The autocovariance of the semidiurnal internal tide (IT) is examined in a 32 d segment of a global run of the HYbrid Coordinate Ocean Model (HYCOM). This numerical simulation, with 41 vertical layers and

Internal tides (ITs) are internal waves generated by the interaction of tidal currents with rough bathymetry. The radiated wave beams can travel thousands of kilometers (e.g.,

At any given position in a stationary medium, tidally forced waves would have a constant phase difference to the astronomical forcing. However, since they propagate within the time-varying ocean circulation, ITs are subject to a variety of mechanisms that cause their phase difference to the tidal forcing at their generation site to shift with time

Only the

Some current high-resolution global ocean circulation models enable the estimation of the barotropic and internal tides concurrently with the ocean circulation

The empirical mapping of ITs from altimetry remains challenging for various reasons

Recently,

The remainder of this work is organized as follows: in Sect.

The different comparisons in this work are all done in terms of vertical displacement of the isotherms. The measurable variables needed to compute vertical isotherm displacement at a fixed depth are the temperature anomaly and vertical temperature gradient at that depth. In this section we briefly describe the temperature time series from each dataset.

We use data from a global collection of Argo Iridium floats deployed by the University of Washington as part of the National Ocean Partnership Program during the period 2004–2022. Between the descending and ascending profiling phases, these floats also record temperature and pressure with an hourly resolution while adrift at 1000 dbar. This so-called park phase typically lasts 10 d. As in

Stitching together data from successive cycles, by filling the time between park phases (typically 6 h) with NaN (not a number) values, one can construct longer time series. In this study, we use segments of 32 d of data (i.e., the duration of the segment of numerical simulation we are using). The sampling period of the park phase can occasionally vary by more than a few seconds. To ensure evenly spaced time series, we linearly interpolate each concatenated record of 32 d onto a time axis with a constant 1 h step. Any interpolated value lying between two original records that are more than 1.5 h apart is replaced by NaN.

The position of the floats can only be determined when they reach the surface. We assume straight trajectories between two successive surfacings (typically 10 d apart). This assumption has been shown to be reasonable by

The Argo dataset used in this work is an updated version of the one used by

The Global Multi-Archive Current Meter Database (GMACMD;

Here, we extracted 331 temperature time series spanning 1972–2010 and meeting the following criteria:

The mooring lies in water deeper than 2000 m.

The record is longer than 64 d, with a sampling interval shorter than 3 h, for adequate resolution of the semidiurnal tidal signals.

The instrument depth is within

One particular mooring is used as an example in Sect.

This study uses 32 d, from 20 May to 20 June 2019, of hourly output at 1000 m depth from a global run of HYCOM, with 41 vertical layers and

A Lagrangian analysis of the simulation is used for a direct model–data validation with Argo floats. The Argo quasi-Lagrangian sampling is mimicked by releasing 41 644 particles randomly across the world oceans. We let the particles be advected by the 2D velocity field at 1000 m for 32 d while sampling temperature with an hourly resolution. This Lagrangian sampling of HYCOM is achieved using the software Parcels

In this section, we present a local example to introduce the methods that are used in Sect.

As in

In the case of Argo floats, Eq. (

For both Argo floats and HYCOM Lagrangian particles, the low-frequency background activity is filtered out from

The HYCOM-derived Lagrangian time series

Example circular geographical patch of 200 km radius (the white star denotes mooring no. 2 at the center). The median positions of the segments of Argo data and of the HYCOM particles are shown by white filled circles and white squares, respectively. The dashed white curves represent the trajectories of the HYCOM particles over the 32 d of numerical simulation. The binned Argo segments were all recorded by the same float; its trajectory is shown by the solid white curve, with white crosses denoting the starting and ending points of the different 32 d segments. In the background we show the amplitude of the

From a finite time series

The sample autocovariances for all HYCOM particles within the circular patch shown in Fig.

A handy tool for monitoring the evolution of the autocovariance of the semidiurnal IT is complex demodulation. Here, it consists of the least squares fitting of

Complex demodulation is just a convenient way of finding the envelope of the sample estimate of an underlying true oscillating function at a given frequency. However, as an estimate of the envelope of the true oscillating function, the complex demodulate can be shown to be biased high (see Appendix

Following

Since HYCOM does not resolve the full spectrum of the oceanic variability, in particular the variability associated with short timescales, the corresponding stochastic noise is expected to differ from the one captured by the in situ data (in terms of both variance and characteristic timescale). The contamination of the first demodulate by this noise would therefore account for a systematic bias when comparing the simulated data with observations. To limit this, we chose to consistently subtract an estimate of the autocovariance of the non-tidal variability from the sample mean autocovariance before computing the complex demodulates. The way we obtain such an estimate is described in Appendix

The result of the complex demodulation at the semidiurnal frequency applied to our two (noise-corrected) mean autocovariance series is presented in Fig.

