# A coarse-grained decomposition of surface geostrophic kinetic energy in the
global ocean

M. Buzzicotti
Department of Physics and INFN
University of Rome Tor Vergata.
&B. A. Storer
Department of Mechanical Engineering
University of Rochester.
&S.M. Griffies
NOAA Geophysical Fluid Dynamics Laboratory
Princeton University Atmospheric and Oceanic Sciences Program.
&H. Aluie
Department of Mechanical Engineering and Laboratory for Laser Energetics,
University of Rochester.
<EMAIL_ADDRESS>

###### Abstract

We apply a coarse-grained decomposition of the ocean’s surface geostrophic
flow derived from satellite and numerical model products. In the extra-tropics
we find that roughly $60\%$ of the global surface geostrophic kinetic energy
is at scales between $100~{}$km and $500~{}$km, peaking at $\approx 300~{}$km.
Our analysis also reveals a clear seasonality in the kinetic energy with a
spring peak. We show that traditional mean-fluctuation (or Reynolds)
decomposition is unable to robustly disentangle length-scales since the time
mean flow consists of a significant contribution (greater than $50$%) from
scales $<500~{}$km. By coarse-graining in both space and time, we find that
every length-scale evolves over a wide range of time-scales. Consequently, a
running time-average of any duration reduces the energy content of all length-
scales, including those larger than $1000~{}$km, and is not effective at
removing length-scales smaller than $300~{}$km. By contrasting our spatio-
temporal analysis of numerical model and satellite products, we show that the
AVISO gridded product suppresses temporal variations less than 10 days for
_all length-scales_ , especially between 100 km and 500 km.

## Plain Language Summary

Traditionally, ‘eddies’ are identified as transient features that vary in time
relative to a background time mean flow. As such, a ‘mean’ flow does not
necessarily imply a large length-scale flow. For example, standing eddies or
stationary meanders due to topographic interactions have little variation in
time, but can still have significant energy at small length scales. Similarly,
‘eddy’, ‘time-varying’ or ’transient’ do not necessarily imply small length-
scale, with examples including the large-scale transient motions from Rossby
waves or fluctuations of the Kuroshio Current. Hence, the traditional time
average approach offers no control over the specific physical length that
divides oceanic flow into ‘large’ and ‘small’. That is, the length-scales
constituting the large-scale flow cannot be varied/controlled by time
averaging in a manner that is consistent with length-scales resolved in a
coarse climate simulation.

We consider a coarse-graining method to studying geostrophic ocean currents
with this method consistent with our notions of ’mesoscale’ as defined by a
length scale. Our approach is directly relevant to scale-aware
parameterization requirements of coarse-resolution simulations, since we are
directly focused on length-scales of the flow fields. To illustrate the
coarse-graining method, and to add understanding to the oceanic flows analyzed
here, we present the first global characterization of kinetic energy content
and its temporal variation as decomposed by coarse-graining according to
precisely defined length-scales.

##  1 Introduction

The oceanic circulation emerges from a suite of linear and nonlinear dynamical
processes that act over a broad range of space and time scales. The flow field
is markedly inhomogeneous and characterized by waves, instabilities, and
turbulent eddies, each of which are subject to a variety of energetic sources
and sinks. The mesoscale defines a key band of spatial scales where ocean
flows are largely geostrophic and where kinetic energy peaks [1].
Correspondingly, it is widely recognized that flow at the ocean mesoscales,
and its response to changes in atmospheric forcing, are fundamental to the
large-scale circulation and central for regional and global transport of heat
and biogeochemical tracers [2].

However, significant gaps remain in our understanding of the mesoscale flows
and their role in ocean circulation and climate. In particular, from a
numerical modeling perspective, despite the ever-increasing ability to conduct
simulations with mesoscale eddy-rich general circulation models (GCMs),
accurately resolving these scales in routine climate-scale (order centuries
and longer) simulations remains prohibitively expensive; e.g., [3]. We are
thus confronted with the need for mesoscale eddy parameterizations for the
foreseeable future [4].

A central question of physical oceanography, and in particular the eddy
parameterization problem, concerns a characterization of flow features
according to length-scale. This question motivates the goal of this paper,
which is to provide a length-scale decomposition of the global ocean
geostrophic kinetic energy, and to study the seasonal variations of this
decomposition. This goal has previously been unavailable due to limitations of
the commonly used Fourier spectral methods, which are unsuited to global ocean
analysis due to the complex geometry of ocean basins. We thus make use of a
coarse-graining method that does not share the limitations of Fourier
analysis. This paper thus serves to detail the use of coarse-graining for the
purpose of decomposing ocean kinetic energy, and in so doing we uncover some
rather novel features of the ocean surface circulation as a function of length
scale.

###  1.1 Limitations of Fourier methods for the ocean

It is common to quantify the spectral distribution of ocean kinetic energy via
Fourier transforms computed either along transects or within regions; e.g.,
[5, 6, 7, 8, 9, 10]. This approach has rendered great insights into the length
scales of oceanic motion and the cascade of energy through these scales [11,
12, 13, 14, 15]. 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 [16]. 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.

###  1.2 Eddy and mean flow decomposition using time averages

A traditional approach to extract the “eddies” from a flow uses time
averaging. This approach is relatively simple operationally and it accords
with the common practice in atmospheric and oceanic sciences of studying long-
term climate means and fluctuations relative to that mean. As part of this
decomposition for turbulent flow, we typically utilize the time averaging
operator as a Reynolds averaging (RA) operator, whereby the average of a
fluctuating quantity vanishes [17]. This assumption is largely based on
practical considerations, with Reynolds averages strictly holding only for
ensemble averages that are unavailable for most applications.

However, within the traditional decomposition, ‘time mean’ flow does not
necessarily imply a large length-scale flow as we shall discuss in this paper.
For example, standing eddies or stationary meanders due to topography [18]
have little temporal fluctuations but can have much structure at length-scales
$\mathcal{O}(100)$km or smaller. Similarly, with a temporal decomposition,
‘eddy’ does not necessarily imply small length-scale. For example, a time
averaging based decomposition would ascribe eddying motion to large-scale
Rossby waves [19] or variations in the Kuroshio Current’s path [20].

Moreover, by construction, an eddy-mean decomposition limits our ability to
analyze temporal variability (from intra-annual to inter-annual, [21, 3]) of
the multiscale coupling and evolution of different length-scales, including
those that need to be resolved/predicted in global climate (coarse-grid)
models. Therefore, it offers limited guidance for coarse-resolution models and
no control over the specific physical length which partitions oceanic flow
into ‘large’ and ‘small.’ In other words, the set of length-scales
constituting the large-scale flow cannot be varied/controlled to be consistent
with those length-scales resolved in a coarse climate simulation. In this
sense, the traditional decomposition cannot help with current efforts to
develop ‘scale-aware’ parameterizations [22, 23, 4, 24].

###  1.3 Coarse-graining

In order to understand the multiscale nature of oceanic flows, while
simultaneously resolving them in space and in time, we use a “coarse-graining”
framework that is relatively new in large-scale physical oceanography [25, 26,
27, 28, 29, 30]. It is a very general approach to analyzing complex flows,
with rigorous foundations initially developed to model [31, 32] and analyze
[33, 34] turbulence. [35] provides a theoretical discussion of coarse-graining
and its connection to other methods in physics. The approach has been recently
generalized to account for the spherical geometry of flow on Earth [36], and
applied to study the nonlinear cascade in the North Atlantic from an eddying
simulation [25].

The coarse-graining framework is very useful from the standpoint of ocean
subgrid scale parameterizations [37, 23, 38, 24, 39, 40, 41]. Namely, it
provides a theoretical basis for constructing subgrid closures that faithfully
reflect the dynamics at unresolved scales. A primary objective in ocean
modeling is practical: an accurate subgrid parameterization that is
numerically stable. Significant advances have been achieved in this regard in
the fluid dynamics and turbulence community [42, 43, 44], and the field of
large-eddy simulation (LES) is well-established [45].

Our use of coarse-graining supports the needs of parameterization, but our
primary objective is to characterize the fundamental dynamics of the flow at
_all_ length scales. Even within the wider fluid dynamics community, much less
work has been done in this regard, i.e. using coarse-graining as a “probe” of
the fundamental scale-physics. For example, LES sub-grid parameterization
studies are seldom concerned with using coarse-graining to probe the energy
pathways across the entire range of scales, such as the cascade [33, 46, 47,
48, 49, 50], forcing [51, 29], dissipation [52], or the range of coupling
between different scales [34, 53].

