Seasonal energy cycle of wind-driven ocean circulation with particular emphasis on the role of bottom topography

ARTICLE IN PRESS

Deep-Sea Research I 53 (2006) 154–168 www.elsevier.com/locate/dsr

Seasonal energy cycle of wind-driven ocean circulation with particular emphasis on the role of bottom topography Toshihiro Sakamotoa,, Isao Umetsua,b a

Department of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan Naval Systems Department, Radio Application Division, Aerospace and Defense Operations Unit, NEC Corporation, Fuchu, Tokyo 183-8501, Japan

b

Received 17 February 2005; received in revised form 12 September 2005; accepted 24 September 2005 Available online 2 November 2005

Abstract We investigate how the bottom slope affects the time-dependent global energy balance of a two-layer subtropical gyre driven by seasonal winds in order to reformulate the concept joint effect of baroclinicity and bottom relief (JEBAR) based on energetics rather than vorticity dynamics. It is shown that the role of JEBAR in this situation is to transfer energy between the barotropic and baroclinic fields. Since a deep current tends to flow in meridional directions along a meridional ridge, the geostrophically balanced pressure-gradient forces can perform work on the zonal barotropic flow over the ridge. The direction of the deep motion, and hence the sign of the work is reversed seasonally because the pressure field in the lower layer exhibits an anticyclonic tendency in winter and a cyclonic tendency in summer. The topographic beta effect strengthens the work on the northwest and southeast sides of the ridge, so that the net contribution from the ridge region is negative in winter and positive in summer. On the other hand, this work must be canceled by enhancing the energy conversion to satisfy the energy equation. As a result, the ridge not only accelerates but also seasonally reverses the sign of the rate of energy conversion. With some modification, a meridional trench and a western continental slope turn out to have qualitatively the same effect on the seasonal transport variation. Therefore, the annual range in the barotropic transport of the gyre is, to varying degrees, reduced irrespective of the details of the large-scale bottom topography. r 2005 Elsevier Ltd. All rights reserved. Keywords: Seasonal variability; Dynamics; Topography; Wind-driven; Subtropical gyre; Kuroshio; North Pacific

1. Introduction In a stratified ocean driven by steady wind forcing, deep layers will be in no-motion in a stationary state if their isostrophes or geostrophic contours are not closed, because the density field is so redistributed as to diminish horizontal pressure Corresponding author.

E-mail address: [email protected] (T. Sakamoto). 0967-0637/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.dsr.2005.09.004

gradients at depths (Rooth et al., 1978). A steady wind-driven circulation, therefore, results in the Sverdrup balance as though there were no bottom topography (Anderson and Killworth, 1977). However, when thermohaline forces, for instance, are taken into account, a pressure torque appears on a sloping bottom as a residual vorticity, yielding a barotropic motion. Sarkisyan and his collaborators interpreted such a phenomenon in a stratified ocean as a result of joint effect of baroclinicity and bottom relief (JEBAR) and represented the whole relevant

ARTICLE IN PRESS T. Sakamoto, I. Umetsu / Deep-Sea Research I 53 (2006) 154–168

physical process in a succinct form in the vorticity equation, i.e., the cross product of the gradient of available potential energy and the gradient of bottom topography; see Sarkisyan (1977) for a review. The concept of JEBAR formulated in this fashion has been applied to slope currents (e.g. Huthnance, 1984; Souza et al., 2001; Blaas and de Swart, 2002), to wind-driven circulations in the Nordic Seas and the Arctic Ocean (Isachsen et al., 2003), to the geostrophic adjustment in the North Atlantic (Dobricic, 2003), and to diagnostic studies of thermohaline circulations ranging from regional to the world oceans (e.g. Lazier and Wright, 1993; Greatbatch et al., 1991; Myers et al., 1996; Mellor et al., 1982; Park and Guernier, 2001), although some practical disadvantages were pointed out (Cane et al., 1998). Transient motions may also prevent lateral pressure gradients in deep layers from vanishing. This idea led Sakamoto and Yamagata (1996) to accommodate JEBAR more appropriately to a time-dependent wind-driven circulation. Consider the seasonal transport variation of the Kuroshio which is known to be smaller than that predicted by the Sverdrup relation with the observed seasonal winds (Blaha and Reed, 1982), similar to the Florida Current. Based on numerical experiments with a two-layer planetary geostrophic model (see also Section 2), they explained the underlying mechanism as follows. The reason why the transport of the Kuroshio becomes larger than the Sverdrup transport in calm summer is that the baroclinic field is relaxed over a bottom slope to produce negative vorticity needed to maintain the anticyclonic barotropic gyre. The winter transport, on the other hand, becomes less than the Sverdrup transport because the water column loses a part of its negative vorticity by strongly interacting with the solid earth via the bottom slope. As a result, the transport of the western boundary current becomes relatively insensitive to the external wind forcing. The winter process is essentially barotropic and is quite analogous to mountain torque in the atmosphere (e.g. Holton, 1992), while the summer process is the core of JEBAR. They also speculated that the alternating barotropic and baroclinic processes are closely associated with continual release and accumulation of available potential energy; during winter, kinetic energy of the barotropic flow is converted to available potential energy, which is in turn liberated in the next summer. In this regard, it is also important that the transformed energy

