An Influence of Bottom Topography
on the Western Boundary Current
 
Yang Haijun1, Qinyu Liu1 and Wei Li1
 
( received 1997/12/30, revised 1998/3/2, accepted 1998/3/31)
 
ABSTRACT

In this paper a linear quasi-geostrophic model with Rayleigh friction for an enclosed oceanic basin is developed to investigate the effects of different types of bottom topography (BT). The numerical method is used to solve the vorticity equation, and twelve experiments are designed to show the patterns of the Western Boundary Current (WBC) when influenced by four types of BT. The compression of vortex tubes by the slope of the BT and the bottom frictional force can generate a strong northward current in the western boundary. The location of the main axis (defined as the position of the two-third maximum northward current) of the BT-induced boundary current (BBC) varies with different BT. Besides acting in the same way as the b -effect, the BT slope is tightly related to the strength of the BBC and its main axis location. For a linear slope, there is a positive correlation between the steepness of slope and the strength of the BBC in the case where the slope width is no less than that of the boundary current. A steeper slope can force a stronger BBC and cause its main axis to shift closer to the western boundary. If the slope varies from flat to steep for a certain BT and if there is a slope break, the location of the slope break can have a remarkable effect on the velocity and width of the BBC. If the slope break is far away from the western boundary, it may well weaken the velocity near the boundary and increase the width of the BBC. As a result, the main axis of the BBC is moved in an eastward fashion gaining closer proximity to the slope break. The results of numerical experiments with the b -effect are qualitatively the same as those without. The combined effect of b and BT makes the WBC slightly narrower and weaker.
 

(Key Words: bottom topography, BT induced boundary current, slope break)
______________________________________________________________________________
1 Institute of Physical Oceanography, Ocean University of Qingdao, Qingdao, P.R.China
 
 
INTRODUCTION
 
Since the pioneering work on westward intensification by Stommel (1948), the western boundary current (WBC) has been a major focus of attention in the field of physical oceanography, which, in turn, has given rise to an enormous number of published articles on the subject.

Among those studies, many papers have theoretically discussed the effects of bottom topography (BT) (eg., Holland,1967; Welander, 1968; Cessi and Pedlosky, 1986; Kawase and Straub, 1991), but they have mainly been centered on the Sverdrup interior with very large horizontal scale of topography. Liu (1990) introduced a frictionless quasi-geostrophic (QG) layered model with a slope-like topography and studied the combined effect of nonlinearity and slope. He found that a reasonably high continental slope can force a very strong northward boundary current known as the continental slope boundary current which can significantly enhance the total transport of the WBC. Recently, Kubokawa and McWillams (1996) investigated the effect of a steep continental rise on WBC by using a QG model with Rayleigh friction, in which the topographic contribution to the potential vorticity gradient was much stronger than the planetary one (i.e.,b ). The analytic solutions derived by them showed there is an equatorward tail over the slope that is notably long when either the slope is gentle or the friction is small. They discussed the effect of the steepness of a slope on the equatorward tail, but they failed to probe into the effects of different types of BT on the WBC.

The aim of this paper is to gain a better understanding of the patterns of the WBC in cases with different BT’s. The QG model is also used here to numerically study this problem. Four types of BT are chosen, and twelve experiments are specifically designed to examine the manifestation of the WBC. The results obtained here are the same as those expected and, they are most useful in revealing the formation mechanism of the South China Sea Warm Current (SCSWC).
 

FORMULATION OF THE PROBLEM
 
A square basin of homogeneous density r , with the pressure gradient proportional to the surface slope is considered here. The vertically averaged linearized equations in steady-state balance of the rotational Earth are: (2.1)

(2.2)

(2.3)

 where (u,v) are the vertically averaged horizontal current component independent of z, and the z-axis is positive upward; f represents the Coriolis frequency; h the sea surface height (SSH) measured from the ocean surface at rest; g the gravitational acceleration; (t x, t y) the wind stress component; R the coefficient of Rayleigh friction; and H represents the ocean depth as a function of x and y, which is much thicker than and independent of h .

In order to simplify the equations, a b -plane (f=f0+b 0y) and simple shapes for topography and wind stress, H=H(x) and t x =0, t y=t y(x), respectively, are assumed in this study. In addition, in order not to exaggerate the effect of bottom frictional force, the bottom friction coefficient, R, is taken as R0H0/H, where R0 is constant and H0 the maximum undisturbed depth.

