vertical eddy energy fluxes in the north atlantic subtropical and subpolar gyres

Vertical Eddy Energy Fluxes in the North Atlantic Subtropical and Subpolar Gyres

XIAOMING ZHAI

School of Environmental Sciences, University of East Anglia, Norwich, United Kingdom

DAVID P. MARSHALL

Atmospheric, Oceanic and Planetary Physics, University of Oxford, Oxford, United Kingdom

(Manuscript received 30 January 2012, in final form 6 September 2012)

ABSTRACT

Eddy energy generation and energy fluxes are examined in a realistic eddy-resolving model of the North

Atlantic. Over 80% of the wind energy input is found to be released by the generation of eddies through

baroclinic instability. The eddy energy generation is located near the surface in the subtropical gyre but

deeper down in the subpolar gyre. To reconcile the mismatch between the depth of eddy energy production

and the vertical structure of the horizontal dispersion of eddy energy, the vertical eddy energy flux is

downward in the subtropical gyre and upward in the subpolar gyre.

1. Introduction

The available potential energy built up by large-scale

wind-driven Ekman pumping of the main thermocline is

believed to be released by the generation of eddies

through instabilities of themean currents (e.g., Gill et al.

1974). This process is parameterized in most coarse-

resolution ocean climate models through an eddy-

induced transport velocity that adiabatically flattens

isopycnals (Gent and McWilliams 1990, hereafter GM).

Although this hypothesized energy route has been sup-

ported by some idealized model studies (Radko and

Marshall 2003, 2004), it is not clear whether results from

these idealized studies (e.g., rectangular basin, flat bot-

tom, simplified surface forcing, etc.) are applicable to

the ocean, or even to realistic ocean simulations.

Previous attempts to estimate the energy released

through baroclinic instability generally have been based

on the GM parameterization, which extracts available

potential energy from themean state at a rate (e.g., Gent

et al. 1995)

›P

›t52

ðððgkGM

j$hrj2rz

dV , (1)

where P is the mean available potential energy (APE),

r is density, kGM is the thickness diffusion coefficient, g is

the gravitational acceleration, and $hr and rz are the

horizontal and vertical density gradients. For example,

Huang and Wang (2003) estimate this energy release

from hydrographic climatology to be;1.3 TW (1 TW51012 W) using kGM 5 1000 m2 s21, a value commonly

adopted for ocean climate models. Wunsch and Ferrari

(2004) obtain values of 0.2 and 0.8 TWusing themodified

eddy closures of Visbeck et al. (1997) and Danabasoglu

and McWilliams (1995), respectively. Although these

estimates seem to suggest that baroclinic instability is

capable of releasing a significant fraction of the wind

energy input to the large-scale ocean circulation, they

are subject to large error bars owing to large uncer-

tainties associated with kGM. Further studies are re-

quired to improve our understanding of where and how

much of the wind energy input to the large-scale ocean

circulation is released by ocean eddies.

There is also the question of how eddy energy is dis-

persed in the ocean once generated. Recent eddy pa-

rameterization schemes proposed for the interior of the

ocean carry the eddy energy as a prognostic variable in

the model equations and have desirable features such

as fluxing potential vorticity downgradient without gen-

erating spurious sources of energy (Eden andGreatbatch

2008;Marshall et al. 2012).However, the new eddy closure

has also been found to be sensitive to parameterizations

Corresponding author address: Xiaoming Zhai, School of Envi-

ronmental Sciences, University of East Anglia, Norwich NR4 7TJ,

United Kingdom.

E-mail: [email protected]

JANUARY 2013 ZHA I AND MARSHALL 95

DOI: 10.1175/JPO-D-12-021.1

� 2013 American Meteorological Society

of the dispersion of eddy energy (Marshall and Adcroft

2010). The parameterized eddy energy will be advected

by the mean flow and diffused, although it is clear from

satellite observations that there is a dominant and

ubiquitous westward propagation of eddy energy in the

ocean interior (e.g., Chelton et al. 2007, 2011), except in

the separated boundary currents.

However, an issue that has received relatively little

attention to date is vertical fluxes of eddy energy. In

general, the horizontal dispersion of eddy energy is

likely to occur at different depths from that of the eddy

energy generation. For example, in the subtropical

ocean the largest lateral density gradients, and hence the

eddy energy generation according to (1), are confined to

surface layers; in contrast the horizontal dispersion of

eddy energy, even if dominated by the first and higher

baroclinic modes, will have a significant component at

depth. Hence vertical eddy energy fluxes are required to

connect the sources of eddy energy and its horizontal

dispersion.

The aim of the present study is threefold:

d examining where and howmuch the wind energy input

to the large-scale ocean circulation is released through

baroclinic instability using a realistic eddy-resolving

model of the North Atlantic;d mapping the vertical eddy energy fluxes in the ocean

model;d thus reconciling the mismatch between the depth of

eddy energy production and the vertical structure of

the horizontal dispersion of eddy energy.

2. A simple eddy energy balance

The momentum and continuity equations for a

Boussinesq, hydrostatic ocean are

DuhDt

1 fk3 uh 1$hp

r05

Fh

r0, (2)

›p

›z52rg , (3)

$h � uh 1›w

›z5 0. (4)

Here, uh andw are the horizontal and vertical velocities,

p is pressure, r is density with r0 its reference value, f is

the Coriolis parameter, k is a vertical unit vector, g is the

gravitational acceleration, z is height and $h is the hor-

izontal gradient operator.

We now split the flow into time-varying and time-

mean components, denoted by primes and overbars re-

spectively. Taking the time-varying component of (3),

multiplying by w0, taking a time-average, and using the

time-varying components of (2) and (4), we obtain

�r0u

0h ��DuhDt

�02u0h � F0

h 1$h � p0u0h�1

›

›zp0w052r0w0g .

(5)

Equation (5) is a simple way of writing the mechanical

energy budget in an incompressible ocean.1 The terms

represent (i) contributions to the eddy energy balance

from the horizontal momentum equation (terms inside

the curly brackets); (ii) the divergence of a vertical eddy

energy flux, p0w0; (iii) the source of eddy energy associ-

ated with dense fluid sinking and buoyant fluid rising in

baroclinic instability, 2r0w0g.The vertical eddy energy flux p0w0 represents the re-

distribution of eddy energy in the vertical but is rarely

discussed in the literature [although the equivalent

vertical energy flux associated with the internal waves/

tides has been widely diagnosed in both observations

and numerical models; for a notable exception in the

context of the mean flow, see Roquet et al. (2011)].2 As

will be shown later in section 4, the vertical eddy energy

flux is predominantly downward in the subtropical gyre

but upward in the subpolar gyre. We propose that these

vertical eddy energy fluxes exist due to mismatches be-

tween the depths at which eddy energy is generated

through baroclinic instability and the horizontal eddy

energy fluxes that disperse the eddy energy. Equation

(5) describes a nonlocal eddy energy balance analogous

to internal wave generation and energy fluxes (e.g., Gill

1982), illustrated schematically in Fig. 1.

Under an assumption of small Rossby number and

Ekman number such that the horizontal momentum

equation can be approximated by geostrophic balance,

the horizontal terms in the eddy energy Eq. (5) reduce to

1 The eddy energy equation can be written in a more conven-

tional way: ›E/›t1$ � ((u1u0)E1 p0u0)2u0h � F0h 1C52r0w0g ,

where u is the three-dimensional velocity vector, E5 r0u0h � u0h/2 is

the eddy kinetic energy, and C5 r0u0h � fu0 � $uhg represents en-

ergy exchange with the mean flow through barotropic instability.

However, since the focus of the present study is the vertical eddy

energy flux p0w0, we distinguish ›p0w0/›z from the remaining terms

on the left-hand side of the above equation and treat the remaining

terms as a single entity as in (5).2 Note that the vertical eddy energy flux is different from the

vertical eddy heat flux r0cpw0u0, where cp is specific heat at constant

pressure and u0 is the eddy potential temperature (see Wolfe et al.

2008). The vertical eddy heat flux is closely related to the source of

eddy energy, 2r0w0g.

96 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 43

�r0u

0h ��DuhDt

�02u0h � F0

h 1$h � p0u0h�5 f$h � p0u0hg

’›

›x

2

b

2r0f2p02!, (6)

that is, the zonal divergence of the westward eddy en-

ergy flux associated with westward propagation of

Rossby waves. Here b is the meridional gradient in the

Coriolis parameter and x is the distance in the zonal

direction. However, the remaining terms are not negli-

gible in the local eddy energy budget, even if the overall

eddy energy flux is westward over much of the ocean

(e.g., Chelton and Schlax 1996; Chelton et al. 2007; Zhai

et al. 2010). For the purpose of this study, we do not

distinguish between the various different contributions

to the eddy energy budget from the horizontal momen-

tum equation and, instead, treat the terms within braces

in (5) as a single entity.

3. The model

The ocean model used in this study is the Massachu-

setts Institute of Technology general circulation model

(Marshall et al. 1997). The model domain extends from

148S to 748N and from 1008W to 208E, with a horizontal

resolution of 1/108 3 1/108. There are 33 geopotential levels

whose thickness increases with depth, ranging from 10 m

at the surface to 250 m at the bottom. The model to-

pography is generated from the 50 Gridded Elevations/

Bathymetry for the World data. Biharmonic operators

are used for horizontal mixing of tracers with back-

ground diffusivity of 13 1010 m4 s21, while a highly scale-

selective biharmonic friction with a Smagorinsky-like

viscosity (Griffies and Hallberg 2000) is adopted for

horizontal mixing of momentum. A linear bottom stress is

used at the ocean floor with drag coefficient of 1.13 1023.

The model is driven by climatological monthly mean

forcing obtained from National Centers for Environ-

mental Prediction–National Center for Atmospheric

Research (NCEP–NCAR) reanalysis (Kalnay et al.

1996). Exchange with the Nordic Seas and the rest of the

South Atlantic Ocean, which lie outside of the model

domain, is crudely taken into account by restoring the

model temperature and salinity fields near the bound-

aries toward the monthly mean climatological values at

all depths with a restoring time scale that varies linearly

from 3 days to infinity over the 48 wide buffer zones.

There is no explicit treatment of sea ice. The model is

FIG. 1. Schematic of the simple eddy energy balance proposed in

section 2 for the (a) the subtropical gyre and (b) subpolar gyre. The

generation of eddy energy is located near the surface in the sub-

tropical gyre but deeper down in the subpolar gyre. To reconcile

themismatch between the depth of eddy energy production and the

vertical structure of the horizontal dispersion of eddy energy, the

vertical eddy energy flux is downward in the subtropical gyre and

upward in the subpolar gyre.

FIG. 2. (a) The annual-mean barotropic transport streamfunction

(Sv) and (b) the wind power input to the North Atlantic Ocean

(W m22).


first spun up for 23 years at the 1/58 resolution, and is then

run for another 9 years at 1/108 resolution, during which

the model variables are saved every 6 days. Results from

the last 7 years are used for this study. The overbars

hereafter denote the time-mean quantities averaged

over the 7-yr period, and primes pertain to the eddy

fields resolved by the model.

4. Results

Figure 2a shows the annual-mean barotropic trans-

port streamfunction. The large-scale pattern compares

well with that from other ocean models with similar

horizontal resolutions (e.g., Smith et al. 2000; Eden et al.

2007). For example, the Gulf Stream does separate

roughly at Cape Hatteras with a cyclonic recirculation

to the north and an anticyclonic recirculation to the

South. However, the path of the North Atlantic Cur-

rent, as well as its eastward turn at the ‘‘Northwest

Corner’’ is less well simulated by the model, a problem

shared also by other models (e.g., Masumoto et al.

2004; Eden et al. 2007). As a consequence, the eddy field

associated with the North Atlantic Current is shifted

somewhat eastward (Fig. 3a).

a. Wind energy input

The rate of wind energy input at the sea surface is

calculated using Wwind 5 t � uo, where t is the surface

FIG. 3. The rate of APE released by baroclinic instability

(2r0w0g, 31025 W m23) at (a) 61 m, (b) 900 m, and (c) 1875 m.

The color scales in Figs. 3–7 are saturated so as to reveal regions of

moderate eddy activity.

FIG. 4. The vertical eddy energy flux (p0w0, 31022 W m22) at (a)

61 m, (b) 900 m, and (c) 1875 m.


wind stress vector and uo is the total ocean surface

velocity. The pattern and magnitude of the wind en-

ergy input (Fig. 2b) are similar to the previous studies

(e.g., Wunsch 1998; Zhai andGreatbatch 2007; Hughes

and Wilson 2008; Scott and Xu 2009). The majority of

the wind energy input occurs in the tropical, western

boundary, and subpolar regions, whereas there is little

wind energy input in the interior of the subtropical gyre.

The atmospheric winds thus seem to spin the subtropical

gyre on its northern and southern edges. Roquet et al.

(2011) suggest that the direct wind energy input to the

ocean, Wwind, is first transported laterally by the Ekman

transport before being pumped into the interior circula-

tion; thus, the energy can be pumped into the ocean in-

terior far from the region of direct surface wind work.

Note that this interpretation is degenerate to the extent

that any rotational energy flux can be added without

modifying the energy budget.

One peculiar feature in Fig. 2b is a hot spot of wind

energy input in the Caribbean Sea, north of Venezuela.

This energy hot spot seems to be a robust feature, as it

also shows up in other studies of wind energy input to

the ocean (e.g., Wunsch 1998; Hughes and Wilson 2008;

Scott and Xu 2009). The sensitivity of the strength of

the subtropical gyre circulation to the wind energy in-

put in the Caribbean Sea is interesting, but left for fu-

ture study. Integrating over the North Atlantic Ocean,

the total wind energy input is estimated to be about

0.14 TW.

In the following subsections we describe how the wind

energy input leads to the generation of eddy energy at

different depths (Fig. 3) and the subsequent vertical

fluxes (Fig. 4), which we explain by relating to the ver-

tical structure of the horizontal dispersion of eddy en-

ergy at three different latitudes (Figs. 5–7).

b. Eddy energy generation

Figure 3 shows the rate at which the eddies release

the APE stored in the mean stratification at different

depths. The eddies are found to release the APE ubiq-

uitously along the whole rim of the subtropical gyre,

even though they are strongly western intensified. The

vertical structure of eddy energy generation is, however,

very different between the subtropical and subpolar

gyres. In the subtropical gyre, the eddy energy genera-

tion, 2r0w0g, is mainly confined to the upper ocean,

FIG. 5. (a) The eddy energy source (2r0w0g,31025 W m23), (b) the vertical eddy energy flux

(p0w0, 31022 W m22), and (c) the balancing eddy energy sources/sinks [defined as the terms

inside the braces in (5), 31025 W m23] at 248N.


especially along the eastern and southern gyre bound-

aries, although it penetrates deeper than 1000 m in the

western boundary region. On the other hand, the eddy

energy generation in the subpolar gyre has a much

deeper structure with its maximum strength at around

2000 m depth on average. This difference in the vertical

structure is readily seen in the zonal transects along

248, 398, and 598N plotted in Figs. 5a, 6a, and 7a, rep-

resenting typical subtropical, western boundary, and

subpolar situations, respectively. The source of eddy

energy is clearly surface-intensified at both 248 and

398N and also strongly western intensified at 398N,

while it peaks at great depths and is also more uni-

formly distributed in the zonal direction at 598N. The

eddy energy generation,2r0w0g, follows the location of

the mean lateral density gradients, as one would expect

from linear instability theory (e.g., Green 1970; Stone

1972).

Along the continental shelf break the plot of 2r0w0greveals regions of blue as well as red, hinting that the

eddies might be fluxing energy back to the mean flow.

We interpret this feature along the sloping bottom

topography as a manifestation of the ‘‘Neptune’’ effect

(e.g., Holloway 1992) or, equivalently, the mixing of

potential vorticity along sloping boundaries (e.g.,

Greatbatch and Li 2000; Adcock and Marshall 2000).

Integrating over the whole North Atlantic Ocean, the

rate of APE extracted by baroclinic instability amounts

to about 0.11 TW, approximately 80% of the wind

energy input at the sea surface. Our results thus confirm

that the wind energy input to the gyre circulations is,

indeed, largely balanced by eddy generation through

baroclinic instability in our model, especially in the

western boundary current and its extension.

c. Vertical eddy energy flux

Figure 4 shows the vertical eddy energy flux, p0w0, atdifferent depths. The most prominent feature near the

surface is the downward eddy energy flux in the North

Brazil Current and Caribbean Sea. The signal is mixed,

although downward on average, in the western bound-

ary region, and weakly upward in the subpolar region.

