Convection Induced by Cooling at One Side Wall in Two-Dimensional Non-Rotating Fluid Applicability to the Deep Pacific Circulation

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1 Journal of Oceanography Vol. 52, pp. 617 to Convection Induced by Cooling at One Side Wall in Two-Dimensional Non-Rotating Fluid Applicability to the Deep Pacific Circulation ICHIRO ISHIKAWA 1, SHIGEAKI AOKI 2, RYO FURUE 1 and NOBUO SUGINOHARA 1 1 Center for Climate System Research, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan 2 National Institute for Resources and Environment, 16-3 Onogawa, Tsukuba-shi, Ibaraki 305, Japan (Received 20 June 1995; in revised form 20 February 1996; accepted 26 February 1996) In recognition that similarity in the density balance leads to resemblance in circulation between the two-dimensional non-rotating and three-dimensional rotating systems which have similar density stratification, we investigate convection induced by cooling at one side wall and heating at the sea surface by using a twodimensional non-rotating model as idealized representation for the deep Pacific circulation. In the model, various vertical profiles are taken for the side wall cooling, which are assumed to correspond to the density structure of the Antarctic Circumpolar Current. In a small diffusivity range, two important features are found to be robust against change in the vertical profile of the side wall cooling. One is that the density stratification is horizontally almost uniform. The other is that the balance in the density equation between the vertical advection and the vertical diffusion holds in the interior. Consequently, the vertical density balance, together with the equation of continuity, determines the circulation pattern for the prescribed vertical profile of the side wall cooling. The multi-layered meridional flow, which is expected to exist in the deep Pacific, is shown to form for certain vertical profiles of the side wall cooling. 1. Introduction Nature of horizontal convection in a two-dimensional non-rotating system driven by differential cooling at the top surface is well known (e.g. Rossby, 1965; Beardsley and Festa, 1972). Two-dimensional here is for the meridional-vertical plane. For the large Rayleigh number, thermocline-like structure near the top surface and very weak stratification at greater depth form associated with upwelling in the broad area and downwelling in the narrow region. These features have strong resemblance to horizontal convection in a three-dimensional rotating system like that studied by Suginohara and Aoki (1991). This resemblance may appear strange when we look at momentum balances which are totally different between the two-dimensional non-rotating system and the three-dimensional rotating system. However, the zonally integrated equation of continuity in the three-dimensional system has exactly the same form as the equation of continuity in the two-dimensional system. Therefore, when the distribution of the vertical velocity in the two-dimensional system is the same as the zonally integrated vertical mass transport in the three-dimensional system, the two systems can be considered to have the identical meridional circulation pattern. Next, we consider the balance in the density equation to estimate the distribution of the vertical velocity. In the twodimensional model similar to that used by Beardsley and Festa (1972), the balance is known to be between the vertical advection and the vertical diffusion except near the boundaries. In the

2 618 I. Ishikawa et al. three-dimensional model of Suginohara and Aoki (1991), the balance in the zonally averaged density equation is also between the vertical advection and the vertical diffusion, except near the bottom. Thus the zonally integrated vertical mass transport in the three-dimensional system and the vertical velocity in the two-dimensional system can be obtained from the vertical density balance when the density stratification is given. Therefore, it can be considered that the similarity of the density balance leads to resemblance in circulation between the two-dimensional nonrotating and three-dimensional rotating systems which have similar density stratification. The circulation in the deep Pacific is not directly induced by the distribution of sea surface heat flux; there is no source for the deep water at its own sea surface. The deep water of the Pacific Ocean, which originates from the North Atlantic Ocean and the Antarctic Ocean, is supplied through the Antarctic Circumpolar Current. The supplied water is not vertically uniform in temperature, salinity, and hence density (Stommel et al., 1973). The vertical distributions of those properties seem to be associated with the vertical structure of the Circumpolar Current. Fiadeiro (1982) tested his Stommel-Arons type abyssal circulation model by comparing with the observed radiocarbon distribution in the deep Pacific. He showed that a uniform increase in upwelling rate from the bottom to the top of the model (the bottom of the thermocline) cannot explain the qualitative features of the radiocarbon distribution, but a model in which the upwelling rate attains a maximum between the bottom and the deep waters can do. This means that the interior flow is poleward in the bottom water, and equatorward in the deep water. This layered structure of the meridional circulation was obtained by Wunsch et al. (1983) from the SCORPIO data and by Johnson and Toole (1993) from the 10 N transpacific data. It is likely that the vertical distribution of density in the Circumpolar Current plays an important role in forming such layered meridional flow in the deep Pacific. Cox (1989) demonstrated the importance of the Circumpolar Current in forming the global-scale water mass structure, using a world ocean model with an idealized topography. With the Drake Passage closed, the depths below the thermocline are dominated by the water formed at the southern boundary. With the Drake Passage opened, on the other hand, the water of the northern origin is allowed to occupy the subthermocline depths. This suggests that the model with a single basin forced by surface differential cooling cannot reproduce the realistic vertical density distribution in the deep Pacific. In other words, to reproduce the realistic density structure in the Pacific by using a single basin model, the vertical distribution of density at the southern wall need to be specified as a boundary condition. Thus, it is not quite unnatural that the Pacific is treated as an ocean thermally driven by heat flux through the side wall boundary. In the present study, we investigate the convection induced by side wall cooling in a twodimensional non-rotating fluid with small diffusivity as a new way to approach understanding of the thermohaline circulation, especially the deep-pacific circulation. In our model, the density flux is imposed in the form of restoring to the prescribed reference density at the left-hand side wall, which is assumed to correspond to that associated with the density structure of the Circumpolar Current. The reference density at the top surface is constant, and the other two boundaries are insulated. The numerical scheme is similar to that of Beardsley and Festa (1972). Several cases changing the vertical distribution of the reference density at the left-hand side wall are calculated, and the resulting steady circulation will be shown to be sensitive to the specified distribution of the reference density. Nature of convection driven by distributed heat flux through one side wall has not been well investigated before. We will show in the following sections that it has a circulation pattern different from that induced by differential cooling at the top boundary, i.e., multi-cell circulation

3 Convection Induced by Cooling at One Side Wall in Two-Dimensional Non-Rotating Fluid 619 or multi-layered meridional flow can take place. This paper is organized as follows. In Section 2, the model used in the present study is described. In Section 3, the results of our experiments are shown. In Section 4, we make dynamic interpretation of the results, and discuss applicability of this model to the three-dimensional rotating system. In Section 5, some concluding remarks are given. 2. Model We consider two-dimensional circulation of an incompressible, viscous Boussinesq fluid in a rectangular container. The Cartesian coordinate system (x,z) is used, where z positive upward from the bottom of the container. The vorticity and density equations are ζ t = J( ψ,ζ ) g ρ 0 x + A 2 ζ H x + A 2 ζ 2 V z, ( 1) 2 t = J( ψ,ρ) + K 2 ρ H x + K 2 ρ 2 V z, ( 2) 2 where ψ is a stream function with u = ψ/ z and w = ψ/ x, ζ = u/ z w/ x = 2 ψ the vorticity, ρ the density deviation from the mean density ρ 0, g the acceleration of gravity, A H the coefficient of horizontal viscosity, A V the coefficient of vertical viscosity, and K H and K V are the coefficients of horizontal and vertical diffusion, respectively. The advection terms are expressed by the Jacobian, J( α,β) = α β x z α β z x. 3 For vorticity, slippery boundary condition is applied all along the boundaries of the container, i.e., 2 ψ = 0 along all the boundaries. ( 4) The bottom and the right-hand side wall are assumed to be insulated, i.e., ( ) K V = 0 at z = 0, 5 z ( ) K H = 0 at x = L, 6 x ( ) where L is the horizontal extent of the model domain. On the top surface and the left-hand side wall the density fluxes are imposed so that the density along the boundaries is adjusted to the prescribed values,

