JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 97, NO. C1, PAGES 687-701, JANUARY
15, 1992
Kinetic Energy Analysis of an Eddy Resolving,Primitive Equation Model of
the North
A.M.
Atlantic
TREGUIER
Laboratoire de Physique des Oceans, IFREMER,
France
As part of the World Ocean Circulation Experiment community modelling effort, a realistic
primitive equation model of the North Atlantic Ocean has been implemented with a resolution
high enough to allow the generation of eddies. In the present paper, the kinetic energy of the
model ocean circulation is analyzed. The eddy kinetic energy distribution in the model shows a
good agreement with data in the western Atlantic but a deficit in the eastern Atlantic, especially
at periods of about 100 days. Kinetic energy balances are computed to allow comparison between
the primitive equation model and previous quasi-geostrophic eddy resolving models. Essential
instability mechanisms are present in both cases,but they are not efficient enough in the primitive
equation model, which is too viscousand lacks the inertial character of quasi-geostrophic solutions.
Our study therefore emphasizes the need for even higher resolution models.
1. INTRODUCTION
still coarsecompared with quasi-geostrophiceddy-resolving
models. With a grid spacing of 37 km at 34øN, the first
The availability of more powerful computersin the recent
Rossbyradius wavelength(2•rx 30 km) is resolvedby only
years has made possiblethe implementation of basin-scMe,
five grid points. The Rossby radii of the higher baroclinic
eddy resolving primitive equation models. "Eddy resolv- modes are not resolved at all. The situation is even worse
ing" in this context means that the spatial resolution of the
at higherlatitudes (at 60øN the zonMgrid spacingis 22 km,
modelis high enoughto allow somevariability ("eddies") the meridional grid spacing 33 km, and the first Rossbyrato be generatedspontaneouslythrough nonlinear instability
dius 10 km). Therefore, one cannot expect the model to
mechanisms,evenwhen the forcingis steady. The first eddy-
fully represent baroclinic instability processeswhich are one
resolvingglobalcirculationmodels(EGCMs) introducedin of the main mechanismsfor eddy generation.
the seventiesused small domainsand/or quasi-geostrophic
equations. They have provided a valuable insight in the
dynamicsof mesoscale
eddies(see Holland [1985]for a review). However,they have beenrun mainly with idealized
wind forcing in simple geometriesand without active thermodynamics. They have not answeredimportant questions
such as the influence of eddies on the global thermohaline
circulation and heat transport, the origin of the eddy activity in relatively quiet oceanic regionslike the northeastern Atlantic, or the. generation and life cycle of important
Despite those limitations, an analysisof the eddy energy
in CME is worthwhile, because this experiment is the first
of its kind and represents the "state of the art" at the beginning of the WOCE years. The present paper addresses
the eddy kinetic energy distribution and the eddy kinetic
energy balances. Our purpose is to understand the eddy kinetic energy cycle in the model, comparing both with data
and simpler models.
Followinga short descriptionof the model (section 2),
the eddy kinetic energy distribution is analyzed in section 3
mesoscale
featureslike "meddies"(Mediterraneaneddies). and related to the location of the mean currents. Special
Eddy-resolvingprimitive equation models begin to ad- attention is given to the eastern Atlantic, where the model
dress those issues. Such a model has been implemented for
kinetic energy is low. Energy balancesare presented in secthe North Atlantic at the National Center for Atmospheric tion 4. Techniques which have been developed mainly in
Research(NCAR) by Bryan and Holland [1989]. The two the context of quasi-geostrophic models are applied to the
experimentsdiscussed
here (hereinafterCME1 and CME2) primitive equation model to give insight into barotropic and
wererun in the contextof the WOCE (World OceanCircu- baroclinic instability processes. Because of the spatiM inlation Experiment) communitymodellingeffort. The model homogeneity of the circulation, spatial maps of the transfer
was initialized with climatologicaldata and integrated for terms and local energy budgets are considered.
25 years with the observed annual cycle of the wind stress
forcing and surface heat fluxes over the North Atlantic.
2. DESCRIPTION
OF THE CME
EXPERIMENTS
Large friction coefficientswere thought necessaryin the first
experiment (CME1), but becausesome current velocities Model and Forcings
were underestimatedin the equatoriM regions,5 more years
wererun with lesshorizontaland verticalmixing(CME2).
The model resolution allows the generation of eddy activity, for example Gulf Stream meandersand rings. However, the meshof 1/3 ø in latitude and 2/5 ø in longitudeis
The modelconfigurationis describedby Spall[1990]and
Schott and Bb'ning[1991] and will be only briefly recalled
here.
The
model
covers the
Atlantic
Ocean
from
65øN. A choice has been made to treat the artificial
15øS to
north
and south boundariesas walls (no inflow or outflow) but
to allow verticM recirculation in "buffer zones" adjacent to
them, by relaxing the temperature and salinity to their climatologicM monthly vMues. The Mediterranean outflow
has been dealt with in the same fashion, since the strait
of Gibraltar is not resolved by the model grid. Relaxation
Copyright 1992 by the American Geophysical Union.
Paper number 91JC02350.
0148-022 7/92/91J C-02350 $05.00
687
688
TREGUIER:
KINETIC
ENERGY ANALYSIS OF ATLANTIC
to climatology has also been added in part of the Labrador
Sea due to the absence of any parameterization of sea ice
CIRCULATION
MODEL
of 3. Although these low coefficientsallow a small amount
of numericalnoiseto developin the equatorialregion(wiggleson the contoursof Plate 2), the equatorialcirculationis
The primitive equationnumericalmodelof Bryan [1969] more realisticin CME2 [Schottand B6ning, 1991]. Thereand Cox[1984]is used. It solvesthe primitiveequationsus- fore the last 4 years of the second experiment have been
formation
in the model.
ing a second-orderfinite difference schemewhich conserves used in the present analysis to calculate mean and eddy kimomentum, energy, tracer content, and tracer variance ex- netic energies. The trend in basin-averagedkinetic energy
cepted for explicit dissipation and forcing. The horizontal during thoseyearsis very small (a few percentof the am-
resolutionis 1/3 ø in latitude from 15øSto 65øN,and 2/5 ø in
longitudefrom 100øWto 15øE.It is necessaryto havea large
numberof verticallevels(30) to allowthe representation
of
mesoscalebottom topography, since the topography is seen
by the model as a series of discrete levels and bathymetry
gradients are not necessarilysmooth when there are too few
levels. The first level is at a depth of 17.5 m, and spacing
between levels grows linearly to reach 250 m at 1250 m and
remain constant deeper down.
The surfaceforcing is calculated from climatologicaldata
sets. The monthly wind stress of Hellerman and Rosenstein
[1983] is used. The heat ttux is calculatedby a bulk formula [Hah, 1984]as a functionof the differencebetweenthe
model-predicted temperature and a prescribed "effective"
atmospheric temperature. The surface salinity is relaxed
plitude of the annumcycle). Of course,this doesnot mean
that the solution is in statistical equilibrium. The density
field is evolving from the initial conditions. The kinetic energy distribution and balancespresentedin the next sections
are stable over a few years but could change on longer time
scales.The model adjustment to the decreasein mixing co-
effiCients
performedat the beginningof CME2 is discussed
in section
4.
3.
