the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 28 Oct 2024
On the hydrostatic approximation in rotating stratified flow
Abstract. Hydrostatic models were and still are the workhorses for realistic simulations of the ocean dynamics, especially for climate applications. The hydrostatic approximation is formally first order in γ = H / L, where H is the vertical and L the horizontal scale of the phenomenon considered. For linear (low amplitude) and unforced stratified rotating flow the dynamics can be separated in balanced flow and wave motion. It is shown that for the linear balanced motion the hydrostatic approximation is exact and for wave motion it is second order, obtaining the leading prefactors. The validity of the hydrostatic approximation therefore also relies on the ratio of the amplitude of wave motion to balanced motion. This ratio adds considerably to the quality of the hydrostatic approximation for larger scale flows in the atmosphere and the ocean.
Imposing the divergenceless condition is a linear projection of the dynamical variables into the subspace of divergenceless vector fields, for both the Navier-Stokes and the hydrostatic formalism. Both projections are local in Fourier space. The projection is followed by a time-evolution operator, which differs in the wave-frequencies, only. Combining the projection and the linear evolution operators in both formalisms leads to the linear projection-evolution operator.
Calculating the difference of the two projection-evolution operators, the expression of the error, scaling and prefactors, done by the hydrostatic approximation is obtained. Analyzing the eigen-space of the projector-evolution operators, it is shown that for rotating-buoyant vortical-flow the hydrostatic-approximation is of third order for buoyant forcing, second order for horizontal and first order for vertical dynamical forcing. Equilibrium solutions are in the kernel of the linear projection-evolution operator and conservation laws are in the kernel of its adjoint.
Using the Heisenberg-Gabor limit it is shown that for large scale ocean dynamics, the difference of the dynamics of the projection-evolution operator between the two formalisms is insignificant. It is shown that the hydrostatic approximation is appropriate for realistic ocean simulations with vertical viscosities larger than ≈10-2 m2 s-1. A special emphasis is on unveiling the physical interpretation of the calculations.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
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Journal article(s) based on this preprint
Interactive discussion
Status: closed
- RC1: 'Comment on egusphere-2024-3307', Anonymous Referee #1, 06 Dec 2024
-
RC2: 'Comment on egusphere-2024-3307', Anonymous Referee #2, 20 Dec 2024
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 ---> divergencelessCitation: https://doi.org/10.5194/egusphere-2024-3307-RC2 -
RC3: 'Comment on egusphere-2024-3307', Anonymous Referee #3, 21 Dec 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-3307/egusphere-2024-3307-RC3-supplement.pdf
-
AC1: 'Comment on egusphere-2024-3307', Achim Wirth, 21 Mar 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-3307/egusphere-2024-3307-AC1-supplement.pdf
Interactive discussion
Status: closed
- RC1: 'Comment on egusphere-2024-3307', Anonymous Referee #1, 06 Dec 2024
-
RC2: 'Comment on egusphere-2024-3307', Anonymous Referee #2, 20 Dec 2024
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 ---> divergencelessCitation: https://doi.org/10.5194/egusphere-2024-3307-RC2 -
RC3: 'Comment on egusphere-2024-3307', Anonymous Referee #3, 21 Dec 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-3307/egusphere-2024-3307-RC3-supplement.pdf
-
AC1: 'Comment on egusphere-2024-3307', Achim Wirth, 21 Mar 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-3307/egusphere-2024-3307-AC1-supplement.pdf
Peer review completion
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Achim Wirth
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(666 KB) - Metadata XML