4 620 I. Ishikawa et al. K H x = αρ ( z) ρ ( ) at x = 0, 7 K V z = ( βρs ρ) at z = H, ( 8) where H is the vertical extent of the model domain. For simplicity, the top surface reference density is taken to be zero, i.e., ρ s = 0 and ρ*(h) = ρ s = 0. Stable stratification for distribution of the side wall reference density is assumed, i.e., the reference density is a monotonically decreasing function with z, in order to avoid occurrence of the Rayleigh-Bénard type instability. Now, we introduce a time scale, T g, by ( ) T g = L g H, 9 ( ) where g = ρg is the reduced gravity and ρ = ρ*(0) ρ*(h). This gives the travelling time for the gravest internal gravity wave to cross the container. If we scale x by L, z by H, ρ by ρ, ψ by HL/T g = H g H, and ζ by L/(HT g ), then the non-dimensional forms of vorticity and density equations become where ( ν H,κ H ) = ζ t = J( ψ,ζ ) x + ν 2 ζ H x + ν 2 ζ 2 V z, ( 10) 2 t = J( ψ,ρ) + κ 2 ρ H x + κ 2 ρ 2 V z, ( 11) 2 1 ( L g H A H, K H ), ( ν V,κ V ) = 1 ( δh g H A V, K V ), ( 12) and δ = H/L is the aspect ratio of the container. On this non-dimensionalization, the vorticity is written as ζ = δ 2 2 ψ x + 2 ψ 2 z. ( 13) 2 Also the non-dimensionalized prescribed density, ρ B, is scaled so that ρ B ( 0) = 1, and ρ B ( 1) = 0. ( 14)

5 Convection Induced by Cooling at One Side Wall in Two-Dimensional Non-Rotating Fluid 621 The non-dimensionalized boundary conditions become κ H = 0 at x = 1, 15 x ( ) κ H x = γ H ρ B ρ ( ) at x = 0, 16 ( ) κ V z = γ Vρ at z = 1, 17 ( ) κ V = 0 at z = 0, 18 z ( ) where γ H = α/ g H and γ V = β/(δ g H ). When we take the typical values of L = 10 9 cm, H = 10 5 cm, g H = 10 2 cm s 1, A H K H 10 8 cm 2 s 1, and A V K V 1 cm 2 s 1, ν V, ν H, κ V and κ H are very small quantities, i.e., of the order of 10 2 or less. This smallness of the parameters implies a singular perturbation problem, and some kinds of boundary layers are expected. For simplicity we assume all of the coefficients are identical, i.e., ν H = ν V = κ H = κ V = κ. The value of κ is taken to be Also we assume the damping factors γ H and γ V take the same value γ. Although there remains ambiguity on this parameter, this factor is taken to be unity. Since κ is small, the density near the left-hand side boundary is forced to be the prescribed value (see Eq. (16)). We consider three vertical distributions of the side wall reference density in the present study, i.e., the exponential profile the hyperbolic-tangential profile ρ ( z) = ρ ( z) = e 5 ( 1 e5 ( z 1) ), 19 ( ) ( ) tanh 5 z 1/2 1, 20 tanh( 5 / 2) which has an inflection point at z = 1/2, and the superposition of the exponential and hyperbolictangential profiles ( ) where ρ 25 z 1 ( z) = ae ( ) + b 2 tanh 25 z c, ( 21 )

6 622 I. Ishikawa et al. Fig. 1. Vertical profiles of the reference density at the left-hand side wall, the exponential density profile (a), the hyperbolic-tangential profile (b), and the superposed profile (c). ( 1 e 25)a +1 b = tanh 25 / 2 ( ), ( 22) c = 1 [ 1 ( 1+ e 25)a], 23 2 which also has an inflection point near z = 1/2. The parameter a is for controlling the ratio b/a of the amplitude of the hyperbolic-tangential profile to that of the exponential profile. For the superposed profile, the depth scale of the stratification if taken to be shallower than the other two, in order to separate clearly the two regions with strong stratification. The exponential profile roughly represents stratification within the main thermocline, and the hyperbolic-tangential profile is used to investigate the effects of the stratification at the deeper levels. These profiles are shown in Fig. 1, in which b/a = 1 for the superposed profile. The numerical scheme is similar to that of Beardsley and Festa (1972). We use the Arakawa Jacobian, with which density, squared density, vorticity, and kinetic energy are conserved. The number of grid points is 80 (horizontal) 50 (vertical). Calculations are carried out until a thermally and dynamically steady balance is established, i.e., to completely cover the diffusion time. 3. Results The stream function fields for the cases of the reference density profiles in Fig. 1 are shown in Fig. 2. The circulation has a one-cell pattern for the case of the exponential profile, the twocell pattern for the case of the hyperbolic-tangential profile, and the three-cell pattern for the case of the superposed profile. Here, a cell is defined as a closed circulation. For the latter two cases, a multi-cell circulation forms, and depths of the boundaries of the circulation cells nearly coincide with those of the inflection points in the profile. Figure 3 depicts density distributions ( )

7 Convection Induced by Cooling at One Side Wall in Two-Dimensional Non-Rotating Fluid 623 Fig. 2. Stream function for cases of exponential density profile (a), the hyperbolic-tangential profile (b), and the superposed profile (c). The contour interval is Positive values by solid curves are for the anti-clockwise circulation and negative values contoured by the dashed curves for the clockwise circulation. Fig. 3. As in Fig. 2 except for density. The contour interval is for the three cases. Although vertical distributions of density are not the same as those of the side wall reference density, the isopycnals are nearly horizontal everywhere except near the left-hand side wall and the bottom. The depth of the inflection point in the profile of the reference density nearly coincides with that in the density profile in the interior. The distributions of the terms in the density equation for the three cases are shown in Fig. 4. The vertical advection and vertical diffusion terms are dominant everywhere except near the left-hand side wall for all of the cases. In the vorticity equation, the vertical viscosity and horizontal density gradient terms are primarily important in the interior (not shown here). To see the overall buoyancy budget, the vertical distributions of the horizontally averaged terms in the density equation are shown in Fig. 5. The dominant terms are the vertical diffusion

8 624 I. Ishikawa et al. Fig. 4. Terms in the density equation for cases of the exponential density profile (a), the hyperbolictangential profile (b), and the superposed profile (c). The contour interval is 0.004, and contours for values out of the range between 0.1 and 0.1 are omitted.

9 Convection Induced by Cooling at One Side Wall in Two-Dimensional Non-Rotating Fluid 625 Fig. 5. Vertical profile of the horizontal averages of the terms in the density equation for cases of exponential density profile (a), hyperbolic-tangential profile (b), and superposed profile (c). and side wall cooling terms. In other words, the buoyancy supplied to each level is lost mainly by the vertical diffusion from that level. The vertical distributions of the side wall cooling (or heating) terms have extrema near the depths of extrema of the curvature of the reference density profile. Two features are found to be common among the three cases. One is that the density stratification in the interior is horizontally almost uniform. The other is that in the interior, the dominant balance in the density equation is between the vertical advection and the vertical diffusion term, which is the same as the three-dimensional rotating system as discussed in introduction. However, the former is not so robust against change in parameter, as will be discussed below. In the case of the superposed profile in Fig. 1(c), stratification at the deeper levels is as strong as that in the thermocline. Such strong stratification may not be expected in the real ocean. Next, cases with weaker stratification in the deep ocean are studied to know whether the multi-cell circulation takes place or not. We consider three profiles shown in Fig. 6, taking 0.5, 0.2, and 0.1 for the ratio b/a in Eq. (21) of the amplitude of the hyperbolic-tangential profile to that of the exponential profile. The circulation patterns for the three cases are shown in Fig. 7. As the stratification at the deeper levels becomes weaker, the middle circulation cell becomes weaker and finally disappears, the circulation having a one-cell pattern. However, as clearly seen in Figs. 7(b) and 7(c) for the one-cell circulation pattern, there remains a leftward flow at mid-depths; the horizontal velocity changes its sign at the deeper levels well below the thermocline. We call this pattern the layered structure. The terms in the density equation for the case of the weakest stratification at the deeper levels (Fig. 6(c)) are shown in Fig. 8. The dominant balance in the interior is between the vertical advection and vertical diffusion terms as found before. For the cases of the one-cell circulation, there is no inflection point in the profile of the interior density at the depth of the inflection point

