حل عددی معادلات آب کم‌عمق دو لایه به‌روش تفکیک مد

نوع مقاله : پژوهشی

نویسنده

استادیار، گروه فیزیک، واحد شوشتر، دانشگاه آزاد اسلامی، شوشتر، ایران

چکیده

در پژوهش حاضر، برای حل عددی دستگاه معادلات آب کم‌عمق دولایه برحسب متغیرهای فشارورد و کژفشار از روش تفکیک مد با گام‌های زمانی متفاوت استفاده شد. برای گسسته‌سازی مکانی، روش‌های مرتبه دوم مرکزی و فشرده مرتبه چهارم به‌ کار گرفته شدند و برای گسسته‌سازی زمانی در هر دو بخش فشارورد و کژفشار از روش لیپ فراگ به‌صورت نیمه‌ضمنی به‌همراه پالایه زمانی روبرت-‌آسلین استفاده شده است. ابتدا مد کژفشار با گام زمانی بزرگ و سپس مد فشارورد با گام زمانی کوچک‌تر حل شده است تا بتوان مقادیر کژفشار موردنیاز در مد فشارورد را از مد کژفشار استخراج کرد. برای این منظور در طی یک گام زمانی کژفشار، مقادیر کژفشار ثابت در نظر گرفته شده یا با درون‌یابی زمانی تعیین شده‌اند. تحلیل منحنی‌های خطا نشان می‌دهد که درون‌یابی زمانی مقادیر کژفشار برای به‌کارگیری در مد فشارورد می‌تواند به نتایجی بهتر منجر شود و شرایط پایدارتری را فراهم کند. همچنین، می‌توان ناپایداری عددی در گام‌های زمانی کژفشار بزرگ را با افزایش ضریب پالایه زمانی کنترل کرد.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Numerical solution of two-layer shallow water equations using mode splitting method

نویسنده [English]

  • Hakim Golshahy
Assistant Professor, Department of Physics, Shoushtar Branch, Islamic Azad University, Shoushtar, Iran
چکیده [English]

In the numerical models that use iterative methods to solve the momentum equations by applying the rigid-lid approximation, the number of iterations increases for high resolution, therefore processing time increases. An alternative method is applying a free surface and splitting equations to barotropic and baroclinic modes. The surface gravity waves that are faster than slow moving internal gravity waves; impose a limitation on the time steps with the CFL condition. Thus, mode splitting method is computationally efficient to handle the multiple time steps separating the barotropic and baroclinic mode equations. In this method, the barotropic mode equations are solved at small time steps consistent with the fast surface gravity wave speeds and the baroclinic mode equations are solved at larger time steps consistent with the slow internal gravity wave speeds. This is used in most of the ocean circulation models and is an unavoidable choice to high resolution models.
In this study, we considered the shallow water equations for two-layer basin with vorticity-divergence formulation using mode splitting method by a small time step of barotropic mode within a larger time step of baroclinic mode. The primary systems of equations that contain both upper and lower layer variables, are rewritten in terms of new (barotropic and baroclinic) variables without any variation or more approximation of primary systems. This procedure can be extended to multi-layer systems so that primary N-layer system of equations is changed to 1 system of barotropic mode equations and N-1 systems of baroclinic mode equations coupled together.
For numerical experiments, a fully baroclinic (non-barotropic) initial condition is considered in a constant depth rectangular domain with 64, 128 and 256 grid points in each direction and periodic boundaries. For spatial differencing, second order centered scheme with low computational cost and fourth-order compact scheme with high computational cost are used. For time integration, a semi-implicit descretization based on leapfrog scheme is implemented with the Robert-Asselin time filter for both barotropic and baroclinic systems of equations, similarly.
Mode splitting method may presents numerical instabilities on the larger baroclinic time steps, in spite of time step limitation based on CFL condition coming from each system of barotropic and baroclinic mode equations taken individually. Here, it is controlled by increasing the coefficient of time filter to some extent.
First, we solve the baroclinic mode equations to derive all baroclinic variables that are necessary to solve barotropic mode equations during a baroclinic time step. In this case, these variables can be taken to be constant up to the next baroclinic time level or determined by time interpolation between two successive baroclinic time levels.
To assess the performance of the numerical method, relative error of energy conservation is calculated. Results show that for the ratio of baroclinic time step to up to 20 times of that of barotropic time step, time evolution of the barotropic and baroclinic variables have appropriate correspondence to the basic state, in which the barotropic mode has the same time step as the baroclinic mode. When this ratio increases, the differences of errors from basic state are presented more clearly. These errors are increased on fourth order compact method insofar as it leads to numerical instabilities so the time filter coefficient had to be increased, while second order scheme is not sensitive and stays stable with small coefficient. Moreover, taking constant for baroclinic variables to solve barotropic mode equations makes the solution on fourth order compact scheme for large baroclinic time step unstable, but on the other hand time interpolation provides more stable condition and has a good performance almost on both spatial schemes.

