On the role of eddy mixing in
the transport of zonal ocean
D I R K O L B E R S ([email protected])
The ocean is turbulent at all scales but the nature
of turbulence depends critically on the scale under consideration and thus turbulence in the ocean comes in many
different species. In the present chapter we are dealing with
planetary-scale currents and the turbulence field interacting with these. The relevant eddy scales are in the so-called
mesoscale range; depending on site – i.e. stratification and
Coriolis parameter – this range is roughly from 10 to 100
km. The intrinsic scale arising from the dynamic equations
is the baroclinic Rossby radius N h/ f , where N is the buoyancy frequency, f the Coriolis frequency, and h the ocean
depth. The Rossby radius is the preferred scale of baroclinic
instability of large-scale currents with vertical shear (see
e.g. Pedlosky [1987]). If the current is unstable (barotropic
instability may combine with the baroclinic instability in
a mixed process), eddies with scales of a few Rossby radii
arise, drawing energy from the potential energy of the shear
flow. Via Reynolds stresses and – as we shall outline in
this chapter – via eddy-induced interfacial form stress, the
turbulent field interacts with the mean flow, reshaping it
by feeding eddy kinetic energy into mean kinetic energy,
whereby the system may come to an equilibrated state, described by the Lorenz energy cycle for zonal flow (Lorenz
A frequently studied object in this area – since precomputer times a paradigm of atmospheric research – is
the large-scale geostrophic, stratified zonal current in the
zonally periodic domain. The model concept was employed for analytical investigations of baroclinic instability
of zonal flows (Charney [1947] and Eady [1949]; see Pedlosky [1987]) and served later in numerous studies of geophysical fluid dynamics, such as the development of parameterization concepts for the turbulent transports achieved by
mesoscale eddies (e.g. Green, [1970]; Stone, 1972; Held,
1978; Held and Larichev, 1996), turbulent shear flow on
the β-plane (e.g. McWilliams et al., 1978; McWilliams and
Chow, 1981; Vallis, 1988; Wolff et al., 1991), and homogeneous β-plane turbulence (e.g. Larichev and Held, 1995;
Pavan and Held, 1996).
The most important examples of zonal mean flow
in the atmospheric circulation are found in the westerly jet
streams in both hemispheres, but there is only one example
of a zonally unrestricted current in the world ocean, namely
the Antarctic Circumpolar Current (ACC). The uniqueness
of this current is manifested by many outstanding properties: it is the only important conduit linking the Atlantic,
Pacific, and Indian Oceans; with its length of roughly 20 000
km it is the longest continuous ocean current; with a transport in the range1 130 Sv (through Drake Passage, Whitworth [1983]) to 150 Sv (between Tasmania and Antarctica,
Rintoul and Sokolov [2001]) it is the largest of the world
ocean’s current systems; its vertical structure is baroclinic
but it does not exhibit significant inversions of the velocity
(e.g. Olbers and Wenzel, 1989) and direction (e.g. Killworth, 1992) with depth as other large currents do; though
eddy activity is present in all large-scale currents in the
world ocean and though marine topography plays a steering role in most of these currents, these two features are
responsible for balance properties of heat and momentum
that are unique to the ACC. The last-mentioned properties
are the concern of the present chapter. Further observational and theoretical concepts and results can be found
in the recent reviews of Olbers (1998) and Rintoul et al.
Eddies are defined here as features in the difference
field of the instantaneous and the time-mean circulation.
These will be called “transient eddies.” There is another
class of eddies in geophysical current that arise if the timemean current is not zonal but undulating due to non-zonal
forcing or topography (both conditions are found in the
ACC). It has become customary to call the difference field of
the time mean and the time-plus-zonal mean the “standing
Most of the ingredients of the physics of zonal currents used here can be elucidated by reference to a simple conceptual model. Consider a zonally unbounded strip
of ocean with the ACC imbedded and split the water column into three layers (which may be stratified) separated
by isopycnals. The upper layer of thickness η1 includes the
Ekman layer, the intermediate layer with base at z = −η2
lies above the highest topography in the Drake Passage belt
(the range of latitudes running through Drake Passage), and
the lower one reaches from z = −η2 to the ocean bottom
at z = −h. On writing the depth and zonally integrated
northward volume flux in each layer as Vi , i = 1, 2, 3, the
1 Sv = 106 m3 s −1 .
Marine Turbulence: Theories, Observations, and Models. Results of the CARTUM Project, ed. Helmut Baumert, John Simpson,
C Cambridge University Press 2005.
and Jürgen Sündermann. Published by Cambridge University Press. 512
Zonal ocean currents
time and zonally averaged balance of zonal momentum
− f V̄1 = −η1∗ p1x
+ τ0 − τ1 − R 1 ,
− η2∗ p2x
+ τ1 − τ2 − R 2 ,
− f V̄2 = η1∗ p1x
− hpbx + τ2 − τb − R3 ,
− f V̄3 = η2∗ p2x
where pi are the pressures in the respective layers, p3 =
pb is the bottom pressure, the overbar denotes the timeand-zonal mean, the star denotes the deviation from this
average (the starred quantities thus include transient and
standing eddies), τ0 is the zonal wind stress, τi are the frictional stresses at interfaces, τ3 = τb is the frictional bottom stress, and Ri are the divergences of appropriate lateral
Reynolds stresses. The meridional circulation is characterized by the pattern of meridional transports V̄i , which, in
this isopycnal framework, are of Lagrangian quality. The
Eulerian parts of V̄i consist of the wind-driven component −τ0 / f (the Ekman transport) in the top layer and a
geostrophic component in the bottom layer, which
is associated with the bottom form stress hpbx . Since i V̄i = 0
by mass balance and assuming that Ri and the bottom frictional stress τb can be neglected, these Eulerian parts of
the meridional circulation must balance, and the same argument states that the overall balance of zonal momentum
is that between the applied wind stress and the bottom form
τ0 hpbx .
This balance has been confirmed for most of the numerical models which include the opposition of submarine
topographic barriers to the zonal flow and have a realistic magnitude of the Reynolds-stress divergence (see e.g.
the POP model in Section 58.3). Eddy effects seem to be
unimportant in the vertically integrated balance unless the
ocean bottom is flat and the neglected Reynolds and frictional terms come into play. Then, simple transport formulas arise: for the familiar diffusive parameterization of
lateral eddy-induced transports of momentum in terms of
a diffusivity Ah , the zonal transport is then proportional to
Y 3 τ0 /Ah for a current of width Y . Models with flat bottoms have a transport of the ACC of a couple of hundred
sverdrups (more than 600 Sv in Bryan and Cox [1972]),
reflecting “Hidaka’s dilemma” (Wolff et al., 1991): with
Reynolds and frictional stresses as the only means for removing the momentum being put into the ACC belt by wind
stress, either unrealistically large transports are obtained
or unrealistically large eddy viscosities have to be considered. In conclusion, realistic models of the ACC must
include topographic effects in order to satisfy the overall
2 We work with the Boussinesq approximation. Pressure and stresses are
divided by a constant reference density.
balance of momentum in the presence of realistic transports.
If, in addition to the assumptions of smallness of Ri
and τi , the flow is adiabatic, then the meridional transport
in each layer must vanish, V̄i = 0, and we find that the
interfacial form stress ηi∗ pi∗x is vertically constant and equal
to τ0 (and the bottom form stress),
ηi∗ pi∗x τ0 ,
i = 1, 2.
Assuming that only transient eddies (denoted by a prime)
contribute, and equating the zonal pressure gradient to the
northward geostrophic velocity, f vg = px , and the layerthickness fluctuation to the (potential) density anomaly,
η = ρ /ρ̄z , we recover the Johnson–Bryden relation
(Johnson and Bryden, 1989)
v ρ = τ0 ,
according to which the northward eddy density flux v ρ (or, loosely speaking, the eddy heat flux), normalized by the
mean density gradient ρ̄z , in the circumpolar belt of the ACC
is of the size of the zonal wind stress τ0 . This simple formula
clearly shows the importance of eddies in the dynamics. It
established one of the most celebrated models of the ACC
transport. On parameterizing the transient density flux by
a gradient form, v ρ = −κρ y , and using the thermal wind,
f Uz = gρ y , the relation does indeed become prognostic for
the zonal shear of the current with current profile U = ū(z),
thus relating the transport T = zr U (z)dz (relative to some
reference level z r ) to the zonally averaged wind stress τ0
and the eddy diffusivity κ, as will be discussed further in
Section 58.2.1
The action of eddies not only is manifested in the
interfacial form stress, but also implies a lateral eddy transport of momentum (the Reynolds-stress term), and these
combine to give the transport of potential vorticity (PV) q.
A more precise formulation of the momentum balance than
(58.4) is thus expressed as a balance between the eddy PV
flux and the vertical divergence of the frictional stress τ in
the water column (Marshall et al., 1993). In a flat-bottom
ocean or the ocean part above the highest topographic barrier along the zonal path of averaging, with adiabatic conditions as manifested in quasi-geostrophic (QG) models, the
balance is written as
∂ ∂ v ρ ∂τ
uv + f
= vq = − ·
∂z ρz
Equation (58.4) is in fact the consequence of (58.5) if the
Reynolds-stress divergence is small and significant frictional effects are absent below the Ekman layer. Notice
that, outside the Ekman layers at the top and bottom of the
ocean, the frictional stress should vanish, thus implying that
v q = 0. The momentum balance in the form (58.5) is the
58.2 Mixing–transport relations
center point of the discussion of QG dynamics and eddy
parameterizations in Section 58.2.2. Incidentally, the eddy
PV flux can be expressed as the divergence of a flux in the
plane spanned by the meridional and vertical directions,
v q = ∇ · F, with the Eliassen–Palm flux vector
F = (−u v , f v ρ /ρ̄z )
used in Section 58.3. This property is a severe hinderance
of parameterizing the PV flux: the parameterized form must
accomplish the integral properties of the divergence in order
to be able to accomplish the balance of momentum. Parameterizations of the momentum flux u v or the density flux
v ρ do not suffer from such constraints. It should be mentioned, however, that no meaningful parameterizations of
u v (with up-gradient transport) are known.
The above concepts neglect many processes that
might be important in numerical simulations of zonal currents or the ACC itself. A more complete theory is developed in Section 58.3, where the incorporation of thermohaline effects (diapycnal mixing) and the influence of
topography are considered. Both effects can be identified
in the simple model above. Topographic effects are seen in
the presence of the standing contribution to the interfacial
stress and in the bottom form-stress term. Furthermore, if
there is exchange of mass between the layers, implying conversion of water masses, V̄i equals the net rate of exchange
with the neighboring layers (the integral of the divergence of
diapycnal transport south of the respective latitude), which
implies diabatic interior effects and surface fluxes of density and, at the same time, a non-zero vertical divergence of
the interfacial form stress. The thermohaline forcing of the
zonal flow is thus hidden in the Coriolis force, or, equivalently, in the meridional overturning streamfunction.
Mixing–transport relations in flat-bottom
The formulation of a complete theory capable of predicting the absolute transport of the ACC is a formidable
challenge. Such a theory would need to account for wind
and buoyancy forcing, stratification, the effects of eddy
fluxes in the momentum and buoyancy budgets, and interactions between the strong deep currents and bottom topography. Some insight can be gained into the factors controlling
the transport of the ACC by appealing to a variety of simpler
58.2.1 Simple Johnson–Bryden models
Disregarding the standing-eddy contribution as in
(58.4) – and other terms that result when topography is
present and the mean flow is not completely zonal – the
transport theory combines the balance of zonal momentum
with parameterizations of the density flux, as indicated
above. This results in
Uz = τ0 .
Apparently, Au = κ f 2 /N 2 defines a diffusivity for vertical momentum transfer achieved by lateral density diffusion. Such an equivalence was previously pointed out by
Rhines and Young (1982), Olbers et al. (1985), and others. Johnson and Bryden (1989) used Green’s
form (Green,
1970) of the diffusivity κ = α2 | f |/ Ri . It is obtained
for a baroclinically unstable flow, where Ri = N 2 /Uz2 is
the local Richardson number,
is a measure of the eddy√
transfer scale, and | f |/ Ri is a growth rate (actually applying to Eady’s model). The constant α measures the
level of correlation between v and ρ in the density flux
(α = 0.015 ± 0.005 according to Visbeck et al. [1997]).
The shear of the zonal flow and wind stress are then related
| f |3 2 2
Uz = τ0 .
Johnson and Bryden’s results are obtained by equating
the turbulence scale to the baroclinic Rossby radius
λ = N h/(| f |π ). For = π 2 λ we obtain their estimate of
the shear:
Uz =
1/2 N
τ0 /ρ0 N (z) 1/2
| f | π 3 αλh
π 2 αh 2 | f |
The first relation was used by Johnson and Bryden (1989),
with λ taken to be a measure of the (vertically constant)
bulk Rossby radius, and shows that the shear is proportional to the local Brunt–Väisälä frequency N (z). More importantly, the shear is proportional to the square root of the
wind-stress amplitude τ0 . In the following we use a local
Rossby radius and an exponential Brunt–Väisälä frequency
profile, N (z) = N0 exp[z/(2d)]. With τ0 = 0.2 m2 s−2 , h =
3500 m, N0 = 1.4 × 10−3 s−1 , d = 2500 m, and a width
Y = 600 km of the ACC, integration of (58.9) yields a transport of 82 Sv relative to the bottom.
Visbeck et al. (1997) suggested that, in the presence
of differential rotation,
√ the eddy transfer may be restricted3
by the Rhines
U/β rather than the Rossby radius.
With = U/β we find a cubic relation between τ0 and
the velocity,
UUz2 =
τ0 β N 3 (z)
α | f |3
For the exponential N (z) this is easily integrated. A transport of 67 Sv relative to the bottom and a total transport
Actually, to be Galileian invariant, the U in the Rhines scale must be a
measure of the square
root of the kinetic energy of fluctuations of the jet,
i.e. the assumption
u 2 ∼ U is hidden within the concept.
Zonal ocean currents
of 124 Sv are obtained for the above set of parameters. In
this model the transport would only mildly increase with
increasing magnitude of the wind stress, as τ0 . Notice,
however, the dependence on β in this regime.
If eddy mixing of PV is down the mean PV gradient, v q = −k q̄ y , the vanishing of the eddy PV flux
implies zero PV gradient, as suggested in the preceding
section on the basis of the balance of momentum. This
results in q̄ y = 0, and thus homogeneous mean PV. Observations do indeed show that isopycnal vorticity gradients
are small (compared with the planetary vorticity gradient
β) in and north of the Antarctic Current regime (Marshall
et al., 1993). Furthermore, in that investigation a linear relation between the large-scale PV and density was found to
exist, namely fρz = a + bρ, with d = | f |/b, the e-folding
scale of the density field. This implies an exponential N (z),
as assumed before, and it also imposes a constraint on the
current shear,
Uzz −
=β 2,
which is obtained by taking the meridional derivative of
fρz = a + bρ and the thermal wind relation f Uz = gρ y .
Vertical integration immediately leads to the velocity profile
and the transport, expressed in terms of the vertical shear at
some level z 0 . A more meaningful interpretation is found if
(58.11) is reformulated by inserting (58.7), which yields a
constraint on the vertical profile of the diffusivity κ,
∂ f2
∂ 1
= .
∂z κ
∂z N 2 Av
Apparently, the assumption of a homogeneous PV state sets
the vertical profile of the lateral diffusivity of density. Notice
that β rules the vertical profile of the diffusivity and that τ0
represents the vertically constant flux of momentum in the
water column which equals the wind stress. In this model
the shear consists of two parts,
Uz =
transient-eddy effects are taken into account in these parameterizations. It is thus not surprising that numerous attempts
with numerical models, whether adiabatic or full potential
energy but with consideration of topography, failed to ver1/2
ify the prediction of the square-root dependence, T ∼ τ0 ,
resulting from a Green parameterization of the diffusivity
κ. Attempts also with the cubic relation, T ∼ τ0 , resulting from the Rhines-scale approach, were without success.
However, there is only a very restrictive range of applicability of (58.4): it is valid only under adiabatic conditions,
only for the water column below the Ekman layer and above
the highest topography in the circumpolar belt, and only if
Reynolds stresses are small in the balance of zonal momentum. It should also be pointed out that, in the above
consideration, the stratification is prescribed by the Brunt–
Väisälä frequency N (z). However, since this is established
by thermohaline forcing, diffusion, and the transports of
heat and salt, it should depend indirectly on the wind stress
as well. This question is posed in Section 58.3.3.
+ β(z − z 0 ) ,
where κ0 = κ(z 0 ). The first contribution is directly winddriven. The second contribution is driven by the eddies
which homogenize the associated PV. The transport (relative to the bottom) of the latter is fairly small and westward (≈ −2 Sv), whereas the first part contributes 39 Sv for
our standard values and a diffusivity of κ0 = 1000 m2 s−1
at z 0 = −1000 m. Following (58.12), κ then increases to
1200 m2 s−1 at depth 3500 m.
As is evident from (58.9), (58.10), and (58.13) the
dependence of the baroclinic transport on the amplitude of
the wind stress and the Brunt–Väisälä frequency is generally governed by the degree of non-linearity of the eddyflux parameterization. It should be kept in mind that only
58.2.2 The quasi-geostrophic view
The relation (58.5) is the QG form of the time-andzonal-mean-momentum balance for zonal flow over a flat
terrain. The QG dynamics are governed by the balance
+ J (ψi , qi ) =
+ ∇ · ui qi = Fi
for the QG potential vorticity (QPV) qi , driven by vorticity
sources Fi . The balance is written for a stack of layers as in
the conceptual model in Section 58.1 (however, with immiscible layers one has Vi ≡ 0), thus ψi is the streamfunction
of the geostrophic current in the ith layer (i = 1, . . ., n),
u i = −ψi y , and vi = ψi x are the velocities, and the QPV is
given by
qi = ∇2 ψi +
(ηi − ηi−1 ) + f
with f = f 0 + βy for flow on a β-plane. As before, the ηi
are the interface depths (in fact, only deviations from a mean
layer thickness Hi matter). They may be expressed in terms
of the streamfunctions; in most applications a rigid-lid condition is used at the top, thus η0 ≡ 0. We discuss properties
of a two-layer model in which the elevation of the interface
is η = η1 + H1 = ( f 0 /g )(ψ2 − ψ1 ). We will examine consequences of parameterizations of the eddy transport vi qi
of QPV and vi η of mass for the zonal transport in a channel
flow that is driven by a zonal wind stress in the top layer,
and has bottom friction but no interfacial friction.
The numerical model applies to a flow in a twolayer channel of length X = 4000 km and width Y = 1500
km on a β-plane the layers have depths H1 = 1000 m and
H2 = 4000 m. The balance of QPV includes friction terms
58.2 Mixing–transport relations
Table 58.1. Some integral quantities of the numerical experiment: Hi Ti is the
transport in the ith layer, and E ikin and E pot are the mean kinetic and potential
energies of the flow
H1 T1
(106 m3 s−1 )
H2 T2
(106 m3 s−1 )
E 1kin
(m3 s−2 )
E 2kin
(m3 s−2 )
E pot
(m3 s−2 )
Fi = k · ∇ × τi /Hi − Ahyp ∇6 ψi where τ1 is the windstress vector, and τ2 is the frictional bottom stress. Anticipating results from a numerical model, subgrid-scale
effects are modeled by biharmonic lateral friction with
hyperviscosity Ahyp = 1010 m4 s−1 . It reflects the parameterization of subgrid momentum transport and serves as
energy and enstrophy dissipation. The channel has a central latitude of 60o S, with f 0 = −1.263 × 10−4 s−1 and
reduced gravity g = 0.02 so that the internal Rossby
radius λ = {g H1 H2 /[ f 02 (H1 + H2 )]}1/2 is 32 km. Lateral boundary conditions ∇2 ψi = 0 and ∇4 ψi = 0 on both
walls (the latter establishes zero momentum flux) and integral auxiliary conditions (McWilliams, 1977) are standard. A fourth-order-accurate formulation of the Jacobian
(Arakawa, 1966) turned out to be necessary in order to compute the second-order eddy balances correctly (Wolff et al.,
1993). The resolution of the model grid is 10 km. The wind
stress is zonal and zonally constant,
π(y + Y/2)
τ1 = τ0 sin
with amplitude τ0 = ±10−4 m2 s−2 for eastward or westward forcing. In the experiments discussed here the frictional stress at the bottom is taken as a linear functional
of the bottom velocity, i.e. τ2 = −H2 u2 , where is the
corresponding coefficient of linear bottom friction (we take
= 10−7 s−1 ).
Numerical solutions obtained with this model have
been described by Wolff and Olbers (1989), Wolff et al.
(1991), Olbers et al. (1997, 2000), and Ivchenko et al.
(1997). Other important similar experiments are reported
in McWilliams et al. (1978). These applications embrace a
large suite of flat-bottom experiments (also with other forcings and non-linear bottom friction) and experiments with
topography in the deep layer (not considered in the QG
dynamics described above). Here, we concentrate the discussion on flat-bottom cases: a typical β-plane case (EFB),
an f -plane case (EB0), both forced by eastward wind, and
a westward-forced β-plane case (WFB). All experiments
were initially integrated for 7 years starting from the state of
rest. At this time the currents in EFB and EB0 are identical
and WFB mirrors exactly the eastward cases: a wide lami-
nar flow matching the meridional scale of the wind stress.
The zonal velocity profile follows the smooth profile of the
applied wind stress in all cases with maximum velocities of
about 40 cm s−1 and the kinetic and potential energies are
identical. The flow field was then perturbed and, because
the states were unstable with respect to this disturbance, the
flow changed dramatically to a fully turbulent field when
the disturbance was advected with the main current and also
propagated. The integration was continued for a total of 110
model years and, judging by the form of the evolution with
time of the kinetic and potential energies, the flow can be
considered to be in a statistically steady state about 4 years
after the introduction of the disturbance. Transports in the
layers and energies in the quasi-steady state can be found
in Table 58.1.
Figures 58.1–58.3 show instantaneous total streamfunction fields and the corresponding eddy streamfunctions
(defined as deviations from the time mean) at the end of
the experiment. The narrow jet in EFB with a characteristic
meridional scale of 400–500 km (which is much larger than
the Rossby radius and much smaller than the wind-stress
scale) is seen to meander significantly, with stronger meanders appearing in the upper layer. Along the entire length
of the basin there are typically five large-scale wave-like
disturbances of the main jet; the wavelength is thus approximately 800 km. The maximum velocity of the mean eastward jet is about 60 cm s−1 . The f -plane experiment EB0
has a much more sluggish flow. Zonal waves are present but
cannot be attributed to a dominant scale. Also, the currents
in the β-plane experiments EFB and WFB differ quite significantly. The westward-flow WFB has a current that is not
a narrow jet, but wide and smooth with a typical meridional
scale equal to the basin width, and significantly smaller
velocities on the current axis.
The EFB eddies form a regular chain in the vicinity
the center of the jet (particularly in the lower layer). The
eddies have an ellipsoidal form, leaning into the direction of
the jet. This pattern causes the Reynolds-stress convergence
and thus the concentration of the jet (Holland and Haidvogel, 1980). The eddy field in EB0 is rather weak compared
with the β-plane cases; the eddies have no dominant scale.
In both cases, the eddies have a strong barotropic vertical
Zonal ocean currents
m s
m s
Fig. 58.1. Instantaneous (upper panels) and eddy streamfunctions (lower panels) for the experiment EFB (flat bottom, β-plane, eastward
wind), the upper layer is on the left and the lower layer is on the right. Contour intervals are 2 × 104 m2 s−1 for the upper panels and
5 × 103 m2 s−1 for the lower-left-hand panel and 2.5 × 103 m2 s−1 for the lower-right-hand panel.
m s
Fig. 58.2. As Fig. 58.1, for the experiment EB0 (flat bottom, f -plane, eastward wind).
m s
58.2 Mixing–transport relations
m s
Fig. 58.3. As Fig. 58.1, for the experiment WFB (flat bottom, β-plane, westward wind).
structure. In WFB the eddy field appears chaotic and eddies
are scattered all over the channel. In contrast to EFB, the
mesoscale eddies in WFB are much stronger in the lower
layer than in the upper layer.
Figure 58.4 shows the profiles of the time-andzonally averaged zonal velocity. Although the total transports of the three cases are similar (see Table 58.1), we find
a significant difference in the structure of the flow. The concentrated jet in EFB with a width of about 400 km is flanked
by side lobes having a similar width. The f -plane flow is
wide and smooth with a typical meridional scale equal to
the basin width – it mirrors almost exactly the sinusoidal
profile of the wind stress. The westward flow has a very
similar structure. The velocities in the center of the jet are
significantly smaller than those in the strong jet in EFB. The
flow is unstable in both eastward cases, whereas for EFB
the mean shear is above the critical value of Phillip’s
inviscid linear instability
criterion (S = ū 1 − ū 2 > β
g H2 / f 02
or S < −β g H1 / f 02 for eastward and westward flow, respectively; the criterion is not exactly applicable since the
current profiles are not meridionally uniform, see e.g. Pedlosky [1987]).
Further time-and-zonal mean fields are presented
in the above-cited articles in which the balances of
the most important properties (momentum, enstrophy,
energies) were investigated as well. The momentum
−Hi vi qi = τ̄i
is of relevance for the following discussion (the overbar
indicates the time-and-zonal average, as before). It is the
layer integral of (58.5), or derived from (58.14) using
Fi = −(1/Hi )∂τi /∂ y (omitting the hyperviscous terms),
τ̄1 (y) is the prescribed wind stress, and τ̄2 = − H2 ū 2 is
the frictional bottom stress. It is immediately evident from
the form of the QPV flux (see Fig. 58.5) that the total QPV
flux, integrated over the depth and width of the channel,
−Y /2
H1 v1 q1 + H2 v2 q2 dy = 0.
This property guarantees the conservation of momentum:
the momentum input by wind in the top layer is extracted
totally by friction at the bottom,
−Y /2
(τ̄1 − H2 ū 2 ) dy = 0.
The constraint (58.18) is a non-local property, which puts a
heavy burden on any parameterization of the eddy QPV flux
since violation would spoil (58.19). Evidently, the relation
(58.19) fixes the transport in the bottom layer entirely in
Zonal ocean currents
0.5 −0.5
u1 0.1
0.5u 0.5
Fig. 58.4. Time and zonal mean profiles of the velocity ū 1 (full), ū 2 (dashed), the barotropic velocity C = (H1 ū 1 + H2 ū 2 )/H (dash–dotted)
and the vertical
shear velocity S = ū 1 − ū 2 (dotted) for EFB, WFB, and EB0 as functions of the scaled meridional latitude y/Y . The critical
phase speeds β g H2 / f 02 for eastward flow and −β g H1 / f 02 for westward flow are included as straight lines for EFB and WFB. Units are
m s−1 .
0.5 −0.5
Fig. 58.5. Eddy diffusivities ki of the potential vorticity flux and κ of layer thickness, as functions of the scaled meridional latitude y/Y , in
units of m2 s−1 . Boundary layers are omitted. EFB (full), WFB (dotted), and EB0 (dash-dotted).
terms of the applied wind stress (see the next section). Thus,
all three experiments should have the same value of the
bottom-layer transport (see Table 58.1; differences are due
to finite-time averaging).
The layer gradients of mean QPV q̄i reveal the
well-known property of the differing signs in the upper
and lower layers in all cases; thus the general condition
of baroclinic instability is satisfied. It is important for the
purposes of the present study that the QPV flux is downgradient in all cases, so the numerically determined diffusivities
ki = −vi qi
∂ q̄i
are positive throughout. The profile of ki is rather complex. A local minimum is observed in the center of the
jet for the eastward cases EFB and EB0 and maximum
values are found on the flanks of the jet. In contrast, the
diffusivities of the WFB case possess a single maximum
in the center; the structure is sinusoidal, as are the current
Another prominent example of a down-gradient flux
is the eddy-induced mass flux (or layer-thickness flux) vi η
such that
κ = −v1 η
∂ η̄
is positive as well. The eddy flux v1 η (which equals v2 η )
is called the “bolus” transport velocity in isopycnal formulations of mean transport equations for tracers (see e.g.
Gent and McWilliams [1990], and Gent et al. [1995]). In
the momentum balance (58.17) the lateral eddy mass transport is equivalent to a vertical momentum transport and a
parameterization of the form (58.21) then implies vertical
transport of horizontal momentum with a diffusion coefficient κHi f 02 /g (see also Eq. (58.7)). In the layer framework
this appears as interfacial friction. In truly large-scale flow
where the respective contributions of relative vorticity can
be neglected in the QPV flux as well as in the mean QPV
gradient, the main difference between QPV diffusion and
thickness diffusion is found in the presence of the eddy
transport of planetary vorticity, ki β, in (58.20).
58.2 Mixing–transport relations
Note that the coefficients for QPV diffusion in the
layers are different and have a clear ordering, k2 > k1 for
eastward flow and k2 < k1 for westward flow on the βplane. In the f -plane case the coefficients have similar
magnitudes (within numerical errors from finite-time averaging, they are equal). The diffusivity κ for layer thickness has a similar structure to the ki , with less prominent
central valleys. It is shown in Olbers et al. (2000) that
known parameterization concepts based upon baroclinic
instability (Green, 1970; Stone, 1972) or homogeneous
β-plane turbulence (Larichev and Held, 1995; Held and
Larichev, 1996) do not explain the double-peak structure
of the eastward cases. We would like to point out that the
following analysis does not consider (58.20) or (58.21) a
physically motivated parameterization. For this task the
diffusivities must be related to mean-field quantities and
regime-robust universal coefficients. We simply investigate
the implication of the positivity of the ki and κ for the channel transport.
The zonal transport based on effective diffusivities
There is a straightforward way to utilize the parameterized
momentum balance (58.5), written now in the form
∂ q̄i
with the help of (58.20), to obtain explicit relations between
the transports in QG models and the external parameters,
i.e. the wind stress, the diffusion and friction coefficients,
and the channel dimensions. On inserting the expression
(58.15) of the QPV into (58.22), we find
∂ 2 ū 1
− 0 (ū 2 − ū 1 ) + β =
g H1
k1 H1
∂ 2 ū 2
− 2 + 0 (ū 2 − ū 1 ) + β = − ū 2 .
g H2
On integrating (58.23) across the channel and utilizing
free-slip conditions for simplicity at the walls, we obtain a set of equations for the layer transports (per unit
Ti =
dy ū i .
These take the form
f 02
(T2 − T1 ) =
− β H1 Y,
f 02
(T2 − T1 ) +
T2 = −β H2 Y,
g K2
with integrated wind stress W and effective (weighted) diffusivities K i , defined by
−Y /2
τ1 dy,
−Y /2
−Y /2
ū 2
The presence of the eddy fluxes requires the conditions of
baroclinic instability to be fulfilled. In terms of the transport
variables Ti , these are written as
T1 − T2
 > g βY H2 / f 02
for eastward flow,
 < −g βY H / f 2
1 0
for westward flow,
which implies that W/(βY H ) > K 1 for W > 0 and |W|/
(βY H ) < K 2 for W < 0, as a consequence of the transport
relations (58.25). These bounds of the effective diffusivities
are satisfied in our experiments.
Surprisingly, the linear set of equations (58.25)
yields solutions for the transports Ti that seem to be nonzero (arising from the β terms) for vanishing forcing τ1 . One
must, however, remember that the diffusivities K 1 and K 2
are not independent but must obey the constraint (58.18),
which, by use of (58.19), is equivalent to the condition of the
total momentum balance of the channel, which determines
the deep transport, T2 = W/( H2 ). On summing (58.25)
and using (58.26) and (58.19), this condition assumes the
particular form
βY H =
where H = H1 + H2 is the total depth of water. This shows
that a proper choice of the diffusivities in fact converts the
β terms in (58.25) into terms proportional to the forcing
W. Notice that (58.28) implies that K 1 < K 2 for eastward
wind stress and K 1 > K 2 for westward wind stress on the βplane, as pointed out before for the numerically determined
ki . In contrast, the f -plane case has K 1 = K 2 , which is
also confirmed by the numerical result. Using (58.28), one
easily finds the solution for the transport in the top layer:
T1 = T2 +
H f 02
where the diffusivities are still subject to the constraint
(58.28). The solution is shown in Fig. 58.6 for the parameters of the three numerical experiments. The figures include
as contours also the values of H1 T1 for the specific cases and
the curves of the constraint (58.28). On extracting the solution K 1 and K 2 from the intersection, we see a good agreement with the eddy diffusivities from Fig. 58.5 in each case.
The depth profile of the QPV diffusivities is thus
a consequence of the balance of momentum. With this
Zonal ocean currents
−2 9
eddy diffusivity layer 1 [m2 s−1]
1500 2000 2500 3000 3500
eddy diffusivity layer 2 [m2 s−1]
1500 2000 2500 3000 3500
eddy diffusivity layer 2 [m2 s−1]
1500 2000 2500 3000 3500
eddy diffusivity layer 2 [m2 s−1]
constraint the planetary conditions enter the theory via dependence on β. Returning to a vertically continuous representation, the relation (58.28) implies
(1/K )z ∼ β/τ0 ,
which is identical to the implication (58.12) of PV homogenization, which was discussed in Section 58.2.1. Notice,
however, that the two diffusivities describe different properties.
This theory can easily be extended to more than two
layers. Notice that the diffusion coefficients of intermediate
layers that are not in contact with the surface or the bottom
do not enter explicitly since these layers are frictionless.
Eddies must, however, be present to homogenize the QPV;
in fact, the momentum balance (58.22) of the intermediate
layers states that the QPV is constant there (see Marshall
et al. [1993] for further demonstration with a three-layer
QG model of the ACC with realistic coastlines and topog-
Fig. 58.6. The transport H1 T1 versus K 1 and K 2 according to (58.29)
for the parameters of the three cases EFB (left), WFB (middle), and
EB0 (right). The dashed curves show the transport contour for the
corresponding numerical solution, and the dotted line is the relation
between the K i given by the constraint (58.28).
raphy). The intermediate layers do, however, contribute to
the transport.
Utilization of the down-gradient property of the
thickness flux, as indicated in (58.21), for determination
of transport is possible only if the Reynolds-stress contribution to the eddy QPV flux in (58.5) or (58.17) is negligible
(as assumed in Section 58.2.1). In this case, the transport
equations corresponding to (58.25) are much simpler – the
coefficients K i are replaced by one value K arising from κ,
and the β terms are absent. Thus, there is no constraint like
(58.28), and T1 − T2 = g W/(K f 02 ), which can be viewed
as the QG form of the Johnson–Bryden relation. This
approach might be applicable to WFB and EB0, but not for
the eastward β-plane case EFB (and a large suite of other
eastward cases presented in Olbers et al. [2000]). In
this most prominent regime, being closest to the ACC
conditions, we find convergence of zonal momentum by
Reynolds stresses an important ingredient of the flow
58.2 Mixing–transport relations
Functional relations The structure of the flow has been
analyzed by considering the functional relationship q¯i =
G i (ψ̄i ) between the mean QPV q̄i and the streamfunction ψ̄i
in each layer.4 Such a relation trivially exists here because
both fields depend only on y. In general a functional relation between q̄i and ψ̄i demands smallness of the frictional
terms compared with the mean advective terms in the QPV
balance (see below). Knowing G i (ψ̄i ) in fact completely
determines the flow: the streamfunctions can of course be
recovered by solving the differential equations
∇2 ψ̄i + (−1)i
f 02
g Hi
(ψ̄1 − ψ̄2 )
+ f = G i (ψ̄i ),
i = 1, 2.
Appropriate boundary conditions are ψ̄i = constant and
∇2 ψ̄i = 0 (in agreement with the numerical model). All
other boundary conditions and constraints are incorporated
into the functionals G i , in particular the values of the transports, as shown below. For a large-scale flow the relativevorticity part can be neglected. Then (58.31) becomes an
algebraic problem.
The numerical experiments indicate that the functional relationship G i (ψ̄i ) is almost linear (see Fig. 58.7
and also McWilliams et al. [1978]), i.e. we may write
q̄i = G i (ψ̄i ) = Ai ψ̄i + Bi + gi (ψ̄i ),
where Ai and Bi are dimensional constants and gi a small
remainder, which, moreover, may be assumed to vanish for
the boundary values of the corresponding streamfunctions.
It is obvious from (58.31) that the coefficients Ai and Bi are
directly related to the four values of the streamfunctions at
the boundaries. Specifically, the Ai are given by the transports in the two layers (or vice versa), as follows from
− 0 (T2 − T1 ) + β H1 Y = −A1 H1 T1 ,
f 02
(T2 − T1 ) + β H2 Y = −A2 H2 T2 .
In this section, the overbar indicates only the time average, i.e. there is
no zonal averaging implied.
characteristics, as manifested in the central jet of the EFB
experiment, shown in Figs. 58.1 and 58.4.
The seemingly linear dependence of layer and total
transports on the applied wind stress W must be taken with
care: the numerically determined diffusivities ki and κ are
not parameterizations; they vary if the forcing or the system
parameters (channel dimensions, coefficients of friction and
stratification, etc.) are changed.
−1.5 −1
−0.5 −1.5
Fig. 58.7. The functional relation between streamfunction and QPV
for the three experiments. The line with smaller ψ range refers to the
lower layer, units are 105 m2 s−1 for ψ̄i and 10−4 s−1 for q̄i .
The presence of the eddy fluxes requires that the conditions of baroclinic instability be fulfilled. In terms of the
transport variables Ti these are given by (58.27). It is an
easy matter to prove A1 < 0 and A2 > 0 from (58.33) and
(58.27). In fact, the general condition of baroclinic instability in a two-layer system may be expressed as ∂G 2 /∂ ψ̄2 > 0
for eastward flow and ∂G 1 /∂ ψ̄1 < 0 for westward flow,
in agreement with the numerical experiments. The Ai
and the diffusivities K i are of course related: A1 K 1 =
− H1 /( H2 )] + [g H1 /(H f 02 )](H2 /K 1 + H1 /K 2 )
A2 K 2 = . We would like to emphasize that all information about the flow beyond the transports is contained in
the deviations gi (ψ̄i ) from the gross linear part Ai ψ̄i + Bi .
Neglecting the gi in (58.31) results in a trivial parabolic
shear flow, which is a bad approximation to our numerical
results of EFB.
The functionals G i (ψ̄i ) result in a complex indirect
way from the forcing at the surface and friction at the bottom
as well as from the eddy fluxes of QPV. A simple view may
be gained by elaborating on the time-mean QPV balance,
J (ψ̄i , q̄i ) = −J (ψi , qi ) + F̄i = −∇ · ui qi
+ k · ∇ × τ̄i /Hi .
For the ideal case of a time-mean – strictly – zonal flow
(as for a flat bottom) the advection of the time-mean QPV
vanishes, J (ψ̄i , q̄i ) = 0, and the divergence of the eddy
QPV flux balances locally the curl of the frictional stress,
as formulated by (58.5) or (58.17) for momentum. For nonzonal flow, e.g. in the presence of topography, the advection
of the time-mean QPV would not vanish in general but the
Zonal ocean currents
integral of the mean advection term of areas bounded by
time-mean streamlines always vanishes, which implies
ψ̄i =constant
ni · ui qi − ti · τ̄i /Hi ds = 0,
where ni and ti are normal and tangential unit vectors with
respect to streamlines (ψ̄i contours) and ds is a line element
along those contours. An approximate functional relation
q¯i G i (ψ̄i ) is found if the terms on the RHS of (58.34) are
small, of the order of a small parameter δ that could be the
angle between contours of ψ̄i and q̄i . Such a scenario of a
PV balance was suggested by Pierrehumbert and Malguzzi
(1984) in a barotropic context. By expanding ψ̄i = ψ̄i0 +
δψ̄i1 + · · · with J (ψ̄i0 , q̄i0 ) = 0, where q̄i0 = G i (ψ̄i0 ), and
assuming a diffusive law for the eddy QPV flux, ui qi =
−ki ∇q̄i0 , we turn (58.35) into
∂G i
∂ ψ̄k0
ψ̄i0 =constant
ki ū i ds +
ψ̄i0 =constant
τ̄i ds = 0, (58.36)
where denotes the component parallel to the ψ̄i0 contour.
Incidentally, (58.36) is the solvability condition for the firstorder equation which determines ψi1 . The zeroth-order flow
ψi0 is, however, determined from (58.31), with the functional G i (ψ̄i0 ) introducing the forcing and the effect of the
eddy fluxes. As can be seen from (58.36), the problem is of
differential-integral type. For the strictly zonal flat-bottom
case we recover the simple problem (58.23).
Instead of parameterizing the eddy flux of QPV we
may consider parameterization of the functionals G i . Then,
in contrast to the case of the eddy coefficients, we do not
search for functions of position y but rather for functionals of ψ̄i depending on integral properties of the forcing,
the parameter of the bottom friction, the channel and layer
dimensions, and the parameters which describe the eddy
fluxes of QPV. It is clear form the preceding analysis that
this task is equivalent to the task of parameterizing the eddy
QPV fluxes.
A general transport theory
After discussing simple conceptual models and results from QG numerical models, we turn to a more complete theory resting on experiments with a global numerical
model of ocean circulation, namely the Parallel Ocean Program (POP) model, which is based on primitive-equation
dynamics and thermodynamics and has eddy resolution
(Smith et al., 1992). The aim is a generalized Johnson–
Bryden transport theory that includes all relevant terms, in
particular the thermodynamic forcing of the zonal current
and topographically induced effects.
The POP model is a global model of Bryan–Cox type
with an implicit free-surface formulation of the barotropic
mode. It uses a Mercator grid, allowing higher resolution
in polar regions (6.5 km at 78o ) than in equatorial regions
(31.25 km at the equator). The vertical grid varies between
25 m at the surface and 550 m in the deep ocean (altogether
there are 20 levels). The surface heat flux is based on Barnier
et al. (1995). In ocean regions where ice is present, the seasurface temperature is restored to −2 o C with a 1-month
time scale. The surface salinity is restored to the monthly
Levitus climatology (Levitus, 1982). In the high latitudes
(poleward of 70o ) the temperature and salinity are relaxed
to the annual Levitus climatology in the top 2 km. The output used in this study is a 5-year time average, taken from
the last of three 10-year simulations using the 1985–1994
European Centre for medium Range Weather Forecasts
winds. Each run was initialized from the final state of the
previous experiment. A 3-day average of winds was used.
