The Contribution of Striations to the Eddy Energy Budget
and Mixing: Diagnostic Frameworks and Results in a
Quasigeostrophic Barotropic System with Mean Flow
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Chen, Ru, and Glenn R. Flierl. “The Contribution of Striations to
the Eddy Energy Budget and Mixing: Diagnostic Frameworks
and Results in a Quasigeostrophic Barotropic System with Mean
Flow.” Journal of Physical Oceanography 45, no. 8 (August
2015): 2095–2113. © 2015 American Meteorological Society
As Published
http://dx.doi.org/10.1175/jpo-d-14-0199.1
Publisher
American Meteorological Society
Version
Final published version
Accessed
Thu May 26 15:12:22 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/101391
Terms of Use
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Detailed Terms
AUGUST 2015
CHEN AND FLIERL
2095
The Contribution of Striations to the Eddy Energy Budget and Mixing:
Diagnostic Frameworks and Results in a Quasigeostrophic Barotropic
System with Mean Flow
RU CHEN
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California
GLENN R. FLIERL
Massachusetts Institute of Technology, Cambridge, Massachusetts
(Manuscript received 29 September 2014, in final form 11 April 2015)
ABSTRACT
Low-frequency oceanic motions have banded structures termed ‘‘striations.’’ Since these striations embedded
in large-scale gyre flows can have large amplitudes, the authors investigated the effect of mean flow on their
directions as well as their contribution to energetics and mixing using a b-plane, barotropic, quasigeostrophic
ocean model. In spite of the model simplicity, striations are always found to exist regardless of the imposed
barotropic mean flow. However, their properties are sensitive to the mean flow. Rhines jets move with the mean
flow and are not necessarily striations. If the meridional component of the mean flow is large, Rhines jets
become high-frequency motions; low-frequency striations still exist, but they are nonzonal, have small magnitudes, and contribute little to energetics and mixing. Otherwise, striations are zonal, dominated by Rhines jets,
and contribute significantly to energetics and mixing. This study extends the theory of b-plane, barotropic
turbulence, driven by white noise forcing at small scales, to include the effect of a constant mean flow. Theories
developed in this study, based upon the Galilean invariance property, illustrate that the barotropic mean flow
has no effect on total mixing rates, but does affect the energy cascades in the frequency domain. Diagnostic
frameworks developed here can be useful to quantify the striations’ contribution to energetics and mixing in the
ocean and more realistic models. A novel diagnostic formula is applied to estimating eddy diffusivities.
1. Introduction
Geostrophic eddies, which are ubiquitous in the
global ocean, dominate the oceanic kinetic energy reservoir (Ferrari and Wunsch 2009). As a key component
of the oceanic circulation, they stir and mix tracers (e.g.,
heat and potential vorticity) and therefore greatly influence the oceanic circulation and climate variability.
Given their importance, much effort has been made
recently toward estimating and interpreting eddy energy
budgets (e.g., Ferrari and Wunsch 2009; Xu et al. 2011;
Chen et al. 2014a) and eddy mixing rates (e.g., Gille et al.
2007; Griesel et al. 2010; Rypina et al. 2012; Klocker
et al. 2012a,b; Abernathey et al. 2013; Chen et al. 2014b;
Chen et al. 2015b) in the global ocean.
Corresponding author address: Ru Chen, Scripps Institution of
Oceanography, University of California, San Diego, 9500 Gilman
Drive, La Jolla, CA 92093-0230.
E-mail: ruchen@alum.mit.edu
DOI: 10.1175/JPO-D-14-0199.1
Ó 2015 American Meteorological Society
The temporal average (i.e., low-frequency component) of eddy motions has banded structures in most of
the ocean that are often termed ‘‘striations’’ (e.g., Cox
1987; Galperin et al. 2004; Maximenko et al. 2005;
Nakano and Hasumi 2005; Richards et al. 2006;
Maximenko et al. 2008; Chen 2013). Spectral analysis of
the output from an eddying global model suggests that
these striations account for more than 10% of the zonal
velocity variability in the upper 1000 m of the central
North Pacific and east North Pacific regions (Chen et al.
2015a). This significant percentage suggests that striations are an important component of the oceanic eddy
field and that their contribution to eddy energetics and
tracer mixing is probably nonnegligible. Yet, while their
origin has received much attention, little effort has
been made toward quantifying these contributions
(e.g., Maximenko et al. 2008; Schlax and Chelton 2008;
Afanasyev et al. 2012; Chen et al. 2015a).
In addition, striations are also embedded in largescale oceanic background flows, such as subtropical and
2096
JOURNAL OF PHYSICAL OCEANOGRAPHY
subpolar gyres. How these background flows influence
the mixing properties and energetics of striations is not
known. Exploring these questions will help us further
understand the consequences of eddies in the ocean and
their impact on the climate system. The focus of this
study is to assess the contribution of striations to eddy
energy budgets and mixing in a barotropic quasigeostrophic (QG) ocean model on a b plane with a background mean flow imposed.
Use of the barotropic QG model, which is one of the
simplest systems that produces banded structures, allows us to focus on both the elementary dynamics and
consequences of striations. In this model, Rhines jets
can arise as a result of the dramatic slowing of the inverse cascade due to the b effect (e.g., Rhines 1975;
Sukoriansky et al. 2007). Recent studies have demonstrated that this model, though highly idealized, is a
useful tool for the theoretical exploration of banded
structures and many related frontiers (e.g., Danilov and
Gurarie 2004; Danilov and Gryanik 2004; Farrell and
Ioannou 2007; Srinivasan and Young 2012). Indeed, this
study shows that striations in this simple system have
rich behaviors and that their effects on mixing and energy budgets are sensitive to the imposed barotropic
mean flow. The simplicity of the model allows ease of
interpretation of these behaviors. We also develop theories relevant to the ocean about the effect of barotropic
mean flows on eddy mixing and energy cascades in the
frequency domain.
This study presents a computational/conceptual
framework to diagnose the contributions that striations
have on mixing and energetics. The computational efficiency and ease of interpretation of the barotropic
model make it a good tool to test the validity of the
computational/conceptual framework. It is important
to reveal the eddy dynamics and thereby allow for the
improvement of eddy parameterization schemes. As
summarized by Abernathey et al. (2013), many
methods have been used to diagnose eddy mixing rates,
such as flux-gradient diffusivities, Nakamura effective
diffusivities, and Lagrangian diffusivities. These
methods essentially estimate mixing rate contributions
from all of the eddies. However, partitioning of the
contributions to mixing from striations and other
eddies has rarely been explored. We present a new
diagnostic framework for this partitioning, based on
special tracers with eddy velocity as their initial condition. Thompson and Richards (2011) presents a diagnostic framework to assess the energetics of jets in
the Southern Ocean by assuming there is a spectral gap
between low-frequency and high-frequency eddies.
This study presents a framework to diagnose the energy
budget of (time dependent) striations without the need
VOLUME 45
for the spectral gap assumption. Our diagnostic
methods can be easily extended to investigating realistic striations.
In summary, this paper 1) develops a diagnostic
framework to evaluate the contributions of striations to
mixing and the energy budget and 2) examines the elementary dynamics of striations and the effect of mean
flow on them using a barotropic QG model on a b plane.
Section 2 describes the model and experimental setup.
Section 3 characterizes and interprets striations in the
model from a spectral perspective. Sections 4 and 5
present the key results: the effect of mean flow on the
contribution of striations to mixing and energy budgets.
Corresponding diagnostic frameworks are described in
detail in these two sections. Section 6 provides the
summary and discusses applications of our work to more
realistic fluid systems.
2. Numerical model and experimental setup
We use the barotropic QG equation
›
›
›
1U 1V
q~ 1 J(c, q~ 1 by) 5 F (x, y, t) 2 rq~ ,
›t
›x
›y
(1)
where (U, V) is the imposed mean flow, c denotes the
eddy streamfunction, q~ is the relative eddy vorticity =2 c,
J is the Jacobian operator, b is the meridional gradient
of the Coriolis parameter, F (x, y, t) is the external
forcing, and r is the friction coefficient. The equation
is solved using the pseudospectral method in a doubly periodic domain (Arbic 2000). The domain is
discretized at 128 3 128 grid points and its size is
3000 km 3 3000 km.
As described in section 1, one of our main goals is to
investigate how the background flow (U, V) influences
the striation properties. Though the background flow
varies spatially in the ocean, here we only consider the
case when (U, V) are constants. The reasons are as
follows. First, in some oceanic regions (e.g., the subtropical gyre away from topography and the oceanic
boundary), the background flow varies slowly in space.
Thus, it can be assumed to be constant, if we examine
striations in a small oceanic patch. Second, this choice
reduces the model [Eq. (1)] to the simplest system and
thus allows us to focus on the basic physics about the
mean flow effect on striations. Third, the constant mean
flow assumption has been widely used in previous
studies about striations, eddies, energetics, and mixing,
leading to results/conclusions relevant to the ocean, at
least to some extent (e.g., Arbic 2000; Killworth 1997;
Flierl and Pedlosky 2007; Smith 2007; Thompson 2010;
AUGUST 2015
Ferrari and Nikurashin 2010; Hughes et al. 2010;
Venaille et al. 2011; Chen et al. 2015a).
