Hydrostatic Consistency in Sigma Coordinate Ocean Models

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1 Hydrostatic Consistency in Sigma Coordinate Ocean Models Peter C. Chu and Chenwu Fan Naval Ocean Analysis and Prediction Laboratory, Deartment of Oceanograhy Naval Postgraduate School, Monterey, California

2 Abstract Truncation error and hydrostatic inconsistency at stee toograhy are two concerns in sigma coordinate ocean models due to the horizontal ressure gradient being difference of two large terms. A consensus is reached in the ocean modeling community on the first concern (truncation error), but not on the second concern (hydrostatic inconsistency). Since the integration of the ressure gradient over a finite volume equals the integration of the ressure over the surface of that volume (always dynamically consistent), dynamical analysis on finite volumes is used to determine the hydrostatic consistency of a sigma coordinate ocean model. A discrete, hydrostatically consistent scheme is obtained for the sigma coordinate ocean models. Comarison between finitevolume and finite-difference aroaches leads to the conclusion that a Boussinesq, hydrostatic, sigma coordinate ocean model with second-order staggered scheme is always hydrostatically consistent. Guidance for imroving numerical accuracy is also rovided. 2

3 . Introduction In regional oceanic (or atmosheric) rediction models the effects of bottom toograhy must be taken into account and a continuous toograhy is imlied in terrainfollowing sigma coordinates. The water column is divided into the same number of grid cells indeendence of deth. We restrict attention to two dimensions. Let (x, z) denote Cartesian coordinates and (x*, σ ) be the sigma coordinates. The conventional relationshi between z- and sigma-coordinates is given by x = x*, z = σ H( x*), () where z and σ increase vertically uward such that z = σ = 0 at the surface and σ =, z = -H at the bottom. The horizontal ressure gradient can be comuted by * σ H * =. (2) x x* H x* σ The horizontal ressure gradient becomes difference between two large terms, which may cause two roblems: () truncation error at stee toograhy [e.g., Gary, 973; Haney, 99; Mellor et al., 994, 998; McCalin, 994; Chu and Fan, 997, 998; Song, 998], and (2) hydrostatic inconsistency [e.g., Mesinger, 984; Haney, 99]. A consensus is reached in the ocean modeling community that the first roblem does exist and several methods have been suggested to reduce the truncation errors to accetable levels: () smoothing toograhy [e.g., Chu and Fan, 200], (2) subtracting a mean vertical density rofile before calculating the gradient [Gary, 973], (3) bringing certain symmetries of the continuous forms into the discrete level to ensure cancellations of these terms such as the density Jacobian scheme [e.g., Mellor et al. 998; Song, 998; 3

4 Song and Wright, 998], (4) increasing numerical accuracy [e.g., McCalin, 994; Chu and Fan, 997, 998, 999, 2000, 200], (5) changing the grid from a sigma grid to a z- level grid before calculating the horizontal ressure gradient [e.g., Stelling and van Kester, 994]. Kliem and Pietrzak [999] claimed that the z-level based ressure gradient calculation is the most simle and effective means to reduce the ressure gradient errors. However, Ezer et al. [2002] found that the density Jacobian scheme is more referable. No consensus is reached on whether the second roblem exists. Based on the earlier work for atmosheric models [e.g., Messiger, 982], Haney [99] ointed out that the vertical discretization in sigma coordinate ocean models ( δσ ) should satisfy the hydrostatic consistency condition, σδ H x r < (3) H δσ to kee the comutational stability. Here r is the hydrostatic consistency arameter; δ H x is the horizontal change in deth of adjacent grid cells; and δσ is the vertical cell size associated with a sigma grid, δxδσ. However, Mellor et al. [994] thought that r is just another measure of the numerical accuracy after conducting a numerical simulation for the North Atlantic Ocean using the Princeton Ocean Model with r = 3. More numerical exeriments with various schemes for the seamount test case [e.g., Ezer et al., 2002] were conducted to show convergent solutions with r = 4.2. These exeriments show that the condition (3) is not the ultimate condition for numerical calculation, but the indication of the first (second) term in the righthand-side of (2) is larger if r < (r > ). Does the hydrostatic inconsistency regarding to the comutational instability really exist? We use the finite volume integration aroach [Lin, 997] to reexamine the concet of hydrostatic consistency (regarding the stability) in this aer. A fully, 4