Deeper down in the water column, the vertical eddy

energy flux is dominated by the downward flux in the



inside the curly brackets in (5), 31025 W m23] at 398N.


western boundary current and its extension and by the

upward flux in the subpolar region, especially in the

Labrador Sea. The vertical eddy energy flux along

the eastern and southern subtropical gyre boundaries is

upward near the surface and downward at depths, but is

too weak to be seen in Fig. 4. This difference in the di-

rection of p0w0 between the subtropical and subpolar

gyres is further illustrated in Figs. 5b, 6b, and 7b. There

are large downward eddy energy fluxes in the ocean

interior underneath regions of strong eddy energy

source at 248 and 398N, for example, the blue patches

between 758 and 608Wand at around 208Win Fig. 5b and

between 758 and 508W in Fig. 6b. This overall pattern of

downward eddy energy fluxes is consistent with the fact

that sources of eddy energy at these latitudes are located

in the upper ocean (Figs. 5a and 6a). Near the sloping

bottom topography in the western part of the transect at

398N, the vertical eddy energy flux appears upward,

possibly due to flow topography interactions, analogous

to the upward planetary wave energy flux in the atmo-

sphere when planetary waves encounter mountains. On

the other hand, the vertical eddy energy flux is mostly

upward in the Labrador Basin and Irminger Basin of

the subpolar gyre, consistent with the deep eddy energy

source there. Some regional features in Figs. 7a and 7b

are also interesting: for example, the eddy energy source

between 258 and 208W is located farther up in the water

column, and the vertical eddy energy flux is conse-

quently upward in the upper 200 m or so and downward

below.

d. Balancing eddy energy sources/sinks

Figures 5c, 6c, and 7c show the balancing eddy energy

sources/sinks inside the braces in Eq. (5) along 248N,

398N, and 598N, respectively. These eddy energy sources/

sinks are dominated by the divergence/convergence of

horizontal eddy energy fluxes (there is a small contribu-

tion from the vertical advection of eddy energy; not

shown). At 248N (Fig. 5) the horizontal eddy energy

fluxes are surface intensified, as one would expect if the

eddies project predominately onto the baroclinic modes,

but, nevertheless, with a nonnegligible component at

depth consistent with both baroclinic and barotropic

modes. This is consistent with the downward eddy energy



inside the braces in (5); 31025 W m23] at 598N.


fluxes found at this latitude. At 398N (Fig. 6) in the vi-

cinity of the separated Gulf Stream, the balancing eddy

energy sources/sinks have substantial magnitude at depth

with downward vertical eddy energy fluxes again required

to connect the shallow eddy energy sources with the

deeper horizontal eddy energy dispersion. In contrast, at

598N (Fig. 7) in the subpolar gyre a more complicated

picture is found, but the balancing eddy energy sources/

sinks are generally more surface intensified than the deep

energy sources, explaining the upward vertical eddy en-

ergy fluxes at this latitude. Thus, our model diagnostics

support the simple conceptual picture presented in Fig. 1

for the eddy energy balance in the interior of the sub-

tropical and subpolar gyres.

5. Concluding remarks

The eddy energy balance in the North Atlantic sub-

tropical and subpolar gyres has been investigated using

an eddy-resolving ocean model, with a particular focus

on the eddy energy generation and vertical eddy energy

fluxes. The major results of the present study are as

follows.

d The majority of the wind energy input to the large-

scale ocean circulation is released by the generation of

eddies through baroclinic instability.d The eddy energy generation is located near the surface

in the subtropical gyre but deeper down in the sub-

polar gyre.d To reconcile the mismatch between the depth of eddy

energy production and the vertical structure of the

horizontal dispersion of eddy energy, the vertical eddy

energy flux is downward in the subtropical gyre and

upward in the subpolar gyre.

Our results suggest that the spreading of eddy energy

by both vertical and horizontal eddy energy fluxes

needs to be taken into account in eddy closures that

carry an explicit eddy energy budget (e.g., Eden and

Greatbatch 2008; Marshall et al. 2012). Future work

should include analysis of the eddy energy balance in

models of higher resolution, with longer simulations,

and surface forcing with higher temporal and spatial

resolutions.

Acknowledgments. XZ thanks Dave Munday for

many helpful discussions about the MITgcm. Financial

support was provided by the U.K. Natural Environment

Research Council. The numerical calculations were

performed at the Oxford Supercomputing Centre

(OSC). We thank two anonymous reviewers for their

many constructive suggestions that led to a much im-

proved manuscript.

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vertical eddy energy fluxes in the north atlantic subtropical and subpolar gyres

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