Equations (2.1)-(2.3) are non-dimensionalized by introducing the horizontal length scale L, vertical scale D (maximum depth of the ocean), velocity scale U, wind stress scale t 0, and the magnitude of the SSH variation h 0, such that:

(2.4)

(2.5)

(2.6)

 where asterisks denote the non-dimensional variables, f*=1+b y* and

 

(2.7) The primary balance in (2.4)-(2.5) is taken to be geostrophic, that is, gh 0/L=f0U. The characteristic scales are taken as follows: L=5´ 105m; U=0.1ms-1; t 0=0.2Nm-2; r =1025kgm-3; H0=300m; D=1000m; g=9.8ms-2; and R0=0.6´ 10-5s-1. Then, the non-dimensional parameter T » 0.039118106.

Equation (2.6) admits the introduction of a stream function defined by

(2.8) After dropping the asterisks for simplification, the vorticity equation can be rewritten as: (2.9) This demonstrates that the stream function is mainly determined by wind stress curl, in which the b -effect and the slope of the BT distort the symmetry of the stream field. Based on Stommel's theory, the westward intensification in oceans is caused by the b -effect. From Eq.(2.9), it can be expected that the stream field should intensify at some place just because of the existence of the slope of the BT even when b =0.
  In the present paper, four different types of BT (Fig.1) are chosen. They are   BT1: 

BT2: 

BT3:  (2.10)

BT4: 

 Here, H0, C0 and C1 are non-dimensional depth scale. H0 and C0 are constant and equal to 0.3 and 0.1 respectively. C1 is given to different values in order to control the slope of BT. k1=(H0-C0)/C1, k2=(1-C0)/C1n, where n is chosen as 5.42.

Wind stress is chosen as the simplest linear relation with x, that is, t y=-x for 0£ x£ 1. Then, its curl dt y/dx=-1=constant. The constant wind stress curl ensures the symmetry of the stream field without taking into account of the b -effect and BT slope.

The impermeability condition is applied at all boundaries, that is,

(2.11) Eq.(2.9) is a classical Poisson problem. It can be discretized by five-point difference scheme, and then the method of successive over relaxation (SOR) is used to get the approximate value at each grid point until the resulting values meet the given precision.
 
 
SCHEMES OF NUMERICAL EXPERIMENTS

Based on different BT’s and whether or not the b -effect is considered, twelve schemes of numerical experiments are designed, with two controlling parameters set: IBETA and ITOPO. IBETA is 0 (for b =0) or 1 (for b >0). ITOPO is 0 for BT1, 1 for BT2, 2 for BT3 or 3 for BT4. The experiments each with different parameters are listed in Table 1:

 

 
EXP.
IBETA
ITOPO
C1
EXP.
IBETA
ITOPO
C1
1
0
0
--
7
1
0
--
2
0
1
--
8
1
1
--
3
0
2
0.3
9
1
2
0.3
4
0
2
0.8
10
1
2
0.8
5
0
3
0.3
11
1
3
0.3
6
0
3
0.5
12
1
3
0.5
The experiments with BT1 are taken as standard ones for comparison with other experiments. The BT2 is used to study the change of the BBC's strength and its main axis location under a step-like topography. The BT3 checks the corresponding variation in the BBC when the linear slope goes from steep (for small C1) to gentle (for large C1). In the case of BT4, the BT slopes dH/dx=nk2xn-1, and dH/dx increases with the increase of x for the same C1. The variation of dH/dx has a maximum, which is matched to the place of the continental slope break in the sea. Thus, the purpose of BT4 is to study whether or not the BBC varies with the movement of the location of the slope break.
 
THE RESULTS OF NUMERICAL EXPERIMENTS

The following expression is defined as an approximation of the mean strength of the meridional current:

(4.1) Here, y1 and y2 are non-dimensional meridional lengths taken as 0.3 and 0.7, respectively.

Experiments with b = 0

Exp.1 is a standard experiment (Fig.2a), and the stream field obtained is completely symmetrical about the central line because of the constant wind stress curl. In Exp.2, a step-like topography (BT2) is added. Fig.2b shows that BT2 has not greatly changed the stream field. However, at x=0.2, i.e., at the step break, the meridional current shows an abrupt increase compared to that in its two sides (Fig.2b, c). The step break hinders the westward current, which results in a sudden uplift of the fluid column. To compensate, the relative vorticity decreases and a strong northward current appears. To the west of the step break, the stream field shows little difference from that of Exp.1 due to zero slope. Thus it can be expected that when the step-like topography is extended eastward, the position of the maximum variation of the meridional current moves eastward accordingly. It implies that this type of BT has the possibility to decide the location of the maximum northward current.
 

