Kuroshio Extension Variability Explored through Assimilation of TOPEX/POSEIDON Altimeter Data into a Quasi- Geostrophic Model

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1 Journal of Oceanography, Vol. 63, pp. 879 to 895, 2007 Kuroshio Extension Variability Explored through Assimilation of TOPEX/POSEIDON Altimeter Data into a Quasi- Geostrophic Model YOUSUKE NISHIHAMA 1 and MOTOYOSHI IKEDA 2 * 1 Division of Ocean and Atmospheric Science, Graduate School of Environmental Earth Science, Hokkaido University, Sapporo , Japan 2 Division of Environmental Science Development, Graduate School of Environmental Science, Hokkaido University, Sapporo , Japan (Received 4 April 2006; in revised form 26 April 2007; accepted 25 May 2007) Mesoscale features in the eastward extension of the Kuroshio were investigated using assimilation of TOPEX/POSEIDON (T/P) data into a three-layer quasi-geostrophic model. The T/P data exhibited an elongated state of the southern recirculation gyre in and 1997, between whose two periods the gyre had a contracted state in A few stationary eddies were located in the southern gyre during the contracted state. The baroclinic instability, which was indicated by the phase shift from the uppermost- to the lowest-layer anomalies toward the downstream side, was evident near the Kuroshio Extension (KE) path. Since the instability never appeared in the artificial model without bottom topography, the topographic barrier for the eastward flow in the lowest layer was a necessary condition for the instability. The instability synchronized with the transition in the western region of the KE axis from the elongated to the contracted states. This evolution was interpreted as if the baroclinic instability played some part in the KE states and was a trigger for the transition from the elongated to the contracted states. Keywords: Kuroshio Extension, quasi-geostrophic model, assimilation, nudging method, TOPEX/ POSEIDON. 1. Introduction The Kuroshio Extension is the lower reach of the Kuroshio in the western North Pacific after it separates from the Japan coast near 35 N, 140 E (Kawai, 1972). This is one of the most energetic regions with mesoscale features, which have typical time scales of a few months. In particular, mesoscale features such as eddy formation, movement and re-absorption into the KE have been investigated in detail in numerous papers. Although the previous papers have indicated interannual and large-scale variabilities of the characteristics of the mesoscale features, their causes have not been fully understood. To explore the main mechanisms of their generation, we have to attempt to reveal the dynamical characteristics associated with the mesoscale variabilities. In addition, the mesoscale variabilities are reconstructed from satellite data, such as T/P altimeter data. Therefore, an assimilation of the T/P data is required to reconstruct the mesoscale features, particularly in the subsurface layer * Corresponding author. mikeda@ees.hokudai.ac.jp Copyright The Oceanographic Society of Japan/TERRAPUB/Springer because the satellite data, by themselves, do not permit us to draw inferences about the ocean interior. In the present paper we focus on the following mesoscale character and large-scale character, which can interact with the mesoscale one, among all the variability in the KE region: viz., (1) many mesoscale eddies and meanders formed in the KE region, and (2) an oscillation between an elongated and a contracted state in the KE region. These two characteristics are the criteria by which the model and the data assimilation technique can be evaluated. A crucial mechanism for the mesoscale features is baroclinic instability, which is indicated by the phase shift from the surface to the subsurface layers toward the upstream side. Following these attempts, we focus on the mesoscale variability and its difference between the elongated and contracted states. The first category of the two characteristics that we focus on has been investigated in previous studies. Based on conventional observation data, the pioneering work of Kawai (1972) allowed him to describe the KE meanders and mesoscale eddies, such as the pinching-off processes of the rings from the KE meanders, their interaction with other rings and their re-absorption by the KE. 879

2 Analyzing temperature structures in TRANSPAC XBT data (in ) and hydrographic data, Mizuno and White (1983) showed that the KE path moved southward from 36 N in to 34 N in , the quasistationary KE meanders became unstable, and eddy activity and the ring formation rate increased. Moreover, they reported the following four findings: (a) two quasistationary meanders existed with their ridges located at 144 E and 150 E in the upstream region of the KE. (b) These meanders had unstable mesoscale perturbation with time scale from two weeks to two months. (c) The maxima in rms differences were located on the two stationary meanders. (d) They interpreted the two quasi-stationary meanders as standing Rossby lee waves similar to meanders induced by the Izu Ridge. Using Geosat sea surface dynamic topography (SSDT) and sea surface temperature (SST) data, Aoki et al. (1995) showed that the westward phase speed of SSDT anomalies was faster than the theoretical phase speed of the first baroclinic mode Rossby wave in the west of the Shatsky Rise and is consistent with the theoretical phase speed in the downstream region. The second category is concerned with the low-frequency variability in the KE region. Satellite altimeter data obtained over the past decade have provided us with a unique opportunity to investigate it in detail thanks to the high resolution and the duration of the data. Using the Geosat and ERS-1 data, Jacobs et al. (1994) showed that the KE path shifted northward in compared to They interpreted this shift as the result of the passage of a westward-moving warm anomaly associated with a Rossby wave, which was originated by the equatorial ENSO event. By analyzing the T/P data ( ) and Geosat data ( ), Qiu (2000, 2002) classified the states in the KE region into elongated and contracted states. He showed that, in the elongated (contracted) state, the KE had a larger (smaller) eastward surface transport, a greater (smaller) zonal penetration, and a more northerly (southerly) zonal-mean path. In addition, the KE adopted the elongated state in , while it changed from the elongated state in to the contracted state in and then back to the elongated state in A forcing mechanism for the low-frequency variability in the KE region was investigated using wind stress data and sea surface height (SSH) satellite data. Using NCEP wind stress data and T/P SSH data, Qiu (2003) suggested that the modulation in the zonal mean jet was remotely forced by wind stress curl anomalies in the eastern North Pacific related to the Pacific decadal oscillations (PDOs). The earlier papers listed above were produced using only satellite data or conventional observation data. Although the satellite remote sensing data are quite useful for monitoring the ocean surface, by themselves the data cannot provide information on the ocean interior. Conventional observation data have generally lower spatial and temporal resolution than satellite data. We therefore need to use a numerical model to create a high resolution data set of the ocean interior to investigate the dynamical character of the mesoscale variability in the KE region. In addition, our model needs to be able to simulate the mesoscale variability listed above: viz. the formation of the eddies and meanders in the KE and the difference in the phase velocity between the east and the west of Shatsky Rise. The model should be accurate enough for baroclinic instability and Rossby wave propagation. Even if the model is perfect, the numerical model cannot reproduce each mesoscale feature by itself, because the correct triggers of meanders and Rossby waves are not introduced without assimilation of observation data. In the last decade, data assimilation methods have been used to investigate the mesoscale features. In twin experiments, Haines (1991) tested the nudging method to see whether the model could reconstruct mesoscale features or not. Judging from the distribution map of the eddy field and the difference between the data and the model, he concluded that the nudging method worked very well, unless barotropic features were dominant. In addition, he carried out assimilation runs with 20 day- and 40 day-insertion frequencies, and showed rapid convergence to the true ocean in all layers. There is no significant improvement after about 550 days. By assimilation into an eddy-resolving model of the Pacific Ocean, the ocean field can be reproduced better than by a model alone (Smedstad et al., 1997). Kubota and Yokota (1999) assimilated the T/P data into a 1.5- layer eddy-resolving model of the Pacific to investigate the effect of the assimilation technique. They showed that the difference between the model and the T/P data decreased rapidly and the data assimilation improved the field during two months from the last update. An operational approach has also been adopted. The operational nowcast system was developed to represent the ocean structures in the mid/high latitudes of the North Pacific (13 N 55 N, 120 E 110 W), and a comprehensive analysis system has been in operation since January 2001 at Japan Meteorological Agency. This consists of an objective analysis system, an assimilative ocean model in the North Pacific (COMPASS-K), and their assimilation by the nudging method. The system has been used to examine the characteristic mesoscale features in the North Pacific, such as the variability of the KE path separation off the Boso Peninsula and the activity of eddies around Japan in the ocean model (Kamachi et al., 1998). Following these previous papers, we decided to use the nudging method as the simplest assimilation technique to reproduce and investigate the flow field of the ocean interior. Furthermore, we are positive about the role of data 880 Y. Nishihama and M. Ikeda

3 assimilation, which aids reproduction of the mesoscale features and makes it possible to investigate the largescale variability in the KE region, upon which the present paper focuses. In the present paper we assimilate the T/P data into the three-layer eddy-resolving QG model of the KE. By the assimilation method we try to reconstruct the flow field of the ocean interior, which cannot be measured by the T/P altimeter. After the reconstruction, we investigate the relation between the temporal change of the mesoscale variability and the large-scale, low-frequency, elongatedcontracted state change in the KE. We then attempt to discover the main cause of this change. The present paper is organized as follows: we describe the numerical model in Section 2 and the assimilating method called the nudging method in Section 3. The T/P satellite altimeter data processing and analysis are explained in Section 4. The results of the data assimilation are reported in Section 5, and the summary and discussion are finally given in Section Basic Equations, Initial and Boundary Conditions and Numerical Method of QG Model A numerical model for data assimilation must be suitable for the particular purposes of research and the characteristics of assimilated data in general. The purpose of the present paper is to investigate the low-frequency variability of the mesoscale features in the KE region under the condition that the mesoscale features are reproduced by the model. We therefore have to choose the model in relation to: (1) the model equation type (primitive (PE) or QG): (2) the vertical resolution: (3) the spatial resolution: and (4) the bottom topography. According to Smedstad and Fox (1994), a QG model does not express the small-scale features (less than internal deformation radius). In contrast, a PE model expresses all-scale features. Therefore, the former expresses the mesoscale features more efficiently than the latter, and we can more effectively extract the mesoscale features from the 4-D data set by using the QG model than the PE one. Moreover, when the T/P data are assimilated into the PE model, the model requires careful treatment to avoid a shock to it (Kamachi et al., 1998). Since the QG model satisfies the geostrophic balance and is not so sensitive to sudden changes in the model fields, the QG model is more suitable for the nudging method. Let us consider vertical resolution: a dynamical ocean field can be decomposed into vertical modes. Various papers have indicated that low-order baroclinic structures are dominant in the mesoscale features. Therefore, a model needs to reproduce a few lowest-order modes, which in the present study are barotropic plus the first and second baroclinic components: i.e., we examine a three-layer model. The baroclinic modes are also necessary for describing baroclinic instability, which is important in the KE region. There are many methods by which we could estimate an ocean interior field from information of the sea surface, such as sea surface dynamic topography (SSDT) by a vertical projection. In a field containing Rossby waves, for example, the higher vertical modes have smaller differences in their phase speeds. Hence, the higher modes are generally suppressed in SSDT. In other words, SSDT data have only low-passed sea surface information (Hsieh, 1985). Consequently, even if the assimilation methods can maintain solutions with high vertical resolution, we cannot rely on a result in high vertical modes. As for the spatial resolution, even if the model resolution is quite a lot higher than that of the T/P data, the model cannot always reproduce the mesoscale features in greater detail. We, however, need to use the higher resolution model, which can resolve the mesoscale features characterized by an internal deformation radius. Lastly, because the bottom topographic effects are essential to the mesoscale features in the KE, the model is supposed to have the bottom topography. Therefore, we do not use a reduced gravity model. The two-layer model usually receives excessively strong bottom topographic effects compared to the three-layer model. For these four reasons, we decided to use the three-layer QG model over the bottom topography with a 1/6 degree spatial resolution. On the assumption of an inviscid fluid, potential vorticity conservation equations for a three-layer QG fluid on a β plane, in non-dimensional form, are: Here: D1 2 η1 H p1 + by+ 0 Dt d =, D2 2 η2 η1 H p2 + by+ 0 Dt d =, D3 2 h 2 H p3 by 0 Dt + + η d =. ( ) η i = Fp i +1 p i, Di = ui vi Dt t + x + y, 2 H 2 2 = x + y 2 2, Kuroshio Extension Variability Explored through Altimeter Data Assimilation 881

4 pi pi ( ui, vi)=,. y x Those for a two-layer model are the same as these equations, excluding the equation for the middle layer. The suffixes denote the layer numbers from the uppermost to the lowest layer. p i is the deviation from the hydrostatic pressure in the no-flow state, and is identical to the streamfunction in the QG model. The horizontal coordinates x and y are eastward and northward, respectively, t is the time, and u and v are the eastward and northward geostrophic velocities, respectively. The horizontal coordinates x and y are nondimensionalized by L, the horizontal velocity components u j and v j by U, and the time by L/U. The nondimensional parameters in the above basic equations are d i, F i, b and ε. These are defined as d i = D i /H, F i = ρ 0 f 0 2 L 2 /(ρ i+1 ρ i )gh, b = βl 2 /U, ε = U/( f 0 L), where D i is the layer thickness, H the total fluid depth, ρ 0 the average density, f 0 the Coriolis parameter at the southern boundary, L the horizontal length scale, ρ i the i-th layer density, g the gravitational acceleration, β the latitudinal gradient of the Coriolis parameter, and U the velocity scale (maximum speed of the initial jet). L becomes an internal Rossby radius of deformation associated with the uppermost and the middle layers, when F 1 = d 1, d is the ratio of the layer thickness to the total thickness, F i is the internal rotational Froude number, and b is the nondimensional β-plane parameter. The location of the interface between the uppermost and the middle layers is given by d 1 + εη 1, and the interface between the middle and lowest layers is at d 1 d 2 + εη 2. The bottom topography is at 1 + h/ε. The scales referred to the KE are L = 21.6 km, u = 1.0 m s 1, T = s, H = 6000 m, D 1 = 300 m, D 2 = 600 m, g = 9.8 m s 2, f 0 = s 1, β = m 1 s 1, ρ 0 = 1000 kg m 3, where L and U are determined so that the velocity profile of the basic jet may represent the KE at the mid-depth between the sea surface and the main thermocline. The non-dimensional parameters are ε = 0.454, d = 0.05, b = 0.006, F 1 = Although the value of ε is not small, the scale of the mesoscale features in the KE is larger than L, and the effective value of ε is about 0.2. The calculation domain is the North Subtropical Pacific including the KE (142 E 180 E, 30 N 40 N) and has two open boundaries on the east and west and two solid boundaries on the north and south. First of all, on the eastern boundary, which is upstream in terms of a Rossby wave, the streamfunction is fixed. In contrast, these quantities are assumed to propagate out through the western boundary at a given speed, C d < 0; ψ/ t + C d ( ψ/ x) = 0, where ψ denotes the streamfunction and potential vorticity. A similar boundary condition was used by Ikeda (1981). However carefully C d may be chosen, the solution could be gradually deformed by the western boundary condition after a large-amplitude meander or a strong eddy reaches that boundary. Nevertheless, the present solutions are essentially not deformed in all the assimilation experiments. Secondly, the northern and southern boundaries are slippery walls on which the perpendicular velocity vanishes, and the streamfunction and potential vorticity are set to be constant. In a numerical model we convert the basic equations to finite difference equations. As time advances, the numerical scheme causes a local instability. To suppress this instability, the potential vorticity at each calculating point is replaced by an average of five points, including the point itself and the four points nearest to it, every 10 days from the beginning. This smoothing scheme is very effective against the smallscale numerical instability, while it scarcely deforms the larger-scale solution. Ikeda (1981) has confirmed that the solutions obtained using this smoothing scheme describe jet meanders and detached eddies. The values of the model parameters influence the solutions. We therefore performed a sensitivity test to their values. As a result, when we choose the values listed above, we confirm that the model can reproduce the essential properties, such as the eddy detachment process from the meanders, the Rossby wave propagation associated with the eddies, and the formation of the recirculation gyre. Thus, the model can reproduce the mesoscale features in the KE. This numerical method is identical to that used in Ikeda (1981). 3. Data Assimilation Technique: Nudging Method Although the satellite altimeter data such as T/P are quite useful for monitoring the ocean surface, the data cannot by themselves provide subsurface fields. The assimilation method for taking the satellite data into a model is an effective way to estimate the subsurface field, provided the surface conditions are influenced by the subsurface conditions. In the present paper, to reproduce the surface and the subsurface fields, we assimilate the T/P (with ERS-1/2) data by the nudging method into the QG models of the KE. The uppermost-layer streamfunction is nudged to the data as, p 1,new = p 1,old + dw(x, y, t)(p 1,obs (p 1,old p 1,mean )) where p 1,old, p 1,new, p 1,obs and p 1,mean denote the uppermostlayer streamfunctions before data insertion, after data insertion, converted from the altimeter SSDT anomaly data, and of the mean state, respectively. dw(x, y, t) is the nudging coefficient. This equation means that the uppermostlayer streamfunction is modified by the quantity that is 882 Y. Nishihama and M. Ikeda

5 Fig. 1. Bottom bathymetry of the western North Pacific (20 N 50 N, 130 E 170 W): contour interval is 1000 m. Letters S, E, and H denote the Shatsky Rise, Emperor Seamounts, and Hess Rise, respectively. Black lines are the boundary of the research domain called the KE region (30 N 40 N, 142 E 180 E). (Lower panel) Location of the domains in the nine subdomains. N, S, E, W and C denote northern, southern, eastern, western and central domains, respectively. proportional to a difference between the SSDT anomaly data and the uppermost-layer streamfunction anomaly. Therefore, as the nudging coefficient dw increases, the T/P data force the model more strongly. In the PE model, the assimilation with larger dw causes unnecessary gravity waves, and hence dw needs to be smaller in order to assimilate the T/P data into the model smoothly. By contrast, if dw is quite small, the T/P data information enters into the model more slowly, meaning that the data may have to be assimilated carefully over many time steps. Using the PE model, Smedstad and Fox (1994) have shown that both methods have a minor difference in the assimilation solutions. Hence, we decided to take the data only at one time step when the data are provided. For the QG model, dw can be larger, while we are still compelled to decide the value of dw carefully. Moreover, dw is an artificial term, which is not determined by a physical consideration. In the present paper, dw is determined through the steps described below. For the first step, dw is set to be uniform in the entire model domain. In many previous studies, dw values are functions of x, y and t. In the present paper, dw is set to be spatially uniform and a function of t only. The reason behind this is that the assimilation data are edited into a 0.5 latitude/longitude regular grid with uniform reliability in almost the entire domain. In addition, we assume that the reliability of the model is also uniform in the entire domain. As in many previous papers, we carried out sensitivity tests with varying dw. The value of the spatially constant dw is varied over in each case. Judging from maps and time series of the difference between the model and the T/P assimilation data, dw is set to be 0.5 in the reference case, although the errors are insensitive to dw as long as it stays in the range The uppermost-layer streamfunction is updated by the altimeter data, while the lower-layer potential vorticity does not change at the nudging time. Therefore, the vertical projection associated with the update on the uppermost layer needs to be estimated. In the present paper, the lower-layer streamfunction is induced through vertical transfer from the uppermost-layer to the lower-layers, as specified by the assumption that the lower-layer potential vorticity is kept unchanged at updating. This is the same assumption as that used by Haines (1991), who showed that this assimilation method works well. Thus, we expect that the assimilation method may reproduce the mesoscale variability reasonably well. 4. T/P Satellite Altimeter Data Performance of data assimilation is generally influenced by the distribution of eddies in the assimilated data, which are often affected by bottom topography, as shown in many previous papers. Therefore, before carrying out Kuroshio Extension Variability Explored through Altimeter Data Assimilation 883

6 Fig. 2. