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# American Institute of Mathematical Sciences

April  2013, 6(2): 401-422. doi: 10.3934/dcdss.2013.6.401

## Asymptotic analysis for the 3D primitive equations in a channel

 1 The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405, United States 2 School of Technology Management/ Mechanical, and Advanced Materials Engineering/ Natural Science, Ulsan National Institute of Science and Technology, San 194, Banyeon-ri, Eonyang-eup, Ulju-gun, Ulsan, South Korea

Received  July 2011 Revised  November 2011 Published  November 2012

In this article, we give an asymptotic expansion, with respect to the viscosity which is considered here to be small, of the solutions of the $3D$ linearized Primitive Equations (EPs) in a channel with lateral periodicity. A rigorous convergence result, in some physically relevant space, is proven. This allows, among other consequences, to confirm the natural choice of the non-local boundary conditions for the non-viscous PEs.
Citation: Makram Hamouda, Chang-Yeol Jung, Roger Temam. Asymptotic analysis for the 3D primitive equations in a channel. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 401-422. doi: 10.3934/dcdss.2013.6.401
##### References:
 [1] A. Bousquet, M. Petcu, C.-Y. Shiue, R. Temam and J. Tribbia, Boundary conditions for limited areas models based on the shallow water equations,, to appear., (). [2] C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267. [3] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations," Oxford Lecture Series in Mathematics and its Applications, 32. The Clarendon Press, Oxford University Press, Oxford, 2006. xii+250 pp. [4] G.-M. Gie, M. Hamouda and R. Temam, Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary, Networks and Het- erogeneous Media (NHM),, to appear., (). [5] M. Hamouda, C. Jung and R. Temam, Boundary layers for the 2D linearized primitive equations, Communications on Pure and Applied Analysis, 8 (2009), 335-359. [6] M. Hamouda, C. Jung and R. Temam, Boundary layers for the 3D primitive equations in a cube,, in preparation., (). [7] M. Hamouda, C. Jung and R. Temam, Regularity and existence results for the inviscid primitive equations in a channel,, in preparation., (). [8] M. Hamouda and R. Temam, "Some Singular Perturbation Problems Related to the Navier-Sotkes Equations," Advances in Deterministic and Stochastic Analysis. (Eds. N. M. Chuong et al.), Springer Verlag, New York, 2007 World Scientific Publishing Co., 197-227. ISBN-13 978-981-270-550-1. [9] M. Hamouda and R. Temam, Boundary layers for the Navier-Stokes equation : The case of characteristic boundary, Georgian Mathematical Journal, 15 (2008), 517-530. [10] A. Huang, M. Petcu and R. Temam, The one-dimensional supercritical shallow-water equations with topography,, Annals of the University of Bucharest, (). [11] G. M. Koblekov, Existence of a solution n the large for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. [12] I. Kukavica, R. Temam, V. Vicol and M. Ziane, Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data, C. R. Math. Acad. Sci. Paris, 348 (2010), 639-645. [13] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753. [14] C. Jung and R. Temam, Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers, International Journal of Numerical Analysis and Modeling, 2 (2005), 367-408. [15] J. L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean. (CAO I,II), Comput. Mech. Adv., 1 (1993), 120 pp. [16] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288. [17] J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. [18] M. C. Lombardo and M. Sammartino, Zero viscosity limit of the Oseen equations in a channel, SIAM J. Math. Anal., 33 (2001), 390-410. (electronic). doi: 10.1137/S0036141000372015. [19] J. Oliger and A. Sundström, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 419-446. [20] M. Petcu and R. Temam, An interface problem: The two-layer shallow water equations,, to appear., (). [21] M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions, Mathematical Methods in the Applied Sciences (MMAS), (2011). doi: 10.1002/mma.1482. [22] J. P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951. [23] A. Rousseau, R. Temam and J. Tribbia, Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity, Discrete Contin. Dyn. Syst., 13 (2005), 1257-1276. [24] A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297-319. [25] M-C. Shiue, J. Laminie, R. Temam and J. Tribbia, Boundary value problems for the shallow water equations with topography, Journal of Geophysical Research, Oceans, 116, CO2015, 006315 (2011). doi: 10.1029/2010JC. [26] R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci., 60 (2003), 2647-2660. [27] R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686. doi: 10.1006/jdeq.2001.4038. [28] R. Temam and X. Wang, Boundary layers for Oseen's type equation in space dimension three,, Russian J. Math. Phys., 5 (): 227. [29] S. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18 (1987), 1467-1511. [30] T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., (1997), 2599-2617.

