Conservation of energy for schemes applied to the propagation of shallow‐water inertia‐gravity waves in regions with varying depth

Espelid, Terje O.; Berntsen, Jarle; Barthel, Knut

doi:10.1002/1097-0207(20001230)49:12<1521::aid-nme9>3.3.co;2-6

Cited by 4 publications

(10 citation statements)

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“…The main results in Reference [19] are given in the following theorem: (3) (a) will be conserved when f = 0 (the depth H (x; y) may vary over ); (b) will be conserved when f = 0 and the depth is constant over .…”

Section: The Methods Of Lines and The C-gridmentioning

confidence: 99%

“…Following the proof found in Reference [19] let us deÿne the following matrices P = (D −1 AD − DAD −1 )=2 and S = (D −1 AD + DAD −1 )=2 + D −1 BD. S is skew-symmetric while the matrix P is symmetric.…”

Section: The Methods Of Lines and The C-gridmentioning

confidence: 99%

“…In a recent paper, Espelid and Berntsen [19] focus on the stability using the C-grid centered di erences in space and using di erent numerical time-stepping methods on problems with varying depth. They demonstrate that varying depth may imply that the stability is lost no matter how small time step one uses.…”

Section: Introductionmentioning

confidence: 99%

“…In the next section we will present some of the results from Reference [19], then we will present the necessary modiÿcations and ÿnally demonstrate both the problem of varying depth and the e ect of the modiÿcation on a number of problems. First we follow [19] and limit the discussion to a model problem which has some general features.…”

Section: Introductionmentioning

confidence: 99%

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Conservation of energy for schemes applied to the propagation of shallow-water inertia-gravity waves in regions with varying depth

Espelid

Berntsen

Barthel

2000

Int. J. Numer. Meth. Engng.

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The linear equations governing the propagation of inertia‐gravity waves in geophysical fluid flows are discretized on the Arakawa C‐grid using centered differences in space. In contrast to the constant depth case it is demonstrated that varying depth may give rise to increasing energy (and loss of stability) using the natural approximations for the Coriolis terms found in many well‐known codes. This is true no matter which numerical method is used to propagate the equations. By a simple trick based on a modified weighting that ensures that the propagation matrices for the spatially discretized equations become similar to skew‐symmetric matrices, this problem is removed and the energy is conserved in regions with varying depth too. We give a number of examples both of model problems and large‐scale problems in order to illustrate this behaviour. In real applications diffusion, explicit through frictional terms or implicit through numerical diffusion, is introduced both for physical reasons, but often also in order to stabilize the numerical experiments. The growing modes associated with varying depth, the C‐grid and equal weighting may force us to enhance the diffusion more than we would like from physical considerations. The modified weighting offers a simple solution to this problem. Copyright © 2000 John Wiley & Sons, Ltd.

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Section: The Methods Of Lines and The C-gridmentioning

confidence: 99%

Section: The Methods Of Lines and The C-gridmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Conservation of energy for schemes applied to the propagation of shallow-water inertia-gravity waves in regions with varying depth

Espelid

Berntsen

Barthel

2000

Int. J. Numer. Meth. Engng.

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“…(See [Espelid et al (2000)], where it is shown that by four-point averaging the transport divided by the square root of the depth gives an energy conserving treatment of the Coriolis force.) This energy can be calculated explicitly, and the last row in Table 2 gives its contribution to the budget region.…”

Section: A Mechanical Energy Budgetmentioning

confidence: 99%

A mechanical energy budget for the North Sea

Barthel

Gade

Sandal

2004

Continental Shelf Research

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A three-dimensional baroclinic numerical model is used to quantify the components of the mechanical energy budget for a semi-enclosed sea, the North Sea. As anticipated, tidal energy is the largest supplier of mechanical energy to the North Sea. Although the energy supply due to winds is responsible for the seasonal variations occurring in the budget, it is one order of magnitude smaller than that due to tides. Most of the input tidal energy is dissipated, specifically in the shallow southern region. Estimates of dissipation and energy flux due to tidal activity are compared to earlier calculations in an attempt to partially validate the results.

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The numerics of hydrostatic structured-grid coastal ocean models: State of the art and future perspectives

et al. 2018

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The state of the art of the numerics of hydrostatic structured-grid coastal ocean models is reviewed here. First, some fundamental differences in the hydrodynamics of the coastal ocean, such as the large surface elevation variation compared to the mean water depth, are contrasted against large scale ocean dynamics. Then the hydrodynamic equations as they are used in coastal ocean models as well as in large scale ocean models are presented, including parameterisations for turbulent transports. As steps towards discretisation, coordinate transformations and spatial discretisations based on a finite-volume approach are discussed with focus on the specific requirements for coastal ocean models. As in large scale ocean models, splitting of internal and external modes is essential also for coastal ocean models, but specific care is needed when drying & flooding of intertidal flats is included. As one obvious characteristic of coastal ocean models, open boundaries occur and need to be treated in a way that correct model forcing from outside is transmitted to the model domain without reflecting waves from the inside. Here, also new developments in two-way nesting are presented. Single processes such as internal inertia-gravity waves, advection and turbulence closure models are discussed with focus on the coastal scales. Some overview on existing hydrostatic structured-grid coastal ocean models is given, including their extensions towards non-hydrostatic models. Finally, an outlook on future perspectives is made.

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Conservation of energy for schemes applied to the propagation of shallow‐water inertia‐gravity waves in regions with varying depth

Cited by 4 publications

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Conservation of energy for schemes applied to the propagation of shallow-water inertia-gravity waves in regions with varying depth

Conservation of energy for schemes applied to the propagation of shallow-water inertia-gravity waves in regions with varying depth

A mechanical energy budget for the North Sea

The numerics of hydrostatic structured-grid coastal ocean models: State of the art and future perspectives

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