Our study includes three main experiments: Exp1 has
no mean flow imposed and represents the original case
presented in Rhines (1975), Exp2 has a southward mean
flow imposed, and Exp3 has eastward mean flow imposed. The choice of southward and eastward mean flow
in Exp2 and Exp3 is based on the fact that both quasizonal and quasi-meridional background flow are prevalent in the ocean (e.g., in the subtropical gyre). The
magnitude of time–mean barotropic velocity in the eddying ocean model, the ECCO2 state estimate,1 varies
spatially, ranging from O(0.01) m s21 in quiescent regions to O(0.1) m s21 in coastal regions and in the
Antarctic Circumpolar Current. Within the relevant
parameter range, we arbitrarily choose the magnitude of
the imposed mean flow in Exp2 and Exp3 to be
0.03 m s21. We also carried out sensitivity experiments
with a range of mean flow magnitudes, to test our
physical interpretations.
The external forcing F (x, y, t) is white noise in time
and has a narrow banded wavenumber spectrum. The
external forcing at time step nDt is
Fn 5
~ 21 fexp[20:01(jKj2 2 K 2 )2 1 iu (k, l)]g
AF
F
n
max(F 21 fexp[20:01(jKj2 2 KF2 )2 1 iun (k, l)]g)
,
(2)
~ is the forcing amplitude, F 21 is the inverse
where A
Fourier transform operator, K 5 ki 1 lj is the wavenumber vector, KF 5 7 3 1023 cycle km21, and un(k, l)
are uniformly distributed random numbers between
0 and 2p. In the wind-driven barotropic model, F (x, y, t)
is essentially equal to curl[t/(r0H)], where t is the wind
stress vector (Pedlosky 1987). Typical values of the
ocean depth H, density r0, and the instantaneous curl(t)
are 1000 m, 1000 kg m23, and 2 3 1026 Pa m21, respec~ to be 2 3 10212 s22.
tively. Therefore, we choose A
The friction coefficient r is 2.9 3 10210 s21, leading to
nonlinear motions with eddy velocity magnitudes of
0.03 m s21. The value 0.03 m s21 falls within the range of
the zonally averaged magnitude of the barotropic eddy
velocity in the ECCO2 state estimate (0–0.05 m s21). We
choose b to be 2 3 10211 m21 s21, which is representative of midlatitude values. All of the experiments start
from rest, and our analysis uses the 200-yr output from
the statistical equilibrium state.
1
2097
CHEN AND FLIERL
ECCO2 refers to ‘‘Estimating the Circulation and Climate of
the Ocean, phase II: High resolution global-ocean and sea-ice data
synthesis’’ (Menemenlis et al. 2008; Chen 2013).
3. Rhines jets versus striations
a. Descriptions of Rhines jets
In a barotropic QG model on a b plane with no mean
flow imposed, elongated structures exist in the instantaneous eddy snapshots (Rhines 1975). We define
such banded structures in the eddy snapshots as ‘‘Rhines
jets.’’ We chose not to use the terminology ‘‘zonal jets,’’
because zonal jets in literature can refer to either bands
in the flow snapshot (e.g., Panetta 1993) or bands in the
temporally averaged flow field (e.g., Maximenko et al.
2005; Richards et al. 2006). Rhines jets show up in the
instantaneous eddy snapshots in all of our experiments
and they are all elongated in roughly the zonal direction
(Fig. 1). However, the Rhines jets in Exp2 are advected
southward by the imposed southward mean flow (Fig. 2).
The averaged width of a single jet, which is either
eastward or westward, can be determined by counting
the number of jets in the spatial domain (Thompson
2010). We examined the zonal averages of zonal eddy
velocities and found that the number of jets is 12 in all of
our experiments. Therefore, the width of a single jet is
250 km, which is roughly consistent with the Rhines
scale (230 km). In this study, our definition of the Rhines
scale, lRhines, follows Thompson (2010):
qffiffiffiffiffiffiffiffiffiffi
(3)
lRhines 5 2p Ve /b ,
where Ve is the temporal and spatial average of eddy
velocity magnitudes. In our experiments, the dominant
meridional wavenumber of Rhines jets, 2p/(2lRhines), is
also consistent with the meridional wavenumber centroid of the eddy kinetic energy (EKE) spectra.
Rhines jets contain 86% of total EKE in our experiments. This percentage estimation is based on the approach from Chen et al. (2015a), which is briefly
described next. The normalized wavenumber spectra of
EKE, SN
EKE , is defined as
SN
EKE 5
(k2 1 l2 )Sc (k, l)
[(k2 1 l2 )Sc (k, l)]max
.
(4)
Here k and l are zonal and meridional wavenumbers,
Sc(k, l) is the wavenumber spectrum of eddy streamfunction c, and [ ]max denotes the maximum value over
the entire available wavenumber space (k, l). We identify an optimum ellipse, which is the smallest one for
which all the wavenumbers with SN
EKE larger than 0.2 are
inside. As shown in Fig. 3, the optimum ellipse is very
narrow and has its major axis aligning with k 5 0, which
is consistent with the fact that zonal elongated structures
dominate the flow field. The percentage of EKE associated with Rhines jets is
2098
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 45
FIG. 1. Snapshots of the eddy streamfunction c (103 m2 s21) in the doubly periodic domain from (a) Exp1, (b) Exp2, and (c) Exp3. Black
arrows denote the direction of the imposed mean flow in each experiment. The x axis denotes zonal distance and the y axis denotes
meridional distance. Rhines jets are robust features in the eddy snapshots of all three experiments.
ðð
Dkl
percentR 5 ðð
V
SN
EKE (k, l, v) dk dl
,
(5)
SN
EKE (k, l, v) dk dl
where Dkl denotes the wavenumber space inside the
optimum ellipse and V denotes the entire available
wavenumber space. More details of this approach are
available in Chen et al. (2015a).
b. Effect of mean flow on Rhines jets
The formation mechanism of Rhines jets has not yet
been fully resolved. Rhines (1975) examined the barotropic decaying turbulence and proposed that Rhines
jets can be formed from the arrest of inverse cascade
on a b plane. Sukoriansky et al. (2007) found that, in the
forced barotropic flow on a b plane, the inverse cascade
cannot be arrested by b, but can be ended by large-scale
dissipation. Other proposed mechanisms include, but
are not limited to, the mixing of potential vorticity by
eddies (Dritschel and McIntyre 2008), zonostrophic instability (Srinivasan and Young 2012), and the shearing
of the eddies by the local shear (Bakas and Ioannou
2013). Our discussion next focuses on the effect of mean
flow on jet characteristics, rather than mechanisms for
the jet origin.
The insensitivity of the existence, width, and zonal
orientation of the Rhines jets to the imposed mean flow
FIG. 2. Hovmöller (time–meridional distance) plots of the zonally averaged eddy streamfunction c (103 m2 s21) in each experiment. This
figure uses the zonally averaged eddy streamfunction as an approximate indicator of Rhines jets. Rhines jets are advected southward by
the imposed southward mean flow in Exp2. Meridional drift of the jets does not appear over the 500-day time scale in Exp1 and Exp3
because of the absence of a meridional component of the imposed mean flow.
AUGUST 2015
2099
CHEN AND FLIERL
FIG. 3. Normalized wavenumber spectrum of EKE (SN
EKE ) on a logarithmic scale in (a) Exp1, (b) Exp2, and (c) Exp3. White contours
denote the optimum ellipse, which denotes the smallest ellipse, which contains all the wavenumbers with SN
EKE larger than 0.2.
(Fig. 1) is due to the Galilean invariance of the barotropic model. Ignoring forcing and friction in Eq. (1), we
obtain
›
›
›
›c ›~
q ›c ›~
q
›c
1U 1V
2
1 b 5 0.
q~ 1
›t
›x
›y
›x ›y ›y ›x
›x
(6)
Equation (6) reduces to the original Rhines jet model
from Rhines (1975), in a moving coordinate system
0
x 5 x 2 Ut,
0
y 5 y 2 Vt
and
c. Effect of mean flow on frequency–wavenumber
spectra
To prepare for the discussion of the dynamics and
consequences of the striations, we first examine the
spectral characteristics of eddies. Define SA (k, l, v) as
the frequency–wavenumber spectra for any arbitrary
variable A(x, y, t),
^ l, v)j2 i .
SA (k, l, v) 5 hjA(k,
0
t 5 t.
In that system, freely evolving eddy characteristics
would be independent of the imposed mean flow: the
dominant eddies become ‘‘zonal’’ when the energy
cascades to low ‘‘wavenumbers’’ and low ‘‘frequencies’’
and Rhines jets are generated and align with the x0 (i.e.,
zonal) direction (note: here the wavenumber and frequency are defined in the moving coordinates).
Even though the imposed white noise forcing is not
Galilean invariant, the above mechanism is still a good
approximation, since the forcing is of small spatial scale
and does not end up affecting the basic statistics of
larger-scale eddies. Basic eddy statistics differ little in
our experiments, indicating Galilean invariance holds
reasonably here. The difference of the domain-averaged
EKE among the three experiments is only 0.3% of the
EKE magnitude. The difference of the domainaveraged vorticity magnitude among the three experiments is only 0.5% of the vorticity magnitude. The
meridional wavenumber centroid of the EKE wavenumber spectra ranges from 0.0020 to 0.0021 cycle km21
in our experiments, and the corresponding zonal wavenumber centroid ranges from 3.0 3 1024 to 3.3 3
1024 cycle km21.