The decay with time lag of the demodulates represented as red crosses in Fig.

In the Eulerian framework, our methods remain practically unchanged. We now define the vertical isotherm displacement at a given location as

For each HYCOM particle, we compute 32 d long time series of

Equation (

The Eulerian sampling serves two main purposes: (i) validating the variance of the IT measured by the Lagrangian particles and (ii) comparing the decorrelation of the IT in the HYCOM data with mooring observations. We illustrate these two aspects for the local example introduced in Sect.

Figure

We bin the global collection of HYCOM particles based on their median position using circular geographical patches of 200 km radius centered on a regular 2.5

As in Sect.

We start by checking how the Lagrangian sampling affects the semidiurnal IT variance. Figure

Semidiurnal IT variance estimated from the Eulerian HYCOM data as a function of the semidiurnal IT variance estimated from the Lagrangian HYCOM data for the unmasked bins in Fig.

We can then map the semidiurnal IT variance (here taken as the first 48 h complex demodulate of

The main patterns visible in Fig.

We investigate this bias by looking at the geographical distribution of the HYCOM-to-Argo semidiurnal IT variance ratio. For presentation purposes, instead of the latter ratio we plot the proxy

The representativity of the zonal-mean variances is smaller north of 40

Semidiurnal IT variance from the autocovariance series plotted in Fig.

In contrast to Argo floats, the Eulerian sampling of moorings allows us to directly monitor the decorrelation of the IT. In a procedure similar to that in Sect.

For all the datasets used in this study, we found the probability distribution of the global collection of local-mean autocovariance at any given time lag to be skewed (not shown). This is not an issue when computing average statistics from the Argo or the corresponding simulated data, since the number of samples (i.e., the number of geographical bins) is very large, and the sample mean is therefore expected to be normally distributed, by virtue of the central limit theorem. In contrast, geographical bins where mooring data are available are fewer. Thus, the influence of the tail of the distribution on the sample mean is larger when analyzing the relatively small collection of bins where mooring data are available than when considering the global collections of Argo and HYCOM data. This precludes the use of statistics that assume a normal distribution when describing regional or even global averages of the autocovariances computed from moorings. To limit the effects of skewness, we discard bins for which either the mooring or the simulated first 48 h demodulate is above the 95th percentile of its observed distribution (here

The moorings may not offer as much spatial coverage as the Argo floats do, but they still provide an opportunity to validate the geographical variability in the semidiurnal IT variance in HYCOM. As in Sect.

To measure the strength of the decorrelation affecting the Eulerian mean autocovariances, we define the semidiurnal coherent variance fraction (

Figure

We investigate a potential latitudinal dependence by plotting the mean autocovariance series computed separately from the geographical bins lying north (149 instruments) and south (11 instruments) of 50

The slower decorrelation of the IT in the simulation can be explained by some decorrelating processes, such as eddies or submesoscale variability, being weaker in HYCOM than in the real ocean. It could also be explained by the time variability in certain decorrelating processes. Our numerical simulation only spans 20 May to 20 June 2019. Therefore, it is potentially missing processes that would specifically occur or intensify at another time of the year (or in a different year). On the other hand, the mooring data span several decades. Thus, our single month of data from HYCOM may not be representative of the broader temporal sampling of the mooring data.

The spatial distribution of the moorings is sparse and tends to be denser in particular areas (e.g., the Gulf Stream region). This is all the more true south of 50

Summarizing numerics of the autocovariance series plotted in Fig.

In Sect.

The HYCOM data allow this to be studied by directly comparing the global-mean autocovariance computed in the Lagrangian and Eulerian frameworks (see Fig.

Taking inspiration from

A constrained least squares fit of the model Eq. (

By dividing

Summary of the parameters estimated from the simulated Eulerian and Lagrangian data. These values are used to compute

We can compare the global-mean values of

Global-mean autocovariance at 1000 dbar computed from the Argo data over the unmasked bins in Fig.

Similarly to

Lastly,

Parameters from the fitting of the model Eq. (

Why is the IT variance lower and IT decorrelation weaker in HYCOM than in the observations, particularly in the Southern Ocean? At the time of writing we cannot think of a particular reason for either the Argo- or the mooring-derived IT variance to be biased high globally. In particular, the correction accounting for the non-tidal variability we systematically subtract from the sample autocovariance precludes any contamination of the first demodulate by the background noise (see Appendix

We also investigated whether the bias in the Southern Ocean could be related to the contamination of the first 48 h demodulate at

As for the model, the horizontal grid spacing limits the number of vertical modes correctly resolved to the first five modes, equatorward of

The mode-

In Fig.

