SuppReferences

# Global Energy Spectrum of the General Oceanic Circulation

Benjamin A. Storer  Department of Mechanical Engineering and Laboratory for
Laser Energetics, University of Rochester, Rochester, New York, USA  Michele
Buzzicotti  Department of Physics, University of Rome Tor Vergata and INFN,
Rome, Italy  Hemant Khatri  Department of Earth, Ocean, and Ecological
Sciences, University of Liverpool, UK  Stephen M. Griffies  NOAA Geophysical
Fluid Dynamics Laboratory and Princeton University Atmospheric and Oceanic
Sciences Program, Princeton, New Jersey, USA  Hussein Aluie
<EMAIL_ADDRESS>Department of Mechanical Engineering and Laboratory for
Laser Energetics, University of Rochester, Rochester, New York, USA

(August 28, 2024)

Since the advent of satellite altimetry, our perception of the oceanic
circulation has brought into focus the pervasiveness of mesoscale eddies that
have typical scales of tens to hundreds of kilometers [5], are the ocean’s
analogue of weather systems, and are often thought of as the peak of the
ocean’s kinetic energy (KE) wavenumber spectrum [7, 19, 23]. Yet, our
understanding of the ocean’s spatial scales has been derived mostly from
Fourier analysis in small “representative” regions (e.g. [16, 14, 4]),
typically a few hundred kilometers in size, that cannot capture the vast
dynamic range at planetary scales. Here, we present the first truly global
wavenumber spectrum of the oceanic circulation from satellite data and high-
resolution re-analysis data, using a coarse-graining method to analyze scales
much larger than what had been possible before. Spectra spanning over three
orders of magnitude in length-scale reveal the Antarctic Circumpolar Current
(ACC) as the spectral peak of the global extra-tropical ocean, at $\approx
10\times 10^{3}\leavevmode\nobreak\ $km. We also find a previously unobserved
power-law scaling over scales larger than $10^{3}\leavevmode\nobreak\ $km. A
smaller spectral peak exists at $\approx 300\leavevmode\nobreak\ $km
associated with the mesoscales, which, due to their wider spread in wavenumber
space, account for more than $50\%$ of the resolved surface KE globally.
Length-scales that are twice as large (up to $10^{3}$ km) exhibit a
characteristic lag time of $\approx 40\leavevmode\nobreak\ $days in their
seasonal cycle, such that in both hemispheres KE at $100\leavevmode\nobreak\
$km peaks in late spring while KE at $10^{3}\leavevmode\nobreak\ $km peaks in
late summer. The spectrum presented here affords us a new window for
understanding the multiscale general oceanic circulation within Earth’s
climate system, including the largest planetary scales.

The oceanic circulation is a key component in Earth’s climate system. It is
both the manifestation and cause of a suite of linear and nonlinear dynamical
processes acting over a broad range of scales in both space and time [7]. The
wavenumber spectrum of the oceanic circulation allows us to understand the
energy distribution across spatial scales throughout the globe, reveals key
bands of scales within the circulation system at which energy is concentrated,
and unravels power-law scalings that can be compared to theoretical
predictions [26]. The spectrum is an important guide to probing (i) energy
sources and sinks maintaining the oceanic circulation at various scales, (ii)
how energy is ultimately dissipated, and (iii) how the ocean at a global
climate scale is coupled to motions several orders of magnitude smaller.

Thanks to satellite observations [10] and high-resolution models and analysis
[12, 16], it is now well-appreciated that the mesoscales, traditionally
thought of as transient eddies of ${\mathcal{O}}(100)\leavevmode\nobreak\ $km
in size, form a key band of spatial scales that pervade the entire ocean and
have a leading order effect on the transport of heat, salt, and nutrients, as
well as coupling to the global meridional overturning circulation [11]. The
mesoscales are generally viewed as forming the peak of the KE spectrum of the
oceanic circulation [7, 10] (e.g. Fig. 5 in [19] or Fig. 5 in [23]). However,
the existence of the mesoscale spectral peak and the length-scale at which it
occurs is not known with certainty [7]. Evidence is often derived from
performing Fourier analysis on the ocean surface velocity [16] or sea-surface
height [14] within regions that are typically 5∘ to 10∘ in extent (nominally
$500\leavevmode\nobreak\ $km to $10^{3}\leavevmode\nobreak\ $km) [22]. The
peak appears in only a fraction of the chosen regions, and spectral energy
tends to be largest at the largest length-scales (smallest wavenumbers), which
are most susceptible to artifacts from the finite size of the chosen regions
and the windowing required for Fourier analysis [7]. To date, there has been
no determination of the oceanic energy wavenumber spectrum at planetary
scales. Do the mesoscales of ${\mathcal{O}}(100)\leavevmode\nobreak\ $km
actually form the peak of the ocean’s KE spectrum? What is the KE content of
scales larger than ${\mathcal{O}}(10^{3})\leavevmode\nobreak\ $km, which
constitute the ocean’s gyres and are directly coupled to the climate system?

Below, we present the first KE spectrum over the entire range of scales
resolved in data from satellites and high-resolution models at the ocean’s
surface, including the spectrum at planetary scales. We find that the spectral
peak of the global extratropical ocean is at ${\approx 10\times
10^{3}\leavevmode\nobreak\ }$km and is due to the ACC. We see vestiges of a
similar peak in the northern hemisphere, which is arrested at a smaller
amplitude and at smaller scales ($\approx 4\times 10^{3}\leavevmode\nobreak\
$km) due to continental boundaries. Another prominent spectral peak is at
$\approx 300\leavevmode\nobreak\ $km, and with an amplitude less than half
that of the ACC. Yet, the cumulative energy in the mesoscales between
$100\leavevmode\nobreak\ $km and $500\leavevmode\nobreak\ $km is very large
($>50\%$ of total resolved energy). We also report the first observation of a
roughly $k^{-1}$ power-law scaling over scales larger than
$10^{3}\leavevmode\nobreak\ $km in both hemispheres, consistent with a
theoretical prediction from a quasigeostrophic model forced by wind [13, 29],
with the power-law scaling extending up to the ACC peak in the southern
hemisphere.