10 626 I. Ishikawa et al. Fig. 6. Three superposed profiles of the reference density at the left-hand side wall. The ratios b/a of the amplitude of hyperbolic tangent profile to that of exponential profile are 0.5 (a), 0.2 (b), and 0.1 (c). Fig. 7. Stream function for cases of superposed profile. The ratios b/a of the amplitude of hyperbolic tangent profile to that of exponential profile are 0.5 (a), 0.2 (b), and 0.1 (c). of the reference density as seen in the distribution of the vertical diffusion term in Fig. 8. In the horizontally averaged density balance shown in Fig. 9, the vertical diffusion and side wall cooling terms become smaller, as the stratification at the deeper levels becomes weaker. These terms change their signs at the depth of the inflection point in the reference density profile for the case of the multi-cell circulation, but the side-wall cooling term has an extremum near the level of the inflection for the cases of the one-cell circulation. This results from the fact that there is no inflection point in the profile of the interior density for the cases of the one-cell circulation as mentioned above. We also studied cases with larger value for the coefficient of diffusivity, κ = The circulation pattern and the distribution of density for one of these cases are shown in Fig. 10. The

11 Convection Induced by Cooling at One Side Wall in Two-Dimensional Non-Rotating Fluid 627 Fig. 8. Terms in the density equation for superposed profile case with b/a = 0.1.

12 628 I. Ishikawa et al. Fig. 9. Vertical profile of the horizontal averages of the terms in the density equation for superposed profile cases with b/a = 0.5 (a), 0.2 (b), and 0.1 (c). Fig. 10. Stream function (a) and density (b) for hyperbolic-tangential profile case with κ V = reference density for this case is the hyperbolic-tangential one shown in Fig. 1(b). The boundary of the circulation cells is not at the depth of the inflection point of the reference density. The density stratification is not horizontally uniform at all. Thus, larger diffusivity tends to destroy the density stratification given at the side wall.

13 Convection Induced by Cooling at One Side Wall in Two-Dimensional Non-Rotating Fluid Discussion The balance between the vertical advection and vertical diffusion terms in the density equation in the interior holds in the wider parameter range. It holds even in the cases for the larger diffusivity mentioned in the previous section. In such a parameter range, when the stratification is known, the vertical velocity can be calculated from w z = κ 2 ρ z ( ) By using this equation and the equation of continuity, the circulation pattern can be predicted. It should be noted that the sign of w is the same as that of 2 ρ/ z 2, as / z is always positive, the sign of w is the same as that of 2 ρ/ z 2. The stratification in the interior is obtained by the overall buoyancy budget found in Fig. 5. The horizontal average of the vertical advection term is almost zero, because the density stratification in the interior is almost the same as that in the sinking region and the horizontal average of vertical velocity must vanish. Thus, the side wall cooling term, instead of the vertical advection term balances against the vertical diffusion, i.e., γ H ( 2 ρ ρ B ρ ) κ V z, 25 2 where an overbar denotes horizontal average. In fact, this was the case for the results shown in Figs. 5 and 9. For the superposed profiles, however, the contribution of the advection terms are not totally neglected, as difference between the density profiles in the interior and in the sinking region is not very small. When diffusivity is small enough, one can predict the interior stratification by using Eq. (25), then the circulation pattern from Eq. (24) where ρ is replaced with ρ and the equation of continuity, given only the reference density. The results of application of this scheme to the three profiles of the reference density in Fig. 1 are shown in Fig. 11. The patterns are similar to those in Fig. 2, except near the boundary, where Eq. (24) does not hold. The number of cells for each profile is correctly predicted. Vertical extent of each cell is nearly the same as in Fig. 2. For the superposed profiles in Fig. 6, this scheme leads to the three-cell pattern for each of the three profiles (not shown here). Discrepancy from Fig. 7 may be due to difference between the density profiles in the interior and in the sinking region. When the vertical distribution of w contains three or more extrema, the layered structure in the circulation takes place as shown in Figs. 7(b) and 7(c). In these figures, there are two extrema below the thermocline. However, it is difficult to count the number of extrema just by looking at the density stratification. It should be noted that the layered structure cannot form when the stratification at the deeper levels is too weak. This is explained as follows. The horizontal density gradient must be large enough for the pressure gradient to change its sign at the subthermocline depths. The horizontal density gradient is in turn formed by the slight tilt of isopycnals, which is caused by vertical diffusion. This means that stratification at the deeper levels is required for the layered structure to be formed. The present model forced by the side wall cooling leads to the formation of the multi-cell circulation and the layered structure, which does not occur in the model forced by the sea surface ( )

14 630 I. Ishikawa et al. Fig. 11. As in Fig. 2 except calculated from Eqs. (24) and (25).

