四维变分同化在三维磁流体湍流小尺度重构中的应用

数据同化（Data Assimilation，DA）方法通过引入实验或观测数据，从而提高已有计算模型的预测效果。DA是气象学中进行数值天气预报的重要方法，并逐步被广泛应用于海洋、水文和地质等不同领域中。近年来，DA在流体力学领域中的应用也日趋广泛。其中，四维变分数据同化（Four-Dimensional Variational Data Assimilation，4DVar）方法引入代价函数从而量化了数据的分析值与真实值之间的差距，通过变分思想将问题转化成了代价函数的极值求解问题，有效降低了预测结果与实际观测之间的误差。目前，关于在瞬时流场的建模和预测中应用4DVar的研究仍比较匮乏。

磁流体动力学（Magnetohydrodynamics，MHD）湍流由于磁场的存在更加复杂，本文对使用4DVar方法重构MHD湍流场的小尺度进行了尝试。将通过直接数值模拟得到的一组Elsässer场的时间序列作为目标场，并将波数低于一定值的Elsässer场的时间序列作为测量数据。应用4DVar重构了低雷诺数下的小尺度三维磁流体湍流场。结果表明，当测量数据的分辨率在略大于阈值$k_c=0.2\eta_{K}^{-1}$时，成功实现了重构，其中$\eta_{K}$是湍流的Kolmogorov长度尺度。若已知至少一个大涡翻转时间的大尺度数据，就有可能通过4DVar方法对小六十四倍甚至更多的尺度进行令人满意的重构。

本文分别评估了$t=0$，即初始场和$t\rightarrow T$时的重构效果，$T$为优化总体时间。
对于初始场，本文评估了它的能谱分布；滤波速度及磁场的梯度张量的不变量的联合概率密度分布（joint PDF）；涡量、电流密度、变形率、磁变形率、涡丝拉伸项等量的系综平均；涡量与变形率张量的对齐情况。发现初始场在一定程度上得到了重构。

$t\rightarrow T$时，瞬时场的重构效果随着$t$的增加而提高。对于$t=T$时的瞬时场，滤波后的拟涡能、电流密度、变形率和磁变形率等小尺度量的重构结果具有10$\%$或更小的归一化逐点误差，以及99$\%$的逐点相关性。重构场和目标场之间的谱相关性在所有波数下都高于95$\%$，差异谱都小于20$\%$。进一步地，为了能定量比较非局部结构的几何形状，我们引入了最小体积闭椭球法（Minimum Volume Enclosing Ellipsoids，MVEE）。结果表明，对于大多数样本，重构场结构的位置和尺寸的误差在10$\%$以内，方向的误差在6$^{\circ}$以内。

[1] BATCHELOR C K, BATCHELOR G. An introduction to fluid dynamics[M]. Cambridgeuniversity press, 2000.
[2] KOLMOGOROVAN. The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers[J]. Proceedings of the Royal Society of London.Series A: Mathematical and Physical Sciences, 1991, 434(1890): 9-13.
[3] PAN H, CHEN X Z, LIANG X F, et al. Cfd simulations of gas-liquid-solid flow in fluidized bed reactors—a review[J]. Powder Technology, 2016, 299: 235-258.
[4] GOSMANA,LEKAKOUC,POLITISS,etal. Mulidimensionalmodelingofturbulent2-phase flows in stirred vessels[J/OL]. Aiche Journal, 1992, 38(12): 1946-1956. DOI: 10.1002/aic.690381210.
[5] 陈坚强, 涂国华, 张毅锋, 等. 高超声速边界层转捩研究现状与发展趋势[J]. 空气动力学学报, 2017, 35(3): 27.
[6] CRAVENS T E, DESSLER A J, HOUGHTON J T, et al. Physics of solar system plasmas[M].Cambridge University Press Cambridge, 1997.
[7] DEKOOLM,BICKNELLGV,KUNCICZ. Magnetic fields in accretion disks[J/OL]. Publications of the Astronomical Society of Australia, 1999, 16(3): 225–233. DOI: 10.1071/AS99225.