(7)
Here hi denotes the ensemble average and ^ denotes the
Fourier transform over the three-dimensional space (x,
y, t). Define fA(x, y, t) as the solution in Exp1, where no
mean flow is imposed. Because of the Galilean invariance, the function
fB (x, y, t) 5 fA (x 2 Ut, y 2 Vt, t)
(8)
describes eddy statistics for cases with an imposed mean flow (e.g., Exp2 and Exp3). Equation (8)
leads to
^ (k, l, v)j2 i
Sf (k, l, v) 5 hjf
B
B
^ (k, l, v 2 kU 2 lV)j2 i
5 hjf
A
5 Sf (k, l, v 2 kU 2 lV) ,
(9)
A
which is confirmed by the comparison of spectra from
Exp1, Exp2, and Exp3 (not shown). Therefore, the
spectra in a case with mean flow are the frequencyshifted versions of those in the case with no mean
flow. This implies that the wavenumber spectra of
eddies do not depend on the imposed spatially uniform mean flow, as
2100
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 45
FIG. 4. Snapshots of the part of the eddy streamfunction c with frequencies lower than the separation frequency VS (103 m2 s21) in the
doubly periodic domain. As described in the main text, VS, which is determined using the method from Chen et al. (2015a), is 0.14, 0.4, and
0.14 cycle yr21 in Exp1, Exp2, and Exp3, respectively. Black arrows denote the direction of imposed mean flow in each experiment.
ð‘
2‘
Sf (k, l, v) dv 5
B
5
ð‘
2‘
ð‘
2‘
Sf (k, l, v 2 kU 2 lV) dv
A
Sf (k, l, v) dv .
A
(10)
d. Effect of mean flow on striation amplitudes
As summarized in section 1, striations in literature
often refer to banded structures in the temporal average
of oceanic fields. Note that the temporal average retains
only low-frequency motions. Following Chen et al.
(2015a), we define striations as banded structures in lowfrequency eddies to be consistent with previous studies.
One natural question is how to determine the separation
frequency between high- and low-frequency motions
(VS). To quantify the amplitude of striations in the
North Pacific, Chen et al. (2015a) proposed an ‘‘optimum
ellipse’’ approach to separate low-frequency eddies from
high-frequency eddies. First, we define the normalized
wavenumber spectrum of the eddy streamfunction c at
frequency v as
SN
c (k, l, v) 5
Sc (k, l, v)
[Sc (k, l, v)]max
,
(11)
where Sc(k, l, v) is the frequency–wavenumber spectrum of c and [Sc(k, l, v)]max is the maximum value of
Sc(k, l, v) at frequency v. Second, for each frequency,
we can determine an optimum ellipse, which denotes the
smallest ellipse containing all the wavenumbers with
SN
c (k, l, v) larger than 0.2. Finally, we diagnose the ratio
between the major and minor axes of this ellipse. If the
ratio between the major and minor axes is larger than
three, the ellipse is considered narrow, the wavenumber
spectrum at this frequency is very anisotropic, and
banded structures dominate. We define VS as the highest
frequency where all the optimum ellipses at frequencies
lower than VS are narrow. Based on this approach, VS is
0.14, 0.4, and 0.14 cycle yr21 in Exp1, Exp2, and Exp3,
respectively.
Striations exist in all three of our experiments, and
they are not necessarily dominated by Rhines jets
(Fig. 4). In Exp2, where the imposed mean flow is in the
meridional direction, the striations are tilted southwestward with amplitudes that are much weaker than
that of Rhines jets (Fig. 1). In Exp1 and Exp3, the direction, width, and magnitude of the striations are
roughly the same as Rhines jets (Fig. 1).
Rhines jets contain most of the eddy energy in this
turbulent system (section 3a). Whether Rhines jets can be
classified as striations depends upon the Rhines jet frequency, vRhines. The spatial structure of Rhines jets, and
thus their dominant wavenumbers, are insensitive to the
imposed mean flow (sections 3a and 3b). In our experiments, the dominant zonal wavenumber of Rhines jet is
approximately zero and the dominant meridional wavenumber of Rhines jets is 2p/(2lRhines), where lRhines is the
Rhines scale for a single eastward or westward jet, defined
in Eq. (3). In the case where no mean flow was imposed
(Exp1), vRhines is roughly zero because of the inverse
cascade (Rhines 1975). Based on Eq. (9) for cases with
mean flow (e.g., Exp2 and Exp3), the Rhines frequency is
vRhines 5 U * 0 1 V * 2p/(2lRhines ) 5 Vp/lRhines .
(12)
If the separation frequency between striations and highfrequency eddies (VS) satisfies
AUGUST 2015
2101
CHEN AND FLIERL
FIG. 5. (a) The percentage of EKE in striations in Exp2 (blue dots), as a function of the
separation frequency VS. The gray line denotes the frequency of Rhines jets vRhines in Exp2. To
calculate vRhines
, defined in Eq. (12), we follow the Rhines scale definition in Eq. (3), that is,
pffiffiffiffiffiffiffiffiffiffi
lRhines 5 2p Ve /b. We choose Ve, the eddy velocity magnitude, to be the spatially and temporally averaged eddy velocity magnitude in Exp2 (0.27 m s21). (b) The percentage of EKE in
striations (blue dots) in a set of sensitivity experiments, whose setup are the same as Exp2 but
with different magnitudes of the imposed southward mean flow. The horizontal axis denotes
the imposed mean flow magnitude in each experiment. For simplicity purpose, we chose VS to
be a fixed value: 0.5 cycle yr21. The ratio between vRhines and VS (green dots) increases as the
mean flow magnitude increases. The gray lines indicates the mean flow magnitude that corresponds to VS 5 vRhines.
VS . vRhines 5 Vp/lRhines ,
(13)
Rhines jets are low-frequency motions and striations
will be dominated by them and will have large amplitudes. For example, in Exp3, V is zero so that vRhines is
zero and striations have large amplitudes. On the other
hand, if Eq. (13) does not hold (e.g., Exp2), striations are
not composed of Rhines jets and they have lower amplitudes and are less energetic.
Figure 5 further illustrates the dependence of striation
amplitudes on VS/vRhines. The percentage of total EKE
that is contained in striations is
ðV ð‘ ð‘
S
SEKE (k, l, v) dk dl dv
2V
2‘ 2‘
percentS 5 ð ‘ S ð ‘ ð ‘
2‘ 2‘ 2‘
,
(14)
SEKE (k, l, v) dk dl dv
where SEKE (k, l, v) is the EKE spectra, that is,
SEKE (k, l, v) 5 (k2 1 l2 )Sc (k, l, v) .
(15)
As shown in Fig. 5, as VS increases or as V decreases,
VS/vRhines increases and a sharp increase of percentS
occurs at
vRhines 5 Vp/lRhines 5 VS .
(16)
Figure 5a shows the case when we keep V fixed but
vary VS. When VS is larger than vRhines, striations are
composed of Rhines jets and percentS is large (Fig. 5a).
Analogously, if we keep VS fixed but vary vRhines by
changing V, a sharp transition of percentS also occurs
when vRhines 5 VS (Fig. 5b).
e. Effect of mean flow on striation directions
Figure 6 shows Sc(k, l, v) in Exp1. In barotropic
turbulence on a b plane, the energy cascades to large
scales and motions with spatial scales larger than a
critical scale (roughly the Rhines scale) behave like
waves (Rhines 1975). Therefore, it is not surprising
that large-amplitude portions of the spectra (red
surface in Fig. 6) roughly lie on the surface of the
Rossby wave dispersion relation (black mesh surface). In the frequency–wavenumber space, 90% of
EKE in our experiments occurs in the Rossby wave
band, defined as
bk
, .
(17)
jv 2 VRossby j 5 v 2 Uk 1 Vl 2 2
2
k 1l 2102
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 45
FIG. 6. The normalized three-dimensional frequency–wavenumber spectrum of the eddy streamfunction in Exp1
viewed from two different angles. Slices show the three-dimensional spectra through the plane v 5 0. Black mesh
surfaces are those of the dispersion relation of the barotropic Rossby waves. The points with the normalized power
larger than 1023.8 are indicated in red. Color bar is on a logarithmic scale.
Here is 0.2 cycle yr21 and VRossby is the Rossby wave
frequency. These eddies can be viewed as a set of weakly
interacting Rossby waves.
Striations are dominated by a group of quasistationary Rossby waves. Because most of the energy
lies near the Rossby wave dispersion relation, and striations are defined as low-frequency motions, wavenumbers with large amplitudes in the striation spectra
lie near the zero Rossby wave frequency curve, defined
as (Fig. 7)
VRossby 5 Uk 1 Vl 2
bk
5 0.
k2 1 l2
(18)
In the case with V 6¼ 0 (Exp2), the zero frequency
Rossby wave curve is shifted from k 5 0 and striations
are nonzonal. Conversely, in the case with V 5 0 (Exp1
and Exp3), the zero frequency Rossby wave curve occurs at k 5 0 and striations are zonal.
c,
V (x, y, t)
ð‘ ð‘ ð‘
^ (k, l, v)ei(kx1ly2vt) H(v, V) dk dl dv,
c
5
2‘ 2‘ 2‘
(19)
where H(v, V) is an arbitrary low-pass filter operator,
such as the boxcar filter:
H(v, V) 5
sin(pv/V)
.
pv/V
(20)
The hKE,
V i budget equation in our doubly periodic
barotropic model is
›
,
2
,
,
,
hKE,
V i 5 hcV [J(c, = c)]V i 1 h2cV [F (x, y, t)]V i
›t
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
PV
FV
1 h2rKE,
Vi ,
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
DV
4. Effect of mean flow on energy cascades and
striations’ energy budget
a. Diagnostic approach
The domain-averaged kinetic energy density from
low-frequency motions is
hKE,
Vi 5
1
,
$c
)
($c,
V
V ,
2
where hi is the spatial average and ,
V denotes the part of
the motion at frequencies lower than V. For example, c,V
is defined as
(21)
where PV denotes the energy flux to the KE,V reservoir
due to eddy–eddy interaction, F V denotes the energy
input into the KE,V reservoir from external forcing, and
DV denotes the dissipation of KE,V due to friction. The
term on the left-hand side represents the temporal rate
of change of KE,V. The mean flow (U, V) does not explicitly enter into Eq. (21) because of the doubly periodic boundary condition. The derivation of Eq. (21) is
provided in appendix A.
