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Tracer Conservation with an Explicit Free Surface Method for Z-Coordinate Ocean Models

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Tracer Conservation with an Explicit Free Surface Method for Z-Coordinate Ocean Models MAY 2001 GRIFFIES ET AL. 1081 Tracer Conservation with an Explicit Free Surface Method for z-Coordinate Ocean Models STEPHEN M. GRIFFIES AND RONALD C. PACANOWSKI NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey MARTIN SCHMIDT Baltic Sea Research Institute WarnemuÈnde, Rostock Warnemunde, Germany V. B ALAJI Cray/SGI, NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey (Manuscript received 2 December 1999, in ®nal form 24 March 2000) ABSTRACT This paper details a free surface method using an explicit time stepping scheme for use in z-coordinate ocean models. One key property that makes the method especially suitable for climate simulations is its very stable numerical time stepping scheme, which allows for the use of a long density time step, as commonly employed with coarse-resolution rigid-lid models. Additionally, the effects of the undulating free surface height are directly incorporated into the baroclinic momentum and tracer equations. The novel issues related to local and global tracer conservation when allowing for the top cell to undulate are the focus of this work. The method presented here is quasi-conservative locally and globally of tracer when the baroclinic and tracer time steps are equal. Important issues relevant for using this method in regional as well as large-scale climate models are discussed and illustrated, and examples of scaling achieved on parallel computers provided. 1. Introduction similar to that considered by Marsigli in the seventeenth century (see Gill 1982), and a global ocean simulation. An explicit free surface method has been developed Before detailing the algorithm and showing examples, for the z-coordinate Geophysical Fluid Dynamics Lab- it is useful to provide some context for the present work. oratory (GFDL) Modular Ocean Model (MOM) (for documentation of the model, see Pacanowski and Grif- ®es 2000). The scheme allows for the use of a free a. Tracer conservation and freshwater forcing surface with substantially improved stability properties For a Boussinesq ¯uid, volume conservation over a over the older approach of Killworth et al. (1991, re- vertical column yields the free surface height equation ferred to as KSWP in the following), as well as for greatly improved conservation of tracers, such as heat ]th 52= ´ U 1 qw, (1) and salt, relative to either the KSWP or Dukowicz and where h is the deviation of the ¯uid surface elevation Smith (1994) approach. The result is an algorithm suit- h able for climate as well as regional ocean models. from its resting position at z 5 0; U 5 #2H u dz is the The purpose of this paper is to present the method vertically integrated horizontal velocity, also known as and to discuss its physical and numerical properties. the transport; and qw 5 P 2 E 1 R represents a volume Solution examples are provided of the Goldsbrough± ¯ux per unit area (with units of a velocity) of freshwater Stommel circulation (see Huang 1993) driven by surface passing across the air±sea interface due to precipitation, freshwater forcing, a baroclinic adjustment problem evaporation, and/or runoff. It is through this balance of volume that dynamical assumptions, which affect h and U, determine the ability of a model to faithfully rep- resent freshwater forcing. Many ocean models employ a constant volume. For Corresponding author address: Dr. Stephen M. Grif®es, NOAA/ Geophysical Fluid Dynamics Laboratory, P.O. Box 308, Forrestal example, the rigid-lid approximation sets ]th 5 0, Campus, Princeton, NJ 08542. thence eliminating fast barotropic waves. With this as- E-mail: [email protected] sumption, volume conservation via Eq. (1) leads to the q 2001 American Meteorological Society 1082 MONTHLY WEATHER REVIEW VOLUME 129 t salinity is constant. For a free surface model, h 5 |z1 | 1 h, where z 5 z1 , 0 is the bottom of the surface tracer cell, and so salt conservation means (|z1 | 1 h)]ts 52s]th. When |z1 | k |h|, this equation is often ap- proximated by |z1 |]ts 52s]th (e.g., KSWP; Dukowicz and Smith 1994; Wolff et al. 1997; Marshall et al. 1997), which generally precludes salt conservation whenever the surface height changes, as through the addition of freshwater [see Barnier (1998) or Gordon et al. (2000) for a discussion of the virtual salt ¯uxes applied to con- stant volume ocean models]. Such limitations are also relevant for heat and other tracers. Our aim here is to relax this approximation and to provide a conservative scheme valid over a wide range of vertical resolutions, including high resolutions where |z1 | ; |h|. In principle, implementation of tracer forcing in ¯ux FIG. 1. Schematic of rectangular surface tracer and velocity cells form is suf®cient to ensure that tracer is globally con- with the free surface. The tracer points