OSOcean ScienceOSOcean Sci.1812-0792Copernicus PublicationsGöttingen, Germany10.5194/os-14-1283-2018Spectral signatures of the tropical Pacific dynamics from model and altimetry: a focus on the meso-/submesoscale rangeSpectral signatures of the tropical Pacific dynamics from model and altimetryTchilibouMichelGourdeauLionellionel.gourdeau@legos.obs-mip.frMorrowRosemarySerazinGuillaumeDjathBughsinhttps://orcid.org/0000-0002-1121-5272LyardFlorentLaboratoire d'Etude en Géophysique et Océanographie Spatiales (LEGOS), Université de Toulouse, CNES, CNRS, IRD, UPS, Toulouse, FranceHelmholtz-Zentrum Geesthacht Max-Planck-Straße, Geesthacht, GermanyLionel Gourdeau (lionel.gourdeau@legos.obs-mip.fr)24October20181451283130120April201828June201818September201828September2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://os.copernicus.org/articles/14/1283/2018/os-14-1283-2018.htmlThe full text article is available as a PDF file from https://os.copernicus.org/articles/14/1283/2018/os-14-1283-2018.pdf
The processes that contribute to the flat sea surface height (SSH) wavenumber
spectral slopes observed in the tropics by satellite altimetry are examined
in the tropical Pacific. The tropical dynamics are first investigated with a
1/12∘ global model. The equatorial region from
10∘ N to 10∘ S is dominated by tropical instability waves
with a peak of energy at 1000 km wavelength, strong anisotropy, and a
cascade of energy from 600 km down to smaller scales. The off-equatorial
regions from 10 to 20∘ latitude are characterized by a narrower
mesoscale range, typical of midlatitudes. In the tropics, the spectral taper
window and segment lengths need to be adjusted to include these larger
energetic scales. The equatorial and off-equatorial regions of the
1/12∘ model have surface kinetic energy spectra consistent with
quasi-geostrophic turbulence. The balanced component of the dynamics slightly
flattens the EKE spectra, but modeled SSH wavenumber spectra maintain a steep
slope that does not match the observed altimetric spectra. A second analysis
is based on 1/36∘ high-frequency regional simulations in the
western tropical Pacific, with and without explicit tides, where we find a
strong signature of internal waves and internal tides that act to increase
the smaller-scale SSH spectral energy power and flatten the SSH wavenumber
spectra, in agreement with the altimetric spectra. The coherent M2 baroclinic
tide is the dominant signal at ∼140 km wavelength. At short scales,
wavenumber SSH spectra are dominated by incoherent internal tides and
internal waves which extend up to 200 km in wavelength. These incoherent
internal waves impact space scales observed by today's along-track
altimetric SSH, and also on the future Surface Water Ocean Topography (SWOT) mission 2-D swath observations, raising
the question of altimetric observability of the shorter mesoscale structures
in the tropics.
Introduction
Recent analyses of global sea surface height (SSH) wavenumber spectra from
along-track altimetric data (Xu and Fu, 2011, 2012; Zhou et al., 2015) have
found that while the midlatitude regions have spectral slopes consistent
with quasi-geostrophic (QG) theory or surface quasi-geostrophic (SQG)
theory, the tropics were noted as regions with very flat spectral slopes
(Fig. 1a). The objective of this paper is to better understand the processes
specific to the tropics that contribute to the SSH wavenumber spectral
slopes observed by satellite altimetry, particularly in the “mesoscale”
range at scales < 600 km and 90 days (Tulloch et al., 2009).
(a) Spatial distribution of altimetric along-track SSH wavenumber
spectral slope calculated in the fixed 70–250 km mesoscale range (from Xu and
Fu, 2011; their Fig. 2). (b) Latidudinal dependence of the altimetric
SSH along-track wavenumber spectra in the Atlantic Ocean (from Dufau et al.,
2016; their Fig. 3). The colors of the spectra refer to the geographical boxes
where along-track data were averaged on the right.
Only a few studies have addressed the tropical dynamics at spatial scales
smaller than this 600 km cutoff wavelength. The tropics are characterized by
a large latitude-dependent Rossby deformation radius (Ld) varying from 80 km
at 15∘ to 250 km in the equatorial band (Chelton et al., 1998). Different studies have
clearly distinguished the tropical regions dominated by linear planetary
waves from the midlatitudes dominated by non-linear regimes (Fu, 2004;
Theiss, 2004; Chelton et al., 2007). Close to the Equator, baroclinic instability
is inhibited, while barotropic instability becomes more important (Qiu and
Chen, 2004), and mesoscale structures arise from the baroclinic and
barotropic instabilities associated with the vertical and horizontal shears
of the upper circulation (Ubelmann and Fu, 2011; Marchesiello et al., 2011).
This distinct regime in the tropics raises many questions on the
representation of the meso-/submesoscale tropical dynamics in the global
analyses of along-track altimetric wavenumber spectra. How are these complex
f-variable zonal currents folded into along-track wavenumber spectra,
calculated in 10×10∘ bins with a dominant meridional sampling in
the tropics? Also, the tropics are characterized by strong ageostrophic
flow, and the representativeness of geostrophic balance from SSH to infer
the tropical dynamics needs to be checked.
Another dynamical contribution that could flatten the SSH wavenumber spectra
in the tropics is associated with high-frequency processes. In altimetric
SSH data, the high-frequency barotropic tides are corrected using global
barotropic tidal models, and in the tropics away from coasts and islands,
these barotropic tide corrections are quite accurate (Stammer et al., 2014).
Altimetric data are also corrected for the large-scale rapid barotropic
response to high-frequency atmospheric forcing (<20 days), the
so-called dynamical atmospheric correction, using a 2-D barotropic model
forced by high-frequency winds and atmospheric pressure (Carrere and Lyard,
2003). With only 10- to 35-day repeat sampling, altimetry cannot track the
evolution of these rapid barotropic processes, and a correction is applied
to prevent aliasing of their energy into lower frequencies. In addition to
these large-scale barotropic corrections which are removed from the
altimetric data, there exist high-frequency SSH signals from internal tides
and internal waves that contribute energy at small-scale (< 300 km)
wavelengths. Their impact on SSH wavenumber spectra has been predicted from
model analyses in different regions (Richman et al., 2012; Ray and Zaron,
2016), and shows that they can dominate in regions of low eddy energy. Dufau
et al. (2016) demonstrated that internal tides can introduce spectral peaks
in the altimetric wavenumber spectra from 100 to 300 km wavelength, especially
at low latitudes (Fig. 1b). Recent results from a high-resolution
1/48∘ model highlight that the tidal and supertidal signals in one
region of the equatorial Pacific greatly exceed the subtidal dynamics at
scales less than 300 km wavelength, and supertidal phenomena are substantial
at scales approximately 100 km and smaller (Savage et al., 2017).
A more technical contribution that can impact the lower spectral slopes
in the tropics concerns the altimetric data processing, the spectral
calculation, and spectral slope estimation. Much attention has been devoted
to the effects of altimetric noise (Xu and Fu, 2012; Zhou et al., 2015; Biri
et al., 2016) which can flatten the calculated spectral slope if the noise
is not removed correctly. Different studies also use different tapering
windows to reduce leakage of non-periodic signals in limited-length data
series, which can also modify the spectral slope. In global studies, a fixed
wavelength band from 70 to 250 km is often used for the spectral slope
calculation (Xu and Fu, 2012; Dufau et al., 2016), which is appropriate for
estimating the spectral slope of the energy cascade at midlatitudes but
may not be well adapted for the tropics where the maximum spectral slope
extends to longer wavelengths, due to the larger Rossby radius there (Fig. 1b).
Thus, the interpretation of altimetric tropical SSH spectra, at spatial
scales smaller than 600 km, remains a matter of debate in terms of ocean
dynamics. This paper aims at filling this gap by studying the dynamical
processes contributing to the small-scale SSH spectra in the tropical
Pacific using modeling and observational data. Two different approaches are
proposed to better understand the contributions to the observed altimetric
flatter spectral slopes. Firstly, we wish to explore the spectral signatures
in SSH and EKE of the tropical Pacific mesoscale dynamics (with periods
greater than 10 days and wavelengths down to 25 km) and we will concentrate
particularly on the tropical “mesoscale” band that varies with latitude.
