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Dynamics of Atmospheres and Oceans
41 (2006) 28–59
Potential vorticity and eddy potential enstrophy in the
North Atlantic Ocean simulated by a global
eddy-resolving model
Mototaka Nakamura ∗ , Takashi Kagimoto
Frontier Research Center for Global Change, 3173-25 Showa-machi,
Kanazawa-ku, Yokohama, Kanagawa 236-0001, Japan
Received 4 December 2004; accepted 25 October 2005
Available online 6 January 2006
Abstract
Eight-year daily mean output of a quasi-global eddy-resolving model is examined with a focus on the
large-scale dynamical characteristics of the North Atlantic Ocean in a framework of potential vorticity (PV)
and its derivatives. The model has reproduced some of the observed features of the mean potential vorticity
field well. The three-dimensional structure of the mean potential vorticity supports baroclinic instability in
most of the basin. Eddies are found to play important roles in the formation and maintenance of the mean
potential vorticity fields. The contribution of relative vorticity to the mean potential vorticity field is found
to be negligible for the most part. However, relative vorticity contribution to the source/sink of potential
vorticity and eddy potential enstrophy is not negligible. We also find that eddies are not necessarily diffusive
even on a basin-scale.
© 2005 Elsevier B.V. All rights reserved.
Keywords: North Atlantic; Potential vorticity; Enstrophy; Eddy dynamics
1. Introduction
During the past two decades, the North Atlantic Ocean has received much attention from
climate researchers. It has been speculated that the critical component of the global thermohaline
circulation, the North Atlantic Deep Water (NADW) formation, may have been absent during
the mini and full glacial periods (e.g. Broecker et al., 1985). Many geological data support this
∗
Corresponding author.
E-mail addresses: [email protected] (M. Nakamura), [email protected] (T. Kagimoto).
0377-0265/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.dynatmoce.2005.10.002
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
29
speculation, while the causal relationship between the collapse of the NADW and glacial climate
is not certain. However, coupled atmosphere-ocean models of various complexity, ranging from
idealized models (e.g. Nakamura, 1996) to coupled GCMs with a realistic topography (e.g. Manabe
and Stouffer, 1993), all suggest a possibility of an instability of the North Atlantic thermohaline
circulation leading to a collapse of the NADW formation, then to a glacial condition in and around
the North Atlantic.
The stability of the thermohaline circulation depends on, to mention only oceanic factors,
transports and mixing of heat and salt that are not represented adequately in coarse-resolution
ocean models because of the difficulty in correctly representing strong transports and mixing
by geostrophic eddies (Nakamura and Chao, 2000a, 2001, 2002). Due to the scarcity of oceanic
data that resolve these eddies on a large-scale with sufficient temporal and spatial coverage, our
knowledge on the crucial processes is rather limited. There have been attempts to investigate the
role of eddies in the large-scale circulation with observational data (e.g. Bower and Hogg, 1992;
Chester et al., 1994; Cronin, 1996; Leach et al., 2002). However, they are limited to relatively
small scales due to the data available. In the advent of a powerful computer in the last several
years, it has become feasible to integrate a global ocean simulation model for many years at
resolutions that are sufficient to resolve geostrophic eddies of the deformation-radius-scale and
larger. Due to the aforementioned scarcity of high-frequency ocean data, output of eddy-resolving
ocean simulation models can be extremely useful, if carefully treated, for studying the large-scale
dynamics of and transport/mixing in oceans. The model output is, needless to say, not perfect.
Nevertheless, it provides us with a set of internally consistent dynamical and tracer fields that
may serve as pseudo-data.
A quantity that has been widely used in dynamical analyses and diagnoses of large-scale
atmospheric phenomena is potential vorticity (PV). PV is essentially the angular momentum per
unit mass and is conserved following a parcel of fluid in an adiabatic-inviscid flow. In oceans,
this property allows us to track a water mass on an isopycnal surface, so long as the effects
of diabatic processes, friction, and diapycnal mixing are weak with respect to those of isopycnal
advection. Also, given a PV field with boundary conditions, one can obtain the associated velocity
field through PV inversion when an appropriate balance condition, such as quasi-geostrophy, is
assumed (Hoskins et al., 1985). Thus, PV is a dynamical quantity with a tracer-like property.
The usefulness of PV for ocean circulation studies has been noted by numerous researchers
in the past and has been demonstrated in theories and idealized numerical models. A series of
pioneering works in the late 1970s and early 1980s demonstrated this usefulness (Holland, 1978;
Rhines and Holland, 1979; Holland and Rhines, 1980; Rhines and Young, 1982a, 1982b). In many
of the idealized numerical model studies in the past, PV was used as the central quantity. Dynamics
in these models were often diagnosed in the framework of PV and eddy potential enstrophy. In
spite of the scarcity of oceanic data to compute PV on a large-scale, there have been some
attempts to obtain large-scale climatological PV fields from hydrological data also (McDowell
et al., 1982; Keffer, 1985; Lozier, 1997, 1999), yielding some valuable insight into large-scale
oceanic circulations. Unfortunately, however, the PV fields calculated from observational data lack
sufficient information to determine the underlying processes. This void is what eddy-resolving
model output may be able to fill if used cleverly.
Nakamura and Chao (2000b, 2001, 2002) extended the aforementioned diagnostic works of
idealized eddy-resolving models to a more realistic model. They attempted to study the large-scale
dynamical characteristics of the North Atlantic Ocean by using 5-year output of an eddy-permitting
North Atlantic model with realistic topography and forcing in a framework of quasi-geostrophic
PV, eddy potential enstrophy, and transient wave activity and its fluxes, all at geometric depths
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M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
defined in the model. They found that eddies play important roles in generating and maintaining
the large-scale quasi-geostrophic PV field and that eddies are not necessarily diffusive. They
also identified transient quasi-geostrophic wave propagation patterns in the simulation. Their
attempt was the first to use output of a realistic eddy-permitting model as pseudo-data to study
the large-scale ocean circulation and produced some new knowledge that may well apply to the
dynamics of the real North Atlantic Ocean. However, due to the limitation on their computational
resources, their attempt suffered from a relatively coarse spatial resolution (approximately 1/6◦
in the horizontal and 37 vertical levels), a short integration period (30-year run), and coarse
output sampling frequency (every 3 days) with loss in the temporal coverage (snapshots, rather
than time-averages). The 1/6◦ horizontal resolution was insufficient to resolve eddies smaller than
60 km, while the 37-level vertical resolution made it difficult to perform diagnoses on interpolated
isopycnal surfaces with reasonable confidence. The 5-year output from the last 5 years of a
30-year integration left doubt in some critics’ minds that the model ocean may have been in
the initial adjustment stage. Finally, the relatively coarse output sampling frequency and the
lost temporal coverage were likely to be responsible for, at least partially, the noisiness of the
results.
