Response of the Separated Western Boundary Current to Harmonic and Stochastic Wind Stress Variations in a 1.5-Layer Ocean Model

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(1)Response of the Separated Western Boundary Current to Harmonic and Stochastic Wind Stress Variations in a 1.5-Layer Ocean Model Jérôme Sirven. To cite this version: Jérôme Sirven. Response of the Separated Western Boundary Current to Harmonic and Stochastic Wind Stress Variations in a 1.5-Layer Ocean Model. Journal of Physical Oceanography, American Meteorological Society, 2005, 35 (8), pp.1341-1358. �10.1175/JPO2752.1�. �hal-00122306�. HAL Id: hal-00122306 https://hal.archives-ouvertes.fr/hal-00122306 Submitted on 11 Feb 2021. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) AUGUST 2005. 1341. SIRVEN. Response of the Separated Western Boundary Current to Harmonic and Stochastic Wind Stress Variations in a 1.5-Layer Ocean Model J. SIRVEN Laboratoire d’Océanographie Dynamique et de Climatologie, UMR:IPSL/MNHN/UPMC/CNRS/IRD, Université Pierre et Marie Curie, Paris, France (Manuscript received 21 April 2004, in final form 9 January 2005) ABSTRACT A time-dependent version of the Parsons model (geostrophic 1.5-layer model of the ventilated thermocline) has been developed to investigate the response of the midlatitude ocean to wind stress variations in a simple configuration. In this model, the total amount of water is kept constant and the eastern boundary thermocline depth can vary in time so as to maintain mass balance. Here, basin modes are not investigated, in contrast to many recent studies, but the emphasis is on the line where the motionless second layer outcrops, which represents the separated western boundary current. It is shown that the position of this line only depends on the wind stress, the earth rotation, and the thermocline interior solution. The position is not influenced by the parameterization of the dissipative processes. This generalizes previous results established in the stationary case. The displacement of the outcrop line in the case of harmonic or stochastic wind stress variations is computed numerically, showing a lag of 0–4 yr that results from a combination of the instantaneous Ekman response and the delayed response due to Rossby wave propagation. Such delay is in satisfactory agreement with observations of Gulf Stream adjustment to wind stress changes, considering the limitations of the model, and is in good agreement with intermediate-resolution OGCM models. Although inertial effects and buoyancy forcing also need to be considered, this suggests that the outcropping mechanism plays a role in the variability of the separated boundary currents and may be dominant in non-eddy-resolving ocean models.. 1. Introduction From several decades of Gulf Stream monitoring, two main modes of variability have been found: the first one exhibits wavelike fluctuations and meandering and is associated with the instability of the jet—it will not be considered here—while the second one shows largescale shifts at seasonal and interannual time scales. At seasonal time scales, the observations show conflicting results. Tracey and Watts (1986) found that the Gulf Stream reaches its northernmost position during the autumn and its southernmost during the spring, whereas Taylor and Stephens (1998) and Frankignoul et al. (2001) did not find any statistically significant signal. At interannual time scales, Taylor and Stephens. Corresponding author address: Jérôme Sirven, Laboratoire d’Océanographie Dynamique et de Climatologie, Université Pierre et Marie Curie, Tour 45, etage 4, CC100, 4 place Jussieu, 75252 Paris Cedex 05, France. E-mail: [email protected]. (1998) found that the Gulf Stream shifts between 1966 and 1996 were correlated with the North Atlantic Oscillation (NAO) during winter, high values of the NAO index favoring a more northern path 2–3 years later. On the other hand, Joyce et al. (2000) used temperature data at 200-m depth to define the yearly position of the Gulf Stream between 1954 and 1990 and found that its correlation with the wintertime NAO was at a maximum for a lag between 0 and 1 yr. Using more extended data, Frankignoul et al. (2001) found that the dominant signal in the Ocean Topography Experiment (TOPEX)/Poseidon altimetry data is a northward (southward) displacement of the Gulf Stream about 11–18 months after a maximum (minimum) of the NAO index. They similarly obtained a dominant lag of 1 yr between the shifts of the Gulf Stream and the NAO from the analysis of the yearly mean position of the Gulf Stream estimated from XBT data; however, the covariability remained significant when the NAO led by up to 7 yr. As a 1-yr delay seemed too short for Rossby wave propagation, they. © 2005 American Meteorological Society. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC. JPO2752.

(3) 1342. JOURNAL OF PHYSICAL OCEANOGRAPHY. suggested that the fast response of the gyre might be influenced by both buoyancy and wind stress forcing within the recirculation gyre, but that linear adjustment theory may become relevant at lower frequency. In an intercomparison of the Gulf Stream shifts and transports in five OGCMs, de Coëtlogon et al. (2004, manuscript submitted to J. Phys. Oceanogr.) also found strong correlations between Gulf Stream variability and the NAO. They distinguished an instantaneous response that they interpreted as the barotropic response, from a delayed baroclinic response, which lags the forcing by 1 to 4 yr. The interannual variability of the Kuroshio Extension and its southern recirculation gyre has been also investigated using the TOPEX/Poseidon data. Qiu (2000) found that the Kuroshio Extension oscillates at large scale between an elongated state (characterized by a larger eastward surface transport, a greater zonal penetration, and a more northerly zonal-mean path) and a contracted state. Analysis of hydrographic and XBT data had already shown a shift from 36°–37°N during 1977–78 to 34°N in 1979–80 (Mizuno and White 1983). Studies from observations (e.g., Deser et al. 1999) and models (e.g., Seager et al. 2001) have shown that the associated changes in the (interior) gyre circulation were consistent with Sverdrup theory, lagging the wind forcing by a few years. However, the mechanisms that drive the lateral shifts of separated western boundary currents are not fully understood. We do not consider here theories for the local separation (through adverse pressure gradient, coastline or shelf geometry, etc.) for which inertial effects are crucial at high Reynolds number, but theories that use large-scale dynamics to predict the larger-scale position of the separated boundary currents. They were initiated by Charney (1955) and Morgan (1956), who suggested that their position could be defined by an isopycnal located at the base of the boundary layer; if this isopycnal outcrops, the stream is detached from the western boundary. Parsons (1969) proposed a simple model based on this mechanism. He considered a stationary 1.5-layer model, where the thermocline is represented by a surface layer with a constant amount of water over a nonmiscible second layer that outcropped if the wind stress was increased over a critical value (or equivalently if the amount of surface water was reduced below a critical value). The path of the separated western boundary current follows