非线性涡旋Rossby波的演变特征及其相互作用
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摘要
本论文根据大气动力学原理,对中尺度动力学方程进行约化摄动,得到了大气内非线性涡旋Rossby波及其相互作用的演变方程,对方程进行了理论研究和实际大气背静下的求解,得到非线性涡旋Rossby波的相速值为10~0m/s,这与螺旋云雨带的实测移速一致,这也是许多学者从涡旋Rossby波说中去寻找台风中螺旋云雨带的形成、维持的理论解释。研究非线性涡旋Rossby波的演变方程、解的存在性、周期波动解及其稳定性和数值格式不仅具有理论意义,而且具有重要的实用价值。通过研究,结果表明:
1.单个非线性涡旋Rossby波的演变过程满足非线性Schr(?)dinger方程:
i(αA/αT)+α(α~2A/αμ~2)+β|A|~2A=0
非线性涡旋Rossby波相互作用的演变方程满足耦合的非线性Schr(?)dinger组:
(?)
2.耦合非线性Schr(?)dinger方程组,若满足条件:
(1)q_j(S_1,S_2)∈C~2,(S_j∈[0,∞))为实值函数,且|q_j(S_1,S_2)|≤M(S_1+S_2),M=const>0
(2)A_(j0)∈H~2 则方程组解存在;
3.对非线性Schr(?)dinger方程及方程组的周期波动解及其稳定性进行了研究,得到了有关稳定和不稳定的判据。
4.对耦合非线性Schr(?)dinger方程组给出了四点隐式格式:
iB_(j(?))+α_jB_(jμ(?))+β_jq_j(|B_1|~2,|B_2|~2)B_j+(c_(gj)~*/4α_j)=0
B_(j|Sh)=0,B_(j|T=0)=e~(i(c_(gj)~*/2α_j)μ)A_(j0)
In the paper, we have investigated the nonlinear Vortex Rossby waves in the atmosphere. It is found by use of the WKB method the wave packet is governed by nonlinear schr(o|¨)dinge equation. It is shown by numerical calculations that the value of its outward-propagating speed is about O(10°m/s), almost to be consistent with observational one of spiralrain bands of tropical cyclone from the Vortex Rossby wave. Thus it is significative to study the evolutionary characteristic of nonlinear Vortex Rossby wave, the existence of the solutions, the periodic waves solutions and the stability and the numerical format. There are five parts in the thesis.First, by the use of WKB method that evolutionary characteristic of nonlinear Vortex Rossby wave is governed by nonlinear schr(o|¨)dinge equation and the interaction between two wave packets is governed by coupled nonlinear Schrodinger equations:Second, for the coupled nonlinear Schrodinger equations, adopted the method of integral estimate, we prove that the condition of the existence for the solutions are: (2) the initial conditions satifyThird, the periodic waves solutions and their stability of the equations are investigated and the criteria for stability and instability are obtained.
Fourth, we give the format of four-point scheme for the coupled nonlinear Schro dinger equat ions:*" )B, +-Z-B, = 0B, =0 , B,AA,J sh J r=oIf the conditions satisfy AJ0 e H\0 < b < \oj■ | < M, and 0 < qj (svs2) < Rfa +s2),Sj e[0,oo), the format of four-point implicit scheme is proved to be stable.Furthermore the error is 0{t + h2).Fifth, it is shown by numerical calculations that the value of its outward-propagating speed is about O(10° m/s), almost to be consistent with observational one of spiralrain bands of tropical cyclone from the vortex Rossby wave. It is the reason that many scholars can find the development and matainance of spiralrain bands from vortex Rossby wave.
1.单个非线性涡旋Rossby波的演变过程满足非线性Schr(?)dinger方程:
i(αA/αT)+α(α~2A/αμ~2)+β|A|~2A=0
非线性涡旋Rossby波相互作用的演变方程满足耦合的非线性Schr(?)dinger组:
(?)
