Journal of Oceanography, Vol. 66, pp. 41 to 60, 2010

Effects of Mass Source/Sink at the Western Boundaryon the Wind-Driven Gyres: Implications for the Ventilation of the North Pacific Intermediate Layer through Convection in the Okhotsk Sea and Tidal Mixing at the Kuril
Straits
TOMOHIRO NAKAMURA1*, TAKAHIRO TOYODA2, YOICHI ISHIKAWA3 and TOSHIYUKI AWAJI3
1

Pan-Okhotsk Research Center, Institute of Low Temperature Science, Hokkaido University,
Sapporo 060-0819, Japan
2
Ocean Climate Change Research Program, Research Institute for Global Change, JAMSTEC,
Yokohama 236-0001, Japan3
Department of Geophysics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan(Received 26 July 2008; in revised form 21 August 2009; accepted 21 August 2009)

A steady quasi-geostrophic 2.5-layer model, forced by both Ekman pumping and amass source/sink situated at the western boundary has been constructed to investigate the effect of diapycnal transport due to convection in the Okhotsk Sea and tidal
mixing at the Kuril Straits on the intermediate layer in the North Pacific. The model
illustrates a combined effect of the wind-driven and mass-driven circulations. First,
net mass input induces a “barotropic” mode inter-gyre flow along the western boundary through the dynamical influence of Kelvin waves. This flow creates characteristic
curves (geostrophic contours) that facilitate inter-gyre communication through thewestern boundary layer from the location of the mass source to the subtropical gyre.
Due to the effect of wind-driven circulation, the offshore part turns eastward into theinterior, encircles the outer rim of the region (which would otherwise be the pool
region in the absence of mass input), and then encounters the western boundary. Eventually, the water fed into the lower layer flows mostly along this path and later flows
away to the equatorial region. Conversely, in the upper layer, water is fed from theequator to the subtropics, and to the subpolar interior region through the westernboundary current. The water then circulates along the outer rim and is absorbed into
the mass sink. The model is controlled mainly by three nondimensional parameters:(1) the ratio of net mass input rate to the maximum Sverdrup transport (Q/TSvmax),which affects the inter-gyre communication by altering the paths of geostrophic contours, (2) the ratio of a mass input rate into the lower layer to that in total (Q2/Q),
which controls the vertical structure of the inter-gyre flow, and (3) the measure of thewind forcing effect relative to the β effect, which determines the horizontal extent of
the area influenced by the mass input. The other parameter regimes with respect to
Q/TSvmax and Q 2/Q are also presented.

1. Introduction
The intermediate layer of the North Pacific Ocean isventilated mainly from the Okhotsk Sea (e.g., Talley,
1993; Warner et al., 1996; Yasuda et al., 1996), a body of

Keywords:⋅ Intermediate layer,⋅ unventilated
thermocline theory,⋅ inter-gyre communication,⋅ North PacificIntermediate
Water,⋅ analytical model,⋅ steady quasigeostrophic model.

water located to the northwest of the Pacific and separated from it by the Kuril Straits (Fig. 1). The ventilation
process involves the formation of the North Pacific Intermediate Water (NPIW), which spreads over most of
the subtropical gyre and intrudes into the tropics (e.g.,
Talley, 1993). The NPIW stores large volumes of greenhouse gases such as CO 2 (e.g., Yamanaka and Tajika,
1996) and considerable quantities of heat and fresh water. A dynamical understanding of how the water influ-* Corresponding author. E-mail: nakamura@lowtem.hokudai.
ac.jp
Copyright©The Oceanographic Society of Japan/TERRAPUB/Springer

41

Fig. 1. Locations of the Okhotsk Sea and the Kuril Straits.

enced by ventilation circulates in the intermediate layer
is important in understanding, predicting, and monitoring the potential response of the North Pacific to longterm climate variability.
In the Okhotsk Sea, the deepest convection in the
North Pacific is associated with the strong, cold wintermonsoon and copious sea ice production (e.g., Kitani,
1973; Alfultis and Martin, 1987; Talley, 1991). Vigorous
vertical mixing is also induced in the Kuril Straits(Yasuoka, 1968; Kitani, 1973; Kawasaki and Kono, 1994;
Gladyshev, 1995; Aramaki et al., 2001), mainly by tides(Nakamura et al., 2000; Nakamura and Awaji, 2001,
2004). The convection ventilates the upper part of theintermediate layer (ⱗ27.0σ θ), whereas tidally induced
vertical mixing is thought to be a main cause of ventilating the lower part (27.0~27.6σθ). This is because observed
distributions of Chlorofluorocarbons (CFCs) indicate thatthe ventilation in the North Pacific extends to much
deeper, denser layers (~27.6σ θ) than the deepest convection (~27.0 σθ), and that the most recently ventilated water in such dense layers lies around the Kuril Straits(Watanabe et al., 1994; Warner et al., 1996; Wong et al.,1998).
The dynamical effects of these processes on the ventilation of the North Pacific intermediate layer have been
investigated using numerical models. By implicitly taking the effects of the above processes into account, restoration of model temperature and salinity in the Okhotsk
Sea to the observed values can drive the Oyashio southward to cross over the wind-driven gyre boundary determined by the Sverdrup flow (Mitsudera et al., 2004), resulting in the freshening of NPIW (Yamanaka et al., 1998).
Comparison of the results of numerical experiments withand without tidally induced vertical mixing at the Kuril
Straits suggests that tidal mixing there can strengthen theventilation in the Okhotsk Sea and the North Pacific(Nakamura et al., 2004, 2006a, 2006b, hereafter the latter reference is referred to as NTIA06). The enhanced
ventilation in the Okhotsk Sea takes place through down-

42

T. Nakamura et al.

ward diffusion and an enhancement of the convection resulting from a preconditioning of salinity values due toan upward salt flux caused by tidal mixing from the saltier
lower layer. The preconditioned (i.e., saltier) water leads
to an increase in the density of the water involved in convection and thus an increase in the volume of the waterreaching the intermediate layer. The ventilated water flows
into the North Pacific and spreads over the entire subarctic
and most of the subtropical gyres, enhancing ventilationof the intermediate layer.
It is interesting that, in association with the inflowof the ventilated water from the Kuril Straits, the flow
pathways in the intermediate layer were also modified tofurther enhance the ventilation in their model results. The
key elements of the dynamics in the change of the flow
pathways were identified by NTIA06 and Nakamura etal. (2004). The forcing is by diapycnal mass transport
from the shallower and deeper layers to the intermediate
layer due to tidally enhanced vertical diffusion and convection. The downward diffusion induces a counterbalancing upwelling from the deeper layers, and thus mainly
excites the first baroclinic mode response (more precisely,
since the vertical profile of density around the Kuril Straits
is concave upwards, vertical diffusion generally decreases
density and hence produces a net mass flux to the lighter,shallower layers). On the other hand, convection brings
surface water into the intermediate layer, and hence excites the second and higher baroclinic mode responses.
The dynamical adjustment to the forcing proceeds
mainly through coastally trapped waves such as Kelvinwaves and eastward-moving (or advected) long-Rossby
waves, which develop as a result of the wind-driven circulation for the second and higher modes. The Kelvinwaves induce a flow towards the equator in the intermediate layer along the western boundary that induces inter-gyre transport of the ventilated water from the Kuril
Straits to the subtropical gyre (other kinds of coastallytrapped waves that are faster than the local currents also
have a similar effect). The transported water turns eastward and flows around the interior region, where the eastward long-Rossby waves carry dynamical information of
the inflow to modify circulation there. The water eventually leaks into the equatorial region and subsequently tothe other oceans. Tatebe and Yasuda (2004) constructed
an analytical model, which relates the Oyashio southward
intrusion (and associated cross gyre transport) to thediapycnal upwelling.
However, in spite of the need for both Kelvin andRossby waves in the intermediate layer adjustment in the
North Pacific, previous theories have not considered thetwo simultaneously. Earlier work on ventilation may be
categorized into the ventilated thermocline type represented by Luyten et al. (1983) and Rhines and Young(1982), which emphasize the effect of wind driven circu-

