916 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
The In¯uence of Negative Viscosity on Wind-Driven, Barotropic Ocean Circulation in a Subtropical Basin
DEHAI LUO AND YAN LU Department of Atmospheric and Oceanic Sciences, Ocean University of Qingdao, Key Laboratory of Marine Science and Numerical Modeling, State Oceanic Administration, Qingdao, China
(Manuscript received 3 April 1998, in ®nal form 12 May 1999)
ABSTRACT In this paper, a new barotropic wind-driven circulation model is proposed to explain the enhanced transport of the western boundary current and the establishment of the recirculation gyre. In this model the harmonic Laplacian viscosity including the second-order term (the negative viscosity term) in the Prandtl mixing length theory is regarded as a tentative subgrid-scale parameterization of the boundary layer near the wall. First, for the linear Munk model with weak negative viscosity its analytical solution can be obtained with the help of a perturbation expansion method. It can be shown that the negative viscosity can result in the intensi®cation of the western boundary current. Second, the fully nonlinear model is solved numerically in some parameter space. It is found that for the ®nite Reynolds numbers the negative viscosity can strengthen both the western boundary current and the recirculation gyre in the northwest corner of the basin. The drastic increase of the mean and eddy kinetic energies can be observed in this case. In addition, the analysis of the time-mean potential vorticity indicates that the negative relative vorticity advection, the negative planetary vorticity advection, and the negative viscosity term become rather important in the western boundary layer in the presence of the negative viscosity. Further, the in¯uence of the resolution and the negative viscosity intensity on the solutions are also examined.
1. Introduction served values or less (Munk 1950). However, as Liu The most striking characters of wind-driven ocean (1990) notes, the observed transport of the Gulf Stream circulation are the intensi®cation and separation of west- still cannot be explained, even if the nonlinear intertial ern boundary current and the establishment of recir- boundary layer is incorporated into the model (Charney culation (Stommel 1948; Munk 1950; Ierley 1987; Moro 1955; Morgan 1956). It can be concluded that the de- 1988; Cessi and Thompson 1990; Haidvogel et al. 1992; ®ciency of these models in modeling the total transports Denng 1993; Chao et al. 1996; OÈ zgoÈkmen et al. 1997). of the Gulf Stream and Kuroshio might arise from the To explore these problems, a lot of numerical investi- neglect of the negative viscosity (the feedback of the gations on wind-driven ocean circulation models had subgrid-scale eddies on the mean ¯ow) (Webster 1961). been made (Bryan 1963; BoÈning 1986; Moro 1988; Ier- More recently, Verron and Blayo (1996) pointed out that ley and Sheremet 1995; Kamenkovich et al. 1995; Sher- in previous models, the model predictions are somewhat emet et al. 1995; Bryan et al. 1995; OÈ zgoÈkmen et al. weak with regard to what we know of the North Atlantic 1997). One of the most important theories of the large- Ocean, especially in Gulf Stream area. In these models, scale ocean circulation is to explain the very large trans- the subgrid-scale eddies were usually parameterized as port measured in the western boundary currents. The turbulent viscosity, which is considered as the dissi- total Sverdrup transport is only about 30 Sv (Sv ϵ 106 pation of the mean motion (Munk 1950). Actually, the m3 sϪ1), while the observed total transport in the western mean motion obtains energy from the eddies in part boundary current can reach more than 100 Sv down- (Webster 1961), which is virtually similar to the at- stream (Holland 1973; Liu 1990; Hogg and Johns 1995). mospheric negative viscosity process (the feedback of In the early linear, frictional models, the theoretically the synoptic-scale eddies on the large-scale waves) ®rst computed transport values are almost half of the ob- proposed by Starr (1968). Unfortunately, the traditional parameterization scheme for subgrid-scale eddies can- not re¯ect this effect (Munk 1950). Though some of the eddies can be resolved in the high-resolution general Corresponding author address: Professor Dehai Luo, Department circulation model (BoÈning and Budich 1992; Bleck et of Atmospheric and Oceanic Sciences, Ocean University of Qingdao, È Qingdao, 266003, China. al. 1995; Chao et al. 1996; OzgoÈkmen et al. 1997), the E-mail: [email protected] contribution of the subgrid-scale eddies to the mean ¯ow
᭧ 2000 American Meteorological Society
Unauthenticated | Downloaded 09/29/21 11:09 AM UTC MAY 2000 LUO AND LU 917 uЈЈץuЈuЈץ h ץ uץ -still cannot be re¯ected completely. In this case, subgrid (١u Ϫ f ϭϪg ϩϪx ϩ , (1´scale eddies should be parameterized in high-resolution ϩ V yץ xץxHץ tץ .models as well
ЈЈץЈuЈץ h yץ ץ In a barotropic wind-driven circulation model, Moro (١ ϩ fu ϭϪg ϩϪ ϩ , (2´examined the importance of negative viscosity ϩ V (1988) yץ xץ yHץ tץ in the intensi®cation and separation of the boundary ץ uץ current and the establishment of the recirculation. In his ϩϭ0, (3) yץ xץ model the negative viscosity was simply regarded as a negative eddy viscosity coef®cient (Welander 1973). Strictly speaking, this assumption is incorrect (Marshall where V ϭ iu ϩ j is the mean velocity, (uЈ, Ј) is the 1981). Moreover, the western boundary current cannot ¯uctuating velocity, f is the Coriolis parameter, h the reach a statistically steady state if the above assumption ocean surface height measured from the ocean surface is allowed. Different from the assumptions made by at rest, H the ocean mean depth, g the gravitational Welander (1973) and Moro (1988), the role of negative acceleration, ( x, y) the wind stress, and ץץ -viscosity in the western boundary current can be re ϭ i ϩ j ١ ected if the second-order term is incoporated into the¯ yץ xץ -Prandtl mixing length theory in developing the param eterization of subgrid-scale eddies (Liu and Liu 1995). For such a parameterization, the contribution of subgrid- the gradient operator. scale eddies to the mean ¯ow may be in part re¯ected A crucial problem is how to deal with the time av- in the ocean circulation model. However, if additional eraging of the ¯uctuating quantities such as ϪuЈЈ in factors are gradually involved into this model, the ob- Eqs. (1) and (2). In atmospheric and oceanic motions, tained results may be to take a step by step approach the eddy viscosity effect of the ¯uctuating quantities on to the observed values. For example, when topography the mean motion (V ϭ iu ϩ j) was only emphasized, similar to that of Denng (1993) and OÈ zgoÈkmen et al. but the effect of the negative viscosity on the mean ¯ow (1997) is used, the ability of the present model in mod- was usually ignored when applying Prandtl's mixing eling the Gulf Stream transport and separation may be