As an important case in point, despite LES being a well established field in
fluid dynamics since the seminal works of [54] and [55], the idea of using
coarse-graining in physical space to extract the energy content at different
scales; i.e., the spectrum, was only recently established and demonstrated by
[16]. This method is central to our calculation here of the spectrum for the
oceanic general circulation. A main advantage of coarse-graining is that it
allows us to decompose different length scales in a flow, at any geographic
location and any instant of time, without relying on assumptions of
homogeneity, isotropy or domain periodicity. This generality makes it ideally
suited for studying oceanic flows with complex continental boundaries over the
entire globe or in any particular regions of interest and at any time.

###  1.4 Key results and outline of this paper

In this paper we make use of the coarse-graining method on a satellite sea
surface product and a global ocean simulation. To directly compare the two
products, we focus on geostrophic components to the horizontal surface
velocity as diagnosed from sea level. We here highlight the key novel results
from this analysis. First, we show that $\approx 60\%$ of the extra-tropical
surface geostrophic kinetic energy, as estimated by satellite sea surface
observations and a global ocean simulation, resides at scales between
$100~{}$km and $500~{}$km, peaking at $\approx 300~{}$km. We also show that
these scales exhibit a clear seasonality, with kinetic energy peaking in
spring. Furthermore, we demonstrate how coarse-graining allows us a new way to
compare observations with models. In particular, comparing data from a
numerical simulation with gridded satellite observations reveals that the
satellite analysis misrepresents the evolution of _all length-scales_ over
time-scales less than $\approx 10$ days, with the mis-representation being
more pronounced at scales $\lesssim 500~{}$km. We show that this
misrepresentation is due to the temporal averaging required to construct the
gridded satellite product.

The paper is organized as follows. In Section 2 we give details on the coarse-
graining and the Reynolds averaging methods used in this work. In Section 3 we
present the data products used in our analysis. In Section 4 we discuss the
main results from this analysis, following the coarse-graining decomposition
such as the measurement of the energy spectrum, and the comparison with the
traditional Reynolds average decomposition. At the end of Section 4 we
introduce the combined spatio-temporal decomposition and compare satellite and
numerical model data. In Section 5 we present our conclusions. A discusses
some technical choices used when coarse-graining.

##  2 Coarse-graining for the ocean

In this section, we discuss the coarse-graining framework and how it is used
to partition energy across length scales. We also discuss the traditional
approach of temporal-based Reynolds averaging, in which the flow is decomposed
into a mean and fluctuating components.

###  2.1 Basics of coarse-graining on the sphere

For any scalar field, $F({\bf x})$, we can calculate its coarse-grained (or
low-pass filtered) version, $\overline{F}_{\ell}({\bf x})$, by convolving
$F({\bf x})$ with a normalized filter kernel $G_{\ell}({\bf r})$,

$\overline{F}_{\ell}({\bf x})=G_{\ell}*F({\bf x})$ (1)

where $*$, in the context of this work, is convolution on the sphere [36],
${\bf x}$ is geographic location on the globe, and the kernel $G_{\ell}({\bf
x})$ can be any non-negative function that is spatially localized (i.e. it
goes to zero fairly rapidly as ${\bf x}\to\pm\infty$). The parameter $\ell$ is
a length-scale related to the kernel’s “width.” We use the notation
$\overline{(\cdots)}_{\ell}$ to denote a coarse-grained field. The kernel is
area normalized for all $\ell$, so that

$\int G_{\ell}({\bf x})\,\mathrm{d}\mathcal{S}=1,$ (2)

where $\mathrm{d}\mathcal{S}$ is the area element on the sphere.
Correspondingly, the convolution (1) may be interpreted as an average of the
function $F$ within a region of diameter $\ell$ centered at location ${\bf
x}$. By construction, at each point in space, ${\bf x}$, the coarse-grained
field, $\overline{F}_{\ell}({\bf x})$, contains information about the scale
$\ell$.

The above formalism holds for coarse-graining scalar fields. To coarse-grain a
vector field on a sphere generally requires more work [36]. However, since we
are concerned only with the surface geostrophic velocity, ${\bf u}({\bf
x},t)$, in this work, it greatly simplifies our analysis. We assume the
geostrophic velocity is non-divergent on the two-dimensional spherical
surface, so that it is related to the geostrophic streamfunction $\psi$ via

${\bf u}=\hat{\boldsymbol{e}}_{r}{\mbox{\boldmath$\times$}}\nabla\psi,$ (3)

with $\hat{\boldsymbol{e}}_{r}$ the radial unit vector in spherical
coordinates, $\psi=\eta\,g/f$, $\eta$ the free sea surface height (SSH), and
the Coriolis parameter, $f=2\Omega\sin(\phi)$, is a function of latitude
$\phi$, where $\Omega$ is Earth’s spin rate.

[36] showed that for non-divergent vector fields such as in eq. (3), coarse-
graining ${\bf u}$ is equivalent to coarse-graining each of its Cartesian
components. We therefore transform the vector from spherical
$(u_{r},u_{\lambda},u_{\phi})$ to planetary Cartesian coordinates
$(u_{x},u_{y},u_{z})$ via:

$\displaystyle u_{x}$
$\displaystyle=u_{r}\cos(\lambda)\cos(\phi)-u_{\lambda}\sin(\lambda)-u_{\phi}\cos(\lambda)\sin(\phi)$
$\displaystyle u_{y}$
$\displaystyle=u_{r}\sin(\lambda)\cos(\phi)+u_{\lambda}\cos(\lambda)-u_{\phi}\sin(\lambda)\sin(\phi)$
(4) $\displaystyle u_{z}$ $\displaystyle=u_{r}\sin(\phi)+u_{\phi}\cos(\phi)$

where $\lambda$, $\phi$ are longitude and latitude, respectively, and
$u_{\lambda}$, $u_{\phi}$ are the zonal and meridional velocity components,
respectively. The radial velocity component, $u_{r}=0$ for the geostrophic
flow. We apply the spherical convolution operation in eq. (1) to each of
$u_{x}$, $u_{y}$, $u_{z}$ as scalar fields to obtain the corresponding coarse-
grained fields $\overline{u_{x}}$, $\overline{u_{y}}$, $\overline{u_{z}}$,
then retrieve the coarse-grained velocity, $\overline{{\bf u}}_{\ell}$ in
spherical coordinates via

coarse-grained radial flow
$\displaystyle=\overline{u_{x}}\cos(\lambda)\cos(\phi)+\overline{u_{y}}\sin(\lambda)\cos(\phi)+\overline{u_{z}}\sin(\phi)=0$
coarse-grained zonal flow
$\displaystyle=-\overline{u_{x}}\sin(\lambda)+\overline{u_{y}}\cos(\lambda)$
(5) coarse-grained meridional flow
$\displaystyle=-\overline{u_{x}}\cos(\lambda)\sin(\phi)-\overline{u_{y}}\sin(\lambda)\sin(\phi)+\overline{u_{z}}\cos(\phi)$

That the ‘coarse-grained radial flow’ (i.e. vertical flow, parallel to
gravity) vanishes is not obvious and was proved in [36] and demonstrated
numerically in [56].

We emphasize that the coarse-graining algorithm we just described is valid
only for non-divergent vectors such as ${\bf u}$ in eq. (3). Significant
errors can arise for a general flow field [56], where the complete coarse-
graining formalism of [36] is necessary (e.g. [25]).

We use the following coarse-graining kernel

$G_{\ell}({\bf x})=\frac{A}{2}\left(1-\tanh\left(10\left(\frac{\gamma({\bf
x})}{\ell/2}-1\right)\right)\right),$ (6)

as shown in Figure 1. It is essentially a top-hat kernel [57] but with graded
edges to avoid numerical artifacts from the non-uniform discrete grid on the
sphere. 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

$\gamma({\bf
x})=R_{\text{{Earth}}}\arccos\Big{[}\sin(\phi)\sin(\phi_{0})+\cos(\phi)\cos(\phi_{0})\cos(\lambda-\lambda_{0})\Big{]}.$
(7)

with $R_{\text{{Earth}}}=6371~{}$km for Earth’s radius. In eq. (6), $A$ is a
normalization factor, evaluated numerically, to ensure $G_{\ell}$ area
integrates to unity as per equation (2). In general, we are not restricted to
this choice of kernel; however, we use it because of its well-defined
characteristic width $\ell$. Indeed, a convolution with $G_{\ell}$ in equation
(6) is a spatial analogue to an $\ell$-day running time-average.

Figure 1: The coarse-graining kernel from equation (6), which is essentially a
top-hat kernel (or moving window) along with graded edges to avoid numerical
artifacts from the non-uniform discrete grid on the sphere. Geodesic distance
$\gamma({\bf x})$ is relative to location ${\bf x}_{0}$ where coarse-graining
is being applied. This plot is for the un-normalized kernel,
$\frac{1}{A}G_{\ell}({\bf x})$, with ${\ell=200~{}}$km. It is a spatial
analogue for an $\ell$-day window for a running time-average.

