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Retrieval of eddy thermal conductivity in the weakly nonlinear Prandtl model for katabatic flows

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Abstract

Because the nonlinearity of actual physical processes can be expressed more precisely by the introduction of a nonlinear term, the weakly nonlinear Prandtl model is one of the most effective ways to describe the pure katabatic flow (no background flow). Features of the weak nonlinearity are reflected by two factors: the small parameter ε and the gradually varying eddy thermal conductivity. This paper first shows how to apply the Wentzel–Kramers–Brillouin (WKB) method for the approximate solution of the weakly nonlinear Prandtl model, and then describes the retrieval of gradually varying eddy thermal conductivity from observed wind speed and perturbed potential temperature. Gradually varying eddy thermal conductivity is generally derived from an empirical parameterization scheme. We utilize wind speed and potential temperature measurements, along with the variational assimilation technique, to derive this parameter. The objective function is constructed by the square of the differences between the observation and model value. The new method is validated by numerical experiments with simulated measurements, revealing that the order of the root mean squre error is 10–2 and thus confirming the method’s robustness. In addition, this method is capable of anti-interference, as it effectively reduces the influence of observation error.

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References

  • Arnold, C. P. Jr, and C. H. Dey, 1986: Observing-systems simulation experiments: Past, present, and future. Bull. Amer. Meteor. Soc., 67, 687–695, doi: 10.1175/1520-0477(1986)067<0687:OSSEPP>2.0.CO;2.

    Article  Google Scholar 

  • Axelsen, S. L., and H. van Dop, 2009: Large-eddy simulation of katabatic winds. Part 2: Sensitivity study and comparison with analytical models. Acta Geophys., 57, 837–856, doi: 10.2478/s11600-009-0042-5.

    Google Scholar 

  • Baklanov, A. A., B. Grisogono, R. Bornstein, et al., 2011: The nature, theory, and modeling of atmospheric planetary boundary layers. Bull. Amer. Meteor. Soc., 92, 123–128, doi: 10.1175/2010BAMS2797.1.

    Article  Google Scholar 

  • Barthélemy, A., H. Goosse, P. Mathiot, et al., 2012: Inclusion of a katabatic wind correction in a coarse-resolution global coupled climate model. Ocean Modell., 48, 45–54, doi: 10.1016/j.ocemod.2012.03.002.

    Google Scholar 

  • Buzzi, M., M. W. Rotach, M. Holtslag, et al., 2011: Evaluation of the COSMO-SC turbulence scheme in a shear-driven stable boundary layer. Meteor. Z., 20, 335–350, doi: 10.1127/0941-2948/2011/0050.

    Article  Google Scholar 

  • De Ridder, K., D. Lauwaet, and B. Maiheu, 2015: UrbClim—A fast urban boundary layer climate model. Urban Climate, 12, 21–48, doi: 10.1016/j.uclim.2015.01.001.

    Article  Google Scholar 

  • Denby, B., 1999: Second-order modeling of turbulence in kata-batic flows. Bound.-Layer Meteor., 92, 65–98, doi: 10.1023/A:1001796906927.

    Article  Google Scholar 

  • Egger, J., 1990: Thermally forced flows: Theory. Atmospheric Processes over Complex Terrain. Blumen W., Ed. Boston, MA, Amer. Meteor. Soc., 43–58.

    Chapter  Google Scholar 

  • Epifanio, C. C., 2007: A method for imposing surface stress and heat flux conditions in finite-difference models with steep terrain. Mon. Wea., Rev., 135, 906–917, doi: 10.1175/MWR3297.1.

    Article  Google Scholar 

  • Gao, J. D., C. J. Qiu, and J. F. Chou, 1995: The sensitivity influence of numerical model initial values on four-dimensional assimilation—Study based on Lorenz system. Acta Meteor. Sinica, 9, 278–287.

    Google Scholar 

  • Grisogono, B., 2003: Post-onset behaviour of the pure katabatic flow. Bound.-Layer Meteor., 107, 157–175, doi: 10.1023/A:1021511105871.

