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Numerical simulations of diffusion in cellular flows at high Péclet numbers
1. | Department of Mathematics, University of Houston, Houston, TX 77204, United States |
2. | Department of Mathematics, Ajou University, Suwon 443-749, South Korea |
3. | Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, United States |
References:
[1] |
R. E. Bank, "PLTMG: A Software Package for Solving Elliptic Partial Differential Equations," SIAM Books, Philadelphia, 1994. |
[2] |
A. Bensoussan, J. L. Lions and G. Papanicoalou, "Asymptotic Analysis for Periodic Structures," North-Holland, 1978. |
[3] |
S. Childress, Alpha-effect in flux ropes and sheets, Phys. Earth Planet Int., 20 (1979), 172-180.
doi: doi:10.1016/0031-9201(79)90039-6. |
[4] |
B. Cushman-Roisin, "Introduction to Geophysical Fluid Dynamics," Prentice-Hall, Englewood Cliffs, NJ, 1994. |
[5] |
C. Cuvelier, A. Segal and A. A. van Steenhoven, "Finite Element Methods and Navier-Stokes Equations," Mathematics and its Applications, 22, D. Reidel Publishing Co., Dordrecht, 1986. |
[6] |
J. Douglas, Jr. and T. F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885.
doi: doi:10.1137/0719063. |
[7] |
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis," Vol. VII, North-Holland, Amsterdam, (2000), 713-1020. |
[8] |
A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54 (1994), 333-408.
doi: doi:10.1137/S0036139992236785. |
[9] |
A. Fannjiang and G. Papanicolaou, Convection-enhanced diffusion for random flows, J. Statist. Phys., 88 (1997), 1033-1076.
doi: doi:10.1007/BF02732425. |
[10] |
J. H. Ferziger and M. Perić, "Computational Methods for Fluid Dynamics," 2nd edition, Springer-Verlag, Berlin, 1999. |
[11] |
D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp., 57 (1991), 597-619. |
[12] |
P. H. Haynes and E. F. Shuckburgh, Effective diffusivity as a measure of atmospheric transport, Part I: Stratosphere, J. Geophys. Res., 105 (2000), 777-794. |
[13] |
P. H. Haynes and E. F. Shuckburgh, Effective diffusivity as a measure of atmospheric transport, Part II: Troposphere and lower stratosphere, J. Geophys. Res., 105 (2000), 795-810.
doi: doi:10.1029/2000JD900092. |
[14] |
S. Heinze, Diffusion-advection in cellular flows with large Péclet numbers, Arch. Ration. Mech. Anal., 168 (2003), 329-342.
doi: doi:10.1007/s00205-003-0256-7. |
[15] |
C. Johnson, "Numerical Solution of Partial Differential Equations by the Finite Element Method," Cambridge University Press, Cambridge, Great Britain, 1987. |
[16] |
R. B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Mathematics of Computation, 32 (1978), 1025-1039. |
[17] |
L. Koralov, Random perturbations of 2-dimensional Hamiltonian flows, Probab. Theory Related Fileds, 129 (2004), 37-62.
doi: doi:10.1007/s00440-003-0320-0. |
[18] |
J. J. H. Miller, E. O'Riordan and G. I. Shishkin, "Fitted Numerical Methods for Singular Perturbation Problems," World Scientific, Singapore, 1996. |
[19] |
K. W. Morton, Numerical solution of convection-diffusion problems, in "Applied Mathematics and Mathematical Computation," Vol. 12, Chapman & Hall, London, 1996. |
[20] |
N. Nakamura, Two-dimensional mixing, edge formation and permeability diagnosed in an area coordinate, J. Atmos. Sci., 53 (1996), 1524-1537.
doi: doi:10.1175/1520-0469(1996)053<1524:TDMEFA>2.0.CO;2. |
[21] |
A. Novikov, G. Papanicolaou and L. Ryzhik, Boundary layers for cellular flows at high Péclet numbers, Comm. Pure Appl. Math., 58 (2005), 867-922.
doi: doi:10.1002/cpa.20058. |
[22] |
P. B. Rhines and W. R. Young, How rapidly is passive scalar mixed within closed streamlines?, J. Fluid Mech., 133 (1983), 135-145.
doi: doi:10.1017/S0022112083001822. |
[23] |
M. N. Rosenbluth, H. L. Berk, I. Doxas and W. Horton, Effective diffusion in laminar convective flows, Phys. Fluids, 30 (1987), 2636-2647.
doi: doi:10.1063/1.866107. |
[24] |
T. A. Shaw, J.-L. Thiffeault and C. R. Doering, Stirring up trouble: Multi-scale mixing measures for steady scalar sources, Phys. D, 231 (2007), 143-164.
doi: doi:10.1016/j.physd.2007.05.001. |
[25] |
J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), 1113-1133.