###  2.2 Partitioning the geostrophic kinetic energy

From the coarse-grained horizontal geostrophic velocity field, $\overline{{\bf
u}}_{\ell}({\bf x},t)$, following equation (1) as prescribed in [36], we
partition kinetic energy (KE) into different sets of length-scales:

$\displaystyle{\mathcal{E}}$ $\displaystyle=\frac{1}{2}|{\bf u}({\bf
x},t)|^{2}$ (bare KE) (8) $\displaystyle{\mathcal{E}}_{\ell}$
$\displaystyle=\frac{1}{2}|\overline{{\bf u}}_{\ell}({\bf x},t)|^{2}$ (coarse
KE) (9) $\displaystyle{\mathcal{E}}_{<\ell}$
$\displaystyle=\frac{1}{2}\left(\overline{|{\bf u}({\bf
x},t)|^{2}_{\ell}}-|\overline{{\bf u}}_{\ell}({\bf x},t)|^{2}\right)$ (fine
KE). (10)

The “bare KE” in equation (8) is the KE per unit mass (m2/s2) of the original
geostrophic flow that includes all scales; “coarse KE” in equation (9)
represents energy of the coarse-grained geostrophic flow at length-scales
larger than $\ell$; and “fine KE” in equation (10) accounts for geostrophic
energy at scales smaller than $\ell$, which we discuss more in the following
two paragraphs. Partitioning geostrophic energy across scales is not trivial
since one needs to ensure that such quantities are physically valid in the
sense described by [55] and [58]. In particular, it is important to ensure
that the partitioned kinetic energy is (i) positive semi-definite ($\geq 0$)
at every ${\bf x}$ and every time, and (ii) that summing the partitions yields
the total energy.

While it is clear that ${\mathcal{E}}_{\ell}\geq 0$ in equation (9), this
property is not obvious for ${\mathcal{E}}_{<\ell}$ in equation (10).
Moreover, it may not be obvious why ${\mathcal{E}}_{<\ell}$ should represent
energy at scales smaller than $\ell$. [58] showed that
${\mathcal{E}}_{<\ell}\geq 0$ if $G_{\ell}\geq 0$, whereas
${\mathcal{E}}_{<\ell}$ can be negative if the coarse-graining kernel
$G_{\ell}$ is not positive semi-definite. A proof using convexity of the
$(\dots)^{2}$ operation illustrates why the first term $\overline{|{\bf
u}({\bf x},t)|^{2}}_{\ell}$ in eq. (10) has an overbar rather than defining
fine KE as $(|{\bf u}({\bf x},t)|^{2}-|\overline{{\bf u}}_{\ell}({\bf
x},t)|^{2})/2$. The proof from [16] is as follows. When using $G_{\ell}\geq
0$, coarse-graining $\overline{(\dots)}_{\ell}$ is a local averaging
operation. From Jensen’s inequality [59], we know that
$\overline{[\mathcal{F}({\bf u})]}_{\ell}\geq\mathcal{F}(\overline{{\bf
u}}_{\ell})$ for any convex operation, $\mathcal{F}$. Since $\mathcal{F}({\bf
u})=|{\bf u}|^{2}$ is convex, we are guaranteed that $\overline{|{\bf u}({\bf
x},t)|^{2}_{\ell}}\geq|\overline{{\bf u}}_{\ell}({\bf x},t)|^{2}$ and,
therefore, ${\mathcal{E}}_{<\ell}\geq 0$ if the kernel $G_{\ell}(r)\geq 0$,
which is the case for our study (see equation (6)).

Regarding condition (ii) on the sum of energy partitions, [36] proved that
(for a normalized $G_{\ell}$) the coarse-graining operation on the sphere in
equation (1) preserves the spatial average of any field,
$\\{\overline{F}_{\ell}({\bf x})\\}=\\{F({\bf x})\\}$, where
$\\{\dots\\}=(Area)^{-1}\int\mathrm{d}\mathcal{S}(\dots)$. Therefore, we have
$\left\\{\overline{|{\bf u}|_{\ell}^{2}}\right\\}=\left\\{|{\bf
u}|^{2}\right\\}$. This property guarantees that the sum of coarse KE and fine
KE yields the total kinetic energy after integrating in space and in the
absence of land,

$\left\\{{\mathcal{E}}\right\\}=\left\\{{\mathcal{E}}_{\ell}\right\\}+\left\\{{\mathcal{E}}_{<\ell}\right\\}.$
(11)

Eq. (11) justifies our interpretation of ${\mathcal{E}}_{<\ell}$ as energy at
scales smaller than $\ell$, since it is the difference between bare and coarse
kinetic energy, on average, while also being positive locally.

###  2.3 Treatment of land-sea boundaries

In the above decomposition of energy, a choice has to be made in the presence
of land. We here discuss three possibilities along with their pros and cons.

#### Deformed kernel

The "deformed kernel" approach is realized by coarse-graining ocean points
near land with a kernel that is deformed or masked to avoid overlapping with
land points. Such a deformed kernel must be renormalized to yield an average
over just ocean points rather than the whole sphere. The main advantage of
this approach is that it treats land as a well-defined boundary that is
separate from the ocean regardless of the coarse-graining length-scale. It is
also familiar to ocean modelers who routinely mask values over land and do not
include such masked values when performing area averages.

However, the deformed kernel has disadvantages that motivate against its use
for coarse-graining ocean flows. First, a kernel that is inhomogeneous (i.e.
changes shape depending on geographic location) does not conserve domain
averages, including the kinetic energy of the flow. The reason for this failed
conservation is detailed in A and demonstrated in Figure 2 (blue plot). This
figure shows how a kernel that is deformed (via masking) to exclude land does
not yield $100\%$ of the total energy, i.e., it does not satisfy equation
(11). As a result, it can yield total energy that is either less than $100\%$
(e.g., over scales larger than $500~{}$km in Figure 2) or _greater_ than
$100\%$ (e.g., between $100~{}$km and $400~{}$km in Figure 2).

For some purposes, the total energy values in Figure 2 are fairly close to
$100\%$ (deviations less than $1\%$) so one might argue that the deformed
kernel is suitable in practice. Nonetheless, a more basic reason to avoid
deformed kernels is that such inhomogeneous kernels (which also include
averaging values at adjacent grid-cells or block-averaging on the sphere) do
not commute with spatial derivatives. Consequently, the coarse-grained field
resulting from a deformed kernel is not guaranteed to satisfy fundamental flow
properties exhibited by the unaveraged flow, such as incompressibility,
geostrophy, and the vorticity present at various scales. These considerations
are further detailed in [25] and [36].

#### Fixed kernel

The "fixed kernel", also shown in Figure 2, is homogeneous so that it
preserves its shape at all locations. When coarse-graining ocean points near
land such that the kernel overlaps land points, we treat land points in a
manner consistent with the boundary conditions between land and ocean. For
example, if we are coarse-graining the velocity, we treat land as water with
zero velocity, which is consistent with the formulation of OGCMs where land is
often treated as a region of zero velocity. Furthermore, we include these zero
land values as part of the coarse-graining operation.

This choice may seem unnatural since we are including unphysical values within
the coarse-graining operation. However, it is helpful to think of coarse-
graining as an operation analogous to removing one’s eyeglasses, rendering an
image fuzzy and boundaries less well-defined. When coarse-graining at a scale
$\ell$, the precise boundary between land and ocean becomes blurred at that
scale and its precise location becomes less certain. The coarse-grained
velocity, $\overline{\bf u}_{\ell}$, can be nonzero within a distance $\ell/2$
beyond the continental boundary over land. Forfeiting exact spatial
localization in order to gain scale information is theoretically inevitable
due to the uncertainty principle, which prevents the simultaneous localization
of data in physical-space and in scale-space [60, 61]. The main advantage of
the “Fixed Kernel” choice is ensuring that coarse-graining and spatial
derivatives commute so that it preserves the fundamental physical properties
of the flow. Further discussion of these issues can be found in [25] and [36].

#### Fixed kernel with or without land

After coarse-graining the velocity field with a fixed kernel, we show in
Figure 2 the level of energy conservation if we include or exclude land points
from the final tally of kinetic energy. We call these, respectively, the
"fixed kernel w/ land" and "fixed kernel w/o land". The latter (orange plot)
highlights how coarse-graining smears energy onto land (within $\ell/2$
distance inland) such that if we exclude land from the final tally, we find
some leakage of energy onto land, which increases as the coarse-graining scale
$\ell$ increases. We find energy leakage of the order of $1\%$ at coarse-
graining scales $<100~{}$km, $\approx 4\%$ for scales $\lesssim 500~{}$km, and
up to $12\%$ at scales of order $2000~{}$km. However, if we choose to include
land in our final tally, we are guaranteed to conserve $100\%$ of the energy
by satisfying equation (11), thus ensuring that the energy budget is fully
closed. After all, in an ocean model on a discrete grid, the land boundary is
only expected to be accurate within a $\Delta x$ distance from any estimate of
the truth, where $\Delta x$ is analogous to our coarse-graining scale $\ell$.