    Article  Google Scholar 

  • Grisogono, B., and J. Oerlemans, 2001a: Katabatic flow: Analytic solution for gradually varying eddy diffusivities. J. Atmos. Sci., 58, 3349–3354, doi: 10.1175/1520-0469(2001)058<3349:KFASFG>2.0.CO;2.

    Article  Google Scholar 

  • Grisogono, B., and J. Oerlemans, 2001b: A theory for the estimation of surface fluxes in simple katabatic flows. Quart. J. Roy. Meteor. Soc., 127, 2725–2739, doi: 10.1002/qj.49712757811.

    Article  Google Scholar 

  • Grisogono, B., and J. Oerlemans, 2002: Justifying the WKB approximation in pure katabatic flows. Tellus, 54, 453–463, doi: 10.3402/tellusa.v54i5.12166.

    Article  Google Scholar 

  • Grisogono, B., and S. L. Axelsen, 2012: A note on the pure katabatic wind maximum over gentle slopes. Bound.-Layer Meteor., 145, 527–538, doi: 10.1007/s10546-012-9746-1.

    Article  Google Scholar 

  • Grisogono, B., T. Jurlina, Ž. Večenaj, et al., 2015: Weakly nonlinear Prandtl model for simple slope flows. Quart. J. Roy. Meteor. Soc., 141, 883–892, doi: 10.1002/qj.2406.

    Article  Google Scholar 

  • Hu, X. M., F. Q. Zhang, and J. W. Nielsen-Gammon, 2010: Ensemble-based simultaneous state and parameter estimation for treatment of mesoscale model error: A real-data study. Geophys. Res. Lett., 37, L08802, doi: 10.1029/2010GL043017.

    Article  Google Scholar 

  • Huang, S. X., J. Xiang, H. D. Du, et al., 2005: Inverse problems in atmospheric science and their application. J. Phys. Conf. Ser., 12, 45–57, doi: 10.1088/1742-6596/12/1/005.

    Article  Google Scholar 

  • Huang, S. X., X. Q. Cao, H. D. Du, et al., 2006: Retrieval of atmospheric and oceanic parameters and the relevant numerical calculation. Adv. Atmos. Sci., 23, 106–117, doi: 10.1007/s00376-006-0011-8.

    Article  Google Scholar 

  • Ingel, L. K., 2011: Toward a nonlinear theory of katabatic winds. Fluid Dyn., 46, 505–513, doi: 10.1134/S0015462811040016.

    Article  Google Scholar 

  • Jeričević, A., L. Kraljević, B. Grisogono, et al., 2010: Parameterization of vertical diffusion and the atmospheric boundary layer height determination in the EMEP model. Atmos. Chem. Phys., 10, 341–364, doi: 10.5194/acp-10-341-2010.

    Article  Google Scholar 

  • Kaipio, J., and E. Somersalo, 2005: Statistical and Computational Inverse Problems. Springer, 357 pp.

    Google Scholar 

  • Lorenc, A. C., 1988: A practical approximation to optimal four-dimensional objective analysis. Mon. Wea. Rev., 116, 730–745, doi: 10.1175/1520-0493(1988)116<0730:APATOF>2.0.CO;2.

    Article  Google Scholar 

  • Mahrt, L., 1982: Momentum balance of gravity flows. J. Atmos. Sci., 39, 2701–2711, doi: 10.1175/1520-0469(1982)039<2701:MBOGF>2.0.CO;2.

    Article  Google Scholar 

  • Mahrt, L., 1998: Stratified atmospheric boundary layers and breakdown of models. Theoret. Comput. Fluid Dyn., 11, 263–279, doi: 10.1007/s001620050093.

    Article  Google Scholar 

  • Mo, R. P., 2013: On adding thermodynamic damping mechanisms to refine two classical models of katabatic winds. J. Atmos. Sci., 70, 2325–2334, doi: 10.1175/JAS-D-12-0256.1.

    Article  Google Scholar 

  • Navon, I. M., 1998: Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography. Dyn. Atmos. Oceans, 27, 55–79, doi: 10.1016/S0377-0265(97)00032-8.

