Isopycnal surfaces may seem the most natural vertical coordinate system for ocean modeling; after all, water mass transports in the ocean occur approximately along isopycnal surfaces. Isopycnic models are designed to facilitate the representation of such transports because, unlike the prior two model classes, there occurs no spurious diapycnal mixing due to the numerical representation of advection. In addition, diapycnal mixing can be added in a controlled form. A recent overview on the philosophy of isopycnal modeling is given by Bleck (1998).

The utilization of an isopycnic coordinate leads naturally to an adaptive vertical grid, which conveniently resolves regions of vertical density gradients (thermocline, surface fronts). This differs from other ocean models in that the vertical grid is time-dependent and can in principle adjust to the dynamic situation of the ocean.

The model description can be found in Bleck *et al.* (1992) and Bleck and Chassignet (1994); a user's manual and the latest references can be obtained from the MICOM web page.

In an isopycnic coordinate system with potential density as the vertical coordinate, , and the hydrostatic equation becomes

Inserted into the pressure gradient terms,

and since is independent of the horizontal coordinate, we have

where the Montgomery potential

is introduced. The transformed hydrodynamic equations then become

Note that the formulation of the momentum advection has different numerical properties than the form used in other models due to a modified averaging procedure.

Strictly speaking, the pressure gradient terms require the use of the *in-situ* density . In that case, density is no longer constant along coordinate lines, and the pressure gradient terms become

This leads to a sum of two terms, reminiscent of those obtained in terrain-following s-coordinates. The consequences of using *in-situ* density have been examined recently (Sun *et al.*, 1999).

Currently, a simplified equation of state, (Friedrich and Levitus, 1972),

is used because it must be efficiently invertible to obtain potential temperature as a function of specific volume.

Inevitably, the isopycnal concept has some inherent limitations. The most obvious is the fact that the use of a single potential density both for layer definition and for baroclinic pressure gradient is dynamically inconsistent. This is a fundamental problem which cannot be cured by an increase in resolution. Since potential density is the vertical coordinate, which must be constant within layers, the inclusion of cabbeling effects is computationally expensive. Therefore, only one of the thermodynamic variables or S is usually computed. This, however, leads to a systematic deviation from the thermal wind relation. It appears that the problem might be reduced to some extent through choice of a different reference pressure (2000 dbar instead of the surface) although such choice may create other problems in the upper ocean. Also, isopycnal coordinate models require special advective treatments in the limit of vanishing layer thickness, and repeated evaluation of the nonlinear equation of state. This may represent a significant computational overhead depending on which algorithmic approaches are used. Thermobaricity is now included in MICOM (Sun *et al.*, 1999).

The model is only available with a split explicit free sea surface scheme. The solution of the barotropic component is shifted in time. It is treated by a forward-backward scheme using the latest update of the continuity equation and the last pressure field.

**Spatial discretization, grids and topography**

The figure below shows two examples of a discretization near the continental margin. Since the layer thickness can change with time, different patterns will evolve. The important features of this discretization are: good representation of the thermocline, poor vertical resolution in areas of small vertical density gradients (well-mixed shelf areas, the mixed layer, the deep ocean) and the need for zero-mass layers where isopycnals outcrop or hit the topography. An unfortunate choice of layer densities may lead to under-resolved regions, especially in long model runs where the stratification drifts away from its initial state. While figure 4a is an example for extremely good resolution in strongly stratified shelf areas and gives a reliable representation of velocity profiles, 4b is more adequate for thermocline and frontal zones.

Horizontally, MICOM is discretized on the Arakawa "C" grid, with curvilinear coordinates like SPEM/SCRUM. Arbitrary topography can be included.

We define the averaging operators as

and the derivative operators as

We also introduce the notation for a vertical sum. The semi-discrete equations for MICOM are then

Finally, for the barotropic mode, we have

MICOM makes use of the time splitting concept; *i.e.*, each individual term of the tendency equations is used to produce an update of the prognostic variable before the next term is evaluated. The computational mode introduced by the leap-frog time-stepping scheme is reduced by an Asselin time filter (see Bleck and Smith, 1990).

The coupling of a *bulk mixed layer model* is a straightforward extension to the layer concept (Bleck *et al.*, 1989). Consequently, MICOM comes with a mixed layer model based on the Kraus and Turner (1967) formulation. Alternatively, a turbulent closure scheme according to Gaspar (1988) can be utilized. The main advantage of the layered model concept is that the depth of the mixed layer can assume any value and is not confined to a fixed grid of limited resolution in the vertical. An inherent drawback, however, is the inability to permit vertical shear in the mixed layer, which may be problematic if it is very deep. Also, special measures in the mixed-layer detrainment algorithm are needed to adequately handle the re-stratification due to both sensible heat and freshwater input.

Following the concept of *zero mass layers* (Bleck and Smith, 1990), each isopycnic layer can disappear or re-appear at each point of the domain. Static instability is treated by *convective adjustment*. Unlike fixed coordinate models, the adaptive formalism has to include mixing of momentum to ensure conservation.

An auxiliary program for *plotting* the prognostic model variables is based on NCAR graphics (Clare and Kennison, 1989). The model is written in *f77* and uses *cgs* units. A model version for parallel computers exists. The code can be obtained via ftp from

Advantages and disadvantages of this model concept are closely related. High resolution in areas of sharp density gradients (both vertically and horizontally) is a highly desirable feature; however, by necessity, vertical shear in homogeneous or near-homogeneous fluid is resolved poorly. This might be important in areas where velocity shear exists in weakly stratified fluid (like in the upper mixed layer).

Also, the diapycnal transports induced by nonlinearities in the equation of state (cabbeling and thermobaricity) are new additions to the code and have not been tested extensively. In the past, attempts have been made to alleviate this by using a different reference level for the density coordinate. The thermodynamic problems mentioned above have been reduced significantly (Sun *et al.*, 1999).

Future developments include a hybrid version of MICOM, which will provide, in contrast to the purely isopycnic version, vertical resolution in the mixed layer and on the shelf. The capability of assigning several coordinate surfaces to the oceanic mixed layer will not only allow for vertical shear near the surface but also make it possible to replace the presently used Kraus-Turner slab mixed layer model with a more sophisticated turbulent closure scheme.

Another isopycnic ocean model is OPYC (Oberhuber, 1993a and 1993b). Differences from MICOM include the use of the "B" grid, and, more importantly, the variation of potential densities in all layers.

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