#### What we use here

While we have implemented all three approaches to coarse-graining (see Figure
14 in A), unless otherwise stated in this work, we choose the fixed kernel w/
land by including land regions that have non-zero velocity (again, as realized
through leakage from nearby ocean values). We have checked that our results in
all figures shown are almost indistinguishable from choosing fixed kernel w/o
land due to the relatively small percentage of energy leakage involved in
Figure 2. We avoid coarse-graining with a deformed kernel to remain consistent
with previous work [25] and with forthcoming studies where we apply coarse-
graining to the dynamical equations where commuting with spatial derivatives
is essential.

Figure 2:  Percentage of total energy recovered by summing the fine and coarse
KE terms in equation (11) obtained by coarse-graining over the full ocean
surface as a function of the filter scale, ${k_{\ell}=1/\ell}$. The three
lines correspond to the three approaches described in section 2.3, namely,
filtering with a fixed kernel shape and excluding/including land (orange/green
lines) when tallying the total energy. We also coarse-grain with a deformable
filter kernel to exclude the filter overlapping land regions (blue line).

###  2.4 The filtering spectrum

[16] showed how coarse-graining can be used to extract the energy content at
different length scales. They do so by partitioning the velocity into discrete
length scale bands rather than the two sets (coarse KE and fine KE) in
equations (9) and (10). The resulting quantity is called the filtering
spectrum. The filtering spectrum is distinct from the traditional Fourier
spectrum, with coarse-graining offering a way to measure energy distributions
without relying on a Fourier transform, thus avoiding the limitations noted in
Section 1.1.

The filtering spectrum is obtained by differentiating in scale the coarse KE

${\overline{E}}(k_{\ell})=\frac{d}{dk_{\ell}}\left\\{{\mathcal{E}}_{\ell}\right\\}=-\ell^{2}\frac{d}{d\ell}\left\\{{\mathcal{E}}_{\ell}\right\\},$
(12)

where $k_{\ell}=1/\ell$ is the ‘filtering wavenumber.’ [16] showed that
${\overline{E}}(k_{\ell},t)\geq 0$ when using certain types of kernels (e.g.,
concave) of which the top-hat kernel is an example. Moreover, [16] identified
the conditions on $G_{\ell}$ for ${\overline{E}}(k_{\ell},t)$ to be meaningful
in the sense that its scaling agrees with that of the traditional Fourier
spectrum (when a Fourier analysis is possible, such as in periodic domains).
Below, we shall sometimes refer to ${\mathcal{E}}_{\ell}$ as the “cumulative
spectrum” following [16] since it accounts for all energy at scales larger
than $\ell$. In contrast, ${\overline{E}}(k_{\ell},t)$, is the spectral energy
_density_ at a specific scale $\ell$.

###  2.5 Reynolds averaging

We close this section by reviewing basic properties of Reynolds averaging (RA)
as realized by time averages.

#### Basics of Reynolds averaging

Time averaging separates the flow into a time-average/‘mean’ and a
fluctuating/‘eddy’ as given by [57]

$\langle{\bf u}\rangle({\bf x})=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}{\bf u}({\bf
x},t)\mathrm{d}t,$ (13) ${\bf u}^{\prime}({\bf x},t)={\bf u}({\bf
x},t)-\langle{\bf u}\rangle({\bf x}),$ (14)

where $\langle{\bf u}\rangle$ is the mean component, ${\bf u}^{\prime}$ the
eddy component, and $T$ represents the entire time record and not just a time
window. Two key properties of the Reynold’s decomposition are

$\langle\langle{\bf u}\rangle\rangle=\langle{\bf
u}\rangle\quad\mbox{and}\quad\langle{\bf u}^{\prime}\rangle=0,$ (15)

so that the mean of a mean returns the mean (idempotence property) while the
mean of the eddy is zero. The resulting mean and eddy kinetic energy
components are respectively given by

$MKE({\bf x})=\frac{1}{2}|\langle{\bf u}\rangle|^{2}({\bf x}),$ (16) $EKE({\bf
x},t)=\frac{1}{2}|{\bf u}^{\prime}|^{2}({\bf x},t).$ (17)

Notice that the sum of mean and eddy kinetic energy is not equal to the total
kinetic energy. Rather, there is an extra cross term, ${\bf
u}^{\prime}\cdot\langle{\bf u}\rangle$, needed to close the budget. However,
the cross term is not positive definite and it has a zero time average,
$\langle{\bf u}^{\prime}\cdot{\bf u}\rangle=0$. Following a RA decomposition,
the total energy can be written as

${\mathcal{E}}({\bf x},t)=EKE({\bf x},t)+MKE({\bf x})+\frac{1}{2}\left({\bf
u}^{\prime}\cdot\langle{\bf u}\rangle\right)({\bf x},t).$ (18)

#### Key differences between Reynolds averaging and coarse-graining

A key difference between coarse-graining and Reynolds-averaging is that within
RA, applying the averaging operation twice on any field yields the same result
whereas that property does not hold for coarse-graining with non-projector
kernels [43]

$\langle\langle F\rangle\rangle=\langle
F\rangle\quad\mbox{whereas}\quad\overline{\overline{F}}\neq\overline{F}.$ (19)

Another important difference is that a Reynolds average does not provide a
control to adjust the partition between the “mean” and “eddy” components. That
is, a Reynolds decomposition is not a length-scale decomposition and this
point is illustrated in section 4.5 (see Figure 11). Consequently, the time-
mean flow is not synonymous with large-scale flow, nor does a Reynolds eddy
fluctuation directly correspond to a characteristic fine-scale.

To help understand the above points, we emphasize the distinction between
time-scale and decorrelation-time for a particular flow feature. While it is
generally true that larger (smaller) scales have slower (faster) time-scale
dynamics, it is not always true that their decorrelation-time follows this
relation. As an example, consider stationary eddies, such as the Mann eddy in
the North Atlantic. Such eddies have a small spatial-scale (relative to the
gyre or basin) but are persistent in time. As a result, even if the timescale
($\sim\ell/u$) for a structure is small when it is associated with the
relatively fast dynamics of eddying flows, it can be highly correlated (or
even stationary) in time, so that its contribution to the $MKE$ is not
completely removed by a time-average. We show this behavior in sections 4.5
and 4.6.

##  3 Satellite and numerical model data

We examine the horizontal geostrophic velocity of surface ocean currents from
a global numerical model simulation and from an analysis of satellite sea
surface altimetry, focusing on regions to the north and south of the tropics,
$[15^{\circ}\text{N}-90^{\circ}\text{N}]$ and
$[15^{\circ}\text{S}-90^{\circ}\text{S}]$, as depicted in Figure 3. We avoid
the tropics since our interest is with the geostrophic flows in the higher
latitudes, and only the surface geostrophic current is available from
satellite altimetry.

Figure 3:  Geographic regions considered in this analysis. The shaded area
between ${[15^{\circ}\text{S}-15^{\circ}\text{N}]}$ is the tropical band that
is neglected in this work. We instead consider the geostrophic flow in the
higher latitudes. Visualization shows a one-year time average of the full
kinetic energy from NEMO for illustration.

Details of the two products are given in the following subsections, and both
were publicly accessed through the Copernicus Marine Environment Monitoring
Service (CMEMS) webpage.

###  3.1 AVISO analysis of satellite altimetry

Geostrophic currents are obtained from the AVISO$+$ analysis of multi-mission
satellite altimetry measurements for sea surface height (SSH) [62]. We used
the Level 4 (L4) post-processed dataset of daily-averaged geostrophic
velocity, gridded at a resolution of ${0.25^{\circ}\times 0.25^{\circ}}$ and
spanning from January 2010 to October 2018. Post processing was performed by
the Sea Level Thematic Center (SL TAC) data processing system, which processes
data from eleven altimeter missions.

The product identifier of the AVISO dataset used in this work is
“SEALEVEL_GLO_PHY_L4_REP_OBSERVATIONS_008_047”, and can be downloaded at
https://marine.copernicus.eu/services-portfolio/access-to-products/.

###  3.2 Numerical simulation

We analyze 1-day averaged surface geostrophic 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 the same grid of $0.25$ degree spacing. We use daily-averaged data that
spans three years, from 2016 to 2019. More details about the coupled data
assimilation system used for the production of the NEMO dataset can be found
in [63, 64]. The specific product identifier of the NEMO dataset used in this
work is “GLOBAL_ANALYSISFORECAST_PHY_CPL_001_015”, and it can be downloaded
from CMEMS at https://marine.copernicus.eu/services-portfolio/access-to-
products/.

##  4 Analysis results

In this section we present results of the coarse-graining analysis along with
a comparison with Reynolds averaging based on time averages. In the second
part of this section we present results from coarse-graining in both space and
in time as a means to characterize the time-scales associated with different
length-scales.