Further details of the POP model and experiments are given
by Maltrud et al. (1998).
The simulated ACC of the POP model is displayed
in Fig. 58.8, showing the path of the current by the concentration of sea-surface-height isolines. As in other highresolution models (e.g. FRAM Group, 1991) and in the observed state of the ocean (from an altimeter, e.g. Le Traon
et al. [1998], or from sea-surface temperature, e.g. Olbers
et al. [1992] and Hughes and Ash [2001]), the current is
not a single zonal jet but a multiple-fragmented-jet system
with a path that is strongly influenced by large-scale topographic features and the gap of Drake Passage, which it
has to cross. The gradual southward shifting of the current
from the Atlantic throughout the Pacific is in accordance
with observations; it is interrupted by the sudden northward
excursion in the lee of Drake Passage.
We apply a zonal averaging to the model fields,
as done in the previous analysis. Obviously, in a zonally
averaged picture many details of such a current system are
lost and the single jets will be smeared out. Most importantly, the zonally integrated state in the Drake Passage band
of latitudes (about 62o S to 56o S) will not represent the local
structure of the current in Drake Passage. Thus, the transport of the averaged current in the Drake Passage band of
latitudes is only about 50 Sv whereas the transport in this
model through Drake Passage is 130 Sv, which is very close
to observations (see Rintoul et al. [2001]).
Figure 58.9 displays the basic Eulerian state of circulation of the POP model. The zonal current in the zonally averaged picture still breaks up into clearly identifiable “jets,” of which two are in the Drake Passage belt, a
further two are just north of Drake Passage latitudes and
must arise from jet features in the Pacific and a moderate eastward turning of the ACC off South America, and
two more at about 45o S and 40o S are associated with the
confluence zone of the Brazil Current and other jet-like features in the South Atlantic (see Fig. 58.8). The zonal mean
potential-density field shows the characteristic features of
58.3 A general transport theory
Fig. 58.8. Sea-surface height in the Southern Ocean of POP. Confidence interval 0.1 m. The contours between −1.0 m and 0 m are full.
the Southern Ocean stratification: strong gradients in the
surface layers (the mixed layer is not resolved), the strong
downward slope of isopycnals from the surface to depths of
roughly 2000 m, starting at the Drake Passage belt latitudes,
and a more or less homogeneous deep layer, extending to
the surface in the Antarctic zone (the area south of the Drake
Passage belt).
The meridional Eulerian transport, zonally integrated and represented by a streamfunction, is characterized
by a fairly strong Deacon cell with a maximum of 27 Sv: the
northward transport in the surface layer (the Ekman transport is mostly in the upper two layers, it has a magnitude
of roughly 16 Sv in the Drake Passage belt) is returned – in
the zonally integrated projection – at great depths, mostly
below 2000 m. The structure and dynamics of the Deacon
cell are extensively discussed in many studies of the Southern Ocean circulation (Döös and Webb [1994], Karoly et
al. [1997], see also Rintoul et al. [2001]). In the free (unblocked) water column, only ageostrophic meridional transports are possible (which are small because the Reynoldsstress divergence is small), whereas at blocked depths
geostrophically balanced transports can appear. In terms of
local particle motion the northward flow induced by Ekman
dynamics in the surface layer is indeed largely returned at
only slightly deeper depths by eddy driving as in QG simula-
tions (the eddy-induced circulation; see below). In terms of
zonally integrated mass transport, however, the return transport must occur in the valleys of the blocking topography,
or, in cases of small or vanishing topography, by an Ekman
bottom boundary flow. In the POP model (and in the real
ACC) it occurs by the action of deep southward geostrophic
currents. This overturning circulation is the most fundamental difference of this realistic simulation from results of the
flat-bottom QG experiments shown above.
Transient eddies result from the ACC jets by instability processes. The POP model resolves the eddy field in
a realistic way with reasonable amplitudes (see the kinetic
energy of the transient eddies in Fig. 58.9 and the discussion in Best et al. [1999]) and thus the mean Eulerian flow
is backed by a significant eddy-induced field of transport. In
a Lagrangian picture there appears the eddy-induced transport of mass on isopycnals (which is the Stokes-like bolus
transport) which, together with residual transport, replaces
the Eulerian transport. This concept is outlined in the next
58.3.1 The transformed-Eulerian-mean framework
Zonal averaging of model equations, as used in the
previous analysis, is a basic tool in the interpretation of
the dynamics of zonal flow. In the atmospheric application
Zonal ocean currents
potential density
zonal current U
−55 −50
Eulerian streamfunction
−55 −50
0. 9
14 10
eddy kinetic energy
−55 −50
−55 −50
Fig. 58.9. Zonally and time-averaged zonal mean velocity U , potential-density field , the Eulerian circulation streamfunction φ, and the
kinetic energy of transient eddies (EKE), as functions of depth and latitude, taken from the POP model. Units are U in m s−1 , φ in Sv, and
EKE in 10−2 m2 s−2 .
this tool is formalized in the transformed-Eulerian-mean
(TEM) framework (e.g. Andrews et al., 1987) and the concept of the residual circulation is frequently invoked. The
following derivation of the TEM, which is appropriate for
the ocean where topography may block the zonal path, can
be found in parts also in McIntosh and McDougall (1996).
For more details see also Olbers and Ivchenko (2001). Define time-averaged and zonally integrated5 variables {φ}
on level surfaces, using the curly-bracket operator, and put
φ ∗ = φ − {φ}/L, with L = L(y, z) being the length of the
circumpolar path at depth z and latitude6 y. The deviation
φ ∗ thus contains transient and standing eddies. The balance
5 These are more convenient than zonally averaged variables in cases of
interruption by topography. For example, the zonal integral of a threedimensional divergence turns into the corresponding two-dimensional divergence of the zonal integral, {∇3 · F3 } = ∇2 · {F2 } if the kinematic
boundary condition holds for F3 .
6 We simplify equations by use of local Cartesian coordinates. All
data evaluations of the POP model are performed in spherical coordinates.
of zonal momentum becomes
− f {v} =
∂{τ }
− {uw} −
{uv} +
δ p̄b
where the overbar is the time average. Lateral subgrid
stresses are ignored for simplicity. The sum of the pressure
differences δ p̄b is extended over all submarine ridges interrupting the integration path at depth z (continents are here
included). Each ridge or continent contributes the difference
between the values on the eastern side and the western side,
i.e. δ pb = p(xE , y, z = −h) − p(xW , y, z = −h). The interfacial form-stress divergence is introduced into (58.37)
by splitting the Eulerian velocity into the eddy-induced –
Stokes-like – part and the remaining residual circulation,
{v} = vres +
{w} = wres −
∂φed ,
where φed = {v ∗ σ ∗ }/z is the eddy-induced streamfunction, with = {σ }/L the zonal mean of the potential density σ (referred to the surface and normalized by a constant
reference density) and {v ∗ σ ∗ } = {vσ } − {v} the eddy flux
(transient and standing) of potential density. From mass
58.3 A general transport theory
Fig. 58.10. Streamfunctions from the POP model as functions of depth and latitude: transient eddy φtrs (upper-left-hand panel), standing eddy
φstd (upper-right-hand panel), eddy total φed (lower-left-hand panel), and residual φres (lower-left-hand panel). Units are Sv. Confidence
interval 10 Sv.
balance it can be shown that the vector ({v} =
−∂φ/∂z, {w} = ∂φ/∂ y) has a streamfunction φ despite
possible interruptions of the zonal path by topography. Thus
there is a streamfunction φres = φ + φed of the residual circulation as well. We split the advective terms in (58.37)
into mean and Reynolds-stress contributions, e.g. {uv} =
{v}U + {u ∗ v ∗ } with U = {u}/L, and express the mean
terms further by the residual and eddy-induced streamfunctions. In the resulting equation the balance (58.37) appears
as residual advection and flux divergences of zonal momentum, driving the zonal mean current U ,
J (φres , U ) +
( f φres − {τ } + F) − ∇ · F = 0,
where J is the y, z-Jacobian and the Eliassen–Palm vector
F = (−{u ∗ v ∗ } + φed Uz , ( f − U y )φed − {u ∗ w ∗ })
and the bottom form stress F(z) for the level interval from
hill top down to level z in the valleys,
F(z) =
δ p̄b dz =
xE (z)
dx p̄b
xW (z)
have been defined. Notice that F(z) ≡ 0 for levels above
the highest topography. For a large-scale flow the Eliassen–
Palm vector simplifies to F = (−{u ∗ v ∗ }, f φed ), the divergence of which equals the eddy PV flux (c.f. Eq. (58.5)).
In the case of negligible contributions from the Reynolds
terms and advection of zonal momentum by the residual
circulation, (58.39) describes a balance of momentum that
is entirely in terms of vertical fluxes: at each level the interfacial form stress due to the eddy field, f φed , is balanced
by the flux due to subgrid motion, {τ }, the flux carried
by the residual circulation, f φres , and the flux due to the
bottom pressure gradient, F. With these contributions to
the momentum balance, the adiabatic flat-bottom regime is
generalized to diabatic and topographic conditions.
The eddy and the residual streamfunctions are
shown in Fig. 58.10. These look very different from the
Eulerian circulation: all have cells with reversed circulations compared with the Deacon cell, and intense cells of
significantly smaller sizes are imbedded. The transient eddies induce a narrow, fairly deep-reaching circulation of
about − 10 Sv in the Drake Passage belt with two imbedded
Zonal ocean currents
smaller cells, one in the upper 300 m of the water column
(actually a double cell of maximum −30 Sv) and one in
the depth range 1500–3500 m (with maximum −25 Sv).
A smaller cell appears in the zone of northward deflection
of the ACC north of Drake Passage and in the confluence
zone further to the north. Standing eddies generate an intensive upper-ocean cell with maximum magnitude −80 Sv
in the deflection zone. All these cells add in the total
eddy streamfunction which presents a deep-reaching reverse eddy-induced flow of −10 to −15 Sv with imbedded
very strong localized cells in the upper ocean and at depth
in the Drake Passage belt. In the resulting residual circulation the Deacon cell is replaced by quite a complicated
pattern: the three small-scale reversed cells of the transientand standing-eddy contributions appear with fairly strong
and compact footprints; they are imbedded in a large-scale
weaker flow that is northward in the surface layers (a remnant of the Ekman flow) and southward at great depths
(a remnant of the Eulerian geostrophic return flow). Notice
that the deep-reaching downward branch of the Deacon cell
north of Drake Passage has completely vanished in φres . The
overwhelming importance of standing eddies in the eddystreamfunction and the residual circulation has been found
in other eddy-resolving (or eddy-permitting) models. For
FRAM (FRAM Group, 1991) this was shown in Karoly
et al. (1997).
The residual streamfunction combines mean Eulerian and eddy-induced transports to give the residual transport of potential density in the zonally integrated balance:
J (φres , ) = −
To derive (58.42), the bottom boundary condition on the
three-dimensional subgrid flux of σ and the kinematic
boundary condition on the three-dimensional velocity are
imposed (as described in footnote 5). Notice that eddies
affect the zonal mean potential density only via J ⊥ =
({v ∗ σ ∗ } y /z + {w ∗ σ ∗ }), representing the (zonally integrated) eddy flux normal to (zonal-mean) isopycnals. It
combines with the diapycnal subgrid flux J d for diffusion
and convection to give J = {J d } + J ⊥ and enters the balance as a vertical divergence.
We note in passing that other forms of eddy streamfunction have been discussed. If J ⊥ = 0 we may obviously write φres = −{w ∗ σ ∗ }/ y , which is the eddy streamfunction of Held and Schneider (1999). It may be used
for non-zero J ⊥ , then ∂ J ⊥ /∂z in (58.42) is replaced
by ∂(−J ⊥ /sσ )/∂ y, where sσ = − y /z is the isopycnal
The connection of φres with subgrid mixing processes and thermohaline surface forcing becomes obvious
on regarding the potential density field (y, z) in (58.42)
Fig. 58.11. A sketch to show the coordinate system oriented at the
isopycnal (y, z) = . Antarctica is to the left and y = YDP is in
the Drake Passage belt.
as given and solving for the residual streamfunction. The
balance (58.42) may be put into the form
1 ∂J
|∇| ∂z
where the coordinate s runs along isopycnals =
(y, z) = constant (see Fig. 58.11). Suppose for simplicity that they crop out at the surface (s = 0), run through a
mixed layer where = ml (y) is vertically constant, and
then are subducted into the interior. We may split the solution of (58.43) into a term from the surface flux-boundary
condition and one from the interior below the mixed layer,
so in the coordinates (, s) we have
φres (, s) = −
H0 () − J (, z ml )
| yml ()|
s=z ml
∂ J/∂z ds , (58.44)
where (, s = 0) is the outcrop point of the isopycnal at the surface and z = z ml is the depth of the base of the
mixed layer at that latitude. The flux at the sea surface,
{J z }(y, s = 0) = H0 (y), is prescribed as a boundary condition. If J ⊥ vanishes, the residual streamfunction φres is
simply given by subgrid fluxes of density and could strictly
be associated with diabatic (thermohaline) physics. Notice,
however, that, even if eddies mix only along isopycnals
and eddy-induced diapycnal fluxes are zero in a time-mean
framework (i.e. the transient-eddy flux vector lies in the
time-mean isopycnal), this would not necessarily imply the
vanishing of J ⊥ . McIntosh and McDougall (1996), however, showed that the difference between geopotential zonal
integration and isopycnal integration is the of the order of
the cube of the perturbation (eddy) amplitude. Thus, contributions to J ⊥ arising from locally isopycnic transienteddy flux are of third order whereas contributions from
locally diapycnal fluxes are of second order. Hence, in this
sense we may associate J ⊥ with diapycnal eddy fluxes and,
58.3 A general transport theory
consequently, φres with diabatic processes of water-mass
conversion. If eddy mixing is along isopycnals, as is usually assumed in eddy-mixing concepts, we may even neglect
J ⊥ altogether.
With φ = 0 at the surface and solid boundaries to
ensure that we have zero normal velocities there, we have
also φres − φed = 0 there and thus (58.39) integrates to
which is the Johnson–Bryden relation in the most complete
form. Here, R(z) collects small terms arising from the integral from the top to level z of the Reynolds stresses, the
residual advection of U, and the φed U terms in the Eliassen–
Palm vector. Apart from the lateral Reynolds term, the other
terms in R will be neglected. Upon inserting (58.44) into
(58.45), both external forcing functions – wind stress and
surface density flux – are implemented in the balance of
zonal momentum and the proportion of direct forcing by
these agents becomes apparent (for QG dynamics this was
derived by Olbers [1993]). Of course, other terms in the balance – the stratification in particular – depend indirectly on
the external forcing, but this dependence cannot be revealed
The eddy transport is further split into transient and
standing components, setting e.g. v ∗ = v̄ ∗ + v , where the
overbar denotes the time average and prime-bearing quantities represent the transient-eddy field. Then
{v ∗ σ ∗ } = v σ + v̄ ∗ σ̄ ∗ (58.46)
and correspondingly for the Reynolds stress. The angle
brackets denote zonal integration. This implies a separation
of the eddy-induced streamfunction into transient and
standing contributions, φed = φtrs + φstd . Notice that
(58.45) is valid not only at latitudes that are zonally
unblocked by continents but also, with the appropriate
integration associated with the zonal integration operators {· · ·} and · · ·, at latitudes that are interrupted by
On taking (58.45) at the maximum depth h m along
the zonal path, the balance of total zonal momentum is
obtained as
{τ0 } − {τb } + F(−h m ) − R(−h m ) = 0,
where τb is the frictional bottom stress and F(−h m ) is the
total bottom form stress. The Reynolds part R is generally
small, and τb can be neglected if the bottom topography is
sufficiently high. Then, the zonal momentum is balanced
by input due to wind stress and removal due to bottom form
stress, as assumed in Section 58.1 and demonstrated for
the POP model in Fig. 58.12. This proved to be applicable to the POP model as well as to other eddy-resolving
{v ∗ σ ∗ }
−fφ = f
− f φres = {τ0 } − {τ } − R + F,
momentum balance
Fig. 58.12. The vertically integrated momentum balance (58.47)
(normalized by f 0 = 1.25 × 10−4 s−1 ), indicating a balance
between wind stress (full) and negative bottom form stress (dashed)
at each latitude (small deviations are due to Reynolds-stress effects).
models of the ACC (see e.g. Olbers [1998] and Rintoul
et al. [2001]).
58.3.2 Transport equations
As in Johnson and Bryden (1989), the transienteddy flux of potential density is parameterized by a downgradient form, v σ = −κ L y , and use of the thermal
wind relation f Uz = g y (here we equate density to potential density) then turns (58.45) into a prognostic equation
for the shear Uz of the zonal current. Thus, the parameterization and the balance of momentum imply
φtrs ,
= {τ0 } − {τ } − R + F + f (φres − φstd ) . (58.49)
κ LUz = −
f φtrs
Obviously, these equations allow the determination of the
zonal current velocity and the associated transport relative
to the bottom (or any other reference level) if the diffusivity
and the terms on the RHS are known.