show all references

##### References:
 [1] A. Bousquet, M. Petcu, C.-Y. Shiue, R. Temam and J. Tribbia, Boundary conditions for limited areas models based on the shallow water equations,, to appear., (). [2] C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267. [3] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations," Oxford Lecture Series in Mathematics and its Applications, 32. The Clarendon Press, Oxford University Press, Oxford, 2006. xii+250 pp. [4] G.-M. Gie, M. Hamouda and R. Temam, Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary, Networks and Het- erogeneous Media (NHM),, to appear., (). [5] M. Hamouda, C. Jung and R. Temam, Boundary layers for the 2D linearized primitive equations, Communications on Pure and Applied Analysis, 8 (2009), 335-359. [6] M. Hamouda, C. Jung and R. Temam, Boundary layers for the 3D primitive equations in a cube,, in preparation., (). [7] M. Hamouda, C. Jung and R. Temam, Regularity and existence results for the inviscid primitive equations in a channel,, in preparation., (). [8] M. Hamouda and R. Temam, "Some Singular Perturbation Problems Related to the Navier-Sotkes Equations," Advances in Deterministic and Stochastic Analysis. (Eds. N. M. Chuong et al.), Springer Verlag, New York, 2007 World Scientific Publishing Co., 197-227. ISBN-13 978-981-270-550-1. [9] M. Hamouda and R. Temam, Boundary layers for the Navier-Stokes equation : The case of characteristic boundary, Georgian Mathematical Journal, 15 (2008), 517-530. [10] A. Huang, M. Petcu and R. Temam, The one-dimensional supercritical shallow-water equations with topography,, Annals of the University of Bucharest, (). [11] G. M. Koblekov, Existence of a solution n the large for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. [12] I. Kukavica, R. Temam, V. Vicol and M. Ziane, Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data, C. R. Math. Acad. Sci. Paris, 348 (2010), 639-645. [13] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753. [14] C. Jung and R. Temam, Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers, International Journal of Numerical Analysis and Modeling, 2 (2005), 367-408. [15] J. L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean. (CAO I,II), Comput. Mech. Adv., 1 (1993), 120 pp. [16] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288. [17] J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. [18] M. C. Lombardo and M. Sammartino, Zero viscosity limit of the Oseen equations in a channel, SIAM J. Math. Anal., 33 (2001), 390-410. (electronic). doi: 10.1137/S0036141000372015. [19] J. Oliger and A. Sundström, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 419-446. [20] M. Petcu and R. Temam, An interface problem: The two-layer shallow water equations,, to appear., (). [21] M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions, Mathematical Methods in the Applied Sciences (MMAS), (2011). doi: 10.1002/mma.1482. [22] J. P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951. [23] A. Rousseau, R. Temam and J. Tribbia, Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity, Discrete Contin. Dyn. Syst., 13 (2005), 1257-1276. [24] A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297-319. [25] M-C. Shiue, J. Laminie, R. Temam and J. Tribbia, Boundary value problems for the shallow water equations with topography, Journal of Geophysical Research, Oceans, 116, CO2015, 006315 (2011). doi: 10.1029/2010JC. [26] R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci., 60 (2003), 2647-2660. [27] R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686. doi: 10.1006/jdeq.2001.4038. [28] R. Temam and X. Wang, Boundary layers for Oseen's type equation in space dimension three,, Russian J. Math. Phys., 5 (): 227. [29] S. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18 (1987), 1467-1511. [30] T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., (1997), 2599-2617.
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When a force of 10 N acts on a particle at 37° east of north then produces a displacement of 5 m South of West. Find work done.

Eshan
askIITians Faculty 2095 Points
4 years ago
Dear student,

Angle between displacement and force vectors=$\dpi{80} 90^{\circ}+45^{\circ}+37^{\circ}=172^{\circ}$

Hence work done=$\dpi{80} Fscos\theta=10\times 5\times cos172^{\circ}$

$\dpi{80} cos172^{\circ}=cos(180^{\circ}-8^{\circ})=-cos8^{\circ}=-cos(45^{\circ}-37^{\circ})$

$\dpi{80} =-(cos45^{\circ}cos37^{\circ}+sin45^{\circ}sin37^{\circ})=-\dfrac{1}{\sqrt{2}}$

Hence work done=$\dpi{80} -25\sqrt{2}J$