the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 23 Jan 2025
Process-based modelling of nonharmonic internal tides using adjoint, statistical, and stochastic approaches. Part II: adjoint frequency response analysis, stochastic models, and synthesis
Abstract. Internal tides are known to contain a substantial component that cannot be explained by (deterministic) harmonic analysis, and the remaining nonharmonic component is considered to be caused by random oceanic variability. For nonharmonic internal tides originating from distributed sources, the superposition of many waves with different degrees of randomness unfortunately makes process investigation more difficult. This paper develops a new framework for process-based modelling of nonharmonic internal tides by combining adjoint, statistical, and stochastic approaches, and uses its implementation to investigate important processes and parameters controlling nonharmonic internal-tide variance. A combination of adoint sensitivity modelling and the frequency response analysis from Fourier theory provides distributed deterministic sources of internal tides observed at a fixed location, which enables assignment of different degrees of randomness to waves from different sources. The wave phases are randomized by the statistical model from Part I, using horizontally varying phase statistics calculated by stochastic models. An example application to nonharmonic vertical-mode-one semidiurnal internal tides on the Australian North West Shelf shows that (i) phase-speed variability primarily makes internal tides nonharmonic through phase modulation, and (ii) important controlling parameters include the variance and correlation length of phase speed, as well as anisotropy of the horizontal correlation of phase modulation. The model suite also provides the map of nonharmonic internal-tide sources, which is convenient for identifying important remote sources, such as the Lombok Strait in Indonesia. The proposed modelling framework and model suite provide a new tool for process-based studies of nonharmonic internal tides from distributed sources.
Competing interests: Dr. Matt Rayson (editor) and other oceanographers at University of Western Australia are involved in an on-going collaborative project with my company (involving myself) on the topic of this manuscript. Also, I have a competitive relationship with them for industry-funded projects on topics related to this manuscript (in Australia).
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this preprint. The responsibility to include appropriate place names lies with the authors.- Preprint
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RC1: 'Comment on egusphere-2024-4193', Anonymous Referee #1, 17 Mar 2025
As part II of a series of manuscripts, this one applies the linear stochastic model proposed in Part II to ocean data to identify sources of non-harmonic internal tides. It is a very good idea to use stochastic modelling to analyze measured data, and since this model is linear, the inversion problem is possible.
My major concern is the validity of this linear modelling applied to a highly nonlinear system. The author did not justify the model in a controlled experiment. Maybe this is done in Part I?
e.g., the impact of the background eddies, which vary in space and time, modify the Matrix A in (25). Whether a simple linear first-order SDE can capture this effect?
Section 3: A suggestion is to reduce this section to provide only the necessary equations that are used in analyzing data.
Also, many assumptions leading to the model are missing.
e.g., in the definition of (3), \Theta' is not a small perturbation of \phi. Does this impact the modelling?(15) is strange; to reach a statistically steady state, damping in this system is required. Is there a damping effect in A?
However, dissipation in the primitive equation happens at small scales, while the forcing appears at large scales, then there must be a nonlinear effect to transfer energy across scales, which is missing in the current linear model.(18a) and (21) look inconsistent.
Line 332: the relation d \theta = ..., should dispersion enter this relation?Line 342, (27) is not an equation.
(32): P_\theta\theta tends to infinity as t tends to infinity, which looks unreasonable.
Citation: https://doi.org/10.5194/egusphere-2024-4193-RC1 -
RC2: 'Comment on egusphere-2024-4193', Anonymous Referee #2, 18 Mar 2025
Review of "Process-based modelling of nonharmonic internal tides using
adjoint, statistical, and stochastic approaches. Part II: adjoint
frequency response analysis, stochastic models, and synthesis"
by Shimizu
The author has developed an original approach to analysis of tidal variability and
applied it to understand observations from the Australian shelf.
He uses the model to quantify sources---source regions, source strengths,
vertical mode, and frequency---of tidal variability observed at a mooring.