lation, and abyssal circulation theories along the lines ofStommel and Arons (1960) and Kawase (1987), which
focus on the effect of mass input/output due tothermohaline forcing.
The ventilated thermocline theory does include theeffect of eastward long-Rossby waves on wind-driven
gyres, but the western boundary is absent in most of these
theories. Such works do not, therefore, consider the effect of Kelvin waves or the inter-gyre flow through thewestern boundary region, even though inter-gyre flows
taking place in the interior have been extensively investigated (e.g., Pedlosky, 1984; Schopp and Arhan, 1986;
Luyten and Stommel, 1986; Schopp, 1988, 1993; Chen
and Dewar, 1993). Although some theories include a western boundary (e.g., Young and Rhines, 1982; Hogg andStommel, 1985; Huang, 1990; Jarvis and Veronis, 1994),
the effect of mass input is absent, thus the inter-gyre flow
is not considered. Huang and Flierl (1987) show that thesemigeostrophic effect allows the subpolar westernboundary current to intrude into the subtropical gyre in
the presence of outcropping. In the absence of mass input, however, all of the water making up this intrusion
should return to the subpolar region, which is not the casefor the North Pacific intermediate layer.
Although the abyssal circulation theory does incorporate the effect of mass-driven Kelvin waves as a deep
western boundary current, the effect of wind-driven circulation is omitted in most of the theories (e.g., Straub
and Rhines, 1990; Pedlosky, 1992; Speer and McCartney,
1992; Straub et al., 1993; Pedlosky and Chapman, 1993;
Ishizaki, 1994; Edwards and Pedlosky, 1995; Cember,1998). Some of these works include the effect of eastward propagating long-Rossby waves due to bottom topography and improve our understanding of the basic
dynamics. Hautala and Riser (1989) consider the effectof wind, but their attention is limited to the shadow region, and hence the effect of eastward long-Rossby waves
is neglected.
In this paper we develop, in a rather heuristic manner, a simple analytical model to investigate the effects
of mass input/output from the western boundary on theintermediate layer, by incorporating the effects of both
Kelvin and eastward long-Rossby waves using a 2.5 layer
model with a western boundary. The focus of the study isthe area of the North Pacific. Our simple model is a firststep toward unifying the two branches of the ventilation
theory and toward a better understanding of the mechanism by which the stratification is maintained in the intermediate density layer between the outcropping component in the subtropics and the portion ventilated by deep
convection. (The former is described in the ventilated
thermocline theory, and the latter in the abyssal circulation theory.)
The paper is organized as follows. The design of themass in/output
from the west+

Ekman pumping (we)in the interior

x=0

x=xe

y=ys

Q1<0
Q2>0

Q=Q1+Q2
idealized
diapycnal transport
in the Okhotsk Sea
and the Kuril Straits

we>0(subarctic gyre)
lay
we<0
er
(subtropical gyre)
1
lay
Q
er
1
2
sta
Q
gn
2
an
t
idealized
North PacificFig. 2. Schematic of the 2.5 layer model.

2.5 layer model is described in Section 2. Solution of the“barotropic” mode and its properties are presented in Section 3. Characteristic curves (geostrophic contours) anda solution for the upper and lower layers (and their properties) are shown in Section 4. The results are summarized and discussed in Section 5. The implications of various parameter regimes are presented in Appendices.
2. Model Design
2.1 2.5-layer QG model
In order to illustrate the essential physics as simply
as possible, we consider a steady quasigeostrophic (QG)
model on a β plane.
In the vertical direction, a 2.5 layer model is chosen.
Crudely speaking, this approach resolves the first andsecond baroclinic modes, and is the simplest model capable of describing the eastward-moving long-Rossby
waves due to wind driven circulation, which play a crucial role in the adjustment of the interior region, as revealed in NTIA06. Although forcing also comes from thehigher modes in the numerical model results of NTIA06,
the higher modes may be considered to represent a quantitative correction to the second mode. This is because
the dynamical role of the higher modes is similar to thatof mode 2, in the sense that all of these modes allow eastward moving long-Rossby waves in the region of a strong
eastward Sverdrup flow.
A schematic of the 2.5 layer model considered hereis shown in Fig. 2. The model domain includes bothsubpolar and subtropical gyres and is open on the equatorial side. The domain is bounded on the east and west
at x = x e and x = 0, respectively, through which no
geostrophic flow is allowed. The upper moving layer(hereafter the upper layer or layer 1) is supposed to surmount the lower moving layer (hereafter the lower layer

Effects of Mass Source/Sink on the Wind-Driven Gyres

43or layer 2) everywhere, so that the lower layer does not
outcrop to the sea surface in the model domain, as is actually the case for the NPIW.
2.2 Forcings
We consider two kinds of external forcing. One is
Ekman pumping, we, imposed at the sea surface. The otherforcing is a mass source associated with the convergence/
divergence of diapycnal transport produced by both theconvection in the Okhotsk Sea and vertical mixing at the
Kuril Straits. The convection causes diapycnal transport
from the lighter, surface layer to the heavier, intermediate layer. Vertical mixing at the Kuril Straits can cause
both enhancement of the convection and upwelling fromthe deeper layers, which balances downward diffusion.
The former works in a similar fashion to the convection
itself, while the latter produces a net mass flux into theintermediate layer from the deeper layers. Associatedvertical mass convergence in the intermediate layer (ordivergence in other layers) induces a mass flux to (orfrom) the North Pacific in the corresponding layer. Thisis idealized as a mass source that feeds fluid of the corresponding density into layer n at a rate Qn from the western boundary at latitude ys (it works as a sink when Qn isnegative).
Note that because the effect of third-layer motion on
the other layers is neglected in a 2.5 layer model, motionof the mass sink from the third layer is also neglected.
Nevertheless, this ineffective mass sink balances the water budget between the North Pacific and the Okhotsk Sea.
Also, any across-interface velocity component is neglected in the domain under consideration. Diapycnaltransport, which includes the cause of the above mass
source/sink and is required to maintain the basic stratification, is supposed to take place out of the domain.
Since the numerical model results of NTIA06 suggest that the Kelvin waves and the associated change inthe flow along the western boundary can remove the mass
fed from the source, we include a western boundary region. This is done by adding a frictional western boundary layer, following Liu et al. (1999). The frictional layer
used here is the Munk layer (Munk, 1950), since this is a
simplest possible theory of the western boundary current
that can represent the effect of both the Kelvin and damping short-Rossby waves. Friction is neglected in the interior.
The mass flux originating from the above source (orsink) is supposed to leave (or enter) the model domain
through the equatorward boundary, so that the volume ofeach layer is held constant (i.e., the mass is conserved).
This assumption is based on the results of the experiments
in NTIA06, which suggest that most of the southward
Kelvin waves and associated mass flux leave the North
Pacific for the Indian Ocean and the South Pacific. The

44

T. Nakamura et al.

Kelvin waves that leave the North Pacific decay as they
circulate in other basins. Consistent with this picture, the
NPIW is actually observed to leak into the IndonesianThrough Flow where it is transformed (Lukas et al., 1991;
Ffield and Gordon, 1992). Thus, the water that left thedomain is supposed to be transformed in other basins andto return through other layers. Although a small fractionof the Kelvin-wave energy can propagate northward along
the east coast of the North Pacific, as seen in the numerical model result (NTIA06), the effect of this is assumed
to be included through the layer thickness at the easternboundary.
2.3 Equations
The vorticity equations, which govern the system,are
J (ψ 1 , q1 ) =f0
we + k ⋅ ∇ × F1 ,H1J (ψ 2 , q2 ) = k ⋅ ∇ × F2 ,

(1)
(2 )

where the subscripts 1 and 2 indicate the upper and lower
layers, respectively (e.g., Pedlosky, 1987, 1996). Here, ψis the streamfunction, and q is the potential vorticity (PV).
Since our focus is on gyre-scale motion, the effect of relative vorticity is negligible in comparison with that of planetary vorticity, so that q can be approximated as
q1 = βy + F1 (ψ 2 − ψ 1 ),(3)

q2 = βy + F2 (ψ 1 − ψ 2 ) − G2ψ 2 ,( 4)

where,Fn =f02,
γ 1 Hn

(5)

Gn =

f02
,
γ 2 Hn(6 )γn =ρ n +1 − ρ n
g,
ρ0

(7)

f0 + βy is the Coriolis parameter on the β plane, Hn is themean thickness of layer n, ρ0 is the reference density, andg is the gravity acceleration. The vector k represents theunit in the vertical direction, and Fn is a horizontal frictional force in layer n which is given in the western boundary region.

3. “Barotropic” Mode (or Mode 1)
3.1 Solution of “barotropic” modeFirstly, the “barotropic” mode roughly corresponding to mode 1 is examined. In the interior, the vertical
average of the vorticity equations yields the Sverdrup
balance:βf∂ψ b= 0 we ,H∂x(8)where H is the total thickness of the moving layers (H =H1 + H2) andψb =H1ψ 1 + H2ψ 2
H( 9)is the barotropic streamfunction. Near the western boundary, a boundary-layer correction in the form of a Munklayer is added:β∂ψ b∂ 4ψ b
,
= Ah
∂x
∂x 4

(10)

where Ah is a horizontal eddy viscosity coefficient. The
Ekman pumping term is neglected, assuming that it issmall in comparison with the friction term.
Since these equations are linear, ψb can be separated
into a wind driven component (ψ bw) and a component
driven by the mass source (ψbm):ψ b = ψ bw + ψ bm .