The low-frequency motions are dominated by banded
structures (Fig. 4). Assuming all the low-frequency motions
AUGUST 2015
CHEN AND FLIERL
2103
FIG. 7. Normalized wavenumber spectrum of the part of the eddy streamfunction with frequency lower than VS on the logarithmic scale
in (a) Exp1, (b) Exp2, and (c) Exp3. The choice of VS is the same as Fig. 4. Large values of the spectra lie on the zero Rossby wave
frequency curve, denoted by the black line.
are striations, and replacing V in Eq. (21) with VS, we
obtain the energy budget equation for the striations:
›
hKEV, i 5 hPV i 1 hF V i 1 hDV i .
S
S
S
S
›t
(22)
Note that any type of low-pass filter H(v, V) can be used
when diagnosing Eqs. (21) and (22).
b. Comparison with previous diagnostic approaches
Our diagnostic framework is distinct from others
presented previously. For example, by assuming a
spectral gap exists between low-frequency and highfrequency motions, Thompson and Richards (2011)
developed an approximate energy budget equation for
studying the low-frequency jets in the Southern Ocean.
However, a spectral gap assumption is not used in our
diagnostic framework for the energetics for the striations [i.e., Eq. (22)].
hPV (x, y, t)i 5
ð‘ ð‘ ð‘
2‘ 2‘ 2‘
›
hKE,
V i 5 hPV i 1 hF V i 1 hDV i ’ 0.
›t
hPV (x, y, t)iArbic
ð‘ ð‘
ð
^
^ (k, l, v)J(2k,
c
2l, 2v) dk dl dv .
5
2‘
(25)
Therefore, hPV (x, y, t)i in their paper matches ours
when one uses a rectangular filter, while ours is more
(23)
Appendix A illustrates that the time mean of the nonlinear energy fluxes (hPV i), which represent energy
cascades in the frequency domain, can be expressed as
an integral over the frequency and wavenumber domain:
^
^ (k, l, v)[H(v, V)]2 J(2k,
c
2l, 2v) dk dl dv ,
where J^ is the Fourier transform of J(c, =2 c). The
barotropic equivalent of the nonlinear energy fluxes in
Arbic et al. (2012) is
jvj,V 2‘
Nonlinear energy cascades in the frequency domain
have recently been discussed in a two-layer model context (Arbic et al. 2012). However, their energy cascade
[Eq. (21) in Arbic et al. (2012)] does not include the local
time rate of change term (›/›thKE,Vi), which is needed to
study the temporal (e.g., seasonal and interannual)
variability of striations. The one-layer version of their
Eq. (21) can be shown to be essentially the time average,
over the entire time series, of our Eq. (21) with a rectangular filter. Averaging Eq. (21) over the entire time
series leads to
(24)
general in that it allows for the use of any type of
filter H.
c. Results
1) EFFECT OF MEAN FLOW ON ENERGY CASCADES
IN THE FREQUENCY DOMAIN
Although the energy cascade in the wavenumber
domain is not sensitive to the mean flow in the barotropic model, the energy cascade in the frequency
domain is [section 5.8.2 in Chen (2013)]. Appendix B
2104
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 45
FIG. 8. The normalized integral energy budget in the frequency domain [Eq. (23)] in each experiment. Red, green, and black curves
denote hPV i, hF V i, and hDV i, respectively. Black boxes are used to highlight the magnitude of each term at the frequency VS. The shaded
area in light red is within the frequency range of striations [0, VS]. Blue dashed lines mark V 5 VS, and VS is the same as that in Fig. 4. In
(b), the magenta dashed line denotes V 5 vRhines in Exp2 and the red dashed line denotes V 5 VM, where hPV i reaches the maximum. The
low-pass filter H(v, V) we chose in this diagnosis is the Butterworth filter.
illustrates that, using the Galilean invariance and
the convolution theorem, nonlinear energy fluxes in
the case with any arbitrary mean flow imposed,
hPV (x, y, t)i 5
ð‘ ð‘ ð‘
2‘ 2‘ 2‘
[hPV (x, y, t)i], can be calculated from the eddy
streamfunction for the case with no mean flow imposed [fA(x, y, t)],
c (k, l, v)Jc (2k, 2l, 2v)H 2 (v 1 kU 1 lV, V) dk dl dv ,
f
A
A
where JA 5 J(fA , =2 fA ). Obviously, because the lowpass filter H depends on the mean flow, so does
hPV (x, y, t)i.
The sensitivity of hPV i to the imposed mean flow, predicted by Eq. (26), can be interpreted as follows. In the
b-plane barotropic QG system with no mean flow imposed,
energy cascades to low frequencies and low wavenumbers;
Rhines jets are low-frequency motions that extract energy
from high-frequency motions (Rhines 1975). Using the
Galilean invariance of the barotropic system, we know that
Rhines jets extract energy from other eddy motions regardless of the imposed mean flow. However, because
the frequency of Rhines jets [Eq. (12)] is sensitive to the
imposed meridional flow, the structure of hPV i is also
sensitive to the imposed meridional flow, as confirmed
through numerical analysis and presented in Fig. 8.
In Fig. 8, a positive (negative) slope of hPV i at the
frequency V means that eddies at that frequency gain
(lose) energy through eddy–eddy interaction. In Exp1
and Exp3, the slope of hPV i is negative at most frequencies higher than VS, which denotes the separation
frequency between striations and high-frequency eddies.
Therefore, energy primarily flows to motions with frequencies lower than VS. However, in Exp2 over the frequency range [VS, VM], hPV i decreases from its
maximum value at VM to roughly zero at VS (Fig. 8).
(26)
Therefore, in this case energy primarily flows to the frequency range [VS, VM]. Note that the Rhines jet frequency in Exp1 and Exp3 is roughly zero, but it lies within
[VS, VM] in Exp2. Consequently, the energy always
moves into the frequency range where Rhines jets reside.
2) ENERGY BUDGET OF STRIATIONS
The energy budget of the striations [Eq. (22)] is also
found to be sensitive to the imposed mean flow. The time
average of the terms in Eq. (22) is highlighted by the
black boxes drawn in Fig. 8. In Exp2, where the Rhines
jets are not classified as striations, external forcing and
eddy–eddy interaction are equally important energy
sources for the striations. However, the amount of energy
extracted by the striations from other eddies is small. In
Exp1 and Exp3, where the striations are dominated by
Rhines jets, the dominant energy source for the striations
is eddy–eddy interaction and the amount of energy extracted by striations from other eddies is large.
5. Effect of mean flow on total mixing and
striations’ contribution
We introduce a diagnostic framework to assess the
role of striations in mixing and then apply this framework to the specific example of barotropic striations.
AUGUST 2015
2105
CHEN AND FLIERL
›
2
1 u $ 2 k= Cj (x, t) 5 0,
›t
a. Diagnostic approach
(30)
1) MIXING BY ALL THE EDDIES
Mixing rates of passive tracers induced by eddies are
often quantified using the eddy mixing tensor Dij,
(Plumb and Mahlman 1987; Bachman 2012; Bachman
and Fox-Kemper 2013; Abernathey et al. 2013), which is
defined as
Fi 5 u0i b0 5 2Dij
›b(x, t)
,
›xj
(27)
where Fi denotes the ensemble average of the eddy
tracer flux, u0i b0 , with the overbar denoting an ensemble
average (the mean) and the prime denoting a deviation
from the mean (eddies). For example, ui is the mean
velocity and u0i is the eddy velocity. Here b denotes the
concentration of a passive tracer, which satisfies
›b
1 u $b 2 k=2 b 5 0,
›t
(28)
where u denotes the velocity vector, which is the sum of
the mean velocity u, the low-frequency eddy velocity u0s ,
and the high-frequency eddy velocity u0h . Here k can
either represent molecular diffusivity or artificial numerical diffusivity to ensure computational stability
(Shuckburgh and Haynes 2003). The symmetric part of
Dij is eddy diffusivity and the antisymmetric part provides information about the transport due to Stokes drift
(Plumb and Mahlman 1987).
Equation (27) is an underdetermined system for one
tracer, and therefore, previous studies solve for Dij by
using the pseudoinversion method, which involves initializing multiple tracers with different initial tracer gradients and then calculating Dij through the least squares
approach (Plumb and Mahlman 1987; Bachman 2012;
Abernathey et al. 2013). However, we chose to use a novel
approach from G. Flierl (2014, unpublished manuscript).
This approach is an extension of the work by Kraichnan
(1987), and it has been successfully applied to mixing estimations in both an idealized barotropic model (Chen
2013) and a realistic regional ocean model (Woods 2013).
He found that, if both $b and Fi vary slowly in space, Dij is
ðt
Dij 5
u0i (x, t)Cj (x, t) dt .
(29)
t0
The term Cj (x, t) denotes the concentration of a ‘‘special
tracer’’ at time t. The special tracer is stirred by the total
flow field and has the same value as the eddy velocity u0j
at time t 5 t0. In other words, Cj (x, t) essentially records
the initial velocity and it can be solved for numerically
by integrating
with the initial condition
Cj (x, t0 ) 5 u0j (x, t0 ) .
(31)
The detailed derivation of this mixing formula [Eqs.
(29)–(31)] can be found in section 5.8.3 of Chen (2013),
and it is very long. Here are some key steps. First, we
obtain the equation for eddy tracer concentration
b0 from the equation for the total tracer concentration
b [Eq. (28)]; second, we rewrite b0 based on the Green’s
function G, satisfying
›
1 u $ 2 k=2 G(x, t j x0 , t0 ) 5 d(x 2 x0 )d(t 2 t0 ) . (32)
›t
We can then rewrite the eddy tracer flux Fi , the eddy
mixing tensor Dij, and the special tracer Cj based on the
Green’s function G as well. The mixing formula can then
be obtained by comparing Dij and Cj .
Compared to the pseudoinversion method, the new
mixing formula [Eqs. (29)–(31)] has several advantages.
First, the pseudoinversion approach can be computationally expensive; for example, six tracers are used in
Abernathey et al. (2013) to infer the eddy mixing tensor
in an idealized channel model. However, to diagnose Dij
using the new formula, we only need N special tracers in
an N-dimensional flow. Second, Dij computed from the
pseudoinversion approach can be sensitive to the choice
of the initial tracer concentration pattern, and improper
choices lead to unusable solutions (Bachman 2012).
However, the initial concentration of the special tracer
field in Eq. (29) is just the eddy velocity. Computation of
Dij in Eq. (29) involves an ensemble average, and it is
insensitive to the initial concentration of the special
tracer in each realization.
2) MIXING BY LOW-FREQUENCY EDDIES
The eddy tracer flux Fi includes contributions from
both low- and high-frequency eddies,
Fi 5 u0i b0 5 u0s,i b0 1 u0h,i b0 ,
(33)
where us,i and uh,i denote the low-frequency and highfrequency eddy velocity, respectively. Thus, u0s,i b0 and
u0h,i b0 represent the contribution of low-frequency and
high-frequency eddies to eddy tracer flux, respectively.
As in section 4, we assume all the low-frequency motions are striations, and thus Ds,ij is approximately
the eddy mixing tensor induced by striations. Similar
to Eq. (27), we can parameterize the two parts as
follows:
2106
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 45
FIG. 9. (top) Spatial distribution of the special tracer C1 (m s21) in Exp1 at (a1) time t0, (a2) 20 days, and (a3) 80 days after time t0.
(bottom) As in the top panels, but for the special tracer C2 (m s21). The term C1 (x, t) satisfies Eq. (30) with the zonal eddy velocity at time
t0 as its initial condition; C2 (x, t) also satisfies Eq. (30) but with the meridional eddy velocity at time t0 as its initial condition. Here t0 is 104
days after the start of Exp1.
u0s,i b0 5 2Ds,ij
Ds,ij 1 Dh,ij 5 Dij .
›b(x, t)
,
›xj
u0h,i b0 5 2Dh,ij
›b(x, t)
,
›xj
(34)
where Ds,ij denotes the eddy mixing tensor induced by lowfrequency eddies and Dh,ij is that induced by high-frequency
eddies. Following the approach in section 5.8.3 from Chen
(2013), but replacing u0i b0 there with u0s,i b0 , we obtain Ds,ij:
ðt
u0s,i (x, t)Cj (x, t) dt .
(35)
Ds,ij 5
t0
Similarly, if we replace u0i b0 in section 5.8.3 from Chen
(2013) with u0h,i b0 , we obtain Dh,ij:
ðt
Dh,ij 5
u0h,i (x, t)Cj (x, t) dt ,
(36)
t0
where Cj is the special tracer defined by Eqs. (30) and
(31). Note that
(37)
Analogous to Dij, the symmetric parts of Ds,ij and Dh,ij
represent the contribution to mixing, and their antisymmetric parts provide information about the Stokes drift.
b. Numerical results
1) A DIAGNOSTIC EXAMPLE: EDDY MIXING
TENSOR IN EXP1
To illustrate the methodology, we compute an estimate of the mixing tensor Dij for Exp1. The first step is to
estimate the special tracer Cj (x, t) defined by Eqs. (30)
and (31). Choose an arbitrary t0 as an initial time and
specify the eddy velocity at that time as the initial value
of the special tracer. Then employ the total velocity field
to advect the special tracer forward to time t. Figure 9
shows representative snapshots of the special tracer
distribution at t0 and two subsequent times. Considering
AUGUST 2015
CHEN AND FLIERL
2107
FIG. 10. The domain-averaged mixing tensor Dij (m2 s21) in Exp1, diagnosed from Eqs. (29)–
(31), as a function of the integration time t 2 t0 (day). The terms D11 and D22 denote zonal and
meridional eddy diffusivities, respectively; D12 and D21 contain information about the Stokes
drift. We used 20 ensembles. The special tracer Cj in each ensemble starts from a different
initial condition, that is, a different choice of t0. The time range that each ensemble covers does
not overlap with each other. Gray area indicates two standard deviations of error. The mixing
tensor levels off as t 2 t0 reaches ;500 days or less. The variable Dij presented in Fig. 11 is the
average of Dij from (t0 1 1500) to (t0 1 2000) days, indicated by the red lines here.
that Dij from Eq. (29) is an integral from time t0 to time t,
we define t 2 t0 as the ‘‘integration time.’’ As the integration time increases, both the magnitude and spatial
gradient of Cj (x, t) decrease because of mixing and
transport. The second step, after obtaining Cj (x, t)
above, is to diagnose Dij using Eq. (29). This two-step
procedure is repeated multiple times to obtain more
ensembles in order to obtain more reliable estimated
values of Dij and its uncertainty (Fig. 10). In addition to
the smoothing of the special tracer, as the integration
time increases, the spatial pattern of the special tracer
gradually decorrelates from its initial pattern (Fig. 9).
Thus, Dij gradually levels off and we use Dij in this
equilibrated time range, indicated by the red lines, for
our final estimate (Fig. 10). The mixing tensor for striations (Ds,ij) can be estimated in a similar way.
2) EXPERIMENTAL RESULTS
Figure 11 show the domain-averaged mixing tensor
for total eddies (hDij i) and that for striations (hDs,ij i) in
the three experiments. For a two-dimensional flow the
tensor is of rank 2. The imposed mean flow does not
affect the domain-averaged mixing in the barotropic
system. The magnitude of hDij i is roughly the same in
the three experiments (Fig. 11) because of Galilean invariance as both tracers and eddies are advected by the
mean flow (appendix C).
Mixing is anisotropic and primarily in the zonal direction regardless of the direction of the imposed mean
flow. In all three experiments, the zonal eddy diffusivity
hD11 i has the largest magnitude among the four components of hDij i (Fig. 11). This result can be understood
by realizing that for the coordinates moving with the
imposed mean flow, the eddy velocity is dominated by
that of the Rhines jets, which are roughly zonal. Therefore, water parcels move in a quasi-zonal direction, and
as a result mixing is predominantly zonal.
Rhines jets are always the dominant contributor to
zonal eddy diffusivity hD11 i. Consequently, striations
contribute significantly to zonal mixing only when striations are dominated by Rhines jets (Fig. 11). In Exp1
and Exp3, where the meridional component of the imposed mean flow is zero, striations are predominantly
Rhines jets and contribute much more to zonal mixing
than the high-frequency eddies. In Exp2, where the
meridional component of the imposed mean flow is
significant, striations are not composed of Rhines jets
and they contribute much less to zonal mixing than the
high-frequency eddies.
Figures 9–11 show results for the intrinsic diffusivity
k in Eqs. (28) and (30) set to 3200 m2 s21. Other values of
k were tried, and it was found that as k decreases, Dij
levels off more slowly as a function of t 2 t0 and its
magnitude increases. However, the effect of mean flow
2108
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 45
FIG. 11. The domain-averaged mixing tensor for total eddies (Dij, blue vertical bars, m2 s21)
and that for striations (Ds,ij, brown vertical bars, m2 s21) in each experiment. Terms D11 and D22
denote zonal and meridional eddy diffusivities, respectively; D12 and D21 contain information
about the Stokes drift. Similarly, Ds,11 and Ds,22 denote zonal and meridional diffusivities induced by striations, respectively. As in Fig. 10, we use 20 ensembles in each experiment to
obtain the results shown here. The error bars, indicated by vertical black lines on the vertical
bars, are two standard deviations. The separation frequency VS used to determine striations is
the same as that in Fig. 4.
on Dij and on the role of striations in the total eddy
mixing was found to be insensitive to k.
6. Summary and discussion
In the low friction, b-plane, barotropic, QG system
driven by small-scale narrow-banded white noise forcing, most of the eddy energy in frequency and wavenumber space is concentrated on the surface of the
barotropic Rossby wave dispersion relation. Therefore,
striations can be viewed as quasi-stationary Rossby
waves. In the case with an imposed meridional mean
flow, Rhines jets move meridionally and have the
frequency Vp/lRhines . If Vp/lRhines is larger than the
separation frequency VS, Rhines jets are not part of lowfrequency motions. As a result, striations are tilted, have
weak amplitudes, contribute little to tracer mixing, and
extract little energy from the high-frequency eddies.
When there is no mean flow imposed or when V is small,
the Rhines jet frequency is smaller than VS. In this case,
striations are zonal, dominated by Rhines jets, contribute significantly to tracer mixing, and extract significant
energy from the high-frequency eddies.
Our experiments summarized above are forced by
white noise, which has zero decorrelation time scale. We
also examined the case when the forcing has the same
narrow-banded wavenumber spectra but has a decorrelation time scale of a few days, as that described in Lilly
(1969), Williams (1978), Maltrud and Vallis (1991), and
Chen et al. (2015a). Our choice is motivated by the fact
that the decorrelation time scale of observed winds is on
the order of a few days (Schlax et al. 2001; Gille 2005;
Monahan 2012). We found that in this case, the striation
properties summarized above still hold. On the other
hand, Srinivasan and Young (2014) found that Galilean
invariance breaks down if the forcing has finite decorrelation time scale. Thus, striation properties in the case
when the forcing decorrelation time scale is very long
could be different from those in the white noise forcing
case. Detailed exploration is left for future work.
Besides forcing, many more regimes about striations in
the background flow are left for future work. For example,
we only consider the case with constant mean flow, which
is a good approximation in some ocean regions (e.g.,
subtropical gyre), but not others (e.g., near topography).
The large horizontal shear of the background mean flow
may lead to barotropic instability and thus eddy–mean
flow interaction. Cases with sheared mean flow, eddy–
mean flow interaction, strong friction, topography, or
multiple vertical models are to be explored. Coastal geometry and eastern boundary current can also greatly influence the generation of striations (e.g., Davis et al. 2014).
Based on the Galilean invariance in the barotropic
QG model, we developed some simple theories for the
energy (appendix B), mixing (appendix C), and spectral
(section 3c) characteristics for the striations. As a result
of these, there are several conclusions that can be drawn.
First, energy cascades in the wavenumber domain are
not sensitive to the imposed mean flow (Chen 2013), but
energy cascades in the frequency domain are (appendix
B). Second, total eddy mixing is insensitive to the
barotropic mean flow (appendix C). Previous theories
suggest that the mixing in the cross-mean flow direction
can be suppressed by the wave propagation relative to
the mean flow (e.g., Green 1970; Ferrari and Nikurashin
2010; Klocker et al. 2012a; Chen et al. 2014b). In other
AUGUST 2015
2109
CHEN AND FLIERL
words, the difference between the mean flow magnitude
and the wave phase speed along the mean flow direction
contribute to the mixing suppression (Chen et al. 2014b).
However, the barotropic mean flow does not affect the
magnitude of the mismatch between mean flow magnitude and wave speed, as waves are Doppler shifted by
the barotropic mean flow. Therefore, we can conclude
from these previous theories that the barotropic mean
flow does not affect the mixing rates in the cross-mean
flow direction, which is consistent with the findings in
this study. Third, we found that at a given wavenumber
vector the barotropic mean flow can shift the spectra in
the frequency domain (section 3c). For example, in an
ocean region, if the barotropic mean flow magnitude is
0.01 m s21 and the dominant eddy length scale is 100 km,
eddy motions with intrinsic frequency of zero will have
the frequency of 1 3 1027 cycle s21, corresponding to a
period of four months, because of the barotropic mean
flow effect. In summary, in this simple barotropic model,
the barotropic mean flow does affect energetics and
mixing of low-frequency structures, the energy cascade
in the frequency domain, and frequency–wavenumber
spectra; however, the barotropic mean flow does not
influence the total mixing rates by all the eddies.
The Galilean invariance also holds in the N-layer QG
model (W. R. Young 2013, personal communication).
The N-layer QG model is shown to be a reasonable
model for realistic oceanic flows in many regions, such as
the Southern Ocean and midlatitude oceanic interior
(e.g., Arbic and Flierl 2004; Thompson 2010; Venaille
et al. 2011). In consequence, the effect of barotropic (not
baroclinic) mean flow on the energy, mixing, and spectral characteristics for the striations in the N-layer QG
model or possibly in the ocean can also be described by
these simple theories summarized in the last paragraph
(i.e., appendixes B and C and section 3c).
Our Exp2 is potentially applicable to the Southern
Ocean from a kinematical perspective. In contrast to
most oceanic regions, banded structures are visible in the
flow snapshot in the Southern Ocean, and these structures
are termed ‘‘Southern Ocean jets’’ (Thompson 2010).
These jets are greatly influenced by topography (e.g.,
Thompson 2008; Ivchenko et al. 2008), and topography
with the scale comparable to the jet width can cause the
meridional drifts of jets in the QG two-layer model
(Thompson 2010). These meridional drifts have also
been identified in some regions of the Southern Ocean
in realistic eddying models (Thompson and Richards
2011). Our Exp2 is analogous to these Southern Ocean
regions in two kinematic aspects: banded structures are
visible in the instantaneous images of the flow field and
they have meridional drifts. Results from Exp2 suggest
that bands in low-frequency motions in the Southern
Ocean might be different from the jets visible in the
snapshots.
One can use our diagnostic framework and their potential descendants to investigate the energy and mixing
consequences of realistic striations. This may be an important result considering that the amplitude of realistic
striations is noticeable (Chen et al. 2015a). Following
the techniques in appendix A, the energy budget equation for striations can be easily generalized to a form
consistent with the governing equations of realistic
models. The framework in section 5 is directly applicable to quantifying the contribution of striations to mixing in selected regions from observations and models of
varying complexity, provided that the local spatial homogeneity assumption is applicable. The separation frequency between realistic striations and high-frequency
eddies can be determined using the method from Chen
et al. (2015a). These efforts can help reveal the role of
striations in the spatial distribution of tracers (e.g., temperature, salinity) and thus the mean state of the climate
and its variability.
We employed a new formula to estimate eddy mixing
tensor [Eq. (29)]. This approach requires fewer tracers to
estimate the mixing tensor than the pseudoinversion approach (e.g., Bachman 2012; Abernathey et al. 2013). Note
that this new formula is based on the assumption that both
the mean tracer gradient and the mean eddy tracer fluxes
vary slowly in space. This assumption is also employed
often in previous mixing studies (e.g., Ferrari and
Nikurashin 2010; Klocker et al. 2012a). Comparison and
reconciliation with other diffusivity diagnostic approaches
are to be explored. Further application of this formula to
oceanic and atmospheric mixing studies is to be carried out.
Acknowledgments. This work, which is part of a Ph.D
thesis in the MIT–WHOI Joint Program (i.e., Chen 2013),
was supported by NASA through Grants NNX09AI87G,
NNX08AR33G, and NNX11AQ12G, as well as by the
NSF through Grant OCE-1459702. Carl Wunsch, as Ru
Chen’s Ph.D. coadvisor, provided insightful suggestions,
generous encouragement, and critical support in many
aspects during the preparation of this work and could
rightfully be coauthor on this manuscript. Suggestions
from two anonymous reviewers greatly improved this
manuscript. We also benefited greatly from Michael A.
Spall’s comments on early versions of this study and discussions with William R. Young.
APPENDIX A
Energy Budget for Low-Frequency Motions
The evolution equation of hKE,Vi is obtained by applying the temporal low-pass filter ,
V to our barotropic
2110
JOURNAL OF PHYSICAL OCEANOGRAPHY
QG model [Eq. (1)], then multiplying it by 2c,V and
integrating over the doubly periodic domain. Using integration by parts, one finds
›
,
2
,
,
,
hKE,
V i 5 hcV [J(c, = c)]V i 1 h2cV [F (x, y, t)]V i
›t
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
FV
PV
1 h2rKE,
Vi ,
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
(A1)
DV
c (k, l, v) 5
P
V
ð‘ ð‘ ð‘
2‘ 2‘ 2‘
VOLUME 45
where hi denotes the domain average. One key to obtaining
Eq. (A1) is noting that the low-pass temporal filter operator
,
V and temporal deviation operator ›/›t can commute
› 2
= c
›t
,
V
›
5 =2 c,
V.
›t
(A2)
The temporal and domain-averaged nonlinear energy
fluxes PV can be written in the frequency–wavenumber
space. Using the convolution theorem, we express the
Fourier transform of PV in terms of the Fourier trans2
,
form of c,
V and [J(c, = c)]V :
^ 2 k0 , l 2 l0 , v 2 v0 )H(v 2 v0 , V) dk0 dl0 dv0 ,
^ 0 , l0 , v0 )H(v0 , V)J(k
c(k
(A3)
where J denotes J(c, =2 c) and H denotes the low-pass filter, as that in Eqs. (19) and (20). Using the definition of
Fourier transform,
c (k, l, v)j
(A4)
hPV (x, y, t)i 5 P
V
k50,l50,v50 ,
where here denotes time mean. Noting that H(2v0 , V) 5 H(v0 , V), Eqs. (A3) and (A4) lead to
ð‘ ð‘ ð‘
0
^
^ 0 , l0 , v0 )[H(v0 , V)]2 J(2k
hPV (x, y, t)i 5
c(k
, 2l0 , 2v0 ) dk0 dl0 dv0 .