For this, we analyze the global 1/12∘ DRAKKAR model in the
tropical Pacific from 20∘ S to 20∘ N, using 5-day
outputs covering the period 1987–2001. In comparison to the altimetric
analyses of Xu and Fu (2012) or Dufau et al. (2016), this model was
specifically chosen to have no high-frequency response to tides, internal
waves or rapid tropical waves, and is not limited at low wavelengths by the
altimetric instrument noise but rather by the horizontal grid resolution.
We will also use this model to explore the effects of using limited segment
lengths or specific windowing when calculating our wavenumber spectra.
In the second part of this paper, we will address the impact on SSH and EKE
of the high-frequency components using a unique modeling experiment: we
will analyze a higher-resolution and high-frequency version of the model: a
1/36∘ regional model of the southwest Pacific (Djath et al.,
2014) with and without tides. These two regional model runs have exactly the
same configuration and high-frequency atmospheric forcing, both versions
include the atmospherically forced internal gravity waves in the tropics.
Careful filtering of the barotropic and coherent internal tides from the
model with tides also allows us to explore the relative impact of the
incoherent tide–ocean circulation interactions, and their signature on the
along-track wavenumber spectra. This two-model configuration allows us to
make a brief investigation of the effects of high-frequency dynamics on the
wavenumber spectra, and to discuss the modeled spectra in comparison with
altimetric wavenumber spectra based on TOPEX/Poseidon, Jason, and
SARAL/ALtiKa altimeter data. These results will help to better understand
the physical content of altimetric observation today, as well as to explore
the finer scales that would be captured using future measurements of the
Surface Water Ocean Topography (SWOT) satellite (Fu and Ubelmann, 2014).
In Sect. 2, the different models and data used are presented. In
Sect. 3, we discuss processing issues for the spectral calculation, particularly
to reduce leakage effects in short tropical segments. In Sect. 4, we
discuss the EKE spectral signature of the dynamics over the tropical Pacific
as simulated by the 1/12∘ resolution model. In Sect. 5, results
are discussed in terms of balanced dynamics and the 1/12∘ model's
SSH spectra are compared to Jason and SARAL/ALtiKa wavenumber spectra.
Finally, the contributions of the high-frequency motions to the SSH spectral
signature are investigated using the 1/36∘ regional resolution
model with and without tides, to illustrate its close match with altimetric
data. Section 6 presents the conclusions of our study.
Models and altimetric dataModels
To study mesoscale and submesoscale activity from an oceanic general circulation model (OGCM), the model has to
properly resolve the corresponding dynamical scales (i.e., be
eddy resolving). The effective resolution for numerical models is that
six to eight grid points are needed to properly resolve dynamical features (Soufflet et
al., 2016). In midlatitudes, numerical convergence requires ∼kilometer
horizontal resolution; however, in the tropics, because of the larger Ld
due the weaker Coriolis force, numerical convergence is obtained from
1/12∘ horizontal resolution, and the increase of resolution to
1/36∘ only seems to displace the dissipative range of the model
toward smaller scale (Marchesiello et al., 2011).
In this paper, we first use a global model at 1/12∘ resolution
from the DRAKKAR consortium based on the Nucleus for European Modelling
of the Ocean (NEMO) code (Madec, 2008; Lecointre
et al., 2011), referenced as G12d5. This model has 46 levels and has been
integrated from 1989 to 2007 using a 3-hourly ERA-Interim reanalysis (Dee et
al., 2011). The 3-D velocities and the 2-D SSH are saved
as 5-day means during the period of integration. This simulation has been
used to document mesoscale variability in the southwest Pacific Solomon Sea
(Gourdeau et al., 2014, 2017). The present study will analyze this simulation
over the tropical Pacific between 20∘ N and 20∘ S.
In the second part of the paper, we use a regional DRAKKAR/NEMO model with
1/36∘ resolution and 75 levels, still with surface forcing from
the 3 h ERA-Interim reanalysis. Two simulations are performed: one without
tidal forcing (R36) over the 1992–2012 period, and one with tidal forcing (R36T)
over the 1992–2009 period (Tchilibou et al., 2018). These different
model configurations are particularly important in this area where internal
tides are active (Niwa and Hibiya, 2011; Gourdeau, 1998), and could modify
accordingly the energy flux for the meso- and submesoscale bands (Richman et
al., 2012). Daily mean model outputs are saved as R36(T)d, as well as
instantaneous fields saved hourly (R36(T)h) during a 3-month period from
January–March 1998. We will use these different configurations to
investigate the impact of high-frequency ageostrophic motions such as
baroclinic tides and internal waves.
Further details on these different model configurations are given in Appendix A.
Altimetric data
Along-track SSH observations from TOPEX/Poseidon covering a period (January 1993
to December 2001) in common with the G12d5 simulations are analyzed
over the tropical Pacific domain. The most recent altimetric missions
(Jason-2 and SARAL/ALtiKa) are also analyzed over the January 2013 to
December 2014 period to compare with the signature of the high-frequency
modeled SSH in R36Th. These data are made available from the Copernicus
Marine and Environment Monitoring Service (CMEMS;
http://marine.copernicus.eu, last access: 22 October 2018). TOPEX/Poseidon and Jason-2 are conventional
pulse-width limited altimeters operating in the Ku band (Lambin et al.,
2010). SARAL/ALtiKa, with its 40 Hz Ka-band emitting frequency, its wider
bandwidth, lower orbit, increased pulse repetitivity frequency, and reduced
antenna beamwidth, provides a smaller footprint and lower noise than the
Ku-band altimeters (Verron et al., 2015). For the different missions, we will
analyze the 1 Hz data, extracted over the same region as our model analysis.
Spectral methods
In the following sections, we present spectral analyses of the modeled SSH
or EKE fields, or the altimetric SSH. The spectral analysis we use is based
on fast Fourier transforms (FFTs) of our signal, which allows us to work with
a limited sampled signal. Longer data records enable a better decomposition
of the variability at each frequency (wavenumber) and thus a better
separation of neighboring frequencies in the spectrum. However, for
wavenumber spectra, long spatial data records can mix information from
different geographical regimes, especially in the tropics where meridional
sections cross the strong zonal currents, making their dynamical interpretation difficult.
Different studies performing spectral analysis of altimetric data or models
over the global ocean use very different data length segments to calculate
the spectrum. Some altimetric studies use data segment lengths of around 500 km
(e.g., Dufau et al., 2016), or 1000 km length tracks averaged in
10 or 20∘ square boxes, with or without overlapping (Xu
and Fu, 2012). Model spectra are mostly calculated in 10 or
20∘ square boxes (e.g., Sasaki and Klein, 2012; Biri et al., 2016;
Chassignet and Xu, 2017). These data segment lengths may be adequate for the
midlatitudes but are not appropriate for the tropics, when the maximum
energy can occur at 600–1000 km wavelengths. Using shorter segments than
this reduces the maximum energy and should increase the leakage from
energetic low wavenumbers to weaker high wavenumbers, thus decreasing the
spectral slope (Bendat and Piersol, 2000).
A wide variety of filter windows are applied in the different studies before
calculating frequency (wavenumber) spectra to reduce the leakage effect.
These include the 10 % cosine taper window or Tukey 0.1 window, referred
hereafter as Tk01 (Le Traon et al., 2008; Richman et al., 2012; Dufau et al.,
2016); the Hanning window, referred to as Hann (Capet et al., 2008; Rocha et
al., 2016); or making the signal double periodic instead of the tapering,
referred as Dbp (Marchesiello et al., 2011; Sasaki and Klein, 2012;
Chassignet and Xu, 2017). In the following, we will also consider a 50 %
cosine taper window (Tk05).
We tested the sensitivity of our G12d5 model's SSH wavenumber spectrum to
the different tapering windows and the double periodic method, using
different data length sizes, and in one or two dimensions. The details are
given in Appendix B.
We find that to safely avoid leakage in the tropics, it is best to use a
long record and an effective taper window. The Tk05 or Hann filters give
convincing results in the equatorial band, with a minimum of 15∘
to 20∘ needed in segment lengths (Fig. B1). We do not advise to
use the Tk01 filter window. In the off-equatorial region, 10∘ data
segments or 10∘× 10∘ boxes are sufficient. We choose to
use the Tukey 0.5 filter for our tropical spectral analyses in this paper.