the line where the second layer outcrops; this line is estimated in such a way that the geostrophic transport between it and the eastern boundary balances the Ekman transport. Gangopadhyay et al. (1992) tested, on the Gulf Stream, whether the (stationary) Parsons model correctly predicts the. VOLUME 35. latitude at which the western boundary currents separate from the coast. They found that theory and observations were consistent when the Ekman transport was integrated over 3–4 years. Other mechanisms, independent of outcropping, have been suggested. For instance, it has been suggested that a balance between forcing and dissipation drives the dynamics of the recirculation gyre (Marshall and Nurser 1988), which then determines the latitude of separation of the Gulf Stream (Cessi 1990); Thompson and Schmitz (1989) linked the separation point of the Gulf Stream to the southward transport of the deep western boundary current, and Spall (1996a,b) showed that such variability could be due to the amount of Labrador Sea Water at the crossover point. The lowfrequency variability of the double-gyre circulation has been also attributed to the internal nonlinear dynamics of the ocean (e.g., Simonnet et al. 2003), independent of the variability of the external forcing. Although it is likely that various effects, indeed, contribute to the meridional shifts of separated western boundary currents, in this paper we only consider the outcropping mechanism, extending the Parsons model to the timedependent case. The response of a separated boundary current (as represented by the outcrop) to wind stress variability is thus investigated, with a focus on how the balance between the geostrophic and the Ekman transport is modified when the wind varies, what time scales are involved, and if the model prediction is consistent with the observed or the hindcast Gulf Stream behavior. The model has obvious limitations. First, the second layer is supposed to be at rest, whereas the motion in the second layer matters for an ocean of finite depth; however, as long as the upper/lower layer ratio is small enough, the analytical theory seems to remain valid (Chassignet and Bleck 1993). We will thus limit our study to the case of a relatively thin upper layer, which ensures a large outcrop area. Second, inertial effects are important (Chassignet and Bleck 1993; Özgökmen et al. 1997), and only ocean general circulation models (OGCMs) with very high resolution (less than 10 km) succeed in correctly representing the Gulf Stream separation and path. Third, surface cooling and heating may induce a net mass transfer across the two layers, which may completely alter the global circulation pattern (Pedlosky 1987; Nurser and Williams 1990). Last, the model, in its present configuration, does not represent the contribution made by the mean overturning circulation to the western boundary current transport and separation. Note, however, that the latter could be included by prescribing a transport at, for example, the northern boundary, mimicking the approach of Johnson and. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(4) AUGUST 2005. 1343. SIRVEN. FIG. 1. Sketch of the main characteristics of the model. (top) Note the existence of a separated western boundary current along the outcrop line (thick arrow) and an isolated western boundary current (dashed arrow). (bottom) Mean wind stress (continuous line) corresponding Ekman pumping we (dash–dotted line), and function (⳵y␶)/f (dotted line).. Marshall (2002). Because of these limitations, this study only contributes to the understanding of some aspects of the basin-scale response to wind stress variability. The time-dependent model is formulated in section 2, and its dynamics near the western boundary are studied analytically in section 3. The interior solution is given in section 4, and the setup for the numerical experiments is given in section 5. The response of the model to harmonic and stochastic forcings is respec-. tively analyzed in sections 6 and 7. The results are summarized and discussed in section 8.. 2. The model We consider (see Fig. 1, upper panel) a timedependent geostrophic 1.5-layer model of the ventilated thermocline, derived from the stationary model described in Salmon (1998, 182–188; see also Huang. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(5) 1344. JOURNAL OF PHYSICAL OCEANOGRAPHY. and Flierl 1987). The domain is rectangular with a Cartesian coordinate system (x, y) such as 0 ⱕ x ⱕ xe and ys ⱕ y ⱕ yn. The lightest layer of density ␳1 and thickness h lies over a motionless layer of density ␳2 and infinite depth. The hypothesis of no motion in the second layer is one of the most severe limitations of the model. However, the thickness of the first layer is assumed to be small in order to reduce the impact of this approximation [Chassignet and Bleck (1993); see also Kamenkovich and Reznik (1972) and Veronis (1973) for analytical studies]. Hydrostatic equilibrium and geostrophy are assumed everywhere in the basin, and a friction proportional to the velocity is applied in the surface layer. For simplicity, the wind stress is zonal and only depends on the latitude y, as given in Fig. 1 (lower panel). With these approximations, the governing equations of the upper layer are ⫺f V ⫽ ⫺⭸x␾ ⫺ ⑀U ⫹ ␶ and. 共1兲. fU ⫽ ⫺⭸y␾ ⫺ ⑀V. 共2兲. for the conservation of momentum and ⭸t h ⫹ ⭸xU ⫹ ⭸yV ⫽ 0. 共3兲. for the conservation of mass, with ␾ ⫽ g⬘h /2. Here g⬘ ⫽ g(␳2 ⫺ ␳1)/␳2 is the reduced gravity, ⑀ is a constant drag coefficient, ␶( y, t) is the wind stress divided by the mean density, U and V are the zonal and meridional transport in the moving layer, and f ⫽ f0 ⫹ ␤y (␤-plane approximation). These equations are essentially the same as those used by Huang and Flierl (1987) except that the representation of the friction has been slightly simplified following Salmon (1998; see p. 187 and the following in his book for more details; the friction ⑀ has here units of a frequency rather than of a velocity), and the temporal variations of the thermocline depth h are now taken into account. The coefficient ⑀ is assumed to be small to obtain sufficiently thin boundary layers. From (1) and (2), the zonal and meridional transports are given by 2. U⫽ V⫽. 1 f2 1 f2. 共⫺⑀⭸x␾ ⫺ f ⭸y␾ ⫹ ⑀ ␶兲. 共4兲. and. 共 f ⭸x␾ ⫺ ⑀⭸y␾ ⫹ f ␶兲,. 共5兲. where the terms of order ⑀2 have been neglected. Replacing in (3) leads to an equation for h:. 冉冊. 2␤⑀ ⑀ ␶ ⭸h ␤ ⫺ ⭸ ␾ ⫹ 3 ⭸y␾ ⫺ 2 共⭸xx ⫹ ⭸yy兲␾ ⫽ ⭸y ⭸t f 2 x f f f. ⫽ ⫺we ,. 共6兲. VOLUME 35. where we ⫽ ⫺⳵y(␶/f ) is the Ekman pumping. This equation, which characterizes the evolution of the system, cannot be easily solved because of the free western boundary (see below). The dynamics in the interior of the gyres are well represented by neglecting terms O(⑀), yielding UI ⫽. ⫺1 ⭸␾, f y I. 1 VI ⫽ 共⭸x␾I ⫺ ␶兲, f ⭸t hI ⫺. ␤ f. 2. ⭸x␾I ⫽ ⭸y. 冉冊. 共7兲 and. ␶ ⫽ ⫺we , f. 