155

propagates westward swiftly in the form of barotropic Rossby waves (Sakamoto and Yamagata, 1997), so that any bump in the open ocean may contribute to mitigating the transport change of the western boundary current in the subtropical and subpolar gyres. The JEBAR concept extended in this way facilitates a global interpretation of the variability of the large-scale ocean circulation and hence may be more useful than the classical modal decomposition analysis which may be applied only locally. It is important to recognize that JEBAR does not represent a true source and sink of potential vorticity but may be regarded as virtual forcing for the barotropic field only if we take into account the changes in the baroclinic field simultaneously. We take advantage of the peculiar form of the JEBAR torque to reach the latter baroclinic processes. We admit, however, that it is not very easy to understand such mode interactions from the potential vorticity equation. Here, we give a new formulation of JEBAR in terms of energy to make it more transparent that JEBAR does accelerate energy conversion between the barotropic and baroclinic fields. Moreover, by investigating the dynamic correspondence between the energy cycle and the flow patterns for particular arrangements of bottom topography, we will also explain the reason why the seasonal reversal of energy conversion takes place, which was unanswered by Sakamoto and Yamagata (1996). To develop an energy theory of JEBAR, we conduct numerical experiments on a wind-driven subtropical gyre with the same twolayer planetary geostrophic model as used by Sakamoto and Yamagata (1996). The present study owes much to Holland’s (1975) energy budget analysis for the baroclinic oceans and we attempt to extend it to time-dependent motions driven by periodic wind forcing. 2. Formulation 2.1. Governing equations We consider a two-layer fluid divided by an immiscible interface. It is contained in a rectangular b-plane basin and bounded by vertical walls at x ¼ 0, Le and y ¼
Ln =2, Ln =2, where x and y are the eastward and northward coordinates, respectively. In the vertical direction the ocean is bounded by a rigid lid at z ¼ 0 and a bottom at z ¼
Hðx; yÞ. We concentrate on slow basin-scale motions which

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may be described by the planetary geostrophic dynamics (e.g. Salmon, 1986, 1992). In this approximation, the Rossby number is much less than unity and at the same time the divergence of horizontal motion may be of order unity after suitable nondimensionalization, so that the inertial terms in the momentum equations can be neglected, whereas the time-dependence of the continuity equations must be retained. Thus, the governing equations are f k  v1 ¼


1 tx rp1 þ i
rv1 , r0 r 0 h1

h1t þ r  ðh1 v1 Þ ¼ 0, f k  v2 ¼


1 rp2
rv2 , r0

h2t þ r  ðh2 v2 Þ ¼ 0,

(1) (2) (3)

As in the derivation of the vorticity equations (Mertz and Wright, 1992), we may obtain various equivalent forms of energy equations. From (1)–(3) we obtain a pair of kinetic and potential energy balance equations, which is a two-layer version of Holland (1975): 0 ¼ W p þ fF; Kg þ tx i  v1 þ D,

(7)

Ft þ r  ð2Fv1 Þ ¼
fF; Kg,

(8)

where K¼

2 X 1 r jvi j2 hi 2 0 i¼1

(4)

It follows from (2) and (4) that the barotropic transport V ¼ h1 v1 þ h2 v2 is always nondivergent. We, therefore, define the barotropic streamfunction c by (6)

The boundary condition (5) indicates that we may set c ¼ 0 at the coasts.

(9)

is the total kinetic energy, F ¼ 12 r0 g0 h21

where the last equation may be replaced by h1 þ h2 ¼ H. In these equations, subscripts 1 and 2 refer to the upper and lower layer, respectively; k is the vertical unit vector; i is the unit vector in the x direction; vi ¼ ðui ; vi Þ is the velocity; hi is the layer thickness which is equal to H i when the interface is flat; f ¼ f 0 þ by is the beta-plane approximation of the Coriolis parameter with b constant; pi is the depth-independent pressure with the relation rp2 ¼ rp1
r0 g0 rh1 ; r0 is a mean density; g0 ¼ gðr2
r1 Þ=r0 is the reduced gravity with g the acceleration due to gravity and ri the density of each layer; tx is the zonal wind stress expressed as a body force in the upper layer. Momentum dissipation is parameterized by Rayleigh friction with a coefficient r, followed by Salmon (1992). The use of this linear damping simplifies the boundary condition into the constraint that there is no normal flow at the coasts, namely, ) ui ¼ 0; x ¼ 0; Le . (5) vi ¼ 0; y ¼
Ln =2; Ln =2

V ¼ k  rc.

