Abstract
The development of a variational method for solving the problem of quasi-geostrophic dynamics in a two-layer periodic channel is considered. The development of the method is as follows. First, the formulation of the variational problem is generalized: the turbulent exchange coefficient of a quasi-geostrophic potential vorticity (QGPV) is included in the control vector. Second, the solution area more accurately describes the size of the Antarctic Circumpolar Current (ACC). Using the selection of linear meridional transport and the expansion of the solution in a Fourier series, the problem is reduced to a nonlinear system of ordinary differential equations (ODEs) in time. The doubly connected domain leads to the fact that the solution of the ODE must satisfy an additional stationary relation that determines the transport of the ACC. The variational algorithm is reduced to solving a system of forward and adjoint equations minimizing the mean squared error of the stationary relation. The QGPV turbulent exchange coefficient is determined in the process of solving the optimal problem. The numerical runs are carried out for a periodic channel simulating the water area of the ACC in the Southern Ocean. The characteristics of stationary current regimes are studied for different values of the model parameters. Sinusoidal circulation in both layers with a linear transfer with the wind, depending on the bottom topography, is typical. In some cases, under the sinusoidal, in the lower layer, a cellular circulation is formed, and sometimes an undercurrent occurs. In this case, the solution of the optimal problem is characterized by a low value of the turbulent viscosity coefficient and a low transport in the lower layer.
REFERENCES
E. A. Volkov, “On the solution of a modified Dirichlet problem on a multiconnected polygon by the rapid block method,” Tr. Mat. Inst. im. Steklova 214, 135–144 (1997).
V. P. Dymnikov and V. B. Zalesny, Fundamentals of Computational Geophysical Fluid Dynamics (Geos, Moscow, 2019) [in Russian].
V. B. Zalesny, “Variation method for solving the quasi-geostrophic circulation problem in a two-layer ocean,” Izv., Fiz. Atmos. Ocean. Phys. 58 (5), 423–432 (2022).
V. O. Ivchenko and V. B. Zalesny, “Diffusion–rotational parameterization of eddy fluxes of potential vorticity: Barotropic flow in the zonal channel,” Izv., Fiz. Atmos. Ocean. Phys. 55 (1), 1–16 (2019).
V. M. Kamenkovich, “On the integration of equations of the theory of sea currents in nonsimply connected domains,” Dokl. Akad. Nauk SSSR 138 (5), 1076–1079 (1961).
G. I. Marchuk, Selected Scientific Works, Vol. 2: Adjoint Equations and Analysis of Complex Systems (RAN, Moscow, 2018) [in Russian].
G. I. Marchuk, Selected Scientific Works, Vol. 3: Models and Methods in Problems of Atmospheric and Oceanic Physics (RAN, Moscow, 2018) [in Russian].
N. I. Muskhelishvili, Singular Integral Equations (Gostekhizdat, Moscow–Leningrad, 1946) [in Russian].
A. A. Pavlushin, N. B. Shapiro, E. N. Mikhailova, and G. K. Korotaev, “Two-layer Eddy-resolving model of wind currents in the Black Sea,” Phys. Oceanogr., No. 5, 3–22 (2015).
V. P. Shutyaev, “Methods for observation data assimilation in problems of physics of atmosphere and ocean,” Izv., Fiz. Atmos. Ocean. Phys. 55 (1), 17–31 (2019).
V. I. Agoshkov and V. M. Ipatova, “Convergence of solutions to the problem of data assimilation for a multilayer quasigeostrophic model of ocean dynamics,” Russ. J. Numer. Anal. Math. Modell. 25 (2), 105–115 (2010).
C. Bernier, “Existence of attractor for the quasi-geostrophic approximation of the Navier–Stokes equations and estimate of its dimension,” Adv. Math. Sci. Appl. 4 (2), 465–489 (1994).
C. Bernier and I. D. Chueshov, “The finiteness of determining degrees of freedom for the quasi-geostrophic multi-layer ocean model,” Nonlinear Anal. Theory, Methods Appl. 42 (8), 1499–1512 (2000).
M. Cai, M. Hernandez, K. W. Ong, and S. Wang, “Baroclinic instability and transitions in a two-layer quasigeostrophic channel model” (2017). https://arxiv.org/pdf/1705.07989.
J. G. Charney, J. Shukla, and K. C. Mo, “Comparison of a barotropic blocking theory with observation,” J. Atmos. Sci. 38, 762–779 (1981).
M. D. Chekroun, H. Dijkstra, T. Sengul, and S. Wang, “Transitions of zonal flows in a two-layer quasi-geostrophic ocean model,” Nonlinear Dyn. 109, 1887–1904 (2022).
Q. Chen, “The barotropic quasi-geostrophic equation under a free surface,” SIAM J. Math. Anal. 51 (3), 1836–1867 (2017).
Q. Chen, “On the well-posedness of the inviscid multi-layer quasi-geostrophic equations,” Discrete Contin. Dyn. Syst. 39 (6), 3215–3237 (2019).
A. Farhat, R. L. Panetta, E. S. Titi, and M. Zian, “Long-time behavior of a two-layer model of baroclinic quasi-geostrophic turbulence,” J. Math. Phys. 53, 115603 (2012).
J. C. Gilbert and C. L. Lemarechal, The modules M1QN3 and N1QN3, Version 2.0c (June, 1995).
V. O. Ivchenko, V. B. Zalesny, and B. Sinha, “Is the coefficient of eddy potential vorticity diffusion positive? Part 1: Barotropic zonal channel,” J. Phys. Oceanogr. 48 (6), 1589–1607 (2018).
J. C. McWilliams, Fundamentals of Geophysical Fluid Dynamics (Cambridge University Press, Cambridge, 2006).
C. Onica and R. L. Panetta, “Forced two layer beta-plane quasigeostrophic flow. Part I: Long-time existence and uniqueness of weak solutions,” J. Differ. Equations 226 (1), 180–209 (2006).
J. Pedlosky, “Finite-amplitude baroclinic waves,” J. Atmos. Sci. 27 (1), 15–30 (1970).
X. Xu, E. P. Chassignet, Y. L. Firing, and K. Donohue, “Antarctic circumpolar current transport through drake passage: What can we learn from comparing high-resolution model results to observations?,” J. Geophys. Res.: Oceans 125 (7), 1–16 (2020).
V. Zalesny, V. Agoshkov, V. Shutyaev, E. Parmuzin, and N. Zakharova, “Numerical modeling of marine circulation with 4D variational data assimilation,” J. Mar. Sci. Eng. 8, 503 (2020).
Funding
This work was supported by the Russian Science Foundation, grant no. 20-11-20057.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares that he has no conflicts of interest.
Rights and permissions
About this article
Cite this article
Zalesny, V.B. Two-Layer Ocean Circulation Model with Variational Control of Turbulent Viscosity Coefficient. Izv. Atmos. Ocean. Phys. 59, 189–200 (2023). https://doi.org/10.1134/S000143382302010X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143382302010X