[8] LAZARIAN A, BERESNYAK A, YAN H, et al. Properties and selected implications of magnetic turbulence for interstellar medium, local bubble and solar wind[J/OL]. Space Science Reviews, 2009, 143(1-4): 387-413. DOI: 10.1007/s11214-008-9452-y.
[9] CHRISTENSEN U R, AUBERT J, HULOT G. Conditions for earth-like geodynamo models[J/OL]. Earth and Planetary Science Letters, 2010, 296(3-4): 487-496. DOI: 10.1016/j.epsl.2010.06.009.
[10] VELIKHOV E, AFONIN A, BUTOV V, et al. A new generation pulsed mhd generator[C]//Doklady Physics: volume 64. Springer, 2019: 238-243.
[11] KALNAY E. Atmospheric modeling, data assimilation and predictability[M]. Cambridge university press, 2003.
[12] BOUTTIER F, COURTIER P. Data assimilation concepts and methods march 1999[J]. Mete-orological training course lecture series. ECMWF, 2002, 718: 59.
[13] STAMMER D, BALMASEDA M, HEIMBACH P, et al. Ocean data assimilation in support of climate applications: status and perspectives[J]. Annual review of marine science, 2016, 8:491-518.
[14] 秦耀军, 周晓勇, 杨亚宾, 等. 基于数据同化技术的地质参数反演分析研究[J]. 水科学与工程技术, 2017(6): 78-82.
[15] 李君妍, 童亚拉. 改进的粒子群算法在太阳能光伏发电资料同化中的应用研究[J]. 华中师范大学学报 (自然科学版), 2021, 55(4): 567-572.
[16] 刘蕴. 基于变分数据同化的核事故源项反演模型研究[D]. 清华大学, 2017.
[17] MINAMIT,NAKANOS,LESURV,etal. A candidate secular variation model for igrf-13based on mhd dynamo simulation and 4denvar data assimilation[J/OL]. Earth Planetsand Space,2020,72(1). DOI: 10.1186/s40623-020-01253-8.
[18] LI Y, ZHANG J, DONG G, et al. Small-scale reconstruction in three-dimensional kolmogorov flows using four-dimensional variational data assimilation[J]. Journal of Fluid Mechanics,2020,885.
[19] DAVIDSON P A. Cambridge texts in applied mathematics: An introduction to magnetohydrodynamics[M/OL]. Cambridge University Press, 2001. DOI: 10.1017/CBO9780511626333.
[20] ALFVéN H. Existence of electromagnetic-hydrodynamic waves[J]. Nature, 1942, 150: 405-406.
[21] IROSHNIKOV P. Turbulence of a conducting fluid in a strong magnetic field[J]. Soviet Astronomy, 1964, 7: 566.
[22] KRAICHNAN R. Inertial-range spectrum of hydromagnetic turbulence[J/OL]. Physics of Fluids, 1965, 8(7): 1385+. DOI: 10.1063/1.1761412.
[23] MASON J, CATTANEO F, BOLDYREV S. Numerical measurements of the spectrum in magnetohydrodynamic turbulence[J]. Physical Review E, 2008, 77(3): 036403.
[24] PEREZ J C, BOLDYREV S. On weak and strong magnetohydrodynamic turbulence[J]. The Astrophysical Journal, 2007, 672(1): L61.
[25] BOROVSKYJE. Contribution of strong discontinuities to the power spectrum of the solar wind[J]. Physical Review Letters, 2010, 105(11): 111102.
[26] GOLDREICH P, SRIDHAR S. Toward a theory of interstellar turbulence. 2: Strong alfvenic turbulence[J]. The Astrophysical Journal, 1995, 438: 763-775.
[27] MÜLLER W C, BISKAMP D, GRAPPIN R. Statistical anisotropy of magnetohydrodynamic turbulence[J]. Physical Review E, 2003, 67(6): 066302.
[28] BOLDYREV S. On the spectrum of magnetohydrodynamic turbulence[J/OL]. Astrophyscial Journal, 2005, 626(1, 2): L37-L40. DOI: 10.1086/431649.
[29] 马建文, 秦思娴. 数据同化算法研究现状综述[J]. 地球科学进展, 2012, 27(7): 747-757.
[30] GILCHRIST B, CRESSMAN G P. An experiment in objective analysis[J]. Tellus, 1954, 6(4):309-318.
