see attached pdf

This paper deals with the validity of the hydrostatic approximation

generally adopted in numerical simulations of ocean dynamics, in

comparison to the resolution of the fully 3D Navier-Stokes (NS)

equations, in the presence of rotation and stratification effects.

The theoretical basis for such an approximation relies on the fact

that the typical scale of vertical processes, say H, is much smaller

than that of horizontal ones, say L : \gamma = H/L is much smaller

than one.

The goal of the paper is to quantify, on a

scale-by-scale/frequency-by-frequency basis, the error that is

associated with the hydrostatic approximation: the hydrostatic vs NS

linearised dynamics are discussed in Fourier space and the error is

quantified power of \gamma.  

I find the paper interesting and worth of publication. However it is

not an easy reading, and I think that its impact could be increased by

improving the notation and the physical explanations. In the

following, I try and share some considerations in the above direction.

My evaluation is thus: Revise and Accept.

-------------------------------------------------------------

0) Generally speaking, the abstract and the introduction section

should be more explicit on the main results of the paper.  What do we

learn? Are the results of interest only for people doing ocean

numerical modeling, or a more general lesson can be learnt?  The

discussion is of course on the linear dynamics, but we know that non

linear terms are crucial: how these impact the validity of hydrostatic

approximation?

1) It seems to me that he content of Sections 3-6 is well known, and

is a little bit tedious. I would probably move details to an Appendix

and discuss the order of the error in terms of \gamma, and possible applications.

2) it would be useful to state from the introduction the range of

variation of the parameter \gamma in the ocean dynamics applications,

and its typical values.

3) the notation is not always reader-friendly. For example, the symbol

"tilde" is used to redefine variables but not with the same meaning,

which results in some confusion.

At line 185, it is used to define the inverse friction time "\nu k^2"

while at line 220 it is used to define the four-components vector

"(\hat{a_x},(\hat{a_y}(\hat{a_z}(\hat{a_b})". So the tilde has not a

unique meaning.  It is better to avoid such confusion.

4) In figure 3, page 8, it is used the notation with "the symbol □ "

that is however introduced only at line 281 at page 12, "the symbol □

is a place holder for “ ” for the Navier-Stokes or “H” for the

hydrostatic formalism. "

The best is probably to avoid it, and use "NS", "H" and "NS,H" when

NS, hydrostatic or both apply, respectively.

5) It is unclear what the author mean with "(real-component) Fourier

sub-space" in the caption of Figure 3, by saying "Schematic view of

the (real-component) Fourier subspace spanned by the velocity

components...".

6) at line 572, the bottom friction coefficient has no physical

dimensions: are these missing?

7) at line 573-574, it is written "shows that bottom friction is

sufficient to hide the difference in resonance frequencies only if the

ocean layer thickness in meters is below "\gamma^{−1}".

Is the author meaning "...in resonance frequencies only if the

ocean layer thickness in meters is below "(\gamma \Delta x)^{−1}"?

8) at line 580, it is written that "The difference in the horizontal

wave velocity, c, between the two formalisms is \Delta c = \gamma c".

Is this obvious?

9) I have a question concerning intermittent burst that are known to

characterise turbulent stratified flows. In PHYSICAL REVIEW E 89,

043002 (2014), it is shown that strong intermittent events

characterizing temporal dynamics come from the direct coupling between

vertical velocity and temperature fluctuations. I wonder if these are

relevant for the ocean dynamics, and if yes which is their fate in the

hydrostatic approximation. 

------

Typos:

line 127 : "from left to right on the l.h.s. of Eq. 3" ---> "from left to right on the r.h.s. of Eq. 3"

line 128 : divegenceless ---> divergenceless