共8兲共9兲. where ␾I ⫽ g⬘h2I /2 and the I index denotes interior solution. Note that (9) is the usual nonlinear equation for the long Rossby waves in the ␤ plane. The interior solution is not valid near the boundaries with the exception of the eastern boundary where the thermocline depth He(t) is set independent of the latitude, so avoiding any mass flux. The dissipation terms O(⑀) in (1) and (2) may then become large and boundary layers must be considered [see, e.g., Cessi and Louazel (2001), or appendix B, for an example of computation in a time-dependent case]. As we will assume that the wind stress curl always vanishes at the northern and southern boundaries [⳵y ␶( ys, t) ⫽ ⳵y␶( yn, t) ⫽ 0; see Fig. 1, lower panel] the transport across them would vanish in the stationary case, even in the absence of dissipation, and there would be no boundary layer. However, in the time variable case, small-amplitude boundary layers are needed to prevent any mass transport because of the variations of He(t). Note also that the solution will depend on the position of the northern and southern boundary layers. Indeed, if the meridional extension of the basin is for example increased, Rossby waves will be radiated over a larger latitude range and consequently the amplitude of the variations of He(t) would be decreased. We will come back to this point in section 7. Along the western boundary, the problem is complicated because, for a sufficiently strong wind stress, the first layer may shoal and the second layer may surface in the northwestern corner of the basin (e.g., Parsons 1969). In this case, a free boundary x ⫽ Xg( y, t) [or equivalently y ⫽ Yg(x, t)], the outcrop line, separates the first layer from the second layer (Fig. 1, top panel). Two main types of circulation can be found, depending on the wind stress pattern and strength. In the singlegyre case, the western boundary current leaves the coast and flows northeastward along the outcrop line if the wind stress is sufficiently strong. Here we consider the double-gyre case, where in addition an isolated. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(6) AUGUST 2005. 1345. SIRVEN. western boundary current flowing southward (dashed arrow in Fig. 1) is found in the northern gyre (Veronis 1973; Huang and Flierl 1987). No mass exchange between the moving layer and the stagnant layer is allowed along the free boundary, so the total volume V0 of the first layer is conserved. This condition reads. 冕. ⍀共t兲. h共x, y, t兲 dx dy ⫽ V0,. 共10兲. where ⍀(t) denotes the area occupied by the moving upper layer. As the boundary layers have a width proportional to ⑀ (see appendix B for details), their contribution to the integral in (10) is at most of order ⑀, and it will be neglected. Consequently, condition (10) simplifies to. 冕. ⍀共t兲. hI 共x, y, t兲 dx dy ⫽ V0.. 共11兲. 3. Solution near the western boundary a. Displacement of the outcrop line We first show that the displacements of the outcrop line can be predicted independently of the detailed structure of the boundary layers. To do this, (3) is integrated over the hatched domain D(t, Y ) ⊂ ⍀(t) (see Fig. 2), which is limited to the north by the outcrop line and the latitude LY (at y ⫽ Y, where the wind stress is ␶Y, and the Coriolis parameter fY ⫽ f0 ⫹ ␤Y ). Using Stokes’s formula, one finds. 冕冕. ⭸t h dx dy ⫹. D. 冖. 共U dy ⫺ V dx兲 ⫽ ⫺␺m共t兲,. ⭸D. 共12兲. where ␺m is introduced to allow for an external supply or removal of mass due to the existence of the isolated western boundary current (Fig. 1), and ⳵D is the boundary of the domain D, taken counterclockwise. Note that this integral constraint, which is verified for any Y, ensures that the variations of water volume south of the latitude y ⫽ Y are balanced by the flows which go across the latitude y ⫽ Y in the interior basin and isolated western boundary current. To compute the first term in (12), we replace ⳵th by ⳵thI (correct to zero order in ⑀) and use the long Rossby wave equation in (9), yielding. 冕冕. D. ⭸t h dx dy ⫽. 冕. Y. ys. ⫹. ␤ f. 冕冋冉冊 0. Here, xw( y, t) defines the longitude of the western boundary including the outcrop line [it is represented by the thick dashed line in Fig. 2: xw ⫽ 0 when y ⱕ yc, and xw ⫽ Xg( y, t) when y ⱖ yc where yc(t) is the latitude where the outcrop line meets the western boundary], and yl(x, t) defines the latitude of the northern boundary including the outcrop line [it is represented by the dash–dotted line in Fig. 2: yl ⫽ Yg(x, t) when x ⱕ xc, and yl ⫽ Y when x ⱖ xc, where xc(Y, t) ⫽ Xg(Y, t) is the longitude where the outcrop line meets LY]. Because ␾I(xe, t) does not depend on y (no flow across the eastern boundary), the integral of ␤␾I(xe, t)/f 2 yields. 冕. 冉冊册. ␶ ␶ 共yl, t兲 ⫺ 共ys, t兲 dx. f f 共13兲. Y. ys. ␤ f. 2. ␾I 共xe, t兲 dy ⫽ ␾I 共xe, t兲. 冉. 冊. 1 1 ⫺ , fs fY. 共14兲. where fs denotes the value of f y ⫽ ys. The integral over x in (13) can be split between 0 and xc ⫽ Xg(Y, t), then between xc and xe, leading to. 冕冋冉冊冉冊册冕冉冊. ␶ ␶ 共yl, t兲⫺ 共ys, t兲 dx f f. xe. 关␾I 共xe, t兲 ⫺ ␾I 共xw, y, t兲兴 dy 2. xe. FIG. 2. Definition of the main notations used when the evolution equation of the outcrop line is established (section 3). Note that xw is a function of y and t, whereas yl is a function of x and t. The following relations are also verified: yc ⫽ yl(0, t) and xc ⫽ xw(Y, t) ⫽ Xg(Y, t).. 0. ⫽. xc. 0. ␶ ␶Y ␶s ⫺ xe , 关Yg共x, t兲, t兴 dx ⫺ 共xc ⫺ xe兲 f fY fs 共15兲. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(7) 1346. JOURNAL OF PHYSICAL OCEANOGRAPHY. where ␶s denotes the value of ␶ at y ⫽ ys. Replacing in (13), one obtains. 冕冕. ⭸t h dx dy ⫽ ␾I 共xe, t兲. D. ␤. f2. ys. ⫹. 冊. 1 1 ⫺ ··· fs fY. 冕冕冉冊 Y. ⫺. 冉. xc. 0. ␾I 共xw, y, t兲 dy. ␶ 关Yg共x, t兲, t兴 dx f. ⫺ 共xc ⫺ xe兲. ␶Y ␶s ⫺ xe . fY fs. 共16兲. The mass transport across ⳵D vanishes except along LY, which is inside the basin; hence the second term in (12) simplifies to. 冕. ⭸D. 共U dy ⫺ V dx兲 ⫽ ⫺. 冕. xc. 共17兲. V dx.. xe. Using (1) and the vanishing of ␾ along the outcrop line, one finds to zero order in ⑀:. 冕. ⭸D. 共U dy ⫺ V dx兲 ⫽. ␾共xe, t兲 ⫺ 共xe ⫺ xc兲␶Y . fY. 共18兲. Because there is no boundary layer at the eastern side, one has ␾(xe, t) ⫽ ␾I (xe, t) so that (12) finally becomes ⫺␺m共t兲 ⫽. ␾I 共xe, t兲 ⫺ xe␶s ⫺ fs ⫹. 冕冉冊. 冕. Y. ys. ␤ f2. 兵␾I 关xw共y, t兲, y, t兴其 dy. ␶ 关Yg共x, t兲, t兴 dx, f. xc. 0. 共19兲. using (16) and (18). The derivative with respect to Y of the first two terms in (19) vanishes, whereas that of the third one is ⫺␤␾I [xw(Y, t), Y, t]/f 2 ⫽ ⫺␤␾I[Xg(Y, t), Y, t]/f 2. The last term depends on Y through xc ⫽ Xg(Y, t). Its derivative is therefore computed taking the limit. lim. 冕. Xg 共Y⫹dY,t兲. Xg 共Y,t兲. 冉冊. dY→0. ⫽ ⫽. 冉冊冋冉冊. VOLUME 35. ⭸Xg ␤ ⫽ ␾I 共Xg , y, t兲, ⭸y f␶. 共20兲. which is valid for every y ⱖ yc and where the time t simply appears as a parameter. Equation (20) depends on ␾I, which contains all the effects associated with Rossby wave propagation; it is explained in section 4 how the first layer depth hI and consequently ␾I are computed using the Rossby wave equation in (9). When ␾I is known, (20) allows one to compute the position of the outcrop line Xg. However, to have a unique solution, it is necessary to set the value of Xg at one latitude. To remove this indeterminacy, we use ␾I[Xg( ycrit), ycrit, t] ⫽ 0, where ycrit is the critical latitude for which the wind stress vanishes. With this condition, ␾I vanishes when ␶ vanishes so that ⳵Xg/⳵y remains everywhere finite and the outcrop line can cross the critical latitude.. b. First-layer changes along the western boundary South of the outcrop line (for y ⱕ yc), the method that we follow to determine the thermocline depth along the western boundary is similar to that used just above. Equation (3) is integrated over a domain D (t, Y ), which is now rectangular, since there is no longer an outcrop line. As there is now no isolated western boundary current, we can set ␺m(t) ⫽ 0 in (12). Equations (13), (14), (15), (16), and (17) remain unchanged except that xc ⫽ xw ⫽ 0 and yc ⫽ yl ⫽ Y. Equation (18) is modified in order to take into account the variations of ␾ along the western side, and thus becomes. 