Our results here open exciting avenues of inquiry into oceanic dynamics,
allowing us to seamlessly probe interactions between motions at scales
${\mathcal{O}}(100)\leavevmode\nobreak\ $km and smaller with planetary scales
larger than $\mathcal{O}(10^{3})\leavevmode\nobreak\ $km relevant to climate.
We are able to do so using a coarse-graining approach developed recently to
probe multi-scale geophysical processes [3, 2, 17].

### Partitioning energy across length-scales

Our methodology, described in the Methods section and in [3, 2], allows us to
coarse-grain the ocean flow at any length-scale of choice and calculate the KE
of the resulting coarse flow. By performing a ‘scan’ over an entire range of
length-scales, we extract the so-called ‘filtering spectrum’ without needing
to perform Fourier transforms [18]. The filtering spectrum and the traditional
Fourier spectrum agree when the latter is possible to calculate, as
demonstrated in [18] and in Fig. E1 of the Methods section. Unlike traditional
Fourier analysis within a box / subdomain, coarse-graining can be meaningfully
applied on the entire spherical planet, including land/sea boundaries, and so
allows us to probe everything from the smallest resolved scales up to true
planetary scales.

#####  Filtering Spectrum

Given a velocity field $\mathbf{u}$ and a filter scale $\ell$, coarse-graining
produces a filtered velocity $\overline{{\bf u}}_{\ell}$ that only contains
spatial scales larger than $\ell$, having had smaller scales removed (see Fig.
1 and the Methods section). Unlike standard approaches to low-pass filtering
geophysical flows, such as by averaging adjacent grid-cells or block-averaging
in latitude-longitude, the coarse-graining of [2] used here relies on a
generalized convolution operation that respects the underlying spherical
topology of the planet, thus preserving the fundamental physical properties of
the flow, such as its incompressibility, its geostrophic character, and the
vorticity present at various scales. The KE (per unit mass, in m2/s2)
contained in scales larger than $\ell$ is

${\mathcal{E}}_{\ell}=\frac{1}{2}\left|\overline{{\bf u}}_{\ell}({\bf
x},t)\right|^{2}\hskip 14.22636pt\mbox{\text{(coarse KE).}}$ (1)

While ${\mathcal{E}}_{\ell}$ quantifies the _cumulative_ energy at all scales
larger than $\ell$, the wavenumber spectrum quantifies the spectral energy
_density_ at a _specific_ scale, similar to the common Fourier spectrum.
Following [18], we extract the KE content at different length-scales by
differentiating in scale the coarse KE:

${\overline{E}}(k_{\ell},t)=\frac{d}{dk_{\ell}}\left\\{{\mathcal{E}}_{\ell}\right\\}=-\ell^{2}\frac{d}{d\ell}\left\\{{\mathcal{E}}_{\ell}\right\\},$
(2)

where $k_{\ell}=1/\ell$ is the ‘filtering wavenumber’ and $\\{\cdot\\}$
denotes a spatial average. Ref [18] identified the conditions on the coarse-
graining kernel for ${\overline{E}}(k_{\ell},t)$ to be meaningful in the sense
that its scaling agrees with that of the traditional Fourier spectrum when
Fourier analysis is possible, such as in periodic domains. Fig. E1 in Methods
shows how the filtering spectrum agrees with the Fourier spectrum performed
within an oceanic box region over length-scales smaller than the box but has
the important advantage of quantifying larger scales without being
artificially limited by the box size and windowing functions to synthetically
periodize the data.

##### Oceanic Gyres and Mesoscales

Figure 1 visualizes the flow from both AVISO satellite data and NEMO
reanalysis model data (see Methods) from a single daily mean at scales larger
than and smaller than $10^{3}\leavevmode\nobreak\ $km, termed “gyre-scale” and
“mesoscale,” respectively. The color intensity illustrates the flow speed and
is consistent with expectations that the large-scale flow is primarily
composed of signals from the western boundary currents, while the small-scales
are dominated by mesoscales fluctuations. In the upper panels of Fig. 1 we can
see clearly several well-known oceanic gyre structures, including the Beaufort
Gyre in the Arctic, the Weddell and the Ross gyres in the Southern Ocean near
Antactica, the subtropical and subpolar gyres in the Atlantic and Pacific
basins, and the ACC. North Atlantic currents are also readily observable,
including the North Atlantic Current, its northward fork to the Norwegian
Atlantic Current, and the southward East Greenland Current. The agreement
between AVISO and NEMO is remarkable.

Figure 1:  Gyre-scale and mesoscale flows  [Colour maps] show the geostrophic
velocity magnitude for length scales [top] larger than
$10^{3}\leavevmode\nobreak\ $km and [bottom] smaller than
$10^{3}\leavevmode\nobreak\ $km for a single day (02 Jan 2015). [Left] shows
the $\nicefrac{{1}}{{4}}^{\circ}$ AVISO dataset and [right] the
$\nicefrac{{1}}{{12}}^{\circ}$ NEMO dataset. [White lines] highlight the
corresponding streamlines, with arrows showing the direction of the flow.
Areas in black include land, and also ice coverage in the AVISO dataset. In
this work we exclude the tropics where velocity from satellite altimetry is
less reliable, and define the northern and southern hemispheres (NH and SH,
respectively) as the ocean poleward of $15^{\circ}$.