MODEL
KINETIC
ENERGY
Eddy Kinetic Energy Distribution
The eddy kinetic energy (EKE) at the first level of the
model (Plate 1) is concentratedin the vicinity of the west-
toward the Levitus[1982] climatology.
ern boundary currents and near the equator, in agreement
with observedEKE patterns. A map derived from Geosat al-
The model has been initialized with temperature and
salinity from the Levitus data set and integrated forward
in time for 25 years. The calculation is entirely prognostic
in the interior: only boundary values are forced toward cli-
ison (Figure 1). Current knowledgeof surfaceEKE is based
on ship drift measurements[Wyrtki et al., 1976], drifting buoys[Richardson,1983],or satellitedata. Thosetech-
timeterdata [Le Traonet al., 1990]is presented
for compar-
matology(at the surfaceand in the buffer zonesnear the niques agree on EKE distribution patterns but not on EKE
open boundaries).However,densityfieldsin the deepocean levels: the maximum in the Gulf Stream is 1800 cm2 s-•
cannot evolve very far from the initial conditions in such a
in Richardsoh's data and 2600 cm • s-• in Geosat data. Le
short simulation. Bryan and Holland [1989]note that the
Traon et al. suggestthat this is due to a differentfiltering of
Gulf Stream rings, becausethe techniquessample different
changesin the averaged thermal structure and heat content
are small. The seasurfacetemperature(SST) remainsclose space and time scales. In the model, the maximum EKE is
slightlybelow1000cm• s-2. This valueis not inconsistent
to the climatological values away from the Gulf Stream region. This shows that the Levitus SST data are relatively
consistent with the surface heat flux parameterization of the
model.
Parameterizations
In experiment CME1, subgrid scale mixing processesare
representedwith (1) a horizontalbiharmonicfriction with
a coefficient of -2.5
x 10•m4s -• for momentum and trac-
ers, (2) a verticalsecondorderdiffusionwith a coefficientof
with
the observations
because the model does not take into
account high-frequencyatmosphericforcingsand has no active mixed layer, both of which could generateeddy activity.
However, the eastward decay of eddy kinetic energy is too
abrupt in the model. Whereas both Geosat and drifting
buoysindicateeddy energylevelsabove1000 cm• s-a at
50øW near the axisof maximumEKE (along38øN), there
are no valueshigherthan 100 cm• s-• in the modelat that
longitude. Along 20øW between40øN and 50øN, EKE lev-
30 x 10-4 for momentumand 0.3 x 10-4 mes-• for tracers, els higherthan 200 cm2 s-a havebeenobserved
by Kraus
(3) a quadraticbottomdragwith coefficient
10-3, and(4)
and K5se[1984]usingdrifting buoydata, whereasthe model
a representationof convectiveadjustment by using a large
EKE does not exceed 5 cm 2 s-•.
(1 m2s-•) verticalmixingcoefficient
for tracersin caseof
A map of EKE at 577 m presented in Plate 1 is qualitatively similar. The values may be compared with estimates derived from current meter measurements, as for
static instability. These parameterizations were made as
simple as possible. It may seem appropriate, for example,
to use vertical mixing coefficients which depend on strati-
examplesummarizedin Dickson[1983]. Contrary to the
fication [Cummins et al., 1990], or dependon Richardson current meter data, model EKE series are not low-pass filnumber[Philanderand Pacanowski,1986]. However,what tered but in the absenceof high frequency atmosphericforcpreciseform shouldbe chosenis still an open question. Since
CME1 was the first eddy-resolvingexperiment of its kind,
conservative
choices were made and friction
coefficients
were
kept constant.
Becauseof numerical constraints,the momentummixing
ing there is very little variability at periods smaller than 5
days. In the areasof LDE (Local DynamicsExperiment)
and POLYMODE (Polygon- Mid-OceanDynamicsExperi-
ment)around30øN,70øW,EKE levelsof 50 to 100cm• s-•
were observed
at 500 m and the
EKE
level
in the model
coefficients
usedin the first experiment(CME1) werelarge. is about20 cm2 s-a. In the EasternAtlantic,valuesof 15
Using the output from the first experiment as initial con-
to 60 cm2 s-a weremeasured
at the NEADS(North-East-
ditions, a secondexperiment (CME2) was run sucessfully ern Atlantic DynamicStudies)and TOURBILLON sitesbefor 5 years, with a biharmonic friction reduced by a factor
of 2.5 and vertical momentum mixing reduced by a factor
tween 30øN and 50øN where the model EKE barely exceeds
1 or 2 cm a s-a.
TREGUIER:
KINETIC
MEAN
ENERGY ANALYSIS OF ATLANTIC CIRCULATION MODEL
KINETIC
ENERGY
-
689
17 M
60.N
2000
1500
50.N
lOOO
40.N
5OO
30.N
½:.,
250
lOO
20.N
5o
10.N
lO
10.S
100.W
80.W
60.W
40.W
20.W
O.
Plate 1. Eddy kineticenergymap at (top) the first modellevel, 17.5m, and (bottom)level 10, 577 m, calculated
over the last 4 years of experiment CME2. Contours levels are in cm2 s-2.
A comparison of the model EKE with the TOPOGULF
(there are only three grid points in the straits of Florida).
experimentmooringsalong 48øN [Colin de Verdiereet al.,
1989] confirms this general pattern and emphasizesthe
The current fails to separate from the coast at Cape Hatteras but rather forms a quasi-permanent eddy north of it
strong difference between the western and eastern Atlantic
with transportof 70 Sv (Figure 3). Boundarycurrent sep-
in the model(Figure 2).
aration is a well known problem in primitive equation experiments, and many explanations have been suggestedin
the case of the CME. For example, the cold slope water to
Relationship Between Eddies and Mean Currents
the north
The distribution of eddy kinetic energy is due to the westward intensification of the mean oceanic circulation, since
the eddies mainly result from the instabihties of the mean
of the Gulf
Stream
because of the low resolution
is absent in the initial
of the Levitus
data
set.
fields
Sur-
face heat fluxes between the ocean and the atmosphere may
also be inadequate owing to the low resolution of available
flow. A detailed examination
of the mean circulation
in the
data. The representation of bottom topography may not
model is a tremendous task. Only a few features revealed be good enough. Finally, the most convincingexplanation
by maxima of mean kinetic energy are examined here, in is probably a lack of inertia of the current due to the inrelationship with patterns of eddy kinetic energy.
sufficient spatial resolution and too strong lateral friction.
The mean kinetic energy MKE of the 4-year-averaged This is suggestedby the low Gulf Stream transport and conflow presentsa lot of mesoscale
structures(Plate 2), which firmed by the energy balances analysis in the next section.
are long-lived since they also appear in a 15-years-averaged The testing of that hypothesisrequiresthe integration of a
MKE for the CME1 experiment(not shown). Just a look higher-resolutionmodel.