Spectral representation of the tropical dynamics
In this section, we analyze the spectral signatures of the tropical dynamics
by first considering the surface velocity fields of the G12d5 simulation
over the open Pacific Ocean. Modeling studies mainly analyze velocity or EKE
fields, and we start our spectral analysis by checking that the model
represents well the main dynamical processes in the tropics. Surface
velocity fields were averaged over the first 40 m depth and include
geostrophic and ageostrophic components. The model resolves a domain of
variability with periods greater than 10 days, and wavelengths exceeding 25 km,
but model dissipation may be active up to 70 km wavelength. Note that
the resonant response to the wind forcing through the 3- to 5-day period,
large-scale equatorially trapped inertia–gravity waves, are not represented
in G12d5 because of the 5-day averaged model outputs.
Snapshot of relative vorticity of the 1/12∘
G12d5 simulation; units are in 1×10-5 s-1. The yellow lines delineate the equatorial
and off-equatorial regions. The dashed lines delineate square boxes for the
different regions to compute wavenumber spectra. The black arrows illustrate
the main zonal tropical currents (SEC: South Equatorial Current, SECC: South
Equatorial Countercurrent, NECC: North Equatorial Countercurrent, NEC: North
Equatorial Current, STCC: Subtropical Countercurrent, HLCC: Hawaiian Lee Countercurrent).
The tropical Pacific is characterized by a series of strong alternate zonal
currents and a large range of ocean variability, in response to the
atmospheric forcing and to the intrinsic instability of the current system.
The main zonal currents spanning the tropical Pacific are shown in Fig. 2:
north of 10∘ N is the westward North Equatorial Current (NEC) and
at its northern edge are the eastward Subtropical Countercurrent (STCC) and
the Hawaiian Lee Countercurrent (HLCC) (Kobashi and Kawamura, 2002; Sasaki
and Nonaka, 2006); between 3 and 8∘ N is the eastward
North Equatorial Countercurrent (NECC); south of 3∘ N, the
westward South Equatorial Current (SEC) straddling the Equator is divided in
two branches by the eastward Equatorial Undercurrent (EUC) that reaches the
surface to the east. The eastward South Equatorial Countercurrent (SECC) in
the southwestern Pacific is between 6 and 11∘ S.
Instabilities of these zonal currents result in meso- and submesoscale
activity illustrated by a snapshot of vorticity (Fig. 2) that illustrates
the description of vortices in Ubelmann and Fu (2011). It is characterized
by structures with a large range of scale and strong anisotropy in the
equatorial band. The largest structures (∼500 km) correspond
to the non-linear tropical instability vortices (TIVs), also associated with
the tropical instability waves (TIWs), and occur north of the Equator
(Kennan and Flament, 2000; Lyman et al., 2007). The off-equatorial regions
(10–20∘ latitude) are characterized by smaller-scale turbulent
structures in Fig. 2.
In order to investigate how these well-known tropical dynamics project into
frequency or wavenumber spectra, we will analyze separately the equatorial
band (10∘ S–10∘ N) and the off-equatorial band
(10–20∘ N and 10–20∘ S)
defined by the different boxes in Fig. 2. The model's representation of the
following diagnostics will be discussed together for each zonal band: the
EKE frequency spectra as a function of latitude and longitude (Fig. 3), the
zonal EKE wavenumber–frequency (k-ω) spectra and meridional EKE
wavenumber–frequency (l-ω) spectra (Fig. 4), and the 1-D
(zonal/meridional) EKE wavenumber spectra (Fig. 5).
Equatorial region
The temporal variability of the tropical EKE signal is shown by EKE
frequency spectra as a function of latitude and longitude in Fig. 3. In the
equatorial band, most of the energy is concentrated within 5∘ of
the Equator (Fig. 3a). The highest EKE occurs in this band at annual to
interannual scales, but there is still significant energy over all periods
greater than the 10 days resolved by this model. EKE spectra averaged in
latitude over 20∘ N–20∘ S are highly influenced by the
energetic equatorial dynamics (Fig. 3b). This band includes the equatorial
wave guide where waves tend to propagate zonally and are organized into a
set of discrete meridional modes (Farrar, 2008). Since zonal
wavenumber–frequency spectra are averaged from a number of latitudes within
the equatorial band, contributions from the different modes may be seen at
once (Fig. 4b). The eastward phase speed (positive wavenumber), due to
fast-moving Kelvin waves at the Equator, is visible even if the strong westward
propagation (negative wavenumber) just off the Equator overpowers the
eastward propagation on the Equator in the averaged spectrum. We have
superimposed on the zonal wavenumber–frequency spectrum the theoretical
dispersion curves of the first baroclinic-Rossby waves in a resting ocean.
Values of wavenumber and frequency for which the EKE power spectrum is
significantly above the background follow relatively well the
variance-weighted mean location of dispersion curves for long equatorial
waves. Meridional wavenumber–frequency (l-ω) EKE spectra were
computed over the 20∘ N to 20∘ S section, in different
longitude bands spanning the Pacific Ocean. Figure 4d shows an example for
the particularly energetic 120–150∘ W band. Other
longitude bands across the Pacific show similar spectral energy patterns
but with lower energy levels. Figure 4b and d illustrate the strong anisotropy
between the zonal (k, ω) and meridional (l, ω) spectra. The
meridional structure of the dominant zonal equatorial waves is well known,
with meridional amplitude decaying away from the Equator over
±5∘ or 550 km. This contributes in the meridional-frequency
EKE spectrum to the fairly constant decrease in spectral energy from long
wavelengths down to 100–250 km wavelength, in both north and south directions (Fig. 4d).
(a) Latitudinal distribution of the EKE frequency power spectra
computed at each model grid point of the G12d5 simulation, and averaged in
longitude. The black line is the critical period from Lin et al. (2008).
(b) Longitudinal distribution of the EKE frequency power spectra
computed at each model grid point of the G12d5 simulation, and averaged between
20∘ S and 20∘ N. Units are in log10 of cm2 s-2 cpday-1.
The ridge of westward variance (Fig. 4b) is nearly vertical, with variance
mainly restricted to large wavelengths but also extending to high
frequencies in relation with TIW activity. In accordance with observations
(Willet et al., 2006; Lee et al., 2018), the modeled TIWs are defined by
periods and zonal wavelengths in the range of 15–40 days and 800–2000 km,
respectively. They have a meridional propagation with northward and
southward motions roughly balanced, which is a hallmark of standing meridional
modes for TIWs as seen in Lyman et al. (2005) and
Farrar (2008, 2011) and earlier work (Fig. 4d). The 33-day TIW variability is triggered by
baroclinic instability of the SEC-NECC system, located between 3–5∘ N
and 160–120∘ W (Fig. 3a and b). They
have an asymmetric structure across the Equator with larger energy north of
the Equator than south of it in accordance with the analysis of
TOPEX/Poseidon sea level data by Farrar (2008). The 20- to 25-day variability,
associated with another type of TIW triggered by barotropic instability of
the EUC-SEC system (Masina et al., 1999), is centered at the Equator, east
of 140∘ W (Fig. 3a and b). Centered at the Equator, from the
background, there is a 60- to 80-day variability extending from 150∘ E
to 130∘ W (Fig. 3a and b) associated with intraseasonal Kelvin waves
(Cravatte et al., 2003; Kessler et al., 1995), as confirmed by eastward
variance and energy centered at l=0 in the zonal and meridional-frequency
spectra, respectively (Fig. 4b and d).
The model represents these tropical signals well, and for wavelengths larger
than 600 km the equatorial waves are the dominant signal (Tulloch et al.,
2009). For wavelengths smaller than 600 km, the variance no longer follows
the Rossby wave dispersion curves, and exhibits a red noise character in
wavelength, and a nearly white noise in frequency. These rapid motions with
250–600 km wavelengths occur in response to wind forcing, wave
interactions, or current instability. The corresponding zonal EKE wavenumber spectrum
(Fig. 5) has a steep slope that continues rising to long wavelengths with a
k-3 relation reaching a peak at 1000 km, reflecting the zonal scales of
the TIWs, before flattening to a k-1 power law at larger scale. Below
70 km, EKE spectra drastically steepen as an effect of model dissipation.
