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## A finite-volume, incompressible Navier–Stokes model for studies of the ocean on parallel computers. (1997)

Venue: | J. Geophys. Res., |

Citations: | 291 - 32 self |

### Citations

1369 |
Numerical Recipes in C
- Flannery
- 1988
(Show Context)
Citation Context ...a multiplier of the d matrix; see (33). Thus when q is set to zero (corresponding to the hydrostatic limit), A3r • is composed only of the blocks D and so can readily be inverted. Thus if we "precondition" (32) by premultiplying it by a matrix which is composed of the inverse of these blocks, that preconditioner will be an exact inverse of A3r• in the hydrostatic limit. These properties of A3r • will be exploited in our chosen method of solution. 4.2. Preconditioned Conjugate-Gradient Solution Method Many standard references to preconditioned conjugategradient methods exist [see, for example, Press et al., 1986, and references therein], but for the sake of completeness we briefly give the "hub" of the method here, emphasizing the use we make of preconditioners. Our problem is to find p given A and f (see (31) and (32)), where np= f (34) and A is a symmetric, positive-definite matrix. In serious ocean modeling applications the size of A is too large for direct methods to be possible in three dimensions and often in two dimensions too. Since A does not change in time, it could be inverted once and stored. However, although A is sparse, its inverse is dense and so operating with its inverse would invol... |

134 | Users guide for a three-dimensional, primitive equation, numerical ocean model - Mellor - 1996 |

93 | The westward intensification of wind-driven ocean currents, Eos Trans. - Stommel - 1948 |

59 |
On the pressure gradient force over steep topography in sigma-coordinate ocean models
- Haney
- 1991
(Show Context)
Citation Context ...iscrete form can no longer be represented by a symmetric, positive-definite matrix. Figure 1. A schematic diagram of an ocean basin showing the irregular geometry, coastlines and islands, in which the Navier Stokes equations are to be solved. The local depth of the ocean is H()t, •), where )t is the longitude and • is the latitude. The rigid lid can be replaced by a free surface, as described in Appendix 2. 2. There are difficulties in accurately representing horizontal pressure gradients near steep topography. The hydrostatic consistency condition is a rather stringent one [see, for example, Haney, 1991] but is not insurmountable [see McCalpin, 1994]' regional models using r/coordinates have been successfully constructed (see Mellor [1992] and Haidvogel et al. [ 1991]) and are now being developed further for global application. 3. It is not straightforward to represent V 2 diffusion terms; they must be transformed to height or density coordinates 4. A pure r/surface converges toward coastlines. Contrary to expectation, there appears to be little gained from using a vertical coordinate that is terrain following near the bottom of the ocean but like z near the surface. This is because the bott... |

58 | A semispectral primitive equation ocean circulation model using vertical sigma and orthogonal curvilinear horizontal coordinates - Haidvogel, Wilkin, et al. - 1991 |

35 | A reformulation and implementation of the Bryan-Cox-Semtner ocean model on the connection machine, - Dukowicz, Smith, et al. - 1993 |

20 | Approximate factorization as a high order splitting for the implicit incompressible flow equations,” - Dukowicz, Dvinsky - 1992 |

18 |
Computational Physics,
- Potter
- 1976
(Show Context)
Citation Context ... in exactly the same manner as the advection terms. The G '• + TM 2 is evaluated using the Adams-Bashforth method (AB2) which makes use of time levels n and n - 1; thus Gn+i/2 = [(37 + x)G n - (1j + X)Gn_i] (19) AB2 is a linear extrapolation in time to a point that is just, by an amount X, on the n + I side of the midpoint n + 1/2. AB2 has the advantage of being quasi-second-order in time and yet does not have a computational mode. Furthermore, it can be implemented by evaluating the G only once and storing them for use on the next time step. The limitation on the time step [see, for example, Potter, 1976, p. 69] is given by I A I At2 2 I At4 4 At < > + - 2 Ivl • A2 2 A4 where v is the fastest propagation velocity anywhere on the mesh of size A. Typically, we set X' to a value of 0.1. It should be noted that if X' = 0, then AB2 is unstable in the inviscid case. If "shaved cells" are being employed, the allowed time step may need to be reduced [see Adcrofi et al., 1996]. In practice, to avoid the need for very small time steps, a minimum size on the volume of a cell is imposed. The prognostic and diagnostic steps of our calculation in HPE, OH, and NH are outlined in the schematic diagram in F... |