Although the results are very specific to this site and the regional setting,
the overall approach is, in principle, more generally applicable. Furthermore,
the approach has enabled a reasonably thorough qualitative discussion of the
mechanisms of refraction along the internal propagation paths, which includes a
nice discussion and analysis of the sensitivity to many of the simplifying
assumptions. Although I suppose there is a narrow audience of readers with interest in
this type of detailed analysis, the creativity and novelty of the approach should
inspire substantial follow-on work. For this reason I think it should be published.
As I understand it, the basic components of described approach are as follows:
(1) There is a deterministic model for the internal tides, which assumes linear
dynamics (based on the author's previous work, and outlined in the Appendix).
This model, and its adjoint, are used to compute the sensitivity of the waves at the
observation site to the distributed sources surrounding it throughout the nearby Eastern
Indian Ocean. The (mean) wave propagation properties are assumed to be constant in time.
(2) There is a model for the second-order statistics of the non-harmonic tide phase
and phase-speed modulations, and their spatial correlations. This model, and its adjoint,
are integrated along ray-paths to describe (map) the statistics of the non-harmonic phase
of the tide as it propagates from source regions (backward along the rays).
(3) There is a model (developed in detail in Part I) which is used to relate the phase
statistics of the individual wave sources to the statistics of the sum of the waves (at the
observation site).
This manuscript (Part II) derives in detail components (1) and (2) above. Then it
estimates the necessary input parameters from first principles (the barotropic-to-baroclinic
tide forcing) and observations (phase speed variance and correlation scales), and
then it proceeds to compare with the observed tidal variability and its sensitivity to
the modeling assumptions (principally, the spatial correlation structure of the phase
speed variability).
Overall, I found the manuscript too long, and rather hard to follow. Due to the complexity
and multi-step development, it is essential for the author to reduce the verbage to the minimum
necessary to communicate clearly. As it is, there are several sets of comments and caveats mentioned,
which, while appropriately nuanced and apparently relevant, make it hard to follow the thread
of the essential analysis.
It seems to me that the use of the adjoint model for the wave linear dyanmics (used to produce
Fig 5) is relatively well-worn in the oceanography literature. The modeling of the along- and
across-path covariance structure of the phase modulations (eqn's (30)-(38)) seems to be totally
new at this level of precision and detail. I would suggest that the author consider breaking this
Part II manuscript into two smaller pieces, one focused on the ray-tracing and modeling of the
phase covariance, and then the other on using this covariance with the adjoint wave model to explain
the observations. I think this manuscript is skillfully using a lot of innovative ideas, and it
will have more impact if the it is broken down into simpler and more digestible pieces.
Of course, this type of re-organization of the presentation will take considerable work.
Given the relatively narrow audience, maybe it is not worth the effort, but I hope the author
will consider it.
In any case, I suggest this article be accepted after significant revisions to address
my detailed comments, below. Some of my questions are answered by text, later in the manuscript,
and reflect my misunderstandings. Nonetheless, I hope my comments will inform the author of
the reaction of an interested reader, and guide him in making the manuscript more comprehensible.
Detailed Comments:
The abstract says that a map of non-harmonic internal waves sources is
identified from data on the Australian North West Shelf.
The abstract could be clearer about exactly what kind of data are used.
l37: "tidal currents" --> "barotropic tidal currents"
l42: Does this sentence make sense? It is comparing "generation" with
"amplitude modulation" and "phase modulation". The "generation" is of a different
category than modulation.
l47: " nonstationary" --> "nonstationary"
l97-l123: This is a long overview, but I don't feel like it has provided me
with specifics needed to understand what is to come. Maybe it can be
shortened or omitted.
Eqn (4a)-(4d): Where are these properties of the wrapped normal proven?
Eqn (4b) and Eqn (5): This is confusing. At line 137, it says that
A_j is deterministic. Doesn't this mean that A'_j is identically zero?
l171: I am not understanding this. I don't understand what distinguishes
\Theta'_j and \Theta''_j.
l195: n is a complex random number, right? I am not knowledgeable enough
about the properties of complex random variables to be certain that
the Cholesky-like decomposition mentioned at this line exists.
