Elsevier

Ocean Modelling

Volume 147, March 2020, 101561
Ocean Modelling

An inverse technique for reconstructing ocean’s density stratification from surface data

https://doi.org/10.1016/j.ocemod.2019.101561Get rights and content

Highlights

  • A simple, robust inverse method for reconstructing ocean’s vertical density profile.

  • The method is based on fundamental principles of oceanic internal tide generation.

  • It only requires sea surface height data to reconstruct the density profile.

  • Will help to obtain a uniform spatial resolution ocean density stratification field.

Abstract

In this article, we propose an inverse technique that accurately reconstructs the ocean’s density stratification profile simply from free surface elevation data. Satellite observations suggest that ocean surface contains the signature of internal tides, which are internal gravity waves generated by the barotropic tides. Since internal tides contain the information of ocean’s density stratification, the latter can in principle be reconstructed from the free surface signature. We consider a simple theoretical model that approximates a continuously stratified ocean as discrete layers of constant buoyancy frequency; this facilitates the derivation of a closed-form dispersion relation. First, we numerically simulate internal tide generation for toy ocean scenarios and subsequently perform Space–Time Fourier Transform (STFT) of the free surface, which yields internal tide spectra with wavenumbers corresponding to the tidal frequency. The density profile is reconstructed by substituting these wavenumbers into the dispersion relation. Finally, we consider a more realistic situation with rotation, bottom topography, shear and density profiles representative of the Strait of Gibraltar. Density reconstruction in the presence and absence of shear is respectively found to be 90.2% and 94.2% accurate. The proposed method can be used to reconstruct climatological mean ocean density field of uniform spatial resolution using only surface elevation data obtained via satellite altimetry.

Introduction

The oceans are by and large stably stratified, that is, the density of ocean monotonically increases with depth. The ocean’s density also varies with latitude and longitude, as well as with seasons. However, variations along the horizontal direction are usually many orders of magnitude (typically in the open ocean O(105)) smaller than that in the vertical (Li et al., 2019). Depending upon the strength of stratification, in general the vertical structure of the ocean’s density is divided into three major layers: (i) top — weakly stratified surface mixed layer, (ii) middle — strongly stratified pycnocline, and (iii) bottom — weakly stratified abyss (Sutherland, 2010).

An accurate knowledge of the ocean’s density field is crucial for ocean and climate modeling (Cummins, 1991). The oceanic density stratification also has a direct impact on the aquatic ecosystem. In oceans and lakes, microbiological activities and accumulation of organisms are strongly affected by the pycnocline (Doostmohammadi et al., 2012). The density stratification influences the formation of spring phytoplankton blooms, which in turns help to maintain a balanced ecosystem (Sherman et al., 1998). In particular, depth of the top mixed layer modulates the interaction between the light availability for photosynthesis and the nutrient supply from the deep oceans (Capotondi et al., 2012). The density gradient at the base of the mixed layer affects the entrainment process, which plays an important role in mixed layer deepening and in supplying nutrients to the photic zone (Capotondi et al., 2012).

The ocean’s density is a function of both temperature and salinity, both of which are measured using CTD (Conductivity, Temperature and Depth) sensors using the ARGO floats (Bradshaw and Schleicher, 1980). These sensors, while descending (or ascending) through the ocean water, collect the necessary information. The vertical profiles of temperature and salinity thus obtained are then substituted into the equation of state to yield ocean’s density profile at a given latitude–longitude. At present, there is a global array of 3800 free-drifting ARGO floats in the global ocean. Indeed, these drifters do provide an accurate measurement of the density field, however, they cannot provide spatially uniform resolution data. Another drawback of this measurement technique is that these floats behave as free-drifters, therefore, their measurement at a particular point of interest cannot be controlled precisely.

The above drawbacks have persuaded us to look into a useful alternative measurement technique. In this article, we propose a strategy that can provide a reasonably accurate estimate of ocean’s density stratification profile (and hence, the pycnocline depth) in a fully non-invasive manner by only analyzing the ocean free surface. To achieve this, we scrutinize one of the most important consequences of ocean’s stable density stratification — the internal tides, which are internal gravity waves (IGWs) forced at the tidal frequency (Wunsch, 1975). Recently, internal tides have been proposed to be a cost-effective tool to infer the change in the upper ocean temperature due to a change in travel time of the low-mode internal tides (Zhao, 2016). Our proposed method provides an optimal compromise between fidelity and simplicity of representation of the ocean density field. To analyze the ocean surface imprint, we invoke a simpler approach by discretizing the vertical variation of density into discrete layers with a linear variation of density (implying the buoyancy frequency is piecewise constant) and construct a theoretical model to estimate the layered density profile. We note here that there are previous studies on the reconstruction of the subsurface flow and density field from the sea surface density and sea surface height using the surface quasigeostrophic (SQG) formalism (Wang et al., 2013). This methodology has been quite successful in the high energetic (shear) regions likes Gulf stream extension and also to an extent in the lesser energetic region. Both SQG and our proposed technique can utilize the satellite altimetry surface data to provide a representation of the climatological density field. Our proposed method, however, can complement the existing SQG methodology in the less energetic regions for which it is ideally suited.

Apart from the internal tides, there is another type of IGW that are generated by the wind-driven flow and have been observed as a prominent peak in the Garrett & Munk continuous internal waves spectrum (Alford et al., 2016). These wind-induced internal waves generated in the ocean mixed layer are commonly known as “near-inertial waves”, and as the name suggests, the frequency of these waves is very close to the Coriolis frequency (D’Asaro, 1989, Garrett, 2001). These waves predominantly undergo downward propagation (Garrett, 2001) and are different from the kind of IGWs the current work is fully based on — the internal tides. From here on, the acronym “IGW” would only represent internal gravity waves generated by the semi-diurnal tides.

