Generalized vertical coordinates

Often the numerics benefit from using a vertical coordinate that is different than and which, in general, even be moving (vary in time).

We denote any such generalized vertical coordinate that evolves with space and time as . The generalized vertical coordinate must vary monotonically with .

Transforming the equations of motions (including their discrete analogues) in this new generalized coordinate involves a series of chain rules for partial differentiation. We have that for any field :

The quantity plays a central role in what follows. We refer to it as "specific thickness" and denote it :

where the subscripts next to denote the quantities that remain constant in the differentiation.

The chain rules for differentiation with respect of , , or become:

The horizontal spatial derivatives of the -coordinate are then rewritten, e.g.,

so that the chain rule above becomes

Similarly, we get equivalent expressions for derivatives as in and .

Mass conservation

With the Boussinesq approximation, the mass conservation reduces to the flow being divergence-less, i.e.,      .

Using the chain rules above, the divergence of the flow in -coordinates becomes:

We can rewrite   and similarly for the direction. After a bit of reordering the above yields

Note that above is the vertical velocity referenced to the coordinate. The vertical velocity of the surface referenced to the coordinate as

Then, the vertical velocity across the surfaces is the difference between and

With the definition of in we get

which implies that the mass conservation is equivalent to:

Tracer equation

The evolution equation for a tracer , which also includes vertical diffusion, reads

Using the same procedure we followed for the continuity equation, the left-hand-side of yields:

We recover the time derivative of the tracer at constant by rewriting the last term using the chain rule for a time derivatives:

As such, the left-hand-side of can be rewritten in -coordinates as

Including the vertical diffusion to the right-hand side we recover the tracer equation:

Momentum equations in vector invariant form

The horizontal momentum equations under the hydrostatic approximation read

where    is the horizontal velocity,    is the three-dimensional velocity, and     is the material derivative.

The above is complemented by the hydrostatic relation

Of the above, the Coriolis term is independent of the vertical frame of reference and the viscous stress is treated similarly to the diffusion of a tracer. In this derivation we focus on:

1. the hydrostatic relation ,

2. the material derivative in the momentum equation , and

3. the horizontal pressure gradient terms in .

Hydrostatic relation

Using the definition of the -derivative in -coordinates

Material derivative in vector invariant form

We set out to transform in -coordinates the material derivative of the horizontal velocity in vector invariant form

where    is the vertical vorticity, and    is the horizontal kinetic energy.

Here, we focus on the component of the velocity; the derivation of the component follows the same steps. Thus, we are transforming

Above, we utilized and repeatedly, e.g., for , , and . Further expanding   , a few terms cancel out and we end up with:

Using the definition of in , we can rewrite as

As done above for the tracer, the last term on the right-hand side, using the chain rule for the time derivative yields

Which completes the derivation of the -momentum equations in -coordinates

We can further split the vertical advection term into a conservative vertical advection and a horizontal divergence term:

and last using the continuity equation to obtain

Horizontal pressure gradient

The horizontal pressure gradients, e.g., , can be transformed using the chain rule :

and combined with the hydrostatic relation :

The last term on the right hand side above describes the pressure gradient associated with the horizontal tilting of the grid. Similarly, the gradient of the free surface transforms to

Finite volume discretization

It is useful to describe how the above equations translate into discrete form in a finite volume staggered C-grid framework.

To do so, we integrate over the cell volume   remembering that in the discrete  . Indices i, j, k below correspond to the x, y, and the vertical directions respectively.

Mass conservation

The mass conservation gives:

Using the notation for cell-averages    and also the divergence theorem, we can rewrite to:

We use diagnose the vertical velocity (in space) given the grid velocity and the horizontal velocity divergence:

where   ,   ,   ,   , and   .

Tracer equation

The tracer equation in discrete form becomes:

leading to

where     ,   , and   .

For an explicit formulation of the diffusive fluxes and a time-discretization using forward Euler scheme, imply:

where is tendency computed on the z-grid.

Note that in case of a multi-step method, e.g., second-order Adams-Bashforth, the grid at different time-steps must be accounted for, and the time-discretization becomes

For this reason, we store the tendencies pre-multiplied by at their current time-level. In case of an implicit discretization of the diffusive fluxes we first compute as in the above equation (where does not contain the diffusive fluxes). Then the implicit step is done on a z-grid as if the grid was static, using the grid at   which includes .