It is worth emphasizing that the flows in Fig. 1 are derived deterministically
from a single daily mean of surface geostrophic velocity data without further
temporal or statistical averaging. Past approaches have used climatological
multi-year averaging (e.g. [1, 21]) or Empirical Orthogonal Function (EOF)
analysis (e.g. [24, 6]), which is a statistical approach that requires
averaging long time-series. Coarse-graining allows us to derive the dynamics
governing the evolution of the flow in Fig. 1 (e.g. [3]), which is not
possible for EOF analysis, and to disentangle length-scales and time-scales
independently and in a self-consistent manner to study interactions between
different spatio-temporal scales that link large-scale forcing, the mesoscale
eddy field, and the global-scale circulation. Such objective disentanglement
of the oceanic circulation by coarse-graining opens the door for analysing the
coupling of the ocean’s mesoscales, on spatio-temporal scales of
$\mathcal{O}$(100 km) and $\mathcal{O}$(30 days), to the global circulation (
$\mathcal{O}$($10^{3}\leavevmode\nobreak\ $km) and $\mathcal{O}$(10 years) )
and the climate system ( $\mathcal{O}$($10^{4}\leavevmode\nobreak\ $km) and
$\mathcal{O}$(100 years) ).

### Global Kinetic Energy Spectrum

Figure 2:  Power Spectral Density Filtering wavenumber spectra (see eq. (2))
of surface geostrophic KE for the global extratropical ocean from AVISO
satellite altimetry and NEMO model re-analysis. Northern and southern
hemispheres (NH and SH, respectively) extend poleward of $15^{\circ}$. Both
panels show the same data, but using [top] lin-log and [bottom] log-log
scales. Plots show the temporal mean, $\langle\cdot\rangle$, of
${\overline{E}}(k_{\ell},t)$ while envelopes show inter-quartile range (25th
to 75th percentiles) of temporal variation. Data markers indicate length
scales at which coarse-graining was performed. The vertical dashed green line
at $40.0\times 10^{3}\leavevmode\nobreak\ $km indicates the equatorial
circumference of the Earth. Dashed black lines provide a reference for
$-\nicefrac{{5}}{{3}}$, $-3$, and $-1$ power-law slopes in the bottom panel.

Figure 2 shows the filtering spectrum for both the northern and southern
hemispheres as obtained from eq. (2) using surface geostrophic velocity data
from both satellite altimetry and a high-resolution model (see Methods). This
is the first spectrum showing the oceanic energy distribution across such a
wide range of scales, from planetary scales ${\mathcal{O}}(10\times
10^{3})\leavevmode\nobreak\ $km down to ${\mathcal{O}}(10)\leavevmode\nobreak\
$km.

The top and bottom panels in Fig. 2 plot the same spectrum in lin-log and log-
log scale, respectively. The top panel highlights the prominent spectral peak
due to the ACC, which is more than twice the mesoscale peak. The bottom panel
highlights the power-law scaling over different $k_{\ell}$ bands.

Note the zero energy content at scales larger than Earth’s circumference and
that energy also decreases precipitously when approaching the smallest scales
resolved by each of the datasets, both of which are physical expectations. It
is not possible for simulation, satellite, or field data to capture all scales
present in the natural ocean, which certainly has scales smaller than
$100\leavevmode\nobreak\ $km. There is excellent agreement between satellite
data and the higher resolution model data used here down to scales $\approx
100\leavevmode\nobreak\ $km, which indicates that all scales larger than
$100\leavevmode\nobreak\ $km are well-resolved by both datasets, whereas
smaller scales (10 – 100 km) are reasonably resolved only in the model data.

##### Antarctic Circumpolar Current and Oceanic Gyres

Unlike previously reported KE wavenumber spectra using Fourier analysis on box
regions (e.g. [19, 23, 16]), some of which show a peak at mesoscales
$O(100)\leavevmode\nobreak\ $km, our Fig. 2 reveals that the largest spectral
peak occurs at scales approximately $100$ times larger, at $\approx 10\times
10^{3}\leavevmode\nobreak\ $km, and only in the southern hemisphere. Indeed,
the ACC at latitude 50∘S has a geodesic diameter of $\approx 8.9\times
10^{3}\leavevmode\nobreak\ $km as measured from the South Pole. This can also
be seen from the yellow color of the ACC in Fig. 1, highlighting its
contribution to KE at large scales. Additional support that this spectral peak
is due to the ACC can be found in Fig. E2, which plots the zonally (east-west)
averaged KE as a function of latitude at various scales larger than
$10^{3}\leavevmode\nobreak\ $km. We can see from Fig. E2 that the dominant
contribution is from latitudes [60∘S, 40∘S], which roughly corresponds with
the ACC. We also see in Fig. E2 a much weaker signal at latitudes [30∘N,
40∘N], which roughly aligns with the Gulf Stream and Kuroshio. Further
corroborating our assertion, the spectral peak in the southern hemisphere seen
in Fig. 2 has no analogous peak in the northern hemisphere. Fig. 2 shows
vestiges of a similar peak in the northern hemisphere, but this is arrested at
a smaller amplitude and at smaller scales ($\approx 4\times
10^{3}\leavevmode\nobreak\ $km) due to continental boundaries.

##### Gyre-scale Power Law

Comparing the KE spectra from both hemispheres in Fig. 2 at scales larger than
$10^{3}\leavevmode\nobreak\ $km, we observe a range of scales that exhibit a
$\sim k^{-1}$ power-law. This scaling has been predicted by [13] (see also
[29]) for baroclinic modes at scales larger than the barotropic deformation
radius, but has not been observed until now. The barotropic deformation radius
is about $2\times 10^{3}$ km in the oceans [26] and the ocean flow tends to be
surface intensified as expected in a baroclinic flow [30]. Thus, the $k^{-1}$
scaling observed in Fig. 2 is consistent with [13]. Previous studies relying
on Fourier analysis within box regions would have had difficulty detecting
such scaling due to the box size artifacts. The $k^{-1}$ extends to larger
scales and peaks at scales $\approx 4\times 10^{3}\leavevmode\nobreak\ $km in
the north, which is the average scale at which the flow starts feeling
continental boundaries and gyres form. This can also be seen from the bright
yellow color of the Gulf Stream and Kuroshio in Fig. 1, highlighting their
contribution to KE at large scales. In the southern hemisphere, on the other
hand, the $k^{-1}$ scaling extends up to the scale of the ACC, which
encounters no continental barriers (in the latitudes of the Drake Passage) as
it flows eastward around Antarctica.