2.2. Energy equations

(10)

is the available potential energy, W p ¼
V  rp2 ¼
r  ðp2 VÞ

(11)

is the work done by the pressure-gradient force in the lower layer,
rp2 , fF; Kg ¼
r0 g0 h1 v1  rh1 ¼
v1  rF

(12)

denotes the rate of conversion from F to K, and D ¼
2rK

(13)

is the dissipation. We note that the conversion, source and sink terms are always balanced in (7) because the velocity tendencies are neglected in the planetary geostrophic approximation as mentioned earlier. It should also be noted that W p may be rewritten as W p ¼
V  rpb þ r2 gV  rH,

(14)

where pb ¼ p1 þ r1 gh1 þ r2 gh2 is the bottom pressure. This identity relates W p with a difference between much larger quantities. Averaging (7) and (8) over the basin and using the boundary condition (5), we have the global energy balance 0 ¼ hF; Ki þ htx i  v1 i þ hDi,

(15)

hFt i ¼
hF; Ki,

(16)

where hi¼

1 Le Ln

ZZ dx dy

(17)

denotes the area-averaging operator over the whole basin and the notation hf gi has been replaced by h i for brevity. From (15) it is obvious that the term

ARTICLE IN PRESS T. Sakamoto, I. Umetsu / Deep-Sea Research I 53 (2006) 154–168

hF; Ki may become a major source of kinetic energy at the expense of potential energy when winds are weak. As will be seen later, W p can play a role in controlling the barotropic field in the presence of variable bottom topography despite hW p i ¼ 0. Since the effect of bottom relief is implicit so far, we divide the kinetic energy following Holland (1975) such that K ¼ K bt þ K bc ,

(18)

where K bt ¼

1 1 r0 jVj2 ¼ r0 j¯vj2 H; 2H 2

v¯ ¼

V , H

(19)

h1 h2 (20) H are the kinetic energy associated with barotropic (i.e. external) and baroclinic (i.e. internal) motions, respectively. The kinetic energy equation (7) is correspondingly divided into K bc ¼ 12 r0 jv1
v2 j2 he ;

he ¼

0 ¼ W p þ fF; K bt g þ tx i  v¯ þ Dbt , 0 ¼ fF; K bc g þ

h2 x t i  ðv1
v2 Þ þ Dbc . H

0 ¼ hHW p i þ hHtx i  v¯ i þ hHDbt i,

(29)

(21) (22)

3. Numerical experiment

(23)

h2 ðv1
v2 Þ  rF (24) H denote the rate of conversion from F to K bt and that from F to K bc , respectively. The dissipation terms are represented by fF; K bc g ¼


Dbt ¼
2rK bt ,

(25)

Dbc ¼
2rK bc .

(26)

We note that the term W p participates only in the barotropic energy equation (21). We can learn more from the basin-wide integration; averaging (21) and (22) over the basin, we obtain another form of the global energy balance: 0 ¼ hF; K bt i þ htx i  v¯ i þ hDbt i,
 h2 x 0 ¼ hF; K bc i þ t i  ðv1
v2 Þ þ hDbc i. H

absence of both winds and variable bottom topography. However, a nontrivial barotropic motion is possible in the presence of varying topography because the conversion term hF; K bt i does not necessarily vanish at an instant of no winds if F is sufficiently accumulated by the wind action at earlier times. In this sense, the global quantity hF; K bt i represents JEBAR from the viewpoint of energetics; it can produce a net energy transfer between the barotropic and baroclinic modes. Although we do not use in later analysis, we may obtain from (21) an alternative form

in which W p enters into the global energy balance in place of any forms of energy conversion. Again, hHW p i ¼ 0 if the bottom is flat everywhere. The correspondence between (27) and (29) is reminiscent of that between the two equivalent expressions for the vorticity balance (e.g. Mertz and Wright, 1992).

Here, 1 fF; K bt g ¼
¯v  rF ¼
r  ðFVÞ, H

157

(27) (28)

The difference between the cases with and without topography now becomes obvious because when the bottom is flat everywhere, hF; K bt i ¼ H
1 hr  ðFVÞi ¼ 0. In particular, from (27) we immediately obtain a trivial solution V ¼ 0 in the

We set Le ¼ 6000 km and Ln ¼ 2000 km and locate a meridional ridge or a trench so that the total depth is prescribed as   H0 ðx
xa Þ2 exp
H ¼ H0  , (30) 2 l2 where the minus (plus) sign corresponds to the ridge (trench), H 0 is the total depth in the flat region, l is the characteristic half width of the ridge and xa is the position of its axis. We assume from the outset that the height of the ridge is H 0 =2. We set H 0 ¼ 4000 m, l ¼ 400 km and xa ¼ 0 km or 1000 km; the former value of xa corresponds to a western continental slope and the latter corresponds to the Izu Ridge in the North Pacific. The seasonal wind forcing which inputs negative vorticity into the whole upper layer is given by   py 2pt þ1 , (31) tx ¼ t0 sin cos Ln T where t0 ¼ 0:1 N m
2 and T ¼ 86400  360 s, corresponding to 1 model year containing 360 days, and hence the amplitude changes between 0 N m
2 on July 1 and 0:2 N m
2 on January 1. The other parameters are set to the values which typify the subtropical gyre in the North Pacific, as listed in Table 1.