Here, we attempt to refine the work of Nakamura and Chao (2001, 2002) by diagnosing 8year daily mean output of a quasi-global eddy-resolving ocean circulation model with realistic
topography and forcing (Masumoto et al., 2004). The 8-year output was taken from the last 8
years of a 97-year integration. Because of the higher resolution (1/10◦ in the horizontal and
54 vertical levels) and higher output sampling frequency with no lost temporal coverage (daily
mean), we use Ertel’s PV as the central quantity and study its three-dimensional structures and
compute the source/sink of PV and eddy potential enstrophy (EPE) on isopycnal surfaces. We
focus on diagnoses on isopycnal surfaces here and report diagnoses of transient wave activity and
other geometric-depth-based quantities elsewhere. We emphasize that our intention in this paper
is to cover large-scale features in a broad manner to provide the reader with an overview of the
diagnostic results. Detailed studies on local features will be conducted and reported elsewhere in
the future. Section 2 describes the model output used as pseudo-data and the diagnostic framework
used for the current work. Section 3 describes and discusses the results of the calculations. Finally,
we offer some remarks in Section 4.
2. Pseudo-data and calculation procedures
2.1. Ocean model
We used a set of 8-year daily mean fields of temperature, salinity, and the horizontal velocity
generated by OGCM For the Earth Simulator (OFES), a quasi-global ocean simulation model
with realistic topography. It is essentially the Modular Ocean Model (MOM) developed at the
Geophysical Fluid Dynamics Laboratory of the National Oceanic and Atmospheric Administration, extensively re-coded to optimize for the Earth Simulator (Sakuma et al., 2003; Masumoto
et al., 2004; Sasai et al., 2004). The model was initialized with the World Ocean Atlas 1998 and
integrated in time for 97 years with climatological forcing: surface fluxes of heat, freshwater,
and momentum derived from the NCEP Reanalyses. The horizontal grid spacing of the model is
1/10◦ in both longitude and latitude. There are 54 unevenly spaced vertical levels in the model,
ranging from 5 m near the surface to some 300 m near the bottom, with 34 of the 54 levels in
the top 1000 m. We used the last 8 years of this integration during which daily mean fields were
archived. For the North Atlantic diagnoses, we extracted 20◦ N–70◦ N and 90◦ W–0◦ W.
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
31
Fig. 1. The mean kinetic energy and eddy kinetic energy at the model’s top level, 2.5 m. The units are 10×−4 m2 s2 .
The model integration reproduced the observed temperature, salinity, and flow fields of the
oceans reasonably well (Masumoto et al., 2004). There is practically no trend in the global mean
or large-scale distributions of salinity, temperature, and, thus, density during the last 47 years
of the integration (Sasai et al., 2004; Sasai and Sasaki, personal communication). Isopycnal
surface locations are, therefore, free of any sustained trend. The simulated Gulf Stream (GS) is
in reasonable agreement with the observed both in terms of the location and strength. One major
problem with the simulated GS is that the observed sharp northward turn toward the northwest
corner and another sharp eastward turn at the northwest corner are not reproduced. Instead, the
simulated GS flows eastward to the mid-Atlantic, 45◦ W or so, then northeastward from there
on. Otherwise, the simulated North Atlantic is in fairly good agreement with the observation in
the vicinity of the GS. Typical instantaneous flow speeds of the simulated Florida Current and
the GS near the surface are in the range of 1.2–2.0 m s−1 , which is close to the observed (e.g.
Richardson et al., 1969; Halkin and Rossby, 1985). The 8-year mean kinetic energy calculated
in the Eulerian framework shows typical values on the order of 0.1 m2 s−2 near the surface in
the vicinity of the GS, whereas eddy kinetic energy shows larger values and broader structures,
both in reasonable agreement with the observed values (e.g. Wyrtki et al., 1976; Schmitz et al.,
1983) (Fig. 1). The mean kinetic energy quickly decreases with depth by two orders of magnitude
from the surface within the top 1000 m, while eddy kinetic energy decreases with depth more
gradually and shows fairly large values in the range of 0.03–0.1 m2 s−2 in the vicinity of the
GS (Fig. 2). Eddy kinetic energy at 700 m level in the model is slightly smaller (by 10% or so)
than the observed values estimated by Owens (1984, 1991) and Richardson (1993). Eddy kinetic
energy (and also the mean kinetic energy to some extent) has a typical vertical structure with
the maximum near the surface in a broad band, decreasing quickly to intermediate depths, then
changing very little toward the near-bottom layer in a manner similar to the observed presented by
Richardson (1983). The simulated North Atlantic shows fairly realistic circulation in the subpolar
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M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Fig. 2. Cross sections of (a) the mean kinetic energy and (b) eddy kinetic energy at 55◦ W. The units are 10×−4 m2 s2 .
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
33
region as well when compared with limited observations (e.g. Lazier and Wright, 1993; Dickson
and Brown, 1994).
2.2. Calculation of potential vorticity, eddy potential enstrophy, and their sources/sinks
From the daily mean temperature, salinity, and the horizontal velocity, PV, denoted by q, was
calculated at all density grid points by:
q=−
1 ∂σ
(f + ζ),
ρ ∂z
(1)
where ρ is density, σ potential density, and f and ζ are planetary and relative vorticities, respectively.
Here, ∂σ/∂z is the local stratification. The horizontal velocity at four corner points were used to
compute ζ at the center density points. For calculation of derivatives, centered finite differencing
(weighted by distance as needed) was used. On solid boundaries, zero velocity and zero normal
gradients were assumed. The vertical density gradient at the very top of the ocean is assumed
to be zero also. The daily three-dimensional q field was linearly interpolated onto 19 isopycnal
surfaces. All the isopycnals were referenced to the sea surface, because of our intention to cover
the entire depth in our diagnoses. The values of σ 0 for the selected 19 isopycnal surfaces are
26, 26.25, 26.5, 26.75, 27, 27.1, 27.2, 27.3, 27.4, 27.5, 27.6, 27.7, 27.76, 27.78, 27.8, 27.825,
27.85, 27.875, and 27.9. They were chosen after examining depths of a larger set of σ 0 surfaces
from the model output to ensure a reasonable depth coverage for the given vertical variations
in q and model level depths. Due to a relatively small vertical variation in water property and
large thickness of each model level below 1000 m, we believe that referencing isopycnals to the
surface at all depths is not likely to result in errors that are larger than those created by the linear
interpolation. At the same time, we note the potential presence of errors that may arise from the
vertical linear interpolation.
To calculate the source/sink of q on the selected σ 0 surfaces, the daily mean horizontal velocity
field was also interpolated onto the aforementioned 19 σ 0 surfaces. Since q is conserved on
isopycnal surfaces in an adiabatic-inviscid flow, the mean source/sink of q, SP , can be computed
by:
Vσ · ∇σ q¯ + Vσ · ∇σ q = SP ,
(2)
where an overbar and a prime indicate, respectively, the time mean and a deviation from the mean.
A subscript σ indicates that the quantities are evaluated on a quasi-horizontal isopycnal surface.
We use the 8-year daily q and velocity interpolated onto σ 0 surfaces for this calculation. Once
the mean source/sink is calculated, it is integrated by the number of days during which the σ 0
surfaces exist at each grid point to obtain the total source/sink. It is this total source/sink that will
be shown and discussed in Section 3.