length theory. Recently, by considering the role of the improved considerably. However, as the ®rst step of our negative viscosity Liu and Liu (1995) investigated at- research we only use a barotropic model without to- mospheric balance motions of the planetary boundary pographic forcing to examine the contribution of the layer and obtained a satisfactory result. In the present negative viscosity to the western boundary current and paper, similar to Liu and Liu (1995), we will apply the recirculation so that the physical signi®cance of the Prandtl's mixing length theory to simplify Eqs. (1) and negative viscosity can be understood. The present model (2), assuming that a lateral mixing length lЈϭx Ϫ x 0 may serve as another theory that offers a possible mech- exists in which the properties of the ¯uid particle are anism for the transport of the western boundary current conservative before arriving at x ϭ x 0 and it interacts and the enhanced recirculation. with another ¯uid particle by turbulent exchange once The present paper is organized as follows: In section it shifts the length lЈ. The average velocities at x ϭ x 0 2, we present a new barotropic wind-driven ocean cir- and x ϭ x are assumed to be (x 0) and (x), respectively. culation model including negative viscosity by using a When the ¯uid particle at x ϭ x 0 arrives at x ϭ x, the modi®ed Prandtl mixing length theory (Liu and Liu ¯uctuating velocity caused by the turbulent mixing may 1995). The analytical solution of the linear Munk model be assumed to be Јϭ(x 0) Ϫ (x) traditionally. Using with weak negative viscosity is obtained in section 3. a Taylor series expansion, it can be approximately ex- In section 4 we present the numerical results of the fully pressed as nonlinear Munk model. In addition, the in¯uence of the 2ץ 1ץ model resolution and negative viscosity intensity on the ЈϭϪ lЈϪ lЈϩ2 ´´´. (4) x2ץ x 2ץ solutions are discussed, and the spatial distribution of the eddy energy associated with the mean ¯ow is also x is only retained in (4), itץ/ץ examined. In section 5 we present the analysis of the Naturally, if the term is the so-called traditional Prandtl's mixing length the- time-mean potential vorticity. The conclusions are given 2 2 x is included, itץ/ ץ in section 6. ory. However, if the second term will be a modi®cation of the traditional Prandtl's mixing length theory. Based on this idea, when the preceding 2. The formulation of the modi®ed barotropic two terms in the right-hand side of (4) are retained, we model obtain 2ץ 1ץ In a uniform depth ocean, when the wind stress is considered as a body force, the averaged equations of ϪЈuЈ ഠ lЈuЈϩlЈ2uЈ . (5) x2ץ x 2ץ the shallow water motion without bottom friction and topography can be written in the form The other terms uЈuЈ, uЈЈ and ЈЈ in Eqs. (1) and
Unauthenticated | Downloaded 09/29/21 11:09 AM UTC 918 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 bץ aץ bץ aץ -can be made likewise. If the difference of the ¯uc (2) J(a, b) ϭϪ, xץ yץ yץ xץ .tuating quantities in various direction is ignored, Eqs (1) and (2) can be rewritten as 2222ץץץץ ϭ4 ϩ ϩ , andٌ 22 u 2222 ץ u ץ h xץ uץ yץ xץ yץ xץ١u Ϫ f ϭϪg ϩϩAH ϩ ´ ϩ V yץx22ץxH ץ tץ 3322ץץץץ .١5 ϭϩ ϩ u ץ 33uץ yץ xץ yץx3322ץϪ ␥ ϩ , (6) yץx33ץ 22 ץ ץ h yץ ץ -5 | must be re ١␥| Note that the relation |A ٌ 4 | k ١ ϩ fu ϭϪg ϩϩAH ϩ H ´ ϩ V -y quired so that the Taylor series expansion in (4) is tenץx22ץyH ץ tץ able. ץ 33ץ Ϫ ␥ ϩ , (7) If is scaled with L, the time t is scaled with 1/(L) 33 y and x and y are scaled with L, while x is scaled withץ xץ HLU (the Sverdrup relation; see Pedlosky 1996), we where AH ϭ lЈuЈ is the horizontal eddy viscosity co- 1 2 obtain ef®cient, and ␥ ϭϪ2 lЈ uЈ is the eddy dispersion co- ef®cient, which represents the role of the negative vis- ץ 2 ␦ ץ cosity for ␥ Ͻ 0. Though the nondimensional eddy vis- ٌ2 ϩ I J(, ٌ2) ϩ xץ tLץ cosity coef®cient is required to be much greater than the nondimensional eddy dispersion coef®cient in Eqs. 34 ␥␦ ␦ץ (١5, (9 ␣ the dispersion term ␥ cannot be neglected. As ϭϪxm ϩ ٌ4 ϩ ,(7)±(6) yL Lץ we will see later, the in¯uence of the negative viscosity term for ␥ Ͻ 0 on the western boundary current is found 1/2 1/3 1/4 to be noticeable in the ®ne-grid model even if it is where ␦I ϭ (U/) , ␦m ϭ (AH/) and ␦␥ ϭ (|␥ |/) . chosen to be rather small. Liu and Liu (1995) had shown Note that we de®ne ␣ ϭ 1.0, ␣ ϭϪ1.0, and ␣ ϭ 0, that the mean kinetic energy is transformed into the eddy respectively, when ␥ Ͻ 0, ␥ Ͼ 0, and ␥ ϭ 0. Here ␦m kinetic energy when ␥ Ͼ 0. This process is called the is the thickness of the so-called Munk boundary layer, positive dispersion process. For this case, the role of the ␦I is the thickness of the inertial boundary layer (Ped- losky 1996), and ␦␥ is called the thickness of the neg- term ␥ is almost the same as the term AH. However, when ␥ Ͻ 0, the eddy kinetic energy may be transformed ative viscosity layer (or the eddy dispersion layer). As into mean kinetic energy. Such a process is called the pointed out by many investigators (Pedlosky 1979, negative viscosity process. It should be pointed out, 1996; Ierley and Sheremet 1995), a very dif®cult prob- however, that the term ␥ should be chosen to be a neg- lem is how to determine the lateral viscosity coef®cient 2 (A ) in the large-scale ocean circulation models. Gen- ative value in terms of ␥ ϭϪlЈ uЈ/2 because AH ϭ lЈuЈ H is usually assumed to be positive in the ocean ¯uid. In erally speaking, parameter AH cannot be rigorously de- previous papers, the negative value of the eddy viscosity termined. Naturally, parameter ␥ is also uncertain. For this reason, in this paper we would rather investigate coef®cent AH (␥ ϭ 0) was considered as the so-called negative viscosity (Welander 1973; Moro 1988). Strict- how the negative viscosity (␥ Ͻ 0) affects the wind- ly, this assumption is incorrect because AH is positive driven western boundary current than determine param- almost everywhere (Marshall 1981, 1984). In the pre- eters AH and ␥. sent paper, this assumption is unnecessary if the term For a symmetric subtropical gyre wind stress forcing, ␥ Ͻ 0 is allowed in the model. For simplicity we assume Eq. (9) can be written as that there is only a zonal component of the wind stress 2 ץ ␦ ץ x, y) ϭ ( x(y), 0) (Munk 1950; Ierley and Sheremet) 2 ϩ I J(, ٌ2) ϩٌ xץ tLץ .(1995 We introduce the streamfunction in Eq. (3) so that 34 ␥␦ ␦ ,x. In this caseץ/ץy and ϭץ/ץit can satisfy u ϭϪ (١5. (10 ␣ ϭϪsin(y) ϩٌm 4 ϩ we can obtain from Eqs. (6) and (7) LL ץץ 22 - ϩ J(, ٌ ) ϩ  When ␥ ϭ 0(␦␥ ϭ 0), Eq. (10) reduces to the non ٌ -x linear Munk models of the wind-driven ocean circulaץ tץ - tion extensively studied by previous investigators (Bryץ 1 x 45 ., (8) an 1963; Ierley and Sheremet 1995; Sheremet et al ١␥ ϭϪ ϩAHٌ Ϫ yץ H 1997). The present model is con®ned in a square region where  is the meridional variation of the Coriolis pa- 0 Յ x Յ 1; 0 Յ y Յ 1.0, which simulates a region of rameter, the subtropical gyre formation.