Figure 2 Distribution of stream function (thin line), vector fields (a,b) and meridional velocity profiles (c, in cm/s). (a) for Exp.1 and (b) for Exp.2. The stream function contour interval is 3´ 103, and the reference vector is 10cm/s. In (c) the crosses are for Exp.1, the open circles for Exp.2 and the open squares for the differences between Exp.2 and Exp.1.
  In Exps.3 and 4, the BT3 is used. The slope is invariable in each experiment, but the one in Exp.3 is steeper than that in Exp.4. Compared with Exp.1, this type of BT changes the symmetry of the stream field so much that a very remarkably westward intensified meridional current, which is called the BT-induced boundary current (BBC), appears (Fig.3). The maximum northward current is close to the western boundary, and away from the boundary, it decreases rapidly. A comparison of Exp.3 (Fig.3a) with Exp.4 (Fig.3b) reveals that the steeper the slope, the stronger the WBC. The location of the BBC main axis (defined as the position of the two-third maximum northward current) is nearer to the western boundary in the case of the steeper slope. The implications of this are two-fold: (1) the BT has the same impact as the b -effect and does intensify current near the western boundary, and (2) both the strength of the BBC and the location of its main axis are tightly related to the steepness of BT, with an obvious positive correlation between them.   Figure 3 Same as Figure 2, but (a) for Exp.3 and (b) for Exp.4. In (c) the crosses are for Exp.3, the open circles for Exp.4 and the open squares for the differences between Exp.4 and Exp.3.   The BT4 is considered in Exp.5 and Exp.6. In these cases, slope is no longer constant but, on the contrary, there is a slope break. The slope is gentle to the west but steep to the east of slope break in both cases. The only difference between the two experiments is the position of slope break (» 0.2 for Exp.5 and » 0.3 for Exp.6). Just like in Exps.3 and 4, the slope breaks the symmetry of the stream field and increases the WBC a great deal (Fig.4). However, in these experiments the main axis of the BBC is not tight near the western boundary, but rather has a tendency to move toward the slope break (from 0.1 for Exp.5 to 0.2 for Exp.6), and the width of the BBC is also enlarged (Fig.4c). To the east of the slope break, the meridional velocity is very weak. Comparing Exp.5 with Exp.6, it can also be concluded that with the eastward movement of the slope break, the maximum BBC decreases, and its zonal width increases. These findings once again show that the BT can not only act in the same way as the b -effect, but also shift the main-axis of the BBC eastward.

Before the remaining experiments are discussed, some explanations on the physical mechanism of the BBC should be clarified. First, it must be emphasized that because of the smaller horizontal scale of the sea basin set in this study (only 500km), the WBC here (far less than 100km) is narrower than that in the large scale ocean. Second, the widths of the BT slope in this paper are all far wider than that of the WBC. It is well known that the westward intensification of ocean circulation is due to aeolotropism caused by the b -effect. From Eq.(2.9), BT can also produce aeolotropism even when b =0 and distort the symmetry of circulation. For a fluid column moving toward the western boundary, its thickness would decrease because of upslope motion. The compression of the vortex tubes would result in a decrease in the relative vorticity so as to maintain the potential vorticity conservation, though the bottom friction increases at the same time. Therefore, the northward current is strengthened. For BT2, the northward current increases only at the step break because of zero slope on its two sides. While for the linear slope of BT3, the variation in relative vorticity is also linear. Hence, the stronger northward flow must be on the inshore side of the slope, and the steeper the slope, the larger is the variation in relative vorticity, thus the larger is the northward current. As to the variable slope of BT4, the relative vorticity changes from large to the east of the slope break to small to the west. As a result, the BBC main axis is not close to the western boundary but moves toward the slope break. If the slope width were much narrower than that of WBC, the WBC would not feel the effect of the slope and the BBC would not exist.
 

Figure 4 Same as Figure 2, but (a) for Exp.5 and (b) for Exp.6. In (c) the crosses are for Exp.5, the open circles for Exp.6 and the open squares for the differences between Exp.6 and Exp.5.
  Experiments with b > 0

Exp.7 is the result of BT1 (Figure omitted). It is well known that the westward intensification must be exhibited because of the b -effect. However, because of the small horizontal scale of the sea basin, the value of b is very small (about 0.22). Thus, little difference is found between Exp.7 and Exp.1. It is now probably argued that the results of the remaining experiments (Exps.8 to 12) based on b >0 should be qualitatively the same as those of Exps.2-6. In fact, this is exactly the case.