The high-mode IGWs often dissipate near their generation site. On the contrary, the low-modes generally travel hundreds and even a thousand kilometers before getting dissipated (Zhao et al., 2016, St. Laurent and Garrett, 2002). An important aspect of low-mode IGWs is that they are very efficient in transporting momentum and energy over large distances, and help in mixing nutrients, oxygen and heat in the oceans (Sarkar and Scotti, 2017). Turbulence and mixing due to IGW breaking play an important role in regulating the global oceanic circulation, and are one of the major factors in the climate-forecast models (Ferrari and Wunsch, 2009).

The signature of the low modes on the ocean surface are detectable by the satellite altimeters (Ray and Zaron, 2011, Ray and Mitchum, 1996); they appear as wave-like perturbations having a frequency of the tidal frequency. In the past two decades, efforts have been made to construct the coherent structure of the stationary low-mode internal-tides using 20 years of sea-surface height (SSH) data from multiple satellite altimeters (Zhao et al., 2012, Zhao et al., 2016). Fig. 1 shows the global estimation of mode-1 SSH using multi-satellite altimetry. The figure also shows the regions of strong internal tide generation sites, which are highly correlated with the large-scale topographic features, shown by the solid black lines. The sampling rate of satellite altimeter is usually larger than the tidal period, but remarkably enough, IGWs can still be studied with a dataset exhibiting sampling only once in every 2060 tidal periods (Zhao et al., 2012). An astounding point comes from these observations that the signature of the IGWs remain both spatially and temporary coherent despite the fact that they are often contaminated by the meso-scale eddies or the sub-surface shear (mostly induced by the wind forcing) (Zhao et al., 2016). The pivotal point of our article is the realization that the ocean surface signature of IGWs (which, as already mentioned, are well detectable via satellite observations) carry the information of ocean’s density stratification, and can in principle be inverted to reconstruct the latter.

We have organized the article as follows. In Section 2 we discuss the closed form dispersion relations for one, two and three-layer of constant buoyancy frequency. Additionally, we have provided a mathematical justification regarding the uniqueness of IGW wavenumbers, and have also highlighted a unique situation where two different density profiles provide the exact same set of wavenumbers. Moreover, sensitivity analysis of density reconstruction has been discussed for one, two and three-layered models. Various aspects of numerical implementation are discussed in Section 3. In Section 4, we first consider toy models with simple density profiles, and then a representative density profile of the Mediterranean sea. In each case, IGWs emanating from the bottom topography impinge on the free surface. Wavenumbers corresponding to the surface signature are substituted in the closed form dispersion relation to reconstruct the underlying density profile. Next we perform semi-realistic simulations of internal tides in the Strait of Gibraltar (the region where Mediterranean Sea meets the Atlantic Ocean). The first case considers real bathymetry, real density profile, Coriolis effect, but no background velocity shear, while the second case includes the effect of velocity shear. The density stratification profile is reconstructed in both cases.

Section snippets

Governing equations and exact solutions

We consider an incompressible, 2D (xz plane), density stratified flow of a Boussinesq fluid on an f-plane. The mean (denoted by overbars) density profile varies in the vertical (z) direction. The governing Navier–Stokes equation in this case is given by (Gerkema, 2001, Gerkema and Zimmerman, 2008, Verma, 2018): ut+(u)u+fzˆ×u=1ρ0pρgρ0zˆ+ν2u,u=0,ρt+uρ=wρ0gN2+κ2ρ+F. Except F, the variables without overbars denote the perturbation quantities. The perturbation velocity field is

Numerical implementation

In order to simulate the internal tides, we numerically solve equation set (1a)–(1c). Following Gerkema (2001), we consider thebarotropic tidal forcing term F=zN2(z)Q0sin(ω0t)h(x)2dhdx,where Q0 is the barotropic flux, h(x) is the local water depth and ω0 is the tidal frequency. The bottom topography has been incorporated; furthermore, the Cartesian coordinate system, xz, has been transformed into a terrain-following coordinate system, xζ with ζzh(x). Therefore, in the terrain-following

One-layer

We first simulate a single-layer flow with N=0.1 s−1 and f=0 s−1. The space–time Fourier Transform (STFT) of the surface elevation field, η, is given by, ηˆ(k,ω)=η(x,t)ei(kxωt)dxdtwhere, ηˆ(k,ω) is the amplitude in wavenumber (k)–frequency (ω) space. For ease of understanding, the solution algorithm is given in Algorithm 1. The STFT of η yields the wavenumbers kn corresponding to the tidal frequency ω0=0.05 s−1.

Fig. 2(a) shows STFT of the free surface elevation; the first vertical-mode (n=1)

Conclusion

Using numerical simulations, we show that ocean’s free surface carries the information of its mean density profile, and lay out the theoretical groundwork towards its reconstruction. The barotropic tides cause the stratified ocean water to move back and forth over the submarine topography, causing the IGW radiation. The IGWs carrying the information of the ocean’s mean density profile impinge on the free surface, the signature of which can be observed using the STFT. The wavenumbers (countably

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors are grateful to G. Saranraj for discussions and many suggestions during the study. A.G. would like to acknowledge the funding support from the Alexander von Humboldt foundation, and the SERB Early Career award ECR/2016/001493.

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