کلیدواژه‌ها [English]

  • Shallow-water equations
  • Two-layer basin
  • Mode splitting method
  • Barotropic and baroclinic modes
  • Numerical instability
قادر، س.، احمدی‌گیوی، ف. و گلشاهی، ح.، 1389، مقایسه عملکرد روش‌های ابرفشرده و فشرده ترکیبی مرتبه ششم در گسسته‌سازی مکانی مدل آب کم‌عمق دولایه‌ای: نمایش امواج گرانی‌-‌لختی و راسبی خطی. مجله ژئوفیزیک ایران، 4(2)، 49-69.
قادر، س.، احمدی‌گیوی، ف. و گلشاهی، ح.، 1391، حل عددی معادلات آب کم عمق با استفاده از روش فشرده ترکیبی مرتبه ششم. مجله ژئوفیزیک ایران، 6(4)، 35-49.
گلشاهی، ح. و قادر، س.، 1396، حل عددی معادلات آب کم عمق دو لایه بر حسب متغیرهای فشارورد و کژفشار با استفاده از روش فشرده مرتبه چهارم. مجله ژئوفیزیک ایران، 11(2)، 1-14.
Bleck, R. and Smith, L.T., 1990, A wind-driven isopycnic coordinate model of the north and equatorial Atlantic Ocean: 1. Model development and supporting experiments. Journal of Geophysical Research: Oceans, 95(C3), 3273-3285.
Bouchut, F., Ribstein, B. and Zeitlin, V., 2011, Inertial, barotropic, and baroclinic instabilities of the Bickley jet in two-layer rotating shallow water model. Physics of Fluids, 23(12), 126601, https://doi.org/10.1063/1.3661995
Chen, C., Liu, H. and Beardsley, R. C., 2003, An unstructured grid, finite-volume, three-dimensional, primitive equations ocean model: application to coastal ocean and estuaries. Journal of Atmospheric and Oceanic Technology, 20(1), 159-186.
Comblen, R., Blaise, S., Legat, V., Remacle, J. F., Deleersnijder, E. and Lambrechts, J., 2010, A discontinuous finite element baroclinic marine model on unstructured prismatic meshes. Ocean Dynamics, 60(6), 1395-1414.
Debreu, L., Marchesiello, P., Penven, P. and Cambon, G., 2012, Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation. Ocean Modelling, 49, 1-21.
Demange, J., Debreu, L., Marchesiello, P., Lemarié, F., Blayo, E. and Eldred, C., 2019, Stability analysis of split-explicit free surface ocean models: Implication of the depth-independent barotropic mode approximation. Journal of Computational Physics, 398, 108875, https://doi.org/10.1016/j.jcp.2019.108875
Dritschel, D. G., Polvani, L. M. and Mohebalhojeh, A. R., 1999, The contour-advective semi-Lagrangian algorithm for the shallow water equations. Monthly Weather Review, 127(7), 1551-1565.
Durran, D. R., 1999, Numerical methods for wave equations in geophysical fluid dynamics. Springer, 465.
Han, L., 2014, A two-time-level split-explicit ocean circulation model (MASNUM) and its validation. Acta Oceanologica Sinica, 33(11), 11-35.
Higdon, R. L., 2020, Discontinuous Galerkin methods for multi-layer ocean modeling: Viscosity and thin layers. Journal of Computational Physics, 401, 109018, https://doi.org/10.1016/j.jcp.2019.109018
Hirsh, R. S., 1975, Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique. Journal of Computational Physics, 19, 90-109.