2.耦合非线性Schr(?)dinger方程组,若满足条件:
(1)q_j(S_1,S_2)∈C~2,(S_j∈[0,∞))为实值函数,且|q_j(S_1,S_2)|≤M(S_1+S_2),M=const>0
(2)A_(j0)∈H~2 则方程组解存在;
3.对非线性Schr(?)dinger方程及方程组的周期波动解及其稳定性进行了研究,得到了有关稳定和不稳定的判据。
4.对耦合非线性Schr(?)dinger方程组给出了四点隐式格式:
iB_(j(?))+α_jB_(jμ(?))+β_jq_j(|B_1|~2,|B_2|~2)B_j+(c_(gj)~*/4α_j)=0
B_(j|Sh)=0,B_(j|T=0)=e~(i(c_(gj)~*/2α_j)μ)A_(j0)
In the paper, we have investigated the nonlinear Vortex Rossby waves in the atmosphere. It is found by use of the WKB method the wave packet is governed by nonlinear schr(o|¨)dinge equation. It is shown by numerical calculations that the value of its outward-propagating speed is about O(10°m/s), almost to be consistent with observational one of spiralrain bands of tropical cyclone from the Vortex Rossby wave. Thus it is significative to study the evolutionary characteristic of nonlinear Vortex Rossby wave, the existence of the solutions, the periodic waves solutions and the stability and the numerical format. There are five parts in the thesis.First, by the use of WKB method that evolutionary characteristic of nonlinear Vortex Rossby wave is governed by nonlinear schr(o|¨)dinge equation and the interaction between two wave packets is governed by coupled nonlinear Schrodinger equations:Second, for the coupled nonlinear Schrodinger equations, adopted the method of integral estimate, we prove that the condition of the existence for the solutions are: (2) the initial conditions satifyThird, the periodic waves solutions and their stability of the equations are investigated and the criteria for stability and instability are obtained.
Fourth, we give the format of four-point scheme for the coupled nonlinear Schro dinger equat ions:*" )B, +-Z-B, = 0B, =0 , B,AA,J sh J r=oIf the conditions satisfy AJ0 e H\0 < b < \oj■ | < M, and 0 < qj (svs2) < Rfa +s2),Sj e[0,oo), the format of four-point implicit scheme is proved to be stable.Furthermore the error is 0{t + h2).Fifth, it is shown by numerical calculations that the value of its outward-propagating speed is about O(10° m/s), almost to be consistent with observational one of spiralrain bands of tropical cyclone from the vortex Rossby wave. It is the reason that many scholars can find the development and matainance of spiralrain bands from vortex Rossby wave.
引文
[1] 余志豪等.地球物理流体动力学.北京:气象出版社,1996.83～92.
[2] 陈联寿等.西太平洋台风概论.北京:科学出版社,1979.48～54.
[3] 伍荣生、余志豪等.动力气象学.上海:上海科学出版社,1983.194.
[4] Anthes R A.(1982,李毓芳等译).热带气旋的发展、结构和影响.北京:气象出版社,1982,46～47.
[5] Elsberry(1986,陈联寿等译).热带气旋全球观.北京:气象出版社,1994.72～75.
[6] MacDabald N J. The evidence for the existence of Rossby-like waves in the hurricane vortex. Tellus, 1968,20: 138～150.
[7] Guinn TA, et al. Hurricane spiral bands. J Atmos Sci, 1993,50: 3380～3403.
[8] Smith Ⅱ G B, Montgomery M T. Vortex axisymmetrization: Dependence on azimuthal wave-umber or asymmetric radial structure changes. Quart J Roy Meteor Soc, 1995,121: 1615～1650.
[9] Montgomery M T, et al. A theory for vortex Rossby-wave and it application to spiral bands and intensity changes in hurricanes. Quart J Roy Meteor Soc, 1997,123: 453～436.
[10] Montgomery M T, et al. Tropical cyclone via consvectively forced vortex Rossby-wave in a three-dimensional quasigeostrophic model. J Atmos Sci, 1998,55: 3176～3207.