A question of interest is how mean currents cross the
at the thin Florida current helps the reader realize the great
need for increased spatial resolution in oceanic models. The
mid-Atlantic ridge. The observedpattern consistsof two
model Florida current transport is underestimated. It is main branches. One turns southeast away from the Gulf
lessthan 25 Sv on averagein CME1 and CME2 (the mean Stream over NewfoundlandRidge and partially recirculates
stream function for the latter experiment is shown in Figure 3), instead of the observed30-31 Sv. This may be due
in the western basin, but part of the branch crossesthe
mid-Atlantic ridge at 35øN to form the Azorescurrentin the
to the wind field [BSninget al., 1991],or to the topography easternbasin. Spall[1990]showsthat an eastwardcurrentis
690
TREGUIER: KINETIC ENERGY ANALYSIS OF ATLANTIC CIRCULATION MODEL
70 W
8O W
60 W
50 W
40 W
30 W
20 W
10 W
I
I
I
I
I
I
60 N
ow
60 N
50 N
50 N
::::::::::::::.::..-
.......
...:....-.:.:...:.:.:.:.:.:.:.:.
.....
...............:::......
.:::::..
.. ===========================================
40 N
40 N
•.. ß
-:..::::::::::::::::::::::::::::::::::::::::::::
30 N
30 N
...-...:..::.:::::::::::::
::::.......... ...................
::::::::::::::::::::::::::::::::::::
.................... ::....============================================================
20 N
20 N
80 W
70 W
I
I
I
60 W
50 W
40 W
30 W
20 W
10 W
0W
LONGITUDE
Fig. 1. Eddy kinetic energymap derived from Geosataltimeter data by Le Traon et al. [1990]. Contour interval
is 100 cm2 s-2; maximum is 2600 cm2 s-2.
presentin CME1, althoughit is situatedto the south(32øN)
and its transport is lower. In CME2 there is a maximum of
surfaceMKE (Plate 2) extendingeastwardin the model at
found. The path of the N AC and the adjacent high pressure
cells associated with recirculations described by Krauss et
latitudes 30ø-35•N, but it is not obviouslyconnectedto the
Gulf Stream system and is more zonal than the observed
al. are present in the model. The model transport (15 to
30 Sv; Figure 3) is smallerthan the valuesof 20 to 40 appearing in Figure 8 of Krauss et al., but transport maxima
Azores
and minima appear at the same locations. Both eastward
front.
This
feature
seems to be concentrated
in the
surface layer: at 577 m there is just a very narrow tongue
jets (at 45•N and 50ON)are associatedwith EKE maxima
in the 5 cm2 -2 contourextendingonly to 30oW.In the (Plate 1) in the westernbasin,but not in the easternbasin.
real ocean, the branch feeding the Azores current is associated with a tongue of maximum EKE clearly visible both
The jets are concentratedin the surfacelayers in the model
so that the net transport into the eastern basin is lower than
in Geosat(Figure 1) and driftingbuoydata. The meanders 10 Sv (Figure 3).
and eddies of the Azores front in the Canary basin are well
The mean flow acrossthe mid-Atlantic ridge along48•N
documented. There is some hint of such a tongue of higher
has recently been evaluated from current meter moorings by
EKE in themodel(Plate1) but it hasvaluesof 10cm2 s-2
Arhan et al. [1989]. During 1 year, three subsurfacemooringswereset in a triangle pattern (about 100 km apart) at
the four sites A, B, C and D along 48øN, and one of those
instead of about 100 cm2 s-2 in the data.
The
other
branch
of mean
flow across the mid-Atlantic
ridge is the North Atlantic Current (NAC) around 50ON, mooringswas renewedfor a secondyear. Those four current
which appearsto be made of severalfronts [Arhan, 1990]. meter per year measurements provided a mixed space-time
The path of the NAC is clearly indicated in the model by a
zone of maximum
of MKE
and a front
of the stream
func-
tion (Plate 2 and Figure 3). The current followsthe continental shelf of the Grand Banks. A tongue of maximum
MKE extends eastwardalong 45•N, at the location where
an intermittent eastward jet has been observedby Krauss et
al. [1987].The main MKE maximumturns eastwardnorth
of Flemish Cap along the Subarctic front and separates in
two branches,one crossingthe ridge at 52øN and another
around 45•N. Deeper, at 577 m, only the northern branch is
average suitable for comparison with the results of a numerical model. For this purpose, mean velocities have been
calculated in the model by averaging 16 data points around
the four TOPOGULF sites during the last 4 years of CME2
(Figure 4). The observedvelocitiesform a coherentpattern which gives Arhan et al. confidencein their estimation,
even though the uncertainty on the observedvaluesis theoretically large. The general eastward direction of the upper
layers is captured by the model, as is the correct amplitude
in the west (mooringsA and B). Arhan et al. attributed
TREGUIER: KINETIC ENERGY ANALYSISOF ATLANTIC CIRCULATIONMODEL
691
in the next section). All values are closeto 1 and never
exceed
2.
Time Scales o] Eddy Variability
The nature of the eddy activity in the model is very different from the west to the east of the mid-Atlantic ridge.
Time series(not shown)at the locationof the TOPOGULF
mooringsrevealthat the kineticenergyvariabilityis high for
periodsof a few monthsin the west, whereasthe dominant
signMin the east is an annual cyclemodulatedby interan-
5
50w
40
30
CLUSTER
A
B
20
C
D
Tourbillon
low
nual variability. A strong annual cycle in the surfacelayer
is found everywherein the eastern Atlantic. It is mainly
due to the kinetic energy associatedwith the Ekman transport, which is concentratedin the first layer of the model.
Averagedoverthe wholeeasternAtlantic (regionR8 in Fig-
Neads 7
ure6), the totalkineticenergyis 16.4cm2 s-2 in layer1
o
and 9.7 cma s-a in layer2, whereasthe kineticenergyof the
348.5©•.,.,•_..2300•100
-•
ß/
Ekmantransport
is Ke = (r/phi)2 : 7 cma s-2 (averaged
over 1 year and assumingit takesplaceoverthe first layer of
depth hi). Examinationof thesetime seriesconfirmsthat
•,• 43.7©
.-/////////A9.0e
13,3e2.1e.•
40
30
20
10w
Fig. 2. Eddy kinetic energyalong 48øN. (a) Model eddy kinetic
energyaveragedover three rowsof points around48øN in the last
4 yearsof CME2. (b) Observedvaluesreproducedfrom Colin de
Verdidreet al. [1989].
most of the shear between layers 1 and 2 is due to the Ekman transport, whichis maximumin Januaryand minimum
in October in the region.
The sharp transition from west to east is reflectedin the
frequency spectra. Model spectra have maximum energy at
periods larger than 150 days at all depths, whereas many
TOPOGULF spectra have a maximum in the "mesoscale"
band between40 and 150 days (H. Mercier, personalcommunication,1990). In the model, the amount of energyfor
periodsbetween40 and 150 days decreases
muchmore (10
to 50 times more) betweensitesA and D than the energy
at periods larger than 150 days, causingthe averagedtime
scale to increase from west to east, which is not the case in
the northward flow at site A to the North Atlantic current,
and the southward
flow at site B to its southern
branch.
In
the model there is southward flow at site A due to a permanent meander of the NAC. The larger deep southward
current
at site B is consistent
with
the fact that
less trans-
port crossesthe mid-Atlantic ridge in the model compared
with observations. The flow amplitude at sites C and D is
underestimated and the tiow reversalis higher in the model
(above600 m). In their analysisof mean velocities,Arhan
the TOPOGULF
data.
To give an idea of how model time scalescompare with
observations, the model results are examined at the location
of the NEADS 1 mooring(33øN,22øW) wherea time series
of 7 yearsexists[Zenk and Milllet, 1988]. Frequencyspectra averagedover the samefrequencybandsas usedby Zenk
and Miiller havebeencalculatedat modellevel 13 (1125 m)
and are plotted together with the data in Figure 5. Both
model and observed spectra have a sizeable amount of en-
et al. emphasizethe topographiccontrol of the ttow revealed ergy at interannuMperiods (larger then 450 days). In the
by the pattern of verticalvelocities(bottom vertical veloci- data most of the energy is found in the so-called"mesoscale"
ties are higher than the Ekman pumpingvelocities).In the band between periods of 46 and 180 days band, whereasthe
model, vertical velocitiesalso tend to be larger in the lower modelEKE is very low in that band (2 ordersof magnitude
layers than in the upper layers, but they are difficult to in- below the observedlevel). Similarly at the location of the
terpret because they are extremely noisy. This is probably TOPOGULF mooringsalong 48øN, the model EKE in the
due to the treatment of the topography as a seriesof stepsin period band 40-150 days is 20 times lower than observedin
the model, which creates numerical noise in the deep layers. the west and 1000 times lower in the east.
In the western basin, the amplitude of the mean currents
is better reproduced by the model than the eddy variabil- Discussion
ity. All the important currents are reproduced and are asAs already mentioned in the introduction, the CME solusociatedwith a strongeddy variability (with the exception
of the branch leading to the Azoresfront). In the east- tions are calculated on a grid which barely resolvesthe first
ern basin, mean currents are qualitatively reproduced in the Rossby radius in the northern latitudes. The model EKE
south[Spall,1990]but they areunderestimated
in the north decreases
northwardin the easternAtlantic (Plate 1), and
(TOPOGULF sites). The relationshipbetweenmean tiow model results compare better with EKE levels measured by
and eddy activity is less clear.
currentmetersin the south(NEADS sites1, 2, and 2.5 listed
Sinceeddy variability is low, the ratio R of eddy and mean by Dickson[1983]than in the north (TOPOGULF sitesC
kinetic energiesin the model is much smaller than the figure and D). Low resolutionis thereforethe most likely explaof at least 10 usually quoted for the open ocean. Regional
nation
for low EKE
levels in the eastern
Atlantic.
This
is
averagesof R have beencalculated(the regionsare defined alsosuggested
by Bb'ningand Budich[1991],who obtained
692
TREGUIER:
KINETIC
EDDY
ENERGY ANALYSIS OF ATLANTIC
KINETIC
ENERGY
-
CIRCULATION
MODEL
17M
60.N
2000
1500
50.N
lOOO
40.N
5OO
30.N
250
lOO
20.N
5o
10.N
lO
5
10.S
i
100.W
80.W
EDDY
I
60.W
40.W
KINETIC
ENERGY
20.W
-
577
0.
M
60.N
2000
1500
50.N
lOOO
40.N
5OO
30.N
250
100
20.N
5O
10.N
10
.
10.S
100,W
80.W
60.W
40.W
20.W
0.
Plate 2. Mean kinetic energymap at the first level of the model (17.5 m) for the last 4 years of CME2.
a doublingof EKE in a idealizedbasin model by increasing and Hua, 1987]. This mechanismis absentin the model,
the resolution. Other factors could play a part:
1. Direct generation of eddy energy by wind fluctuations
since monthly averages of the wind field are used. CME
experiments are underway at Institute Fiir Meereskinde
is thoughtto be importantin the easternAtlantic [Treguier (IrM), Kiel (C. BSning,personalcommunication,
1991) to
TREGUIER: KINETIC ENERGY ANALYSIS OF ATLANTIC CIRCULATION MODEL
MEAN
693
STREAMFUNCTION
60.N
50.N
40.N
ZO.N
10.N
10.S
100M
80.W
60.W
40.W
20.W
0
Fig. 3. Mean streamfunctionfor CME2. Contourintervalis 10 Sv; negativevaluesof the streamfunctionare
shaded.
TOPOGULF
DATA
Frequencyspectra
350
10
600
4.9
1500
2500
2.1
3000-
40004300 -
(m)
Site A
Site B
Site C
35øW
30ow
25ow
1.1
Site
D Scale
1.5
20øW
1•
o
o.oool
,
0.3 0.2q_
,........... ,
, , , ,',,,,
0.001
0.02
, , , ,;-,•¾---¾
,0.001
, , ,,,,,
0.01
0.1
0.1
'
0.1
,
, , ', ,,,,
1
frequency(cpd)
Neads I ---
1500•..•
250030004000-
(m)
MODEL
Model
Fig. 5. Frequency spectrum of kinetic energy at the location of
the NEADS 1 mooring. The observedspectrum at 1000 m is reproducedfrom Zenk and M611er[1988],with the model spectrum
at level 10 during the last 4 yearsof CME2. The energycontent
of each frequency band is indicated in cm2 s-2.
RESULTS
Fig. 4. Time-mean velocitiesat the 4 TOPOGULF sites. (a)
Model velocitiesfor the last 4 yearsof CME2. (b) Observedvelocitiesreproducedfrom Arhan ef al. [1989].
3.Radiation or advectionof eddy kinetic energyfrom the
west may be deficientin CME becauseof the position of the
Gulf Stream too far north, insufficient Gulf Stream transport, and/or the lack of a strong permanentbranch linked
investigatethe influenceof high-frequency
weathervariabil- to the Azoresfront. Also,the mid-Atlanticridgemaybe too
ity on the model.
stronga barrier. The representationof the topographyas a
2. The variability of the upper mixed layer in responseto •,.,;,• ,,f •,,,,,• ;• •h,-,,•,. , ..... , .... •-]"'• in certain cases;
seasonaland transient atmosphericforcing, which is not pa- it is difficult to measure whether that detail of the numerics
rameterized,may be essentielfor the easternAtlantic circu- has or not an effect on the flow.
lation.
4.The Mediterraneanwater in the easternAtlantic may be
694
TREGUIER: KINETIC ENERGY ANALYSIS OF ATLANTIC CIRCULATION MODEL
a sourceof eddyenergy(generationof meddiesfor example). anced, because the transient behavior is dominated by the
The influence of this water massis underestimated in CME,
seasonalcycle which is adequately resolved with 1 year of
as wasshownby Spall[1990].
data.
5.Mixing coefficientsfor momentumwere thought to be too
high in CME1, and experiment CME2 was run with lower
mixing coefficients. Although the circulation is improved
in equatorial regions, as was shown by Schott and BSnin9
Despite the decrease in vertical friction and horizontal
friction imposed at the beginning of the CME2 experiment,
time seriesof kinetic energy exhibit little trend during the
last 4 years. The adjustment to a changein vertical friction
[1991],the decreasein friction coefficients
has surprisingly is expected to be rapid, since 94% of the total kinetic enlittle effect at higher latitudes. The increase in volume- ergy sink by vertical friction takes place between15øSand
averaged eddy kinetic energy between CME1 and CME2 is 15øN in the model and the equatorial adjustmenttime is of
1 cm•' s-•', (from4.3 to 5.3overthewholedomain)but all
order of a few months. On the other hand, it is not clear
the growth occursin the equatorial region, North Brazil cur-
what the adjustment time to a changein horizontal friction
should be. Anyhow, the relatively small differencesbetween
CME1 and CME2 suggestthat the CME solution outside
the equatorial band does not depend much on the precise
rent, and Gulf of Mexico. The EKE even decreasesin the
surface layers of the eastern Atlantic. This suggeststhat
instability processesare inhibited by low resolution rather
than explicit friction in the model. At low resolution, the
numerical schemedoes not accurately describe the solution
of the continuousequations, and baroclinic instability tends
value of the friction
coefficients.
In a large-scalemodel like CME including western boundary currents and equatorial regions as well as shallow seas,
to be underestimated
aswasshownby Barnier et al. [1991]. basin-averagedstatistics are almost meaningless. SubdoTwo model solutions are not sufficient to discriminate bemains have been defined to take into account the spatial
tween those possible factors. All of them will have to be inhomogeneityof the flow (they are labelled R1 to RI0 in
consideredcarefully in the next experiments.
Figure 6). They are as large as possiblein order to filter
out mesoscalevariability while keepingkinetic energystatistics homogeneousenough in each region. The northern and
4. KINETIC ENERGY BALANCES
southern regionsR1 and RI0 are strongly influencedby the
Kinetic energy balanceshave been computed in order to boundary conditions at the north and south walls and thereunderstand how eddy variability is generatedin the model. fore will not be discussed.Area averagesof EKE at levels 1
There are two sources: the wind stress seasonalvariability (17.5 m) and 13 (1125 m) of the modelhavebeenplottedin
and the instabilities of the mean flow. In quasi-geostrophic Figure 6 to help characterizethe subdomains. The highest
models with steady forcing, the transients generatedby in- EKE levels are found in the western boundary currents, and
stabilities have long time scales,and energy budgetsmust it is therefore important to define special regions distinct
be averagedfor 10 yearsor more. In contrast, kinetic energy from the interior. A "North Brazil current"region(R3) has
budgetscalculated over 4 years for CME2 are quite well bal- been defined, in which the dynamics are very different from
R10
EKE•
=
8.3
=
5.7
•---
1.
EKE
R6
EKE•
EKEx
=
209
EKExs
=
11
R9
=
59
EKE•s
=
R7
2.4
R8
EKE•
=
35
EKE•
EKE•s
=
2.8
EKE•s
R5
R3
EKE•
=
97
EKE•s
=
1.7
EKE•.
R4
EKE•
=
EKE•s
=
17
=
EKE•s= 11
R2 EKE•
R1
=260
EKE•s
=7l
EKE•
=
27
EKE•s
=
1.7
Fig. 6. Map of the regions(numberedR1 to RI0) usedfor the energyanalysis.In eachregion,the area averages
of EKE at modellevel1 (17.5m) andat modellevel13 (1125m) areindicatedin cm•' s-•'.
TREGUIER: KINETIC ENERGYANALYSISOF ATLANTICCIRCULATIONMODEL
695
the equatorial dynamics of region R2. The "Florida cur-
locities. The opposite behavior is found when the vertical
rent" region(R6) alsoincludesthe locationnorth of Cape
heat diffusioncoefficientis large [Bryan, 1987].
Hatteras, where a permanent eddy is found in the model.
Region R8 east of the mid-Atlantic ridge is the most quies-
Kinetic energy is directly exchanged between the external and internal modes through nonlinear and topographic
transfers N and T. Quasi-geostrophictheory explains a
cent.
Note the rapid decreaseof EKE with depth (difference nonlinear transfer from the internal mode to the external
betweenEKE• and EKE•3). In the westernboundarycur- mode by the concept of inverse cascadein the vertical direcrents, EKEa3 is low becausestrong flows are found on the tion [Charney,1971]. In CME2, the nonlineartransferN is
continental shelves where the ocean depth is smaller than
1000 m. The study of B6ning [1989] suggeststhat the
surface intensification in the eastern Atlantic region of the
model(R8) is due to bottom topography.The averagera-
small, showingthat the inversecascadeis absent or compensated by other nonlinear processes.On the other hand, topographic energy exchangesare expected to render the flow
more baroclinic, either by counteracting the inverse cascade
tio of EKE at levels 10 and 24 is 7.5 in region R8. Ob- in QG turbulence[Rhines,1977; Treguierand Hua, 1988]or
servedKE(500 m)/KE(4000 m) ratiosat the NEADS moor- by generating bottom intensified mean currents. However,
ingsrangefrom about3 to 100,asplottedby Dickson[1983] this latter theory is valid in homogeneousturbulence or for
specialmean flow patterns (e.g., an equivalentbarotropic
in his figure 22a.
The subdomainsof Figure 6 are usedin two fashions:first, jet in a periodicchannel)and doesnot apply to a complex
to study the spatial dependencyof energy transfer terms,
since those terms are usually too noisy to be plotted without any spatial averaging, and second,to evaluate regional
energy balances.
basin circulation as the CME solution. Topography is found
to generateexternal mode energy,even more than the wind
stress(Figure 7). It is a sourceof meanrather than eddyenergy: the mean and eddy contributions to T are T=1.6 and
T'=0.1 respectively.B6ning[1989]showsthat in basin-scale
Global Balance:
Internal
and External
Mode
Before consideringthe instability processes,let us look
at the partition of energy between the external mode com-
P E models, T is due mainly to the effect of the continental
shelf on the western boundary current. This is the case in
CME2 too, and the positive basin-averaged T is essentially
due to the contributionfrom the Caribbeanbasin(T=l.7 in
ponent(depth-integratedvelocities)and the internal mode regionR5) and the Gulf Stream (T=0.8 in regionR9). The
component. Energy exchangesbetweenthe internal and the expression
of T (appendix)showsthat a positivecontribuexternal
mode have been studied in turbulence
theories and
models, and such analysis has been introduced by Holland
[1975]for primitive equationmodels. Our diagram(Figure 7) slightly differsfrom his (energytransfer terms are
definedin the appendix).
Let us discussfirst the energy source and sinks due to
nonconservativeprocesses.The wind forcing r is larger on
the internal mode than on the external mode, as is usual in
tion for the external mode arises from a positive Vc.•YH,
e.g., external mode currents flowing down the topographic
slope. This happens in the Gulf Stream region, where the
western boundary current leaves the continental shelf. The
picture in the Caribbean is less clear, due to the complex
coastline and topography there.
Topographic effectsin the Caribbean and Gulf of Mexico
being the main source of barotropic energy for the whole
quasi-geostrophic
(QG) and primitiveequation(PE) wind- North Atlantic model, numerical methods should be care-
forced gyres where the external mode is more intensified fully designedto resolve better those effectsin future experalong the western boundary than the internal mode, and iments.
receiveslessforcingenergyon averageover the domain. In
CME2, kineticenergyis removedmainlyby horizontal(Fh) Global Balance: Mean and Eddy Kinetic Energy.
and verticalfriction (F•). This contrastswith QG models,
The energy balance of Figure 8 showsexchangesbetween
wherethe main energysinkis bottomfriction (F b).
Transfer
terms are similar to
The buoyancyterm B is a net generationof potentialen- MKE and EKE in CME2.
those
of
Figure
7,
the
overbar
representing
transfersof MKE
ergywhichwouldbe absentin a QG modelwith no external
sourceor sink of potential energy. The sign of B means and the prime transfers of EKE. There is no topographic
that the wind does work to maintain the mean stratificatransfer in such a diagram, as can be readily seen from the
tion. This has been shown to happen in most wind- and expressionof T,
buoyancy-forced
PE modelsas soon as the typical winddriven velocitiesare larger than typical buoyancy-drivenve-
T=-//(pc-pt,)V•y.
dx
dy,
.9
6.3
Fv 2.8
El
I.I
N
0.1
T
1.7
INTERNAL
8.1
•H -I.4
Kœ
EXTERNAL
f a - 1.3
3.7
Fig. 7. Global energybalance for CME2 (internal and external
mode). All valuesare integratedover the basin and normalized
where pc and Vc are the external mode pressureand velocity
and p• the bottom pressure. Topography is constant in time;
therefore all mixed terms involving eddy velocity and timemean pressure or vice-versa vanish in the time average. Of
coursetopography has an indirect effect on the eddy-mean
interactions by modifying the phaserelationshipsin the flow,
thereby affecting the N transfer.
For CME2 (Figure 8a), EKE is generatedhalf by the seasonalwind fluctuations(r') and half by nonlinearkinetic
...............
;---- N [ ......
]]-- ;--* .....
*'•
-o •---'-*-'---;--
in-
stability). There is a much smaller nonlinearconversion
by the total oceanvolume.Units are cm2 s-2 for energylevels from potential energyB', which may be due to baroclinic
and 10-6 cma s-3 for energytransfers.
instability.
696
TREGUIER:
KINETIC
ENERGY ANALYSIS OF ATLANTIC
CIRCULATION
MODEL
of the wind stressenergyinput, and (3) the importanceof
i
Fv 1.8
N
MKE
I.
6.6
EKE
Fv
0.9
FH
0.9
5.3
i
f•
B
0.2
B
I,
bottom friction, which is the main energy sink.
None of those features are present in the CME2 experiment. Although the EKE box is directly forced by the seasonal variations of the wind stress, it contains less energy
than the MKE
box. The smallness of the nonlinear
transfer
N as well as the low level of mean energy relative to the QG
experiments showsthat the circulation is both lessunstable
and lessinertial (the freejet and recirculationin QG models
are a resonantinertial response
to the windforcing). Energy
0,1
dissipation is also different in CME2. Vertical friction, which
is absentin QG models,accountsfor 43% of the total kinetic
energydissipationand bottom friction representsonly 21%.
Thosepercentages,
however,vary widelyoverthe basin(Figure 9). Verticalfrictionis dominantin the tropicsand small
in the westernAtlantic, and represents28% of the kinetic
8.85
F.
0.•5
N
MKE
Fs
7.7
27
EKE
71
energy dissipation in the eastern Atlantic. Bottom friction
I.
f a 6.3
is dominant(49%) in the Florida currentregionbecausethe
oceandepth is small and the bottom velocitiesare large (the
bottom friction is quadratic in CME2, whereas it is linear
in QG models). Lateral friction is larger in CME2 than in
-•
MKE
18
•
EKE
27
FB 0.4
>
Fs 1.6
QG modelsbecauseboth the coefficientand the grid size are
larger. A more efficient horizontal friction means that there
is no requirement for energy to be transferred downward, as
would be the case in QG models where bottom friction is
the only possiblesink for mesoscaleand large-scaleeddies.
Finally, in CME2 the conversion between mean kinetic
and potential energiesis comparatively larger than in QG
experiments. Only part of this energy returns as a trans-
fer B' from eddy potential energyto eddy kinetic energy.
Maps of thoseterms (Figure 10) showthat the averaged
B is a rather small differencebetweenlarge negative (in
Fig. 8. Global energy balances for the time mean and eddy kinetic energy.(a) CME2 primitive equationexperiment.(b) Hol- the west) and positive(in the east) values,reflectinga spaland and Schmitz[1985] QG model. (c) Barnlet et al. [1991] tial redistributionof energy by the pressurefluxes. B' is
B
B
I.
I.
QG model. All values are integrated over the basin, divided by
total oceanvolume,andexpressed
in the sameunits(cm2 s-2 for
energylevelsand 10-6 cm2 s-3 for energytransfers).
To compare CME2 with quasi-geostrophicsolutions,similar balances have been reproduced from Holland and
positive in most regions, excepted in the North Equatorial
current region, the Gulf of Mexico, and the Florida current,
and is larger in the Gulf Stream/North Atlantic current region and the recirculation where baroclinic instability is expected to be active. It seemstherefore reasonableto assume
that in thoseregionsthe B' transferresultsfrom the baroSchmitz[1985](three-layerexperiment3L-4) (Figure 8b) clinic instability of the mean flow. However,B' cannot be
and from a more recent QG model with 6 layers and a grid
simply interpreted as the signature of baroclinic instability'
can generatea B' transfer. For instance,
size of 10 km [Barnlet et al., 1991] (Fig 8c). Let us con- other processes
one notesthat B' decreases
by a factor of 5 betweenexpercentrate on the two QG models, labelled QGa and QGb.
iments
CME1
and CME2
when the friction
coefficients
are
The strength of the forcing is different, owing to a different basin size and wind stress and MKE varies roughly in
the same proportion. In both cases the main sink for the
MKE box is the nonhnear generation of eddy kinetic energy
through barotropic instability. The net transfer is smaller in
by a factor of 2. The decreaseof B' occursalmostentirely
in the westernpart of the basin (North Brazil current,Gulf
of Mexico, Florida current and Gulf Stream/North Atlantic
QGb, becausebaroclinicinstabihty(expressed
by the trans-
current). In the first threeregions,B', whichwaspositive
decreased,whereas the transfer from MKE to EKE increases
fer • = -B' ) is largerin that experiment
andtendsto in CME1, becomesnegative in CME2. This suggeststhat
inducea nonlineartransferfrom EKE to MKE [Barnier et
al., 1991]. Anotherdifferencebetweenthe two casesis the
(relatively) much larger lateral friction in QGb. This is a
commonfeatureof high vertical resolutionmodels(seethe
eight-layerexperimentof Hollandand Schmitz[1985])probably due to the excitation of higher baroclinic modes with
smaller
horizontal
scales which are more vulnerable
to lat-
eral friction. In summary, the main features shared by both
there is in the model a mechanismgenerating eddy potential
energy from the eddy kinetic energy, a mechanismwhich is
enhanced when friction is decreased. It may be linked to
a topographic effect rendering eddies either more surface
intensified or bottom trapped, thereby increasingtheir potentiM energy. A more detailed analysis would be necessary
to understand all those competing processes,but the model
spatial resolutionin western boundary currents is probably
not good enough to allow such an analysis. At the present
stage, one cannot rule out the possibility that specific fea-
QG models(and probablyall the other QG EGCMs run so
far) are (1) an eddy kinetic energylevel significantlylarger
than the mean, (2) a nonlineartransferfrom meanto eddy turesof the kineticenergybalance(the signof B', for exkineticenergyrepresenting
a largeportion(morethan 50%) ample) dependon the numerics.
TREGUIER:
KINETIC
ENERGY
ANALYSIS OF ATLANTIC
RI0
-
47
MODEL
•, -- 84 F,•-- 0
R6
Ft
CIRCULATION
R9
F•
=
65
F,,
=
5
F•
--
29
F,• --- 4
F•, -
49
R7
R8
=
80
F•
=
58
F,• =
15
F,•
--
28
F•
5
F•
--
14
F•
F,
=
60
F•
=
•S
F•
=
25
=
R4
R3
•, = 15
F•=4 k•
R2F,=9
F,•
=87F•=4
l
F,,
=
13
F,• =
F•
52
F•
35
=
=
8•
=
32
F,• =
F•
54
R1
•, ----14
Fig. 9. Importanceof the friction termsFl (lateral), Fv (vertical), and Fb (bottom) in the variousregions.Figures
are percentagesof the total friction (eddy and mean) in eachregionfor the last 4 yearsof CME2.
B=-
1.2
B • = 0.
R6
B
B
=
-1.05
B'
=
-0.03
R9
=
-1.83
$' = 0.06
R7
R8
0.26
0.94
0.05
0.02
R5
R3
B
=
--0.43
B• =
-0.05
B'
B
-
-0.26
B'
=
-0.03
=
R2
-0.10
•=0.1
B
RI
B'=0.12
=
B' =
0.70
o.m
697
698
TRmaumR:
KINETIC
ENERGY ANALYSIS OF ATLANTIC
CIRCULATION
MODEL
potential energy. This contrasts again with the results of
H86, who found that in the eastern part of the jet the main
The large spatial inhomogeneity in the model response
sourceof energywas the nonlinearenergyflux, representing
suggeststhat local energy balances would be more useful
the decelerationof the jet (more energyadvectedinto the
than global ones. Local energy budgets are displayed in
Regional Balances
regionthan advectedout). This decelerationwas balanced
Figure 11 for two regions: the Florida current (R6) and mainly by a negative pressureflux and also to some extent
the North Atlantic current (R9). Those balancesinclude
by a conversioninto mean potential energy. The eddy kinetic energy on the other hand is forcedby conversionfrom
are difficult to interpret (definitionsare providedin the apeddy potential energy as well as eddy pressurefluxesin the
pendix). However, the comparisonwith the resultsof a
primitive equation model. This can be attributed to barosimilar analysis in a quasi-geostrophicbasin model proves
clinic instability, which is expectedin that region. Fluxesof
interesting[Hall, 1986](hereinafterreferredto as H86).
the same sign and similar magnitude are found in the QG
large pressurefluxes •P through the region boundarieswhich
Let us considerregionR6 (Figure 11a). The main en-
model(for examplein the regionslabelled"recirculation"
in
ergy sourcefor the mean flow is the pressureflux, consistent
H86), but the eddy energylevelsare comparativelysmaller
with the results of H86 for the western part of the central in the PE model.
jet. However, in the QG model this energy input served to
Local energy balancesdemonstrate that the redistribution
acceleratethe jet and was therefore balanced by a negative
of time-mean kinetic energy within the basin is performed
flux of kinetic energy (more kinetic energy left the region
by pressurefluxes and exchangesbetween mean kinetic and
than entered). In CME2 the nonlinearflux is very small
potential energy. Potential energy balanceswould be necesand the kinetic energy provided by the pressure fluxes is
sary to fully understandthe energycycle,but this is beyond
either convertedinto potential energy(B) or dissipatedby
the scopeof the presentpaper. Contrary to QG models,the
friction, which plays a much more important part in CME2
than in the QG model. Note that the pressure flux terms nonlinear advectivefluxes of energy are everywhereof secondary importance, confirmingthe lack of inertia of western
in Figure 11 are of the same order of magnitude as those
boundary currents in the CME experiment.
calculated by H86 in the QG model, whereas the nonlinear
advective fluxes are at least one order of magnitude smaller,
5. CONCLUSIONS
showingthe lack of inertia of the western boundary current
in CME2.
The main source of EKE in the model Florida
The CME experiment has often been presented as the
current is a conversionfrom MKE, which may be interpreted convergenceof two kinds of models: the low resolutionbasinas barotropic instability.
scale PE models with active thermodynamicson one hand
In regionR9 (Figure lib), the pressureflux is still an im- and the high resolution,eddy-resolvingQG modelson the
portant source of MKE but is smaller than the input from other hand. The presentanalysisshowsthat although we
the wind stress. Both are balanced by a conversion B into are on the path of sucha convergence,it is not yet achieved.
For the first time, eddy kinetic energy(EKE) levelshave
beencalculatedin a numericalmodelwith enoughrealismto
Fv 0.0
0.1
MKE
1.05
0.01
allow direct comparison with observations. We have shown
that EKE levels are reasonablein the vicinity of western
boundary currents but are too low in the eastern At]antic.
0.0::3
Spall [1990]quoteda factor of 5 in the Azoresfront, but
0.32N'_•EKE
a5
1.3
Fa 0,24
Fa' 0.06
•0.13
'
time spectra show that EKE may be too low by 2 orders of
magnitude when only the frequencyband around 100 days
is considered. Overall, the lack of eddy activity east of the
mid-At]antic ridge is about 1 order of magnitude. Although
the absenceof high-frequencywind variability and the coarse
representation of bottom topography may play a part, we
suggestthat the low eddy activity results mainly from the
low level of instabilityin the model. It is the (still) coarse
horizontal resolution rather than the value of mixing co1.2
1.8
o Ol
Fv 0.01
F.0.5
Fa 0.1:3
efficients which is responsible for the viscous character of
a 06
Fv 0.0•
MKE
0.9
N 0.07
EKE 099
•0.04
>
Fa 0.01
0.87
10,15 0.04
•0.02
the solution. In experiment CME2, momentummixing coefficients have been decreasedcompared with CME1. The
new coefficientsare just sufficient to prevent noisefrom developing in the equatorial region and yet the eddy kinetic
energy in the eastern Atlantic doesnot increasefrom CME1
to CME2. Becauseof the numericalconstraints,a higher
resolution
will be needed in order to allow further
decrease
of the mixing coefficientsand a better representationof instability processes.
Better solutionsmay also be obtained if the oceaninterior
Fig. 11. Local energybalancesin regions(a) R6 and (b) R9.
Transfers inside the region are indicated by simple arrows and can be made less viscousby choosingspace-dependentfricfluxes through the region boundary by thick arrows for pressure tion coefficients. However, complicated parameterizations
fluxes •P, and double arrows for nonlinear fluxes .T. Only the
are difficult to test in the absenceof "convergent"numerimost significant transfers are indicated to improve readability.
Unitsare cm2 s-2 for energylevelsand10-6 cm• s-s for energy cal solutions,i.e., solutionswhichdo not dependanymoreon
transfers.
the spatial resolutionof the model. A recentstudy [Barnlet
TREGUIER:KINETICENERGYANALYSIS
OF ATLANTICCIRCULATION
MODEL
699
et al., 1991] suggeststhat quasi-geostrophic
basin models effort must be devoted to improve the models' numerics,
"converge"for a meshof 10 km. With that resolution,the their spatial resolution,and the analysisof their solutions.
flow dependencyon the biharmonicfriction coefficientbeAPPENDIX
comessmall and the conservationof potential vorticity along
particle trajectoriesis verifiednumerically.Consideringthe
Kinetic energydiagnostics
in the frameworkof the Coxcomplexgeometryo•[oceanbasins,it is possiblethat even
Bryan primitiveequationmodelare classical,sincethey are
smallergrids will be neededfor primitive equation, "realistic" models.
In CME as in previousbasin scale PE models, the topo-
graphicconversion
of baroclinicenergyis the main sourceof
barotropicenergyfor the wholedomain. This shouldmotivate an improvement of the numerical treatment of topography, especiallyin the region of western boundary current
formation.
Our physicalinterpretation of the model kinetic energy
cycle is unfortunately limited. A potential energy analysis
would have been valuable, but there is no simple potential
energyequation becausethe nonlinearityof the equationof
built in the code (which usesan energy-conserving
algorithm). They are discussed
in detail for exampleby Holland
[1975]. For the presentanalysisthe diagnosticshave been
extended in two fashions: first, they can be calculated for
the mean and eddy energy separately and the exchangesbetween eddy and mean are computed. Second, it is possible
to calculate energy exchangesin open regions. The definitions of the energy transfer terms are straightforward and
will not be recalled
here.
For the global energy diagnostic separating the external and the internal mode, we have adopted a presentation
which is differentfrom that introducedby Holland [1975]
stateintroducesextra effects(e.g., cabelling).The develop- and subsequentlyused by many authors. In that now clasment of a completepotential energydiagnosticis a difficult
task which would have to be done before an experiment is
run, so that the various terms needed for the energy cal-
culation are properly stored (especiallyin the caseof the
convectiveadjustment). Another difficulty of the analysis
comesfrom the complex behavior of energy transfer terms,
which have differentsignsin different regionsof the domain.
In order to understand processesat work it is necessary to
combinea global analysislike the presentone with regional
analysis[Spall,1990]. However,regionalanalysisof the most
energeticregionsof CME2 doesnot seempossiblebecauseof
the lack of resolution in key regionslike the Florida current
or the Gulf of Mexico. The estimate of vorticity balances,
for instance, involves derivatives which cannot be accurate
with a second-orderfinite differenceschemewhen the region
of interest is only a few grid points wide.
Other numerical problems may prevent an extension of
the presentstudy. The numericalmethod which ensuresen-
ergy conservationin the GeophysicalFluid DynamicsLaboratory (GFDL) model(namely,the specialformof the continuity equationusedto calculatevertical advectionof kinetic
energy)introducesnumericalnoisein the presence
of highresolutiontopography. In CME2 we found that the noise
greatly increasedthe vertical advectionof KE, since the
basin averagedvertical and horizontal advectionof kinetic
energyare comparablewhereasin a stronglygeostrophic
circulation their ratio should equal the Rossbynumber. The
sical representation, three transfer terms are plotted: the
external mode pressure work PW•, the internal mode pressure work PWi, and the buoyancy B:
PW•= -///V•Vp
dzdgt
dz
PWi
= -///l,•.Vp
dz
dgt
dz
B = -///wpgdzd•tdz.
V is the horizontal velocity vector, p is the pressure and
subscripts e and i refer to the external and internal mode,
respectively. Cartesian coordinates are used here for sim-
plicity (the model usessphericalcoordinates).In a closed
domain, those three terms are related:
PW• + PW• = B,
(2)
reflectingthe conservation
of total energy(pressurework
termstransformkineticenergyinto potentialenergy). Plottingall threetermsona diagramisclearlyredundant.Moreover,the choiceusuallymadeto representPW• as an energy
transferbetweenpotential energyand barotropickinetic en-
ergyis arbitraryand obscures
its naturalmeaning.In fact,
PW•
= - /// V•Up
dz
d•t
dz
noise does not affect first-order quantities like tracers or ki-
netic energies(Plates I and 2 are relativelysmooth)and
appearsonly when vertical velocitiesin the lower layersare
considered.However,this numerical problem still may affect
instability processes.The only way to find out is through
mode
comparisonof similar experimentsrun with other codes(es- w•e•e p• •s t•e •ottom p•essu•e,p, •s t•e exte•
peciallycodesusingan ArakawaC grid, wherethis problem
fi•t te•m • (3) • •e•o w•e• •teg•ted ove•ß dosed••.
with vertical advectionis not found).
There is no doubt that basin-scaleeddy-resolvingnumer- ß •e •eco• te•m • t•e topog•pMc t•s•e•
ical experiments will bring a better understanding of the
ocean. Direct observationswill remain costly and of limited
geographical
extent (or poorspatial-temporalresolution)in
the near future, and satellites seeonly the surface. Therefore
•.'-',-1;
•.,v.,...,,....,.
•..•..
•,,,.,.•.,
,,..•.,....,,,
,.,v
external •
t•e •te•
mo•e; •t w•s•es w•e• t•e oce•
•ottom •s fi•t. •s
te•m •dse• o•]y • t•e •efic e•e•gy
••ces
•
•t seems
•etwee• t•e •te•
•
t•e exte•
mo•e •efic e•e•gy,
&,•.l•.
damental questionsas the origin and fate of eddy activity on
a global scale,or the quantitative importance o• wind-forced
and buoyancy-forcedmotions. Before we get there, a large
is avoidedby plotting only T and B insteadof PWe, PWi.,
and B. The resultingdiagram(Figure7) is moreconsistent
than the cl•sicM
one.
700
TREGUIER:
KINETIC
ENERGY
ANALYSIS OF ATLANTIC
Note that T is not calculated in the original Cox-Bryan
diagnostics,since it is equal to PFV½averagedover the domain. However, the equality is not true at each grid point
and for the local diagnosticsit is necessaryto know T independently. A finite-difference form has been derived which
preservesthe equality T - PFV½as well as the properties
of T to vanish when there is no topography or no depthdependency of the flow.
In order to calculate regional energy budgets,one has to
take into accountfluxes through the region boundaries. Let
us consider the external and internal mode pressure work
terms PI/V• and PFVi. In a closed domain, those terms are
linked to the topographic transfer T and the buoyancy work
B by the relationships PW• - T and PW½ -[- PWi -- B. In
an open region,
PW½
=
T + •½
PWi+ PW½= B+ P½
+ Pi.
(4)
•P• is the external mode pressureflux (product of the external modevelocity and the pressure)integratedalongthe
region boundary and 7•i the internal mode pressure flux.
Thosefluxescan be deducedfor eachregionfrom (4), since
all the other terms are calculated independently.
Similarly, the nonlinear advection terms summed over the
region can be interpreted as fluxes through the boundary
and/or conversionsbetweenmean and eddy or external and
internal energy. Two kinds of nonlinear terms can be distinguished. Some terms cannot produce an exchange,e.g.,
they vanish when integrated over a closed basin and they
only reflect a redistribution of energy in the domain through
advection. Other nonlinear terms can generate exchanges
between vertical modes or between mean and eddy energy.
In that case for instance, two terms are defined' N, which
CIRCULATION
MODEL
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appearsin the equationfor the mean KE, and N', which Hah, Y.J., A numerical world ocean circulation model, Part II, A
appears in the equation for the eddy KE. Integrated over a
closed
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•' being an energy flux.
Harrison and Robinson[1978]discussregionalenergydiagnosticsin a primitive equation model and argue that an
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Acknowledgments. I thank Frank Bryan and Bill Holland for
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(ReceivedDecember3, 1990;
revised July 23, 1991;
acceptedJuly 23, 1991.)