(11)

The wind driven component is the same as that obtained in the absence of the mass source. It is hence givenasψ bw = ψ bi

( x, y)− ψ bi( x, y)φ m ( x ),(12)

where ψbi is the streamfunction of the Sverdrup flow in
the interior,ψ bi = −f0 x ewe ( x ′, y)dx ′,
βH ∫x

(13)

and φm represents the zonal structure of the Munk layer,
 3x  
1x    3x 
φ m = exp −sin+
 cos .
3  2δ m  
 2δ m    2δ m (14)Here, δm is the thickness scale of the Munk layer (δ m =(Ah/β)1/3), and a no-slip condition is imposed at the western boundary.
The mass-driven component of the barotropic mode
is contained in the western boundary layer, where the mass
source is located and friction is present. In fact, the massdriven barotropic flow is zero in the interior, similar to
the model of Kawase (1987). This is because it is governed by β∂ ψbm/∂x = 0 in the interior, which results inψbm of an arbitrary constant when combined with the nonormal flow condition at the eastern boundary(∂ψ bm/∂y = 0). Accordingly, ψbm can be set to be zero in
the interior.
The mass-driven component carries the mass flux
from the source, and thus the zonally integrated meridional volume transport of this component is Q, where Q =Q1 + Q 2. Such a mass flux takes place equatorward of themass source through the Kelvin wave dynamics (since
the effect of Kelvin waves propagating around the basin
is not explicitly considered). Since the mass flux leaves
the domain through the equatorward boundary, the massconservation constraint is assured by the combination of
the conditions for zonally integrated volume transport due
to ψ bw and ψbm.
Under the above two conditions—(i) ψbm approaches
to zero away from the western boundary and (ii) the zonally integrated meridional volume transport is Q
equatorward of ys—together with no-normal flow and noslip conditions at the western boundary, ψ bm is obtained
from (10) in a similar way to the Munk solution asψ bm =

Qφ ys ( y)φ m ( x ).
H

(15)Here, the meridional structure function φys(y) is definedas

φ ys ( y) = 1, for y ≤ ys ,= 0, for y > ys ,(16)

and the zonal structure is that of the Munk layer. Taking
these two structures together, (15) reflects the fact that
Kelvin wave activity redistributes the input mass along
the western boundary on the equator side of the mass
source, leading to the formation of a mass-driven western boundary current. The amplitude of the latter is determined from the net mass influx divided by the total
mean layer thickness.
Combining the two components, we obtain thestreamfunction of the barotropic mode,Qψ b = ψ bi ( x, y) − ψ bi ( x, y) − φ ys ( y)φ m ( x ).
H



(17)

Effects of Mass Source/Sink on the Wind-Driven Gyres

453.2 Properties of the “barotropic” mode solution
Prior to considering the “baroclinic” mode solution,
the properties of the barotropic mode solution are examined, since these determine the characteristic curves ofψ 2 in the QG dynamical framework under consideration.
The solution (17) consists of the Sverdrup interior,
its western boundary layer correction, and the mass driven
component. Since the last two components are confinednear the western boundary, the interior flow is not affectedby the mass input for the barotropic mode. We thus examine the western boundary currents in the presence ofthe mass source, before presenting an overall pattern ofthe barotropic mode flow. It should be noted that thisunchanged interior flow is different from the results usually obtained by abyssal circulation theories. The difference arises from the present model settings, in which both
temporal change in layer thickness and mass transport
across the layer interface are absent.
3.2.1 Confluence point of the western boundary currents
A schematic of how the western boundary current iscomposed of the wind- and mass-driven components ispresented in Fig. 3(a). The wind-driven component(– ψbi(x, y)φ m) compensates the convergence (or divergence) of the Sverdrup transport through its variation intransport. Although the mass driven component, (Q/H)φm,
merely adds a constant transport, its contribution is important since it enables a western boundary current tocross the boundary of the subpolar and subtropical gyres.
The latter is defined by the isoline ψbi = 0 (e.g., Schopp
and Arhan, 1986) and becomes a zonal line y = ygb near
the western boundary in the present case. When Q = 0,
the gyre boundary is collocated with the edge of theboundary of the western boundary currents. In contrast,
when Q ≠ 0, the solution (17) indicates that the meridional velocity of the Munk layer is no longer zero at ygb as
it now disappears at meridional positions whereHψ bi ( x, y) = Q.

(18)

Accordingly, the confluence point of the western boundary currents of the subpolar and subtropical gyres, if itexists, moves to the meridional position, ycwbc, which satisfies (18) and is closest to ygb for the same x, or moves toy s when Q << 0. From the dynamical point of view, theconfluence moves to the latitude where the given net-massflux is compensated by the Sverdrup flow. This result isessentially equivalent to that obtained by Tatebe andYasuda (2004).
To consider the spatial shift of ycwbc in more detail,the relationship between the mass influx Q, the Sverdruptransport at the western boundary TSv, and ycwbc is shown
schematically in Fig. 3(b). When Q ≥ 0, the position ofy cwbc can be categorized into three cases:

46

T. Nakamura et al.

Fig. 3. (a) Schematic of the “barotropic” mode western boundary current driven (left to right) by a mass source, by wind,
and in total for a moderate mass input (i.e., case (ii)).
(b) Schematic of the relation between Sverdrup transport at
the western boundary (T Sv), the net mass influx (Q), andthe latitudinal position of the confluence of the subpolarand subtropical western boundary currents (y cwbc). A schematic distribution of T Sv is shown as a function of y. ycwbc isdetermined from Q and T Sv, as indicated as an example bydashed lines. ycwbc can thus be categorized according to Q
and TSv into five cases as shown in the left. In each case,ycwbc is located as shown at the bottom.

(i) ycwbc = ygb, when Q = 0: The confluence ycwbc isat the gyre boundary, when the net mass input is zero.
(ii) ycstg < ycwbc < ygb, when 0 < Q < T Svmax: The confluence ycwbc is in the northern part of the subtropical gyre
for a moderate mass input. The quantity T Svmax denotes
the maximum Sverdrup transport in the subtropical gyre,
whereas ycstg, its latitude, represents the center of boththe subtropical gyre and the pool region there if it exists.
This case corresponds to that of the North Pacific.
(iii) ycwbc is absent in the domain, when Q ≥ TSvmax:
When the net mass influx is larger than the maximum
Sverdrup transport, the western boundary current flows
equatorward throughout the subtropical gyre, and reaches
the equatorial region. In other words, a confluence of theQ=0.2TSvmax

Q=0

barotropic streamfunction ψb

ycwbc

^

geostrophic contours q2

ycwbc

Ekman pumping

subtropicalpool region

communication belt

threshold value of Q, which dictates whether ycwbc is at ys
or further to the equator, becomes T Svmin when ys > ycspg
or T Svys when y s < y cspg . Here, T Svmin is the minimum
Sverdrup transport taking place at y cspg, the center of the
subpolar gyre, and TSvys is the Sverdrup transport at y s.
Thus, writing the condition for the latter case in parentheses, the position of ycwbc can be categorized into thefollowing two additional cases:(iv) ygb < ycwbc < ycspg (or ys), when T Svmin (or TSvys) <Q < 0: The confluence ycwbc is in the southern part of the
subpolar gyre, when the mass source works, on the whole,as a moderate sink. This situation could be realized by
the occurrence of strong deep convection, which brings
the upper layer water to the deep layer.
(v) ycwbc = ys, when Q ≤ TSvmin (or T Svys): The confluence ycwbc is located at ys for such a large mass output.
In the following, we focus on case (ii) together withcase (i) for comparison. A discussion of the other cases isgiven in Appendix 1.
3.2.2 Flow pattern of the “barotropic” mode
In order to show a typical flow pattern of thebarotropic mode, we consider here a simple case in which
we is a function only of y with a form,
we = −W0 sin= 0,Fig. 4. (Top) “Barotropic” mode streamfunction, ψ b , and(middle) geostrophic contours, q̂2 . (Left) The case (i) whenQ = 0 and (right) the case (ii) when 0 < Q < TSvmax. T Svmax isthe maximum Sverdrup transport. ψ b, q̂2 , x, and y are
nondimensionalized by TSvmax/H, βL y, L x, and Ly, respectively. Shaded areas denote negative values. The Ekman
pumping used is shown at the bottom of the figure. The
thick lines in the case with mass input represent the isolinesof ψ b = Q/H and q̂2 = βy cwbc + F̂ Q/H. Details are written
in the text.

subpolar and subtropical western boundary currents does
not exist. Such strong diapycnal transport rarely occurs
in major basins.
When Q < 0, the net mass flux works as a sink, andhence the position of ycwbc moves poleward toward themass sink. In this case the position also depends on thelocation of the sink, because the mass-driven flow is induced only on the equator side of ys through the Kelvinwave dynamics. Accordingly, the northern limit of theposition of y cwbc is fixed at ys regardless of the magnitudeof the mass output. (The confluence could move further
north if we allowed a portion of the Kelvin wave to circulate around the domain.) Associated with this, theπy, for − Ly ≤ y ≤ Ly ,Lyfor y < − Ly , Ly < y.