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basin, which is about 3.4 years.1 Thus, we conduct integration for 7 model years and the last seventhyear data are used for the analysis. The energy budgets are computed every 36 time steps corresponding to 1 day.

Table 1 Parameter values used in the numerical experiment Parameter

Value

Le Ln H0 H1 f0 b r0 g0 r

6000 km 2000 km 4000 m 1000 m 7:3  10
5 s
1 2  10
11 m
1 s
1 1000 kg m
3 0:02 m s
2 1  10
6 s
1

4. Results

For numerical calculations, we solve each velocity component by eliminating the pressure gradient terms from the momentum equations (1) and (3) and using (6) as 1 f h2;1 u1;2 ¼
cy  2 2 H f þr H    r tx 0 0 
g h1y þ
g h1x , f r0 h1 v1;2 ¼

1 f h2;1 c  H x f 2 þ r2 H  x   t r 0 0 
g h1x þ g h1y , f r0 h1

ð32Þ

ð33Þ

where the upper (lower) sign corresponds to the upper (lower) layer. From these we derive the vorticity equation in the ‘‘Sarkisyan form’’    0    f g 2 1 1 tx h1 ; ¼J
J c; H H r0 H y 2   1 rc , ð34Þ
rr  H where the first term on the right-hand side is known as the JEBAR term in the original sense. We adopt (2), (32), (33) and (34) as the model equations to be integrated numerically. We apply finite-difference schemes to the basic equations. The discretized variables are arranged on a 150  50 C-grid (e.g. Haltiner and Williams, 1980), so that the boundary condition (5) can naturally be introduced. Time integration of (2) is carried out by the leap-frog scheme with the time step Dt ¼ 40 min from the initial condition h1 ¼ H 1 , ui ¼ vi ¼ 0 at all grid points. The spin-up time in the present model may be estimated in advance as the time needed for baroclinic long Rossby waves to transit the entire width of the

Fig. 1 shows the numerical solutions of c and h1 for each season. When the bottom is flat (Fig. 1a), the gyre consists of a Sverdrup interior and a western boundary layer of Stommel type. The transport is large in winter and zero in summer in harmony with the seasonal wind forcing adopted here. This is because the nontopographic Sverdrup relation always holds because of the infinite speed of barotropic Rossby waves under the planetary geostrophic approximation. In contrast, the bottom slope tends to split the gyre into western and eastern parts and the former maintains the western boundary current in summer (Fig. 1b–d). Fig. 2a shows the variations of the barotropic transport defined as the maximum value of c. We confirm that the annual range in the transport variation is reduced when the bottom topography is included. Discontinuities in the slope of each curve, typically seen in February and September, are associated with a shift in the peak of c (Fig. 1), caused by the exchange in dominance between the wind-driven and the JEBAR-driven transports (Sakamoto and Yamagata, 1996). Fig. 2b and c show the variations of the conversion rates, hF; Ki and hF; K bt i, respectively. We confirm that both the positive conversion in summer and the negative conversion in winter are enhanced in the presence of topography. The curve of hF; K bt i for the continental slope seems to be offset in the positive direction, so that it is slightly positive even in January. Probably, this is partly because the bottom slope is just underneath the western boundary current. Box diagrams for (15) and (27) are shown in Fig. 3. Fig. 3a verifies the topographic enhancement of hF; Ki, by which the amount of potential (kinetic) energy is increased (decreased) in winter, and vice versa in summer. We 1 In the numerical experiment, we find that an equilibrium state is established faster than expected from this estimate when a ridge is included. One possible explanation is that Rossby waves excited over the ridge accelerate the adjustment process west of the ridge (Barnier, 1988; Sakamoto and Yamagata, 1997; Tailleux and McWilliams, 2000). Another interpretation based on energy conversion is made in Section 5.

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0

h1

winter

winter

1 60

y

y

1

159

0

1060 1000

-1

2

0

4

-1

6

2

0

summer

0 -1

0

2

4

1060

0

2




h1

winter

winter

50 40

0

2

4

0 -1

6

1060

0

1000

2

summer

(b)

0

40

0

2

6

4 x

6

summer

1 y

y

1 0

4 x

x

-1

6

1

0 -1

4 x

y

y

-1

6

1000

0

x 1

6 summer

1 y

y

1

(a)

4 x

x

-1

1000

1070

0

0

4

2

6

x

Fig. 1. Contour lines of barotropic streamfunction c and upper-layer thickness h1 in the case of (a) the flat bottom, (b) the ridge, (c) the western continental slope, and (d) the trench. The maps are drawn every 3 months beginning with January 1. Contour intervals are 5 Sv for c and 10 m for h1 . Both the abscissa and the ordinate are scaled by 1000 km. The axis of the ridge and the trench is located at x ¼ 1000 km. In (a), there is no barotropic flow in summer.

note that hF; Ki is positive and hence nonzero in summer even when the bottom is flat, indicating that the summer circulation is also supported by the baroclinic energy transfer hF; K bc i.2 On the other hand, Fig. 2c and Fig. 3b clearly indicate that the seasonal change in hF; K bt i is consistent with 2 In the North Atlantic, the potential energy stored in the warm water is sufficient to maintain the subtropical gyre for 1000 days or more without wind forcing (Stommel, 1965, p. 148). To confirm this we have made a spin-down experiment with the present model by turning off the winds on January 1. The result was that after 5 years, hFi ¼ 3:95  10
3 kJ m
2 and hKi ¼ 9:41  10
5 kJ m
2 for the ridge case, whereas hFi ¼ 1:37  10
2 kJ m
2 and hKi ¼ 2:72  10
4 kJ m
2 for the flatbottom case. This indicates that JEBAR accelerated the spindown process as expected. Even after 10 years, motion was discernible in both cases.

rectifying the annual range in the barotropic transport. Fig. 4 shows the horizontal distributions of each term in the energy balance (21). We note that another energy balance (7) displays very similar patterns. With and without bottom relief, the wind action simply reflects the wind fields and the dissipation takes place exclusively in the narrow western boundary layer. We also see that active energy conversion takes place over the sloping bottom, and possibly just outside the western boundary layer. In fact, the value of fF; K bt g over the bottom slope is one order of magnitude greater than the maximum wind action and also far exceeds the global means hF; K bt i (Fig. 2c and 3), indicating that the bottom topography amplifies the basinwide energy cycle. For example, in the ridge case

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160




h1

winter

winter

1

0

y

y

1

55

0

1060 1000

-1

-1 0

2

4

6

0

2

x

4

summer

summer

-1

1 20

0

1000 0

y

y

1 0

4

2

(c)

-1

6

1060

0

2

winter

y

50

0

1060

40

0

1000

4

2

-1

6

0

2

x summer

-1

0

0

2

6

4

summer

1 y

y

20

0

4 x

1

(d)

6

winter

h1

1

0

4 x




1 y

0

x

-1

6

x

6

0 -1

1070

0

1000

1040

2

4

6

x

x

Fig. 1. (Continued)

(Fig. 4b), the distribution of fF; K bt g is dominant on the northwest and southeast sides of the ridge. We discuss this asymmetry in the next section. We here show in Fig. 2d the seasonal cycle of fF; K bt g averaged separately over a strip of 400 km in x and 1000 km in y which occupies the northern or southern half of the ridge. We see that the contribution from the northern part of the ridge is larger than that from the southern half, especially in winter. Finally, we observe that the pressure work W p appears to balance the conversion rate over the sloping region in all topographic cases. This correspondence encourages us to seek a mechanism for energy conversion by looking into the bottom processes around the relief. 5. Dynamics We have confirmed that the reduced seasonal variation of the barotropic transport is associated

with the enhanced seasonal cycle of hF; K bt i. Here, we explain the mechanism responsible for the latter enhancement to offer another interpretation of JEBAR rectification, without resorting to the vorticity balance. 5.1. Energy conversion over the flat bottom Under the seasonally varying wind forcing, the pressure gradient in the lower layer rp2 tends to be undercompensated in winter and overcompensated in summer so that the lower layer is centered by a high (low) pressure in winter (summer). In fact, our numerical result shows that rp2 points toward the central latitude in winter and away from that in summer (Fig. 5a). This is because the baroclinic adjustment by which the lower layer tends toward a state of isostasy is much slower than the barotropic adjustment which is accomplished instantaneously in the present model. The baroclinic field, thus,

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1.5

70

50

flat bottom ridge continental slope trench

40 30 20

1.0 Conversion rate (mW m-2)

Volume transport (Sv)

60

161

0.5 00 -0.5 -1.0

10 -1.5

flat bottom ridge continental slope trench

0 -1.0 J F MAM J J A S O N D J Month

(a)

0.6

12 88

0.2 0 -0.2 flat bottom ridge continental slope trench

Conversion rate (mW m-2)

Conversion rate (mW m-2)

0.4

-0.4

north

44 00 south

-4 -8 -12

-0.6 (c)

J F MAM J J A S O N D J Month

(b)

-16 J F MAM J J A S O N D J Month

(d)

J F MAM J J A S O N D J Month

Fig. 2. Annual variations in (a) the barotropic transport in Sv, (b) the rate of conversion from potential to total kinetic energy, hF; Ki, in mW m
2 , (c) the rate of conversion from potential to barotropic kinetic energy, hF; K bt i, in mW m
2 , and (d) fF; Kg averaged over the southern half of the ridge ð800 kmpxp1200 km,
1000 kmpyp0 kmÞ and the northern half of the ridge (800 kmpxp1200 km, 0 kmpyp1000 km) in units of mW m
2 .