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4 43 3. The modi®ed linear Munk model and its where (␦␥/␦m) ϭ ␦␦␥ k/( mL) is a measure of the strength analytical solution of the negative viscosity. In this section, in order to study the effect of the Strictly speaking, it is rather dif®cult to obtain an negative viscosity on the structure of the western bound- accurate solution of Eq. (12). However, if parameter 4 ary current, as a simple example we deal with the linear (␦␥/␦m) is small, then the perturbation expansion meth- Munk model (Munk 1950). In this model, the nonlinear od can be applied to solve this equation. If the solution term in the boundary layer is ignored. Consequently, to Eq. (12) can be expanded as a power series of [ 4 when the nonlinear term in Eq. (10) is ignored, for ϭ (␦␥/␦m) ] steady ¯ow the equation reduces to 2 34 ϭ 0 ϩ1 ϩ 2 ϩ ´´´, (13) ␥␦␦ץ (١5. (11 ␣ ϭϪsin(y) ϩٌm 4 ϩ xLL then when (13) is substituted into Eq. (12), we obtainץ 1ץ 4ץ ,(In order to obtain the analytical solution of Eq. (11 00Ϫϭsin(y), (14a) xЈ k 4ץxЈץ 5 may be considered as a perturbation term ١ the term of the linear Munk model if the negative viscosity is ץ ץ 45ץ weak. In deriving the analytical solution of Eq. (11), 11ϪϭϪ␣ 0, (14b) xЈץ45xЈץxЈץ the boundary conditions considered here are still the same as those used by Munk (1950). On the other hand, ץ ץ 45ץ to emphasize the importance of the negative viscosity, 22ϪϭϪ␣ 1. (14c) xЈץ45xЈץxЈץ the subboundary layer has been ignored in the western boundary current. It should be pointed out that when higher order terms When ␦␥ ϭ 0, Eq. (11) is the well-known linear Munk model. In this case, Eq. (11) can be naturally considered n (n ϭ 1,2,´´´) in(13) are retained, the series ex- as an extension of the linear Munk model, which is pansion solution (13) is also correct even if is not called the modi®ed linear Munk model. In most previous very small. model studies, the lateral viscosity coef®cient AH was According to Munk et al (1950) and Sheremet et al. assumed to be varied over several orders of magintude (1995), the solution to Eq. (14a) can be easily expressed (Moro 1988; Kamenkovich et al. 1995). In this section, as in order to allow a comparison between the analytical Munk solutions without and with negative viscosities, xЈ ͙3 1 ͙3 Ϫ(xЈ/2) A may be chosen to be ®xed. For example, ␦ may be 0 ϭ 1 ϪϪe cos xЈϩ sin xЈ H m []k 22͙3 chosen to be ␦m ϭ L/20. (y can be generally sat- ϫ sin(y). (15ץ/ץ x kץ/ץ Since the relation is®ed in the western boundary layer, when the stretched boundary layer variable xЈϭkx (k ϭ L/␦m) is used, Eq. Equation 15 is the well-known Munk solution. Once we (11) can be rewritten as substitute (15) into Eq. (14b), its solution can be de-
454 termined. Also, we can determine the solution for 2 in 1ץЈ ץ ␥␦ ץ ϩ ␣ Ϫϭsin(y), (12) terms of Eq. (14c). In such way, the solutions of all the 45 xЈ k equations can be obtained asץxЈץ xЈ ␦mץ
␣ ͙3 1 ͙32͙3 ͙3 Ϫ(xЈ/2) Ϫ(xЈ/2) 1 ϭϪ xЈe cos xЈϪ sin xЈϩ␣ sin xЈe sin(y), (16a) []32͙3 292 2␣2 ͙3 112͙3 ͙3 ␣2 Ϫ(xЈ/2) 2 2 ϭ e Ϫ(1 ϩ xЈ) cos xЈϩ (1 Ϫ xЈ) sin xЈϩxЈ sin xЈϩ sin(y). (16b) Ά·92[ ͙3 229͙3 ]
Obviously, Eqs. (16a,b) are the ®rst- and second-order rather convenient to start with the case of ®xed ␦m and modi®cations of the linear Munk solution, which re¯ects varying ␦␥. Because the solution to Eq. (12) only con- the contribution of the subgrid-scale eddies to the west- tains parameter , it can be considered as a free param- ern boundary current. Webster (1961) observed that the eter. As an example, we may choose parameter ␦m ϭ transfer of kinetic energy from the meander (eddies) to L/20 (k ϭ 20). Figure 1 shows the streamfunction so- 2 the mean ¯ow is rather strong. In other words, the eddies lutions for ( ഠ 0 ϩ1 ϩ 2) of Eq. (12) for can strengthen the western boundary current. Perhaps, three cases ϭ0 (without negative viscosity), (, ␣) (13) can indicate this point. To illustrate the dependence ϭ (0.4, 1) (negative viscosity) and (, ␣) ϭ (0.4, Ϫ1) of the modi®ed Munk solution on parameter ␦␥,itis (positive viscosity).
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2 FIG. 1. The isoline patterns of streamfunction solutions ( ഠ 0 ϩ1 ϩ2) for the modi®ed linear Munk model (12) for ϭ0.4 (a) without negative viscosity (␣ ϭ 0), (b) with negative viscosity (␣ ϭ 1), and (c) with positive viscosity (␣ ϭϪ1). Contour interval (CI) corresponds to 0.1.
It is found from Fig. 1 that when ϭ0, the stream- sent the the northward velocity [V(x)] of the western function solution is similar to the linear Munk solution boundary current in Fig. 2 for the three cases above, in ,xЈϭV(x) sin(y) is de®nedץ/ץx ϭ kץ/ץobtained by Sheremet et al. (1997). However, when which ϭ ϭ 0.4, the intensi®cation of the northward western where is the northward velocity in the western bound- boundary current can be detected. Even so, no recir- ary layer. It is not dif®cult to ®nd that the negative culation can be found in the northwest corner of the viscosity is able to strengthen the northward western basin in the absence of nonlinearity. In a nonlinear bro- boundary current, but slightly alter its width. In contrast, tropic model, Moro (1988) found that the existence of the positive viscosity can only make the western bound- the recirculation region is due to a weak dissipation. ary current weaker. In addition, we note that when ϭ 43 Furthermore, he found that the negative viscosity favors 0.4,␦␦␥ /( mL) ϭ 0.02 is found for k ϭ 20 (or ϭ 0.02 the establishment of recirculation and is essential for in the next section). On the other hand, when the neg- the separation of the boundary current. Actually, the ative viscosity is retained in the nonlinear Munk model, establishment of the recirculation is attributed to the how does the negative viscosity affect both the western nonlinearity, while the negative viscosity only favors it. boundary current and the recirculation? This problem However, for a positive viscosity the northward western deserves further study. In the next section, we will pre- boundary current weakens. In order to further under- sent the numerical solutions of Eq. (10) with the neg- stand the importance of the negative viscosity in the ative viscosity (␣ ϭ 1). intensi®cation of the western boundary current, we pre- 4. The modi®ed nonlinear Munk model and its numerical solution a. The ®nite difference scheme of the modi®ed nonlinear Munk model In the above section, we have presented the analyt- ical solution of the linear Munk model with negative viscosity. In this linear model, the recirculation cannot be detected. Perhaps, the negative viscosity is not es- sential for the establishment of the recirculation even though it can favor the recirculation. In this section, the problem we must answer is focused on studying how the negative viscosity affects both the western boundary current and the recirculation for different Reynolds numbers. It is convenient to introduce the Reynolds number Re for the boundary layer (Sheremet et al. 1997)
␦ 3 I 3/2 Re ϭϭR /EL, (17) FIG. 2. The pro®le of the meridional velocity V(x) for ϭ0.4. ␦m The long dashed curve corresponds to the case with negative viscosity 2 3 (␣ ϭ 1). The point dashed curve correspods to the case with positive where R ϭ (␦I/L) and EL ϭ (␦m/L) . viscosity (␣ ϭϪ1). The solid curve corresponds to the case without Since the structure of the solution of Eq. (10) mainly negative viscosity (␣ ϭ 0). depends on Reynolds number Re, in the real compu-
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tation holding R ®xed and varying EL would be more Note that the boundary conditions for the ®fth-order appropriate because the lateral friction coef®cient AH is derivatives considered here are added in comparison known very poorly (Sheremet et al. 1997). Certainly, with those used by Ierley and Ruehr (1986) and Ierley in some papers the inertial term R was assumed to be (1987). The ®fth-order equation has ®ve boundary con- varied, while the lateral friction coef®cient EL was con- ditions in the x or y direction. Because the ®eld be- sidered as a ®xed value (BoÈning 1986). However, be- comes rather weak near the eastern and southern bound- cause the lateral friction coef®cient A cannot be easily H aries, it is reasonable to only consider (18c,d) on the determined, the lateral friction coef®cient E may be L western and northern boundaries. considered as a free parameter. Similar to Sheremet et When 0 and / t 0, Eq. (10) reduces to the ϭ ץ ץ al. (1997), when holding R ®xed we can discuss the ␦␥ ϭ dependence of the solution of Eq. (10) on the Reynolds steady nonlinear Munk model studied by many inves- 4 tigators (Munk et al. 1950; Moore 1963; Ierley and number Re. In addition, because E␥ ϭ (␦␥/L) is un- certain, it is convenient to introduce a small parameter Ruehr 1986; Ierley 1987; Moro 1988). In some special
ϭ E␥/EL. If parameter EL is prescribed, then E␥ can cases, the steady solution of this equation can be ob- be known when holding ®xed. Therefore, the Rey- tained analytically (Ierley and Ruehr 1986). Unfortu- nolds number Re can be considered as a varying pa- nately when the Reynolds number exceeds some certain rameter if and R are ®xed. On the other hand, it must value, the steady solution becomes unstable, and the be pointed out that in previous numerical studies the time-dependent solution is very chaotic (Moro 1988; slip and no-slip conditions were usually applied. There Kamenkovich et al. 1995; Sheremet et al. 1995). In the are two reasons for using the no-slip boundary condi- preceding section, we have obtained analytical steady tion. First, Kamenkovich et al. (1995) pointed out that solutions of Eq. (10) for three different cases in the the no-slip condition is more relevant for western boundary currents than the slippery condition from a absence of nonlinearity. Here, we will solve Eq. (10) physical standpoint. Second, Verron and Blayo (1996) numerically and discuss the dependence of the structure found that the no-slip boundary condition leads to a of the boundary layer solution on the Reynolds number satisfactory western boundary current separation. For Re in the presence of negative viscosity. these reasons, in the present paper we only use the no- In previous studies, ®nite-difference schemes were slip conditions to solve equation (10) (Ierley and Ruehr proposed to solve the two-dimensional, wind-driven 1986; Ierley 1987). Of course, we should also examine ocean circulation models (Bryan 1963; BoÈning 1986; how the negative viscosity affects the western boundary Foreman and Bennett 1988; Moro 1988; Denng 1993; current under slippery boundary condition. This case Kamenkovich et al. 1995). In this paper, a ®nite-differ- will be discussed in another paper. ence scheme is constructed to solve Eq. (10) numeri- In the square domain considered here, the boundary cally, in which the continuous variables are replaced by conditions are (Bryan 1963; Ierley and Ruehr 1986; Ierley 1987) the discrete variables, x ϭ (i Ϫ 1)⌬x (i ϭ 1,2,´´´, I), y ϭ (j Ϫ 1)⌬y (j ϭ 1,2,´´´,J), and t ϭ n⌬t (n ץ ϭ 0, ϭ 0atx ϭ 0, x ϭ 1, (18a) ϭ 1,2,´´´,N). The ®nite-difference form of Eq. (10) x can be written asץ ץ ϭ 0, ϭ 0aty ϭ 0, y ϭ 1, (18b) nϩ1 nn i,jץy i,jϪ i,jץ ϩ2 RJ( n , ٌ2 n ) ϩٌ xץ 4 ⌬t i,j i,jץ ϭ 0atx ϭ 0orx ϭ 1, (18c) x4 4 n 5 nץ (i,j, (19 ١␥ϭϪsin[⌬y( j Ϫ 1)] ϩ ELi,jٌ ϩ E 4ץ ϭ 0aty ϭ 0ory ϭ 1. (18d) y4 whereץ
nnϪ nץnnϪ nץ ⌬x ϭ⌬y ϭ D, i,jϭ iϩ1,jiϪ1,ji,ji,j, ϭ ϩ1 i,jϪ1 , y 2⌬yץ x 2⌬xץ
nnnn n i,jϩ1 ϩ i,jϪ1 ϩ iϩ1,jiϩ Ϫ1,ji,jϪ 4 , 2 n ϭٌ i,j D2
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FIG. 3. The isoline patterns of time-averaged streamfunction without negative viscosity for several Reynolds numbers: Re ϭ 0.1, 0.5, 1, 2, 5, 10. CI ϭ 0.1.
nnnn Ϫ1,jץiϩ1,jiץ i,jϪ1ץ i,jϩ1ץ 2 n Ϫٌ2 n Ϫٌ2 n ϩٌ2 nٌ yץ y Ϫ1,jץx iϩ1,jiץ x i,jϪ1ץ i,jϩ1 , J( n , ٌ2 n ) ϭ , ٌ4 nnnϭ Px ϩ Py i,j i,j 2D i,j i,j i,j 22n 2 n 2 n 2 n 22n 2 n 2 n 2 n i,jٌ i,jϩ1 ϩٌi,jϪ1 Ϫ 2ٌ i,jٌץi,jٌ iϩ1,jiϩٌ Ϫ1,jϪ 2ٌ i,jٌץ Pxn ϭϭ , Pyn ϭϭ , yD 22ץ xD22i,jץ i,j nnnn Pxiϩ1,jiϪ Px Ϫ1,ji,jϩ Py ϩ1 Ϫ Pyi,jϪ1 . ١5 n ϭ i,j 2D
Equation (19) can be solved with a SOR method if preserving computational ef®ciency. This size does not is speci®ed at all points initially and at the boundaries affect substantially the system's nonlinear dynamics be- for each subsequent time step (Denng 1993). Note that cause both the nonlinear and negative viscosity terms x 2 at the western and eastern bound- themselves are almost negligible outside the boundaryץ/2ץ the values of aries are prescribed according to the difference scheme current regions (Jiang et al. 1995). Initially, we consider proposed by Bryan (1963). In the present paper, we a state of rest. The response of this system to the wind carry out a series of numerical experiments holding pa- stress can be determined by the numerical method and rameter R ®xed while varying the Reynolds number Re its numerical results are reported in detail in the fol- (Kamenkovich et al. 1995). Without the loss of gener- lowing several subsections. ality, we choose R ϭ 10Ϫ3 (the forcing of ®xed wind Ϫ3 stress), which is slightly smaller than R ϭ 1.28 ϫ 10 b. The numerical results of model taken by Bryan (1963). In addition, the time and space steps of the model integration are chosen to be ⌬t ϭ In this section, to further examine the importance of 0.025 and ⌬x ϭ⌬y ϭ D ϭ 1/(I Ϫ 1) (I ϭ J ϭ 61). In the negative viscosity, we shall give the numerical so- this paper, we choose a smaller basin size, similar to lutions of Eq. (10) for two cases without and with neg- that of Holland (1978), to maximize resolution while ative viscosity, respectively. Because both EL and E␥
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FIG. 4. As Fig. 3 but for case with negative viscosity ( ϭ 0.02). CI ϭ 0.1. are unkown and R is easily determined from observa- lations (basin modes) generated by the initial impact of tions, we can know E␥ through ϭ E␥/EL when holding the wind stress. For such a long time-averaging the basin ®xed if EL can be determined. Generally speaking, modes can almost be ®ltered out. Figure 3 shows the because E␥ is rather small compared to EL in Eq. (10), time-averaged streamfunction ®elds without negative must be considered as a small free parameter. As we viscosity ( ϭ 0.0) for several Reynolds numbers, Re shall see later, the negative viscosity is found to have ϭ 0.1, 0.5, 1, 2, 5, and 10. a remarkable in¯uence on the western boundary current For Re ϭ 0.1 the solution is very similar to the linear and the establishment of recirculation even if parameter Munk solution (Munk 1950; Pedlosky 1996). However, is chosen to be rather small. In future work, an im- when Re ϭ 0.5 nonlinearity breaks the symmetry of the perative probelm is how to determine the negative vis- circulation relative to y ϭ 0.5 and the maximum of the cosity parameter E␥ from the real observational data. streamfunction is displaced to the north of y ϭ 0.5. For The discussion of this problem is beyond the scope of Re ϭ 1.0 a weak recirculation appears in the northwest our study. In this paper, we may choose ϭ 0.02 or corner of the basin. However, when the Reynolds num- ϭ 0.04 as a measure of the negative viscosity. Though ber increases further, the size and intensity of the re- this parameter is chosen to be rather small, its role can- circulation gyre will increase (Re ϭ 2.0, 5.0, 10), and not be neglected, especially in the ®ner-grid model. it ®nally ®lls the whole basin for Re → ϱ (®gures omit- However, the choice of ϭ 0.1 cannot keep the model ted). On the other hand, we ®nd that the recirculation stable for a very long time integration when Re is rather still can be established in the northwest corner of the large. This may be why parameter is assumed to be basin for slightly large Reynolds numbers even if the very small. negative viscosity is neglected. Naturally, the nonline- arity is likely to be one of the dominant factors for the onset of recirculation gyre (Ierley and Ruehr 1986; Ier- 1) THE ANALYSES OF THE TIME-AVERAGED ley 1987). Under the same conditions as in Fig. 3, Fig. STREAMFUNCTION FIELDS 4 shows the case with negative viscosity for ϭ 0.02. In this subsection, we present the numerical solutions It is found from this ®gure that for Re ϭ 0.1, when the of model (10) without and with negative viscosity, re- negative viscosity is included, the streamlines in the spectively. The time-dependent solutions are averaged western boundary current become more crowded than over a suf®ciently large time interval ⌬t ϭ t2 Ϫ t1 ϭ in Fig. 3. This indirectly indicates further intensi®cation 850 (t1 ϭ 400, t 2 ϭ 1250) so as to ®lter out the oscil- of the western boundary current by the negative vis-
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FIG. 5. The total transport of time-averaged western boundary current Q(y) as a function of latitude y for several values of Re ϭ 0.1, 0.5, 1, 2, 5, 10. The solid curve corresponds to the case without negative viscosity. The dashed curve corresponds to the case with negative viscosity (␣ ϭ 1, ϭ 0.02).
cosity. In addition, we observe that the negative vis- 2) THE TOTAL TRANSPORT OF THE TIME-MEAN cosity favors the establishment of the recirculation in WESTERN BOUNDARY CURRENT the northwest corner and slightly increases the penetra- To highlight the signi®cance of negative viscosity in tion scale of the jet near the northern boundary. As the total transport of the western boundary current, in shown by Marshall and Marshall (1992), the enhanced this subsection we will give the total transport of the recirculation can increase the zonal penetration scale of time-mean western boundary current (x, y) for differ- the northern boundary current. To some extent, the neg- ent Reynolds numbers. According to Sheremet et al. ative viscosity can be seen as a possible mechanism for (1995), the total transport of the western boundary cur- the enhanced recirculation. This result was ®rst sug- rent can be de®ned as gested by Moro (1988), who found that the establish- ment of recirculation requires negative viscosity. How- x0 ever, even if the negative viscosity enhances the recir- Q(y) ϭ ͵ (x, y) dx, (20) culation in the northwest corner of the basin, it is not 0 essential for the establishment of recirculation. Only the where x 0 ± 0 is the ®rst point at which (x, y) is zero, nonlinearity is essential for the establishment of the re- and the integral is taken from the western coast (x ϭ circulation (Ierley 1987). Therefore, it can be concluded 0) to x 0. that the negative viscosity may strengthen the western In the wind-driven ocean circulation, an important boundary current and the recirculation. More recently, problem is the so-called intensi®cation of western Verron and Blayo (1996) pointed out that in the prevoius boundary current. Here, the value of Q(y) is considered models, the model predictions are too weak with regard as a measure of the western boundary current intensi- to what we know of the North Atlantic Ocean, especially ®cation. in the Gulf Stream area. The jet penetration and total For the numerical model considered here (61 ϫ 61), transport are too low with respect to observations even the total transports of the time-mean western boundary though the Gulf Stream separation is satisfactory. Based currents without ( ϭ 0.0) and with ( ϭ 0.02) negative on such facts, it is not dif®cult to obtain such a con- viscosity are shown in Fig. 5. clusion that the failure of the previous model predictions We can see in Fig. 5 that the maximum transport of might arise out of the lack of the negative viscosity in the western boundary current without negative viscosity the wind-driven ocean circulation models. does not increase noticeably with the increasing of Re.