13 | On the use of pressure as vertical coordinate in modeling convection, - Miller - 1974 |

9 |
Numerical integration of the three-dimensional Navier Stokes equations for incompressible flow,
- Williams
- 1969
(Show Context)
Citation Context ...% - Vh' (HV•p•7 •/'-•) (30) Here -r• is the discrete analogue of (I/H) fø_r• ( ) dz, a vertical integral over the whole depth of the ocean; we sum over the vertical faces of the zones each of depth Azlk making up the column of ocean. The "diVh" and "gradh" operators are horizontal components of (21) and (26), respectively. Note that on the right-hand side of (29) we have retained a IVy. (H•)] "+•- IVy. (H•)]" at At [Vh' (H•")]• which ensures that [Vh ß (H•"] " 1 _• 0 as the model steps forward. This "relaxation" technique is often used in solving the Navier Stokes equations [see, for example, Williams, 1969]. In the present context it obviates the need to step forward barotropic equations for v-•"separately. In our method, only the prognostic equations for interior velocities, (12), are stepped forward and the horizontal divergence of the depthintegrated horizontal velocities at each time step evaluated and used to modify the source function to the 2-D elliptic problem accordingly, so "tying together" the velocity and the pressure field. Since solid boundaries always coincide with the faces of zones, the imposition of boundary conditions in the formulation of the elliptic problem presents no pro... |

8 | Numerical algorithms for use in a dynamical model of the ocean, - Adcroft - 1995 |

6 | Computational design of the basic dynamical processes of - Arakawa, Lamb - 1977 |

4 | The relation of sea-floor voltages to ocean transports in North Atlantic Circulation models; model results and practical considerations for transport monitoring, - Flosadottir, Larsen, et al. - 1996 |

2 |
Implicit free surface for the CoxBryan-Semtner ocean model,
- Dukowicz, Smith
- 1994
(Show Context)
Citation Context ... rapidly. Appendix 1 In spherical coordinates, zones are defined by intersecting surfaces every (A/X, zX&, zXr) of (longitude, latitude, height), and (except when they abut irregular topography) the volumes of the zones and the surface areas of their faces are given by /t• •: rA&Ar; Ay ø = r cos &A/XAr; A• ø = r 2 COS &A&A/• I/ø= r 2 COS qbAqbAXAr Analogous expressions can be written down in other coordinate systems. Appendix 2: Implicit Free Surface The rigid-lid condition can be readily relaxed and the surface of the model ocean treated as a free surface. If implicit methods are used [e.g., Dukowicz and Smith, 1994], the same time step can be employed as for all other model variables. Moreover, the resulting modified 2-D elliptic equation forps is more easily inverted. Let us suppose that the elevation of the free surface h about Acknowledgments. This research was supported by grants from ARPA, TEPCO, and the Office of Naval Research. The model was developed on the CM5 housed in the Laboratory for Computer Science (LCS) at MIT as part of the SCOUT initiative. Much advice and encouragement on computer science aspects of the project was given by Arvind of LCS. Jacob White of the Department of Electrical E... |

1 | Treatment of topography in ocean models using finite-volumes, - Adcroft, Hill, et al. - 1997 |

1 | A small-scale dynamical model using a terrain-following coordinate transformation, - Clarke - 1977 |

1 | Time depended viscous flow, - Harlow, Welch - 1965 |

1 | A non-hydrostatic mesoscale ocean basin model, I, Well-posedness and scaling, - Mahadevan, Oliger, et al. - 1996 |

1 | Elimination of the Helmholtz Equation associated with the semi-implicit scheme in a grid-point model of the shallow water equations, - Tanguay, Robert - 1986 |

1 | Center for Meteorology and Physical Oceanography, Department of Earth, Atmospheric and Planetary Sciences, - Adcroft, Heisey, et al. - 1995 |