Non-hydrostatic pressure in σ-coordinate ocean models

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1 Non-hydrostatic pressure in σ-coordinate ocean models Eirik Keilegavlen and Jarle Berntsen May, 7 Abstract In many numerical ocean models, the hydrostatic approximation is made. This approximation causes a considerable saving in computing time. However, if one wants to increase the grid resolution and resolve phenomena on small scale, the approximation becomes questionable. A description of how to extend a σ-coordinate hydrostatic C-grid model to include non-hydrostatic dynamics is given. The test cases are.. Introduction With improvements in computing power, it becomes possible to use increasingly finer meshes in numerical ocean models. At some limit, depending on the problem studied, the hydrostatic assumption made in most ocean models will no longer be a good approximation for all the time and length scales resolvable by the model. Further refinements of the grid should therefore include nonhydrostatic physics. Several numerical models have been built that includes nonhydrostatic physics, see Mahadevan et al. (99a), Mahadevan et al. (99b), and Stansby and Zhou (99). These models are built with focus on mesoscale and small scale studies, but because of their nonhydrostatic nature, they probably are too expensive to use on a larger model area. As far as we know, these models cannot be run hydrostatically. This paper describes a relatively easy way to extend the existing σ - coordinate Bergen Ocean Model (BOM) (Berntsen, ) to include nonhydrostatic physics through a velocity - correction term. A similar extension of an atmospheric model has been implemented by Dudhia (99), but we have not seen anyone make a similar extension of an ocean model. The velocity

2 Eirik Keilegavlen and Jarle Berntsen - correction can be switched on and off by the user. With local grid refinement implemented, an idea is that the the nonhydrostatic correction should be switched on when the grid resolution is fine enough to resolve nonhydrostatic phenomena. This will save much computing time, since running the whole model area with nonhydrostatic physics included will be very expensive and unnecessary as the hydrostatic approximation probably is good for the horizontal length scales resolvable by the coarsest grid. This switch also makes it easy to study the effect of nonhydrostatic physics on a given test case.

3 Non-hydrostatic pressure in σ-coordinate ocean models Numerical studies of flow over a sill in a fjord The set up of the numerical experiments is described in (Berntsen et al., 7), and is very similar to that used in (Xing and Davies, b; Xing and Davies, a). It is also recently documented that non-hydrostatic pressure effects are very important when studying super-critical flows over sills (Xing and Davies, 7). The topography and the initial stratification are given in Fig.. The initial buoyancy frequency N is.s. The depth profile H(x) in m is specified according to { + +(x/w H(x) = sill, x < ) + +(x/w sill, x > ) assuming x = m at the top of the sill and W sill = m is the width of the sill. The Coriolis frequency f =. s. In this study there will be flow in the cross fjord direction due to the Coriolis effect, but there will be no cross fjord variation in this flow. There is no flow through the sea bed or the closed right end of the fjord. Initially the water elevation is η, and there is no flow. The M tide is forced into the fjord by specifying a velocity in the x-direction u as u = u tide c(η η tide )/H I, z at the left boundary. In the above equation H I is the depth at the inflow point, c = gh I, η is the water elevation just inside the left open boundary, η tide is the forced tidal water elevation, and u tide is the corresponding forced tidal velocity. The tidal forcing is computed from a standing wave solution given in (Gill, 9) p. and η tide = η cos(kl x ) cos(ω M t), u tide = cη H I sin(kl x ) sin(ω M t), where ω M =.rad s, k = ω M /c, and L x the length of the channel (L x = m). The value of η is.m giving super-critical flow over the sill at maximum inflow as in (Xing and Davies, b). The standing wave solution is strictly for a channel with constant depth. However, by forcing with the above open boundary condition the tidal signal propagate with very little disturbances into the domain and can radiate through the