responds slowly to the uncompensated deep pressure field. As seen in Fig. 1a, right, the bowl shaped by the density interface deepens in winter and shallows in summer. As a result, the potential energy hFi increases in winter and decreases in summer, as already confirmed in Fig. 3. It follows from the potential energy equation (16) that hF; Ki must be negative in winter and positive in summer, which is consistent with the numerical result (Fig. 2b). On the other hand, by geostrophy, the deep current is anticyclonic in winter and cyclonic in summer over the whole basin (Fig. 5a) and hence is

almost collinear with V which is always nearly zonal away from the western boundary (cf. Fig. 1a, left). Thus, W p is negligible in the Sverdrup interior, as confirmed in Fig. 4a. It follows from (21) that the energy conversion fF; K bt g is merely comparable with the wind action. 5.2. The effect of a meridional ridge The numerical results needed for the following argument are shown in Fig. 5b. In the presence of the meridional ridge of sufficient height, the f =h2 contours are deviated from the latitude circles so

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162

4. 76 4. 87 4. 99 4. 86

winter xi.v1 Φ 11.9 13.1 13.5 12.8

K 1.67 1.55 1.70 1.55

Φ, K 1.41 1.77 1.59 1.75 D

xi.v1

Φ, Kbt 0 0.46 -0.03 0.27 Dbt (b)

xi.v1 Φ 10.3 9.73 9.42 10.0

1.44 1.43 1.46 1.23 Kbt 0.72 0.48 0.75 0.48 1.44 0.97 1.49 0.96

K 0.35 0.44 0.40 0.45

Φ, K 0.70 0.87 0.79 0.89

3.35 3.10 3.40 3.11

(a)

winter

summer

D

summer xi.v1

Φ, Kbt 0 0.18 0.24 0.16 Dbt

0 0 0 0

0.70 0.87 0.79 0.89

0 0 0 0 Kbt 0 0.09 0.13 0.08 0 0.18 0.24 0.16

Fig. 3. Basin-scale energy diagrams corresponding to January 1 and July 1; (a) shows the total energy and (b) shows the barotropic part. The generation, conversion and dissipation rates are in mW m
2 and the energy storages are in kJ m
2 . The value for the state at rest is subtracted from the potential energy. The four numbers in each column correspond (from top to bottom) to the case of the flat bottom, ridge, western continental slope, and trench.

that they are oriented almost meridionally on both sides of the ridge crest. Since deep currents tend to follow the f =h2 contours, they flow nearly meridionally along the ridge. More specifically, because of the uncompensated deep pressure field, v2 directs southward (northward) on the western (eastern) side of the ridge in winter, and in the opposite directions in summer. Thus, rp2 directs westward (eastward) on the western (eastern) side of the ridge in winter and reverses the direction in summer. On the other hand, V has a large eastward (westward) component in the northern (southern) half of the basin (cf. Fig. 1b). Therefore, the pressure-gradient forces can now perform work on V over the ridge; in winter, for instance, W p is positive in the northwest and

southeast quadrants and negative in the other quadrants over the ridge. However, as mentioned earlier, jW p j is dominant on the northwest and southeast sides of the ridge both in winter and in summer. This asymmetry may be understood by virtue of the topographic beta effect; both velocity and pressure fields in the lower layer in Fig. 5b actually reveal ‘‘westward’’ intensification if we regard the direction of rðf =h2 Þ as a ‘‘northward’’ direction. The pressure work averaged over the ridge region is, therefore, positive in winter and negative in summer. On the other hand, since the energy balances (7) and (21) require that the pressure work must be balanced by fF; Kg and fF; K bt g, respectively, the net energy conversion over the ridge is thus negative in winter and positive in summer. In this way, the ridge activates fF; K bt g such that K bt is converted to F in winter and vice versa in summer as preferred. An equivalent explanation is possible solely in terms of the baroclinic adjustment. Fig. 1b shows that another bowl of the h1 field is produced over the ridge, corresponding to the separated barotropic gyre. In winter, this new bowl partly cancels the anticyclonic pressure field in the lower layer. Thus, the basin-wide anticyclonic tendency in the lower layer is weakened, leading to a decrease in the anticyclonic barotropic transport. In summer, to accelerate the release of the available potential energy the bowl must be more flattened than in the flat-bottom case, producing a high pressure center above the ridge. Since this high pressure cancels and even reverses the overcompensated cyclonic pressure field in the lower layer, the anticyclonic barotropic transport arises in summer. The above argument is, of course, consistent with the experimental fact that hF; K bt i is negative (positive) in winter (summer). In other words, the ridge accelerates the tendency toward isostasy, so that the subtropical gyre in winter (summer) is less barotropic (baroclinic) than that over the flat bottom. 5.3. The effect of a western continental slope As seen in Fig. 5c, the configuration of the f =h2 contours implies that dynamically speaking, the western continental slope may be regarded as the eastern half of the ridge. In fact, the patterns of the deep flow and the deep pressure gradient over the continental slope resemble those on the eastern side of the ridge. Since the motion in the lower layer runs along the f =h2 contours in the northward

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beta effect that the main contribution to the energy conversion comes from the southern part of the continental slope. This is the reason why the seasonal variation of hF; K bt i for the continental slope (Fig. 2c) is qualitatively very similar to that for the southern half of the ridge (Fig. 2d).

