Abstract
Based on the theory of stratification, the well-posedness of the initial value problem for the linear shallow-water equations on an equatorial beta-plane was discussed. The sufficient and necessary conditions of the existence and uniqueness for the local solution of the equations were presented and the existence conditions for formal solutions of the equations were alsoven. For the Cauchy problem on the hyper-plane {t=0}, the local analytic solution were worked out and a special case was discussed. Finally, an example was used to explain the variety of formal solutions for the ill-posed problem.
Similar content being viewed by others
References
HAN Wei, HUANG Si-xun. Equatorial trapped waves in the mean state with slow variation: inhomogenous equatorial duct [J]. Journal of Hydrodynamics, Ser. B, 2004, 16(2):182–185.
HUANG Si-xun, XU Ding-hua, LAN Wei-ren et al. Generalized variational optimization analysis for 2-D flow field [J]. Journal of Hydrodynamics, Ser. B, 2005, 17(4):459–465.
MCPHADEN M. J., BUSALACCHI A. J., CHENEY R. et al. Tropical ocean-global atmospheric observing system: A decade progress [J]. J. Geophs. Res., 1998,103:14169–14240.
DU Hua-dong, HUANG Si-xun, SHI Han-qing. Theoretical analysis and numerical experiment for 1-D semi-geostrophic shallow water model [J]. Journal of Hydrodynamics, Ser. A, 2004, 19(1):38–45.(in Chinese)
MATSUNO T. Quasi-geostrophic motions in the equatorial area [J]. J. Met. Soc. Japan, 1966, 44(1):25–43.
GILL A. E. Some simple solutions for heat-induced tropical circulation [J]. Quart. J. R. Met. Soc., 1980,106:447–462.
GILL A. E., PHLIPS P. J. Nonlinear effects on heat-induced circulation of the tropical atmosphere [J]. Quart. J. R. Met. Soc., 1986,112:69–91.
CHAO Ji-ping, LIN Yong-hui. The foundation and movement of tropical semi-geostrophic adaptation [J]. Acta Meteor Sinica, 1996,10(2):129–141.
CHAO Ji-ping. On the semi-geostrophic adaptation and evolution motions in the tropical atmosphere and ocean [J]. Acta Meteorologica Sinica, 2000,58(3):257–264. (in Chinese)
SHI Wei-hui, CHEN Da-duan, HE You-hua. The theory of stratification and foundation of non-linear partial differential equation [M]. Shanghai: Shanghai University Press, 2001. (in Chinese)
SHI Wei-hui. Solutions analytiques de quelques equations aux derives partielles en mecanique des fluides [M]. Paris: Hermann, 1992.
CHEN Da-duan, HE You-hua. On the well-posedness of the initial value problem of non-static rotating fluid [J]. Applied Mathematics and Mechanics (English Edition), 2004, 25(3):288–296.
CHEN Da-duan, HE You-hua. Stabilities of Boussinesq approximate equations for non-static rotating fluids [J]. Chinese Journal of Atmospheric Sciences, 2002, 26(3):293–298.
SHEN Chun, WANG Yue-peng, SHI Wei-hui. Stability of system of two-dimensional non-hydrostatic revolving fluids [J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(3): 317–325.
SHEN Chun, SUN Mei-na. Initial and boundary value problems for two-dimensional non-hydrostatic Boussinesq equations [J]. Journal of Shanghai University(English Edition), 2005,9(2):114–119.
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (Grant No: 90411006, 40175014) and the Key Foundation of Shanghai Municipal Commission of Science and Technology (Grant No: 02DJ14032).
Biography: SHEN Chun (1975-), Male, Ph.D.
Rights and permissions
About this article
Cite this article
Shen, C., Shi, Wh. General Cauchy Problem for the Linear Shallow -Water Equations on an Equatorial Beta-Plane. J Hydrodyn 18, 303–309 (2006). https://doi.org/10.1016/S1001-6058(06)60007-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1016/S1001-6058(06)60007-3