###  4.1 Coarse-graining the surface geostrophic flow from AVISO

We split the geostrophic kinetic energy from AVISO into its fine and coarse-
grained components following equations (9) and (10). For a qualitative
appreciation of this decomposition, Figure 4 displays maps of the kinetic
energy just over the Atlantic using two different filter scales,
$\ell=100~{}\mbox{km}$ in the top row and $\ell=400~{}\mbox{km}$ in the bottom
row. From left to right, panels in Figure 4 show the total kinetic energy,
${\mathcal{E}}$, the coarse energy, $\mathcal{E}_{\ell}$, and the fine energy,
${\mathcal{E}}_{<\ell}$. The fine scale kinetic energy,
${\mathcal{E}}_{<\ell}$, represents kinetic energy at scales less than $\ell$,
as represented (or projected) on a grid of resolution ${\Delta x\sim\ell}$.
Notably, as seen in Figure 4, ${\mathcal{E}}_{<\ell}$ does not have small
scale features, which results since there is a filter applied to both terms in
equation (10) defining ${\mathcal{E}}_{<\ell}$. This definition ensures that
${\mathcal{E}}_{<\ell}$ is positive semi-definite at each point in space and
time.

Visualization of fine kinetic energy, ${\mathcal{E}}_{<\ell}$, is still useful
to identify the regions where it is dominant in the ocean. Even so, one may
wish to view the alternative quantity

${\mathcal{E}}-{\mathcal{E}}_{\ell}=\frac{1}{2}\left(|{\bf u}({\bf
x},t)|^{2}-|\overline{{\bf u}}_{\ell}({\bf x},t)|^{2}\right),$ (20)

which is shown in the right-most column of Figure 4. This quantity reveals
more fine scale features since only the second term on the right hand side is
filtered. However, as discussed in Section 2.1, the energy difference,
${{\mathcal{E}}-{\mathcal{E}}_{\ell}}$, can be negative locally in space, and
so it does not serve our purposes for decomposing the energy into non-negative
terms.

Figure 4:  Maps of the coarse-grained decomposition of kinetic energy
performed on a single-day from the AVISO analysis at two different filter
scales, ${\ell=100}$ km (top row) and ${\ell=400}$ km (bottom row). Here the
bare KE, ${\mathcal{E}}({\bf x},t)$, is compared with coarse KE,
${\mathcal{E}}_{\ell}({\bf x},t)$, and fine KE, ${\mathcal{E}}_{<\ell}({\bf
x},t)$. The right-most column shows the resulting fine scale term defined by
equation (20), which can yield values that are negative.

###  4.2 Seasonality of the fine scale geostrophic kinetic energy

In Figure 5, we present the seasonality of the fine scale geostrophic kinetic
energy in the North (solid lines) and South (dashed lines). We show results
just from AVISO, though note that similar results hold for the NEMO output.
Figure 5 shows the calculation of fine scale kinetic energy with four
different filter scales, $\ell=77~{}$km, 129 km, 215 km, and 464 km. We choose
these length-scales due to their equal spacing on a logarithmic scale.

Figure 5:  Top panel: Time-series of fine scale geostrophic kinetic energy,
${\mathcal{E}}_{<\ell}$, representing energy at scales smaller than $\ell$, in
the North (solid lines) and South (dashed lines) obtained from the AVISO
analysis. Different colors indicate different scales $\ell$. Middle panel: The
same time-series, but now normalized by the original kinetic energy,
${\mathcal{E}}$, to show the $\%$ of kinetic energy residing at scales
${<\ell}$. Bottom panel: deviation of ${\mathcal{E}}_{<\ell}$ from the time-
mean value to emphasize seasonal variability. Vertical grid lines indicate the
start of each quarter-year (01Jan, 01Apr, 01Jul, 01Oct). We see that the
mesoscales exhibit a seasonal oscillation that generally peaks in the spring,
and that they account for a dominant fraction of the total geostrophic kinetic
energy. Here, we show only 6.5 years of the 9-year record analyzed with a
4-day sampling, but averages are based on a 1-day sampling.

The fine scale geostrophic kinetic energy in Figure 5 reveals a clear
seasonality across all length scales, generally peaking in the spring and
attaining a minimum in the autumn of both hemispheres. The study of [65] and
[66] arrive at a similar conclusion about seasonality using different methods.
The cause for such seasonality is an ongoing topic of research, requiring the
analysis of various mesoscale sources and sinks. [29] recently showed that
eddy-killing is a significant global mesoscale sink with a seasonal cycle that
peaks in winter, thereby offering a potential explanation for mesoscale
seasonality. Another possible explanation was put forth by [65], who showed
evidence for seasonality in the upscale energy cascade in the North Pacific.
While it is beyond the scope of this paper, it is worth noting that, in
addition to highlighting seasonal variations, the bottom panel of Figure 5
reveals a general increase in mesoscale energy over the plotted timeframe.
This long-term increase is present in both the North of Tropics and South of
Tropics regions and is consistent with recent analysis by [67].

In the middle panel in Figure 5 we show the fine scale kinetic energy
normalized as a percentage of the total energy,
$\left\\{{\mathcal{E}}_{<\ell}\right\\}/\left\\{{\mathcal{E}}\right\\}(t)$.
From this plot we can see that along the full time series, more than $90\%$ of
the total geostrophic kinetic energy resides at scales ${\lesssim 500~{}}$km
and a large percentage (${\approx 60\%}$) is contained between ${\approx 100}$
and ${\approx 500}$ km, which can be identified as the bulk of the geostrophic
flow.

###  4.3 The filtering spectrum

Figure 6:  Left panel: Cumulative surface geostrophic kinetic energy spectra,
${\mathcal{E}}_{\ell}$, as a function of scale $\ell$, obtained from both the
AVISO and NEMO products in the North and South. Right panel: Filtering spectra
obtained as a derivative with respect to ${k_{\ell}=1/\ell}$ of the
corresponding cumulative energy spectra showed in the left panel (see equation
(12)). Envelopes show inter-quartile range (25th to 75th percentiles) of
temporal variation. Note that the bulk of surface geostrophic kinetic energy
resides at scales within the 100 km to 500 km range.

In Figure 6 we show the filtering spectrum for the global ocean surface
geostrophic kinetic energy as obtained from equation (12) for both AVISO and
NEMO. In the left panel we show the cumulative energy spectra,
${\mathcal{E}}_{\ell}$, as a function of coarse-graining scale, $\ell$. These
results highlight those from Figure 5, revealing that the overwhelming
contribution to geostrophic kinetic resides at length-scales $\ell<500~{}$km.
Based on prior characterization of ocean energetics [2], we assume that this
length scale is dominated by mesoscale features such as geostrophic
turbulence, boundary currents, and fronts. Hence, our analysis provides
further compelling evidence for dominance of the ocean surface kinetic energy
by mesoscale flows.

In the right panel of Figure 6, we show the actual filtering spectrum, which
is the derivative (in scale) of the corresponding plot in the left panel. We
can see a peak centered at ${\ell\approx 300~{}}$km, with the bulk of the
geostrophic kinetic energy residing between scales 100 km and 500 km. From the
analysis shown in Figure 6, we find that ${60\pm 0.4\%}$ of the kinetic energy
in the extra-tropics from the AVISO and NEMO products lies within the
$100~{}$km to $500~{}$km scale-band. Again, Figure 6 provides compelling
evidence that the mesoscales in the extra-tropical latitudes constitute the
most energetic component of the oceanic circulation.

If we focus on each hemisphere separately, we find that the $100~{}$km to
$500~{}$km scale-band comprises ${62\pm 1\%}$ of the North’s kinetic energy
and ${58\pm 1\%}$ of the South’s kinetic energy. Indeed, one can notice slight
differences between the North’s and South’s spectra in Figure 6 (right panel),
where the South has relatively more energy at scales larger than $1000~{}$km.
This bias can be attributed to the large-scale contribution from the Antarctic
Circumpolar Current. On the other hand, the North has slightly more kinetic at
scales within the 100 km to 1000 km range, which can be attributed to boundary
currents (Kuroshio and Gulf Stream). In support of our assertion, consider
Fig. 7, which shows the zonally-averaged kinetic energy for selected length-
scale bands. Scales larger than 1000km (blue plot in Fig. 7) have a dominant
contribution from latitudes [60∘S, 40∘S], which roughly corresponds with the
Antarctic Circumpolar Current. However, these latitudes are no longer dominant
when considering the band of smaller scales: 215 km ${<\ell<}$ 1000 km. These
scales (orange plot in Fig. 7) show a distinct signal at latitudes [30∘N,
40∘N], which roughly aligns with the Gulf Stream and Kuroshio. There is also a
weaker signal at latitudes [40∘S, 35∘S], with roughly aligns with the Agulhas
and the Brazil-Malvinas currents.