5 hydrostatically consistent (i.e., hydrostatically stable) grid scheme is develoed for sigma coordinate ocean models. This scheme rovides a criterion for the identification of hydrostatic consistency for various finite difference schemes. The outline of this art is as follows: Descrition of the hydrostatic consistency is given in section 2. A hydrostatically consistent staggered scheme for horizontal ressure gradient is given in Section 3. Evidence of second-order staggered sigma ocean model is always hydrostatically consistent is given in Sections 4 and 5. In section 6, the conclusions are resented. 2. Hydrostatic Consistency Let the flow field change in x z lane only (Fig. ). A finite volume (traezoidal cylinder) is considered with the length of L y (in the y-direction) and the cross-section reresented by the shaded region (traezoid) in Figure. The resultant ressure force (F) acting on the finite volume is comuted as follows: F= L n ds (4) y C where is the ressure, C reresents the four boundaries, n denotes the normal unit vector ointing inward, and ds is an element of the boundary. The contour integral is taken counter-clockwise along the eriheral of the volume element. The ressure force exerts on boundaries of the finite-volume with w, e, u, and l on the west, east, uer, and lower sides. The horizontal (F x ) and vertical (F z ) comonents of the resultant ressure force are comuted by Fx L = y dz l + edz+ udz+ wdz, (5)

6 2 4 Fz L = y dx l + udx, (6) 3 where oints, 2, 3, and 4 are the four vertices of the finite volume. For hydrostatic balanced models, the following condition must hold Fz = g m, (7) where g is the gravitational acceleration, m is the mass of the finite volume. Equation (7) states that the vertical comonent of the resultant ressure force acting on the finite volume exactly balances the total weight of the finite volume. For a Boussinesq, hydrostatic ocean model, the ressure field is given by 0 = + ρ0 gη+ g ρ( x, z', t) dz ', (8) atm z where atm is the atmosheric ressure at the ocean surface, ρ 0 is the characteristic density, and η is the surface elevation. Substitution of (8) into (6) leads to Fz = gly ρ( x, z ', t) dz ' dx + ρ( x, z ', t) dz ' dx z 3 z = gl ρ( x, z ', t) dz ' dx = g m, y S (9) where S is the area of the traezoid (Fig. ) comuted by S = ( xi+ xi) ( zi, k + zi+, k zi, k+ zi+, k+ ), zik, = Hi σ k. (0) Eq.(9) indicates that the finite-volume discretization guarantees the hydrostatic balance in Boussinesq, hydrostatic ocean models. Using (5) the horizontal ressure gradient is comuted by 6

7 2 3 4 F x = ldz + edz + udz + wdz x Ly S S () If the horizontal ressure gradient is reresented by (), the model is conserved and hydrostatic inconsistency does not exist. Thus, deviation from the hydrostatic consistency becomes deviation of the horizontal ressure gradient comutation from (). 3. Staggered Grid The staggered grid is reresented in Figure with the velocity at the center of the volume and ressure at the four vertices. Discretization of the horizontal ressure gradient with the finite-volume consideration () is given by = l( zi+, k+ zi, k+ ) + e( zi+, k zi+, k+ ) + u( zi, k zi+, k) + w( zi, k+ zi, k x S ), (2) where l, e, u, w are the mean values of ressure at the four sides of the traezoid. Equation (2) is the criterion for justifying the hydrostatic consistency for ocean model with staggered grid. If the horizontal ressure gradient in sigma coordinates (2) can be reresented by (2), the model is hydrostatically consistent. Otherwise the model may be hydrostatically inconsistent. For ocean models with C-grid, the two consecutive finitevolumes are considered as one volume (Fig. 2). The hydrostatic consistency can be easily evaluated on these finite-volumes. 4. Second-Order Accuracy For the second-order staggered grid, l, e, u, w, are taken as the arithmetic means of ressure at the two vertices, w = + ik, ik, +, 2 e = + i+, k i+, k +, 2 7

8 l = + ik, + i+, k+ ik, i+, k u 2 = + 2. (3) Substitution of (3) into (2) leads to ( i+, k+ i, k) ( Hi+ σk Hiσk+ ) + ( i+, k i, k+ ) ( Hiσk Hi+ σk+ ) = x δx δσ H + H ( ) ik, i k i i+, (4) where δ xi = xi+ x i and δσ k = σ k σ k +. Equation (4) is the discretization of the horizontal ressure gradient with the finite-volume consideration. 5. Finite Difference Scheme Finite difference schemes are commonly used in sigma coordinate ocean models. Using the central difference scheme, the horizontal ressure gradient (2) is discretized by δ + σ + σ H H + = δx 2 δx H + H δx 2 δσ = i+, k i+, k+ i, k i, k+ k k+ i+ i i, k i+, k i, k+ i+, k+ ik, i i i+ i k ( i+, k+ i, k) ( Hi+ σk Hiσk+ ) + ( i+, k i, k+ ) ( Hiσk Hi+ σk+ ) ( ) δx δσ H + H i k i i+, (5) which is exactly the same as (4). This means that the sigma coordinate ocean models with second-order staggered grid is always hydrostatically consistent. This confirms Mellor et al. s [994] claim that the hydrostatic consistency is irrelevant any way in the sigma coordinate ocean models and that the hydrostatic consistency arameter r is just another measure of the numerical errors. 6. Conclusions 8