##### Mesoscale Eddies

In Fig. 2, we find a second spectral peak between $100\leavevmode\nobreak\ $km
and $500\leavevmode\nobreak\ $km, centered at ${\ell\approx
300\leavevmode\nobreak\ }$km, that is associated with the mesoscale flow.
While we can see from Fig. 2 that the mesoscales do not form the largest peak
of the KE spectrum, their cumulative contribution between scales
$100\leavevmode\nobreak\ $km and $500\leavevmode\nobreak\ $km greatly exceeds
that of scales larger than $10^{3}\leavevmode\nobreak\ $km. This is because
the mesoscale flow populates a wider range of wavenumbers compared to the
gyre-scale flow (note the logarithmic x-axis). Indeed, integrating the energy
spectrum in Fig. 2 within the band $100\leavevmode\nobreak\ $km to
$500\leavevmode\nobreak\ $km yields more than $\approx 50\%$ of the total
energy resolved by either satellites or the mesoscale eddying model in the
extratropics. Coarse-graining allows us to determine this fraction of KE
belonging to the mesoscales _in the global ocean_. This is because integrating
the filtering spectrum over all $k_{\ell}$ in Fig. 2 yields the total KE (as
resolved by the data), which was not possible in past studies using Fourier
analysis in regional boxes.

The power-law spectral scaling at mesoscales and smaller scales has been the
focus of many previous studies (e.g. [22, 25, 9, 16]) using Fourier analysis
within box regions. While this is not our focus here, we observe that the
overall mesoscale spectral scaling lies between $k^{-5/3}$ and $k^{-3}$ in
Fig. 2, consistent with previous studies [25, 31]. Note that mesoscale power-
law scaling is more clearly seen in smaller regions (e.g. Fig. E1 in Methods)
as the mesoscale power-law and the corresponding wavenumber range change
significantly depending on the geographical location (see Fig. 15 in [22]).

#####  Characteristic Velocity and Energy Content within Key Scale Bands

From the spectra in Fig. 2, we partition the energy _conservatively_ into four
bands of interest: ${\ell\leq 100\leavevmode\nobreak\ }$km, 100 km to 500 km,
500 km to $10^{3}$ km, and ${\ell\geq 10^{3}\leavevmode\nobreak\ }$km such
that the sum of their energy equals total KE. From KE within a scale band,
$\mathrm{KE}_{\mathrm{band}}$, we can infer a characteristic root-mean-square
(RMS) velocity, $u_{\mathrm{rms}}=\sqrt{2\times\mathrm{KE}_{\mathrm{band}}}$
at those scales. The results are summarized in Table 1. The mesoscale band
(100–500 km) has the highest RMS velocity, between 15 and 16 cm/s, and
accounts for more than $50\%$ of the total energy in the model data.
Mesoscales are slightly more energetic in the NH than SH. Table 1 allows us to
also infer a characteristic timescale,
$\tau_{\mathrm{meso}}=\ell/u_{\mathrm{rms}}={\mathcal{O}}(25)\leavevmode\nobreak\
$days. The RMS velocity decreases significantly for larger scales, with
hemisphere-asymmetries becoming more prominent. Within the ACC-containing band
of $\ell>10^{3}\leavevmode\nobreak\ $km, the NH and SH RMS velocities are
approximately 4.2 and 5.4 cm/s, respectively, and with an associated
characteristic timescales,
$\tau_{\mathrm{gyre}}=\ell/u_{\mathrm{rms}}={\mathcal{O}}(\mathrm{few})\leavevmode\nobreak\
$years.

$\ell$-band |  $\nicefrac{{1}}{{12}}^{\circ}$ NEMO
---|---
RMS Vel. | % of
[cm/s] | Total KE
NH | SH | NH | SH
$\ell\leq 100\leavevmode\nobreak\ $km | 13.15 | 13.29 | 37.8 | 39.7
100 to 500 km | 15.48 | 15.00 | 53.2 | 50.2
500 to 1000 km |  04.64 |  04.08 |  04.7 |  03.7
$1000\leavevmode\nobreak\ \mathrm{km}\leq\ell$ |  04.26 |  05.31 |  04.0 |  06.2
|  $\nicefrac{{1}}{{4}}^{\circ}$ AVISO
$\ell\leq 100\leavevmode\nobreak\ $km | 11.21 | 11.24 | 28.9 | 30.8
100 to 500 km | 16.32 | 15.36 | 61.9 | 57.7
500 to 1000 km |  04.57 |  04.05 |  04.9 |  04.0
$1000\leavevmode\nobreak\ \mathrm{km}\leq\ell$ |  04.16 |  05.53 |  04.0 |  07.4
Table 1:  Energy content of scale ranges The RMS velocity in separate $\ell$
bands for each hemisphere, as well as the percent of total KE contained within
each $\ell$ band, for both the [upper half] $\nicefrac{{1}}{{12}}^{\circ}$
NEMO and [lower half] $\nicefrac{{1}}{{4}}^{\circ}$ AVISO datasets. See Table
E1 in Methods for uncertainty estimates.

