As discussed in Section 6.11 of Gill (1982), the linearized primitive equations for a stratified fluid can be partitioned into a countably infinite (i.e., discrete) number of orthogonal eigenmodes, each with a different vertical structure. Gill denotes the zeroth vertical eigenmode the barotropic mode, and the infinity of higher modes are called baroclinic modes. Because of the weak compressibility of the ocean, wave motions associated with the barotropic mode are weakly depth dependent, and so correspond to elevations of the sea surface (see Hidgon and Bennett 1996 for a proof of the weak depth dependence in a flat bottomed ocean). Consequently, the barotropic field goes also by the name external mode. Barotropic or external waves constitute the fast dynamics of the ocean primitive equations. Baroclinic waves are associated with undulations of internal density surfaces, which motivates the name internal mode. Baroclinic waves, along with advection and planetary waves, constitute the slow dynamics of the ocean primitive equations.
For a flat bottom ocean, the vertical eigenmode problem is straightforward to solve, and many important ideas can be garnered from its analysis. For a free surface with a flat bottom, Gill shows that the barotropic mode has a vertical velocity which is approximately a linear function of depth, with the maximum vertical velocity at the free ocean surface and zero velocity at the flat bottom. In contrast, for a rigid lid and flat bottom ocean, the barotropic mode is depth independent and the vertical velocity identically vanishes. The baroclinic modes, as they are associated with movements of the internal interfaces, are little affected by the surface boundary condition. Therefore, the baroclinic modes in the free surface are quite similar to those in the rigid lid. Note that nonlinearities and nontrivial bottom topography generally couple the barotropic and baroclinic modes.
By construction, the depth averaged momentum equations only have solutions which depend on the horizontal directions. Consequently, the depth averaged mode of a rigid lid ocean model corresponds directly to the barotropic mode of the linearized rigid lid primitive equations. Additionally, the rigid lid model's depth dependent modes correspond to the baroclinic modes of the linearized rigid lid primitive equations. Therefore, depth averaging in the rigid lid model provides a clean separation between the linear vertical modes.
Just as for the rigid lid, the baroclinic modes are well approximated by the depth dependent modes of the free surface ocean model, since the baroclinic modes do not care so much about the upper surface boundary condition. In contrast, the ocean model's depth averaged mode cannot fully describe the free surface primitive equation's barotropic mode, which is weakly depth dependent. Therefore, some of the true barotropic mode spills over into the model's depth dependent modes. In other words, a linearized free surface ocean model's depth averaged mode is only approximately orthogonal to the model's depth dependent modes. It turns out that the ensuing weak coupling between the ocean model's fast and slow linear modes can be quite important for free surface ocean models, as described by Killworth et al. (1991) and Higdon and Bennett (1996). The coupling, in addition to the usual nonlinear interactionas associated with advection, topography, etc., can introduce pernicious linear instabilities whose form is dependent on details of the time stepping schemes.
Regardless of the above distinction between vertically averaged and barotropic mode for free surface models, it is common parlance in ocean modeling to refer to the vertically integrated mode as the barotropic or external mode. This terminology is largely based on the common use of the rigid lid approximation, for which there is no distinction. With the above discussion kept in mind, there should be no confusion, and so the terminology will be used in this manual for both the rigid lid and free surface formulations. Since there is little difference between the rigid lid and free surface baroclinic modes, it is quite sensible to use this term to refer to the ocean model's depth dependent modes.