9 () Using the finite-volume integration aroach, a hydrostatically consistent, discrete scheme [equation (2)] is obtained to comute horizontal ressure gradient. For the second-order accuracy, this scheme is exactly the same as the commonly used sigma coordinate ocean models (staggered grids) with the second-order central difference scheme. This indicates that the current sigma coordinate ocean models with second order staggered scheme are always hydrostatically consistent. (2) Deviation of discretization schemes for comuting the horizontal ressure gradient from equation (2) can be taken as a measure for hydrostatic inconsistency. The larger the deviation, the larger the hydrostatic inconsistency is. (3) Equation (2) rovides the guidance for establishing hydrostatically consistent schemes for horizontal ressure gradient. More accurate schemes should be develoed on the base of accurate estimate of mean ressure at four sides of the finite-volume (i.e., l, e, u, w). Acknowledgements This study was suorted by the Office of Naval Research, the Naval Oceanograhic Office, and the Naval Postgraduate School. 9

10 References Beckmann, A., and D. B. Haidvogel, Numerical simulation of flow around a tall isolated seamount. Part : Problem formulation and model accuracy. J. Phys. Oceanogr., 23, , 993. Chu, P.C., and C.W. Fan, Sixth-order difference scheme for sigma coordinate ocean models. J. Phys. Oceanogr., 27, , 997. Chu, P.C., and C.W. Fan, A three-oint combined comact difference scheme. J. Comut. Phys. 40, , 998. Chu, P.C., and C. Fan, A three-oint sixth-order nonuniform combined comact difference scheme. J. Comut. Phys., 48, , 999. Chu, P.C., and C.W. Fan, A staggered three-oint combined comact difference scheme. Math. Comut. Modeling, 32, , Chu, P.C., and C.W. Fan, An accuracy rogressive sixth-order finite- difference scheme. J. Atmos. Oceanic Technol., 8, , 200. Ezer, T., H. Arango, A.F. Shcheetkin, Develoments in terrain-following ocean models: intercomarisons of numerical asects. Ocean Modelling, 4, ,

11 Gary, J.M., Estimate of truncation error in transformed coordinate rimitive equation atmosheric models. J. Atmos. Sci., 30, , 973. Haney, R.L., On the ressure gradient force over stee toograhy in sigma coordinate ocean models. J. Phys. Oceanogr., 2, 60-69, 99. Kliem, N., and J.D. Pietrzak, On the ressure gradient error in sigma coordinate ocean models: A comarison with a laboratory exeriment. J. Geohys. Res., 04, , 999. Lin, S.-J., A finite volume integration method for comuting ressure gradient force in general vertical coordinates. Q. J. R. Metoerol. Soc., 23, , 997. McCalin, J.D., A comarison of second-order and fourth-order ressure gradient algorithms in a sigma-coordinate ocean model. Inter. J. Num. Methods in Fluids, 8, , 994. Mellor, G.L., T. Ezer, and L.-Y. Oey, The ressure gradient conundrum of sigma coordinate ocean models. J. Atmos. Oceanic Technol.,, 26-34, 994. Mellor, G.L., L.-Y. Oey and T. Ezer, Sigma coordinate ressure gradient errors and the seamount roblem. J. Atmos. Oceanic Technol., 5, 22-3, 998.

12 Song, Y.T., A general ressure gradient formulation for ocean models. Part : Scheme design and diagnostic analysis. Mon. Wea. Rev., 26, , 998. Song, Y.T., and D.G. Wright, A general ressure gradient formulation for the ocean models. Part 2: Momentum and bottom torque consistency. Mon. Wea. Rev., 26, , 998. Stelling, G.S., and J.A.T.M. van Kester, On the aroximation of horizontal gradients in sigma coordinates for bathymetry with stee bottom sloe. Int. J. Numer. Methods Fluids, 8, ,

13 Figure. Finite-volume discretization and staggered grid in terrain-following coordinates with two cells reresenting r > and r <. Figure 2. Double finite-volumes for C-grid. 3

Hydrostatic correction for sigma coordinate ocean models

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 08, NO. C6, 306, doi:0.09/00jc00668, 003 Hydrostatic correction for sigma coordinate ocean models Peter C. Chu and Chenwu Fan Naval Ocean Analysis and Prediction Laboratory,