The magnification of the contributions from modes 2 and 3 to the variance at 1000 dbar in the Southern Ocean only affects how propagating ITs are perceived at the Argo parking depth. This has no connection with the generation and dissipation processes that set the underlying modal partitioning of the IT energy. Additional explanations might be found by examining whether the main parameters affecting the generation of ITs (namely the bottom topography, barotropic tidal forcing, and stratification) are less accurate in this region than in the rest of the globe.

The pattern of the enhanced discrepancies between Argo and HYCOM in the Southern Ocean (darker blue in Fig.

In the recent SRTM15+ bathymetry

Efforts have been made to improve the accuracy of the

Lastly, to assess the stratification in HYCOM, we compare the phase speed of a mode-1 gravity wave in the model with the phase speed determined from climatology. The phase speed of a mode-

In Fig.

In this work we compare a 32 d segment of a global run of the HYCOM model, including realistic tidal and atmospheric forcing, with in situ observations of the semidiurnal IT around 1000 dbar. First, a Lagrangian sampling of the simulation was compared to park-phase data from Argo floats to validate the geographical variability in the semidiurnal IT variance in HYCOM (see Sect.

The main spatial patterns of the simulated IT variance at 1000 dbar broadly agree with Argo observations, with energy radiating away from low-mode IT generation hotspots (see Fig.

While the difference between the model and Argo data appears reasonably homogeneous across most of the world ocean, it steadily increases towards the poles (see Fig.

The mooring data support the above results for the semidiurnal IT variance. Additionally, we found that the decorrelation affecting the semidiurnal IT in HYCOM over a 32 d window is weaker than observed in the mooring records, on average (see Fig.

We also investigated the effects of the Lagrangian sampling inherent to the Argo floats. When comparing autocovariances computed from the HYCOM data sampled in the Lagrangian and Eulerian frameworks, respectively, we found the IT variance to be unaffected in the mean (see Fig.

Finally, we discuss the potential sources of bias. We could not think of a particular reason for the IT variance obtained from either the Argo or the mooring data to be biased high, particularly in the Southern Ocean. However, HYCOM is subject to various limitations. First and foremost, the model can only correctly resolve vertical modes up to 5 in most of the global oceans. Approaching the poles, the reduced number of layers further limits the number of resolved modes. While mode-1 ITs supposedly account for most of the tidal variability at 1000 dbar on average globally

We consider the function

Let

We now show the inequality Eq. (

Define a scalar product

Two aspects of the processing originally used in

We can investigate this further using a simple model, adapted from

We start by varying

In Fig.

First 48 h demodulate of the mean autocovariance computed from 1000 synthetic time series generated following the model Eq. (

We repeat the same experiment for

Theoretical autocovariance of the AR1 process

First 48 h demodulate of the mean autocovariance computed from 1000 synthetic time series generated following the model Eq. (

Hence, for the typical

It is not straightforward to understand why the contribution of the non-tidal variability to the value of the first demodulate can be negative. Because of the way we defined

As mentioned in Sect.

In the present appendix, we derive a model for the autocovariance of a tidal variability on top of a high-pass-filtered stochastic noise. In Appendix

Assume that the non-tidal variability

The Butterworth filter is linear in amplitude but not in phase. As a workaround, we applied a second-order filter twice, once forward and once backward, to recover a fourth-order filter with zero phase. The transfer function of a second-order high-pass Butterworth filter in the

Hence, we can evaluate Eq. (

As mentioned in Sect.

We fit the full model Eq. (

The inclusion of a tidal variability in the model Eq. (

Argo data were obtained from US GDAC (

GG: conceptualization, formal analysis, project administration, software, visualization, writing – original draft, writing – review and editing. JN: conceptualization, funding acquisition, writing – original draft, writing – review and editing. MCB: project administration, writing – review and editing. JFS: formal analysis, resources, software. BKA: writing – review and editing.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank Shane Elipot and one additional anonymous reviewer who greatly helped improve this paper.

Gaspard Geoffroy and Jonas Nycander are supported by grant number 2017-04623 from the Swedish Research Council. The computations and data handling were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre (NSC), partially funded by the Swedish Research Council through grant agreement no. 2018-05973. Maarten C. Buijsman and Brian K. Arbic are supported by the Office of Naval Research (ONR) grant numbers N00014-19-1-2704 and N00014-19-1-2712, respectively, which both fall under the project “Modeling, characterizing, and predicting effects of internal gravity waves on acoustic propagation on basin to global scales”. Jay F. Shriver was supported by ONR grant number N0001422WX01919, which falls under the project “Diagnosis and validation of the time and spatial variability of remotely generated internal waves in global ocean simulations”. The article processing charges for this open-access publication were covered by Stockholm University.

This paper was edited by Ilker Fer and reviewed by Shane Elipot and one anonymous referee.