doi: doi:10.1137/S0036142999362936. |
[26] |
B. I. Shraiman, Diffusive transport in a Rayleigh-Bénard convection cell, Phys. Rev. A, 36 (1987), 261-267.
doi: doi:10.1103/PhysRevA.36.261. |
[27] |
E. Shuckburgh, H. Jones, J. Marshall and C. Hill, Quantifying the eddy diffusivity of the Southern Ocean I: Temporal variability I, J. Phys. Oceanogr., to appear (2010). |
show all references
References:
[1] |
R. E. Bank, "PLTMG: A Software Package for Solving Elliptic Partial Differential Equations," SIAM Books, Philadelphia, 1994. |
[2] |
A. Bensoussan, J. L. Lions and G. Papanicoalou, "Asymptotic Analysis for Periodic Structures," North-Holland, 1978. |
[3] |
S. Childress, Alpha-effect in flux ropes and sheets, Phys. Earth Planet Int., 20 (1979), 172-180.
doi: doi:10.1016/0031-9201(79)90039-6. |
[4] |
B. Cushman-Roisin, "Introduction to Geophysical Fluid Dynamics," Prentice-Hall, Englewood Cliffs, NJ, 1994. |
[5] |
C. Cuvelier, A. Segal and A. A. van Steenhoven, "Finite Element Methods and Navier-Stokes Equations," Mathematics and its Applications, 22, D. Reidel Publishing Co., Dordrecht, 1986. |
[6] |
J. Douglas, Jr. and T. F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885.
doi: doi:10.1137/0719063. |
[7] |
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis," Vol. VII, North-Holland, Amsterdam, (2000), 713-1020. |
[8] |
A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54 (1994), 333-408.
doi: doi:10.1137/S0036139992236785. |
[9] |
A. Fannjiang and G. Papanicolaou, Convection-enhanced diffusion for random flows, J. Statist. Phys., 88 (1997), 1033-1076.
doi: doi:10.1007/BF02732425. |
[10] |
J. H. Ferziger and M. Perić, "Computational Methods for Fluid Dynamics," 2nd edition, Springer-Verlag, Berlin, 1999. |
[11] |
D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp., 57 (1991), 597-619. |
[12] |
P. H. Haynes and E. F. Shuckburgh, Effective diffusivity as a measure of atmospheric transport, Part I: Stratosphere, J. Geophys. Res., 105 (2000), 777-794. |
[13] |
P. H. Haynes and E. F. Shuckburgh, Effective diffusivity as a measure of atmospheric transport, Part II: Troposphere and lower stratosphere, J. Geophys. Res., 105 (2000), 795-810.
doi: doi:10.1029/2000JD900092. |
[14] |
S. Heinze, Diffusion-advection in cellular flows with large Péclet numbers, Arch. Ration. Mech. Anal., 168 (2003), 329-342.
doi: doi:10.1007/s00205-003-0256-7. |
[15] |
C. Johnson, "Numerical Solution of Partial Differential Equations by the Finite Element Method," Cambridge University Press, Cambridge, Great Britain, 1987. |
[16] |
R. B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Mathematics of Computation, 32 (1978), 1025-1039. |
[17] |
L. Koralov, Random perturbations of 2-dimensional Hamiltonian flows, Probab. Theory Related Fileds, 129 (2004), 37-62.
doi: doi:10.1007/s00440-003-0320-0. |
[18] |
J. J. H. Miller, E. O'Riordan and G. I. Shishkin, "Fitted Numerical Methods for Singular Perturbation Problems," World Scientific, Singapore, 1996. |
[19] |
K. W. Morton, Numerical solution of convection-diffusion problems, in "Applied Mathematics and Mathematical Computation," Vol. 12, Chapman & Hall, London, 1996. |
[20] |
N. Nakamura, Two-dimensional mixing, edge formation and permeability diagnosed in an area coordinate, J. Atmos. Sci., 53 (1996), 1524-1537.
doi: doi:10.1175/1520-0469(1996)053<1524:TDMEFA>2.0.CO;2. |
[21] |
A. Novikov, G. Papanicolaou and L. Ryzhik, Boundary layers for cellular flows at high Péclet numbers, Comm. Pure Appl. Math., 58 (2005), 867-922.
doi: doi:10.1002/cpa.20058. |
[22] |
P. B. Rhines and W. R. Young, How rapidly is passive scalar mixed within closed streamlines?, J. Fluid Mech., 133 (1983), 135-145.
doi: doi:10.1017/S0022112083001822. |
[23] |
M. N. Rosenbluth, H. L. Berk, I. Doxas and W. Horton, Effective diffusion in laminar convective flows, Phys. Fluids, 30 (1987), 2636-2647.
doi: doi:10.1063/1.866107. |
[24] |
T. A. Shaw, J.-L. Thiffeault and C. R. Doering, Stirring up trouble: Multi-scale mixing measures for steady scalar sources, Phys. D, 231 (2007), 143-164.
doi: doi:10.1016/j.physd.2007.05.001. |
[25] |
J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), 1113-1133.
doi: doi:10.1137/S0036142999362936. |
[26] |
B. I. Shraiman, Diffusive transport in a Rayleigh-Bénard convection cell, Phys. Rev. A, 36 (1987), 261-267.
doi: doi:10.1103/PhysRevA.36.261. |
[27] |
E. Shuckburgh, H. Jones, J. Marshall and C. Hill, Quantifying the eddy diffusivity of the Southern Ocean I: Temporal variability I, J. Phys. Oceanogr., to appear (2010). |
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