    Article  Google Scholar 

  • Nielsen-Gammon, J. W., X. M. Hu, F. Q. Zhang, et al., 2010: Evaluation of planetary boundary layer scheme sensitivities for the purpose of parameter estimation. Mon. Wea. Rev., 138, 3400–3417, doi: 10.1175/2010MWR3292.1.

    Article  Google Scholar 

  • Salvador, N., N. C. Reis Jr, J. M. Santos, et al., 2016: Evaluation of weather research and forecasting model parameterizations under sea-breeze conditions in a North Sea coastal environment. J. Meteor. Res., 30, 998–1018, doi: 10.1007/s13351-016-6019-9.

    Article  Google Scholar 

  • Shapiro, A., B. Burkholder, and E. Fedorovich, 2012: Analytical and numerical investigation of two-dimensional katabatic flow resulting from local surface cooling. Bound.-Layer Meteor., 145, 249–272, doi: 10.1007/s10546-011-9685-2.

    Article  Google Scholar 

  • Shapiro, A., E. Fedorovich, and S. Rahimi, 2016: A unified theory for the Great Plains nocturnal low-level jet. J. Atmos. Sci., 73, 3037–3057, doi: 10.1175/JAS-D-15-0307.1.

    Article  Google Scholar 

  • Smith, C. M., and E. D. Skyllingstad, 2005: Numerical simulation of katabatic flow with changing slope angle. Mon. Wea. Rev., 133, 3065–3080, doi: 10.1175/MWR2982.1.

    Article  Google Scholar 

  • Stiperski, I., I. Kavčič, B. Grisogono, et al., 2007: Including Coriolis effects in the Prandtl model for katabatic flow. Quart. J. Roy. Meteor. Soc., 133, 101–106, doi: 10.1002/qj.19.

    Article  Google Scholar 

  • Sun J. L., D. H. Lenschow, L. Mahrt, et al., 2013: The relationships among wind, horizontal pressure gradient, and turbulent momentum transport during CASES-99. J. Atmos. Sci., 70, 3397–3414, doi: 10.1175/JAS-D-12-0233.1.

    Article  Google Scholar 

  • Tan, Z. M., J. Fang, and R. S. Wu, 2006: Nonlinear Ekman layer theories and their applications. Acta Meteor. Sinica, 20, 209–222.

    Google Scholar 

  • Tosaka, N., K. Onoshi, and M. Yamamoto, 1999: Mathematical Approach and Solution Methods for Inverse Problems: Inverse Analysis of Partial Differential Equation. University of Tokyo Press, 294 pp. (in Japanese)

    Google Scholar 

  • van den Broeke, M. R., 1997: Structure and diurnal variation of the atmospheric boundary layer over a mid-latitude glacier in summer. Bound.-Layer Meteor., 83, 183–205, doi: 10.1023/A:1000268825998.

    Article  Google Scholar 

  • Yan, B., and S. X. Huang, 2014: Variational regularization method of solving the Cauchy problem for Laplace’s equation: Innovation of the Grad–Shafranov (GS) reconstruction. Chinese Phys. B, 23, 650–655, doi: 10.1088/1674-1056/23/10/109402.

    Google Scholar 

  • Zhao, X. F., and S. X. Huang, 2012: Estimation of atmospheric duct structure using radar sea clutter. J. Atmos. Sci., 69, 2808–2818, doi: 10.1175/JAS-D-12-073.1.

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Meteorology and Oceanography, National University of Defense Technology, Nanjing, 211101, China

    Bing Yan, Sixun Huang & Jing Feng

  2. State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, State Oceanic Administration, Hangzhou, 310012, China

    Sixun Huang

Authors
  1. Bing Yan
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  2. Sixun Huang
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  3. Jing Feng
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Correspondence to Sixun Huang.

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Supported by the National Natural Science Foundation of China (41575026).

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Yan, B., Huang, S. & Feng, J. Retrieval of eddy thermal conductivity in the weakly nonlinear Prandtl model for katabatic flows. J Meteorol Res 31, 965–975 (2017). https://doi.org/10.1007/s13351-017-7025-2

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