(19)

The gyre boundary is set on y = ygb = 0, and the subpolarand subtropical gyres range from 0 < y < L y and –Ly < y <0, respectively.
Further, ψ b is nondimensionalized by TSvmax/H as,Qψ˜ b = ψ˜ bi ( x˜ , y˜ ) − ψ˜ bi ( x˜ , y˜ ) − max φ ys ( y˜ )φ m ( x˜ ),
T
Sv



(17′)

where
TSvmax
f LW= 0 x 0.
βH
H

(20)

Here, indicates a nondimensional variable, x and y are
nondimensionalized by L x and Ly, respectively, and xe is
set to Lx. According to (17′), the barotropic mode solution is controlled by only two nondimensional parameters,
Q/TSvmax and δm/Lx, once ys is specified. The former isintroduced in this study and measures the relative contribution to the western boundary current due to mass input
and wind forcing. The latter is the ratio of the Munk layer
thickness and the basin width.
Figure 4 shows the distribution of ψb for cases (i)Effects of Mass Source/Sink on the Wind-Driven Gyres

47

Table 1. Parameters.
Notationsδ m /Lx
Q/TS vma x
α
Q2 /Q
H2 /H
F̂ /G2δm
Ah
Lx
Ly
TS vma x
Q
Q2
F̂

β
H
H2
γ1
G2

Parameters

ValuesNondimensional parameters
nondimensional Munk layer width

0.0025inter-gyre boundary and communication
extent of pool regions

0.2
2

vertical structure of mass-driven flow
background stratification
background stratification

1.5
0.6
0.3

Dimensional parameters
Munk layer width

20 km

horizontal viscosity coefficientzonal extent of the basin
meridional extent of a gyre
maximum Sverdrup transport
in the subtropical gyrenet mass flux into layers 1 and 2
mass flux into the lower layer
defined in (24)

2 × 10 2 m2 s – 1
8,000 km2,000 km30–50 Sv
1–8 (or ~3) Sv~6 Sv
3 × 10 – 9 m– 2

total thickness of layers 1 and 2
thickness of the lower layer
reduced gravity

2 × 10 – 1 1 m– 1 s – 1
1,000 m
600 m
0.015 m s – 1

defined in (6)1 × 10 – 8 m– 2

gradient of Coriolis parameter

Values of nondimensional parameters used in case (ii) and rough estimates of dimensional parameters for the North Pacificintermediate layer.

and (ii). The values of Q/T Svmax and δ m/L x are set to be 0.2
and 2.5 × 10–3, respectively, and are shown in Table 1
with values of dimensional parameters. Although the valueof Q/T Svmax would be larger than a realistic value*, it ischosen for the clarity of the figure and because the solution does not change qualitatively for 0 < Q/T Svmax < 1, as
is argued in the previous section.
Although the streamfunction in the interior remains
the same, the mass-driven flow affects the starts and endsof the streamlines. Those paths emanating from the mass
source intrude from the subpolar region into the subtropics along the western boundary, enter the interior to cir-*The annual mean Sverdrup transport in the subtropicalgyre of the North Pacific is in the range of 30–50 Sv (1 Sv =10 6 m 3 s–1; e.g., Aoki and Kutsuwada, 2008). Changes in thestrength of the meridional overturning due to tidal mixing at
the Kuril Straits, which may give an estimate of Q, are 1–8 Sv
and ~3 Sv for all experiments and the main experiment in
NTIA06, resulting in Q/T Svmax of around 0.02–0.27 or 0.06–
0.1, respectively.

48

T. Nakamura et al.

culate through the outer rim of the subtropical gyre, andfinally leave the subtropics to move to the equatorial region along the western boundary. Note that the above path
is different from that of the Kelvin waves which cause
dynamical adjustment to the mass input, as expected from
Fig. 3.
4. Lower Layer Circulation
4.1 Equations of ψ2
Given ψ b in the previous section, ψ 1 can be rewritten as,

ψ1 =Hψ b − H2ψ 2
.
H1

(21)

This enables the lower layer equations to be written interms of ψ2, whose solution in turn yields ψ 1.
4.1.1 In the interior
In the interior, approximated here as an inviscid region, the governing equation becomesJ (ψ 2 , qˆ 2 ) = 0,

(22)

where
qˆ 2 = βy + Fˆ ψ b ,

(23)

f02 H
,
γ 1 H1 H2

(24)

Fˆ =

(

)

q2 = qˆ 2 − Fˆ + G2 ψ 2 .

(25)

As reviewed by Pedlosky (1996), isolines of q̂2 arecharacteristic curves of (22) and are often called
geostrophic contours, along which both the geostrophic
streamfunction and PV in layer 2 (i.e., ψ2 and q2) are conserved in the nondissipative limit. Since this dynamical
aspect of the geostrophic contours is usually understoodin terms of long-Rossby waves, we call this inviscid component a “long-Rossby wave” in the following and write
the streamfunction and PV associated with this component as ψ2lr and q2lr, respectively.
In order to allow the lower layer in the interior to be
in motion, there must be a so-called pool region, which isshielded from the geostrophic contours emanating fromthe eastern boundary (e.g., Rhines and Young, 1982).
Otherwise, the shadow zone, where the geostrophic contours transmit the eastern boundary condition, covers thewhole interior, and hence motion in the lower layer, ifany, is confined near the western boundary regardless ofthe presence of mass source under the present QG approximation. Our interest is thus in the case with a pool
region. The condition of the presence of a pool region isthat a point exists which satisfies−∂ψ b β
>
∂y
F̂

(26)

somewhere in the gyres, thus an eastward Sverdrup flow
can arrest westward propagating Rossby waves (e.g.,
Pedlosky, 1996). The following discussion concentrates
on the case in which this condition is met.
4.1.2 Near the western boundaryFirstly, let us consider a situation without mass input. Even in this case, the absence of adequate theoriesof the western boundary current makes it very difficult togive a completely convincing argument for closing thecirculation through the western boundary region. Nevertheless, in order to gain an insight which could be helpful
for future development of more rigorous arguments, we
make a simplifying assumption that friction can be neglected in the lower layer even in the western boundary

region. This is associated with the assumption that friction in the upper layer is given by Ah(∂4ψ1/∂x4) + Ah (H2/
H1)(∂4ψ2/∂x4).
In other words, we consider the case in which friction in the lower layer is so small that the characteristic
curves are not greatly altered although the informationpassing through the characteristics would be slightly
modified. This condition may be met when c·∇q2 >>A2∇ 4ψ 2 where c is the phase velocity of long Rossby
waves of the second or higher modes and A2 is the horizontal eddy viscosity coefficient in layer 2. Scaling thewidth of the western boundary region l as ~50 km, thedeformation radius Ld as ~50 km, the meridional scale L y
as 2000 km, and the meridional component of c, cy as 2
cm s–1, A2 should satisfy A2 << cy(l4/L yL d2) ~ 2500 m 2s –1,
which may not be a very unrealistic value. A similar value
may be obtained from a comparison of the viscous and β
terms. Note that a coarse resolution model of NTIA06,
whose results motivated this study, set Ah as 1000 m2s –1
and narrowly satisfied this condition, and more recent
models having higher resolutions usually meet the condition.
If we assume that friction is negligible, the lower
layer circulation can be closed, since there is no PV input
in the lower layer if thermohaline forcing is absent. In
fact, since ψ b takes the same value at the eastern andwestern boundaries (due to the constraint that no net meridional volume transport is induced by the wind), (23)indicates that the isolines of q̂2 in a pool region must be
closed before reaching the western boundary, and hence
those of ψ 2lr are closed. The western boundary is outside
the pool region but in the shadow zone, and the fluid isstagnant there. Hence, a frictional western-boundary current is not required in the lower layer in this case.
When Q2 ≠ 0 (i.e., the mass source is present in thelower layer), a western boundary current is required tocarry the additional mass flux along the western boundary, as seen in the numerical model result (NTIA06). Thus,
we also introduce the Munk layer in the lower layer as in
the barotropic mode and assume that the lower layer circulation consists of the long-Rossby wave component andthe Munk layer component (ψ2wbc), with the latter including the component influenced by friction or representing
the flow induced by the Kelvin-wave adjustment. Thatis,

ψ 2 = ψ 2lr + ψ 2wbc ,(27)

where ψ 2lr and ψ 2wbc are governed, respectively, by()

J ψ 2lr , qˆ 2 = 0,(28)

Effects of Mass Source/Sink on the Wind-Driven Gyres

49

and

β

∂ψ 2wbc∂ 4ψ 2wbc
= Ah
.
∂x
∂x 4

(29)

Note that, similarly to the effect of viscosity, the presence of ψ 2wbc may also distort the characteristics of ψ2lr.
When the current velocity associated with ψ2wbc is weak,the characteristics are not significantly affected. We shallconsider this case further below. On the other hand, whenthe velocity component associated with ψ2wbc is stronger
than that associated with ψ 2lr, ψ2lr obtained from (28) is
no longer valid in the western boundary region. Instead,the current is dominated by the ψ2wbc component in thiscase, and thus the dominant feature would be captured.
4.2 Geostrophic contours
The distribution of geostrophic contours is examined
to obtain an insight into the response of the lower layer
to the mass source.
4.2.1 Parameter settings
Examples of the distribution of q̂2 for cases (i) and(ii) are shown in Fig. 4. The settings are common to thoseof ψb (the other cases are shown in Appendix 1).
For these figures, a third nondimensional parameter
related to a pool region ( α ) appears through thenondimensionalization of (23) by βL y as,q˜ˆ 2 = y˜ + αψ˜ b ,

(30)

where

α=

f 3L WFˆ TSvmax= 2 0 x 0 .
βLy Hβ Lyγ 1 H1 H2(31)This parameter measures the relative contribution to potential vorticity of the thickness variation due to the wind
forcing and the planetary vorticity gradient, and thus determines the area of a pool region (e.g., Pedlosky, 1996).
This is set to 2, which is relevant to the situation of the
NPIW.
4.2.2 Properties of q̂2
To illustrate the effect of mass input, the wind-onlycase is shown first. When Q = 0, the pool regions in the
subpolar and subtropical gyres are symmetric due to the
QG approximation. The geostrophic contours emanatingfrom the eastern boundary meet the western boundary at
the same latitude and make the boundary the shadow zone,
as argued in the previous section.
In contrast, when Q ≠ 0, geostrophic contours are
deflected further due to the mass-driven barotropic flow.
Firstly, due to the “barotropic” mass influx, geostrophic
contours near the western boundary turn to the interior at
y = ys, and then towards the south. Although the deflected50

T. Nakamura et al.

contours in the nearshore region soon hit the westernboundary, some of those in the offshore region cross thegyre boundary at y = ygb = 0, and subsequently penetrate
into and circulate around the outer rim of the region (corresponding to the subtropical pool region when Q = 0)
and finally encounter the western boundary, thereby forming a belt-like shape. These contours therefore connect
the subpolar to subtropical gyres in a region we define as
the “communication belt”. Associated with this, the subtropical pool region, in which geostrophic contours areclosed, undergoes a contraction, while the subpolar poolremains almost the same.
More precisely, the communication belt represents
the zone covered by the family of geostrophic contours
that pass through the line y = y s between the westernboundary and the subpolar pool region and includes thenearshore part, where geostrophic contours do not reachthe interior. This definition of the communication belt israther convenient for obtaining a solution. Also, the lower
layer current in the nearshore part is able to flow into thesubtropics when the component ψ 2wbc is added, as shownin Subsection 4.4.2.
The communication belt is thus adjacent to thesubpolar pool region and to the subtropical shadow zone.
The boundary on this side is given by the isolines,qˆ 2 = βygb ,(32)

as in the Q = 0 case, since ψb and hence the geostrophic
contours in the interior remain the same in the present
QG dynamics. On the other side, the communication belt
is bounded by the subtropical pool region and by the western boundary. The former boundary is given by thegeostrophic contours connected to the point (x = 0, y =ycwbc) where ψ b = Q/H, that is,Qqˆ 2 = βycwbc + Fˆ ,H

(33)

for the following reason. On the line y = y cwbc , thebarotropic western boundary current has no meridional
component (because of the neglect of Ekman pumping in
the western boundary region), and thus the geostrophic
contour extending along y = ycwbc from the western boundary reaches the interior. This contour then connects withthe geostrophic contour in the northern part of the region
that would define the subtropical pool region in the caseQ = 0, since we are considering case (ii) in which ycwbc islocated between ygb and ycstg (Fig. 3). The contour thus
encircles the subtropical pool region and forms its boundary with the communication belt, as shown in Fig. 4 using the thick line.
Note also that as the barotropic flow leaks out to theequatorial region due to the mass-driven component, theshadow zone also extends into the equatorial region near
the western boundary. Thus, the effects of the easternboundary condition are communicated to the equatorial
region.

C=

4.3 Determination of ψ2
Similar to the case of ψb, ψ2 can be obtained asψ 2 = ψ 2lr + ψ 2wbc
Q
= ψ 2lr +  2 − ψ 2lr ( x = 0)φ ys ( y)φ m ( x ),H 2
on the line segment y = y s, 0 ≤ x ≤ x ( q̂2 = 0), where C is
an arbitrary constant, and x ( q̂2 = 0) is the most westerly
longitudinal position at which q̂2 is zero. Here, we set(34)

where ψ2lr (x = 0) is the value at the western boundary
and we use the condition that the zonally-integrated meridional transport in layer 2 is Q 2 equatorward of ys. Notethat the above condition ensures the mass conservation
relation in layer 2 (and thus in layer 1 when combinedwith the corresponding condition for the barotropic mode).
The component ψ2lr is written from (25) as

Q2 Fˆ + G2
H2 βy + Fˆ Qs
H

so that the mass input firstly forces the component of ψ 2lr
at y = ys, that is ψ2 = ψ2lr at x = 0, y = y s.
The above condition yields ψ2lr in the communication belt asψ 2lr =Q2qˆ 2,
H2 βy + Fˆ Q
s
H

(35)

ψ 2lr = Cqˆ 2,
ˆF + G2

(36)



 Q
F + G2 .
q2lr = qˆ 2 1 − 2Q H2 βys + Fˆ ˆwhere q2lr is the PV associated with ψ2lr, and we used theassumptions that both ψ2lr and q2lr are constant along thegeostrophic contours except at the entry of the mass inflow, and that these are identical to ψ2 and q2 in the interior, respectively. The component ψ2lr is then determinedusing the approximation of PV homogeneity in the pool
regions and an inflow condition at y = y s.
It should be noted that, theoretically, the PV homogeneity requires both closed characteristics and a uniform
PV distribution on the outermost closed characteristics(e.g., Pedlosky, 1996). These conditions may be disturbed
in the western boundary region by either friction or theψ2wbc component. Nevertheless, if a PV flux into the pool
region taking place in the interior region is significantly
greater than that in the western boundary region, PV values in a large part of the pool region will be close to thevalue on the pool boundary on the interior side. In thissense, PV homogeneity would be useful as a rough approximation, at least outside the western boundary region.
We assume that this condition is met in the following argument in order to obtain an idealized picture of the lower
layer circulation as a first step.
The component ψ2lr can be determined, with an inflow condition of the form

(38)

because the functional relationship between q̂2 and ψ 2lr
holds in the communication belt, and q2lr becomesqˆ − q2lr
ψ 2lr = 2,
Fˆ + G2

(37)

(39)

H

From the approximation of PV homogeneity, q2lr ina pool region is equal to its boundary value. For the subtropical pool region, q2lr on the boundary is equal to q2lr
at x = 0, y = ycwbc given as




Q  Q
F + G2 ,
q2lr =  βycwbc + Fˆ  1 − 2H   H2 βy + Fˆ Q 
s
ˆ



(40)

H

from (33). Thus, ψ2lr in the subtropical pool region becomes, from (35),QQ
β ( y − ycwbc ) + Fˆ ψ b − 
βycwbc + FˆH  Q2lrH.
ψ2 =+H2 βy + Fˆ Q
Fˆ + G2s
H(41)
Similarly, ψ2lr in the subpolar pool region is given asψ 2lr =

βy + Fˆ ψ b,
Fˆ + G(42)

2

Effects of Mass Source/Sink on the Wind-Driven Gyres

51because q2lr on the boundary is zero.
Beyond these three regions (i.e., the communication
belt and the two pool regions), lies the shadow zone, andthus,ψ 2lr = 0,(43)which is the value at the eastern boundary.
For comparison, ψ2 in the pool regions when Q = 0
is readily obtained from the PV homogeneity approximation, namely,ψ 2 = ψ 2lr =βy + Fˆ ψ b
qˆ 2
=
.
ˆ
F + G2Fˆ + G2(44)This is derived from the facts that q̂2 = q2 = 0 on theboundaries of the both pool regions, and that ψ2wbc is zerofrom the condition of a zonally-integrated meridionaltransport. In other regions, ψ2 = 0.
4.4 Properties of ψ2 and ψ1
4.4.1 Nondimensional form
Before giving examples of ψ 2 lr , wenondimensionalize ψ 2lr and ψ2wbc by TSvmax/H2, to highlight the influence of external parameters in controlling
lower layer transport. This yields
 Q   Q  α −1 y˜ + ψ˜ b,
ψ˜ 2lr =  2   max  Q   TSv  α −1 y˜ + Q
sTSvmax(38′)

52

T. Nakamura et al.

α −1 y˜ + ψ˜ b,
 G2 
1+  
 Fˆ 

(34′)

In the communication belt, a new nondimensionalparameter,

Q2
,
Q

(45)

is introduced to represent the ratio of the mass influx tothe lower layer to the total mass influx to the moving layers. The lower layer transport through the communication belt is proportional to this ratio (or, more precisely,
to Q2/T Svmax).
The ψ2lr component in the pool regions further introduce two more nondimensional parameters, H2/H andF̂ /G2 = γ2H/ γ1H1. These two parameters represent theeffect of the background stratification and also appear in
the case without mass input, which has a nondimensional
form as in (42′). In particular, the parameter F̂ /G2 arises
through the PV homogeneity approximation used in thepool regions.
The
upper
layer
streamfunctionisnondimensionalized by TSvmax/H1 to yieldψ˜ 1 = ψ˜ b − ψ˜ 2 .