冕. ⭸D. 共U dy ⫺ V dx兲 ⫽. ⫺␾ 共0, Y, t兲 ⫹ ␾ 共xe, t兲 ⫺ xe␶Y . fY 共21兲. With these simplifications, (19) is eventually replaced by. ␾ 共0, Y, t兲 ⫽ fY. ␾I 共xe , t兲 ⫺ xe␶s ⫺ fY fs. 冕. Y. ys. ␤ f2. 关␾I 共0, y, t兲兴 dy,. 共22兲. ␶ 关Yg共x, t兲, t兴 dx f. which determines ␾ (or h) along the western boundary.. dY. Xg共Y ⫹ dY, t兲 ⫺ Xg共Y, t兲 ␶ 共Y, t兲 lim dY→0 f dY. 册. ␶ ⭸Xg . 共Y, t兲 f ⭸Y. In conclusion, differentiating (19) with respect to Y leads to the first-order differential equation. 4. Interior solution Far from the boundary layers (e.g., for distances from the western boundary or the outcrop line larger than ⑀/␤; see appendix B), dissipation can be neglected and (7), (8), and (9) together with suitable initial and boundary conditions suffice to characterize the evolution of the ocean interior. For a stationary Ekman. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(8) AUGUST 2005. pumping w0( y) ⫽ ⫺d(␶0/f )/dy, where ␶0 is the mean wind stress, the solution is hI2␴ ⫽ H20 ⫹. 2f 2共x ⫺ xe兲 w0, ␤g⬘. 共23兲. where H0 denotes the thermocline depth along the eastern side and is determined by the approximate relation (11). More precisely, (11) becomes. 冕公 ⍀. 1347. SIRVEN. H20 ⫹ 关2f 2共x ⫺ xe兲w0兴Ⲑ共␤g⬘兲 dx dy ⫽ V0.. Consequently, if the total amount of water V0 is given, this equation can be numerically solved and provides H0; on the other hand, if H0 is given, this equation determines V0. The solution for a time dependent forcing of the form we( y, t) ⫽ w0( y) ⫹ S(t)wa( y) requires initial and boundary conditions. We assume that (i) at t ⫽ 0, the state of the ocean is given by (23) in the domain ⍀(t) and (ii) at t ⬎ 0, the thermocline depth at the eastern boundary He(t) (see Fig. 1) remains independent of latitude (no mass flux condition). It may vary with time, however [as in the stationary case, (11) is used to compute at each instant the value of He(t); see also appendix A]. Equation (9) is integrated by the method of characteristics, hence replaced by the system of differential equations: dt ⫽ ds, dhI ⫽ ⫺weds, dx ⫽ ⫺. and. ␤g⬘hI f2. 共24兲. ds,. where s represents the abscissa along the characteristics. The characteristics issuing at t ⫽ 0 from a point of abscissa x␴ in the interior basin are given by hI ⫺ hI␴ 共x␴ , y兲 ⫽ ⫺w0 t ⫺ wa共 y兲. 再. 冕. t. S共u兲 du. and. 0. x ⫺ x␴ ⫽ ⫺共␤g⬘Ⲑf 2兲 hI␴ 共x␴ , y兲t ⫺ w0 t2Ⲑ 2 ⫺ wa共 y兲. 冕冋冕 t. ␷. 0. 0. 册冎. S共u兲 du d␷. 共25兲. [the subscript ␴ is added to a variable in order to recall that it is considered at t ⫽ 0; thus hI␴(x␴, y) is obtained doing x ⫽ x␴ in (23)]. Elimination of x␴ defines hI as a function of x, y, and t. The characteristics issuing from the eastern boundary at a time t0 ⬎ 0 are given by. hI ⫽ He 共t0兲 ⫺ w0共t ⫺ t0兲 ⫺ wa x⫽⫺. ␤g⬘ f2. ⫺ wa. 冋. 冕. t. S共u兲 du. and. t0. He 共t0兲共t ⫺ t0兲 ⫺ w0. 冕冕 t. ␷. t0. t0. 共t ⫺ t0兲2 2. 册. S共u兲 du d␷ .. 共26兲. The novelty of our approach mainly lies in (20) and (22), which provide an easy way to compute the position of the outcrop line in a time variable case. The shape of the outcrop line, derived from (24) and (20), only depends on the interior solution and the wind stress, and is thus independent of the dissipative mechanisms that determine the structure of the boundary current. It has avoided the traditional boundary layer techniques (stretched coordinate and matching with interior solution) to compute the position of the outcrop line, but it does not provide the thermocline depth in the boundary layer nor the boundary layer structure. However, boundary layer theory has been implicitly used in our analysis by assuming the existence of a boundary layer whose thickness decreases with ⑀ (thereby replacing ⳵th by ⳵thI inside the surface integrals). Note that the boundary layer solution could be calculated, for example, as in Cessi and Louazel (2001; such a computation is sketched in appendix B). To solve the problem considered here, three equations are needed [(11), (20) or (22), and (25) or (26)]; one of which is an integral constraint [(11)]. This is more complicated than the solution given by Johnson and Marshall (2002) for a similar model, but used in a different context. In particular, a time-delay equation characterizing the thermocline depth along the eastern side could not be established in our case because the outcropping prevents the linearization of the Rossby wave equation. However, the time delays are represented here since the Rossby wave propagation is taken into account in (20). This leads us to compute He(t) following an iterative algorithm as explained in appendix A. To summarize, the changes in He(t) are propagated westward, modifying the position of the outcrop line with a time delay depending on the Rossby wave propagation, and adding to the effect of the wind stress variations. The changes of hI and of the outcrop line induces in turn an adjustment in He(t), in order to keep the amount of water constant.. 5. Setup for the numerical computations The experimental setup aims at crudely mimicking the North Atlantic Ocean. Hence, the ocean basin is a. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(9) 1348. JOURNAL OF PHYSICAL OCEANOGRAPHY. 6000-km square centered at 40°N ( f0 ⫽ 9.35 ⫻ 10⫺5 s⫺1 and ␤ ⫽ 1.74 ⫻ 10⫺11 m⫺1 s⫺1). The first layer depth at t ⫽ 0 along the eastern side is H0 ⫽ He (0) ⫽ 650 m. The solution is computed on a regular grid with a resolution of 50 km. However, the function Xg( y, t), which defines the position of the separated western boundary current (the “model Gulf Stream”), is computed to a much better accuracy using a 10-km latitudinal grid size. To compute the steady-state response of the model (Fig. 3), we use the algorithm described in appendix A; it rests upon (20). Appendix B describes how the transports in the western boundary layer are calculated using traditional boundary layer techniques. It is only used to plot Figs. 2 and 4. Figure 3 shows that, as expected, there is a strong current along the western boundary and the outcrop line [maximum mass transport of 49 Sv (Sv ⬅ 106 m3 s⫺1)], which crosses the zero wind stress curl line. An isolated southward western boundary current (not represented here, see Fig. 1) closes the circulation in the first layer and transports about 17 Sv, keeping the volume constant. In the time-dependent experiments, a wind stress fluctuation ␶aS(t) is added to the mean wind stress ␶0. It coarsely aims at representing the wind stress changes associated with the NAO (the dominant mode of variability on the North Atlantic). The condition ⳵yy␶ ⫽ 0 when ␶ ⫽ 0 is always fulfilled so that hI is never negative on the outcrop line (Huang and Flierl 1987). Figure 4 shows the total wind stress in a positive NAO phase ␶0 ⫹ ␶a (thick line) and in a negative NAO phase ␶0 ⫺ ␶a (thin line). The zero line of wind stress curl (or of Ekman pumping) shifts northward when the wind intensifies; the anomaly of Ekman pumping vanishes at about the latitude of the separation point and is notably weaker southward than northward, as observed (Marshall et al. 2001). To facilitate the analysis of the timedependent experiments, Fig. 5 shows the stationary state of the model for a steady forcing corresponding to a low (top) and a high (bottom) NAO index. When the NAO index is low, the outcrop line is located