Internal pressure errors in sigma-coordinate ocean models—sensitivity of the growth of the flow to the time stepping method and possible non-hydrostatic effects
Introduction
Sigma coordinate ocean models, or models based on more generalised topography following coordinate systems, are presently widely used in oceanographic studies. From the home page for the Princeton Ocean Model (POM) (Blumberg and Mellor, 1987) we for instance find that this model alone has more than 1100 users from 59 countries in year 2002. The abilities of such models to resolve the bottom and surface layers are attractive. However, the controversy over internal pressure errors in sigma coordinate ocean models is still worrying to some of the users. Geophysical flow is to a large extent determined by the balance between internal pressure and the Coriolis force. If there are large errors in the estimates of internal pressure, the estimated circulation will be wrong.
To our knowledge Haney, 1991 was the first to focus on the problem with the pressure gradient force over steep topography in sigma coordinate ocean models and hydrostatic consistency. The literature in numerical oceanography traces back to earlier publications in numerical meteorology on pressure gradients in topography following coordinates, and Haney also summarises some of this literature. See for instance Sundquist, 1975, Sundquist, 1976, Janjic (1977), Mesinger (1982) and Mesinger and Janjic (1985).
The source of the problem is that in -coordinates, the x-component of the internal pressure is writtenwhere x is the horizontal coordinate, z the vertical coordinate, the density, H the depth, and . Near steep topography the two terms on the right may be large, comparable in magnitude, and often opposite in sign. This may cause large errors in the estimates of the internal pressure.
Introducing the buoyancy where g is the gravity constant and a constant reference density Mellor et al. (1994) using Taylor expansions studied the discretisation error for the 2nd order internal pressure method used in POM. The error is to leading orderThe error term (2) shows that the error will decrease as and both tend to zero. It also shows that for a given and the error will be limited as goes to zero. On the other hand to add more layers does not improve the quality when the term already is the limiting factor. The error term (2) give hope in the sense that as computers get more powerful and we may reduce and , we may sooner or later get to the point where we do not have to worry about internal pressure errors. However, for the foreseeable future the horizontal resolution in models covering large geographical areas and containing steep shelf slopes and underwater sea-mounts will not be sufficient and the internal pressure errors may still be worrying. For cases where the buoyancy is a function of z only, and for idealised density profiles, the internal pressure errors may be translated to erroneous geostrophic velocities. This is done for instance in Haney (1991) and in Slørdal (1995). It is not hard to demonstrate errors greater than in geostrophic velocities.
Mellor et al. (1994) showed that in diagnostic experiments, the pressure errors are maintained in time as an input of potential energy and the mean kinetic energy does not tend to zero even if the system is not externally forced. In prognostic experiments, the density field is allowed to be advectively adjusted, and it is demonstrated that for this case the numerical potential energy may go to zero and that the mean kinetic energy accordingly will die out.
Beckmann and Haidvogel (1993) studied flow trapped to a seamount. For the non-forced case and horizontal stratification, eight eddies around the seamount were growing for larger Burger numbers and they did not die out prognostically. Thus for 3D cases allowing vorticity to develop, it was demonstrated that the effects of initial internal pressure errors may even grow in time.
The seamount case was addressed in Mellor et al. (1998). They categorised the errors reported in 1994 as sigma errors of the first kind (SEFK). These errors are associated with 2D () flow. They categorised the vorticity errors reported by Beckmann and Haidvogel as sigma errors of the second kind (SESK). The internal pressure errors create first errors in the velocities, which again create a compensating error in the density field. In the SESK case the velocities driven by the perturbed density field do not die out. In their experiments with apparently comparable viscosities Mellor et al. reported much smaller erroneous velocities than Beckmann and Haidvogel and they did not grow in time. Mellor et al. (1998) applied a harmonic viscosity of which was scaled to conform approximately to the biharmonic coefficient applied in Beckmann and Haidvogel (1993) at the grid size of 6 km. The vortices, however, have approximately the same length scales as the seamount, which is 25 km, and errors at the 25 km scale are more strongly suppressed by harmonic viscosity. If the vortices are allowed to grow, as they are in Beckmann and Haidvogel (1993), there will be vertical exchanges of water masses that create real and much stronger internal pressure and associated geostrophic flow. To avoid that this happens, the viscosity must be large enough to suppress the strong initial growth of the vortices, see Mellor et al. (1998) and Berntsen (2002).
Many experiments so far, for instance the experiments described in Mellor et al. (1998), are performed with large values of horizontal viscosities. For the large viscosity case, the initial growth of velocities due to internal pressure gradient errors may be controlled by viscosity, and therefore the erroneous velocities may be kept at an acceptable level. On the other hand, in coarse grid size experiments with viscosities at a level that is reasonable from physical considerations, the errors in the velocities may grow to unacceptable levels, see Berntsen (2002) and Shchepetkin and McWilliams (2003a).
