrendorfer, M., 2000: The total energy norm in a quasigeostrophic model. J. Atmos. Sci., 57, 34433451, https://doi.org/10.1175/1520-0469(2000)057<3443:NACTEN>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ehrendorfer, M., and R. M. Errico, 1995: Mesoscale predictability and the spectrum of optimal perturbations. J. Atmos. Sci., 52, 34753500, https://doi.org/10.1175/1520-0469(1995)052<3475:MPATSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ehrendorfer, M., and J. Tribbia, 1997: Optimal prediction of forecast error covariances through singular vectors. J. Atmos. Sci., 54, 286313, https://doi.org/10.1175/1520-0469(1997)054<0286:OPOFEC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ehrendorfer, M., J. J. Tribbia, and R. M. Errico, 1995: Mesoscale predictability: An assessment through adjoint methods. Seminar on Predictability, Reading, United Kingdom, ECMWF, 157–183.

  • Ehrendorfer, M., R. M. Errico, and K. D. Reader, 1999: Singular-vector perturbation growth in a primitive equation model with moist physics. J. Atmos. Sci., 56, 16271648, https://doi.org/10.1175/1520-0469(1999)056<1627:SVPGIA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eriksson, K.-E., and K. Lindgren, 1987: Structural information in self-organizing systems. Phys. Scr., 35, 388397, https://doi.org/10.1088/0031-8949/35/3/026.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eriksson, K.-E., K. Lindgren, and B. Å. Månsson, 1987: Structure, Context, Complexity, Organization: Physical Aspects of Information and Value. World Scientific Publishing Co., 446 pp.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Errico, R. M., 2000: Interpretations of the total energy and rotational energy norms applied to determination of singular vectors. Quart. J. Roy. Meteor. Soc., 126, 15811599, https://doi.org/10.1256/smsqj.56602.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Errico, R. M., and M. Ehrendorfer, 1995: Moist singular vectors in a primitive-equation regional model. Preprints, 10th Conf. on Atmospheric and Oceanic Waves and Stability, Big Sky, MT, Amer. Meteor. Soc., 235–238.

  • Errico, R. M., K. D. Reader, and M. Ehrendorfer, 2004: Singular vectors for moisture-measuring norms. Quart. J. Roy. Meteor. Soc., 130, 963987, https://doi.org/10.1256/qj.02.227.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gauthier, P., M. Tanguay, S. Laroche, S. Pellerin, and J. Morneau, 2007: Extension of 3DVAR to 4DVAR: Implementation of 4DVAR at the Meteorological Service of Canada. Mon. Wea. Rev., 135, 23392354, https://doi.org/10.1175/MWR3394.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gelaro, R., and Coauthors, 2017: The Modern-Era Retrospective Analysis for Research and Applications, Version 2 (MERRA-2). J. Climate, 30, 54195454, https://doi.org/10.1175/JCLI-D-16-0758.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gelaro, R., R. H. Langland, S. Pellerin, and R. Todling, 2010: The THORPEX observation impact intercomparison experiment. Mon. Wea. Rev., 138, 40094025, https://doi.org/10.1175/2010MWR3393.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holdaway, D., R. Errico, R. Gelaro, and J. G. Kim, 2014: Inclusion of linearized moist physics in NASA’s Goddard earth observing system data assimilation tools. Mon. Wea. Rev., 142, 414433, https://doi.org/10.1175/MWR-D-13-00193.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holdaway, D., R. Errico, R. Gelaro, J. G. Kim, and R. Mahajan, 2015: A linearized prognostic cloud scheme in NASA’s Goddard earth observing system data assimilation tools. Mon. Wea. Rev., 143, 41984219, https://doi.org/10.1175/MWR-D-15-0037.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Honerkamp, J., 1998: Statistical Physics: An Advanced Approach with Applications. Springer-Verlag, 410 pp.

  • Janisková, M., and C. Cardinali, 2017: On the impact of the diabatic component in the Forecast Sensitivity Observation impact diagnostics (ECMWF Tech. Memo. 786). Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications, S. K. Park and L. Xu, Eds., Vol. III., Springer International Publishing, 483–511, https://doi.org/10.1007/978-3-319-43415-5_2.