4 Eirik Keilegavlen and Jarle Berntsen open boundary on outflow. For the density a Neumann boundary condition is applied at the open boundary. In the vertical equidistant σ-layers are used, and horizontally the grid applied is equidistant and the grid size is m. The time step used is.s, and external time steps are used for each internal step. The experiments are run over two tidal cycles. The experiments are performed with constant values of viscosity and diffusivity as in (Xing and Davies, b), that is: horizontal viscosity A h = m s, vertical viscosity A V = m s, horizontal diffusivity K h = 7 m s, and vertical diffusivity K V = 7 m s. In Figs.,, and the non-hydrostatic pressure, and u, and w, at maximum inflow during the second tidal cycle are given. The velocity fields are generally very consistent in all experiments. The sensitivity to the method for estimating the non-hydrostatic pressure is smaller than the corresponding sensitivity to horizontal grid size and sub-grid closure, see (Berntsen et al., 7). The non-hydrostatic pressures are more sensitive to the choice of surface boundary condition, p = or p =, than to the choice of method, (?) n or (?). The non-hydrostatic pressures for p = are also consistent with the n pressures computed with the z-coordinate model MITgcm, see (Xing and Davies, 7). When using p = as surface boundary condition, convergence or divergence of the depth integrated flow may occur, and the surface elevation will adjust accordingly, see Fig.. The compensating adjustments of the gradients of the surface elevations may partly explain that even if the nonhydrostatic filed are different, the velocity fields may still be consistent. It is the gradients of the total pressure that drive the flow.

5 Non-hydrostatic pressure in σ-coordinate ocean models 7 P (ci=. N/m**) P (ci=. N/m**) P (ci=. N/m**) 7 9 (c) 7 9 P (ci=. N/m**) 7 9 (d) Figure : The non-hydrostatic pressure fields after t = / T for a) P as in BF, b) P as in BF with P = at surface, c) P as in KM, and d) P as in KM with P =

6 ... Eirik Keilegavlen and Jarle Berntsen RHO (ci=. ) RHO (ci=. ) RHO (ci=. ) (c) RHO (ci=. ) (d) Figure : The density fields after t = / T for a) P as in BF, b) P as in BF with P = at surface, c) P as in KM, and d) P as in KM with P =

7 Non-hydrostatic pressure in σ-coordinate ocean models U (ci=. cm/s) U (ci=. cm/s) U (ci=. cm/s) 7 9 (c) 7 U (ci=. cm/s) (d) Figure : The U velocities after t = / T for a) P as in BF, b) P as in BF

8 7 Eirik Keilegavlen and Jarle Berntsen 7 9 W (ci=. cm/s) W (ci=. cm/s) W (ci=. cm/s) (c) W (ci=. cm/s) (d) Figure : The W velocities after t = / T for a) P as in BF, b) P as in BF

9 Non-hydrostatic pressure in σ-coordinate ocean models 9.. Surface elevation [m] Surface elevation [m] Figure : The surface elevations after t = / T for a) P as in BF, b) P as in KM. Solid lines for dp/dn = at the surface, and dashed lines for P= at the surface.

10 Eirik Keilegavlen and Jarle Berntsen Surface pressure [N/m**] 7 7 Surface pressure [N/m**] 7 7 Figure : The surface pressures after t = / T for a) P as in BF, b) P as in KM. Solid lines for dp/dn = at the surface, and dashed lines for P= at the surface.

11 Non-hydrostatic pressure in σ-coordinate ocean models References Berntsen, J. (). USERS GUIDE for a modesplit σ-coordinate numerical ocean model. Technical Report, Dept. of Applied Mathematics, University of Bergen, Johs. Bruns gt., N- Bergen, Norway. p. Berntsen, J., Xing, J., and Davies, A. (7). On the sensitivity of internal waves and mixing to horizontal resolution and sub-grid scale mixing. Manuscript in preparation for Ocean Modelling. Dudhia, J. (99). A nonhydrostatic version of the Penn State-NCAR mesoscale model: Validation tests and simulation of an Atlantic cyclone and cold front. Montly Weather Review, :9. Gill, A. (9). Atmosphere-Ocean Dynamics. Academic Press. ISBN Mahadevan, A., Oliger, J., and Street, R. (99a). A Nonhydrostatic Mesoscale Ocean Model. Part I: Well posedness and scaling. Journal of Physical Oceanography, :. Mahadevan, A., Oliger, J., and Street, R. (99b). A Nonhydrostatic Mesoscale Ocean Model. Part II: Numerical implementation. Journal of Physical Oceanography, : 9. Stansby, P. and Zhou, J. (99). Shallow-Water Flow Solver With Nonhydrostatic Pressure: D Vertical Plane Problem. International Journal for Numerical Methods in Fluids, :. Xing, J. and Davies, A.. Influence of stratification and topography upon internal wave spectra in the region of sills. Geophysical Research Letters, :L doi:.9/gl. Xing, J. and Davies, A.. Processes influencing tidal mixing in the region of sills. Geophysical Research Letters, :L doi:.9/gl. Xing, J. and Davies, A. (7). On the importance of non-hydrostatic processes in determining tidally induced mixing in sill regions. Submitted to Continental Shelf Research.