Huang, H., Chen, C., Cowles, G. W., Winant, C. D., Beardsley, R. C., Hedstrom, K. S. and Haidvogel, D. B., 2008, FVCOM validation experiments: Comparisons with ROMS for three idealized barotropic test problems. Journal of Geophysical Research, 113(C7), C07042, https://doi.org/10.1029/2007JC004557.
Kang, H. G., Evans, K. J., Petersen, M. R., Jones, P. W. and Bishnu, S., 2021, A scalable semi-implicit barotropic mode solver for the MPAS Ocean. Journal of Advances in Modeling Earth Systems, 13(4), e2020MS002238.
Kantha, L. H. and Clayson, C. A., 2000, Numerical models of oceans and oceanic processes. Academic Press, 940.
Karsten, R. H. and Swaters, G. E., 1999, A unified asymptotic derivation of two-layer, frontal geostrophic models including planetary sphericity and variable topography. Physics of Fluids, 11(9), 2583-2597.
Lazure, P. and Dumas, F., 2008, An external-internal mode coupling for a 3D hydrodynamical model for applications at regional scale (MARS). Advances in Water Resources, 31(2), 233-250.
Lele, S. K., 1992, Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103, 16-42.
Madala, R. V. and Piacsek, S. A., 1977, A semi-implicit numerical model for baroclinic oceans. Journal of Computational Physics, 23, 167-178.
Mellor, G. L., 2004, Users guide for a three-dimensional, primitive equation, numerical ocean model (January 2004 version) In: Program in Atmospheric and Oceanic Sciences. Princeton University, Princeton, NJ 08544-0710, 56.
Morel, Y., Baraille, R. and Pichon, A., 2008, Time splitting and linear stability of the slow part of the barotropic component. Ocean Modelling, 23(3-4), 73-81.
O'Brien, J. J. and Hurlburt, H. E., 1972, A numerical model of coastal upwelling. Journal of Physical Oceanography, 2, 14-26.
Qiang, W., Zhou, W. and Wang, D., 2014, Implementation of new time integration methods in POM. Ocean Dynamics, 64(5), 643-654.
Shchepetkin, A. F. and McWilliams, J. C., 2005, The regional oceanic modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model. Ocean Modelling, 9(4), 347-404.
Simonnet, E., Ghil, M., Ide, K., Temam, R. and Wang, S., 2003, Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part I: Steady-state solution. Journal of Physical Oceanography, 33(4), 712-728.
Smith, R., Jones, P., Briegleb, B., Bryan, F., Danabasoglu, G., Dennis, J., Dukowicz, J., Eden, C., Fox-Kemper, B., Gent, P., Hecht, M., Jayne, S., Jochum, M., Large, W., Lindsay, K., Maltrud, M., Norton, N., Peacock, S., Vertenstein, M. and Yeager, S., 2010, The parallel ocean program (POP) reference manual ocean component of the community climate system model (CCSM) and community earth system model (CESM). LAUR-01853, 141, 1-140.
Spydell, M. and Cessi, P., 2003, Baroclinic modes in a two-layer basin. Journal of Physical Oceanography, 33(3), 610-622.
Tanaka, Y. and Akitomo, K., 2010, Alternating zonal flows in a two-layer wind-driven ocean. Journal of Oceanography, 66(4), 475-487.
Zhuang, Z., Yuan, Y. and Yang, G., 2018, An ocean circulation model in σ S-z-σ B hybrid coordinate and its validation. Ocean Dynamics, 68(2), 159-175.