Off-equatorial regions
Poleward of 10∘, the equatorial trapped waves become insignificant,
and most of the energy is concentrated at periods greater than 60 days (Fig. 3a).
This corresponds to results by Fu (2004) showing a decreasing
frequency range with latitude, where the maximum frequency at each latitude
corresponds to the critical frequency of the first-mode baroclinic waves
that varies from 60 days at 10∘ S to 110 days at 20∘ S
(Lin et al., 2008). The zonal wavenumber–frequency spectrum strongly differs
from those in the equatorial belt (Fig. 4a and c), and is closer to the
midlatitude spectra (Wunsch, 2010; Wakata, 2007; Fu, 2004) with smaller energy
in the south tropics than in the north as also reported by Fu (2004). The
theoretical dispersion curves for midlatitude first baroclinic Rossby waves
are shown for the case of meridional wavenumbers corresponding to infinite
wavelengths. At low wavenumbers (i.e., long wavelengths > 600 km),
the motions follow the baroclinic dispersion curves.
Although linear Rossby wave theory provides a first-order description of
the EKE spectra, in both hemispheres energy extends to higher frequencies
(Fig. 3a), and as the wavenumber and frequency increases, significant
deviations from the baroclinic dispersion curves occur (Fig. 4a and c). Much of
the energy lies approximately along a straight line called the
“non-dispersive line” in wavenumber–frequency space as it implies non-dispersive
motions. The wavenumber dependencies along the “non-dispersive line” could
be the signature of non-linear eddies (Rhines, 1975). The westward
propagation speed is estimated at ≈10 cm s-1, close to the eddy
propagation speed found in this latitudinal range by Fu (2009) and Chelton
et al. (2007). But these regions are defined as a weakly non-linear regime
(Klocker and Abernathey, 2014). In this region of mean zonal currents, the
dispersion curves experience Doppler shifting by the zonal flow which makes
the variability nearly non-dispersive (Farrar and Weller, 2006). So, the
non-dispersive line could account for coherent vortices and more linear
dynamics as Rossby waves or meandering jets propagating westward (Morten et al., 2017).
Zonal wavenumber–frequency EKE spectra averaged over the (a) 10–20∘ N
region, (b) 10∘ S–10∘ N region, and (c) 10–20∘ S
region. (d) Meridional wavenumber–frequency EKE spectra covering
20∘ S–20∘ N averaged over the 120–150∘ W region.
Superimposed on panels (a) and (c) are the theoretical dispersion
curves for the first-mode baroclinic waves. Superimposed on panel (b) are
the theoretical dispersion curves for the first three baroclinic wave modes, and
the Kelvin wave mode. Units are in log10 of cm2 s-2 cpday-1 cpkm-1.
The zonal EKE wavenumber spectra (Fig. 5) in the off-equatorial regions
exhibit a standard shape with a long-wavelength plateau and a spectral break
at about 300–400 km, followed by a drop in energy close to a
k-2/k-3 relation (Stammer, 1997). These steep spectral slopes
correspond to an inertial range characteristic of mesoscale turbulence (Xu
and Fu, 2011). These different spectra confirm that the northern tropics are
more energetic than the southern part with a mesoscale range extending to
larger scale. It quantifies the more active turbulence in the Northern
Hemisphere, as illustrated in Fig. 2.
Anisotropic EKE spectra
Classically, wavenumber spectra are investigated throughout an oceanic basin
by dividing the basin in square boxes where spectra are calculated to take
account of the regional diversity of QG turbulence properties (Xu and Fu,
2011; Sasaki and Klein, 2012; Biri et al., 2016; Dufau et al., 2016). Here,
the spectra analysis of the equatorial and off-equatorial bands described
above is revisited in 10∘× 10∘ boxes for the
off-equatorial region, and in 20∘× 20∘ boxes for the
equatorial region that are suited to recover the shape of the mesoscale
range in the tropics (e.g., Sect. 3). Within each equatorial or
off-equatorial latitude band, spectra in the different boxes are similar (not
shown). Therefore, spectra are averaged over all the boxes and we present one
mean spectrum representative of the square boxes for each band (equatorial
and off-equatorial). In geostrophic turbulence, which is non-divergent to
leading order, isotropy implies that 1-D (zonal/meridional) and 2-D
azimuthally integrated wavenumber spectra (or wavenumber magnitude spectra)
are identical and follow the same power law. In the tropics, there is a
stronger anisotropic component of the dynamics, which will be explored in Fig. 6.
Zonal wavenumber EKE spectra averaged over the equatorial (orange line)
and off-equatorial latitude bands (north: green; south: blue). Units are
in cm-2 s-2 cpkm-1.
When we concentrate on the 20∘× 20∘ equatorial box, we
are limited to wavelengths smaller than 2000 km, and the meridional EKE
spectrum has a higher level of energy than the zonal one (Fig. 6b). It
reflects that a given level of energy corresponds to higher zonal than
meridional wavelengths. It is consistent with the widely held notion that
scales of variability near the Equator tend to be larger in the zonal
direction than in the meridional direction for many kinds of variability
(mean currents, inertia–gravity waves, Kelvin waves, Yanai waves, TIWs). The
magnitude EKE spectrum is mostly representative of the meridional one. Note
that since along-track altimetry is mainly orientated in the meridional
direction in the tropics, altimetric SSH measurements are particularly well
suited to account for the dominant meridional variability, within the limit
of the geostrophic hypothesis. Despite the anisotropy at every scale, the
different EKE spectral components have a similar shape, with a continuous
k-3 slope between 100 and 600 km wavelength. The peak of the EKE
spectra corresponds to a wavelength of 1000 km. These modeling results
compare relatively well with the analysis of the submesoscale dynamics
associated with the TIWs by Marchesiello et al. (2011). They observe a peak
of energy around 1000 km corresponding to the TIW wavelength and a linear
decay of the spectrum with a slope shallower than -3. It is doubtful to
define an inertial band in the equatorial region, but we can say that at
wavelengths from 100 to 600 km, the EKE spectral slope of k-3 is
consistent with a QG cascade of turbulence.
In the 10∘× 10∘ off-equatorial boxes, the energy at long
wavelengths is greatly reduced compared to the equatorial band. The peak of
the EKE spectra corresponds to a wavelength of 300 km. Yet the zonal,
meridional, and magnitude EKE spectra are similar for wavelengths up to 250 km
(Fig. 6a and c). So, poleward of 10∘, the hypothesis of isotropy
seems to be relevant for scales up to 250 km even if the flow is supposed to
be weakly non-linear, and sensitive to the beta effect (Klocker and Abernathey,
2014). The EKE slope over the redefined mesoscale range from 100 to 250 km
is between -2 and -3, which lies between the prediction of SQG and QG turbulence.
Our modeled zonal frequency–wavenumber spectra differ strongly across the
equatorial and off-equatorial regions. They show a good representation of
the tropical wave and TIW/TIV dynamics. The slope of the ridge of westward
variance in the zonal k-ω spectrum in Fig. 4 increases towards the
Equator. As the slope becomes steeper, more power is concentrated at lower
wavenumbers. The change in slope of the ridge itself is mainly related to
the change in deformation radius, and expresses linear or non-linear
variability propagating non-dispersively (Wortham and Wunsch, 2014). The
equatorial region differs from the off-equatorial regions in having strong
anisotropy with mainly zonally oriented structures (Fig. 6), higher energy
at long wavelength due to the strong activity of long equatorial waves, and
an overlap between geostrophic turbulence and Rossby wave timescales that
produces long waves and slows down the energy cascade to eddies with scales
consistent in the tropics with a generalized Rhines scale (Lr) (Theiss,
2004; Tulloch et al., 2009; Klocker et al., 2016; Eden, 2007). Moreover, our
modeled spectral analysis shows the contrasts between the equatorial and
off-equatorial regions for the wavenumber range where a steep slope is
observed. In the weakly non-linear regime of the off-equatorial regions, we
find spectral slopes of k-2/k-3 over a short 100–250 km
wavenumber range. The equatorial dynamics are characterized by a peak of
energy at 1000 km due to TIWs, and a large “mesoscale” range over 100–600 km
wavelength with a k-3 spectral slope.