Figure 7:  Time- and zonally-averaged kinetic energy computed from AVISO
within selected length-scale bands (see in-set legend) as a function of
latitude. We can see that the Antarctic Circumpolar Current has significant
energy at scales ${>1000~{}}$km, while the North has significant energy within
${\approx 30^{\circ}}$N-$40^{\circ}$N where the Western Boundary Currents are
located. Note that the latitude axis is broken to exclude the band
[$15^{\circ}$S, $15^{\circ}$N].

###  4.4 Reynolds averaging decomposition

In this subsection and the next, we show that the time-mean flow consists of
an entire range of length scales with substantial contributions from the
mesoscale. Figure 8 shows the mean-fluctuation decomposition following the
Reynolds averaging approach. The maps are focused on the Atlantic region to
help reveal details and we show just those obtained from AVISO.

Figure 8:  Decomposition of geostrophic kinetic energy from AVISO for the
Atlantic basin from a time averaging (Reynolds) decomposition. Left panel:
total energy, ${\mathcal{E}}({\bf x},t)$ at a single day. Left middle panel:
9-year time mean, $MKE({\bf x})$. Right middle panel: fluctuating eddy term,
$EKE({\bf x},t)$. Right panel: the cross term required to recover the total
geostrophic energy as defined in equation (18). Note that $MKE({\bf x})$
contains small length-scales and $EKE$ contains a large-scale component of the
flow.

The time mean is obtained by averaging the velocity over the whole time series
available, spanning nine years. From left to right we show the total energy at
a single day, the time mean energy, $MKE({\bf x})$, the fluctuating eddy term,
$EKE({\bf x},t)$, and the cross term, $1/2({\bf u}^{\prime}\cdot\langle{\bf
u}\rangle)$.

Having used a relatively long time series for averaging, the mean energy in
Figure 8 is rather depleted away from major current systems, so that the Gulf
Stream and the Antarctic Circumpolar Current are quite pronounced relative to
the gyre interiors. We appreciate from this figure that the mean flow retains
a substantial contribution from structures with a variety of sizes. In the
same way, the ‘eddy’ (or temporally fluctuating) flow in Figure 8 contains
most of the small scale fluctuations but also a substantial contribution from
large-scale structures. The cross term shown on the right panel of Figure 8
has strong fluctuations around zero, which make its contribution almost (but
not exactly) zero after a spatial-average. The blending of length scales
revealed by these figures reflects the inability of time averaging to
decompose the kinetic energy according to length-scales.

To further investigate the role of the three Reynolds average energy terms,
Figure 9 shows their temporal variability in both hemispheres.

Figure 9:  Top panel: Time-series of total geostrophic kinetic energy,
$\left\\{{\mathcal{E}}(t)\right\\}$ (blue), and the fluctuating component,
$\left\\{EKE(t)\right\\}$ (orange), in the North (solid line) and South
(dashed line) from the AVISO analysis. Vertical grid lines indicate the start
of each quarter-year (01Jan, 01Apr, 01Jul, 01Oct). Bottom panel: Time-series
of the cross term (blue) and kinetic energy of the 9-year mean, $\\{MKE({\bf
x})\\}$ (orange), in the North (solid line) and South (dashed line). $EKE$
constitutes a substantial portion of the total energy and with an almost
indistinguishable temporal variation. Here, we show only 6.5 years of the full
9-year record. Plots shown use a 4-day sampling frequency, but averages are
based on a 1-day sampling of the 9-year record.

In the first row, we see that $EKE$ constitutes a substantial portion of the
total energy ${\mathcal{E}}$ ($80\%$) and their temporal evolution is almost
indistinguishable. Both $EKE$ and ${\mathcal{E}}$ tend to peak during the
spring-summer. The bottom row of Figure 9 shows $MKE$, which is independent of
time, and the cross term, which has a zero average. These two quantities are
much less energetic, with the mean term ${\approx 20\%}$ of the total and the
cross term fluctuates about its zero average without a clear seasonal signal.

###  4.5 Spatio-temporal decomposition

In this section we present results from coarse-graining in both space and time
to reveal the time-scales of various length-scales, including length-scales
present in the 9-year temporal mean. Our analysis demonstrates a new way for
comparing data from satellite analysis (AVISO) and numerical models (NEMO).

The approach consists of measuring the filtering spectrum of a temporally-
smoothed version of the original velocity field. The latter is obtained from a
running window time-average,

$\langle{\bf u}\rangle_{\tau}({\bf
x},t)=\frac{1}{\tau}\int_{t-\tau/2}^{t+\tau/2}{\bf u}({\bf
x},t^{\prime})\,\mathrm{d}t^{\prime},$ (21)

with $\tau$ the size of the time window. Note that a running window time-
average in equation (21) is similar to spatial coarse-graining (equation (1))
since

$\langle\langle F\rangle_{\tau}\rangle_{\tau}\neq\langle F\rangle_{\tau},$
(22)

so that it does not satisfy the Reynolds averaging idempotent property,
${\langle\langle F\rangle\rangle=\langle F\rangle}$.

Combining equation (12) with equation (21) allows us to measure the filtered
energy spectrum of the time smoothed field

${\overline{E}}(k_{\ell},\tau)=\langle\frac{d}{dk_{\ell}}\\{\frac{1}{2}|\langle\overline{{\bf
u}}_{\ell}\rangle_{\tau}|^{2}\\}\rangle=\langle\frac{d}{dk_{\ell}}\\{{\mathcal{E}}_{\ell,\tau}\\}\rangle,$
(23)

where we introduced

${\mathcal{E}}_{\ell,\tau}({\bf x},t)=\frac{1}{2}|\langle\overline{\bf
u}_{\ell}\rangle_{\tau}|^{2},$ (24)

which is the cumulative spectrum of the temporally-smoothed field. As
indicated, ${\mathcal{E}}_{\ell,\tau}({\bf x},t)$ is a function of both the
size of the time-window, $\tau$, and the inverse coarse-graining scale,
$k_{\ell}=\ell^{-1}$.

We show the time-smoothed energy map, ${\mathcal{E}}_{\ell=0,\tau}$, in Figure
10 from AVISO. Here, the two columns compare results from the North and the
South regions, while different rows compare results with different time
windows, $\tau$. From these maps we can see that increasing $\tau$ from one
day to 1093 days reduces the energy down to ${\approx 21\%}$ (${\approx
25\%}$) of the original total energy in the North (South). Hence, averaging
over three years brings the energy down to values comparable to those over the
full nine years obtained in the previous section by the Reynolds averaging
decomposition, where we found that $MKE$ accounts for ${\approx 20\%}$ of the
total energy in the extra-tropics. This result indicates that temporal
averaging converges quickly for the geostropphic kinetic energy, and using
longer time records does not significantly alter the partitioning between the
temporal mean and fluctuating components of the surface geostrophic ocean
flow.

Figure 10:  The surface geostrophic kinetic energy from the temporally coarse-
grained flow, ${\mathcal{E}}_{\ell=0,\tau}$, in the North (left column) and
South (right column) from AVISO. The top row shows the original 1-day averaged
flow. The middle and bottom rows show the kinetic energy from the flow when
averaged with a ${\approx 6~{}}$months time window and a ${\approx 3~{}}$years
time window, respectively, with the kinetic energy decreasing with an
increasing time window. Each panel indicates the $\%$ of kinetic energy
remaining relative to the 1-day top row.

In Figure 11 we show the filtering spectra of the temporally-smoothed flow,
${\overline{E}}(k_{\ell},\tau)$.

Figure 11:  Top panel: Kinetic energy spectra as a function of spatial scale,
${k_{\ell}=1/\ell}$, of the temporally smoothed velocity field,
${\overline{E}}(k_{\ell},\tau)$, in the North from AVISO. Colours indicate the
time-window size $\tau$ used in the smoothing procedure, ranging from 1 day
(pale yellow colour) up to 1000 days (dark green). Bottom panel: The same
spectra but now normalized by ${\overline{E}}(k_{\ell},\tau=0)$, which is the
spectrum of the flow without temporal smoothing. In both panels the dashed
lines represent the spectra of the mean energy obtained via the Reynolds
averaging decomposition, from which we find that $70\%$ of energy in the time-
mean flow in the North is at scales ${<500~{}}$km ($55\%$ in the South, not
shown).

The top panel shows the filtering spectra ${\overline{E}}(k_{\ell},\tau)$ as a
function of $k_{\ell}$ for various values of $\tau$. Bottom panel shows the
same quantities, but normalized by ${{\overline{E}}(k_{\ell},\tau=0)}$, which
is the spectrum of the flow without temporal coarse-graining. From this
analysis, we find that the main effect of the temporal coarse-graining is to
make the energy peak around ${\ell\approx 300~{}}$km less pronounced, with the
consequence of having a spectral energy distribution that is more evenly
distributed across length-scales than in the original flow. Indeed, we can see
from Figure 11 (bottom panel) that time averaging removes energy at all
scales. As we increase the time window $\tau$, ${\overline{E}}(k_{\ell},\tau)$
at scales ${\lesssim 300~{}}$km appears to mostly shift downward in equal
proportion. This behavior is counter to the notion that smaller scales
decorrelate faster in time and are removed more efficiently with time-
averaging.