[31] GANDIN L S. Objective analysis of meteorological field[J]. Gidrometeorologicheskoe Izdatestvo, 1963, 286.
[32] KALMAN R E. A new approach to linear filtering and prediction problems[J]. J Basic Eng,1960, 82.
[33] FUJII K. Extended kalman filter[J]. Refernce Manual, 2013: 14-22.
[34] EVENSEN G. The ensemble kalman filter: Theoretical formulation and practical implementation[J]. Ocean dynamics, 2003, 53(4): 343-367.
[35] EVENSEN G. Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics[J]. Journal of Geophysical Research: Oceans,1994, 99(C5): 10143-10162.
[36] EVENSEN G, VAN LEEUWEN P J. An ensemble kalman smoother for nonlinear dynamics[J]. Monthly Weather Review, 2000, 128(6): 1852-1867.
[37] COURTIER P, ANDERSSON E, HECKLEY W, et al. The ecmwf implementation of three-dimensional variational assimilation (3d-var). i: Formulation[J]. Quarterly Journal of the Royal Meteorological Society, 1998, 124(550): 1783-1807.
[38] COURTIER P, TALAGRAND O. Variational assimilation of meteorological observations with the adjoint vorticity equation. ii: Numerical results[J]. Quarterly Journal of the Royal Meteorological Society, 1987, 113(478): 1329-1347.
[39] RABIER F, JÄRVINEN H, KLINKER E, et al. The ecmwf operational implementation of four-dimensional variational assimilation. i: Experimental results with simplified physics[J].Quarterly Journal of the Royal Meteorological Society, 2000, 126(564): 1143-1170.
[40] HOUTEKAMER P L, MITCHELL H L. Ensemble kalman filtering[J]. Quarterly Journal of the Royal Meteorological Society: A journal of the atmospheric sciences, applied meteorology and physical oceanography, 2005, 131(613): 3269-3289.
[41] BLAYO É, BOCQUET M, COSME E, et al. Advanced data assimilation for geosciences: Lecture notes of the les houches school of physics: Special issue, june 2012[M]. OUP Oxford,2014.
[42] WINIAREK V, BOCQUET M, SAUNIER O, et al. Estimation of errors in the inverse modeling of accidental release of atmospheric pollutant: Application to the reconstruction of the cesium-137 and iodine-131 source terms from the fukushima daiichi power plant[J]. Journal of Geophysical Research: Atmospheres, 2012, 117(D5).
[43] STOHL A, SEIBERT P, WOTAWA G, et al. Xenon-133 and caesium-137 releases into the atmosphere from the fukushima dai-ichi nuclear power plant: determination of the source term,atmospheric dispersion, and deposition[J]. Atmospheric Chemistry and Physics, 2012, 12(5):2313-2343.
[44] SAUNIER O, MATHIEU A, DIDIER D, et al. An inverse modeling method to assess the source term of the fukushima nuclear power plant accident using gamma dose rate observations[J].Atmospheric Chemistry and Physics, 2013, 13(22): 11403-11421.
[45] WUNSCHC,HEIMBACHP. Practicalglobaloceanicstateestimation[J]. PhysicaD:Nonlinear Phenomena, 2007, 230(1-2): 197-208.
[46] STAMMER D, UEYOSHI K, KÖHL A, et al. Estimating air-sea fluxes of heat, freshwater, and momentum through global ocean data assimilation[J]. Journal of Geophysical Research:Oceans, 2004, 109(C5).
[47] BINTANJA R, VAN DE WAL R S, OERLEMANS J. A new method to estimate ice age temperatures[J]. Climate Dynamics, 2005, 24(2): 197-211.
[48] BONANB,NODETM,RITZC,etal. An etkf approach for initial state and parameter estimation in ice sheet modelling[J]. Nonlinear Processes in Geophysics, 2014, 21(2): 569-582.
[49] LEMIEUX-DUDON B, BLAYO E, PETIT J R, et al. Consistent dating for antarctic and greenland ice cores[J]. Quaternary Science Reviews, 2010, 29(1-2): 8-20.
[50] FOURNIER A, NERGER L, AUBERT J. An ensemble kalman filter for the time-dependent analysis of the geomagnetic field[J]. Geochemistry, Geophysics, Geosystems, 2013, 14(10):4035-4043.