Modeled and altimetric SSH wavenumber spectraContribution from low-frequency dynamics
The SSH is a measure of the surface pressure field, an important dynamical
variable, which may be balanced in the tropics by both geostrophic and
ageostrophic motions. The ocean circulation is classically inferred from
altimetric SSH through the geostrophic equilibrium. Here, we consider how
the wavenumber spectra of geostrophic currents (EKEg) differ from those of
the total currents analyzed in Sect. 4. Close to the Equator, as
f approaches zero, the geostrophic current component can still be calculated
using the beta approximation, following Picaut et al. (1989). Figure 6 shows
the difference between the wavenumber spectra calculated from the total EKE
averaged over the upper 40 m, and from the geostrophic component of EKE
estimated at the surface.
Zonal (orange), meridional (green), and magnitude (blue) EKE wavenumber
spectra averaged over (a) 10–20∘ N, (b) 10∘ S–10∘ N,
and (c) 10–20∘ S regions. The magnitude geostrophic EKE
wavenumber spectrum is also shown (EKEg, blue dashed line). The
vertical green dashed lines delineate the fixed 70–250 km mesoscale range. For
reference, k-2 and k-3 curves are plotted (black lines). Units are
in cm2 s-2 cpkm-1.
In the equatorial band at scales from 300 to 1000 km, the ageostrophic EKE
is more energetic, with a stronger contribution to the total EKE than the
geostrophic component (Fig. 6b). In the off-equatorial bands (Fig. 6a and c),
the geostrophic and total EKE spectra are similar at larger wavelengths.
However, in all regions, the total EKE is steeper than the geostrophic EKE
at scales from 250 km down to the 20 km resolved by the model. In
midlatitude regions, Ponte et al. (2013) also noted stronger geostrophic EKE at
small wavelengths (and weaker spectral slopes) compared to upper ocean EKE
spectra, associated with wind-driven mixed layer dynamics. In terms of
spectral slope in the equatorial region, using the geostrophic EKE rather
than the total EKE tends to flatten the spectra in the 600–110 km mesoscale
range, and changes the spectral slope from k-3 to k-2. In the
off-equatorial regions, the geostrophic EKE has a slightly flatter spectral
slope between -2 and -3 in the 100–250 km band.
Meridional SSH wavenumber spectra averaged over the equatorial (orange)
and off-equatorial latitude bands (north: green, south:blue) for the
G12d5 simulation (line). TOPEX/Poseidon along-track altimetric SSH wavenumber spectra
are averaged over the same latitude bands (dashed). Units are in cm2 cpkm-1.
Since the altimetric ground tracks have a more meridional orientation in the
tropics, the altimetric SSH spectra should be like the model's meridional
SSH spectra that are shown in Fig. 7. SSH meridional wavenumber spectra
(Fig. 7) confirm that in the off-equatorial regions, the northern zone has
higher spectral power over all wavelengths, as expected from the EKEg
spectra. Within the wavelength band from 100 to 250 km, both off-equatorial
regions have SSH spectral slopes between k-4 and k-5 (equivalent
to k-2 and k-3 in EKE) similar to QG dynamics. The modeled SSH
spectra show a similar anisotropy in the equatorial zone as the EKE spectra,
with a more energetic meridional SSH spectrum than the zonal spectrum (not
shown). It is notable that although the level of energy is higher in the
equatorial region than in the off-equatorial regions, the SSH variability is
lower for wavelengths smaller than 500 km. This reduced SSH variability of
the G12d5 model is not in agreement with the higher small “scale” SSH
levels altimetry to be discussed in the next section (Sect. 5.2). From
100 to 600 km, the SSH spectral slopes in the equatorial region are close
to k-4, consistent with the k-2 spectral slopes in EKEg. The fixed
wavelength band used by previous studies (70–250 km) can be compared to this
longer wavelength band. Using the fixed wavelength band leads to a slight
reduction in the low-frequency SSH spectral slope estimate but without a
drastic modification. These results indicate that if the internal balanced
dynamics of our 1/12∘ model were the main contribution to the
altimetric SSH, then we would expect a k-4 (SQG) slope in the
equatorial band and closer to k-5 (QG) in the off-equatorial band.
Figure 7 also shows the along-track TOPEX/Poseidon SSH spectra over the same
region and period as the G12d5 simulation. The altimetric data are selected
with the same segment lengths and with the same pre-processing and spectral
filtering as in the model. In the equatorial and off-equatorial zones, the
altimetric SSH wavenumber spectra clearly exhibit the weaker
k-2/k-1 spectral slopes in the 70–250 km mesoscale range as
described in previous studies (Xu and Fu, 2011, 2012; Zhou et al., 2015). At
scales larger than our spectral slope range (600 km in the equatorial
region, 200 km in the off-equatorial zones), the model–altimeter spectra
have similar shapes, although the altimeter data have higher spectral power.
Potentially, the high-frequency < 10-day rapid equatorial waves, with
longer wavelengths not included in the model, may contribute to these
differences. The spectral peaks in the altimetric data at 120–150 km
wavelength are indicative of internal tides, as noted by Dufau et al. (2016),
Savage et al. (2017), and others. In addition to the internal tide
peaks, the general higher spectral energy in the altimetry data at
wavelengths < 200 km has been proposed to be due to high-frequency
internal gravity waves (e.g., Richman et al., 2012; Savage et al., 2017) but
may also include altimetric errors from surface waves and instrument noise
(Dibarboure et al., 2014). We will investigate the high-frequency
contribution to the altimetric SSH spectra in the next section.
Contributions from high-frequency dynamics including internal tides
To investigate the contribution of the high-frequency SSH variations, we
include an analysis of the meridional SSH spectra from a small region east
of the Solomon Sea in the southwest Pacific. This spectral analysis is
derived from the 1/36∘ model with high-frequency atmospheric
forcing and instantaneous snapshots saved once per hour during a 3-month
period, and run in the two configurations, with and without tides (see
Sect. 2). The model has been validated and analyzed (Djath et al., 2014),
and a companion paper will address the model with tides more in detail
(Tchilibou et al., 2018). Here, we consider specifically the impact of the
different high-frequency tides and non-tidal signals on the meridional SSH spectra.
The internal tide can be broken down into a coherent component that is
predictable and can be separated with harmonic and modal analysis, and an
incoherent component that varies over time, due to changing stratification
(Zaron, 2017) or interaction with the mesoscale ocean circulation (Ponte and
Klein, 2015). The coherent baroclinic (internal) tide and the barotropic
tide are calculated in our study using a harmonic and modal decomposition
(Nugroho, 2017) which separates the barotropic mode and nine internal tide
modes, and provides a more stable energy repartition between the baroclinic
and barotropic components (Florent Lyard, personal communication, 2017). Previous
studies have addressed the internal tide and high-frequency components in
the tropics by careful filtering of a model with tides (e.g., Richman et al.,
2012; Savage et al., 2017). Aside from the issues of artifacts introduced by
the tidal filtering, it is often tricky to cleanly separate the spectral
contributions coming from the mesoscale ocean circulation and the incoherent
component of the internal tides. The advantage of using our two-model
configuration is that we can specifically calculate the high-frequency
non-tidal components of the SSH spectra from the first model, and the
component due to the interaction of the internal tide and the model's
eddy–current turbulence with the second model.
SSH variability of the 1/36∘ regional model with explicit
tides (R36Th) over the 3-month simulation for (a) the mesoscale signal
and (b) the internal waves and internal tides defined by a 48 h
cutoff period. Units are in cm2. The SARAL/ALtiKa (black line) and Jason-2
(dashed line) tracks used to compute the altimetric spectra in Fig. 9 are superimposed.
Figure 8 shows the geographical distribution of the standard deviation of
SSH for the model including the tidal forcing for the low-frequency
(>48 h) component of the ocean (mesoscale) dynamics and for the
high-frequency component (<48 h) due mainly to internal waves and
internal tides. The large mesoscale variability (up to 6 cm) east of the
Solomon Sea in Fig. 8a is similar to the model without tides (not shown),
and well documented as current instability from the SECC-SEC current system
(Qiu and Chen, 2004). It is notable that the high-frequency variability from
the model with tides in Fig. 8b is as high as the mesoscale variability,
especially in the Solomon Sea, and comes mainly from the M2 baroclinic tide.