In several recent papers new algorithms that attempt to reduce the size of the internal pressure errors, and thereby the growth of the erroneous flow, are given. The use of higher order approximations is an obvious method for reducing the local errors, see McCalpin (1994) and Chu and Fan, 1997, Chu and Fan, 2003. Weighted Jacobian methods have been suggested, see Song (1998), Song and Wright (1998) and Shchepetkin and McWilliams (2003a) also suggest a more accurate method for estimating the internal pressure. It has been suggested to interpolate the density back to z-levels to calculate the pressure gradient force. In this category we find the method suggested in Stelling and Van Kester (1994). This is to our knowledge the only method that guarantees that if the gradients are zero, the estimated internal pressures also are zero. In his Ph.D. thesis Slørdal (1995) stated that the pressure gradient force may be underestimated using this approach and suggested linear interpolation in the vertical. Kliem and Pietrzak (1999) compare alternative algorithms focusing on geostrophic currents, and as a general conclusion they favours the use of z-level based methods. In the evaluation of horizontal gradients in sigma-coordinate shallow water models given in Fortunato and Baptista (1996) they state that overall, the sigma coordinates are considered to be the best approach. To the present authors it is still not evident that it is preferable to compute the internal pressure gradient in terrain following models in z-coordinates, even if it is used in many model studies, see for instance Xing and Davies (2001).
It is important to reduce the errors in the estimated forces. However, the stability of the solution method may also be essential. If local errors are growing in time, it may be of little value that they are initially small. If local errors are damped by the solution technique, one may be able to produce accurate results even if locally the internal pressure errors are non-negligible. Therefore, there should also be focus on how growth or damping of the internal pressure errors is affected by other algorithmic or model choices made in terrain following models. In the present paper it is investigated how the time stepping method may affect the response of the flow to the artificial pressure.
As the grid size is reduced, the internal pressure errors will tend to zero. However, with finer resolution the validity of the hydrostatic assumption becomes questionable. In Marshall et al. (1997) it is suggested that for length scales from somewhere between 1 and 10 km and downwards, non-hydrostatic effects may become important. Introducing non-hydrostatic pressure into an ocean model may strongly affect the internal wave propagation and overturning, see Legg and Adcroft (2003). In the present paper it is investigated how non-hydrostatic pressure may affect the growth of internal pressure errors.
Section snippets
The seamount case
The seamount problem was defined by Beckmann and Haidvogel (1993). In the present study, we want to compare our results to the more recent results presented in Shchepetkin and McWilliams (2003a), and the specifications are therefore as in their study.
The bottom topography is defined aswhere and . The seamount is placed in the centre of a channel that is 320 km long in the -direction and 320 km wide in the -direction. The channel is closed at
Numerical methods
In the horizontal, a uniform grid with grid cells is used in all experiments to be described. The grid spacing is . The -coordinate ocean model is described in Berntsen (2000) and available from www.mi.uib.no/BOM/. The governing equations are basically the same as for the POM (Blumberg and Mellor, 1987, Mellor, 1996), but the numerical methods are different. The variables are discretised on a C-grid. In the vertical, the standard -transformation, , where z is the
Stability analysis
Stability analysis of ordinary differential equations (ODEs) is a classic topic in numerical analysis. This sections contains a brief application of that theory to some selected methods. See Dahlquist (1985), Henrici (1962), Gear (1971) Hairer et al. (1987) and/or Hairer and Wanner (1991) for further analysis of numerical methods for ODEs.
Let us consider the ODE for some . A numerical scheme applied to this ODE is defined to be absolutely stable if it produces bounded solutions as
Sensitivity to the time stepping method and the time step
The solution is propagated forward in time with the following methods:
- (i)
the Euler forward method,
- (ii)
the Euler forward predictor, trapezoidal method corrector pair, and
- (iii)
the Euler forward predictor, Euler backward corrector pair
Sensitivity to non-hydrostatic pressure corrections
For the seamount case described in Section 2 and spatial resolution described in Section 3, it could be expected that the response of the model to the internal pressure errors could be affected by non-hydrostatic pressure effects, see Marshall et al. (1997). In Heggelund et al. (2004) a method to estimate the non-hydrostatic pressure in a -coordinate model is described. During each full model time step a sequence of operations are performed to ensure that all physical effects are taken into
Concluding remarks
Errors in the estimates of the internal pressure in -coordinate ocean models may create artificial flow. A number of recent papers have addressed this problem, and there have been many attempts to reduce the errors and the corresponding growth of the flow by reducing the local errors by introducing for instance higher order estimates of the gradients.
There has been less attention to other aspects of the ocean models that may be essential to the growth of the errors. In the present paper it is
Acknowledgements
This work has been supported by Norsk Hydro grant NH-5203909, and has also received support from The Research Council of Norway through a grant of computing time (Programme for Supercomputing). Two anonymous referees are thanked for useful remarks.
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