Dear Yilmaz

> 1)  "s(k) is the dimensionless coordinate at vertical gridpoint (k)"
>      Does this variable refer to itself, i.e. whose "formula_terms" attribute
>      is being parsed? If this is the case, why do we still include this 
> element
>      in the "formula_terms" attribute?

Yes, it does refer to itself. The idea is that the formula identifies all the
variables which are needed to compute the dimensional vertical coordinate. It
happens that s is one of the terms in the formula, not all kinds of vertical
coordinate include themselves as one of the terms.

> 
> 2) do variables which correspond to "depth" and "eta" need to have the same
>     units? That is, can one of them be in meters and the other in centimeters?
>     If they can be different, then doesn't the variable which
> corresponds to "depth_c"
>     also need to have a unit? Do we need to expect a unit for it?

The standard does not make this clear, but I agree, they all need units, and
generic software should do appropriate units conversion.

> Because since it is
>    a scalar variable, it is not required to have one according to CF
> conventions(?).

I think that's a confusion of the two meanings of "dimensionless". One sense
of "dimensionless" means a pure number, without units. Quantities like that,
such as sea-ice area fraction, do not need units. That's a different meaning
of "dimensionless" meaning zero-dimensional i.e. scalar. Scalar quantities
with physical dimensions do need units.

> 3) We know that a dimensional coordinate variable is either
> monotonically increasing
>     or decreasing. Will it be the case for dimensionless ones after
> the formula is applied?
>     That is, do these formulas and associated variable contents
> guarantee that, when the
P
>     formula is applied, we will obtain a vector of pressure or height
> values, which are monotonically
>     increasing or decreasing? (I would expect so since they are levels
> of depth or pressure)

I don't think that is guaranteed, though I agree it would be physically
sensible to arrange it. But it's not a requirement of CF.

> 4) "level", "layer", and "sigma_level" are deprecated according to CF
> conventions, however, still
>     allowed. So how would it be possible for a generic application to
> interpret them?

They should all be interpreted as dimensionless, equivalent to 1.

> 5) And finally, under section "2.4 Dimensions" it says that
> "dimensions other than those of space
> and time may be included". If this is the case, is it still possible
> for a generic application to work
> without being able to interpret those additional dimensions? That is,
> how a could a generic application
> decide on which index value to pass for that dimension without recognizing it?

I don't really understand that question. The dimensions correspond to the array
subscripts. You can subscript an array without knowing the physical meaning of
the dimensions, can't you. Non-spatiotemporal dimensions should be identified
by standard name, although that is not a requirement of CF.

Cheers

Jonathan
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