We note that the M2 barotropic tide amplitude within the Solomon Sea is
relatively weak (not shown), and the largest internal tide amplitudes are
close to their generation sites, particularly where the barotropic tide
interacts with the northern and southern Solomon Islands and the
southeastern Papua New Guinea (PNG) extremities (Tchilibou et al., 2018).
For the model without tides, the high-frequency variability due to the
atmospherically forced internal gravity waves is very low (∼1 cm)
compared to the model with tides, and shows a relatively uniform distribution (not shown).
The region used for our spectral analysis (2–13∘ S,
163–165∘ E; Fig. 8b) is outside the Solomon Sea with its strong
regional circulation delimited by the islands and bathymetric gradients, and
is more representative of the open Pacific Ocean conditions analyzed in the
previous sections. The latitude band from 2 to 13∘ S
lies mostly the equatorial band defined in our previous analyses, and it is
mainly representative of the SECC region (Fig. 2).
Meridional SSH wavenumber spectra averaged over 163–165∘ E
for the hourly outputs of the 1/36∘ resolution regional model without
tides (R36h, green) and 5-day averaged outputs (R36d5, orange). Meridional SSH
spectra of the G12d5 simulation are in cyan. SSH meridional wavenumber spectra
for the hourly outputs of the 1/36∘ regional model with explicit tides
once the barotropic tides has been removed (R36Th-BT, in blue) are shown. The spectrum of
the coherent baroclinic tides has been added to the spectrum of the model without
tides (R36h + BC, purple); the contribution of the only M2 coherent baroclinic
tide is in red (R36h + M2BC). The difference between the blue and purple
curves corresponds to the incoherent internal tides. The corresponding
along-track SSH altimetric spectra for SARAL/ALtiKa (line) and Jason-2 (dashed) are in
black. Units are in cm2 cpkm-1.
The meridional SSH spectrum from the 1/36∘ model run with no tides (R36h)
with hourly outputs is shown in Fig. 9 (in green). The SSH from this
version with no tides but averaged over 5 days is also shown (in orange),
i.e., with equivalent temporal sampling to our 1/12∘ model
analysis. The difference between these curves represents the non-tidal
high-frequency component of the circulation (<10 days) due to rapid
tropical waves and internal gravity waves forced by the atmospheric forcing
and current–bathymetric interactions. Also shown is the spectrum calculated
at the same location from our open ocean G12d5 1/12∘ model (in
cyan) with similar spectral slope to the 5-day averaged version of our
regional R36h 1/36∘ model, though with slightly lower energy at
scales less than 70 km wavelength as expected but also in the 180 to 600 km
wavelength band. So, the 1/36∘ model with no tides, when filtered
to remove the high-frequency forcing, is quite close to the 1/12∘
model in this equatorial band. The main point is that the additional
high-frequency dynamics in R36h increase the spectral SSH power from 300 km
down to the smallest scales from 0.4 to 0.5 cm2 and reduce
the spectral slope calculated in the fixed 70–250 km range from k-5
with the 5-day average (in orange) to k-4 for the full model with no
tides (in green).
The 1/36∘ model with tides (R36Th) is also shown in blue but with
the barotropic tide removed. The additional meridional SSH spectral power is
due both to the coherent and incoherent internal tides, with a large
increase in variance up to 300 km wavelength from 0.5 cm2 for R36h to
2.8 cm2 for R36Th. So, the main contributors to the high wavenumber SSH
spectral power are from the baroclinic tides compared to
atmospherically forced high-frequency dynamics (green curve). To illustrate
the respective part of coherent and incoherent baroclinic tides, the
coherent baroclinic tide signature based on the nine tidal constituents
summed over the first nine internal modes is calculated, and this signal is
added to the model without tides (purple curve). The coherent baroclinic
tides explain most of the tidal signature in the 300–30 km wavelength range,
and the difference with the raw signal (blue curve) exhibits the signature
of incoherent tides. The contribution of the incoherent component increases
significantly at scales smaller than 30 km and explains 30 % of the SSH
variance. The most energetic coherent internal tide component comes from the
M2 tide, and the large increase in amplitude centered around 120–140 km
wavelength corresponds to the first baroclinic mode (not shown). The other
peaks around 70 km, and 40 km could be due to higher modes, and similar
peaks are found in the tidal analysis of MITGCM model data by Savage et al. (2017)
in the central equatorial Pacific. At the main M2 internal tide
wavelengths, the incoherent internal tide has 1.6 times the SSH energy of
the coherent tide, indicating that even at the main internal tide
wavelengths, the incoherent internal tide is energetic.
We note that at wavelengths from 70 to 250 km used in the global altimetry
spectral analysis, this 1/36∘ model with the full tidal and
high-frequency forcing has a flat spectral slope of around k-1.5, quite
similar to the analysis of along-track spectral from Jason-2 (in dashed
black) and SARAL (in solid black), in the same region but over the longer
2013–2014 period. We note that the barotropic tide has also been removed
from the altimetric data, using the same global tide atlas applied at the
open boundary conditions for our regional model (FES2014, Lyard et al.,
2018). If we use the “mesoscale” range defined for the global model
analysis in the equatorial band over 100–600 km wavelength, we still have a
weak spectral slope of k-2 for both the model with tides and altimetry.
Jason-2 has a higher noise level than SARAL at scales less than 30 km
wavelength (Dufau et al., 2016); the small differences in spectral energy
between Jason-2 and SARAL over wavelengths from 150 to 450 km may be
influenced by the different repetitive cycles of the very few tracks
available (one track for Jason-2 and three tracks for SARAL/ALtiKa) between both
missions and their slightly different track positions.
This regional analysis provides a number of key results. The high-frequency,
high-resolution regional model confirms our open ocean 1/12∘
analysis. The dynamics at scales > 10 days, with no tidal
forcing, give rise to SSH spectral slopes from 70 to 250 km of around k-5
in this equatorial band in accordance with the G12d5 simulation. Note that
it differs from the k-4 slope typical of the equatorial region
discussed above. It reflects modulation associated with low-frequency
variability. This 3-month period corresponds to an El Niño event
characterized by relatively low mesoscale activity in this region of the
southwest Pacific (Gourdeau et al., 2014). Including the high-frequency but
non-tidal forcing increases the smaller-scale energy, and flattens the SSH
spectra with slopes of around k-4. This non-tidal high-frequency
(<10-day) component increases the SSH spectral energy out to scales
of 200 km wavelength, suggesting a dominance of rapid small-scale
variability of internal gravity waves (Garrett and Munk, 1975). But the
higher-frequency atmospheric forcing and ocean instabilities alone cannot
explain the very flat altimetric spectral slopes in this equatorial region.
When coherent and incoherent internal tides are included, the spectral slope
in the 70–250 km wavelength band becomes very close to that observed with
altimetric spectra. This confirms the recent results presented by Savage et
al. (2017) for a small box in the eastern tropics, and previously proposed
by Richman et al. (2012) and Dufau et al. (2016). The separation of the
coherent M2 internal tide demonstrates that it clearly contributes SSH
energy in the 50–300 km wavelength band, but the incoherent tide, and its
cascade of energy into the supertidal frequencies, is the dominant signal at
scales less than 50 km. The incoherent and coherent internal tides have
similar energy partitioning within the 50–300 km wavelength band.
Discussion and conclusion
The processes that could contribute to the flat SSH
wavenumber spectral slopes observed in the tropics by satellite altimetry
have been examined in the tropical Pacific. This study has used two
complementary approaches to better understand how the equatorial and
off-equatorial dynamics impact the SSH wavenumber spectra. In the first
part of this study, we have concentrated on the low-frequency (>10-day)
tropical dynamics to better understand how the complex zonal
current system and dominant linear tropical waves affect the mainly
meridional altimetric SSH wavenumber spectra. In the second part of the
study, we have used a high-frequency, high-resolution regional modeling
configuration, with and without tides, to explore the high-frequency
contributions to the meridional SSH wavenumber spectra.