more southerly and the main current separates from the outcrop line farther south where the Ekman pumping vanishes. This corresponds to the observations (e.g., Frankignoul et al. 2001), which indicate a southward shift of the Gulf Stream during low NAO period. To calculate the time-dependent solution the interior equations are first integrated, as indicated in section 4. The outcrop line is obtained by integrating (20) using a first-order predictor corrector scheme on the refined grid, where ␾(t, y) is computed by linear interpolation. As the shifts of the outcrop line remain small and the amount of water near the outcrop line is also small, the first layer volume of water is computed using (11) and. VOLUME 35. FIG. 3. Response of the model to the stationary mean wind stress: (top) thermocline depth (m) and (bottom) transports (U, V ) computed using (4) and (5).. the first layer depth along the eastern boundary He(t) is adjusted to keep this volume constant. For more details about the numerical algorithm, see appendix A. Two cases are investigated: in the first one, the time variations S(t) are harmonic (a 3-month time step is then used); in the second one, they follow the time series of the Hurrell NAO index (section 7). For comparison with observational studies, we focus below on the shifts. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(10) AUGUST 2005. SIRVEN. FIG. 4. Extreme states of the wind stress (continuous line) and the Ekman pumping (dash–dotted lines).. of the separated western boundary current only over the westernmost 2000 km, where the current remains close to the outcrop line.. 6. Response to a harmonic forcing The longitude–time diagram of the interior solution hI at midbasin (40°N) is shown in Fig. 6 for a forcing period of 10 yr. The adjustment takes about 20 years. Then a cyclostationary state is established. The vertical thick line indicates the mean position of the outcrop line. Westward of this longitude, the solution has no physical significance since h vanishes along the outcrop line. The interior solution exhibits one important aspect of the dynamics of the system: Rossby wave propagation. Rossby waves—satisfying (9)—respond to the wind stress variations and propagate across the basin to the outcrop line in about 5 years, deepening or shoaling the thermocline. This time of propagation increases to about 8 years at the latitude of the separation point because the waves cross the whole basin (not shown). Figure 7 (continuous line) shows the time evolution of the point where the western boundary current separates from the western coast for a 60-yr integration. After transient adjustment, a regular oscillation appears that lags the forcing (dashed line) by about 6 months in the high NAO phase and 24 months in the low NAO phase. The response is thus strongly nonlinear [following (20), it varies as the inverse of the wind. 1349. stress]. The time evolution of the function Xg( y, t), which gives the trajectory of the model Gulf Stream, is illustrated in Fig. 8. The upper panel shows the evolution Xg( y, t) at different latitudes [the meridional coordinate increases from 3700 km (lower continuous curve) to 4100 km (upper continuous curve)]. A decrease of Xg corresponds to a northward shift of the current. As already suggested by Fig. 5, the model Gulf Stream shifts northward when the NAO is high, as far as 2500 km from the western coast. For y ⫽ 3700 km, the distance of the Gulf Stream to the western coast Xg( y, t) is about 500 km. The amplitude peak to peak of the oscillation is about 150 km after the adjustment phase. It reaches a maximum of about 350 km when the model Gulf Stream is at about 2000 km from the coast [middle curve: Xg( y, t) ⬃ 2000 km for y ⫽ 3850 km], then decreases. These variations correspond to meridional shifts of the Gulf Stream of about 50 km. One cycle of the evolution of Xg( y, t) is represented for various latitude ( y increases from 3600 to 3900 km) in Fig. 8 (lower panel). When y ⫽ 3600 km, the mean distance of the Gulf Stream to the western coast is 91 km [i.e., the mean value of Xg( y, t)]. It is about 1850 km for y ⫽ 3900 km. The time lag between response and forcing increases with the mean distance (or, equivalently, latitude). After high NAO phase, it ranges from about 6 months in the west to about 4.5 yr farther to the northwest. After low NAO phase, it similarly increases from 2 to 4 yr. When the forcing has a 5-yr period (Fig. 9) the adjustment is faster and is followed by a symmetrical, periodic signal with a 20-km amplitude, which lags the forcing with a time lag increasing from 6 months along the western boundary to 1.5 yr at 2500 km from the boundary. For a forcing with a 1-yr period, the response of the Gulf Stream is nearly in phase with the forcing (not shown). Last, we consider a forcing with 20-yr period and 0.5 times the amplitude (Fig. 10). The adjustment now takes about 25 years and is followed by a periodic signal of large amplitude. Indeed, although the amplitude of the forcing has been divided by 2, the amplitude of the oscillation of the separation point is increased by a factor 1.5–2, reaching 40 km (Fig. 10, top). The nonlinearity of the response has nearly disappeared, as the time lag is about 2 yr, independent of the phase of the NAO. In the cyclostationary state, Xg(t, y) shows a northward shift of the model Gulf Stream when the model NAO is high (Fig. 10, bottom). As previously, the time lag increases with distance from the western coast, ranging from 2 yr close to the coast to 4 yr at 2500 km from the coast. The time lag between forcing and response results from the instantaneous response of the Ekman trans-. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(11) 1350. JOURNAL OF PHYSICAL OCEANOGRAPHY. VOLUME 35. FIG. 5. Thermocline depth (m) and transports of the model in response to the extreme states of the wind stress. (top) This state corresponds to a low NAO index (slight weakening of the wind and southward shift of the zero wind stress curl line). (bottom) This state corresponds to a high NAO index (slight strengthening of the wind and northward shift of the zero wind stress curl line).. port and the delayed response to the variations of the Ekman pumping via Rossby waves. At latitude 3900 km, where the current separates from the western coast, the variations of the Ekman pumping remains very small (Fig. 4), whereas those of the wind stress are large. The dynamics are thus dominated by the Ekman. transport and the boundary current response is fast. At latitude 4200 km, the Ekman pumping variations are larger and their effects become comparable with those caused by the variations of the Ekman transport. Two responses of comparable amplitude thus add: the first one is delayed by about 8 years (the propagation time. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(12) AUGUST 2005. 