Is R^{1/2} always defined for complex n? Is it complex-valued or real-valued?
l206-l220: This section points out the non-uniqueness of the matrix square
root. Is R always real-valued?
l221-l227: I'm afraid I don't understand this..
l253: Usually the "objective function" is a quadratic expression in data
assimilation, so this is a little confusing. Why not just refer to J as
an arbitrary linear function of x?
Fig 2: Please put panel labels in the same relative location in each
panel, i.e., at the top left.
Fig 2d: Some of the structure depicted in this figure looks like it could
be caused by spurious bottom topography data, e.g., such as the
linear features around 119E, 19S and the apparent correlation of the
forcing function with the 100m 200m and 500m isobaths.
I would be curious to see a histogram of depths from this region to see if
it exhibits peaks at 100m intervals.
l306: "same reasoning" refers simply to treating the finite-dimensional
linear sum (eqn (12)) as an approximation of the integral?
I am not convinced that the arguments about the matrix R translate directly
to the function R, here.
l310: Regarding the non-uniqueness: Aside from the explanation in 3.1, isn't it
more fundamental that s_{nh} is not unique? s_{nh} is a function (it has inifintiely
many degrees of freedom), while E(A'^2) is a scalar. Therefore it is not possible
to uniquely determine s_{nh} from E(A'^2).
This is distinct from the non-uniqueness of R^{1/2} discussed in 3.1.
Fig 3a: Does this figure contain a spurious pink line? There is a straight
line at about 116.5E from 9S to 17S which looks out of place and does not appear
to represent a ray path.
Sect 3.3: While this section makes true statements, equation (26) is do general that
I'm having trouble seeing how this will be used. Why not simply present (26) in the more
specific context, where components of P, B, and Q are defined.
l350: Rather than refer to this as Lorentzian, which is usually applied to spectra
of narrowband process with phase modulations, I wonder if it would be better to call it
a first-order autoregressive process? I guess it is Lorentzian, but centered at the
zero frequency.
l354: P_{cc} is "stationary" -- do you mean P_{cc} constant in space and time?
Eqns (31a-b): Are these derived from (27) and (28), which are the explicit form of
equation (25)?
l363: The spatial variability of \overline{c} and L_C is included in (31a-b), not in
equation (32), right?
l368: This is a little confusing. I think you are saying that for each source, j, there
is an associated P_\theta\theta, the value of which is determined by the path between the
source j and the observation location where P_\theta\theta (\sigma^j) is evaluated.
l377-l382: This is an interesting point. Presumably, though, there is an high-frequency
cutoff. The whole ray-tracing and propagation paradigm only makes sense if c' is slowly varying
compared to \omega. Right?
Eqn (33): The last row of A, does it represent the evolution of
\Delta\theta'', the phase evolution along the two paths? Is that why the
terms are opposite signs, because \theta'' = \theta'(path 1) - \theta'(path 2)>
l419: The assumption of "time-independent" also mean "space-independent" in this context, becauase
time is measured along ray paths. Is that correct?
Eq (39): Please clarify: even though \overline{c}, L_C and \Delta\eta/l are
assumed to be time-independent, P_\theta\theta is not time-independent and varies
with t according to equation (32). But there is an oddity: the observation is at
a signle point, and the path separation, \Delta\eta, presumably linearly
decreases to 0 approaching this point. I wonder if there is a cancellation of the
linear grown with time (equation (32)) and the linear decrease with time (\Delta\eta),.
Aha: l425-l440: It appears that you have already thought-through the consequences
of my comment about Eq (39).
Overall comment on this section: I found the development a little hard to follow.
I wonder if it might be more direct to state the covariance evolution equations all at
once, eqns (31)a-b and (38)a-c, and explain how these describe the along-path and across-path
phase covariances (and c'). I'm not sure of the best approach. Perhaps it would be sufficient
to add a short paragraph after eqn (26) with an overview of the approach to follow,
so that the reader is prepared to accept the notation for along-path and acorss-path
phase variations and their covariances. Maybe change the header of 3.4 to mention
the "along-path phase difference", which would make it more parallel to the header
of section 3.5.
l449-l456: I'm sorry, but I don't understand what the phrase "were included as
forcing of nonharmonic internal tides in the semidiurnal frequency band" means.