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FIG. 6. Time variation of the total kinetic energy in the whole basin for several Reynolds numbers: Re ϭ 0.1, 0.5, 1, 2, 5, 10. The dashed curve corresponds to the case with negative viscosity. The solid curve corresponds to the case without negative viscosity.
In particular, the variation of the maximum transport is modeling realistic Gulf Stream separation and behavior rather small in the range from Re ϭ 2.0 to Re ϭ 10. It (Verron and Blayo 1996). However, the theoretical is very interesting that one can ®nd Q(y) ϭ 0 near y ϭ transports are always less than the observed estimates. 0.94 for Re ϭ 2ϳ5, but Q(y) ± 0atReϭ 10 in the This indirectly indicates that lack of the negative vis- interval of 0 Ͻ y Ͻ 1.0. If the point of Q(y) ϭ 0is cosity may be one of the reasons that various models seen as a separation point of the western boundary cur- cannot correctly represent the transports. rent, the jet separation (premature separation) from the western boundary can be observed (Haidvogel et al. 3) THE TIME VARIATIONS OF THE KINETIC 1992). However, this behavior cannot be observed in a coarse-grid model even if the negative viscosity is re- ENERGIES IN THE WHOLE BASIN AND IN THE tained (not shown). As demonstrated by many investi- WESTERN BOUNDARY LAYER gators, in some numerical models the low grid resolution In this subsection, to explore the contribution of neg- has been shown to affect the jet separation from the ative viscosity to the western boundary current, we western boundary to some degree (Bryan and Holland should see the time variations of the kinetic energies in 1989). In addition, it can be shown that the negative the whole basin and in the western boundary layer, re- viscosity increases the maximum transport of western spectively. Here, we de®ne boundary current. For Re ϭ 2ϳ5, the negative viscosity 11 is found to enhance the jet separation from the western E (t) ϭ (u 22ϩ ) dx dy (21) boundary, and to have some in¯uence on the separation T ͵͵ 00 latitude of western boundary current. Thus, we conclude that the jet separation requires the ®ne-grid resolution 1 x0 E (t) ϭ (u 22ϩ ) dx dy (22) and negative viscosity. Holland (1967) and Liu (1990) W ͵͵ found that the Gulf Stream transport is strongly related 00 to topographic effects, and suggested that the topgraphic as the kinetic energies in the whole basin and in the effect could be seen as a possible mechanism for the western boundary layer respectively, where x 0 is the enhanced transport of the western boundary current and same as in (20). the enhanced recirculation. In this paper, the negative For ϭ 0 and ϭ 0.02, the time variations of the viscosity term incorporated in Eq. (10) is seen as a total kinetic energies in the whole basin and in western tentative subgrid-scale parameterization. This term is boundary layer are shown in Figs. 6 and 7, respectively. found to have a noticeable effect on the western bound- Figure 6 shows the time variations of the total kinetic ary current. Thus, the negative viscosity proposed here energies in the whole basin for ϭ 0 and ϭ 0.02. may be regarded as a possible mechanism for the en- It is found from Fig. 6 that in the absence of negative hanced western boundary current, recirculation and jet viscosity the total kinetic energy in the whole basin penetration near the northern boundary. In a ®ne-grid increases with time and reaches a steady state rapidly model BoÈning and Budich (1992) found a tendency for for Re ϭ 0.1ϳ1.0. However, when Re exceeds 2.0, the energy transfer preferentially from eddies to the mean total kinetic energy exhibits either periodic or aperiodic ¯ow, while the use of the coarse-grid model leads to oscillation depending on the Reynolds number. Even so, severe distortion of the mean ¯ow. The accuracy of the these total kinetic energies are still found to oscillate numerical scheme for the no-slip boundary condition near an equilibrium state. However, in the presence of has been shown to constitute one of the ingredients for negative viscosity the total kinetic energy in the whole
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FIG. 7. Time variation of the total kinetic energy in the western boundary layer for several Reynolds numbers: Re ϭ 0.1, 0.5, 1, 2, 5, 10. The dashed curve corresponds to the case with negative viscosity. The solid curve corresponds to the case without negative viscosity.
FIG. 8. The time-averaged total transports and streamfunction ®elds of the western boundary currents without negative viscosity and with varying negative viscosity for grids 31 ϫ 31 and 61 ϫ 61. (a) The total transport for grids 31 ϫ 31, where the solid curve indicates the case without negative viscosity, the long dashed curve corresponds to ϭ 0.02, and the point dashed curve corresponds to ϭ 0.04. (b) The time-mean streamfunction ®eld for grids 31 ϫ 31 with negative viscosity ( ϭ 0.04), CI ϭ 0.1. (c) Same as Fig. 8a but for grids 61 ϫ 61; (d) same as Fig. 8b but for grids 61 ϫ 61.