    • Crossref
    • Export Citation
  • Joly, A., 1995: The stability of steady fronts and the adjoint method: Nonmodal frontal waves. J. Atmos. Sci., 52, 30823108, https://doi.org/10.1175/1520-0469(1995)052<3082:TSOSFA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Joly, A., and A. J. Thorpe, 1991: The stability of time-dependent flows: An application to fronts in developing baroclinic waves. J. Atmos. Sci., 48, 163183, https://doi.org/10.1175/1520-0469(1991)048<0163:TSOTDF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Karbou, F., E. Gérard, and F. Rabier, 2010: Global 4DVAR assimilation and forecast experiments using AMSU observations over land. Part I: Impacts of various land surface emissivity parameterizations. Wea. Forecasting, 25, 519, https://doi.org/10.1175/2009WAF2222243.1.

    • Search Google Scholar
    • Export Citation
  • Karlsson, S., 1990: Energy, entropy and exergy in the atmosphere. Ph.D. thesis, Institute of Physical Resource Theory, Chalmers University of Technology. Göteborg, Sweden, 121 pp., https://core.ac.uk/download/pdf/70599863.pdf.

  • Kleeman, R., 2002: Measuring dynamical prediction utility using relative entropy. J. Atmos. Sci., 59, 20572072, https://doi.org/10.1175/1520-0469(2002)059<2057:MDPUUR>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kleist, D. T., D. F. Parrish, J. C. Derber, R. Treadon, W.-S. Wu, and S. Lord, 2009: Introduction of the GSI into the NCEP global data assimilation system. Wea. Forecasting, 24, 16911705, https://doi.org/10.1175/2009WAF2222201.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kullback, S., 1959: Information Theory and Statistics. John Wiley & Sons, Inc., 409 pp.

  • Kullback, S., and R. A. Leibler, 1951: On information and sufficiency. Ann. Math. Stat., 22, 7986, https://doi.org/10.1214/aoms/1177729694.

  • Langland, R., and N. Baker, 2004: Estimation of observation impact using the NRL atmospheric variational data assimilation adjoint system. Tellus, 56A, 189201, https://doi.org/10.3402/tellusa.v56i3.14413.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1955: Available potential energy and the maintenance of the general circulation. Tellus, 7A, 157167, https://doi.org/10.3402/tellusa.v7i2.8796.

    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1978: Available energy and the maintenance of a moist circulation. Tellus, 30A, 1531, https://doi.org/10.3402/tellusa.v30i1.10308.

    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1979: Numerical evaluation of moist available energy. Tellus, 31A, 230235, https://doi.org/10.3402/tellusa.v31i3.10429.

    • Search Google Scholar
    • Export Citation
  • Mahfouf, J.-F., and R. Buizza, 1996: On the inclusion of physical processes in linear forward and adjoint models: Impact of large-scale condensation on singular vectors. ECMWF Newsletter Number, No. 72, ECMWF, Reading, United Kingdom, 2–6, https://www.ecmwf.int/node/14652.

  • Mahfouf, J.-F., and B. Bilodeau, 2007: Adjoint sensitivity of surface precipitation to initial conditions. Mon. Wea. Rev., 135, 28792896, https://doi.org/10.1175/MWR3439.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mahfouf, J.-F., R. Buizza, and R. M. Errico, 1996: Strategy for including physical processes in the ECMWF variational data assimilation system. Workshop on Non-linear Aspects of Data Assimilation, Reading, United Kingdom, ECMWF, 595–632, https://www.ecmwf.int/node/10924.

  • Majda, A. J., R. Kleeman, and D. Cai, 2002: A mathematical framework for quantifying predictability through relative entropy. Methods Appl. Anal., 9, 425444, https://doi.org/10.4310/MAA.2002.V9.N3.A8.

    • Search Google Scholar
    • Export Citation
  • Margules, M., 1910: The mechanical equivalent of any given distribution of atmospheric pressure, and the maintenance of a given difference in pressure (Translation by C. Abbe of a lecture read at the meeting of the imperial academy of science, Vienna, July, 11, 1901). Smithson. Misc. Collect., 51, 501532, https://www3.nd.edu/~powers/ame.20231/gibbs1873b.pdf.