(21′)

(41′)

in the subtropical pool region, andHψ˜ 2lr =  2  H Q   Q ψ˜ 2wbc =  2   max  − ψ˜ 2lr ( x˜ = 0)φ ys ( y˜ )φ m ( x˜ ).
 Q   TSv 

4.4.2 Flow patterns within the lower and upper layers
Examples of ψ̃ 2 and ψ̃ 1 distributions are shown in
Fig. 5 for cases (i) and (ii). The values of parameters are
summarized in Table 1. The parameter Q 2 is set greaterthan Q (Q2/Q = 1.5)*, to reflect the fact that downward
diapycnal volume transport to the intermediate layer is
induced by the convection in the Okhotsk Sea (NTIA06).
The parameters H2/H and F̂ /G2 are set to 0.6 and 0.3,
respectively, since these values lie within the parameter
range of the NPIW.
When Q = Q 2 = 0, no inter-gyre flow occurs. This isdue to the absence of both the communication belt and ain the communication belt,Q 
α −1 ( y˜ − y˜ cwbc ) + ψ˜ b − max 
T

H
Sv 
ψ˜ 2lr =  2  H G2 
1+  
 Fˆ 
Q
α −1 y˜ cwbc + max
TSv Q  Q + 2   max  Q   TSv  α −1 y˜ + Q
sTSvmaxin the subpolar pool region, and gives(42′)*The Oyashio water transport that contributes to the NPIW
formation may give a rough estimate of Q 2, and is about 5–6
Sv (e.g., Shimizu et al., 2003). Although the Oyashio transport
includes the influence of the Bering Sea, this may be regarded
as mass input from the northwest corner, the effects of which
are similar to that of the input from the western boundary. WhenQ ~ 3 Sv, Q2/Q thus becomes about 2, whereas it is about 1
when Q/T svmax = 0.2 and T svmax = 30. We take the average of thetwo estimates for the value of Q2/Q.

Q/TSvmax=0.2, Q2 /Q=1.5
lower layer streamfunction ψ 2

Q=Q2=0upper layer streamfunction ψ 1

Fig. 5. (Top) Layer 2 streamfunction, ψ2, and (bottom) layer 1
streamfunction, ψ 1. (Left) When Q = Q2 = 0, (right) when
Q/T Sv max = 0.2 and Q 2 /Q = 1.5. ψ 2 and ψ 1 are
nondimensionalized by T Svmax /H 2 and T Sv max /H 1 , respectively. The contour interval is 0.05 for ψ2 and 0.1 for ψ1.

mass-driven western boundary current. Accordingly, thelower layer circulation is confined to the pool regions,and both ψ̃ 2 and ψ̃ 1 are symmetric about the gyre boundary.
In contrast, when Q ≠ 0 and Q2 ≠ 0, the water supplied from the subpolar western boundary to the lower
layer crosses the gyre boundary and moves toward thesubtropics. This water separates from the western boundary and circulates in the interior. It then returns to thewestern boundary and eventually leaves the domain toenter the equatorial region. Overall, the flow path is similar to that of the NPIW and to the spreading route of the
response to tidal mixing at the Kuril Straits, as impliedby the impact on salinity distribution (NTIA06).
Distinct features are the inter-gyre transport near thewestern boundary and the entrainment into the interior.
The former transport is considered to be important for
NPIW formation (e.g., Talley, 1993; Yasuda et al., 1996),
and is due to both the communication belt and the mass
driven western boundary current in the lower layer,
whereas the latter entrainment spreads water to an areathat would be called the pool region in the absence of anet mass source (i.e., the communication belt and the pool
region in the subtropics). Associated with the entrainment,
the PV supplied from the mass source is carried throughthe communication belt and affects the PV value in thesubtropical pool region.
In the subpolar region, the circulation in the pool
region is scarcely changed in this solution since here thefunctional form of the solution becomes the same as thatin the case without mass input. Although the model results in NTIA06 suggest that the streamfunction and PV
values in the subpolar region are also modified, the QG
approximation should be relaxed to express such an effect.
In contrast to the lower layer, the upper layer waterflows from the equatorial to the subtropical regions
through the western boundary current, and to the subpolarinterior. The water then circulates along the outer rim of
the subpolar gyre and leaves the domain by entering themass sink, where the water is assumed to subduct into
the lower layer.
4.4.3 Vertical structure of the inter-gyre flow
In the above case, the direction of the inter-gyre flowin the upper layer is opposite to that in the lower one.
This kind of vertical structure in the inter-gyre transport
through the western boundary region is determined by
the ratio Q2/Q in the present solution.
In the lower layer, the inter-gyre flow is equatorward
when Q2 > 0, absent when Q2 = 0, and poleward when
Q2 < 0. This is because ψ 2 = Q 2/H2 at x = 0 and for y ≤ ysfrom (34). (Note that the equatorward flow corresponds
to that away from the source and the poleward flow tothat toward the sink.) In the upper layer, the direction of
the inter-gyre flow is determined by Q1 in the same manner, because ψb = Q/H at x = 0 for y ≤ ys from (17), andhence ψ1 = (Q – Q2)/H1 = Q1/H1. Writing this conditionin terms of Q 2 and Q, it follows that the upper layer intergyre flow is equatorward when Q 2 < Q, absent when Q2 =Q, and poleward when Q2 > Q.
Accordingly, the ratio Q2/Q can be used to describe
various combinations of upper and lower inter-gyre flow
directions which determine the vertical structure of thewestern boundary current at the gyre boundary. WhenQ > 0, the following five cases can be considered (withflow paths shown in Appendix 2).
Case (1) when Q 2/Q > 1 (Q1 < 0, Q 2 > 0): The intergyre flow is equatorward in the lower layer but poleward
in the upper one. This is the case shown in the preceding
sections.
Case (2) when Q 2/Q = 1 (Q1 = 0, Q 2 > 0): The intergyre flow is equatorward in the lower layer but absent in
the upper one. This situation could be realized when theconvection from the upper to lower layer is either absent
or compensated by upwelling from the lower to the upper
layer.
Case (3) when 0 < Q2/Q < 1 (Q1 > 0, Q 2 > 0): Theinter-gyre flow is equatorward in both lower and upper
layers. This situation occurs when the upwelling to theupper layer is larger than the convection to the lower layer

Effects of Mass Source/Sink on the Wind-Driven Gyres

53

Table 2. Vertical structure for various Q2/Q when Q > 0.

Case

Range

Lower layer flow

Upper layer flow

Case (1)Case (2)Case (3)Case (4)Case (5)Q2 /Q > 1
Q2 /Q = 1
0 < Q2 /Q < 1
Q2 /Q = 0
Q2 /Q < 0equatorward
equatorward
equatorward
vanishes
poleward

poleward
vanishesequatorward
equatorward
equatorward(i.e., the net mass flux is upward) and when this net massflux to the upper layer is smaller than the upwelling fromthe abyssal layer.
Case (4) when Q 2/Q = 0 (Q1 > 0, Q2 = 0): The intergyre flow does not occur in the lower layer but isequatorward in the upper one. This situation occurs only
when the upwelling to the upper layer exactly balances
the sum of the upwelling and convection to the lower
layer.
Case (5) when Q 2/Q < 0 (Q1 > 0, Q2 < 0): The intergyre flow is poleward in the lower layer but equatorwardin the upper one. This situation occurs when the net massflux between the upper and lower layers is upward and islarger than the upwelling from the abyssal layer.
The above cases are summarized in Table 2. The condition Q < 0 yields five similar cases, but the direction of
the inter-gyre flow is opposite to those when Q > 0.
When Q = 0, both the lower and upper layer intergyre flows change their directions at Q2 = 0, and hence
the above five cases degenerate into three cases: Q2 = 0
and Q2 ⭵ 0. (Note that we should use Q2 or a parameter
proportional to Q2 instead of Q 2/Q.) The case Q2 = 0 corresponds to the wind-only scenario discussed in the previous section, into which the cases (2), (3) and (4) above
degenerate. Here inter-gyre communication is minimal,
either in terms of water circulation or geostrophic contours. The other two possibilities, Q2 > 0 and Q2 < 0, correspond to the cases (1) and (5) above, respectively. Interestingly, even if Q = 0, an inter-gyre flow does occur
when Q2 ≠ 0, although geostrophic contours do not crossthe gyre boundary. The flow paths for Q = 0 are also presented in Appendix 2.
5. Summary and Discussion
Although the key dynamical elements in the intermediate layer adjustment were identified as coastal
Kelvin waves driven by mass input and eastward-advectedlong-Rossby waves in NTIA06, the effect of the adjustment through such Kelvin waves was not considered inventilated thermocline theory, while that of the eastward
moving long-Rossby waves within the wind-driven circulation has not been included in the abyssal circulation
theory. As a first step to combine these two effects, we
54