Maybe you could start by clarifying what you mean by "in the modeling". Does this
refer to ray tracing (modeling internal waves per se), or does it refer
to some version of equation (26) (modeling phase/phase speed covariance), or some
combination of these?
l460: omit "in this feasibility study" --- We are 18 pages into this manuscript, and
a number of new concepts have been introduced. You need to simplify the language as much as
possible to make this story comprehensible.
l464: "amplitude" --- Need to be careful here. The amplitude, sqrt(A^2 + B^2), is not a linear functional
of the time series A cos(omega t) + B sin(omega t). I'm not sure if the model you
are describing is for the waves, or for their statistics. If it is the latter,
then the amplitude could very well be a linear functional of the state.
l470: The "model" here clearly describes the model for the wave propagation.
l479-l480: I don't understand what this means. How does the assumption that the
barotropic and VM1 kinetic energies are the same relate to bottom friction?
Or, is all of l479-l484 to be interpreted together (but this seems to contradict the simpler
explanation in l478)?
l495: Using the language of Bennett, it sounds like you computed representer functions
and their adjoints (which you referred to as the sensitivities) for VM1-4 and
the 4 tidal frequencies, assuming a measurement at PIL200.
l502-l508: I would suggest moving this to the discussion.
l509-l516: How were \sigma_C^2 and L_c were chosen to integrate (31)?
Aha -- I see later.
l522: Are you saying that the non-M2 tides will use the same along-path (P_\theta\theta, P_cc,
and P_c\theta) statistics, as determined for M2? You haven't mentioned yet the
across-path statistics, but I assume those a included also?
l551: "length scale larger than eddies" --- Indeed. You might want to cite Buijsman's study
of the near-equator variability in HYCOM.
l588-l589: I don't understand the parenthetical comment about "one dimensional".
l592-594: Why is this other estimate "more accurate"? Simplify if possible. And
don't consider all the caveats (or move them to the Discussion).
l597-l608: I don't understand the significance of all these details. Perhaps
this could be expanded somewhat to explain, or maybe all this could be pushed
into the discussion or an appendix.
l616-l625: Hmmm. It seems like you should have sorted out the significance of these
processes with some wave model runs using variable phase speed modulations.
It sounds like you have made some arbitrary assumptions about the nature of the
phase variability.
l655: "sound nothing" --> "sound like nothing"; but you should probably rephrase this
in the positive: "This observation is significant because ..."
l660: Remind us what is the "source function". Which term in which equation is it?
Fig 8: Sorry if I have fogotten: why is it necessary to make a Gaussian fit to the covariance
functions? Is it because the Weaver diffusion model is used with the wave evolution
equation model? Or, do you just need to estimate the correlation length?
l705-l728: This is nice qualitative discussion.
Also, the following discussion of phase correlations is good, too.
l783-l785: I don't understand the distinction between the "total modelled
variance" and the "VM1 M2 component". Does the total mean that all VM1-4 and
the 5 harmonic constants are included? The amplitudes of the VM2-4 and non-M2 are
so much smaller than VM1 M2, though, I'm not sure of the usefulness of the degrees of
freedom concept for these smaller components.
l821: Didn't Rainville and Pinkel's study find that the variability of propagation
paths is "large", in some sense. At that time, there was some discussion of how this
invalidated some aspects of the ray-tracing approach, but I cannot reconstruct the
arguments. In any case, you proposed model (without the path variability) yields
plausible results. But maybe this is just another aspect of the non-observabilkity of the
detailed mechanisms in this problem.
l922: Do you know if this system is equivalent to Sam Kelly's Coupled Shallow Water (CSW) representation?
It looks equivalent except he used z (rather than sigma) as the vertical coordinate.
Anyway, it might be worth mentioning the equivalence. Also, his formulation admits
somewhat simpler expressions for the coupling coefficients (L_{nm}) than
appears to be the case here (documented in an appendix of Zaron, Musgrave, and Egbert).
Some earlier work by Lahaye has similar expressions for the mode-coupling terms.Citation: https://doi.org/10.5194/egusphere-2024-4193-RC2
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