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The Influence of Negative Viscosity on Wind-Driven, Barotropic Ocean

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The Influence of Negative Viscosity on Wind-Driven, Barotropic Ocean 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 [ 106 pation of the mean motion (Munk 1950). Actually, the m3 s21), 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 q 2000 American Meteorological Society Unauthenticated | Downloaded 09/29/21 11:09 AM UTC MAY 2000 LUO AND LU 917 still cannot be re¯ected completely. In this case, subgrid- ]u ]h t ]u9u9]u9y9 scale eddies should be parameterized in high-resolution 1 V´=u 2 fy 52g 12x 1 , (1) models as well. ]t ]xH12]x ]y In a barotropic wind-driven circulation model, Moro ]y ]h ty ]y9u9]y9y9 (1988) examined the importance of negative viscosity 1 V´=y 1 fu 52g 12 1 , (2) ]t ]yH ]x ]y in the intensi®cation and separation of the boundary 12 current and the establishment of the recirculation. In his ]u ]y 150, (3) model the negative viscosity was simply regarded as a ]x ]y negative eddy viscosity coef®cient (Welander 1973). Strictly speaking, this assumption is incorrect (Marshall where V 5 iu 1 jy is the mean velocity, (u9, y9) 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, (t x, t y) the wind stress, and viscosity in the western boundary current can be re- ]] ¯ected if the second-order term is incoporated into the = 5 i 1 j Prandtl mixing length theory in developing the param- ]x ]y 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 2u9y9 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 5 iu 1 jy) 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 l95x 2 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 5 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 l9. The average velocities at x 5 x 0 2, we present a new barotropic wind-driven ocean cir- and x 5 x are assumed to be y(x 0) and y(x), respectively. culation model including negative viscosity by using a When the ¯uid particle at x 5 x 0 arrives at x 5 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 y95y(x 0) 2 y(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 ]y 1 ]2y model resolution and negative viscosity intensity on the y952 l92 l912 ´´´. (4) solutions are discussed, and the spatial distribution of ]x 2 ]x2 the eddy energy associated with the mean ¯ow is also examined. In section 5 we present the analysis of the Naturally, if the term ]y/]x is only retained in (4), it is the so-called traditional Prandtl's mixing length the- time-mean potential vorticity. The conclusions are given 2 2 in section 6. ory. However, if the second term ] y/]x is included, it 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 In a uniform depth ocean, when the wind stress is ]y 1 ]2y considered as a body force, the averaged equations of 2y9u9 ø l9u91l92u9 .
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