T. Nakamura et al.

developed, in a rather heuristic manner, an idealized
model of intermediate layer circulation and investigatedthe effects of mass input/output from the western boundary on the wind-driven circulation.
The model is a steady quasi-geostrophic (QG) 2.5layer model. It is forced both by Ekman pumping, which
produces subpolar and subtropical gyres, and by a mass
source/sink, which feeds fluid from the subpolar westernboundary at a rate Qn into layer n with net mass input of
Q = Q1 + Q 2. A frictional western boundary layer is incorporated to represent the effect of Kelvin waves, whichcarry the mass away from the source, while the interior isapproximated as inviscid and adiabatic.
The model describes the following ventilation process. Firstly, a net mass input induces a “barotropic” modeinter-gyre flow along the western boundary through the“Kelvin wave” dynamics. The “barotropic” inter-gyreflow then enables the geostrophic contours to make inter-gyre communication from the location of the mass
source to the subtropical gyre. Some of these geostrophic
contours turn eastward and extend into the interior owing to the effect of the wind-driven flow. These contours
go around the outer rim of a zone that would represent
the pool region in the absence of the mass input, and thenreturn to the western boundary. Potential vorticity travels along the above path into the subtropical interior andaffects the circulation. Eventually, the water supplied to
the lower layer flows into the subtropics near the westernboundary, circulates in the interior region along the above
path, and then flows away toward the equatorial region.
This is similar to the fate of the water ventilated in the
Okhotsk Sea or the Kuril Straits. Conversely, in the upper layer, water is fed from the equator to subtropics, andto the subpolar interior through the western boundary
current. The water then moves along the outer rim of the
subpolar gyre and is absorbed into the mass sink (to betransformed around the Okhotsk Sea). In this way, themodel illustrates a basic mechanism through which tidal
mixing at the Kuril Straits and convection in the Okhotsk
Sea ventilate the North Pacific intermediate layer.
The properties of the model are controlled by six
nondimensional parameters, namely, Q/TSvmax, Q2/Q, α,δm/L x, H2/H, and G2/ F̂ . The first two are introduced here,
while the other four have been introduced in previous
theories. Parameter Q/TSvmax is the ratio of net mass input
rate to the maximum Sverdrup transport. This parameter
determines the confluence point of “barotropic” westernboundary currents (together with T Svmin or TSvys when Q <0), and thus affects the inter-gyre communication throughthe western boundary layer by altering the paths of thegeostrophic contours. Parameter Q2/Q is the ratio of rateof mass input into the lower layer to that in all moving
layers. This parameter determines the vertical structureof the western boundary current and controls the direc-

tion and volume transport of the inter-gyre flow in theupper and lower layers. Parameter α is a measure of theeffect of wind forcing and determines the size of the pool
region when Q = 0 (e.g., Pedlosky, 1996), and hence it
determines the horizontal extent of the interior flow paths
originating from the lower layer mass source in the present
solution. The other parameter regimes with respect to
Q/TSvmax and Q2/Q are presented in Appendices 1 and 2.
It should be noted that although the confluence pointof the subtropical western boundary current is determined
only by Q/TSvmax, Q2/Q, and Q/TSvmin (or Q/T Svys) in thismodel, a variety of other causative factors has been suggested in previous studies. Any slight shift of the separation point induced by these mechanisms would not significantly change the basic scenario, however. In addition, the result is not sensitive to the choice of a Munk or
Stommel (1948) layer or to the use of a non-slip or slip
condition, since these only modify the zonal structure
function φm(x).
Although the present model assumes the input/output of mass leaks out of the model region, it is a matterfor discussion whether the model agrees with the limiting case, in which the model region is closed and there
are compensating mass source and sink at the westernboundary in the model region. When the source is situated poleward of the sink, Kelvin waves carry the massequatorward within the western boundary region from thesource to the sink. The difference from the present modelis the change in the meridional structure function, φ ys,
which now becomesφ ys ( y) = 1, for ysink ≤ y ≤ ys ,= 0, for y > ys and y < ysink ,(46)

where ysink is the latitude of the sink.
On the other hand, when the sink is situated polewardof the source, Kelvin waves carry the mass around thebasin counterclockwise in the northern hemisphere until
the waves reach the sink. Rossby waves will then emanate from the eastern boundary to conduct an adjustmentin the interior. As is shown in Subsection 3.1, thebarotropic mass-driven flow should vanish in the interior
in the present model configuration. However, input mass
must be carried away from the western boundary region,otherwise the layer thickness there continues to increase.
The present model resolves this problem by allowing themass to leak out of the domain through the equatorward
boundary. But for the closed basin, some processes neglected in the present model would be required. One possible solution may be the introduction of both a frictionalboundary layer on the all closed boundaries and diapycnal
mass transport from layer 2 to layer 3 in the model region. The former allows a flow along the boundaries, theFig. 6. Streamfunction in layer 2 in a parameter range corresponding to the deep western boundary current in the North
Atlantic in the density range of the upper Labrador Sea
Water. The parameter values are set, following Spall (1996),
as Q/T Svmax = 10/41, Q 2/Q = 1, α = 0.6, H2/H = 900/1750,
and G2/ F̂ = 0.06. The other settings are the same as thosein Fig. 5.

southern boundary in particular, and the latter allows thedamping Rossby waves propagating from the easternboundary (in other words, non-zero ψ bm is possible in theinterior owing to the diapycnal transport). The situation
then becomes similar to the lower layer of Kawase’s(1987) model. Note that, different from the present model,
the interior dynamics is not Sverdrupian in such a case.
The extension to such a model is left for future work.
Within a certain parameter range, the model showssome similarities to the results of the numerical experiments of Spall (1996) and Katsman et al. (2001) (Fig. 6).
For example, the lower layer flow originating from themass source is split into two mean paths, one moving to
the equator along the western boundary and the other into
the interior. The interior path merges with the other pathalong the boundary after circulating within the interior.
This feature is also similar to the observed deep westernboundary current (DWBC) in the density range of theupper Labrador Sea Water (e.g., Talley and McCartney,
1982; Pickart and Smethie, 1993; Bower and Hunt, 2000).
Spall (1996) also showed that the presence of the DWBC
does not preclude the formation of a large region of homogenized PV, but that it does affect the interior value of
PV. In addition, the horizontal extent of the interior pathincreases as the Gulf Stream or wind forcing is enhanced.
These three features are consistent with the following
three features of the present model, respectively: the approximation of PV homogeneity, the transport of PV fromthe mass source to the subtropical interior along thegeostrophic contours, and the fact that the interior flow
path lies along the communication belt and hence its horizontal extent is controlled by the parameter α. These similarities exist between our steady and rather simple modeland the numerical results, despite the fact that Spall (1996)
used an eddy-permitting primitive model. The present
model is therefore expected to incorporate the essential

Effects of Mass Source/Sink on the Wind-Driven Gyres

55

physics required to understand intermediate layer ventilation and thus we hope that the model may provide an
insight useful for the construction of a theory of ventilation from the ventilated thermocline to the abyss.
Acknowledgements
We wish to acknowledge Dr. J. P. Matthews for his
critical reading. The first author thanks Dr. H. Mitsuderafor his encouraging comment. Thanks are extended to thereviewers for their useful comments. The comment ofReviewer 1 is particularly helpful to improve the solution of ψ2lr. This study was supported by the Category 7
of MEXT RR2002 Project for Sustainable Coexistenceof Human, Nature and the Earth and by a Grant-in-Aid
for the 21st Century COE Program (Kyoto University,
G3). T.N. was partly supported by the JSPS Research
Fellowships for Young Scientists and by a Grant-in-Aid
for Young Scientists.
Appendix 1: Barotropic Flow and Geostrophic Contours for Various Q/TSvmax
Changes in the paths of the “barotropic” flow andgeostrophic contours from the case Q/T Sv max = 0 to
Q/T Svmax = 0.2 are discussed in Subsection 3.2. Here, pathchanges resulting from variation in the Q/TSvmax parameter are discussed. Figure A1 shows maps of ψb and q̂2
for various values of Q/T Svmax. The other settings are thesame as those in Fig. 4.
Noticeable path changes for ψb and q̂2 take place at
Q/TSvmax = 0, 1, and T Svmin/TSvmax (or T Svys/T Svmax) associated with the shift of ycwbc, the confluence point of thewestern boundary currents in the subpolar and subtropical gyres discussed in Subsection 3.2.1. The paths are thus
characterized into seven types when the three cases of
Q/TSvmax = 0, 1, and TSvmin/TSvmax are considered separately.
For simplicity, the case TSvmin = TSvys (i.e., ycspg = ys) ispresented here. Note also that TSvmin/TSvmax = –1 in all thecases presented here, since T Svmin = –TSvmax from (19).
Type (i) when Q/TSvmax = 0 (i.e., no net mass input/
output). The circulation is driven only by wind, and there
is no inter-gyre transport for the barotropic mode, as discussed in Section 3 (Fig. 4).
Type (ii) when 0 < Q/TSvmax < 1 (i.e., a moderate mass
input). The confluence ycwbc is in the subtropical gyre, sothat “barotropic” inter-gyre transport occurs and that thecommunication belt appears, as described in Subsection
4.2. Variations in this range are given from (18), (17),and (33), which indicate that an increase in Q/T Sv max
causes an equatorward shift of ycwbc from the gyre boundary toward the center of the subtropical gyre, an increasein volume transport of the barotropic inter-gyre flow,
broadening of the communication belt, and shrinking of
the subtropical pool region.
Type (iii) when Q/T Svmax = 1 (i.e., the mass input rate
56