1351. SIRVEN. FIG. 6. Time–longitude diagram showing the evolution of the thermocline depth (interior solution hI) in midbasin in response to a harmonic forcing (units: m). The dashed lines correspond to the area where hI has no longer physical significance (west of the outcrop line where h ⫽ 0). The wind stress forcing has a 10-yr period.. of the Rossby waves across the basin to the outcrop line), while the second one is instantaneous. This leads to a time lag of about 4 yr when the period of the forcing is sufficiently long (longer than 10 yr). The delays also depend on the frequency of the forcing. Indeed, at low frequency the wave dynamics, which are slow, can produce large variations of the thermocline depth, whereas it becomes impossible at high frequency. The lag is thus dominated by longer time scales at low frequency and shorter time scales at high frequency. To test whether the zonal size of the basin has a large influence on the model response, we made an experiment with the same setup as in the 10-yr period experiment except that the width of the basin is 10 000 km. In this setting, the area where the second layer surfaces increases notably and the western boundary current separates from the western coast more southward. However, the time scale associated with the shifts of the separated current remains quite similar to those observed for a basin of 6000-km width. This agrees with the mechanism described in the previous paragraph: the response is dominated by the Ekman transport as the outcrop line is located more southward, compensating the increase of the propagation time of the Rossby wave across the basin. The meridional size of the basin may also be ex-. FIG. 7. Time evolution of the latitude where the model Gulf Stream leaves the western coast and function S(t). The first curve lags the second one with a time lag between 6 and 25 months. The wind stress forcing has a 10-yr period.. pected to impact on the response of the western boundary current in this model. Indeed, in a basin extending for example to 50°S, Rossby waves would be radiated off the eastern side on a largest latitudinal range, and consequently their role and the variability of He(t) would be both reduced. As the model has been developed in the ␤ plane, such an experiment was impossible. However, the impact of such a change would probably remain moderate. Indeed, in this model, the large thermocline variations (about 100 m) are mainly due to the variability of the wind stress. The time evolution of He(t), which is a consequence of this main variability, induces thermocline variations, which does not exceed some meters. This situation differs from those considered in studies about the variability of the thermohaline circulation (e.g., Johnson and Marshall 2002). In this case, the wind stress variability is neglected, and the only source of Rossby waves is the variations of He.. 7. Response to an “NAO-like” forcing For a more realistic case, the function S(t) is derived from the monthly anomaly time series of Hurrell’s (1995) NAO index from 1950 to 1999. Its power spectrum is nearly white. The function S(t) has been interpolated every 5 days as the time step for the numerical integration has been reduced to 5 days. Frankignoul et al. (2001) used two datasets to esti-. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(13) 1352. JOURNAL OF PHYSICAL OCEANOGRAPHY. VOLUME 35. FIG. 8. (top) Time evolution of Xg( y, t) at different latitudes [the meridional coordinate y goes from 3700 km (lower continuous curve) to 4100 km (upper continuous curve)] when the forcing S(t) has a 10-yr period. (bottom) Time evolution of the anomalies of Xg for different latitudes (meridional coordinates going from 3600 to 3900 km) when the model is in a cyclostationary state. The mean positions indicated in the figure represent the mean distance of the model Gulf Stream to the western coast at the considered latitudes. They correspond also to the mean value of Xg(Y, t) at these latitudes. Negative values of the continuous curves correspond to northward shifts.. mate the changes in the Gulf Stream location: monthly data derived from the TOPEX/Poseidon (hereinafter T/P) measurements between October 1992 and November 1998 and yearly data derived from temperature measurements between 1954 and 1998. They performed an analysis into empirical orthogonal functions (EOF) and principal components (PC) of these datasets. Here, a similar analysis is performed on the time series of Xg,. respectively averaged with a 1-month averaging filter (Figs. 11 and 12) and a 1-yr averaging filter (Fig. 13). Figure 11 shows the first two EOFs (top) and their associated PCs (bottom) of the monthly averages of Xg. They represent more than 50% and 40% of the variability, respectively. The second PC indicates an instantaneous response of the model to the wind stress variations, being very similar to the time series of the forc-. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(14) AUGUST 2005. 1353. SIRVEN. FIG. 9. (top) Time evolution of the latitude where the model Gulf Stream leaves the coast and function S(t) when the wind stress forcing has a 5-yr period. (bottom) Time evolution of the anomalies of Xg for different latitudes (meridional coordinates going from 3600 to 3900 km) when the model is in a cyclostationary state. The mean positions indicated in the figure represent the mean distance of the model Gulf Stream to the western coast at the considered latitudes. They correspond also to the mean value of Xg(Y, t) at these latitudes. Negative values of the continuous curves correspond to northward shifts.. ing. The first PC presents substantial variability at low frequency and the associated EOF shows a uniform meridional shift, a northward shift being associated with a high NAO period, as in the observations. The cross correlation between the first PC and S(t) is shown in Fig. 12. The correlation is largest at zero lag, but remains large for lags exceeding several years. There is some consistency with the analysis derived from the. monthly T/P data by Frankignoul et al. (2001), who found that the dominant signal is a northward (southward) displacement of Gulf Stream axis 11–18 months after the NAO reaches positive (negative) extrema. However, strong in-phase correlations were not seen in T/P data. When the time series of Xg are filtered at 1 yr, the EOFs remain similar to the previous ones (not shown) and a significant correlation between the first. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(15) 1354. JOURNAL OF PHYSICAL OCEANOGRAPHY. VOLUME 35. FIG. 10. As in Fig. 9 but for a 20-yr period.. PC and S(t) is found from 0 to 6 yr (Fig. 13). Frankignoul et al. (2001), analyzing yearly XBT data, found a narrower peak, ranging from 0 to 3 yr. However, they also found significant covariability up to 7 yr, using more powerful statistical methods. Interestingly, a better consistency is found with OGCM simulations. De Coëtlogon et al. (2004, manuscript submitted to J. Phys. Oceanogr.) have analyzed the Gulf Stream variability in five global OGCMs of intermediate resolution which all show a similar behavior, illustrated here for the ocean model of the Nansen Environmental and Remote Sensing Center (NERSC) in Bergen. The NERSC model is derived from the Mi-. ami Isopycnic Coordinate Ocean Model (MICOM). It was forced in 1° ⫻ 1° resolution with air–sea fluxes estimated from the National Centers for Environmental Prediction (NCEP)–National Center for Atmospheric Research (NCAR) reanalysis (Kalnay et al. 1996) for the 1948–2000 period. As the resolution is too coarse to monitor the shift in the Gulf Stream axis, we focus on the (largely baroclinic) mass transport between 200 and 600 m in the vicinity of the Gulf Stream from 50° to 70°W, which is highly correlated to the temperature fluctuations at 200 m along the mean axis of the Gulf Stream, an intensification of the transport being associated to a temperature increase and high. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(16) AUGUST 2005. 1355. SIRVEN. FIG. 13. Lagged covariance of the first PC of Xg (thin line) and of the first PC of the 17° isotherm at 200 m (thick discontinuous line; Frankignoul et al. 2001) with Hurrell’s (1995) NAO index. For positive time NAO leads. The thin dashed line gives the 5% level for no-correlation and independent samples. The response of the model is filtered at 1 yr to match with the Hurrell index.. FIG. 11. (top) First and second EOFs of Xg( y, t), which determines the separated western boundary current position. (bottom) Corresponding PCs.. NAO periods. As intermediate-resolution OGCMs do not correctly represent the mean position of the Gulf Stream, the interpretation of the results will have to remain cautious. Note, however, that the variability of the temperature along the mean axis of the Gulf Stream has been used in studies based on observations to estimate Gulf Stream shifts. It is therefore reasonable to analyze the same variable—or any other which is highly correlated to it—as a “proxy” of the Gulf Stream po-. FIG. 12. Lagged covariance of the first PC of Xg( y, t) with the function S(t). For positive time S(t) leads.. sition in a model, especially as in the NERSC model, where outcropping is well represented. Figures 14 and 15 show the cross correlations between the NAO index and the second PC, which represents 31% of the variance of the signal, of this baroclinic mass transport [the first PC (not shown) primarily reflects the rapid barotropic response of the model to the wind stress variations]. There are significant correlations at lags varying from 0 to 4 or 5 yr with a peak at zero lag and a maximum between 2 and 3 yr, so our predicted response is closer to the OGCM one than to that of the observations. Moreover, this suggests that the role of the Rossby waves is still overrepresented. Whether the better agreement at lag 0 only reflects the lack of eddy dynamics in the OGCM remains unknown.. 8. Summary and conclusions The Parsons model has been extended to the timedependent case in order to investigate the response of a. FIG. 14. Lagged covariance of the second PC of the baroclinic transport between 200 and 600 m in the Gulf Stream region with the NAO index. For positive time NAO leads. The dashed lines give the 5% level for no-correlation and independent samples.. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(17) 1356. JOURNAL OF PHYSICAL OCEANOGRAPHY. FIG. 15. Lagged covariance of the second PC of the baroclinic transport between 200 and 600 m in the Gulf Stream region with the NAO index filtered at 1 yr. For positive time NAO leads. The dashed line gives the 5% level for no-correlation and independent samples.. separated boundary current (as represented by the outcrop) to wind stress variability in an idealized two gyre basin. The wind stress and the amount of upper-layer water were chosen to be consistent with the gross features of the oceanic circulation in the North Atlantic. In particular, the lower layer outcrops in the northwestern part of the basin without invading the whole subpolar gyre, so that the circulation is asymmetric with respect to the line of zero wind stress curl and a strong northeastward current separates from the western coast and flows across this line into the subpolar gyre. The two other cases discussed in the stationary case by Huang and Flierl (1987) [(i) no outcropping corresponding to very weak wind stress or large volume of upper-layer water and (ii) very large outcropping filling the whole subpolar basin corresponding to very large wind stress or small volume of upper-layer water] were not investigated because they correspond to less realistic conditions. The wind stress anomaly applied in all the experiments has a spatial pattern that crudely mimics that of the NAO. In response to this anomaly, we observe that a northward (southward) displacement of the axis of the separated boundary current follows a strengthening (weakening) of the wind stress. The response of the separated boundary current lags the forcing by 6 months (close to the western coast) to 4 yr (farther north) if a harmonic forcing is used; correlations between response and forcing are significant for lags ranging from 0 to 5 yr if a stochastic forcing is used, and the largest correlations are found at lag zero. These results complement the stationary study of Gangopadhyay et al. (1992), who suggested a response with a time delay only between 3 and 4 yr. In our model, the time lags between forcing and response results from the combination of two effects: 1) in. VOLUME 35. response to the variations of the wind stress forcing there is an instantaneous change of the surface Ekman transport, which immediately induces variations of the separated boundary current and 2) as the Ekman pumping is altered, long baroclinic Rossby waves are generated, leading to a delayed adjustment of the thermocline. They combine in such a way that, at any time, the flux of water across a given latitude balances the variations of the volume in the moving layer, south of this latitude. This leads to relation (20), which allows us to determine the position of the separated boundary current from the variable interior solution, independently of the detailed structure of the boundary currents, even though they must exist. The spatial shifts of the separated boundary current are in agreement with the observed response of the Gulf Stream to a positive (negative) phase of the NAO (Taylor and Stephens 1998; Frankignoul et al. 2001) but their time lags with the forcing differ from the observed ones. Indeed the monthly T/P data do not show significant correlations with the forcing, neither at lag 0 nor at low frequency but mainly around 1 yr, and yearly XBT data have a maximum correlation with the forcing around 1–2 yr. The agreement is better with intermediate-resolution OGCMs: they show the largest correlations with the forcing at lag 0 and significant correlations in a range larger than the observations. The time lags found with our model are also in broad agreement with the 4-yr lags found by Seager et al. (2001) for the gyre adjustment in an OGCM of the northern Pacific. A more quantitative agreement between our idealized model and the observations was not expected because of model limitations. First, the model is completely adiabatic and fluid never crosses an isopycnal surface. Pedlosky (1987) extended the Parsons model by assuming heat or buoyancy fluxes at the surface in order to allow conversion of fluid from one density to another, hence fluid to cross the isopycnals in the upper mixed layer. In the limit considered by Pedlosky, there was no longer separation in the subtropical gyre and the outcrop line only appeared in the subpolar gyre, being simply due to Ekman suction. Note, however, that Nurser and Williams (1990), considering an intermediate case between the adiabatic Parsons model and Pedlosky’s diabatic model, showed that heating along the outcrop line shifts the path of the separated boundary current northward. This displacement is in the same direction as in the observations since a high NAO index is associated with a heating of the ocean around 35°N. Taking this heating effect into account would reinforce the amplitude of the wind driven responses that were reinvestigated here. More worrisome is our neglect of eddy dynamics,. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(18) AUGUST 2005. 1357. SIRVEN. which would drive the progression of the stream, its separation from the western boundary, and its links with the recirculation gyre. The associated time scales must therefore be studied in models that correctly represent these dynamics and recirculation gyres. There is a large literature about this, including both shallow water models and quasigeostrophic models. These models predict realistic values of the mean transport by the separated boundary current, which is intensified by the recirculation gyre. Because they come from a rather complex model (21⁄2-layer shallow water model), the results of Simonnet et al. (2003) seem to be the most complete today. These authors do not force their model with a stochastic forcing but follow a dynamical system approach, pointing to the importance of global bifurcation. They attribute the low-frequency variability of the double-gyre circulation to the internal nonlinear dynamics of the ocean. Because of their approach, they obviously do not consider the problem of the lags between forcing and response. To summarize this discussion, a correct description of the mechanisms linked to the dynamics of the outcrop lines and the separated boundary currents certainly requires a model representing the recirculation gyre and eddies dynamics. However, as our model captures some features found in the observations and in moderateresolution OGCMs, it usefully complements the studies based on eddy-only dynamics. It also suggests that outcropping is an essential ingredient in the shifts of separated western boundary currents. Acknowledgments. We thank S. Février, C. Frankignoul, and C. Herbaut for numerous discussions and G. de Coëtlogon who provided Figs. 13 and 14. Reviewers are also acknowledged for their constructive suggestions. This research was supported in part by IUF and a grant from PNEDC.. APPENDIX A Numerical Algorithms Used to Compute the Solutions The technique that we used to compute the solutions is new since it does not require the computation of the boundary layers. We first explain it in the stationary case. The interior solution hI is computed on a regular rectangular grid using (23) for given Ekman pumping and thermocline depth along the eastern boundary H0 (H0 ⫽ 650 m). Then hI, hence ␾I, are known and the differential equation in (20) is integrated with an implicit Euler scheme. The integration is initiated at the point where hI vanishes, (x ⯝ 4000 km and y ⫽ 6000 km), and is performed both northward and southward.. Hence, the position of the outcrop line Xg is known and the volume V0 can then be computed. Last, for completeness, h has also been computed in the boundary layers as indicated in appendix B. However, this step is only used to plot Figs. 2 and 4 and not elsewhere. Note that, if V0 was prescribed rather than H0, then H0 should be computed iteratively until the numerical scheme converges to the volume V0. The variable case needs a more complex numerical algorithm because the initial volume V0 must be kept constant. We suppose that the variables He(t), hI(t, x, y), and Xg(t, y) are known on a grid (xi, yj) at time t. To compute them at time t ⫹ ␦t, (25) are first used where hI␴ is replaced by hI(t, xi, yj), x␴ by xi, y by yj, and the integrals are computed between t and t ⫹ ␦t. The interior thermocline depth hI is thus known at t ⫹ ␦t, west of the longitude given by the second equation in (26) where t – t0 is set equal to ␦t (for a time step of 3 months, this corresponds to a distance from the eastern boundary ␦x that never exceeds 400 km at midlatitudes). Obviously hI is no longer known on the initial grid, but an interpolation by spline allows us to compute its values on the initial grid when it is needed. As hI is known in the western part of the basin, the differential equation in (20) can be integrated following the scheme described for the stationary case. The position of the outcrop line Xg is thus known at the time t ⫹ ␦t. It remains to compute the interior thermocline depth hI in the extreme east part of the basin (east of ␦x). As the field hI is very smooth, it is sufficient to compute He(t ⫹ ␦t); if there are missing values on the grid (xi, yj), they are obtained by interpolation. To compute He(t ⫹ ␦t), (11) is solved by an interative algorithm [note that ⍀(t ⫹ ␦t) is known since Xg(t ⫹ ␦, y) is known]. Iterations are stopped when the relative variation of the volume with respect to the initial volume is smaller than 0.003. With this algorithm, He is thus adjusted to keep the volume nearly constant. This leads to variations of He(t) smaller than 8 m.. APPENDIX B The Western Boundary Layer To complete the solution presented in sections 3 and 4, we determine h in the two narrow boundary layers along the outcrop line and the western coast. We first consider the outcrop line. In the neighborhood of a point P0 of coordinates [x0 ⫽ Xg( y0), y0] and at a time t, we introduce the stretched coordinates x̃ ⫽ (x ⫺ x0)/⑀ and ỹ ⫽ (y ⫺ y0)/⑀ as in Huang and Flierl (1987) and assume that the solution has the form h ⫽ hb(x̃, ỹ, t) ⫹ ⑀h1(x̃, ỹ, t) ⫹ · · · . Replacing in (6) and keeping only the largest terms (of order ⑀⫺1), we obtain. Unauthenticated | Downloaded 02/11/21 09:50 AM UTC.

(19) 1358. JOURNAL OF PHYSICAL OCEANOGRAPHY. 共⭸x̃x̃ ⫹ ⭸ỹỹ兲␾b ⫹ ␤⭸x̃␾b ⫽ 0,. 共B1兲. where ␾b ⫽ g⬘h2b/2. Along the outcrop line, which is well approximated near P0 by x̃ ⫽ ␣ỹ with ␣ ⫽ ⳵yXg(x0, y0, t), ␾b must vanish, whereas it must match ␾I outside the boundary layer. These two conditions lead to the approximate solution. 再. 冋. ␾b ⫽ ␾I 共x0, y0, t兲 1 ⫺ exp. ⫺␤共x̃ ⫺ ␣ỹ兲 1 ⫹ ␣2. 册冎. .. 共B2兲. The same method leads to the solution along the western boundary:. ␾b ⫽ ␾I 共0, y0, t兲关1 ⫺ exp共⫺␤x̃兲兴.. 共B3兲. The complete solution ␾ is equal to ␾(x, y, t) ⫽ ␾b(x, y, t) ⫹ ␾I(x, y, t) ⫺ ␾I[xb( y), y, t] since ␾I[xb( y), y, t] was counted twice [xb( y) ⫽ 0 if y ⬍ yc and xb( y) ⫽ Xg( y) if y ⬎ yc]. The boundary flows in Figs. 2 and 4 are derived from (B2) and (B3). We did not compute the analytical expression of the “corner flow” (e.g., where the western boundary current separates from the western wall), but used a smoothing procedure as in Huang and Flierl (1987). The northern and southern boundary layers can be studied similarly. For the isolated western boundary current we refer to Huang and Flierl (1987). Note that the position of the separated boundary current does not depend on the solutions in the boundary layers and, in particular, on ␾b when ␧ is small. REFERENCES Charney, J. G., 1955: The Gulf Stream as an inertial boundary layer. Proc. Natl. Acad. Sci., 41, 731–740. Chassignet, É., and R. Bleck, 1993: The influence of layer outcropping on the separation of boundary currents. Part I: The wind-driven experiments. J. Phys. Oceanogr., 23, 1485–1507. Cessi, P., 1990: Recirculation and separation of boundary currents. J. Mar. Res., 48, 1–35. ——, and S. Louazel, 2001: Decadal oceanic response to stochastic wind forcing. J. Phys. Oceanogr., 31, 3020–3029. Deser, C., M. A. Alexander, and M. S. Timlin, 1999: Evidence for a wind-driven intensification of the Kuroshio Current extension from the 1970s to the 1980s. J. Climate, 12, 1697–1706. 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