Our 1/12∘, 5-day averaged model confirms the results from
previous modeling studies that at seasonal to interannual timescales the
most energetic large-scale structures tend to be anisotropic and governed by
linear dynamics. At intraseasonal frequencies and in the tropical
“mesoscale” band at scales less than 600 km wavelength, one major question
was how the cascade of energy is affected by the expected high level of
anisotropy and the weak non-linear regimes. Within the “mesoscale” range,
the EKE wavenumber spectra are isotropic in the off-equatorial regions
between 10 and 20∘, and it is more anisotropic in the
equatorial region between 10∘ N and 10∘ S, with a higher level
of energy for the meridional EKE spectrum than for the zonal one that
reveals larger scales of variability in the zonal direction than in the
meridional direction, as expected. In the off-equatorial range, EKE peaks at
around 300 km wavelength, and the steep EKE decrease at smaller wavelength
is characterized by spectral slopes between k-2 and k-3, which lie
between the regimes of SQG and QG turbulence. These weakly non-linear
off-equatorial regions thus have a similar structure to the non-linear
midlatitudes within the range from 100 to 250 km. In the equatorial band from
10∘ S to 10∘ N, the total EKE is more energetic than the
off-equatorial region, and the EKE spectral slope approaches k-3 over a
large wavenumber range, from 100 to 600 km, consistent with QG dynamics,
even though there is a strong ageostrophic component here. Using the fixed
wavelength (70–250 km) band to estimate “mesoscale” spectral slope leads
to a slight reduction in the low-frequency spectral slope estimate but
without a drastic modification. When geostrophic velocities (rather than the
total surface flow) are used to calculate EKE, there is similar spectral
energy in the off-equatorial regions at longer wavelengths. In the
equatorial band 10∘ N–10∘ S, the ageostrophy is more
evident with a more marked change in spectral slope based on geostrophic
velocities and the beta approximation at the Equator. At large scales in the
equatorial band, the ageostrophic equatorial currents are more active,
related to the energetic zonal currents. In all regions, at wavelengths
shorter than 200 km, the geostrophic spectra become more energetic and the
small-scale ageostrophic components counteract the balanced
geostrophic flow, as found at midlatitudes (Klein et al., 2008; Ponte and
Klein, 2015). This gives a slightly flatter spectral slope over the 70–250 km
wavelength, but the regime remains between k-2 and k-3 in the
off-equatorial region, approaching k-2 (and k-4 in SSH) in the
equatorial band. So, using SSH and geostrophic currents slightly flattens the
EKE wavenumber spectra, but the modeled SSH wavenumber spectra maintain a
steep slope that does not match the observed altimetric SSH spectra.
The choice of regional box size and filtering options also impacts on the
spectra. Previous global altimetric studies have calculated along-track SSH
wavenumber spectra in 10∘× 10∘ boxes, and with varying
segment lengths (512 km for Dufau et al., 2016; around 1000 km for Xu and
Fu, 2011, Chassignet et al., 2017, etc.), and with different tapering or
filtering applied (see Sect. 3). In the equatorial band where the EKE peak
extends out to 600 km wavelength, it is important to have segment sizes and
filtering that preserve this peak and shorter scales. The combined effects
of a 10 % cosine taper and the short segment lengths lead to a much
flatter altimetric SSH spectra, reaching k-1 in the Dufau et al. (2016)
study. We find that the double periodic spectra, the Hanning and
Tukey 50 % taper filter, all give similar results in the tropics, but it is
necessary to extend the box size to a minimum of 15 to
20∘ in segment length or box size in the equatorial band. In the
off-equatorial band, these filtering options with a 10∘ segment
length or box size are sufficient. Even with the preferred pre-processing
for the altimetric data, and larger segment lengths in our analyses, the
altimetric SSH spectra remain quite flat (k-2 in the off-equatorial
zone, k-1.3 in the equatorial band), and do not reflect the steeper
spectral slopes predicted by the model.
The regional high-resolution models with both high-frequency atmospheric and
tidal forcing and high-frequency hourly outputs provide the last pieces of
the puzzle. In contrast to previous results based on global ocean models
with tidal forcing (Richman et al., 2012; Savage et al., 2017), this two-model
configuration with and without tides has the same atmospheric and boundary
forcing, which allows us to clearly separate the internal tide signals from
the high-frequency dynamical component. Even though only a small region of
the tropical Pacific is available for this analysis, the regional model and
the global 1/12∘ model show similar QG spectral slopes when they
are compared over the same domain and with 5-day averaged data. Using hourly
data and no tides increases the SSH spectral power at scales smaller than
200 km, possibly due to internal gravity waves in the tropics (Farrar and
Durland, 2012; Garrett and Munk, 1975). We note that Rocha et al. (2016)
found a similar increase in their detided along-track model runs in Drake
Passage but at scales less than 40 km wavelength, far below the noise level
of our present altimeter constellation. In the tropics, this contribution of
high-frequency non-tidal SSH signals out to 200 km wavelength will also
impact today's along-track altimeter constellation, whose noise levels
block ocean signals at scales less than 70 km for Jason class satellites,
and 30–50 km for SARAL and Sentinel-3 SAR altimeters (Dufau et al., 2016).
So, non-tidal internal gravity waves will partially contribute to the higher
small-scale SSH variance and flatter spectral slopes in today's altimetric SSH data.
The regional model with tides shows the very important contribution of
internal tides to the flat SSH slopes in the tropics. We have separated out
the predictive part of the barotropic tide and internal tides, since open
ocean barotropic tides are well corrected for in altimetric data today
(Lyard et al., 2018; Stammer et al., 2014), and corrections are becoming
available for the coherent part of the internal tide (Ray and Zaron, 2016).
In this open ocean tropical region east of the Solomon Sea, when coherent
and incoherent internal tides are included, the spectral slope in the 70–250 km
wavelength band becomes very close to that observed with altimetric
spectra. This confirms the recent results presented by Savage et al. (2017)
for a small box in the eastern tropics, and previously proposed by Richman
et al. (2012) and Dufau et al. (2016). The separation of the coherent
M2 internal tide demonstrates that it clearly contributes significant SSH
energy in the 50–300 km wavelength band, but around the main internal tide
wavelengths, there is a strong signature of M2 incoherent internal tide. The
incoherent tide, and its cascade of energy into the supertidal frequencies,
is the dominant signal at scales less than 50 km. This strong incoherent
internal tide is consistent with recent studies that suggest that internal
tides interacting with energetic zonal jets can generate a major incoherent
internal tide (Ponte and Klein, 2015), and may explain the reduction of the
coherent internal tides in the equatorial band in global models (Shriver et
al., 2014) and altimetric analyses (Ray and Zaron, 2016). Our model
highlights that the internal tide signal is strong in this equatorial
region, and the incoherent tide accounts for 35 % of the SSH spectral
power in the 50–300 km wavelength band, and is not predictable.
These results have important consequences for the analyses of along-track
altimetric data today, and for the future high-resolution swath missions
such as SWOT. Today's constellation of satellite altimeters have their
along-track data filtered to remove noise at scales less than 70 km for all
missions (Dibarboure et al., 2014; Dufau et al., 2016), and these data are
now being used with no internal tide correction in the global gridded
altimetry maps of SSH and geostrophic currents. The imprint of these
internal tides is evident in the along-track data (see Fig. 1b from Dufau et
al., 2016) but is also present in the gridded maps (Richard D. Ray, personal
comunication, 2017). In the future, a coherent internal tide correction may be
applied to the along-track data based on Ray and Zaron (2016), to reduce some
of this non-balanced signal. It is particularly important to remove the
unbalanced internal wave signals from SSH before calculating geostrophic
currents. But it is clear that the incoherent internal tide and internal
gravity waves reach scales of 200 km in the tropics, and their signature in
SSH remains a big issue for detecting balanced internal ocean currents from
along-track altimetry and the future SWOT wide-swath altimeter mission.
Removing this signal to detect purely balanced motions will be challenging,
since filtering over 200 km removes much of the small-scale ocean dynamics
of interest in the tropics. On the other hand, there will also be a great
opportunity to investigate the interaction of the internal tide and ocean
dynamics in the tropics in the future, with both models and fine-scale
altimetric observations.
Data is available upon request by contacting the correspondence author.