(A5)
2‘ 2‘ 2‘
APPENDIX B
where J(x, y, t) and JB(x, y, t) denote the Jacobians:
Effects of Mean Flow on Energy Cascades
J(x, y, t) 5 J[c(x, y, t), =2 c(x, y, t)]
The eddy streamfunction in the experiment with mean
flow c(x, y, t) has the same statistical characteristics as
fB(x, y, t), where
fB (x, y, t) 5 fA (x 2 Ut, y 2 Vt, t)
(B1)
with fA (x, y, t) denoting the eddy streamfunction
in the experiment with no mean flow (section 3c).
Thus, the nonlinear energy fluxes in the experiments
with mean flow can be represented as a function of
fB (x, y, t):
hPV (x, y, t)i 5
ð‘ ð‘ ð‘
2‘ 2‘ 2‘
JB (x, y, t) 5 J[fB (x, y, t), = fB (x, y, t)] .
(B2)
JB (x, y, t) 5 JA (x 2 Ut, y 2 Vt, t) ,
(B4)
where JA (x, y, t) 5 J[fA (x, y, t), =2 fA (x, y, t)]. Thus,
cB and c
JB satisfy
f
c (k0 , l0 , v0 2 k0 U 2 l0 V) and
c (k0 , l0 , v0 ) 5 f
f
B
A
(B5)
Therefore, Eq. (A5) becomes
c (k0 , l0 , v )Jc (2k0 , 2l0 , 2v )[H(v 1 k0 U 1 l0 V, V)]2 dk0 dl0 dv ,
f
A
0 A
0
0
0
where v0 5 v0 2 k0 U 2 l0 V. Therefore, nonlinear energy
fluxes in the case with mean flow [i.e., Eq. (A5)] can be
(B3)
Equation (B1) leads to
0 0
0
0
0
c 0 0 0
Jc
B (k , l , v ) 5 JA (k , l , v 2 k U 2 l V) .
,
PV (x, y, t) 5 [c(x, y, t)],
V [J(x, y, t)]V
5 [fB (x, y, t)],
[J (x, y, t)],
,
V B
V
and
2
(B6)
obtained from the eddy field in the case with no mean
flow (i.e., fA).
AUGUST 2015
APPENDIX C
Comparing Eqs. (C8) and (C9) with Eqs. (C2) and
(C3) leads to
Effects of Mean Flow on Total Mixing
Here we discuss, in our barotropic model, why the
imposed mean flow does not influence the spatially averaged total mixing rate. First, consider the eddy mixing
tensor Dij in a system with no mean flow imposed. We
define the eddy velocity and the mixing tensor in this
case as u0A and DAij , respectively. Using Eq. (29) and
setting t0 in Eq. (29) to be zero without loss of generality,
we know that
DAij 5
ðt
0
u0Ai (x, t)CAj (x, t) dt .
(C1)
Here the special tracer CAj (x, t) satisfies
›
1 u0A $ 2 k=2 CAj (x, t) 5 0
›t
CAj (x, t)jt50 5 u0Aj (x, t)jt50 .
(C2)
(C3)
Next, consider the case with mean flow u imposed. We
define the eddy velocity and the mixing tensor in this
case as u0B and DBij. Again, we know from Eq. (29) that
DBij 5
ðt
0
u0Bi (x, t)CBj (x, t) dt .
(C4)
Here the special tracer CBj (x, t) satisfies
›
1 u $ 1 u0B $ 2 k=2 CBj (x, t) 5 0
›t
(C5)
with the initial condition
CBj (x, t)jt50 5 u0Bj (x, t)jt50 .
(C6)
We know from the Galilean invariance that, from a
statistical point of view,
u0B (x, t) 5 u0A (x0 , t) 5 u0A (x 2 ut, t) .
(C7)
The term (x0 , t) denotes the moving coordinates, where
x0 5 x 2 ut and t0 5 t. Using Eq. (C7), we can write Eqs.
(C5) and (C6) in the moving coordinates,
›
0
0
2
1 uA (x , t) $ 2 k= CBj 5 0
›t0
(C8)
with the initial condition
CBj jt50 5 u0Bj (x, t)jt50 5 u0Aj (x0 , t0 )jt50 .
CBj (x, t) 5 CAj (x0 , t0 ) 5 CAj (x 2 ut, t) .
(C10)
Equations (C7) and (C10) lead to
u0Bi (x, t)CBj (x, t) 5 u0Ai (x 2 ut, t)CAj (x 2 ut, t) .
(C11)
Therefore, at each time step in the doubly periodic domain, u0Bi (x, t)CBj (x, t) has the same spatial pattern as
u0Ai (x, t)CAj (x, t). Consequently, the spatially averaged
mixing tensor is independent of the imposed mean flow,
that is, hDAij i 5 hDBij i.
REFERENCES
with the initial condition
2111
CHEN AND FLIERL
(C9)
Abernathey, R., D. Ferreira, and A. Klocker, 2013: Diagnostics of
isopycnal mixing in a circumpolar channel. Ocean Modell., 72,
1–16, doi:10.1016/j.ocemod.2013.07.004.
Afanasyev, Y. D., S. O. Leary, P. B. Rhines, and E. G. Lindahl,
2012: On the origin of jets in the ocean. Geophys. Astrophys.
Fluid Dyn., 106, 113–137, doi:10.1080/03091929.2011.562896.
Arbic, B. K., 2000: Generation of mid-ocean eddies: The local
baroclinic instability hypothesis. Ph.D. thesis, Massachusetts
Institute of Technology and Woods Hole Oceanographic Institution, 290 pp.
——, and G. R. Flierl, 2004: Baroclinically unstable geostrophic turbulence in the limits of strong and weak bottom
Ekman friction: Application to midocean eddies. J. Phys.
Oceanogr., 34, 2257–2273, doi:10.1175/1520-0485(2004)034,2257:
BUGTIT.2.0.CO;2.
——, R. B. Scott, G. R. Flierl, A. J. Morten, J. G. Richman, and
J. F. Shriver, 2012: Nonlinear cascade of surface oceanic
geostrophic kinetic energy in the frequency domain. J. Phys.
Oceanogr., 42, 1577–1600, doi:10.1175/JPO-D-11-0151.1.
Bachman, S., 2012: A diagnostic suite of models for the evaluation
of oceanic mesoscale eddy parameterizations. Ph.D. thesis,
University of Colorado, 246 pp.
——, and B. Fox-Kemper, 2013: Eddy parameterization challenge
suite I: Eddy spindown. Ocean Modell., 64, 12–28, doi:10.1016/
j.ocemod.2012.12.003.
Bakas, N. A., and P. J. Ioannou, 2013: On the mechanism underlying the spontaneous emergence of barotropic zonal jets.
J. Atmos. Sci., 70, 2251–2271, doi:10.1175/JAS-D-12-0102.1.
Chen, R., 2013: Energy pathways and structures of oceanic eddies
from the ECCO2 state estimate and simplified models. Ph.D.
thesis, Massachusetts Institute of Technology and Woods
Hole Oceanographic Institution, 206 pp.
——, G. R. Flierl, and C. Wunsch, 2014a: A description of local and
nonlocal eddy-mean flow interaction in a global eddypermitting state estimate. J. Phys. Oceanogr., 44, 2336–2352,
doi:10.1175/JPO-D-14-0009.1.
——, J. L. McClean, S. T. Gille, and A. Griesel, 2014b: Isopycnal
eddy diffusivities and critical layers in the Kuroshio Extension
from an eddying ocean model. J. Phys. Oceanogr., 44, 2191–
2211, doi:10.1175/JPO-D-13-0258.1.
——, G. R. Flierl, and C. Wunsch, 2015a: Quantifying and interpreting striations in a subtropical gyre: A spectral perspective.
J. Phys. Oceanogr., 45, 387–406, doi:10.1175/JPO-D-14-0038.1.
2112
JOURNAL OF PHYSICAL OCEANOGRAPHY
——, S. T. Gille, J. L. McClean, G. R. Flierl, and A. Griesel,
2015b: A multiwavenumber theory for eddy diffusivities
and its application to the southeast Pacific (DIMES)
region. J. Phys. Oceanogr., 45, 1877–1896, doi:10.1175/
JPO-D-14-0229.1.
Cox, M., 1987: An eddy-resolving numerical model of the
ventilated thermocline: Time dependence. J. Phys. Oceanogr.,
17, 1044–1056, doi:10.1175/1520-0485(1987)017,1044:
AERNMO.2.0.CO;2.
Danilov, S., and V. M. Gryanik, 2004: Barotropic beta-plane turbulence in a regime with strong zonal jets revisited. J. Atmos.
Sci., 61, 2283–2295, doi:10.1175/1520-0469(2004)061,2283:
BBTIAR.2.0.CO;2.
——, and D. Gurarie, 2004: Scaling, spectra and zonal jets in betaplane turbulence. Phys. Fluids, 16, 2592–2603, doi:10.1063/
1.1752928.
Davis, A., E. D. Lorenzo, H. Luo, A. Belmadani, N. Maximenko,
O. Melnichenko, and N. Schneider, 2014: Mechanisms for the
emergence of ocean striations in the North Pacific. Geophys.
Res. Lett., 41, 948–953, doi:10.1002/2013GL057956.
Dritschel, D. G., and M. E. McIntyre, 2008: Multiple jets as PV
staircases: The Phillips effect and the resilience of eddytransport barriers. J. Atmos. Sci., 65, 855–874, doi:10.1175/
2007JAS2227.1.
Farrell, B. F., and P. J. Ioannou, 2007: Structure and spacing of jets
in barotropic turbulence. J. Atmos. Sci., 64, 3652–3665,
doi:10.1175/JAS4016.1.