[11] Redekopp, L.,G. J. Fluid Mech, 1997, 82: 725～745.
[12] Malguzzi, P.,Malanotte Rizzoli, J. Atmos. Sci., 1984,41: 2620～2628.
[13] 刘式适、刘式达.中国科学.B辑.1983,(3):279～289.
[14] Flierl, G. et al.,Dyn. Ahtmos. Oceans, 1980,(5): 1～41.
[15] McWilliams, J.,Dyn. Atmos. Oceans, 1980,(5): 43～66.
[16] Benney. D. J.,Stud. Appl. Math., 1979,60: 1～10.
[17] Yamagata, T., J. Meteor. Soc. Japan, 1980,58: 160～171.
[18] Boyd, J.P.,J. Phys. Oceanogr.,1983,13: 428～449.
[19] 罗德海、纪立人.中国科学.B辑,1989,(1):103～112.
[20] Carney, T. Q. and D. O. Vincent, Meso-synoptic scale interaction during AVE/SESAME1, 10～11 Aprill 1979, Part1: theoretical development of interaction equations, Alon. Wea. R, V.,1986,114: 344～352.
[21] 谭本馗、伍荣生.非线性Rossby波及其相互作用—Ⅰ.Rossby包络孤立波的碰撞.中国科学(B辑),1993,23(4):437～448.
[22] 谭本馗、伍荣生.强迫耗散作用下的Rossby包络孤立波及其相互作用.大气科学.Vol:19,No:3,289～300.
[23] 罗哲贤.热带气旋复杂结构的时间演变问题.见:第十二届全国热带气旋科学讨论会摘要文集,2002.10.
[24] 余志豪.台风螺旋雨带—涡旋Rossby波[J].气象学报,2002,60(4):502～507.
[25] A.H.奈佛.摄动方法[M].上海科学技术出版社,1984.
[26] 徐祥德、张胜军、陈联寿、魏凤英.台风涡旋螺旋波及其波列传播动力学特征:诊断分析[J].地球物理学报,2004,47(1):33-41.
[27] 罗哲贤.台风环流区域内中尺度涡量传播特征的研究[J].气象学报,2003,61(4):396-405.
[28] 高守亭、杨惠君.多维约化摄动和大气中的非线性波[J].大气科学,1986,10(1):35-45.
[29] 罗德海.旋转正压大气中的非线性Schrodinger方程和大气阻塞[J].气象学报,1990,48(3):265～274.
[30] 许习华、丁一汇.中尺度大气运动中孤立重力波特征的研究[J].大气科学,1991,15(4):58-68.
[31] Maslowe. S.A.,J. Fluid Mech.,1997,79: 689-702.
[32] Jeffrey, A.,Kawahara, T., Asymptotic methods in Nonlinear Wave Theory., [M]. Pitman Publishing Inc., 1982.
[33] 郭柏灵.一类非线性Schrodinger方程及其方程组的数值计算问题.计算学报,1981,No3:211～223.
[34] Whitham, G. B.,Linear and nonlinear waves, John Wiley and Sons, Inc., 1974.
[35] Newton, P. K. and J .B. Keller, SIAM J. Appl. Math. 47,959～964(1987).
[36] Benjamin, T. B.and J. E. Felt, J. Fluid Mech. 27, 417～430(1967).
[2] 陈联寿等.西太平洋台风概论.北京:科学出版社,1979.48～54.
[3] 伍荣生、余志豪等.动力气象学.上海:上海科学出版社,1983.194.
[4] Anthes R A.(1982,李毓芳等译).热带气旋的发展、结构和影响.北京:气象出版社,1982,46～47.
[5] Elsberry(1986,陈联寿等译).热带气旋全球观.北京:气象出版社,1994.72～75.
[6] MacDabald N J. The evidence for the existence of Rossby-like waves in the hurricane vortex. Tellus, 1968,20: 138～150.
[7] Guinn TA, et al. Hurricane spiral bands. J Atmos Sci, 1993,50: 3380～3403.