T. Nakamura et al.

equals the maximum Sverdrup transport in the subtropical gyre). The poleward western boundary current vanishes, and the Sverdrup flow into the subtropical interior
is compensated solely by mass input from the westernboundary. This results in the absence of both a closed
barotropic streamline and a pool region in the subtropical gyre.
Type (iv) when Q/TSvmax > 1 (i.e., a large mass input). An equatorward western boundary current flows
through the subtropics. The streamlines from the mass
source diverge to form both this boundary current and aflow circulating in the interior. They later converge toflow into the equatorial region.
Type (v) when TSvmin/T Svmax < Q/T Svmax < 0 (i.e., amoderate mass output). The confluence ycwbc is in the
subpolar gyre. Interestingly, the “barotropic” cross-gyretransport flows from the equatorial region through thesubtropical western boundary current to the subpolar interior, circulates around the outer rim of the subpolar gyre,and eventually flows into the mass sink. Associated with
this, geostrophic contours originating from the subtropical eastern boundary and/or the equatorial region extend
through the subtropical western boundary current to thesubpolar interior. These contours correspond to a subpolar
counterpart of the communication belt discussed in Subsection 4.2. Associated with this subpolar communication belt, shrinkage takes place in the subpolar pool region rather than the subtropical pool.
Variations in this range are similar to those in thecase Q > 0. An increase in |Q/TSvmax| causes a poleward
shift of ycwbc from the gyre boundary toward the mass
sink, an increase in volume transport of the barotropicinter-gyre flow, broadening of the subpolar communication belt and shrinkage of the subpolar pool region.
Type (vi) when Q/TSvmax = TSvmin/TSvmax (i.e., the mass
output rate equals to the minimum Sverdrup transport at
the western boundary). In contrast to the case Q/TSvmax =1, the subpolar western boundary current vanishes only
on the equator side of ys because of the Kelvin wave dynamics. In addition, both closed streamlines and the pool
region remain in the subpolar gyre on the poleward sideof ys. This is due to the oscillatory structure of the Munklayer. It should be noted that when ys < y cspg and Q =TSvys, such closed paths are present even for the Stommel
layer. Accordingly, when Q < 0, both the latitudinal position of the mass sink and the longitudinal structure of thewestern boundary current affect the subpolar gyre circulation.
Type (vii) when Q/TSvmax < TSvmin/TSvmax (i.e., a large
mass output). The streamlines extending from the equatorial region diverge to form the poleward, subpolar western boundary current and a flow circulating within the
subpolar interior. They later converge to flow into themass sink.

barotropic streamfunction ψb^geostrophic contours q2

Q/TSvmax=1.5

Q/TSvmax=1

Q/TSvmax=-0.2

Q/TSvmax=-1

Q/TSvmax=-1.5

Fig. A1. (Left) Barotropic mode streamfunction and (right) geostrophic contours for various values of Q/TSvmax, which are selected so that the seven types discussed in Appendix 1 are represented (the cases 0 < Q/T Svmax < 1 and Q/T Svmax = 0 are shownin Fig. 4). The other settings are the same as those in Fig. 4.

Appendix 2: Upper
and
Lower
Layer
Streamfunctions for Various Q2/Q
Flow paths of the “baroclinic” mode for various Q2/
Q are shown in Fig. A2. The baroclinic-flow paths change
as the vertical structure of the western boundary current
changes, which is categorized into the five cases (Table2). The values of Q2 are selected to represent these cases.
The values of Q are set as 0.2TSvmax and 0, which
represent the two barotropic flow types considered in
Sections 3 and 4 (i.e., those when 0 < Q/TSvmax < 1 and
Q = 0). The other regimes are omitted for the following
reasons. When Q ≥ 1, streamlines exhibit interesting features associated with the absence of the subtropical pool
region (the corresponding region is occupied by the communication belt) and with the presence of a flow fromthe mass source to the equatorial region along the western boundary (or from the equatorial region to the sink).
This regime is omitted, however, because it rarely occurs
in major basins. When Q < 0, geostrophic contours ema-

nating from the western equatorial region are present(Appendix 1), so that an additional boundary conditionon the equator side of the subtropical gyre is required toobtain a solution. Since such a condition depends on thesituations in the western equatorial regions, which differ
considerably between basins, we do not include this regime in the figure.
A2.1 0 < Q/TSvmax < 1
When Q > 0, the lower layer circulation changes at
Q2/Q = 0 as the lower layer inter-gyre flow changes its
direction, while that in the upper layer changes at Q2/Q =1. We thus describe the circulation in these two layers
separately.
The lower layer inter-gyre flow is equatorward, when
Q2/Q > 0. This includes the case shown in Subsection
4.4.2. The water fed from the mass source flows away
through the subpolar western boundary region to the subtropical interior, and to the equatorial western boundary
Effects of Mass Source/Sink on the Wind-Driven Gyres

57

(a) Q/TSvmax=0.2
lower layer streamfunction ψ 2upper layer streamfunction ψ 1

Q2/Q
=1

Q2/Q
=0.5

Q2/Q
=0

Q2/Q
= -1.5(b) Q/TSvmax=0
lower layer streamfunction ψ2

upper layer streamfunction ψ1

Q2/‘Q’=1.5

Q2/‘Q’=-1.5

Fig. A2. Streamfunctions of (left) the lower and (right) upper layers for various values of Q 2/Q, when (a) Q/TSvmax = 0.2 and(b) Q/T Svmax = 0. In the case (b), Q 2 is normalized by “Q”, which is the value of Q in the case (a), in order to facilitate thecomparison. The other settings are the same as those in Fig. 5.

region. Although such qualitative features remain the samein this range, the inter-gyre transport increases with increasing Q 2/Q.
When Q2/Q = 0, the lower layer inter-gyre flow vanishes. This is because the absence of a mass input in thelower layer works to restrict the flow in the communication belt. Thus, in terms of water circulation, inter-gyrecommunication vanishes in the lower layer, although theinformation travels through the communication belt. The
subtropical pool region is still in motion, but the PV value
58

T. Nakamura et al.

is different from that in the wind only case. When
Q/T Svmax ≥ 1, the whole subtropical gyre becomes stagnant in the lower layer as the subtropical pool region disappears.
When Q2/Q < 0, the lower layer inter-gyre flow is
directed poleward to the mass sink: Water flows from theequatorial region to the subtropics, splits into two paths(one along the western boundary and one encircling thesubtropical interior), enters the subpolar western boundary region, and finally flows into the mass sink.

The upper layer inter-gyre flow is poleward, when
Q2/Q > 1, which is the case shown in Fig. 5. The watercrossing gyre boundaries flows from the equatorial region to the subtropics through the western boundary region, and to the subpolar interior (the outer rim of the
subpolar gyre). This flow path is similar to that of thebarotropic mode in the case Q/TSvmax < 0, rather than thatof the lower layer in the case Q2/Q < 0.
When Q 2/Q = 1, the upper layer inter-gyre flow vanishes. Thus, all of the mass input from the western boundary is transported through the lower layer.
When Q2/Q < 1, the upper layer inter-gyre flow is
directed equatorward. As a result, the input water flows
from the subpolar western boundary region into the outerrim of the subtropical gyre, and to the equatorial region.
A2.2 Q = 0
When Q = 0, both the lower and upper layer circulation paths change at Q2 = 0, as the inter-gyre flows inboth layers change their directions. When Q 2 > 0, the inter-gyre flow is equatorward in the lower layer but
poleward in the upper one. Thus, the water fed into thelower layer flows from the subpolar region to the subtropics, and to the equator, similar to the cases when Q >0 and 0 < Q 2/Q < 1. Conversely, in the upper layer, thewater coming from the equatorial region flows throughthe subtropics to the subpolar gyre, similar to when Q > 0
and Q2/Q > 1.
When Q2 < 0, the inter-gyre flow in both layers reverses its direction (i.e., it becomes poleward in the lower
layer and equatorward in the upper layer). Associated with
this, the lower layer circulation becomes similar to thecase when Q > 0 and Q2/Q < 0, while the upper layer
flow is similar to when Q > 0 and Q 2/Q < 1.
It should be noted that because geostrophic contours
in the interior are not affected by mass input/output, theflow paths in the interior region remain the same in thepresent model, even though an inter-gyre flow is inducedby the mass input/output.
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