### Seasonality and Spectral Lag Time

Figure 3 shows the seasonality in surface KE as a function of length-scale
from both satellite and model data, which exhibit similar trends. The most
striking feature of Fig. 3 is the approximately constant lag time between
length-scales of the same ratio as they attain seasonal maxima (red) and
minima (blue). Going from 10 km up to $10^{3}\leavevmode\nobreak\ $km, length-
scales that are $\times 2$ larger experience a lag of $\approx
40\leavevmode\nobreak\ $days in their seasonal cycle, such that in both
hemispheres KE at $100\leavevmode\nobreak\ $km peaks in late spring while KE
at $10^{3}\leavevmode\nobreak\ $km peaks in late summer. A detailed regression
analysis is in the Methods section. These results agree with and extend
previous analysis [15, 25, 20] within regional boxes, which found that scales
between 50–100 km have maximal KE in the spring while scales larger than
$200\leavevmode\nobreak\ $km (but smaller than the box) tend to peak with a
delay of one–two months. Possible explanations for the seasonal variation in
KE at different scales include the increased eddy-killing from winter’s high
winds [17], and an inverse energy cascade from the submesocales which
energizes mesoscales in spring months [25, 20]. While Fig. 3 is suggestive of
an inverse cascade, in which seasonal variations propagate up-scale at the
rate we observe, it alone is not sufficient evidence (see [32]) and a direct
measurement of the cascade as in [3] is required but is beyond our scope here.

#####  Gyre-scales

At gyre-scales the surface flow is influenced directly by continental
boundaries, wind, and buoyancy forcing. Indeed, at scales
$>10^{3}\leavevmode\nobreak\ $km in Fig. 3, there is a noticeable break in the
seasonal trends we discussed in the previous paragraph. In the SH, where the
ACC is not impeded by continents, we see from Fig. 3 a pronounced winter peak
at $\approx 10\times 10^{3}\leavevmode\nobreak\ $km, which correlates with
maximal wind forcing [17]. Scales between $10^{3}\leavevmode\nobreak\ $km and
$3\times 10^{3}\leavevmode\nobreak\ $km in both hemispheres peak in autumn,
consistent with previous analysis showing an autumn maximum in the surface
flow of western boundary currents [27, 28, 8] due to the upper ocean seasonal
heating cycle. At NH scales larger than $5\times 10^{3}\leavevmode\nobreak\
$km, the KE is too small to be meaningful (Fig. 2).

Figure 3:  Seasonality Normalized deviation (cf. Methods section) of the
60-day running average of surface kinetic energy in [left] NH and [right] SH
for both [top] satellite and [bottom] model datasets. Horizontal axis shows
time binned into months. Vertical axes show filtering wavenumber
$k_{\ell}=\ell^{-1}$. The green line in the bottom left panel shows a 100-fold
scale increase over 8 months.

### Conclusion and Outlook

Our spectral characterization of the ocean’s surface velocity over the entire
range of length-scales resolved by satellites and models revealed the ACC as
the spectral peak of the extratropical ocean at $\approx 10\times
10^{3}\leavevmode\nobreak\ $km, with gyres in the northern hemisphere yielding
a smaller peak that is arrested at $\approx 4\times 10^{3}\leavevmode\nobreak\
$km due to continental boundaries. By partitioning kinetic energy across
length-scales in a manner that conserves energy and covers the global ocean,
we showed that length-scales ${\ell<500\leavevmode\nobreak\ }$km make an
overwhelming contribution to surface kinetic energy due to populating a wide
range of wavenumbers, despite not forming the most prominent spectral peak.
Based on prior characterization of ocean energy [7], we reason that these
length scales are dominated by mesoscale features such as geostrophic
turbulence, boundary currents, and fronts. Our analysis also revealed a
characteristic lag time, with length-scales that are twice as large
experiencing a lag of $\approx 40\leavevmode\nobreak\ $days in their seasonal
cycle.

The expanded spectral analysis spurs new questions and lines of inquiry. We
hope future investigations will shed light on the dynamic coupling between the
spectral peaks at the gyre- and meso-scales, determine if the $k^{-1}$ slope
between the two peaks is indeed due to baroclinic modes [13, 29], and whether
the characteristic spectral lag-time is caused by an inverse cascade.

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  1. Acknowledgements We thank D. Balwada, M. Jansen, and S. Rai for valuable discussions and comments. We also thank Bo Qiu for kindly sharing the data used in Fig. E1. This research was funded by US NASA grant 80NSSC18K0772 and NSF grant OCE-2123496. HA was also supported by US DOE grants DE-SC0014318, DE-SC0020229, DE-SC0019329, NSF grant PHY-2020249, and US NNSA grants DE-NA0003856, DE-NA0003914. Computing time was provided by NERSC under Contract No. DE-AC02-05CH11231 and NASA’s HEC Program through NCCS at Goddard Space Flight Center. This work was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 882340).

  2. Competing Interests The authors declare that they have no competing financial interests.

  3. Correspondence Correspondence and requests for materials should be addressed to H.A.
Email<EMAIL_ADDRESS>
## Methods

### Description of datasets

For the geostrophic ocean surface currents, we use Level 4 (L4) post-processed
dataset of daily-averaged geostrophic velocity on a
$\nicefrac{{1}}{{4}}^{\circ}$ grid and spanning January 2010 to October 2018
(except for the seasonality analysis, where we use 2012-2016). The data is
obtained from the AVISO$+$ analysis of multi-mission satellite altimetry
measurements for sea surface height (SSH) pujol2016Supp. The product
identifier of the AVISO dataset used in this work is
“SEALEVEL_GLO_PHY_L4_REP_OBSERVATIONS_008_047”.

We also analyze 1-day averaged surface SSH-derived currents from the NEMO
numerical modeling framework, which is coupled to the Met Office Unified Model
atmosphere component, and the Los Alamos sea ice model (CICE). The NEMO
dataset consists of weakly coupled ocean-atmosphere data assimilation and
forecast system, which is used to provide 10 days of 3D global ocean forecasts
on a $\nicefrac{{1}}{{12}}^{\circ}$ grid. We use daily-averaged data that
spans four years, from 2015 to 2018. More details about the coupled data
assimilation system used for the production of the NEMO dataset can be found
in gmd-4-223-2011Supp,lea2015assessingSupp. The specific product identifier of
the NEMO dataset used here is “GLOBAL_REANALYSIS_PHY_001_030”.