From Figure 11, we see that as the time window ${\tau\to\infty}$,
${\overline{E}}(k_{\ell},\tau)$ converges to the time-mean spectrum (dashed
plot in Figure 11), which highlights that the 9-year mean flow consists of an
entire continuum of length-scales. The spectrum of the 9-year mean flow shows
a reduction of ${\approx 20}$% of the total at the largest scales of
${>1000~{}}$km (bottom panel in Figure 11). Our analysis of Figure 11 (dashed
line), allows us to infer that roughly $60\%$ of the kinetic energy in the
9-year time-mean is at scales ${<500~{}}$km outside the
${[15^{\circ}\text{S}-15^{\circ}\text{N}]}$ band, underscoring the poor
association between temporal averaging and length-scales.

Figure 12:  Combined spatio-temporal coarse-graining shows space-time 2D
spectra, $-\partial_{\tau}\partial_{k_{\ell}}{\mathcal{E}}_{\ell,\tau}$,
(central panel) from AVISO (blue isolines) and NEMO (red isolines). Right
(top) panel shows energy spectra as a function of time-scale $\tau$ (length-
scale ${k_{\ell}=1/\ell}$). Spatio-temporal spectral isolines from AVISO are
concentric circles with a peak around ${\tau\approx 20~{}}$days and
${\ell\approx 300~{}}$km while those from NEMO (red isolines) are horseshoe-
like with a peak encompassing smaller time-scales. AVISO mis-represents time-
scales ${\lesssim 10~{}}$days over all length-scales.

###  4.6 Spatio-temporal comparison of AVISO and NEMO

We now demonstrate a new method to compare data from satellite analysis
(AVISO) and numerical models (NEMO) by using a spatio-temporal coarse-graining
to identify inconspicuous flow properties or artifacts, and may complement
current efforts to disentangle balanced from unbalanced motions in SSH-derived
flows. Figure 12 presents space-time 2-D spectra,
$-\langle\frac{d}{d\tau}\frac{d}{dk_{\ell}}\\{{\mathcal{E}}_{\ell,\tau}\\}\rangle$,
which decomposes the energy as measured from AVISO and NEMO. In the main panel
of Figure 12 we show the isolevels of space-time spectra from AVISO (blue
lines), superposed onto those from NEMO (red lines). Here, the color intensity
is proportional to the energy as indicated by the colorbar in Figure 12. At
the side of the main panel we show the one dimensional energy spectra as a
function of time, $\tau$, and scale, $\ell$. The most striking difference is
that AVISO isolines (blue) are concentric circles with a peak at ${\tau\approx
20~{}}$days and ${\ell\approx 300~{}}$km, while NEMO isolines (red) resemble
horseshoes with a peak that encompasses shorter time-scale ${\tau\leq
20~{}}$days and a wider range of length-scales. The right panel in Figure 12
plots $-\langle\frac{d}{d\tau}\\{{\mathcal{E}}_{\ell=0,\tau}\\}\rangle$,
underscoring the difference between AVISO and NEMO spectra, which disagree
significantly over time-scales smaller than ${\approx 10}$ days. Note that the
top panel in Figure 12, which compares spectra of spatial scales, shows very
good agreement and, without a temporal decomposition, it fails to detect the
disagreement in time-scales that exist over a wide range of spatial scales,
from ${\ell\approx 100~{}}$km to 1000 km (main panel in Figure 12).

Remember that for the entire analysis in this paper, we are using 1-day
averages of SSH to derive velocity from the NEMO data. While the SSH from
AVISO is also available daily, it is effectively averaged over longer periods
of time to produce gridded SSH maps from along-track altimeter data. We
hypothesize that the difference between isocontours from AVISO and NEMO in
Figure 12 comes from the optimal interpolation used to produce the gridded
AVISO product [62], which is necessary to construct the global maps from
satellite altimeters’ along-track data. To support this hypothesis, in Figure
13, we show the spectra as a function of $\tau$ measured from AVISO and NEMO.
In this plot we have repeated the analysis of the NEMO spectra after passing
the data through a 7-day running time-average (green line), which reproduces
the time average over the satellite orbits. We can see that the green curve
overlaps the AVISO measurement (blue) very closely, supporting our hypothesis.
This is similar to what was done in [15, 8] who were comparing the cascade
from AVISO and model data and determined that AVISO’s spectral fluxes can be
reproduced from model data after filtering the latter in both space and time.

What component of the flow could be yielding the discrepancy between NEMO and
AVISO? The most obvious possibility is unbalanced motion present in the 1-day
mean SSH fields of NEMO that is absent from AVISO due to the effective weekly
averaging required for gridding the satellite measurements. However,
unbalanced motion had been believed to be important mostly over length-scales
$\lesssim 100$ km and time-scales $\lesssim 2$ days (e.g. [68, 69]). If our
conjecture is correct, it would imply that unbalanced motion is present at all
scales and is significant even at scales between $200~{}$km to $1000~{}$km,
requiring averaging over a few days to be removed. Isolating balanced from
unbalanced motions (e.g. [70]) is an active research topic that is beyond the
scope of this work.

Figure 12 shows the importance of performing a combined spatio-temporal
decomposition to access all information in the data. Our method is similar to
frequency-wavenumber analysis performed within Fourier boxes by several recent
studies: [15] were interested in mesoscale-driven intrinsic low-frequency
variability, while [71, 69, 72] were primarily motivated by isolating the
unbalanced motions from SSH-derived velocities. Our Figure 12 is analogous,
for example, to Figure 4 in [15] and to Figure 3 in [72]. However, as we
mentioned in the introduction, the coarse-graining approach gives us access to
the _global_ energy budget and, moreover, frees us from the limitations of
Fourier boxes and the required tapering and detrending. As such, the approach
here complements previous frequency-wavenumber analysis by allowing us to
access much larger length-scales.

A common feature between our Figure 12 and those in previous studies is a
slight elongation of iso-contours along the diagonal from small to large
spatio-temporal scales in the main panel of our Figure 12. Such elongation is
most prominent in Figure 3 of [72], who were probing scales $<100~{}$km and
from roughly $3~{}$hours to $40~{}$days. The diagonal elongation of
isocontours represents a slight tendency for larger length-scales to have
longer time-scales.

However, we emphasize that unlike in [72], such tendency is only _slight_ over
the larger scales we analyze here. In fact, an important take-away from Figure
12 is that all length-scales evolve over a wide range of time-scales,
consistent with plots in Figure 11. Consider, for example, ${\ell\approx
500~{}}$km in the main panel of Figure 12 at different $\tau$ values. We see
that the (red) isoline is almost vertical over ${\tau\approx 5~{}}$days to
${\tau\approx 50~{}}$days, indicating that flow at $500~{}$km has an equal
contribution from all these time-scales. We also see that both AVISO and NEMO
isolines get flatter (stretched horizontally) as $\tau$ increases, such that
at ${\tau\approx 300~{}}$days, there is almost equal energy at all scales
between ${\ell\approx 100~{}}$km and ${\ell\approx 1000~{}}$km.

Figure 13: Evidence that the disagreement between AVISO and NEMO over time-
scales ${\lesssim 10~{}}$days is due to temporal averaging used in generating
the gridded AVISO product. Here, we show temporal spectra from AVISO (blue)
and NEMO (red) in the North (solid lines) and South (dashed lines), which
disagree over ${\tau\lesssim 10~{}}$days as in Figure 12. However, the
temporal spectra from NEMO agree with those from AVISO after applying a 7-day
temporal smoothing to the original NEMO velocities (green). This result
supports our hypothesis that AVISO is missing dynamical information at time-
scales less than 10 days due to temporal smoothing over all length-scales.

##  5 Conclusions

###  5.1 Summary of the main results

In this paper, we have exhibited a suite of analysis insights available from a
coarse-graining approach that is relatively new in physical oceanography. As
part of our coarse-grained analysis of numerical model and satellite analysis
products, we found that the surface geostrophic kinetic energy is dominated by
the mesoscale flow, thus supporting our understanding that it is the most
energetic component of the general circulation [2]. More precisely, our use of
coarse-graining to measure the _global_ spectrum [16], reveals that ${\approx
60\%}$ of the surface geostrophic kinetic energy resides at scales between 100
km and 500 km. We also found that the kinetic energy has a clear seasonality,
peaking in the spring of both hemispheres, thus supporting analysis using
different methods by [65] and [66]. Furthermore, results of the global energy
spectrum from both AVISO satellite analysis and NEMO model are consistent.