are denoted by a solid dot served. However, with time-dependent cell thicknesses, and the tracer cells are enclosed by solid lines; a velocity point is details related to the discrete time stepping scheme, in- denoted by an 3 and is enclosed by dashed lines. These points have cluding the use of time ®lters necessary with the three- ®xed vertical position z 5 z1/2 , 0, regardless of the value for the t t time-level leapfrog scheme, serve to preclude strict con- free surface height. The thickness of a tracer cell is h 52z1 1 h , t servation. Additionally, global conservation is neither where zk50 5 h is the prognostic surface height determined through volume conservation [i.e., the vertically integrated continuity equa- necessary nor suf®cient to ensure locally conservative u u tion (1)]. The thickness of a velocity cell is h 52z1 1 h , where behavior. For example, consider an ocean at rest with hu is determined by taking the minimum height of the four surround- globally constant tracer concentration. Then allow a ing tracer cells, as prescribed by the partial cell approach of Paca- nowski and Gnanadesikan (1998). Thicknesses of the cells are as- wind to blow so to initiate velocity and surface height t sumed to be positive, so that 2z1 1 h . 0. In models without an ¯uctuations, yet do not introduce any tracer surface ¯ux- explicit representation of tides, this constraint in practice means that es. In addition to global tracer content remaining ®xed, |z1 | must be greater than roughly 2 m. For models with tides, |z1 | may the tracer concentration locally at every point in the need to be somewhat larger, depending on the tidal range considered. ocean will maintain the same constant value. This be- havior provides an example of ``local'' tracer conser- balance qw 5 = ´ U, which forms the basis for the work vation, which is not a necessary result of global con- of Huang (1993). The corresponding barotropic dynam- servation. Instead, it arises so long as there is compat- ics require both a streamfunction and velocity potential, ibility between tracer and volume budgets. thus introducing two elliptic problems. Elliptic problems are generally inef®cient to solve, especially in the pres- b. Splitting between fast and slow dynamics ence of realistic geometry, topography, and surface forc- ing, and moreso on parallel computers (see section 5). For computational ef®ciency, primitive equation Bryan (1969) made the further assumption that the ver- ocean models aim to exploit the timescale split between tically integrated velocity has zero divergence, resulting the fast barotropic gravity waves, which can propagate in a zero vertical velocity at the ocean surface: w(z 5 at some 200 m s21, and the much slower (some 100 0) 52= ´ U 5 0. This assumption eliminates the ve- times slower) remaining dynamics. Such models ap- locity potential, yet at the cost of precluding a direct proximate the barotropic mode by the depth-averaged freshwater forcing. motion, and the baroclinic mode is approximated by the A free surface model, in principle, should trivially al- deviation from the depth average. In a rigid-lid ocean low for the introduction of freshwater, since it only re- model, vertical averaging provides a clean split between quires the addition of qw to the free surface height equa- the fast and slow modes. In contrast, vertical averaging tion (1). However, doing so consistently and conserva- does not completely separate out the fast dynamics in tively requires some changes to the baroclinic and tracer a free surface model, since in this case the barotropic equations that are not commonly made. To illustrate this mode is weakly depth dependent. The papers by KSWP, point, consider what conservation means for total salt in Dukowicz and Smith (1994), Higdon and Bennett a single grid cell in the special case of zero salt ¯uxes, (1996), Higdon and de Szoeke (1997), and Hallberg but generally nonzero fresh water ¯uxes. With rectan- (1997) provide discussions of these issues, which can gular model cells (Fig. 1), salt is conserved if be quite subtle yet important for the purpose of provid- ing a stable split between the fast and slow modes in ] (hts) 5 0, (2) t realistic ocean models. Additionally, details of this split where ht is the thickness of a model tracer cell. Salt have crucial implications for the tracer conservation conservation with a rigid lid, in which ]th 5 0, means properties of the model. MAY 2001 GRIFFIES ET AL. 1083 c. Modeling context and summary of key results [though advances are available, as documented by Gu- yon et al. (2000)]. MOM has traditionally been a rigid-lid ocean model, where the streamfunction algorithm of Bryan (1969) was used to solve for the barotropic component of the d. Contents of this paper dynamics. In the 1990s, important alternatives have This paper consists of the following sections. Section been added, including the explicit free surface of KSWP, 2 describes the model's discrete tracer equations.
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