In Fig. 58.13 we compare a linear model (i.e. vertically constant κ) with the non-linear Green–Stone and
Visbeck concepts (see Section 58.2.1), using the POP data
at latitude 59o S (left-hand panels). The linear model is apparently most appropriate for the POP data below 500 m
depth. In the right-hand panel of Fig. 58.13 a vertically
constant diffusivity is applied at each latitude. This linear
model obviously breaks down south of the Drake Passage
belt and in the confluence zone, where negative values of κ
result from the fit.
If the terms on the RHS of (58.49) and κ are given,
the equations yield by double vertical integration the trans-
Zonal ocean currents
transient eddy flux
Green–Stone Visbeck et al.
1 2
(du/dz) 2
1 2
u*(du/dz) 2
[m2 s-1]
Fig. 58.13. Left-hand panels: three fits to parameterization of the transient-eddy flux of potential density at latitude 59o S (left: linear, middle:
Green–Stone, right: Visbeck et al. ). For the fit, indicated by the straight line, only data below 500 m depth were used (diamonds). The axes
are scaled. Right-hand panel: diffusivity κ(y) for the linear fit of the eddy flux to the mean gradient y . Units are m2 s−1 .
port of the Johnson–Bryden model relative to the bottom,
1 0
z N2
[({τ0 } − {τ }
κ −h
L f2
− R + F) + f (φres − φstd )] ,
TJB − Tb = −
where Tb = hU (−h) and a local Brunt–Väisälä frequency
N 2 (y, z) = −gz . The complete transport of the zonal
mean current is thus expressed as a sum of seven terms,
TJB = Tb + T0 + Tfr + TR + Tform + Tres + Tstd .
Here, Tfr represents the contribution fron the subgrid stress
τ in the water column, TR is the Reynolds-stress forcing,
and Tform results from the bottom form stress. Some of the
contributions to the total transport can be expressed more
explicitly in terms of the applied external forcings, namely
the wind stress and the flux of buoyancy at the sea surface.
The contribution from the Eulerian mean terms deriving from τ0 and F involves Ekman and geostrophic transports. Here, the abbreviation τ0 = {τ0 }/L is used. They can
be expressed as
T0 =
τ0 N02 2
b ,
Tform =
1 Nb2
(h 2 − h 2b )F(−h). (58.52)
2 κL f 2
The vertical scale b characterizes the first moment of
the2lo2 2
z N dz
cal Brunt–Väisälä
and bN02 = N 2 dz, and Nb refers to some mean value in
the deep layer. The parameter h b arises from the first moment of the bottom form stress in (58.50); it lies in the
interval D ≤ h b ≤ h, where D is the minimum depth of
topographic barriers on the zonal path. The balance of total
momentum, Eq. (58.47), may be used to replace the bottom
form stress F(−h) by −Lτ0 .
We assume that the subgrid stress τ is effective only
in Ekman layers of thickness δ close to the surface and
bottom. Then, assuming a linear decrease of the stress in
these layers, we obtain
Tfr ≈
τfr Nfr2 2
where the index fr refers either to the frictional surface or
to the bottom layer. Thus, Tfr is very small relative to T0 ,
of order (δ/b)2 , and, even if the bottom is flat and the wind
stress must be balanced by the frictional stress at the bottom,
i.e. τfr ≈ τ0 , the frictionally induced zonal transport Tfr is
The contribution from φres ,
Tres = −
φres ,
describes the thermohaline forcing of the zonal mean current. We consider the solution (58.44) of φres in terms
of the surface-flux and interior-conversion terms. Suppose
now for simplicity that all isopycnals crossing the latitude
y = YDP of Drake Passage – where (58.54) is considered –
outcrop to the south (see Fig. 58.11 for a schematic view
and Fig. 58.9 for the POP model) so that the condition
(YDP , z) = ml (y = y ) maps the depth interval at YDP
on to the latitude band south of Drake Passage, resulting in
y = y (YDP , z) or z = z (YDP , y). If we insert (58.44) into
(58.54), two contributions to the ACC transport arise from
thermohaline fluxes in the system, one from fluxes across
the sea surface south of Drake Passage and one from fluxes
across the zonal mean isopycnals in the interior, also south
of Drake Passage.
In the following analysis, the above analytical
forms for T0 , Tfr , and Tform are not used; instead, they
are evaluated from the POP data directly. All transport
terms in (58.51) are functions of latitude and the volume
58.3 A general transport theory
POP zonal transports in DP belt
JB model of zonal transport
Tfrm + Tfr
transport [m2 s –1]
transport [m2 s –1]
term (which is not small but beyond parametrical treatment), the zonal balance is expressed as7
κUz =
transport of the zonal mean ACC is then obtained by
integration across the Drake Passage latitudes. Figure
58.14 shows the T terms obtained from POP data. Here,
T and Tb use the simulated profile U (z) of the zonal mean
current. The left-hand panel also shows the transport TJB
of the Johnson–Bryden model. It is revealed here that the
constant-κ fitting procedure, using only data from below
500 m, is incapable of reproducing the upper-level fluxes
(see Fig. 58.13), leading to a discrepancy between T
and TJB . The other T terms are obtained using the above
expression (58.50) and data from the POP model’s output.
They are displayed in the right-hand panel of Fig. 58.14.
A few outstanding signatures are readily extracted
from Fig. 58.14. It is obvious that the part of transport relative to the bottom, T − Tb , carries most of the transport:
although the bottom transport is locally a 20% contribution,
its alternating sign yields a very low value in the integral.
With respect to forcing mechanisms the direct wind forcing (T0 ) is clearly dominant. Together with the thermohaline forcing (Tres ) it establishes eastward flow. Retarding
mechanisms, yielding westward transports, are the bottom
form-stress component Tform and the standing eddies Tstd ,
the first slightly leading in magnitude. The Reynolds stress
forcing is of alternating sign and is insignificant.
58.3.3 Scaling theory
Scaling may be applied to extend limited knowledge from limited numerical experiments. We investigate
the extended Johnson–Bryden model from Section 58.3 in a
broader context. The stratification is represented by a buoyancy jump B = −g, occurring over a depth scale
d in the vertical and over Y in the meridional direction
(the latter scale is assumed given). Neglecting subgrid and
Reynolds stresses (which are small) and the standing-eddy
The effect of bottom form stress shielding part of the wind
stress according to (58.52) is assumed to be included in the
wind-stress term. The residual streamfunction is related to
the density flux H0 by φres B/(LY ) ∼ Q0 = −gH0 /L,
using the density balance in the integrated form (58.44).
The zonal balance must be consistent with the meridional
balance, or, in terms of the thermal wind relation, with Uz =
B/( f Y ) = U/d. Hence, we arrive at
Fig. 58.14. Left-hand panel: mean zonal transport in the Drake
Passage belt. Right-hand panel: estimates of various transport terms
in the separation (58.51).
with the zonal mean buoyancy flux Q0 . This equation relates the slope of the isopycnals to the forcing and eddy
effects appearing here in terms of the diffusivity κ. Remember that τ0 is a measure of the zonal mean wind stress in
the Drake Passage belt whereas Q0 , according to (58.44),
is collected from the isopycnal outcrops in the Antarctic
A Green–Stone√ parameterization (see Section
58.2.1) has κ = α f 2 / Ri , with = π 2 λ proportional to
the internal Rossby radius λ = N d/( f π ) (which is based
on d). The diffusivity becomes κ = απ 2 (d B)3/2 /( f 2 Y )
and the balance (58.56) may be written as
γ U 5/2 ∼
τ0 + Q0
with γ = απ 2 / Y f . Without additional assumptions this
relation yields for weak wind stress8 γ U 5/2 ∼ Q0 / f . In this
regime U would thus be a weak function of the buoyancy
flux. For strong wind we obtain γ dU 3/2 ∼ τ0 / f , which,
however, does not completely remove all internal scales
since d = d(τ0 , Q0 , . . .). This also happens to the transport T = dU . The magnitude of the observed wind field
places the interest more on the limiting case of strong
The lesson from this scaling attempt is that the zonally averaged equations are insufficient to gain a complete
scaling theory. Both momentum balances and the buoyancy
balance have been used to determine the dependences of the
four unknown quantities Uz , φres , B, and d on the forcing
parameters τ0 and Q0 . An additional phenomenological relation must be found from numerical modeling or physical
f is the modulus of the Coriolis parameter in the following.
With Q0 = 5 × 10−9 m2 s−3 , Y = 2000 km, and B = 5 × 10−3
m s−2 , weak wind stress is τ0 2 × 10−4 m2 s−2 .
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