[51] COLTICE N, ROLF T, TACKLEY P J, et al. Dynamic causes of the relation between area and age of the ocean floor[J]. Science, 2012, 336(6079): 335-338.
[52] BOCHER M, COLTICE N, FOURNIER A, et al. A sequential data assimilation approach for the joint reconstruction of mantle convection and surface tectonics[J]. Geophysical Journal International, 2016, 204(1): 200-214.
[53] HAYASE T. Numerical simulation of real-world flows[J]. Fluid Dynamics Research, 2015, 47 (5): 051201.
[54] GRONSKIS A, HEITZ D, MEMIN E. Inflow and initial conditions for direct numerical simulation based on adjoint data assimilation[J/OL]. Journal of Computational Physics, 2013, 242:480-497. DOI: 10.1016/j.jcp.2013.01.051.
[55] MONS V, CHASSAING J C, GOMEZ T, et al. Reconstruction of unsteady viscous flows using data assimilation schemes[J/OL]. Journal of Computational Physics, 2016, 316: 255-280. DOI:10.1016/j.jcp.2016.04.022.
[56] KATO H, OBAYASHI S. Approach for uncertainty of turbulence modeling based on data assimilation technique[J]. Computers & Fluids, 2013, 85: 2-7.
[57] KATOH,YOSHIZAWAA,UENOG,etal. Adataassimilationmethodologyforreconstructing turbulent flows around aircraft[J]. Journal of Computational Physics, 2015, 283: 559-581.
[58] LI Z, ZHANG H, BAILEY S C, et al. A data-driven adaptive reynolds-averaged navier–stokes k–𝜔 model for turbulent flow[J]. Journal of Computational Physics, 2017, 345: 111-131.
[59] MELDI M, POUX A. A reduced order model based on kalman filtering for sequential data assimilation of turbulent flows[J]. Journal of Computational Physics, 2017, 347: 207-234.
[60] FOURES D P, DOVETTA N, SIPP D, et al. A data-assimilation method for reynolds-averaged navier–stokes-driven mean flow reconstruction[J]. Journal of fluid mechanics, 2014, 759: 404-431.
[61] PROTAS B, NOACK B R, ÖSTH J. Optimal nonlinear eddy viscosity in galerkin models of turbulent flows[J]. Journal of Fluid Mechanics, 2015, 766: 337-367.
[62] BEWLEY T R, PROTAS B. Skin friction and pressure: the “footprints”of turbulence[J].Physica D: Nonlinear Phenomena, 2004, 196(1-2): 28-44.
[63] CHEVALIER M, HŒPFFNER J, BEWLEY T R, et al. State estimation in wall-bounded flow systems. part 2. turbulent flows[J]. Journal of Fluid Mechanics, 2006, 552: 167-187.
[64] COLBURN C, CESSNA J, BEWLEY T. State estimation in wall-bounded flow systems. part3. the ensemble kalman filter[J]. Journal of Fluid Mechanics, 2011, 682: 289-303.
[65] HEITZ D, MÉMIN E, SCHNÖRR C. Variational fluid flow measurements from image sequences: synopsis and perspectives[J]. Experiments in fluids, 2010, 48(3): 369-393.
[66] MONS V, CHASSAING J C, SAGAUT P. Optimal sensor placement for variational data assim-ilation of unsteady flows past a rotationally oscillating cylinder[J]. Journal of Fluid Mechanics,2017, 823: 230-277.
[67] MONS V, CHASSAING J C, GOMEZ T, et al. Is isotropic turbulence decay governed by asymptotic behavior of large scales? an eddy-damped quasi-normal markovian-based data assimilation study[J]. Physics of Fluids, 2014, 26(11): 115105.
[68] FOURNIER A, EYMIN C, ALBOUSSIERE T. A case for variational geomagnetic data assimilation: insights from a one-dimensional, nonlinear, and sparsely observed mhd system[J/OL].Nonlinear Processes in Geophysics, 2007, 14(2): 163-180. DOI: 10.5194/npg-14-163-2007.