The equation for the time–mean eddy potential enstrophy (EPE), q 2 /2 is given by:
∂
∂t
q 2
2
+ Vσ · ∇ σ
q 2
2
+ Vσ q · ∇σ q + Vσ q · ∇σ q¯ = s q ,
(3)
where s denotes the instantaneous source of q (e.g. Rhines and Holland, 1979). After some
manipulations, one obtains a set of equations that describe the budget of EPE on an isopycnal
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
34
surface, relating eddy q fluxes to the mean source of EPE, SE :
∗
(Vσ q )d · ∇σ q¯ = SE ,
(4)
and
∗
(Vσ q )r
· ∇σ q¯ = −Vσ · ∇σ
q 2
2
,
(5)
∗
where the modified divergent eddy flux or the generalized residual flux, (Vσ q )d , and the modified
∗
rotational flux or the mean balancing flux, (Vσ q )r , are defined by,
∗
∗
(Vσ q )d = (Vσ q ) + Fσ − (Vσ q )r = (Vσ q )d + Fσ
(6)
and
∗
(Vσ q )r
= γ kˆ × ∇σ
q 2
2
D
−
∇σ
|∇σ q¯ |2
q 2
.
2
(7)
Here, Fσ is an effective eddy flux that represents the effect of isopycnal eddy self advection,
defined by,
Fσ =
(Vσ q · ∇σ q )∇σ q¯
(Vσ · ∇σ q 2 )∇σ q¯
=
,
|∇σ q¯ |2
2|∇σ q¯ |2
(8)
and D and γ are defined by:
D = Vσ · ∇σ q¯ ,
(9)
and
γ=
ˆ σ × ∇σ q¯ )
−k(V
.
|∇σ q¯ |2
(10)
(see Nakamura, 1998 and Nakamura and Chao, 2002 for the details of the derivation and more
∗
discussion on the implications of these equations.) As evident in Eqs. (4) and (5), (Vσ q )r represents a part of the isopycnal eddy flux that balances the advection of eddy enstrophy by the
∗
mean flow, whereas (Vσ q )d represents a part of the isopycnal eddy flux that balances source/sink
∗
of eddy enstrophy on an isopycnal surface. Therefore, by calculating (Vσ q )d and taking a dot
product with ∇σ q¯ , one would find the source/sink of EPE and the anti-diffusive or diffusive nature
of eddies on isopycnal surfaces. Again, we interpolated the 8-year daily q and velocity onto σ 0
surfaces for this calculation. As in the calculation of q source/sink, the mean EPE source/sink
computed with the above equations is integrated by the number of days during which the σ 0 surfaces exist at each grid point to yield the total source/sink of EPE. It is this total EPE source/sink
that will be shown and discussed in Section 3.
To assess the significance of ζ contribution to the daily q, q¯ , and generation/dissipation of q and
EPE, the above calculations were repeated with only the f component first. Then we calculated the
ζ contribution as the residual. An exception is the q¯ calculation. We computed the ζ contributions
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
35
to q¯ in two separate terms. Note that q¯ consists of three components:
q¯ = −
1 ∂σ
1 ∂σ
f−
ζ¯ −
ρ ∂z
ρ ∂z
1 ∂σ
ρ ∂z
ζ.
(11)
First, the total contribution (i.e. the sum of the second and third terms on the right-hand-side of
Eq. (11)) of ζ was calculated as the residual, by subtracting q¯ calculated with only the f component
from q¯ calculated with both relative and the planetary components. Second, we calculated the
contribution of ζ¯ (the second term on the right-hand-side of Eq. (11)). We then subtracted the
contribution of ζ¯ from the total ζ contribution to assess the eddy contribution (the third term on
the right-hand-side of Eq. (11)) of ζ.
As described above, the calculation of q and the EPE source/sink involves summation of
second- and third-order terms on selected σ 0 surfaces. There are various potential sources of
errors introduced during the calculation. Two major sources are the calculation of q and the
interpolation of q and Vh onto σ 0 surfaces. Errors associated with calculation of q at model’s
grid points are probably negligible, since the computation method used here has a second-order
accuracy. More problematic is the vertical interpolation of q and Vh onto σ 0 surfaces. First of all, an
insufficient vertical resolution of a model makes it meaningless to examine detailed dynamical or
tracer characteristics on σ 0 surfaces. Also, σ 0 surfaces selected for diagnoses should be separated
by one or more model levels. At the same time, for the projection of interpolated values onto
σ 0 surfaces at a given longitude and latitude to be meaningful, the ratio of the vertical level
thickness to the horizontal grid length must be greater than the slope of σ 0 surfaces. Furthermore,
the projection of Vh onto σ 0 surfaces for calculation of Vσ yields large errors when the isopycnal
slope is large. We selected σ 0 surfaces for this diagnosis in such a way that they tend to lie in
separate model layers for a given longitude and latitude. We also found that the isopycnal slope is
generally smaller than the ratio of the vertical to the horizontal grid length, except in the immediate
vicinity of side boundaries and in convective regions where the slope can exceed 0.002 often and
1.0 in extreme cases. It is practically impossible to assess the errors arising from the interpolation
in this study, simply because the σ 0 surfaces are not defined in the model to begin with. The reader
should keep this limitation in mind when interpreting the results to be presented. Nevertheless, we
believe that the qualitative aspect of the results are robust, mainly because the vertical variation
length scales of q and Vh are significantly larger than the model’s level thicknesses in general
and the isopycnal slope is generally 0.001 or smaller. It is unlikely for errors associated with the
interpolation to accumulate systematically in such a way that it changes the qualitative aspect of
the results.
2.3. Trajectory calculation
The daily mean horizontal velocity interpolated onto σ 0 surfaces was used to calculate trajectories of particles released at various locations. Trajectory calculations were performed on only
six σ 0 surfaces that represent characteristic q¯ structures observed on the 19 surfaces. The six σ 0
surfaces selected for the trajectory calculations and the number of runs performed for them are
as follows: 26.5σ 0 : 72 runs; 27.1σ 0 : 97 runs; 27.4σ 0 : 86 runs; 27.7σ 0 : 85 runs; 27.825σ 0 : 78
runs; and 27.9σ 0 : 41 runs. In each of the trajectory runs, 100 uniformly distributed particles were
released from a box of 2◦ × 2◦ and advected by the full daily isopycnal velocity for the first set
and by the 8-year mean isopycnal velocity for the second set. The locations of particle release
were chosen to cover the σ 0 surfaces more or less evenly, roughly with a mesh of 5◦ × 5◦ , but
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M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
with some adjustments to place particles on interesting features. Although the daily velocity field
is available for only 8 years, we extended the runs to 32 years by using the velocity time series 4
times. Since we are interested in the overall pattern and strength of transport and mixing, not the
exact paths of particles, we believe that the repeated use of the velocity time series is not an issue
here. We employed the standard fourth-order Runge-Kutta method with a time step of 1 h for the
time integration of the particle path and archived instantaneous particle locations once a day.