Model configurations used in this studyGlobal model at 1/12∘
The model used is the ORCA12.L46-MAL95 configuration of the global
1/12∘ OGCM developed and operated in the DRAKKAR consortium
(https://www.drakkar-ocean.eu/, last access: 22 October 2018) (Lecointre et al., 2011). The numerical code is
based on the oceanic component of the NEMO system (Madec, 2008). The model formulation is based on
standard primitive equations. The equations are discretized on the classical
isotropic Arakawa C grid using a Mercator projection. Geopotential vertical
coordinates are used with 46 levels with a 6 m resolution in the upper layers
and up to 250 m in the deepest regions (5750 m). The “partial step”
approach is used (Adcroft et al., 1997) to allow the bottom cells thickness
to be modified to fit the local bathymetry. This approach clearly improves
the representation of topography effects (Barnier et al., 2006; Penduff et
al., 2007). The bathymetry was built from the GEBCO1 dataset
(https://www.gebco.net/, last access: 22 October 2018)
for regions shallower than 200 m and from ETOPO2
(https://www.ngdc.noaa.gov/mgg/global/relief/ETOPO2/, last access: 22 October 2018) for regions deeper than 400 m
(with a combination of both datasets in the 200–400 m depth range). Lateral
boundary conditions for coastal tangential velocity have a strong impact on
the stability of boundary currents (Verron and Blayo, 1996). Based on
sensitivity experiments, a “partial-slip” condition is chosen, where the
coastal vorticity is not set to 0 (“free-slip” condition) but is weaker
than in the “no-slip” condition. The atmospheric forcing (both mechanical
and thermodynamical) is applied to the model using the CORE bulk-formulae
approach (Large and Yeager, 2004, 2009). The simulation started from rest in 1978
with initial conditions for temperature and salinity provided by the
1998 World Ocean Atlas (Levitus et al., 1998). It was spun up for 11 years using
the CORE-II forcing dataset and then integrated from 1989 to 2007 using a
3-hourly ERA-Interim forcing (Dee et al., 2011).
Regional model at 1/36∘ with and without tides
As part of the CLIVAR/SPICE program, regional simulations of the Solomon Sea
in the southwestern tropical Pacific have been performed (Ganachaud et al.,
2014). The numerical model of the Solomon Sea used in this study has a
1/36∘ horizontal resolution and 75 vertical levels. It is based
on the same oceanic component as the NEMO system presented above. This
1/36∘ resolution model is embedded into the global 1/12∘
ocean model presented above and one-way controlled using an open boundary
strategy (Treguier et al., 2001). Its horizontal domain is shown in
Fig. 8. The bathymetry of the high-resolution Solomon Sea model is based on the
GEBCO08 dataset. Atmospheric boundary conditions, consisting in surface
fluxes of momentum, heat, and freshwater, are diagnosed through classical bulk
formulae (Large and Yeager, 2009). Wind and atmospheric temperature and
humidity are provided from the 3-hourly ERA-Interim reanalysis (Dee et al.,
2011). A first version of the regional model with 45 vertical levels has
been initialized with the climatological mass field of the World Ocean Atlas
(Levitus et al., 1998) and was integrated from 1989 to 2007. More technical
details on this configuration may be found in Djath et al. (2014). The new
version used here is distinct from the former version by the number of
vertical levels (75 levels in the new version) but above all by its ability
to take account realistic tidal forcing (Tchilibou et al., 2018). The model
is forced at the open boundary by prescribing the first nine main tidal
harmonics (M2, S2, N2, K2, K1, O1, P1, Q1, M4) as defined from the global
tides atlas FES2014 (Lyard et al., 2018) through a forced gravity wave
radiation condition. The model is initialized by the outputs from the ORCA
1/12∘ version.
Spectral sensitivity tests
We tested the sensitivity of our G12d5 model's SSH wavenumber spectrum to
different tapering windows and the double periodic method, using different
data length sizes, and in one or two dimensions. The following steps were
performed for these test spectra, evaluated within 10∘ S–10∘ N/160–120∘ W: the model data are
extracted meridionally and zonally in fixed segment lengths of 5, 10, and 20∘
and within a 20∘× 20∘
square box; the mean and linear trend (fitted plane for two-dimensional
case) were removed from each data segment or box; the filter window (Tk01,
Tk05, Hann) or Dbp are applied; temporal and spatial (longitude, latitude)
series spectra are calculated and averaged in Fourier space. The results are
shown in Fig. B1.
Tk01 meridional spectra in the tropics are the most perturbed by the short
segment lengths (Fig. B1a). In the 70–250 km range commonly used to define a
global mesoscale band (delimited by the green vertical lines), the spectral
slope flattens as the data segment length decreases. The 5∘ segment
spectra with a Tk01 window have a k-1.3 slope, which explains the very
shallow slope in the tropics observed by Dufau et al. (2016) who applied this
short data segment size and a Tk01 window. Meridional spectra differ
primarily at larger scales from 100 to 500 km, when short segment lengths are
used (Fig. B1a). A comparison of the meridional spectrum using 20∘
segments and different windows (Tk01, Tk05, Hann, and Dbp) is shown in
Fig. B1b. Even with the 20∘ segments, Tk01 is distorted. On the
other hand, the Tk05, Hann, and Dbp match well, with a near-linear cascade of
energy over the 30–1000 km wavelength range, and are more adapted for the
tropics since they capture the main range of SSH mesoscale dynamics,
particularly the spectral energy peaks around 1000 km wavelength.
Similar calculations were performed for the zonal spectra (not shown) and
confirm that the Tk01 method deforms the zonal spectra and flattens the
spectral slope within the 70–250 km wavelength band as the data segment size
decreases. Tk05, Hann, and Dbp 20∘ segment spectra match, although
the Dbp has more noise at small scale.
We also conducted a sensitivity test in the off-equatorial region (not
shown): flattening and deformation of the spectrum by Tk01 persist, but the
10∘ segments or 10∘ square box are long enough to
capture the off-equatorial dynamics.
Sensitivity experiments for different spectral processing techniques
applied to meridional SSH wavenumber spectra representative of the equatorial
region. (a) SSH wavenumber spectra using a Tukey 0.1 window (blue) and
a Tukey 0.5 window (red) depending on segment lengths: 5∘ (dots),
10∘ (dashed), and 20∘ (line). (b) SSH wavenumber spectra
using different windowing over a 20∘ segment length: Tukey 0.1 window
(Tk01, blue), Tukey 0.5 window (Tk05, red), and Hanning window (Han, green). The
double periodic method (Dbp, black) is also tested. For reference, k-1 and
k-5 curves are plotted. Units are in cm2 cpkm-1.
The particular sensitivity of spectra in the tropics to the choice of
spectral segment length and windowing is linked to energetic EKE and SSH
signals extending out to longer wavelengths, and their spectral leakage from
low to high wavenumbers. Tk01 gives the worst performance in the tropics,
and the distortion of spectra is amplified for short data segments. Both the
Tk05 and the Hann windowing are a good compromise for preserving much of the
original signal and reducing leakage, but they need to be applied over larger segments.
MT is a PhD student under the supervision of LG and RM.
GS participated in the calculation and interpretation of the spectra. BD ran
the 1/36∘ model. FL participated in computing and analysing the baroclinic tides.
The authors declare that they have no conflict of interest.
Acknowledgements
The authors wish to acknowledge Ssalto/Duacs AVISO who produced the altimeter
products, with support from CNES (http://www.aviso.altimetry.fr/duacs/,
last access: 22 October 2018). The authors would like to thank the
DRAKKAR team for providing them with the high-resolution global ocean
simulation, and especially Jean Marc Molines for his support. This work benefited
from discussions with Julien Jouanno, Frederic Marin, and Yves Morel from LEGOS. We
particularly thank Tom Farrar (WHOI) and an anonymous reviewer for their
constructive comments, and Jacques Verron (IGE), Claire Menesguen (LOPS), and Xavier Capet
(LOCEAN) for their time, and their fruitful comments. Michel Tchilibou is funded
by Université de Toulouse 3. Lionel Gourdeau and Guillaume Sérazin are funded by
IRD; Rosemary Morrow is funded by CNAP; and Bughsin Djath was funded by CNES. This work
is a contribution to the joint CNES/NASA SWOT project “SWOT in the tropics”
and is supported by the French TOSCA programme.
Edited by: John M. Huthnance
Reviewed by: Tom Farrar and one anonymous referee
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