    • Search Google Scholar
    • Export Citation
  • Marquet, P., 1991: On the concept of exergy and available enthalpy: Application to atmospheric energetics. Quart. J. Roy. Meteor. Soc., 117, 449475, https://doi.org/10.1002/qj.49711749903.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marquet, P., 1993: Exergy in meteorology: Definition and properties of moist available enthalpy. Quart. J. Roy. Meteor. Soc., 119, 567590, https://doi.org/10.1002/qj.49711951112.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marquet, P., 2003: The available-enthalpy cycle. I: Introduction and basic equations. Quart. J. Roy. Meteor. Soc., 129, 24452466, https://doi.org/10.1256/qj.01.62.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marquet, P., and T. Dauhut, 2018: Reply to “Comments on ‘A third-law isentropic analysis of a simulated hurricane.”’ J. Atmos. Sci., 75, 37353747, https://doi.org/10.1175/JAS-D-18-0126.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Palmer, T. N., R. Gelaro, J. Barkmeijer, and R. Buizza, 1998: Singular-vectors, metrics, and adaptative observations. J. Atmos. Sci., 55, 633653, https://doi.org/10.1175/1520-0469(1998)055<0633:SVMAAO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pearce, R. P., 1978: On the concept of available potential energy. Quart. J. Roy. Meteor. Soc., 104, 737755, https://doi.org/10.1002/qj.49710444115.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Procaccia, I., and R. D. Levine, 1976: Potential work: A statistical-mechanical approach for systems in disequilibrium. J. Chem. Phys., 65, 33573364, https://doi.org/10.1063/1.433482.

    • Search Google Scholar
    • Export Citation
  • Putman, W., 2007: Development of the finite-volume dynamical core on the cubed-sphere. Ph.D. thesis, Florida State University, 91 pp., https://diginole.lib.fsu.edu/islandora/object/fsu%3A168667.

  • Rabier, F., E. Klinker, P. Courtier, and A. Hollingsworth, 1996: Sensitivity of forecast errors to initial conditions. Quart. J. Roy. Meteor. Soc., 122, 121150, https://doi.org/10.1002/qj.49712252906.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rivière, O., G. Lapeyre, and O. Talagrand, 2009: A novel technique for nonlinear sensitivity analysis: Application to moist predictability. Quart. J. Roy. Meteor. Soc., 135, 15201537, https://doi.org/10.1002/qj.460.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Shannon, C. E., 1948: A mathematical theory of communication. Bell Syst. Tech. J., 27, 379423, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Szargut, J., and T. Styrylska, 1969: Die exergetische Analyse von Prozessen der feuchten Luft (An exergetic analysis of processes for damp air). Heiz.-Lüft.-Haustechn., 20 (5), 173178.

    • Search Google Scholar
    • Export Citation
  • Talagrand, O., 1981: A study of the dynamics of four-dimensional data assimilation. Tellus, 33A, 4360, https://doi.org/10.3402/tellusa.v33i1.10693.

    • Search Google Scholar
    • Export Citation
  • Thépaut, J.-N., and P. Courtier, 1991: Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model. Quart. J. Roy. Meteor. Soc., 117, 12251254, https://doi.org/10.1002/qj.49711750206.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thomson, W., 1853: On the restoration of mechanical energy from an unequally heated space. Philos. Mag., 5 (30), 102105, https://doi.org/10.1080/14786445308562743.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Trémolet, Y., 2008: Computation of observation sensitivity and observation impact in incremental variational data assimilation. Tellus, 60A, 964978, https://doi.org/10.1111/j.1600-0870.2008.00349.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wu, W.-S., R. J. Purser, and D. F. Parrish, 2002: Three-dimensional variational analysis with spatially inhomogeneous covariances. Mon. Wea. Rev., 130, 29052916, https://doi.org/10.1175/1520-0493(2002)130<2905:TDVAWS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Xu, Q., 2007: Measuring information content from observations for data assimilation: Relative entropy versus Shannon entropy difference. Tellus, 59A, 198209, https://doi.org/10.1111/j.1600-0870.2006.00222.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zadra, A., M. Buenner, S. Laroche, and J.-F. Mahfouf, 2004: Impact of the GEM model simplified physics on extratropical singular vectors. Quart. J. Roy. Meteor. Soc., 130, 25412569, https://doi.org/10.1256/qj.03.208.

    • Crossref
    • Search Google Scholar
    • Export Citation
1

The original notations of Shannon and Kullback using p(pj) and q(qj) are replaced here to avoid confusion with the pressure p and the specific water content quantities qt, qυ, ql, qi.

Save