Formulation and implementation of a “residual-mean” ocean circulation model
Introduction
In the ocean interior, mesoscale eddies are thought to be quasi-adiabatic, in the sense that eddy buoyancy fluxes tend to be directed along mean buoyancy surfaces (Gent and McWilliams, 1990; hereafter GM). The eddy transport of mean buoyancy in the ocean interior can then be represented by an eddy-induced velocity. Thus, mean buoyancy is advected by the sum of the Eulerian and eddy-induced velocities, known as the “residual flow”. However, close to the surface, eddy fluxes develop a diabatic component because isopycnal surfaces are maintained vertically by strong mixing while motions are constrained to be horizontal (see Marshall and Radko, 2003, Kuo et al., 2005, Ferrari and McWilliams, submitted for publication). Then, in addition to advection, mesoscale eddies can contribute a cross-isopycnal flux (a “residual flux”) to the buoyancy equation as the surface is approached.
One can fully adopt a residual-mean framework by choosing to make the residual velocity a prognostic variable rather than the Eulerian velocity. Following Andrews and McIntyre, 1976, Ferreira et al., 2005 (hereafter FMH) derive a residual-mean momentum equation for the 3-d large-scale ocean circulation. They show that, in the limit of small Rossby number, it has the same form as the primitive equations; forcing terms in the horizontal momentum equation now include an eddy contribution which has the form of a wind stress except it exists in the body of the fluid and is zero at top and bottom. This eddy stress represents the property of mesoscale eddies to redistribute momentum vertically between isopycnal layers through “form drag”, correlations between interface displacements and pressure fluctuations. Its vertical divergence drives the eddy-induced part of the residual flow. Under certain assumptions, this forcing can be related to an eddy flux of potential vorticity (see for example Rhines and Young, 1982, Greatbatch, 1998, Wardle and Marshall, 2000). For a quasi-geostrophic zonally averaged system, the correspondence is straightforward because the eddy stress and Reynolds stress make up the components of the Eliassen–Palm flux whose divergence is the eddy potential vorticity flux (Andrews et al., 1987).
The residual-mean framework encourages us to revisit the representation of mesoscale eddies in coarse-resolution ocean climate models and provides a framework in which to improve them. We formulate and implement a simple closure for the eddy stress and residual flux. Study of the eddy stresses estimated in FMH (who used a least-squares procedure to ‘solve’ for the eddy stresses which brought a global ocean model into consistency with the observations) suggests that they can be represented as a down-gradient vertical momentum flux with a constant vertical viscosity. The residual eddy buoyancy flux, which is often neglected in coarse-resolution models, is parameterized as a down-gradient horizontal buoyancy flux acting over a surface diabatic layer.
A significant computational advantage of the residual-mean formulation studied here is that it is numerically more robust and requires less horizontal viscosity than traditional approaches. As pointed out by Griffies et al. (2000), the use of lateral momentum mixing in ocean models is not motivated by physical principles but only by practical contingencies, i.e. to suppress grid-point noise, maintain western boundary layers, and provide a sink of momentum at grid-scale. For example, Laplacian momentum friction – the form used in the present study – is not a realistic representation of the effects of unresolved Reynolds stresses on the resolved large-scale circulation (see, for example, the discussion in Marshall, 1981). Thus it is desirable to reduce horizontal viscosity as much as possible to maintain stability. Appealingly, the residual-mean formulation employed here allows one to significantly reduce lateral mixing of momentum.
In Section 2, we briefly derive the residual-mean equations on which our model is based. In Section 3, we describe the closures assumed for the eddy stress and the residual flux. In Section 4, numerical simulations with the residual-mean model are carried out using the MIT general circulation model (GCM) with a realistic configuration and 2.8° resolution. The model climatologies are compared with the observations and those obtained from Eulerian simulations employing the GM parameterization scheme. The effects of reduced horizontal viscosity in the residual framework are detailed in Section 5. Finally, the role of the residual flux is illustrated in Section 6. Conclusions are given in Section 7.
Section snippets
Residual-mean framework
The residual-mean theory used here is identical to that of FMH which was itself mainly inspired by Andrews and McIntyre, 1976, Treguier et al., 1997. In this section, we briefly review key elements. For more details, the reader is referred to FMH and references therein.
Eddy closure
To close the TEM system, we must parameterize the eddy stress in the ocean interior, the depth of the surface diabatic layer hs, the tapering μ within that layer and the residual flux. We must also describe the eddy parameterization employed for tracers other than buoyancy.
Before going further, we make some simplifying assumptions. As discussed in the introduction, we first assume that the residual flux is negligible in the ocean interior: that is, eddy fluxes are well approximated by their
The MITgcm
To test the proposed parameterization, we carry out a series of experiments using the MITgcm (Marshall et al., 1997a, Marshall et al., 1997b). The model has a horizontal resolution of 2.8° and 15 levels in the vertical. The geometry is ‘realistic’ except for the absence of the Arctic Ocean; bathymetry is represented by partial cells (Adcroft et al., 1997). The model is forced by observed monthly mean climatological surface wind stresses from Trenberth et al. (1990), and observed monthly mean
Choice of horizontal viscosity
As noted in the introduction, horizontal friction is introduced in ocean models largely for pragmatic reasons. In coarse-resolution GCMs, it is mainly required to suppress grid-point noise and maintain western boundary currents (Griffies et al., 2000). Therefore, lateral mixing should be reduced to the minimum required for numerical integrity of the solution.
A yardstick to guide the choice of horizontal viscosity is to require that the width of the Munk (frictional) boundary layer be resolved
Role of diabatic eddy fluxes
Finally, we carry out an experiment in which the residual flux is incorporated. We only discuss the results in the case of low viscosity (ReslAhF) because the global effects of the residual flux are almost null if the reference viscosity is used. At low viscosity, they are still relatively small (compare ReslAhF and ReslAh in Table 1) but more apparent, perhaps, because of the presence of stronger meridional gradients.
The climatology of the model is degraded, both in temperature and salinity,
Conclusions
The residual-mean framework was used to revisit the parameterization of mesoscale eddies in coarse-resolution ocean climate models. In the residual-mean formulation, the resolved circulation is the residual circulation, the sum of Eulerian and eddy-induced circulations. The residual-mean momentum equation has an additional forcing term, the vertical divergence of an eddy stress. In the adiabatic limit, appropriate to the ocean interior, the residual flow advects all tracers. However, near
Acknowledgments
We acknowledge useful discussions with Raffaele Ferrari and Alan Plumb and the help of Jean-Michel Campin with numerical aspects. The calculations were carried out on the ACES computer at MIT.
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