[8] Smith Ⅱ G B, Montgomery M T. Vortex axisymmetrization: Dependence on azimuthal wave-umber or asymmetric radial structure changes. Quart J Roy Meteor Soc, 1995,121: 1615～1650.
[9] Montgomery M T, et al. A theory for vortex Rossby-wave and it application to spiral bands and intensity changes in hurricanes. Quart J Roy Meteor Soc, 1997,123: 453～436.
[10] Montgomery M T, et al. Tropical cyclone via consvectively forced vortex Rossby-wave in a three-dimensional quasigeostrophic model. J Atmos Sci, 1998,55: 3176～3207.
[11] Redekopp, L.,G. J. Fluid Mech, 1997, 82: 725～745.
[12] Malguzzi, P.,Malanotte Rizzoli, J. Atmos. Sci., 1984,41: 2620～2628.
[13] 刘式适、刘式达.中国科学.B辑.1983,(3):279～289.
[14] Flierl, G. et al.,Dyn. Ahtmos. Oceans, 1980,(5): 1～41.
[15] McWilliams, J.,Dyn. Atmos. Oceans, 1980,(5): 43～66.
[16] Benney. D. J.,Stud. Appl. Math., 1979,60: 1～10.
[17] Yamagata, T., J. Meteor. Soc. Japan, 1980,58: 160～171.
[18] Boyd, J.P.,J. Phys. Oceanogr.,1983,13: 428～449.
[19] 罗德海、纪立人.中国科学.B辑,1989,(1):103～112.
[20] Carney, T. Q. and D. O. Vincent, Meso-synoptic scale interaction during AVE/SESAME1, 10～11 Aprill 1979, Part1: theoretical development of interaction equations, Alon. Wea. R, V.,1986,114: 344～352.
[21] 谭本馗、伍荣生.非线性Rossby波及其相互作用—Ⅰ.Rossby包络孤立波的碰撞.中国科学(B辑),1993,23(4):437～448.
[22] 谭本馗、伍荣生.强迫耗散作用下的Rossby包络孤立波及其相互作用.大气科学.Vol:19,No:3,289～300.
[23] 罗哲贤.热带气旋复杂结构的时间演变问题.见:第十二届全国热带气旋科学讨论会摘要文集,2002.10.
[24] 余志豪.台风螺旋雨带—涡旋Rossby波[J].气象学报,2002,60(4):502～507.
[25] A.H.奈佛.摄动方法[M].上海科学技术出版社,1984.
[26] 徐祥德、张胜军、陈联寿、魏凤英.台风涡旋螺旋波及其波列传播动力学特征:诊断分析[J].地球物理学报,2004,47(1):33-41.
[27] 罗哲贤.台风环流区域内中尺度涡量传播特征的研究[J].气象学报,2003,61(4):396-405.
[28] 高守亭、杨惠君.多维约化摄动和大气中的非线性波[J].大气科学,1986,10(1):35-45.
[29] 罗德海.旋转正压大气中的非线性Schrodinger方程和大气阻塞[J].气象学报,1990,48(3):265～274.
[30] 许习华、丁一汇.中尺度大气运动中孤立重力波特征的研究[J].大气科学,1991,15(4):58-68.
[31] Maslowe. S.A.,J. Fluid Mech.,1997,79: 689-702.
[32] Jeffrey, A.,Kawahara, T., Asymptotic methods in Nonlinear Wave Theory., [M]. Pitman Publishing Inc., 1982.
[33] 郭柏灵.一类非线性Schrodinger方程及其方程组的数值计算问题.计算学报,1981,No3:211～223.
[34] Whitham, G. B.,Linear and nonlinear waves, John Wiley and Sons, Inc., 1974.
[35] Newton, P. K. and J .B. Keller, SIAM J. Appl. Math. 47,959～964(1987).
[36] Benjamin, T. B.and J. E. Felt, J. Fluid Mech. 27, 417～430(1967).