(southward) direction in winter (summer), a low (high) pressure anomaly is produced over the continental slope. The original high (low) pressure center near the western boundary is thus weakened in winter (summer), so that the barotropic anticyclonic gyre diminishes in winter and does not disappear in summer (Fig. 1c). The anomalous pressure over the bottom slope in the lower layer may be inferred from the baroclinic field (Fig. 1c). In winter (summer), the subtropical bowl, which is just above the continental slope, is more bent (flattened) than in the flat-bottom case (cf. Fig. 1a), leading to a further increase (decrease) in potential energy. It follows from the topographic

5.4. The effect of a meridional trench We use Fig. 5d for the following argument. A meridional trench also deflects the deep motion in meridional directions, so that the geostrophically balanced pressure-gradient forces can exert work on a zonal component of the barotropic flow. Thus, the

 x i.v

{Φ,Kbt}

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Dbt winter

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2

1 x

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2 x

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2 x

Fig. 4. Horizontal distribution of (from left to right) the work done by the lower-layer pressure, the rate of conversion from potential to barotropic kinetic energy, the wind action, and the dissipation rate, in the energy Eq. (21), in the case of (a) the flat bottom, (b) the ridge, (c) the western continental slope, and (d) the trench. Contour interval is 1 mW m
2 for the wind action and 10 mW m
2 for the other quantities. Regions of negative values are shaded. Both the abscissa and the ordinate are scaled by 1000 km. The axis of the ridge and the trench is located at x ¼ 1000 km.

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164

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Fig. 4. (Continued)

effect of the trench on the seasonal transport variation is expected to be qualitatively the same as that of the ridge and, in fact, the difference in the transport variation between both cases is almost indiscernible (Fig. 2a). However, it is not straightforward to explain this similarity in terms of W p by the simple analogy of the ridge case. This is because under the planetary geostrophic approximation which permits an Oð1Þ displacement of the bottom, the f =h2 contours in the ridge and trench cases are not antisymmetric latitudinally. First, jrf =h2 j over the trench is much weaker than that over the ridge of the same height, so that the trench does not act as a perfect barrier against the deep flow. In fact, the velocity field in the lower layer reveals a northern boundary current that connects both sides of the trench. Thus, mass

conservation implies that v2 and hence rp2 are intensified far east of the trench, in which jW p j is correspondingly large (Fig. 4d). Second, since rf =h2 always directs eastward in the immediate vicinity of the western boundary because of the northward flowing western boundary current, the f =h2 contours are convex southward west of the trench (Fig. 5d), whereas they are weakly convex northward or nearly zonal west of the ridge (Fig. 5b). This is the reason why the velocity fields outside the western boundary layer are very different in both cases, even though there is always an anticyclonic (cyclonic) tendency in winter (summer). This difference reflects the distribution of W p near the western coast and we clearly see in Fig. 5d that just outside the western boundary layer, W p is strongly positive (negative) in winter

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(summer), which must be canceled by the net negative (positive) fF; K bt g in winter (summer). As a result, the seasonal energy cycle becomes similar to the ridge case, as verified numerically in Figs. 2 and 3. We cannot demonstrate with the present experiment about the effect of a much deeper trench like the Izu–Ogasawara Trench and the Mariana Trench. At present, we only anticipate that the effect of the Izu–Ogasawara Trench on the transport variation in the western basin might be overwhelmed by the Izu Ridge and the continental slope.

6. Discussion and conclusions Observations suggest that the seasonal variation of the Kuroshio transport is substantially smaller than expected from the Sverdrup relation. The mechanism for the weak seasonality was explained by Sakamoto and Yamagata (1996) using the concept of JEBAR. In short, JEBAR produces negative vorticity needed to maintain the anticyclonic barotropic gyre throughout the year, at the expense of the baroclinic field over variable bottom topography. Since, however, the baroclinic processes that accompany with such internal produc-

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Fig. 5. Horizontal distribution of (from left to right) f =h2 , v2 , the zonal pressure gradient px , and the meridional pressure gradient py , in the case of (a) the flat bottom, (b) the ridge, (c) the western continental slope, and (d) the trench. Contour intervals are 2  10
9 m
1 s
1 for f =h2 and 0:2 mPa m
1 for both px and py . The scale of the velocity vector is shown by the arrow below each panel. Regions of negative values are shaded. Both the abscissa and the ordinate are scaled by 1000 km.