Fig.5 shows the BBC’s in Exps.9-12 are slightly narrower and weaker than those without the b -effect. This implies that though the b -effect and BT both contribute to the generation of a strong current in the western boundary, their joint influence on the WBC is not to make it stronger, but rather to make it weaker. The b -effect counteracts the effect of the BT slope to some extent, and this is particularly true for the BT chosen here because dH/dx is positive in the x-direction and |b -(G /H)dH/dx| is less than (G /H)dH/dx. Thus the aeolotropism of the stream field is weaken.
 

Figure 5 Meridional velocity profiles (unit: cm/s). (a) The crosses are for Exp.9, the open circles for Exp.10, the closed circles for Exp.11 and the open squares for Exp.12. (b) The crosses are for Exp.3, the open circles for Exp.9 and the open squares for the differences between Exp.9 and Exp.3. (c) the crosses for Exp.5, the open circles for Exp.11 and the open squares for the differences between Exp.11 and Exp.5.
 
CONCLUSIONS AND DISCUSSION

In the present paper, the linear, barotropic, wind-driven ocean circulation in a square basin are studied from the perspective of four different types of BT. The three main results are listed here:

1) When b = 0, the slope of the BT can act as the b -effect and generate westward intensified current. The steepness of the slope has a notable influence on the strength of the BBC and the location of its main axis. In short, there is a positive correlation: the steeper the slope, the stronger the BBC. The steeper slope also shifts the main axis of BBC closer to the western boundary.

2) For varying slope, the location of the slope break exerts a remarkable influence on the meridional velocity and the width of the BBC. If the slope break is far away from the western boundary (BT4), it can weaken the velocity near the boundary and increase the width of the BBC. The resulting effect is to move the main axis of BBC eastward.

3) When b is small, the results of numerical experiments are qualitatively the same as those without the b -effect. The combined effect of b and BT makes the BBC narrower and weaker.

This work is a preliminary study on the formation mechanism of the South China Sea Warm Current (SCSWC). The SCSWC is a northeastward current against the wind direction over the northern shelf of the SCS in wintertime. Its main axis is often found between 200m and 400m isobaths. Based both on observational data and the results of numerical modeling, several theories have previously been proposed to explain its formation. For example, Guan (1978) think the SCSWC can be qualitatively viewed as a "subtropical counter current" in the SCS and the baroclinic effect determined by the vertical structure of the temperature field might be the primary factor generating it. However, some barotropic models driven by climatological mean wind can successfully reproduce this current (Zeng et al., 1989; Li et al., 1994). Even if the Luzon Strait is closed in models, this current can also be simulated out. This suggests that the baroclinicity is not a necessary requirement for the formation of the SCSWC. So the second theory thinks that the SCSWC is a compensatory current of wind-generated current restrained by coastlines or hindered by the Xisha and Nansha Islands (Zeng et al., 1989). The third theory, as proposed by Chao (1995), explains that the SCSWC is a return current, which is triggered by a reduction in the sea level gradient built up by the monsoon-driven southwestward coastal current along the northwestern boundary of the SCS. But the fact is often that the stronger the northeast wind, the stronger is the SCSWC (Guan, 1985). All of these impel us to put forward a new explanation.

We think the SCSWC might be a WBC in the local sea region. From the Hellerman and Rosenstein wind stress atlas (1983), the wind stress curl over the northern shelf of the SCS in winter is negative, though the wind is strong and directs southwestward. Negative wind stress curl can generate an anticyclonic circulation in an enclosed sea basin, so in the west of the sea, the current is northward. Because the main axis of the SCSWC is between 200m and 400m isobaths, which exactly corresponds to the continental shelf break, we think the shelf break might determine the position of the SCSWC. The b -effect does not play a key role because of the small horizontal scale of the SCS. In the present work, the BT4 for C1=0.3 is similar to that in the north of the SCS. The results here show the main axis of the WBC really moves toward the slope break. The role of the baroclinicity might not be the reason but rather the consequence of the SCSWC formation. It may indeed be helpful to strengthen the SCSWC. In future papers we will apply Eq.(2.9) to the whole SCS and check whether the SCSWC is manifest or not by inputting realistic wind stress and BT data.
 

ACKNOWLEDGMENTS
 
This work is jointly supported by the Space-time Structure of South China Sea Circulation and its Formation Mechanism (49636230) and the Study of the Relation between the Kuroshio Loop Current and Subtropical Current in the Luzon Strait (49476269) of the National Nature Science Foundation.]
 
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