By coarse-graining in both space and time, we have shown that every length-
scale evolves over a wide range of time-scales. This result makes temporal
averaging, which is traditionally used to decompose oceanic flow into a mean
and fluctuating components, unable to decompose the flow according to length-
scales. Indeed, we showed that temporal averaging reduces the energy at all
length-scales and not just the mesoscales. We found that the mean flow from a
9-year average has over $50\%$ of its energy residing at length-scales smaller
than $500~{}$km. This result makes us appreciate the significance of
temporally coherent forcing mechanisms acting on the mesoscales, such as
bottom topography and continental boundaries.

An important contribution of this work is to demonstrate how a combined
spatio-temporal coarse-graining analysis in Section 4.6 is able to expose
hidden properties in the data. We did so by showing that the gridded AVISO
product misrepresents the SSH-derived velocity over time-scales less than
${\approx 10~{}}$days at _all length-scales_ , including at ${\approx
1000~{}}$km. The misrepresentation, however, was most severe over scales
${\lesssim 500~{}}$km. We showed evidence supporting our hypothesis that this
deficiency in AVISO is due to the temporal averaging (optimal interpolation)
required for generating a gridded product from altimeter along-track
measurements [62]. This deficiency is unravelled from a combined spatio-
temporal analysis but not by a spatial scale analysis alone. In fact, AVISO
has a spatial resolution [73] comparable to that of the NEMO model we analyzed
here and their spatial spectra agree well at all resolved spatial scales. We
conjecture that the discrepancy between AVISO and NEMO is due to unbalanced
motions in NEMO that is absent from AVISO, including at scales larger than 500
km.

###  5.2 Coarse-graining and the filtering spectrum

The coupling between different length- and time-scales and also between
different geographic regions presents a major difficulty in understanding,
modeling, and predicting oceanic circulation and mixing. Indeed, the oceanic
kinetic energy budget is estimated to suffer from large uncertainties [2]. A
major reason behind these difficulties is a lack of scale-analysis methods
that are appropriate in the _global_ ocean.

In this paper, we have demonstrated the versatility of coarse-graining in
serving as a robust scale-analysis method for the global ocean circulation
that complements existing methods. The approach is very general, allows for
probing the dynamics simultaneously in scale and in space, and is not
restricted by assumptions of homogeneity or isotropy commonly required for
traditional methods such as Fourier or structure-function analysis. Therefore,
coarse-graining offers a way to probe and to quantify the energy budget at
different length-scales globally while maintaining local information about the
heterogeneous oceanic regions.

Here, we have also demonstrated how the recently developed _filtering
spectrum_ [16], which relies on coarse-graining, can be used to quantify the
energy spectrum in the ocean. The method frees us from the limitations of
Fourier boxes, and allows us to extract the spectrum globally. We view this
work as a necessary step toward constructing a scale-aware global Lorenz
Energy Cycle for the ocean circulation.

## Acknowledgments

We thank D. Balwada, M. Jansen, H. Khatri, and S. Rai for valuable discussions
and comments. This research was funded by US NASA grant 80NSSC18K0772. 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 study has been
conducted using E.U. Copernicus Marine Service Information. The product
identifier of the AVISO dataset used in this work is
“SEALEVEL_GLO_PHY_L4_REP_OBSERVATIONS_008_047”, and can be downloaded at
https://marine.copernicus.eu/services-portfolio/access-to-products/. The
product identifier of the NEMO dataset is
“GLOBAL_ANALYSISFORECAST_PHY_CPL_001_015”, and is available at
https://marine.copernicus.eu/services-portfolio/access-to-products/.

##  Appendix A Deforming the kernel around land

As outlined in section 2.1, filtering with a constant kernel while treating
land as zero-velocity water and including land cells (“Fixed Kernel w/ Land”)
in the final tally is guaranteed to conserve $100\%$ of the energy, while
excluding land cells and integrating only over water cells (“Fixed Kernel w/o
Land”) leads to a loss of about 11% of the total kinetic energy at a filter
scale of 2000 km (see Figure 2). This result follows since some of the kinetic
energy ‘smears’ onto the land cells, which are then excluded from the spatial
integrals.

An alternative approach is to deform the kernel around land (“Deforming
Kernel”) so that only water cells are incorporated in the filtering operation.
This approach has the advantage of not needing to treat land as water, yet we
have shown in Figure 2 that this choice still does not conserve $100\%$ of the
energy, sometimes even yielding larger values, albeit still within 1% error.
Here, we explain why a deforming a kernel cannot be expected to yield $100\%$
of the energy, unlike the “Fixed Kernel w/ Land.”

To illustrate how the loss of energy conservation can happen with the
Deforming Kernel method, consider a one-dimensional domain with five equally
spaced points and a simple kernel that has a weight of 2 at the target point,
1 at neighbouring points, and 0 otherwise.

If the domain were periodic then the filtering operation could be represented
as the matrix

$G:=\left[\begin{array}[]{ccccc}\nicefrac{{1}}{{2}}&\nicefrac{{1}}{{4}}&0&0&\nicefrac{{1}}{{4}}\\\
\nicefrac{{1}}{{4}}&\nicefrac{{1}}{{2}}&\nicefrac{{1}}{{4}}&0&0\\\
0&\nicefrac{{1}}{{4}}&\nicefrac{{1}}{{2}}&\nicefrac{{1}}{{4}}&0\\\
0&0&\nicefrac{{1}}{{4}}&\nicefrac{{1}}{{2}}&\nicefrac{{1}}{{4}}\\\
\nicefrac{{1}}{{4}}&0&0&\nicefrac{{1}}{{4}}&\nicefrac{{1}}{{2}}\end{array}\right]$

such that ${\overline{\mathrm{KE}}=G\cdot\mathrm{KE}}$, where $\mathrm{KE}$ is
a column vector. Note that the sum of each row of $G$ is 1, a result of
normalizing the kernel (assuming a grid spacing of 1 for simplicity).

Domain integrating in this scenario is simply left-multiplying by the row
vector $S:=[1,1,1,1,1]$, which is equivalent to taking a column-wise sum.
Since $S\cdot G=S$, $S\cdot\overline{\mathrm{KE}}=S\cdot
G\cdot\mathrm{KE}=S\cdot\mathrm{KE}$, and so the domain integrated kinetic
energy is conserved.

However, if the domain is non-periodic (such as if the edges were ‘land’),
then the resulting filtering kernel according to the Deformed Kernel approach
that excludes anything outside the boundaries would be

$G:=\left[\begin{array}[]{ccccc}\nicefrac{{2}}{{3}}&\nicefrac{{1}}{{3}}&0&0&0\\\
\nicefrac{{1}}{{4}}&\nicefrac{{1}}{{2}}&\nicefrac{{1}}{{4}}&0&0\\\
0&\nicefrac{{1}}{{4}}&\nicefrac{{1}}{{2}}&\nicefrac{{1}}{{4}}&0\\\
0&0&\nicefrac{{1}}{{4}}&\nicefrac{{1}}{{2}}&\nicefrac{{1}}{{4}}\\\
0&0&0&\nicefrac{{1}}{{3}}&\nicefrac{{2}}{{3}}\end{array}\right]$

In this case, $S\cdot
G=[\nicefrac{{11}}{{12}},\nicefrac{{13}}{{12}},1,\nicefrac{{13}}{{12}},\nicefrac{{11}}{{13}}]\neq
S$, and so in general $S\cdot\overline{\mathrm{KE}}\neq S\cdot\mathrm{KE}$.
Moreover, there is no guarantee that $S\cdot\overline{\mathrm{KE}}\leq
S\cdot\mathrm{KE}$, and so it may be that the total filtered kinetic energy
exceeds the total unfiltered kinetic energy.

As observed, in general the error arising from deforming the kernel will be
much smaller than that of treating land as zero-velocity water and only
integrating over true water cells, especially for large filter kernels.
However, again, it is worth recognizing that deforming the kernel does not
guarantee energy conservation. To fully conserves energy and maintain
commutativity with differentiation, we choose the “Fixed Kernel w/ Land”
option, which treats land as zero-velocity water and include land cells in
spatial integrals to compute total energy. Figure 14 shows the spectra
obtained from implementing the three different coarse-graining possibilities.

Figure 14:  Comparison of three land treatment methods: deforming around land
(blue), fixed kernel excluding land (orange), and fixed kernel including land
(black). Vertical labels describe what is plotted in each panel. Left column
corresponds to coarse KE and right column to fine KE. Top row presents
cumulative KE spectra, middle row the $k$-derivative of the cumulative KE, and
bottom row the $\log(k)$-derivative of cumulative KE. Note that the y-axis is
common within each row. Data corresponds to North of the Tropics. Envelope
shows inter-quartile range (25th to 75th percentiles) of temporal variation.

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