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Fig. 5. (Continued)

tion of vorticity cannot clearly be seen from the potential vorticity equation, we have developed the energetic aspects of JEBAR in the present study. From the energy equation in which more emphasis is put on the barotropic component, we have ensured that bottom topography in general can help net energy transfer between the barotropic and baroclinic modes. We may regard this global property as JEBAR from the energetics points of view; JEBAR is not a source or sink of energy but represents conversion between available potential energy and kinetic energy of the barotropic field. We also expect that the energy formulation of JEBAR will be useful when we include the effect of atmospheric thermal forcing (e.g. Sakamoto, 2005) and when we extend the analysis to the ocean–solid

earth system which exchanges heat and mechanical energy across the sea floor. Although Sakamoto and Yamagata (1996) pointed out that the annual energy cycle could be modified by JEBAR, they did not answer the question as to how conversion between potential and kinetic energy is reversed seasonally. The basis of our approach is the notion of imperfect compensation of horizontal pressure gradients in the lower layer which may occur if the large-scale winds evolve at a shorter period than the baroclinic adjustment. This is in striking contrast with Dobricic (2003) who conducted numerical experiments to establish the general principle that steady wind forcing pushes the North Atlantic circulation into equilibrium when the potential energy reaches

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its minimum value. Under seasonal winds, a high (low) pressure center, and hence anticyclonic (cyclonic) tendency exists in the lower layer in winter (summer). The uncompensated pressure gradients are almost perpendicular to the barotropic flow when the bottom is flat, but may be deflected over a bottom slope because the deep flow tends to follow geostrophic contours. In the latter case, the pressure-gradient forces can perform work on the barotropic flow.3 The energy equation in the planetary geostrophic approximation tells us that this work must be balanced by the conversion between potential and kinetic energy. In the presence of a meridional ridge, the energy conversion is always dominant on the northwest and southeast sides of the ridge because of the topographic beta effect. The sign of the work (and hence of the energy conversion) changes seasonally because the direction of the deep flow is reversed seasonally. Consequently, the ridge accelerates negative (i.e. from kinetic to potential energy) conversion in winter and positive conversion in summer, as desired. The western continental slope may simply be regarded as the eastern half of the ridge. On the other hand, the trench produces the north–south asymmetry of the pressure work not only over the trench but also near the western boundary. However, we have verified numerically that the energy cycle in the presence of the trench is, in basin average, almost the same as that in the presence of the ridge. In conclusion, each topographic feature contributes to the reduction of the seasonal transport variation in quite a similar manner. It seems straightforward to extend our results to a case in which the topography consists of some ridges and trenches, because it is rather easy to predict deep motions and associated deep pressure anomalies over such kinds of bottom relief. A question arises as to what happens in a basin with much more realistic topography including smallscale bumps and hollows. It is not certain at present whether globally averaged energy conversion systematically affects the basin-scale circulation or has little effect simply because of the randomness of the topography. Furthermore, the inclusion of eddies

167

due to instabilities is an essential extension and the interaction between baroclinic eddies and bottom topography may be more than a simple analogy of the present large-scale picture. The statistical properties of subgrid-scale random eddies over a sloping bottom is described in terms of the Neptune effect (Holloway, 1992) provided that the system tends to reach an equilibrium barotropic state, although this transition is possible only when the scale of the mean motion is larger than the deformation radius (Salmon et al., 1976). Progress has been made in understanding geostrophic eddies over variable bottom topography based on a different principle (e.g. Adcock and Marshall, 2000). At any rate, another framework might be needed to relate mesoscale baroclinic activities with energy transfer due to JEBAR. The present results need to be verified by observations. The key ingredient of the summer overcompensation or the seasonal reversal of bottom pressure gradients seems to be an interesting research target in the field study. Recently, Kawabe et al. (2003) carried out full-depth CTDO2 measurements in the northwestern Pacific to obtain new maps of deep geostrophic flow in winter at a depth around the Lower Circumpolar Water. They found a volume transport of the order of 1 Sv around the Izu Ridge, whose direction is not inconsistent with our flow pattern in winter. Johnson (1998) summarizes observed deep currents over several trenches in the Pacific, showing that the mean circulation over the north end of the Izu–Ogasawara Trench is cyclonic. To our knowledge, however, there is no observational evidence that might verify the seasonal variability of the deep circulation. On the other hand, the bottom pressure has occasionally been measured by seismologists, but there seem to be very few who attempt to relate it with the atmospheric changes through the ocean circulation. It would be interesting if we could detect small changes in the bottom pressure against the ridge flank by monitoring anomalous changes in some elements associated with the subtropical gyre, and vice versa.

Acknowledgements 3

Topographically induced pressure-gradient force was considered as a mechanism for decelerating the Antarctic Circumpolar Current (so-called form drag) to solve Hidaka’s dilemma (Munk and Palme´n, 1951) and, later on, has been developed as mountain torque to explain the atmospheric circulation (e.g. Widger, 1949; Newton, 1971).

The authors wish to thank three anonymous reviewers for the useful comments which greatly helped us clarify some of the issues and improve the manuscript.

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