[69] MORZFELD M, FOURNIER A, HULOT G. Coarse predictions of dipole reversals by low-dimensional modeling and data assimilation[J/OL]. Physics of the Earth and Planetary Interiors,2017, 262: 8-27. DOI: 10.1016/j.pepi.2016.10.007.
[70] SVEDIN A, CUELLAR M C, BRANDENBURG A. Data assimilation for stratified convection[J/OL]. Monthly Notices of the Royal Astronomical Society, 2013, 433(3): 2278-2285. DOI:10.1093/mnras/stt891.
[71] HAYASHI K, ABBETT W P, CHEUNG M C, et al. Coupling a global heliospheric magnetohydrodynamic model to a magnetofrictional model of the low corona[J]. The Astrophysical Journal Supplement Series, 2021, 254(1): 1.
[72] BISWAS A, HUDSON J, LARIOS A, et al. Continuous data assimilation for the 2d magnetohydrodynamic equations using one component of the velocity and magnetic fields[J/OL].Asymptotic Analysis, 2018, 108(1-2): 1-43. DOI: 10.3233/ASY-171454.
[73] HUDSON J, JOLLY M. Numerical efficacy study of data assimilation for the 2d magnetohydrodynamic equations[J]. Journal of computational dynamics, 2019, 6(1).
[74] MENDOZA O, DE MOOR B, BERNSTEIN D. Data assimilation for magnetohydrodynamics systems[J/OL]. Journal of Computational and Applied Mathematics, 2006, 189(1-2): 242-259.DOI: 10.1016/j.cam.2005.03.030.
[75] YOSHIDAK,YAMAGUCHIJ,KANEDAY. Regeneration of small eddies by data assimilation in turbulence[J/OL]. Physical Review Letters, 2005, 94(1). DOI: 10.1103/PhysRevLett.94.014501.
[76] ELMEGREEN B G, SCALO J. Interstellar turbulence i: observations and processes[J]. Annu.Rev. Astron. Astrophys., 2004, 42: 211-273.
[77] SCALO J, ELMEGREEN B G. Interstellar turbulence ii: implications and effects[J]. Annu.Rev. Astron. Astrophys., 2004, 42: 275-316.
[78] TAYLORGI. Statistical theory of turbulence-ii[J]. Proceedings of the Royal Society of London.Series A-Mathematical and Physical Sciences, 1935, 151(873): 444-454.
[79] SREENIVASAN K R. On the scaling of the turbulence energy dissipation rate[J]. The Physics of fluids, 1984, 27(5): 1048-1051.
[80] HEISENBERG W. Zur statistischen theorie der turbulenz[M]. Springer Berlin Heidelberg,1985: 82-111.
[81] ONSAGER L. Statistical hydrodynamics[J]. Il Nuovo Cimento (1943-1954), 1949, 6(2): 279-287.
[82] 龚建东. 同化技术: 数值天气预报突破的关键–以欧洲中期天气预报中心同化技术演进为例[J]. 气象科技进展: 英文版, 2013(3).
[83] ESTERM,KRIEGELHP,SANDERJ,etal. Adensity-based algorithm for discovering clusters in large spatial databases with noise.[C]//kdd: volume 96. 1996: 226-231.
[84] KHACHIYAN L G. Rounding of polytopes in the real number model of computation[J]. Mathematics of Operations Research, 1996, 21(2): 307-320.
[85] TODD M J. Minimum-volume ellipsoids: Theory and algorithms[M]. SIAM, 2016.
[86] MOSHTAGH N. Minimum volume enclosing ellipsoid. matlab central file exchange[J]. URLhttps://www. mathworks. com/matlabcentral/fileexchange/9542-minimum-volume-enclosing-ellipsoid, 2020.
[87] VIEILLEFOSSE P. Internal motion of a small element of fluid in an inviscid flow[J]. Physica A: Statistical Mechanics and its Applications, 1984, 125(1): 150-162.
[88] CANTWELL B J. Exact solution of a restricted euler equation for the velocity gradient tensor[J]. Physics of Fluids A: Fluid Dynamics, 1992, 4(4): 782-793.
[89] DALLAS V, ALEXAKIS A. Structures and dynamics of small scales in decaying magnetohydrodynamic turbulence[J/OL]. Physics of Fluids, 2013, 25(10). DOI: 10.1063/1.4824195.