Ferrari, R., and C. Wunsch, 2009: Ocean circulation kinetic energy:
Reservoirs, sources and sinks. Annu. Rev. Fluid Mech., 41,
253–282, doi:10.1146/annurev.fluid.40.111406.102139.
——, and M. Nikurashin, 2010: Suppression of eddy mixing across
jets in the Southern Ocean. J. Phys. Oceanogr., 40, 1501–1519,
doi:10.1175/2010JPO4278.1.
Flierl, G. R., and J. Pedlosky, 2007: The nonlinear dynamics of
time-dependent subcritical baroclinic currents. J. Phys.
Oceanogr., 37, 1001–1021, doi:10.1175/JPO3034.1.
Galperin, B., H. Nakano, H.-P. Huang, and S. Sukoriansky, 2004:
The ubiquitous zonal jets in the atmospheres of giant planets
and Earth’s oceans. Geophys. Res. Lett., 31, L13303,
doi:10.1029/2004GL019691.
Gille, S. T., 2005: Statistical characterization of zonal and meridional ocean wind stress. J. Atmos. Oceanic Technol., 22, 1353–
1372, doi:10.1175/JTECH1789.1.
——, K. Speer, J. R. Ledwell, and A. C. N. Garabato, 2007: Mixing
and stirring in the Southern Ocean. Eos, Trans. Amer. Geophys. Union, 88, 382–383, doi:10.1029/2007EO390002.
Green, J. S. A., 1970: Transport properties of the large-scale
eddies and the general circulation of the atmosphere.
Quart. J. Roy. Meteor. Soc., 96, 157–185, doi:10.1002/
qj.49709640802.
Griesel, A., S. T. Gille, J. Sprintall, J. L. McClean, J. H. LaCasce,
and M. E. Maltrud, 2010: Isopycnal diffusivities in the Antarctic Circumpolar Current inferred from Lagrangian floats in
an eddying model. J. Geophys. Res., 115, C06006, doi:10.1029/
2009JC005821.
Hughes, C. W., A. F. Thompson, and C. Wilson, 2010: Identification of jets and mixing barriers from sea level and vorticity
measurements using simple statistics. Ocean Modell., 32, 44–
57, doi:10.1016/j.ocemod.2009.10.004.
Ivchenko, V. O., S. Danilov, and D. Olbers, 2008: Eddies in numerical models of the Southern Ocean. Ocean Modeling in an
Eddying Regime, Geophys. Monogr. Ser., Vol. 177, Amer.
Geophys. Union, 177–198.
VOLUME 45
Killworth, P. D., 1997: On the parameterization of eddy transfer.
Part 1: Theory. J. Mar. Res., 55, 1171–1197, doi:10.1357/
0022240973224102.
Klocker, A., R. Ferrari, and J. H. LaCasce, 2012a: Estimating
suppression of eddy mixing by mean flows. J. Phys. Oceanogr.,
42, 1566–1576, doi:10.1175/JPO-D-11-0205.1.
——, ——, ——, and S. Merrifield, 2012b: Reconciling float-based
and tracer-based estimates of eddy diffusivities. J. Mar. Res.,
70, 569–602, doi:10.1357/002224012805262743.
Kraichnan, R. H., 1987: Eddy viscosity and diffusivity: Exact formulas and approximations. Complex Syst., 1, 805–820.
Lilly, D. K., 1969: Numerical simulation of two-dimensional turbulence. Phys. Fluids, 12, II-240, doi:10.1063/1.1692444.
Maltrud, M., and G. K. Vallis, 1991: Energy spectra and coherent
structures in forced two-dimensional and geostrophic turbulence. J. Fluid Mech., 228, 321–342.
Maximenko, N. A., B. Bang, and H. Sasaki, 2005: Observational
evidence of alternating zonal jets in the world ocean. Geophys.
Res. Lett., 32, L12607, doi:10.1029/2005GL022728.
——, O. V. Melnichenko, P. P. Niiler, and H. Sasaki, 2008: Stationary mesoscale jet-like features in the ocean. Geophys. Res.
Lett., 35, L08603, doi:10.1029/2008GL033267.
Menemenlis, D., J. Campin, P. Heimbach, C. Hill, T. Lee,
A. Nguyen, M. Schodlock, and H. Zhang, 2008: ECCO2: High
resolution global ocean and sea ice data synthesis. Mercator
Ocean Quarterly Newsletter, No. 31, Mercator Ocean,
Ramonville Saint-Agne, France, 13–21. [Available online at
http://ecco2.org/manuscripts/reports/ECCO2_Mercator.pdf.]
Monahan, A. H., 2012: The temporal autocorrelation structure of
sea surface winds. J. Climate, 25, 6684–6700, doi:10.1175/
JCLI-D-11-00698.1.
Nakano, H., and H. Hasumi, 2005: A series of zonal jets embedded
in the broad zonal flows in the Pacific obtained in eddypermitting ocean general circulation models. J. Phys. Oceanogr.,
35, 474–488, doi:10.1175/JPO2698.1.
Panetta, R. L., 1993: Zonal jets in wide baroclinically unstable regions:
Persistence and scale selection. J. Atmos. Sci., 50, 2073–2106,
doi:10.1175/1520-0469(1993)050,2073:ZJIWBU.2.0.CO;2.
Pedlosky, J., 1987: Geophysical Fluid Dynamics. Springer-Verlag,
710 pp.
Plumb, R. A., and J. D. Mahlman, 1987: The zonally averaged
transport characteristics of the GFDL general circulation/
transport model. J. Atmos. Sci., 44, 298–327, doi:10.1175/
1520-0469(1987)044,0298:TZATCO.2.0.CO;2.
Rhines, P. B., 1975: Waves and turbulence on a beta-plane. J. Fluid
Mech., 69, 417–443, doi:10.1017/S0022112075001504.
Richards, K. J., N. A. Maximenko, F. O. Bryan, and H. Sasaki,
2006: Zonal jets in the Pacific Ocean. Geophys. Res. Lett., 33,
L03605, doi:10.1029/2005GL024645.
Rypina, I. I., I. Kamenkovich, P. Berloff, and L. J. Pratt, 2012:
Eddy-induced particle dispersion in the near-surface North
Atlantic. J. Phys. Oceanogr., 42, 2206–2228, doi:10.1175/
JPO-D-11-0191.1.
Schlax, M. G., and D. B. Chelton, 2008: The influence of mesoscale
eddies on the detection of quasi-zonal jets in the ocean. Geophys. Res. Lett., 35, L24602, doi:10.1029/2008GL035998.
——, ——, and M. H. Freilich, 2001: Sampling errors in wind fields
constructed from single and tandem Scatterometer datasets.
J. Atmos. Oceanic Technol., 18, 1014–1036, doi:10.1175/
1520-0426(2001)018,1014:SEIWFC.2.0.CO;2.
Shuckburgh, E., and P. Haynes, 2003: Diagnosing transport and
mixing using a tracer-based coordinate system. Phys. Fluids,
15, 3342, doi:10.1063/1.1610471.
AUGUST 2015
CHEN AND FLIERL
Smith, K. S., 2007: The geography of linear baroclinic instability in
Earth’s oceans. J. Mar. Res., 65, 655–683, doi:10.1357/
002224007783649484.
Srinivasan, K., and W. R. Young, 2012: Zonostrophic instability.
J. Atmos. Sci., 69, 1633–1656, doi:10.1175/JAS-D-11-0200.1.
——, and ——, 2014: Reynolds stress and eddy diffusivity of
b-plane shear flows. J. Atmos. Sci., 71, 2169–2185, doi:10.1175/
JAS-D-13-0246.1.
Sukoriansky, S., N. Dikovskaya, and B. Galperin, 2007: On the
arrest of inverse energy cascade and the Rhines scale.
J. Atmos. Sci., 64, 3312–3327, doi:10.1175/JAS4013.1.
Thompson, A. F., 2008: The atmospheric ocean: Eddies and jets in
the Antarctic Circumpolar Current. Philos. Trans. Roy. Soc.
London, A366, 4529–4541, doi:10.1098/rsta.2008.0196.
——, 2010: Jet formation and evolution in baroclinic turbulence
with simple topography. J. Phys. Oceanogr., 40, 257–278,
doi:10.1175/2009JPO4218.1.
2113
——, and K. J. Richards, 2011: Low frequency variability of
Southern Ocean jets. J. Geophys. Res., 116, C09022,
doi:10.1029/2010JC006749.
Venaille, A., G. K. Vallis, and K. S. Smith, 2011: Baroclinic turbulence in the ocean: Analysis with primitive equation and
quasigeostrophic simulations. J. Phys. Oceanogr., 41, 1605–
1623, doi:10.1175/JPO-D-10-05021.1.
Williams, G. P., 1978: Planetary circulations: 1. Barotropic representation of Jovian and terrestrial turbulence. J. Atmos.
Sci., 35, 1399–1426, doi:10.1175/1520-0469(1978)035,1399:
PCBROJ.2.0.CO;2.
Woods, N. W., 2013: Physical controls on copepod aggregations in the
Gulf of Maine. Ph.D. thesis, Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 213 pp.
Xu, C., X. Shang, and R. X. Huang, 2011: Estimate of global eddy
energy generation/dissipation rate from altimetry data. Ocean
Dyn., 61, 525–541, doi:10.1007/s10236-011-0377-8.