15 Convection Induced by Cooling at One Side Wall in Two-Dimensional Non-Rotating Fluid 631 differential cooling. Occurrence of this feature depends on the reference density profiles prescribed at the side wall. The observed density structure of the Circumpolar Current suggests the possibility of occurrence of this feature, at least, the layered structure. Obata et al. (1995) show that the vertical density balance of the zonally integrated density equation actually holds in the deep levels in their three-dimensional rotating ocean model driven by uniform heating at the sea surface and vertically distributed body cooling at the south-west corner of the model ocean, and clearly demonstrate that the layered structure forms for a realistic distribution of body cooling. The present model can be used as a tool to interpret qualitative features of such circulation, although it should be noted that applicability of the present model is limited in the way that it does not precisely reproduce the zonally averaged density distribution of three-dimensional rotating systems, as pointed out by Obata et al. (1995). 5. Concluding Remarks Circulation induced by side wall cooling with various vertical profiles has been investigated by using the two-dimensional non-rotating model. The circulation pattern is found to be very sensitive to the vertical profile of the side wall cooling. However, two robust features in the small diffusivity range are found. One is horizontally almost uniform density stratification and the other is the vertical balance in the density equation. These features make it possible to predict the circulation pattern for a given vertical profile of the side wall cooling, i.e., a prescribed vertical profile of the reference density. The vertical velocity is obtained by substituting the reference density into the vertical density balance, and then the meridional velocity from the equation of continuity. There is a possibility of formation of the multi-cell pattern, which has zeros in the vertical distribution of the vertical velocity, and the layered structure, which contains extrema at the subthermocline depth, depending on the reference density profile prescribed at the side wall. The present results may be applicable to the three-dimensional rotating system because of the similarity of the balance in the density equation. How well the present model is applicable to the deep Pacific depends on how well the vertical density balance holds in the deep Pacific. This problem is studied by Obata et al. (1995) in their three-dimensional model. Acknowledgements We would like to thank Drs. K. Nakajima and Y. Yamanaka for their fruitful and pleasant discussions. All the figures in this paper are drawn with graphic routines in GFD-Dennou Library, developed by Dennou Club, with Dr. Y.-Y. Hayashi, at University of Tokyo, Drs. M. Shiotani and S. Sakai, at Kyoto University. References Beardsley, R. C. and J. F. Festa (1972): A numerical model of convection driven by a surface stress and non-uniform horizontal heating. J. Phys. Oceanogr., 2, Cox, M. D. (1989): An idealized model of the world ocean. Part I: The global-scale water masses. J. Phys. Oceanogr., 19, Fiadeiro, M. E. (1982): Three-dimensional modeling of tracers in the deep Pacific Ocean: II. Radiocarbon and the circulation. J. Mar. Res., 40, Johnson, G. and J. M. Toole (1993): Flow of deep and bottom waters in the Pacific at 10 N. Deep-Sea Res., 40, Obata, A., R. Furue, S. Aoki and N. Suginohara (1995): Modeling layered structure in deep Pacific circulation. J. Geophys. Res., 101,

16 632 I. Ishikawa et al. Rossby, H. T. (1965): On thermal convection driven by non-uniform heating from below: an experimental study. Deep-Sea Res., 12, Stommel, H., E. D. Stroup, J. L. Reid and B. A. Warren (1973): Transpacific hydrographic sections at lats. 43 S and 28 S: the SCORPIO expedition I. Preface. Deep-Sea Res., 20, 1 7. Suginohara, N. and S. Aoki (1991): Buoyancy-driven circulation as horizontal convection on β-plane. J. Mar. Res., 49, Wunsch, C., D. Hu and B. Grant (1983): Mass, heat, salt and nutrient fluxes in the South Pacific Ocean. J. Phys. Oceanogr., 13,