3. Diagnostic results
3.1. Evolution of PV fields
Time series of the q fields exhibit very large fluctuations due to the seasonal cycle in areas
that experience outcropping and perturbations associated with geostrophic waves (eddies, from
here on). Careful visual examination finds slow variations associated with those in larger-scale
flows and the GS also. The contribution of ζ, to the instantaneous q is generally negligible, a
few percent or less, except along boundary currents and in the vicinity of the GS, where ζ can
be on the same order as f. Strong stratification just below the mixed layer during warm months
is associated with extremely large q values. The areas where these extreme values occur on σ 0
surfaces exist only a part of the year since these isopycnals outcrop and disappear during cold
months. On the other hand, there are regions in which the outcropping does not generate large
q, but small q due to weak stratification associated with generation of a relatively thick layer of
water mass of 12–19 ◦ C by heat loss during winters. These areas appear as pools of small q¯ on σ 0
surfaces discussed in Section 3.2. Generally speaking, on surfaces of small σ 0 values, the area of
the surfaces extend northward beginning in March and reaching the peak in August.
In areas of no outcropping, q fluctuations are predominantly due to waves of various frequencies. PV perturbations associated with these waves in non-outcropping areas are much smaller
than those observed in outcropping areas. They typically have vortex-like structures, some of
which are highly distorted, and appear to propagate along the background large-scale q¯ contours.
Eastward propagation of waves is visible along the core and to the immediate north of the GS in
the model, while slow westward propagation is seen to the south of it. The simulated GS exhibits
meanders that occasionally achieve large amplitudes and even break off its path. As shown later,
these waves that are omnipresent throughout the basin play important roles in the maintenance of
the large-scale q¯ structures. Large wave-induced q fluctuations occur where the q¯ gradient is large,
but do not necessarily reflect the amplitude of the waves. Diagnoses based on quasi-geostrophic
transient wave activity, which is a measure of the wave amplitude, will be reported in another paper.
3.2. Mean PV fields
The 8-year mean PV on nine of the selected σ 0 surfaces and the corresponding mean depth will
be shown and discussed in the following subsections. Note that q¯ and depth shown are the averages
over the period during which the σ 0 surfaces exist for each grid point. Thick contours indicate the
border between the area in which the σ 0 surface exists every day during the 8 years and the area
in which the surface disappears a part of the period due to outcropping or “collision” with the
topography. The first impression of these figures is the large-scale nature of the overall structures.
The scale is that of the atmospheric forcing and the large-scale topography, not that of local oceanic
motions. Most of the small-scale structures are due to topography. Only the strong currents along
parts of the western and northern boundaries appear in clear mean-current-generated signals.
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
37
3.2.1. Thermocline
From 26.0σ 0 to 27.4σ 0 , q¯ fields show some common characteristics that we refer to as the
thermocline type here. The mean PV on 26.0σ 0 , 26.5σ 0 , 27.1σ 0 , and 27.4σ 0 , and the mean depth
of these σ 0 surfaces are shown in Fig. 3. Very large values generated during warm months in areas
of outcropping are clearly reflected in q¯ . Also, the GS advects highly stratified water along the
western boundary in the top few hundred meters, resulting in a narrow band of very large q¯ there.
Although the outcropping areas are generally characterized by very large q¯ , there are regions of
exception. Particularly notable are pools of low q¯ in the middle latitudes that sweep the subtropics
eastward first, then northward to the Iceland Basin, with increasing σ 0 from 26 to 27.5. Daily q
fields show the outcropping lines reaching the pools of low-q water during cold months, indicating
that the low q is created by surface cooling and, then, advected into the interior. As indicated later,
using Fig. 7, trajectory calculations show that the waters in these pools tends to circulate within
the pools and their vicinity in a turbulent manner, slowly spreading along the contours of the
large-scale q¯ . Cross sections of q¯ and temperature (not shown) at various longitudes show that
these pools of low q¯ are a manifestation of relatively thick water masses (Fig. 4).1 At 65◦ W, a
pool of low q¯ is observed between 22◦ N and 35◦ N sandwiched by 25.5σ 0 and 26.5σ 0 (Fig. 4a).
The temperature of the water mass of this pool ranges from 18 to 22 ◦ C At 45◦ W, a low-¯q pool is
located between 30◦ N and 40◦ N at σ 0 values slightly larger than those seen at 65◦ W (Fig. 4b).
Here, the temperature range of the water mass is from 17 to 19 ◦ C. A low-¯q pool shifts northward
to between 35◦ N and 50◦ N at 25◦ W (Fig. 4c), and between 42◦ N and 55◦ N at 15◦ W (Fig. 4d).
At the same time, the σ 0 values between which a low-¯q pool exists increase to 26.75 and 27.0 at
25◦ W (Fig. 4c), and to 27 and 27.2 at 15◦ W (Fig. 4d). The water mass temperature of the pool
at 25◦ W ranges from 14 to 16 ◦ C, whereas it ranges from 11 to 13 ◦ C at 15◦ W. Examinations of
daily q fields and trajectory calculations suggest that these water masses are produced when the
pertinent σ 0 surfaces outcrop and are spread by eddy mixing.
Another major feature in q¯ is an east-north-eastward-elongated tongue or a band of nearly
homogeneous q¯ of moderately large values along the path of the simulated GS for the layers
26.5σ 0 to 27.4σ 0 . This feature is found in the climatological PV calculated by Keffer (1985) from
the Levitus, 1982 Ocean Atlas and also in the eddy-permitting model diagnoses by Nakamura
and Chao (2001). The tongue penetrates only to 50◦ W at 26.5σ 0 , but to the eastern boundary
at 27.4σ 0 . Trajectory calculations reveal strong advection by the GS and vigorous mixing by
eddies in the vicinity of the GS responsible for most of this feature from the western boundary to
approximately 20◦ W. Nakamura and Chao (2001) found the same mechanism behind the feature.
The eastern end of the feature is created by relatively weak advection and mixing of water that
originates at around 45◦ N near the eastern boundary. Examples of these trajectory runs are shown
in Figs. 5 and 6. Fig. 5 shows the time-integrated particle density in a box of 1◦ × 1◦ at the
beginning, year 5, year 10, and year 30. In the run shown in Fig. 5, trajectories of 100 particles
released in a 2◦ × 2◦ box centered at 74◦ W and 35◦ N were calculated by the full daily isopycnal
flow. Fig. 6 shows the integrated particle density of the same 100 particles advected by the mean
flow. The time-integrated particle density in 1◦ × 1◦ boxes is calculated by simply counting and
adding up the number of days during which a particle exists in each box. For example, if one
particle was found in a particular box 10 times and another particle was found in the same box
20 times during the first 5 years of a trajectory run, then the integrated particle density of the
1 The mean PV and σ shown in the cross sections are the averages of the non-interpolated three-dimensional values
0
during the entire period. Thus, q¯ along the mean σ 0 shown here does not equal q¯ on σ 0 surfaces shown in Fig. 3.
38
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Fig. 3. The mean PV on (a) 26.0σ 0 , (b) 26.5σ 0 , (c) 27.1σ 0 , and (d) 27.4σ 0 surfaces (upper panels), and the mean depth
of these σ 0 surfaces (lower panels). The units are PVU = 1 × 10−12 m−1 s−1 for the mean PV and meters for the mean
depth.
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
39
Fig. 4. Cross sections of the mean PV (color shade) and the mean σ 0 (contours) at (a) 65◦ W, (b) 45◦ W, (c) 25◦ W, and (d)
15◦ W. The units are PVU = 1 × 10−12 m−1 s−1 .
40
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Fig. 4. (Continued ).
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Fig. 5. The time-integrated particle density of a trajectory run on the σ 0 = 27.1 surface at the beginning, year 5, year 10, and year 30. The full daily mean flow was used for the
trajectory calculation. The units are particles per box of 1◦ × 1◦ . See text for the details of the calculations.
41
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Fig. 6. Same as in Fig. 5, but for trajectories calculated with the mean flow.
42
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
43
box is 30 at year 5. We choose to show this quantity rather than usual trajectory paths because
we are primarily interested in the spread of particles, not the exact paths of the particles. For our
purpose, the integrated particle density is much easier to examine visually and to grasp the rate,
direction, and extent of the particle movement. Particles remain in a narrow path along the mean
flow and travel long distances when they are advected by only the mean flow as shown in Fig. 6.
On the other hand, when they are advected by the full flow that contains eddies, they spread out
to a much larger area, although the travel distances are shorter. Although the travel distances
are shorter when the eddy advection is included, the travel path length is greater with the eddy
advection. Clearly, the area painted by particles advected only by the mean flow and the q¯ field
do not show any structural similarity, indicating eddies’ critical roles in forming the q¯ structure.
Streaklines of the smoothed mean flow and the ratio of eddy kinetic energy (KEE ) to the mean
kinetic energy (KEM ), KEE /KEM , are shown in Fig. 7 to suggest transport and mixing patterns
observed in trajectory runs. The mean flow and variance fields used for these figures are smoothed
by re-gridding the original fields into 1◦ × 1◦ boxes by simple averaging. The streaklines are
plotted using the streamline setting of GrADS software, and are given only to show the direction
of the mean flow, since showing the mean flow in vectors would make a figure extremely hard to
see. Generally, particle trajectories on σ 0 surfaces show spreads of particles in various directions,
but wider spreads along the q¯ contours or within areas of weak q¯ gradients. They also show a
Fig. 7. Streaklines of the mean flow and the ratio of eddy kinetic energy to the mean kinetic energy, KEE /KEM (color
shade), on surfaces of 26.5σ 0 , 27.1σ 0 , 27.825σ 0 , and 27.9σ 0 . The σ 0 values are shown on the top of each frame. Both the
mean flow and velocity variances used here have been smoothed by simple averaging in 1◦ × 1◦ areas.
44
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Table 1
The mean daily path length of particles in meters per day while they are in motion
σ0
No. of runs
Full flow (m day−1 )
Mean flow (m day−1 )
Full/mean
26.5
27.1
27.4
27.7
27.825
27.9
All
72
97
86
85
78
41
459
277.1
146.8
130.8
98.7
95.9
118.6
144.2
70.1
30.7
23.0
25.5
47.4
46.9
38.8
3.95
4.78
5.69
3.87
2.02
2.53
3.72
tendency to spread faster into the direction of the mean flow, as shown in Figs. 5 and 6. Such a
tendency is not visible, however, in areas of extremely weak (with respect to eddies) mean flow. In
areas of small KEE /KEM , eddy mixing is inhibited, allowing particles to move more or less along
the mean flow in a cluster. On the other hand, vigorous eddy mixing moves particles around in a
chaotic manner in areas of very large KEE /KEM . Extremely large particle spread is accomplished
easily when the mean flow field has saddle points in areas of large KEE /KEM . When areas of
large KEE /KEM are separated by a band of small KEE /KEM , the band of small KEE /KEM acts
as a mixing barrier. On the surfaces of 26.5σ 0 , 27.1σ 0 , and 27.4σ 0 , areas of small KEE /KEM
are limited to the vicinity of boundaries, exhibiting barrier-free strong turbulent mixing in the
interior. The mean daily path length (only while particles are in motion) of all particles for all
runs for each σ 0 surface is listed in Table 1. The ratio of the daily mean path length by the full
flow advection to that by the mean flow advection is also given in Table 1. Nakamura and Chao
(2001) computed a similar ratio for their trajectory runs and found the ratio to be about 2.7. In the
present work, the overall ratio is about 3.7, indicating that eddy mixing is resolved significantly
better with the higher spatial resolution, a 0.1◦ × 0.1◦ horizontal mesh with 54 levels, used in this
model. The ratio of 3.7 suggests that eddies are almost three times as strong as the mean flow in
terms of advective mixing.
3.2.2. Intermediate layers
Fig. 8 shows q¯ on 27.7σ 0 , 27.78σ 0 , and 27.825σ 0 , and the mean depth of these σ 0 surfaces.
The mean PV on 27.5σ 0 , 27.6σ 0 , and 27.7σ 0 surfaces shows a transitional character from the
thermocline type to what we refer to as the intermediate-water type. On these surfaces, the band
of relatively large q¯ at upper layers is no longer robust, eroded by injection of low-q water from
the Labrador Basin through the deep western boundary current. At the same time, high-q water
of the Norwegian Sea begins to show up spilling over in the northern part of West European
Basin. From 27.76σ 0 to 27.825σ 0 , the q¯ field shows nearly uniform small values to the west
of the Mid Atlantic Ridge (MAR), under the tongue of relatively large q¯ in the thermocline.
This layer of nearly homogeneous low-¯q occupies varying depths of water from 500 to 2500 m
as shown in Fig. 4(a and b). Its thickness generally decreases southward, as it can be seen to
decrease from 2000 m at 45◦ N to only 500 m or so at 32◦ N in Fig. 4b. This feature, the pool of
nearly homogeneous low-¯q, was found in hydrographic data by Lozier (1997, 1999)2 and in the
The simulated pool of nearly homogeneous low-¯q is located a few to several hundred meters above the observed pool.
Relatively coarse vertical resolution below the thermocline in the model and referencing σ to the surface at all depths are
likely to be responsible for this discrepancy.
2
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
45
Fig. 8. The mean PV on (a) 27.7σ 0 , (b) 27.78σ 0 , and (c) 27.825σ 0 surfaces (upper panels), and the mean depth of these
σ 0 surfaces (lower panels). The units are PVU = 1 × 10−12 m−1 s−1 for the mean PV and meters for the mean depth.
46
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
eddy-permitting model output by Nakamura and Chao (2001), but not found in the Levitus ocean
atlas. Trajectory runs show that the origin of the low-q water is likely to be, again, the Labrador
Basin (not shown). The low-q water is generated there by deep convection during winters, then
advected southward and spread vigorously throughout the pool of low-¯q, eventually exiting the
pool at around 30◦ N through the western boundary current after circulating within the pool for a
very long time. The area of this low-q water generation is seen as an island of minimum q¯ on the
27.825σ 0 surface in the Labrador Basin (Fig. 8). The low-q water generated in the Labrador Basin
is contained within the pool of low q¯ to a remarkable degree. Similarly clear separation of the
low-q water to the west of the MAR from the surrounding waters was reported by Nakamura and
Chao (2001) also. An example of trajectory runs that show this characteristic is shown in Fig. 9.
This containment may be deduced from bands of small KEE /KEM (i.e. weak eddy mixing) and
weak divergence in the mean flow streaklines along the edge of the pool (Fig. 7). This separation
of low-q water from the comparatively high-q water to the east and south of the low-¯q pool is
most likely due to the MAR.
In the intermediate layer, high- to moderate-q water originates from the Norwegian Sea and the
Mediterranean Sea in the model. However, the model produces only very weak Mediterranean
Outflow and, thus, is likely to substantially underestimate the rate of high-q water production
by the outflow. Trajectory calculations suggest that most of the high- and moderate-q water to
the east of the MAR has its origin in the weak overflow from the Norwegian Sea into the West
European Basin (not shown). The high-q water enters the eastern North Atlantic through the
canyon on the southeastern side of the Rockall Plateau, spreads southward then southwestward
along the MAR while mixing vigorously with low-q water arriving from the north. Nakamura
and Chao (2001) also found this high- to moderate-q water to the east of the MAR, but attributed
it to the Mediterranean Outflow. We note that Nakamura and Chao (2001) mistakingly identified
the Mediterranean Outflow to be the source of the high-q water due to the smaller area coverage
in their diagnoses. Given the location of high-q water and flow patterns in their model output, we
believe that their model output also had the high-q water in the intermediate layers originating
from the Norwegian Sea. We find moderate-q water of Mediterranean Outflow origin between
27.7σ 0 and 27.8σ 0 spreading and mixing very slowly along the eastern boundary, but not along
the MAR.
3.2.3. Deep layers
Below 27.85σ 0 , the pool of nearly homogeneous low-¯q to the west of the MAR and moderately
high q¯ along and to the east of the MAR disappear. The mean PV on 27.875σ 0 and 27.9σ 0 surfaces,
and the mean depth of these σ 0 surfaces are shown in Fig. 10. Moderate values of q¯ returns to
the western side of the MAR. In these deep layers, eddy mixing is still vigorous to the west of
the MAR. Trajectory runs show a slow southward outflow of water near the western boundary
at around 30◦ N after chaotically circulating in the pool of moderately high q¯ underneath the
simulated GS for a long time (not shown). Farther below the level of the largest σ 0 selected in our
calculations, q¯ values decrease toward the bottom in general, as seen in cross sections presented in
Fig. 4. The structure and values of q¯ in these deep layers in the model are similar to those calculated
by Lozier (1997, 1999), in spite of the difference in the reference for which σ is calculated.
3.2.4. Baroclinically unstable conditions
The quasi-horizontal structures of q¯ presented above suggest that most of the basin is prone to
baroclinic instability. The quasi-horizontal q¯ gradient reverses its sign in the vertical at some depth
in most places on a large-scale in an organized manner. Cross sections of the meridional gradient
Fig. 9. Same as in Fig. 5, but for a trajectory run on the σ 0 = 27.825 surface.
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
47
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M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Fig. 10. The mean PV on (a) 27.875σ 0 and (b) 27.9σ 0 surfaces (upper panels), and the mean depth of these σ 0 surfaces (lower panels). The units are PVU = 1 × 10−12 m−1 s−1
for the mean PV and meters for the mean depth.
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
49
of q¯ at 65◦ W and 35◦ W are shown in Fig. 11 as examples of a typical vertical structure. Near
the surface, the gradient of q¯ is predominantly meridional because of differential solar heating
and is almost entirely positive. The thickness of this layer changes noticeably from the south to
north near the simulated GS. Below this thin near-surface layer of positive meridional gradient,
there is a layer of small negative gradient. In plots of q¯ , this layer of small negative meridional q¯
gradient appears to have a nearly homogeneous q¯ field within the area of the negative meridional
gradient. Farther below, there is another layer of positive gradient, although the magnitude is much
smaller than that near the surface. Roughly speaking, the layer of negative meridional gradient is
thinner at lower latitudes, and vice versa, indicating the latitudinal dependence of the penetration
depth and scale height of active baroclinic waves. Effects of major topographic features can also
be seen when details are examined. Nakamura and Chao (2001) suggested that this q¯ structure
is a manifestation of the large-scale oceanic state approaching baroclinic neutrality in a fashion
similar to the atmosphere (Lindzen, 1993). If the theory is relavent here, then the difference in the
thickness of the layer of nearly homogeneous q¯ reflects the difference in the shortest wave that
can grow in the given large-scale state.
3.2.5. Contribution of relative vorticity
Fig. 12 shows the total ζ contribution to q¯ on the 26.0σ 0 and 27.0σ 0 surfaces. The magnitude
of ζ¯ contribution ranges from 10 to 50% in the core of the Florida Current and boundary currents,
but is generally less than 1% even at upper levels. The ζ term is, on the other hand, a robust
second-order contributor in the vicinity of the GS where eddy kinetic energy is large, and is in
the range of a few to 10% down to the surface of 27.3σ 0 , roughly 300 m in depth. Elsewhere,
the ζ term is less than 1%. One may conclude, therefore, that the mean large-scale PV can be
calculated fairly accurately from the mean isopycnal thickness data.
3.3. Sources/sinks of PV
Water on an isopycnal surface gains q when it receives cyclonic vorticity through friction or
when the isopycnal thickness decreases due to buoyancy gain on its upper part or buoyancy loss on
its lower part, and vice versa. Thus, one would expect q gain where the surface wind stress curl is
cyclonic, and vice versa. PV gain through friction can occur along boundaries also. Additionally,
one would expect q gain when an area is receiving buoyancy forcing (i.e. heat input or freshwater
input) at the surface, and vice versa. On a large-scale, on isopycnal surfaces, we expect the net
input of q into the ocean in areas that outcrop, the net transport of q from the areas of outcropping
to areas of no outcropping, and the net q dissipation in the areas of no outcropping. Due to the fact
that areas of outcropping experience both q gain and q loss during surface heating and cooling,
respectively, through a year, and also because the number of days during which a particular locale
experiences outcropping varies from location to location, the net q gain or loss may be observed
in small areas within the area of outcropping.
Plots of the total q source/sink on σ 0 surfaces show large-scale structures, despite noisiness on
small scales. An example is shown in Fig. 13.3 Because of the variable duration of the period of
outcropping and highly variable advective strength, the net effect of surface heating and cooling on
3 As mentioned in Section 2, the source/sink is calculated from the total q change due to the mean flow and eddy
advection. We note that contributions by the mean flow and eddies are of the same order of magnitude, and that they tend
to cancel out each other, making their sum less noisy than each of them. This tendency was also found by Nakamura and
Chao (2001).
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M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Fig. 11. Cross sections of the meridional gradient of the mean PV (color shade) and the mean σ 0 (contours) at (a) 65◦ W
and (b) 35◦ W. The units are PVU per kilometers.
Fig. 12. The contribution of ζ to the mean PV on two σ 0 surfaces in percent.
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
51
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M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Fig. 13. The 8-year total production of PV on the σ 0 = 26.5 surface. The units are PVU.
q shows large areas of net gain and loss even within the area of outcropping. A typical structure
in the outcropping areas is a large band of net q gain surrounded by large areas of net q loss,
suggesting that the net q gain in the area of particularly strong heating effect is distributed to the
surrounding by advection. We computed the q source/sink due to ζ and compared with that due
to f. We found that the f contribution is somewhat larger than the ζ contribution, but not by an
order of magnitude in general. We also found that the fields are noisier when these two terms are
plotted individually, indicating their tendency to cancel out each other locally. This may appear
surprising since the ζ contribution to q¯ was found to be generally negligible. However, when one
realizes that viscous dissipation operates on ζ, not f, it seems quite reasonable.
Clearly evident in Fig. 13 is the presence of many small features that are not necessarily noise,
but due to topographic features and to the meanders of the mean flow. When the area-averaged
total q source/sink is calculated for the outcropping and non-outcropping areas separately, much
simpler and comprehensible pictures emerge. The area-averaged total q source/sink due to f and
ζ for the outcropping area, non-outcropping area, and the entire area on all calculated σ 0 surfaces
is given in Tables 2–4, respectively. In the outcropping area, on average, both f and ζ show net q
gain, except for small net loss through f on 27.76σ 0 and 27.825σ 0 surfaces. The total, the sum of
f and ζ contributions, is a net gain on all σ 0 surfaces. Note that the ζ contribution is not negligible
generally even when averaged for the entire outcropping area. In the non-outcropping area, on
average, both f and ζ terms show the net q loss with some exceptions in the upper layers. Again, the
ζ term is not negligible in general. These numbers are in agreement with a broad picture of net q
gain in outcropping areas and net q loss in non-outcropping areas. In interpreting the q source/sink
by ζ, one has to keep in mind that the sense of ζ contribution depends on the vorticity forcing
through the surface, i.e. the sense of wind stress. Broadly speaking, the sense is cyclonic in the
northern half of the basin, where σ 0 surfaces tend to outcrop more readily, and anti-cyclonic in
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
53
Table 2
Total PV source due to f (central column) and ζ (right column) averaged over outcropping areas on σ 0 surfaces
σ0
26.0
26.25
26.5
26.75
27.0
27.1
27.2
27.3
27.4
27.5
27.6
27.7
27.76
27.78
27.8
27.825
27.85
27.875
27.9
SP−f (PVUm−2 )
+2860
+2954
+3114
+3183
+3290
+2631
+2357
+2183
+2013
+4329
+2212
+1159
−206.2
+22.52
+973.2
+638.8
+516.6
−24.49
+114.6
SP−ζ (PVUm−2 )
+229.5
+338.1
+846.0
+1043
+1269
+1098
+1122
+943.7
+783.3
+449.2
+448.6
+491.4
+380.4
+351.1
+250.0
+240.0
+383.4
+378.1
+488.4
Note that 1PVU = 1 × 10−12 m−1 s−1 .
Table 3
Total PV source due to f (central column) and ζ (right column) averaged over non-outcropping areas on σ 0 surfaces
σ0
26.0
26.25
26.5
26.75
27.0
27.1
27.2
27.3
27.4
27.5
27.6
27.7
27.76
27.78
27.8
27.825
27.85
27.875
27.9
Note that 1PVU = 1 × 10−12 m−1 s−1 .
SP−f (PVUm−2 )
+1397
+1421
+673.7
+370.0
−390.1
−107.0
−132.5
−388.8
−443.3
−983.0
−377.7
−161.3
−59.98
−49.14
−72.69
−85.86
−123.6
−211.6
−234.0
SP−ζ (PVUm−2 )
−359.5
+489.7
+81.39
−102.6
−176.9
−86.12
−132.3
−122.9
−100.5
−52.98
−90.59
−61.76
−12.83
−15.44
−17.06
−12.52
−18.25
−7.359
−31.79
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M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Table 4
Total PV source due to f (central column) and ζ (right column) averaged over the entire areas on σ 0 surfaces
σ0
SP−f (PVUm−2 )
SP−ζ (PVUm−2 )
26.0
26.25
26.5
26.75
27.0
27.1
27.2
27.3
27.4
27.5
27.6
27.7
27.76
27.78
27.8
27.825
27.85
27.875
27.9
+2552
+2454
+2031
+1677
+1028
+853.2
+627.4
+318.8
+157.2
+87.37
+65.17
+26.62
−78.36
−40.83
+48.98
−14.70
−50.47
−186.5
−178.5
+106.3
+387.8
+498.3
+430.1
+380.4
+329.2
+250.7
+170.5
+115.5
+48.35
+12.56
+16.95
+36.33
+28.08
+13.87
+12.24
+27.55
+45.98
+51.10
Note that 1PVU = 1 × 10−12 m−1 s−1 .
the southern half of the basin, where the surfaces tend to remain shielded from the surface. When
the entire area is considered, the ζ term shows a net q gain on all σ 0 surfaces, while the f term
shows a net gain from 26.0σ 0 to 27.7σ 0 and on 27.8σ 0 , and a net loss on σ 0 : 27.76, 27.78, 27.825,
27.85, 27.875, and 27.9. The net loss on the larger σ 0 surfaces is attributed to the Norwegian Sea;
non-outcropping layers/areas in the Norwegian Sea are not covered even at 27.9σ 0 and, thus, the
net q loss in the Norwegian Sea is not included in the values given in Table 4.
3.4. Source/sink of EPE
In the equations derived by Nakamura (1998), the net source/sink of eddy potential enstrophy
(EPE) on an isopycnal surface is related to the up-/down-gradient effective eddy q flux on the
surface. They reveal, therefore, the anti-diffusive or diffusive nature of eddies on isopycnal surfaces. In practice, one has to be cautious in areas of outcropping when interpreting SE , since, as
discussed above, q perturbations arising from the seasonal cycle are very large.
The total source/sink of EPE on σ 0 surfaces exhibits, while significant noisiness is present,
clearly identifiable large-scale structures. The structures vary noticeably with depth, particularly
to the west of the MAR. Examples are shown in Fig. 14. Generally, on the large-scale, the net sink
or a diffusive character of eddies is observed in relatively quiet areas in the subtropics and east
of the MAR. In areas of vigorous eddy activity, e.g. in the vicinity of the GS, there tends to be
more of the net source or an anti-diffusive character of eddies. However, there are exceptions to
these general rules. For example, the relatively quiet area in the subtropics on the 27.7σ 0 surface
shows a robust anti-diffusive character of eddies on a large scale. When contributions from f and ζ
are compared, they show generally similar orders of magnitude and tend to cancel out each other
locally.
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
55
Fig. 14. The 8-year total production of EPE on the σ 0 = 26.5 and 27.1 surfaces. The units are PVU2 .
To obtain a simple picture, we computed the area-weighted averages for the f and ζ contributions
on all σ 0 surfaces separately for the outcropping area and non-outcropping area. The area-averaged
total EPE source/sink due to f and ζ in the outcropping area, non-outcropping area, and the entire
area is given in Tables 5–7, respectively. As one may expect, the ζ contribution is almost entirely
negative or diffusive on average both in outcopping and non-outcropping areas. On the other hand,
the f contribution shows a reversal in its sign, from diffusive to anti-diffusive, at mid levels in
both outcropping and non-outcropping areas. Even the sum of the two is anti-diffusive on 6 σ 0
surfaces even in the non-outcropping area where most of q perturbations is due to eddies. This is
somewhat surprising and casts further doubt on the uniformly diffusive parameterization of eddy
Table 5
Total EPE source due to f (central column) and ζ (right column) averaged over outcropping areas on σ 0 surfaces
σ0
SE−f (103 PVU2 m−2 )
SE−ζ (103 PVU2 m−2 )
26.0
26.25
26.5
26.75
27.0
27.1
27.2
27.3
27.4
27.5
27.6
27.7
27.76
27.78
27.8
27.825
27.85
27.875
27.9
−2877
−4017
−10100
−11573
−9070
−10302
−9670
−9227
−7649
+3854
+2962
+2262
+1139
+109.2
+2863
+8907
+7458
+3248
+5502
−2294
−2184
−1365
−792.1
−964.3
−1228
−1164
−1956
−1885
−2253
−1120
−77.94
−581.7
−533.7
−587.2
−657.6
−375.6
−90.53
+15.50
Note that 1PVU = 1 × 10−12 m−1 s−1 .
56
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
Table 6
Total EPE source due to f (central column) and ζ (right column) averaged over non-outcropping areas on σ 0 surfaces
σ0
SE−f (103 PVU2 m−2 )
SE−ζ (103 PVU2 m−2 )
26.0
26.25
26.5
26.75
27.0
27.1
27.2
27.3
27.4
27.5
27.6
27.7
27.76
27.78
27.8
27.825
27.85
27.875
27.9
−15403
−7017
−1736
−1286
−652.9
−688.0
−627.4
−388.8
−276.2
−2000
−534.4
+43.87
+133.4
+185.8
+267.7
−4.554
−1.049
+149.9
+208.0
−4190
−534.7
−850.5
−801.3
−608.1
−395.6
−533.2
−312.9
−216.8
−114.4
−152.0
−30.32
−45.96
−46.23
−26.78
−32.07
−40.30
−59.71
−79.41
Note that 1PVU = 1 × 10−12 m−1 s−1 .
Table 7
Total EPE source due to f (central column) and ζ (right column) averaged over the entire areas on σ 0 surfaces
σ0
SE−f (103 PVU2 m−2 )
SE−ζ (103 PVU2 m−2 )
26.0
26.25
26.5
26.75
27.0
27.1
27.2
27.3
27.4
27.5
27.6
27.7
27.76
27.78
27.8
27.825
27.85
27.875
27.9
−5516
−5000
−6388
−6068
−3898
−4060
−3387
−2820
−2078
−815.9
+61.28
+359.6
+259.2
+294.0
+568.2
+870.6
+850.9
+578.6
+1051
−2694
−1644
−1136
−797.0
−745.4
−687.3
−725.8
−764.8
−624.7
−546.8
−316.8
−140.1
−113.0
−104.1
−91.70
−93.50
−78.60
−63.98
−41.95
Note that 1PVU = 1 × 10−12 m−1 s−1 .
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
57
mixing that has already received strong criticism from several works (Nakamura and Chao, 2000,
2002; Tansley and Marshall, 2000; Solovev et al., 2002). When the entire area is considered, this
anti-diffusive role of eddies and seasonal q variations on surfaces of larger σ 0 shows up clearly,
in spite of the robust diffusive role of the ζ contribution. Note that the ζ contribution is somewhat
smaller than that of f, but certainly not negligible on most of the σ 0 surfaces considered here.
4. Concluding remarks
We presented here an overview of the large-scale q fields and source/sink of q and EPE on σ 0
surfaces in the simulated North Atlantic. Overall, the results are qualitatively the same as those
presented in Nakamura and Chao (2001, 2002), in spite of the substantially improved spatial
resolution of the model and output amount available for the diagnoses. One major exception
is the origin of relatively large q in the intermediate layers. We identified the source to be the
Norwegian Sea, while Nakamura and Chao (2000a,b) mistakingly identified the source to be the
Mediterranean Sea. We find very important roles of eddies in the formation and maintenance of
large-scale structures. In the current work, the role of eddies in advective mixing is significantly
higher than that found in Nakamura and Chao (2000a,b), as expected from the stronger eddies
simulated in OFES due to the higher spatial resolution. There is a possibility of even higher
significance of eddies in advective mixing in numerical models of higher spatial resolution, say
a 0.01 × 0.01◦ horizontal resolution with 100 vertical levels. As in Nakamura and Chao (2000a,
2001, 2002), we find that eddies are not necessarily diffusive locally. A somewhat surprising
finding in the current work is that eddies can be anti-diffusive in the ocean interior even on basinaverage on some σ 0 surfaces. It implies eddy-induced generation of basin-scale available potential
energy in the intermediate and deep layers. The output sampling frequency and time-coverage
used in the current work (daily mean) are significantly better than those used by Nakamura and
Chao (2001, 2002) (every 3-day snapshots) in terms of reducing aliasing when the q and EPE
source/sink are calculated, giving us a higher confidence in the results. This finding casts stronger
doubt on the down-gradient eddy mixing parameterizations widely used in coarse-resolution
models and will need to be examined further in the future.
While most of the findings presented here suggest that the use of coarse-resolution models
is highly questionable, at least for studying the mid- and high-latitude phenomena, it strongly
supports the calculation of large-scale background PV fields with hydrographic data on a relatively
coarse mesh only. We find that the ζ contribution to q¯ is completely negligible in most of the basin.4
Thus, the first-order large-scale background dynamical characteristics can be deduced from the
climatological hydrographic data only. Only if the ζ contribution in the vicinity of the boundary
currents is available, in addition to the hydrographic data, we would have an accurate background
PV field that can be used for various dynamical analyses. Such a reliable reference state will be
extremely useful for constraining realistic numerical models that, in turn, will provide us with
better pseudo-data for various research works.
The large-scale nature of the q¯ fields presented here is a manifestation of the scale of the
external forcing—the atmosphere. Most likely because of the reasonable accuracy of the forcing
used to drive the ocean here, the three-dimensional q¯ field is in fairly good agreement with that
calculated from hydrographic data. The mean kinetic energy and eddy kinetic energy in the model
are also in fairly good agreement with the observational velocity data. This gives us confidence
4
The ζ term can be significant in the instantaneous q, however.
58
M. Nakamura, T. Kagimoto / Dynamics of Atmospheres and Oceans 41 (2006) 28–59
in performing the same diagnoses on the North Pacific where observational data are much more
scarce than they are in the North Atlantic. We will report the results of the diagnoses on the North
Pacific in the future. We will also examine the dynamical characteristics in the simulated southern
oceans in the future and report the results.
Acknowledgment
Model integration and diagnoses were done on the Earth Simulator maintained by the Earth
Simulator Center. We are grateful to Drs. H. Sakuma, Y. Sasai, and H. Sasaki for providing the
model output to us and preprocessing the output.
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