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FIG. 9. The horizontal plots of the mean and eddy kinetic energies for the cases without negative viscosity and with negative viscosity ( ϭ 0.04) in a ®ne-grid model (61 ϫ 61). (a) Mean kinetic energy without negative viscosity (CI ϭ 10). (b) Eddy kinetic energy without negative viscosity (CI ϭ 2). (c) Mean kinetic energy with negative viscosity (CI ϭ 10). (d) Eddy kinetic energy with negative viscosity (CI ϭ 2). basin is much larger than that without negative viscosity. Re ϭ 5.0ϳ10.0 the total kinetic energy of the western Within the range of Re ϭ 0.1ϳ0.5 the total kinetic boundary current with negative viscosity is sometimes energy goes into a stationary state that has the same less than that without negative viscosity, but its mean character as that without negative viscosity, but for Re value is still larger than that without negative viscosity. ϭ 1.0ϳ2.0 the total kinetic energy exhibits an oscil- This indirectly con®rms that the negative viscosity fa- lation that has a statistically steady state. When Re ex- vors the intensi®cation of western boundary current. ceeds 5.0, the total kinetic energy seems to be ever Because the negative viscosity proposed here can rep- increasing in the time integration interval (t ϭ n⌬t, n resent the contribution of subgrid-scale eddies to the ϭ 50 000) considered here. Probably, it can reach a western boundary current, it may be seen as a tentative statistically steady state through a very long time in- parameterization of subgrid-scale eddies. tegration. This problem is beyond the scope of our study here. It appears, therefore, that the negative viscosity 4) THE IMPACT OF THE RESOLUTION AND NEGATIVE (the role of subgrid-scale eddies) has an important effect on the total kinetic energy in the whole basin. Figure 7 VISCOSITY ON THE WESTERN BOUNDARY shows the time variations of the total kinetic energies CURRENT without and with negative viscosity in the western In this subsection, in order to further show the in¯u- boundary layer. It is obvious that for Re ϭ 0.1ϳ2.0 the ence of negative viscosity on the western boundary cur- total kinetic energy without negative viscosity in the rent, the comparisons between the results obtained in western boundary layer gradually goes into a stationary the coarse-grid (31 ϫ 31) and ®ne-grid (61 ϫ 61) mod- state. But in the range of Re ϭ 3.0ϳ10.0 it exhibits an els should be made for different values of the negative aperiodic oscillation. Interestingly, in the presence of viscosity. For the two resolution models, Fig. 8 shows negative viscosity, for Re ϭ 0.1ϳ2.0 the total kinetic the numerical results for different values of the negative energy of the western boundary current is always larger viscosity. than that without negative viscosity. However, for For the coarse-grid case (31 ϫ 31), the total transports
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FIG. 10. The numerical results for grids 61 ϫ 61 without negative viscosity. (a) the time- mean relative vorticity (CI ϭ 100). Contributions of (b) RVA (CI ϭ 10), (c) BETA (CI ϭ 2), and (d) VISC (CI ϭ 5) to the time-mean potential vorticity budget. of western boundary currents for three cases of ϭ 0, western boundary current. For ϭ 0.04 the time-mean 0.02, and 0.04 are shown in Fig. 8a. It can be seen that stream function ®elds for the coarse-grid (31 ϫ 31) and for the coarse-grid case the maximum value of Q(y) ®ne-grid (61 ϫ 61) models are shown in Figs. 8b and corresponds in turn to 0.9, 1.0, and 1.2 when equals 8d, respectively. This comparison between the two ®g- 0.0, 0.02, and 0.04. This indicates that in the presence ures also indicates the importance of negative viscosity of negative viscosity the increase of the total transport in the enhancement of recirculation in the ®ne-grid mod- is not noticeable in a coarse-grid model. However, for el. Figure 9 shows the horizontal distributions of the the ®ne-grid model (61 ϫ 61) the increase of the total mean (Em) and eddy (Ee) kinetic energies in the ®ne- meridional transport becomes rather noticeable. For ex- grid model (61 ϫ 61) for both cases of ϭ 0 and 2 2 2 2 ample, when ϭ 0 and ϭ 0.02, the maximum value ϭ 0.04 [Em ϭ (u ϩ )/2 and Ee ϭ (uЈ ϩ Ј )/2 are of Q(y) corresponds to 1.1 and 1.45, respectively, but de®ned as the mean and eddy kinetic energies respec- it can reach 1.9 for ϭ 0.04. Therefore, the resolution tively, where the overbar and prime represent the time- has an important in¯uence on the importance of the mean and eddy components, respectively]. One can see negative viscosity in the intensi®cation of a western that in the absence of negative viscosity the mean kinetic boundary current. In particular, in the ®ner-grid model energy exhibits a narrow tongue that has a maximum the negative viscosity becomes more important. On the amplitude of 130.8 (Fig. 9a). But in the presence of the other hand, it can be found that in the coarse-grid model negative viscosity, this energy tongue becomes broader
Q(y) ϭ 0 does not exist between y ϭ 0 and y ϭ 1.0 for and extends toward the southeast where the Em maxi- the three cases discussed above. But for the ®ne grid mum value reaches 232.8 (Fig. 9c). It is, therefore, nat- model (61 ϫ 61) the locations of Q(y) ϭ 0 for both ural to draw a conclusion that the drastic increase of ϭ 0 and ϭ 0.02 are basically the same and near y ϭ the mean kinetic energy arises from the negative vis- 0.94. When ϭ 0.04, Q(y) ϭ 0 is located at y ϭ 0.88. cosity. Interestingly, we notice that in the absence of Therefore, the negative viscosity does also have some negative viscosity, the eddy energy is weak (Fig. 9b). in¯uence on the location of Q(y) ϭ 0. On the whole, However, in the presence of negative viscosity the eddy in the ®ner-grid model the negative viscosity can lead energy increases considerably and the eddy activity pre- to the drastic increase of the meridional transport of vails over a broad region in which highest Ee value
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FIG. 11. The numerical results for grids 61 ϫ 61 with negative viscosity ( ϭ 0.04). (a) the time-mean relative vorticity (CI ϭ 100). Contributions of (b) RVA (CI ϭ 10), (c) BETA (CI ϭ 2), (d) VISC (CI ϭ 5), and (e) NVT (CI ϭ 5) to the time-mean potential vorticity budget.
occurs in a core to the northern side of the Em maximum, of the time-mean vorticity balance equation. It would that is, in the region of high lateral mean shear. These be interesting to present all the balance terms in the results suggest that the negative viscosity might not only time-mean vorticity balance equation. We decompose increase the mean energy but also the eddy energy. This the dependent variable in Eq. (10) (␣ ϭ 1) into the point is easily explained. BoÈning et al. (1991) had dem- time-mean and eddy components, designated by an ov- onstrated that a source of eddy kinetic energy can be erbar and a prime, respectively. The time-mean form of provided by the barotropic instability, which was also the potential vorticity equation can then be written as suggested by Sheremet et al. (1995), who found that the (Haidvogel et al. 1992; OÈ zgoÈkmen et al. 1997) appearance of the eddies in the boundary layer is a result ץץ -of an instability of the boundary current for high Rey 222 ϭϪRJ(, ٌ ) Ϫ RJ(Ј, ٌ Ј) Ϫٌ xץ tץ nolds numbers. In this paper, it has been shown that the negative viscosity can lead to an energy transfer from ABC the eddies (unresolved) to the mean ¯ow (Webster E 45E sin( y), (23) Ϫ ١␥ In such a process, both the mean ¯ow and its ϩ Lٌ ϩ .(1961 lateral shear are intensi®ed by the eddies (®gures omit- DE F ted). However, as mentioned earlier, the barotropic in- stability occurs easily in the strong lateral shear region where A ϩ B is the time-mean relative vorticity ad- of the mean ¯ow and causes the growth of disturbances vection (RVA), C the planetary vorticity advection (resolved eddies) (Cessi and Ierley 1993). Therefore, it (BETA), D the vorticity dissipation by the lateral fric- is not surprising to obtain such a conclusion that the tion (VISC), E the negative viscosity term (NVT), and negative viscosity could not only increase the mean en- F the vorticity input by wind forcing. Note that the t ϭץ/2ٌץ ergy but also the eddy energy. statistically steady state is de®ned such that 0. However, the statistically steady state is not easily achieved if the negative viscosity is chosen to be slightly 5. The time-mean potential vorticity analysis larger. If the spatial distributions of all the balance terms In this section, we wish to examine how the negative in Eq. (23) can be obtained, one can detect the contri- viscosity affects the western boundary current in terms bution of the negative viscosity to the western boundary