### Coarse-graining on the sphere

For a field $\phi({\bf x})$, a “coarse-grained” or (low-pass) filtered field,
which contains only length-scales larger than $\ell$, is defined as

$\overline{\phi}_{\ell}({\bf x})=G_{\ell}*\phi,$ (M-1)

where $*$, in the context of this work, is a convolution on the sphere as
shown in aluie2019convolutionsSupp and $G_{\ell}({\bf r})$ is a normalized
kernel (or window function) so that $\int d^{2}{\bf r}\leavevmode\nobreak\
G_{\ell}({\bf r})=1$. Operation (M-1) may be interpreted as a local space
average over a region of diameter $\ell$ centered at point ${\bf x}$,
analogous to a moving time average. The kernel $G_{\ell}$ that we use here is
essentially a graded top-hat kernel:

$G_{\ell}({\bf x})=\frac{A}{2}\left(1-\tanh\left(10\left(\frac{\gamma({\bf
x})}{\ell/2}-1\right)\right)\right).$ (M-2)

We use geodesic distance, $\gamma({\bf x})$, between any location ${\bf
x}=(\lambda,\phi)$ on Earth’s surface relative to location
$(\lambda_{0},\phi_{0})$ where coarse-graining is being performed, which we
calculate using

$\gamma({\bf
x})=R_{\text{\tiny{E}}}\arccos\Big{[}\sin(\phi)\sin(\phi_{0})+\cos(\phi)\cos(\phi_{0})\cos(\lambda-\lambda_{0})\Big{]},$
(M-3)

with $R_{\text{\tiny{E}}}=6371\leavevmode\nobreak\ $km for Earth’s radius. In
eq. (M-2), $A$ is a normalization factor, evaluated numerically, to ensure
$G_{\ell}$ area integrates to unity. A convolution with $G_{\ell}$ in equation
(M-2) is a spatial analogue to an $\ell$-day running time-average.

The above formalism holds for coarse-graining scalar fields. To coarse-grain a
vector field on a sphere generally requires more work
aluie2019convolutionsSupp, particularly for vector fields that need not be
toroidal (2D non-divergent) or potential (2D-irrotational). However, as this
work focuses on SSH-derived 2D non-divergent velocity fields, these concerns
do not apply here. More details can be found in buzzicotti2021Supp.

#### Comparing coarse-graining to Fourier analysis

It is common to quantify the spectral distribution of ocean kinetic energy via
Fourier transforms computed either along transects or within regions; e.g.,
FuSmith1996Supp,Chenetal2015Supp,Rochaetal2016Supp,khatri2018surfaceSupp,ORourke_etal2018Supp,CalliesWu2019Supp.
This approach has rendered great insights into the length scales of oceanic
motion and the cascade of energy through these scales
ScottWang05Supp,ScottArbic07Supp,arbic2012nonlinearSupp,Arbicetal13Supp,arbic2014geostrophicSupp.
However, it has notable limitations for the ocean where the spatial domain is
generally not periodic, thus necessitating adjustments to the data (e.g., by
tapering) before applying Fourier transforms. Methods to produce an
artifically periodic dataset can introduce spurious gradients, length-scales,
and flow features not present in the original data sadek2018extractingSupp. A
related limitation concerns the chosen region size, with this size introducing
an artificial upper length scale cutoff. In this manner, no scales are
included that are larger than the region size even if larger structures exist
in the ocean. Furthermore, the data is typically assumed to lie on a flat
tangent plane to enable the use of Cartesian coordinates. However, if the
region becomes large enough to sample the earth’s curvature, then that puts
into question the use of the familiar Cartesian Fourier analysis of sines and
cosines. The use of spherical harmonics, common for the atmosphere, is not
suitable for the ocean, again since the ocean boundaries are complex. These
limitations mean that in practice, Fourier methods are only suited for open
ocean regions away from boundaries, and over a rather limited regional size.

As a demonstration of both the validity and advantages of coarse-graining for
energy spectra, consider Figure E1. This figure reproduces the energy spectrum
from Figure 3 of Qiuetal2018jpoSupp, and includes both the coarse-graining,
and traditional Fourier energy spectra measured from the
$\nicefrac{{1}}{{12}}^{\circ}$ NEMO dataset. Spectra are calculated for the
$5^{\circ}\times 5^{\circ}$ box centred at
$164^{\circ}\mathrm{E},\leavevmode\nobreak\ 37^{\circ}\mathrm{N}$, which
corresponds to the Kuroshio current.

Figure E1:  Energy Spectra. Comparison of the filtering spectrum (orange),
traditional Fourier spectrum (blue), and the Fourier spectrum from [Supp14]
(green, data kindly provided by authors of [Supp14]) for the ${5^{\circ}\times
5^{\circ}}$ box region centred at $164^{\circ}\mathrm{E},\leavevmode\nobreak\
37^{\circ}\mathrm{N}$ (roughly the Kuroshio extension). The dashed black line
provides a reference for a $-\nicefrac{{5}}{{3}}$ slope. Note that the $\ell$
axis (top) is in $100$ km.

For length scales ${\lesssim 200\leavevmode\nobreak\ }$km, the three spectra
generally agree very well, and all produce close to a $-\nicefrac{{5}}{{3}}$
spectrum. Qiuetal2018jpoSupp used a higher resolution dataset, and so the
spectra diverge for very small scales. However, coarse-graining does not
require tapering, and so the spectrum at scales ${\gtrsim
200\leavevmode\nobreak\ }$km are not contaminated by the shape of the tapering
window. As a result, coarse-graining is able to detect that the spectrum for
this region peaks at ${\sim 250\leavevmode\nobreak\ }$km, with a minimum near
$600\leavevmode\nobreak\ $km.

#### ACC as the spectral peak

In Figure E2 we provide a visualization of the zonally-averaged kinetic energy
for selected filtering scales. Scales larger than $10^{3}$ km have a dominant
contribution from latitudes [60∘S, 40∘S], roughly corresponding with the ACC,
and another contribution over [30∘N, 40∘N], roughly corresponding to the NH
boundary currents. Scales larger than $5\times 10^{3}$ km continue to show a
clear ACC signal, with no NH signal since this filter scale is just beyond the
NH gyre spectral peak. Finally, scales larger than $12\times
10^{3}\leavevmode\nobreak\ $km have no distinct ACC signal, showing that the
ACC has been fully removed by this scale. Combined, these provide further
support for our claim that the $10\times 10^{3}\leavevmode\nobreak\ $km
spectral peak corresponds to the ACC.

Figure E2:  Energy by Latitude Time- and zonally-averaged kinetic energy
computed from AVISO as a function of latitude for a selection of filter scales
(see legend). Note that the latitude axis is broken to exclude the band
[$15^{\circ}$S, $15^{\circ}$N].

#### Land Treatment

When coarse-graining near land, it is necessary to have a methodology for
incorporating land into the filtering kernel (c.f. buzzicotti2021Supp for more
in-depth discussion of land treatments). In the work presented here, we make
the choice of treating land as zero velocity water. Since coarse-graining is
essentially a ‘blurring’ (analogized with taking off ones glasses to have a
blurrier picture), the land-water division itself also become less well-
defined, and so treating land as zero-velocity water is both conceptually
consistent and aligns with no-flow boundary conditions. Additionally, this
land treatment allows for a ‘fixed’ (or homogeneous) filtering kernel at all
points in space, and as a result allows for commutativity with derivatives
(i.e. divergence-free flows remain divergence-free after coarse-graining)
aluie2019convolutionsSupp. Note, however, that only the true water area is
used as the denominator when computing area averages (e.g. the NH area-average
energy is the coarse energy summed over all NH cells, including land, divide
by the water-area of NH). The ocean areas used for the area-averaging are
$\approx 104\times 10^{6}\leavevmode\nobreak\ \mathrm{km}^{2}$ for NH and
$\approx 155\times 10^{6}\leavevmode\nobreak\ \mathrm{km}^{2}$ for SH.

##### Deforming kernel approach

An alternative choice is to deform the kernel around land, so that only water
cells are included, at the cost of losing the homogeneity of the kernel. The
benefit to this approach is that it does not require conceptually treating
land as zero velocity water. However, it has the significant drawback that
coarse-graining no longer commutes with differentiation and, as a result, does
not necessarily preserve flow properties such as being divergence-free.
Additionally, a kernel that is inhomogeneous (i.e. changes shape depending on
geographic location) does not necessarily conserve domain averages, including
the kinetic energy of the flow, and has the potential to both increase or
decrease the domain average. More details are provided in buzzicotti2021Supp.

##### Comparing land treatments

Figure E3 presents the energy spectra, similar to Fig. 2 in the main text,
using both deforming and fixed kernels for the single day 02 Jan 2015. The
deforming-kernel spectra agree remarkably well with the non-deforming (fixed)
kernel spectra, in that they present the mesoscales, ACC, and gyre peaks in
similar locations. There are some quantitative differences, such as the
deforming kernel SH spectra presents a slightly broader and higher-magnitude
ACC peak.

Figure E3:  Filtering spectra, analogous to Fig. 2, using both deforming and
fixed kernels on the AVISO dataset for a single day (02 Jan 2015).

####  Isolating Hemisphere Spectra

In this work, we are primarily concerned with the extra-tropical latitudes:
$[90^{\circ}\mathrm{S},15^{\circ}\mathrm{S}]$ and
$[15^{\circ}\mathrm{N},90^{\circ}\mathrm{N}]$. However, at very large length
scales information from the equatorial band and opposing hemisphere can become
introduced through the expanding filter kernel. To resolve this issue, we use
a ‘reflected hemispheres’ approach, wherein one hemisphere is reflected and
copied onto the other hemisphere, essentially producing a world with two
north, or two south hemispheres. It is worth noting that reflected hemispheres
and equatorial masking would not be necessary in a context where ageostrophic
velocities are also considered and a global power spectrum is desired. They
are used here because we wish to disentangle the power spectra of the extra-
tropical hemispheres separately.

Figure E4:  Same layout as Fig. 2, but without hemisphere reflections. That
is, NH and SH spectra are extracted from a single global coarse-graining. Note
that the NH spectra has a large-scale peak near $\ell=20\times
10^{3}\leavevmode\nobreak\ $km, which is not observed using reflected
hemispheres. Produced using the same four-year period of NEMO data, sub-
sampled to every fourth day.

Figure E4 shows the filtering spectra from NEMO without relying on hemisphere
reflection, and is to be compared to Fig. 2 in the main text. The two are in
qualitative agreement, with an ACC peak in the SH and mesoscale peaks in both
hemispheres. Unsurprisingly, the spectra only deviate for very large filtering
scales, where an increasing amount of extra-hemisphere information is captured
by the large kernels. Specifically, the NH spectra has a third peak at scales
$\ell>10\times 10^{3}\leavevmode\nobreak\ $km that is not present when using
reflected hemispheres. This very large-scale peak is a result of the NH
kernels capturing the ACC. It is worth noting, however, that the main ACC peak
is still present in the SH spectra, as is the NH gyre peak at approximately
$\ell=3\times 10^{3}\leavevmode\nobreak\ $km.

### Seasonality

A 60-day running mean is applied to remove higher frequencies and allow us to
better consider the longer-time trend, and individual years are averaged onto
a ‘typical’ year for the purpose of comparison. Seasonality results using the
5 years spanning 2012-2016 for AVISO, and the 4 years spanning 2015-2018 for
NEMO.

A useful statistical method for comparing signals is to compare the normalized
deviation, or z-scores, of the signal. For a set of points
$\\{x_{i}\,|\,i=1\ldots N\\}$, each point $x_{i}$ is transformed into a
corresponding z-score $z_{i}$ via $z_{i}=(x_{i}-\mu_{x})/\sigma_{x}$, where
$\mu_{x}$ and $\sigma_{x}$ are the mean and standard deviation of the $x_{i}$.
As a result, data points that are larger than the mean produce a positive
z-score, while those smaller than the mean produce a negative z-score. Note
that the normalized deviation (z-scores) in Fig. 3 are computed independently
for each $k_{\ell}$, and so comparing magnitudes between scales is non-
trivial.

####  Regression Analysis of Phase Shift

Fig. 3 presents a clear phase shift in the seasonal cycle as a function of
length-scale. In order to quantify the phase shift, we need to first extract a
meaningful set of $(k_{\ell},\mathrm{time})$ points. To that end, we extract,
for each $k_{\ell}$, the i) times corresponding to the lowest 10%, ii) middle-
most 10%, and iii) highest 10% of the normalized deviations presented in Fig.
3. Heuristically, this is extracting the
$(k_{\ell},\mathrm{time})$-coordinates for the line of darkest red, darkest
blue, and the two white lines, results in a total of four regression sets.
Where necessary, periodic phase adjustments are applied to maintain
monotonicity in time, and the $k_{\ell}$ grid is truncated to focus on regimes
with a clear linear trend. The extracted data points are shown as the dots /
vertical bars in Figure E5, along with their corresponding regression fits.

Figure E5:  Extracted data points and regression fits for the seasonal drift
exhibited in Fig. 3. Panel layout ordering is identical to Fig. 3. Dots (
which appear as vertical bars due to density ) show the extracted
$k_{\ell}$-time points, and the lines show the corresponding regression fit.
In the legend, ‘min’ means the lowest 10% z-score, ‘mid’ the two middle-most
10% groups, and ‘max’ the highest 10%.

Figure E6 presents the linear regression slope analysis for the data shown in
Fig. E5. The different regression analyses generally agree well, with 12 of
the 16 regression sets indicating at 35–45 day time-lag per octave of spatial
scale. From this analysis, we conclude that length-scales that differ by a
factor of two (i.e. $\ell_{1}/\ell_{2}=2$) have seasonal cycles that are off-
set by $41\pm 3$ days. Scales that differ a decade ($\ell_{1}/\ell_{2}=10$)
would correspondingly have a phase shift of $136\pm 10\leavevmode\nobreak\
$days, or roughly 4.5 months.

Figure E6:  Regression slope estimates and confidence intervals for each of
the datasets shown in Fig. E5. Vertical dashed lines separate the data
sources, with the text along the top indicating the source data and
hemisphere. Diamonds indicate the regression slope estimate, dark envelopes
the 75% confidence interval, and light envelopes the 95% confidence interval.
The r-value for each regression fit is printed alongside the corresponding
slope distribution. The left-most illustration, separated by a thick black
line, presents the mean (diamond) and confident intervals (envelopes) across
the 16 regression analysis. In the legend, ‘min’ means the lowest 10% z-score,
‘mid’ the two middle-most 10% groups, and ‘max’ the highest 10%.

###  Uncertainty estimates of $\ell$-band Values

Table 1 presented median values of the RMS velocity and percentage of total KE
contained within various $\ell$-bands. Supplemental Table E1 presents the
interquartile range (25th to 75th percentiles) to provide an estimate for the
sensitivity of those values.

$\ell$-band |  $\nicefrac{{1}}{{12}}^{\circ}$ NEMO
---|---
RMS Vel. | % of
[cm/s] | Total KE
NH | SH | NH | SH
$\ell\leq 100\leavevmode\nobreak\ $km | 12.77 to 13.56 | 12.98 to 13.78 | 36.3 to 40.5 | 38.2 to 41.0
100 to 500 km | 14.97 to 16.16 | 14.57 to 15.46 | 50.8 to 54.6 | 49.0 to 51.5
500 to 1000 km |  04.47 to 04.79 |  03.99 to 04.16 |  04.3 to 05.1 |  03.4 to 03.9
$1000\leavevmode\nobreak\ \mathrm{km}\leq\ell$ |  04.11 to 04.43 |  05.27 to 05.36 |  03.6 to 04.4 |  05.9 to 06.6
|  $\nicefrac{{1}}{{4}}^{\circ}$ AVISO
$\ell\leq 100\leavevmode\nobreak\ $km | 10.89 to 11.55 | 11.04 to 11.57 | 28.3 to 30.2 | 30.1 to 31.7
100 to 500 km | 15.62 to 16.83 | 15.02 to 15.79 | 60.7 to 62.8 | 56.7 to 58.5
500 to 1000 km |  04.46 to 04.68 |  03.98 to 04.12 |  04.6 to 05.2 |  03.8 to 04.2
$1000\leavevmode\nobreak\ \mathrm{km}\leq\ell$ |  04.06 to 04.26 |  05.47 to 05.57 |  03.7 to 04.4 |  07.0 to 07.7
Table E1:  Energy content of scale ranges Interquartile range of RMS velocity
values (left half) and percent of total kinetic energy in separate $\ell$
bands in each hemisphere, for both the [upper half]
$\nicefrac{{1}}{{12}}^{\circ}$ NEMO and [lower half]
$\nicefrac{{1}}{{4}}^{\circ}$ AVISO datasets.

## Data availability

All other data that support the findings of this study are available from the
corresponding author on reasonable request.

plain OceanSupp