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FIG. 12. As in Fig. 11, but for ϭ 0.02. current. For the same parameter conditions as in Fig. 9, negative VISC region is mainly located in the northwest Fig. 10 shows the spatial distributions of the time-mean corner of the basin. Further, when the negative viscosity relative vorticity, RVA, BETA, and VISC in the absence is retained, VISC in the western boundary layer be- of negative viscosity, while the numerical results for comes slightly stronger, but the basic pattern almost ϭ 0.04 and Re ϭ 2 are shown in Fig. 11. remains the same (Fig. 11d). Figure 11e shows a region Figure 10a indicates a strong positive vorticity region of strong NVT in the western boundary layer, similar near the western wall of the midbasin and a negative to the spatial distribution of RVA in Fig. 11b. It can be vorticity tongue in its east ¯ank. This tongue extends found that these negative RVA, BETA, and NVT terms toward the northern boundary. In the northwest corner will intensify the western boundary current. However, of the basin, the positive vorticity region is very narrow, we notice that the premise of the derivation (negative but in the presence of negative viscosity the positive viscosity term small relative to lateral eddy viscosity) vorticity near the western wall becomes very strong, does not hold in the western boundary layer if ϭ 0.04 and a weak negative vorticity region begins to appear is allowed. Mathematically, the model proposed here in the northwest corner (Fig. 11a). In addition, we ob- cannot be acceptable. However, this model is permitted serve that the negative vorticity tongue has an eastward physically. This is because, when is allowed to be a extension, and shifts away from the northern boundary. smaller value, the derivation of Eq. (10) is acceptable. In the absence of negative viscosity, Fig. 10b shows a weak, negative relative vorticity advection (RVA) in the Even so, the negative viscosity is still found to have an western boundary layer near y ϭ 0.4 and a strong pos- important in¯uence on the western boundary current. itive RVA in the northwest corner. In the presence of Such a small choice for parameter does not negatively negative viscosity the negative RVA is intensi®ed con- affect the basic results obtained in this paper. Figure 12 siderably, but the postive RVA in the northwest corner shows the case for ϭ 0.02. It is found that the basic becomes weaker and is shifted to the south. On the other results in this ®gure are almost the same as in Fig. 11, hand, we see that in the presence of negative viscosity but the absolute value of NVT is smaller than that of the negative planetary vorticity advection becomes VISC, which supports the validity of Eq. (10) including stronger and has an eastward and southward extension. the ®fth-order derivatives (negative viscosity term small In Fig. 10d, in the south of y ϭ 0.83 VISC is always relative to lateral eddy viscosity) in the western bound- positive in the western boundary layer, but a narrower ary layer. At least, in the premise of a very small the
Unauthenticated | Downloaded 09/29/21 11:09 AM UTC MAY 2000 LUO AND LU 931 modi®ed model presented here is acceptable for a wind- Bryan, F., and W. R. Holland, 1989: A high resolution simulation of driven ocean circulation. the wind- and thermohaline-driven circulation in the North At- lantic Ocean. ``Aha Huliko'': A Parameterization of Small-Scale Processes, P. Muller and D. Henderson, Eds., Hawaii Institute 6. Conclusions of Geophysics, 99±115. ,W.BoÈning, and W. R. Holland, 1995: On the mid-latitude In the present paper, a new tentative subgrid-scale circulation in a high-resolution model the North Atlantic. J. Phys. Oceanogr., 25, 289±305. parameterization of the boundary layer is proposed by Bryan, K., 1963: A numerical investigation of a nonlinear model of retaining the second-order terms (negative viscosity) in a wind-driven ocean. J. Atmos. Sci., 20, 594±606. the Prandtl mixing length theory. In an academic ocean Cessi, P., and L. Thompson, 1990: Geometrical control of the interial box model with a single-gyre wind stress, we have dis- recirculation. J. Phys. Oceanogr., 20, 1867±1875. cussed the in¯uence of the negative viscosity on the , and G. R. Ierley, 1993: Nonlinear disturbance of the western western boundary current from many aspects (time-av- boundary currents. J. Phys. Oceanogr., 23, 1727±1735. Chao, Y., A. Gangopadhyay, F. O. Bryan, and W. R. Holland, 1996: eraged streamfunction ®eld, meridional transport, ki- Modeling the Gulf Stream system: How far from reality? Geo- netic energy, time-mean potential vorticity balance phys. Res. Lett., 23, 3155±3158. terms, and so on). It is found that the negative viscosity Charney, J. G., 1995: The Gulf Stream as an inertial boundary layer. is likely to play an important role in the intensi®cation Proc. Natl. Acad. Sci. USA, 41, 731±740. of the western boundary current and the establishment Denng, J., 1993: The problem of Gulf Stream separation: A barotropic approach. J. Phys. Oceanogr., 23, 2182±2200. of the recirculation. Especially, the impact of the neg- Foreman, M. G. G., and A. F. Bennett, 1988: On no-slip boundary ative viscosity becomes more noticeable in the ®ne-grid conditions for the incompressible Navier±Stokes equations. Dyn. model. In the western boundary layer, the negative vis- Atmos. Oceans, 12, 47±70. cosity is found to strengthen the negative vorticity ad- Haidvogel, D. H., J. C. McWilliams, and P. R. Gent, 1992: Boundary vections (RVA and BETA), which enhance the western current separation in a quasigeostrophic, eddy-resolving ocean circulation model. J. Phys. Oceanogr., 22, 882±902. boundary current and the recirculation. The enhanced Hogg, N. G., and W. E. Johns, 1995: Western boundary currents. reicirculation allows the eastward jet near the northern Rev. Geophys., 33 (Suppl.), 1311±1334. boundary to have a long penetration distance. Further, Holland W. R., 1967: On the wind-driven circulation in an ocean we observe the drastic increase of the mean and eddy with bottom topography. Tellus, 19, 582±600. kinetic energies in the presence of negative viscosity. , 1973: Baroclinic and topographic in¯uences on the transport in western boundary currents. Geophys. Fluid Dyn., 4, 187±210. It should be pointed out that the present model does , 1978: The role of mesoscale eddies in the general circulation not contain other physical processes (coastline orien- of the oceanÐNumerical experiments using a wind-driven, qua- tation, bottom topography and so on) in the wind-driven sigeostrophic model. J. Phys. Oceanogr., 8, 363±392. ocean circulation. If these factors are considered, our Ierley, G. R., 1987: On the onset of inertial recirculationn in brotropic results may be improved considerably and consistent general circulation models. J. Phys. Oceanogr., 17, 2366±2374. , and O. G. Ruehr, 1986: Analytic and numerical solutions of a with the observations. Another important problem is nonlinear boundary-layer problem. Stud. Appl. Math., 75, 1±36. how to determine the negative viscosity coef®cient ␥ , and V. A. Sheremet, 1995: Multiple solutions and advection- from the observed data. This is a very dif®cult problem. dominated ¯ows in the wind-driven circulation, Part I. Slip, J. On the other hand, the sublayer problem of western Mar. Res., 53, 703±734. boundary current is not discussed in this paper. These Jiang, S., F. Jin, and M. Ghil, 1995: Multiple equilibria, periodic, and problems need further study. aperiodic solutions in a wind-driven, double-gyre, shallow-water model. J. Phys. Oceanogr., 25, 764±786. Kamenkovich, V. M., V. A. Sheremet, A. R. Pastushkov, and S. O. Acknowledgments. The authors wish to thank Dr. N. Belotserkovsky, 1995: Analysis of the barotropic model of the Hogg, Dr. Z. 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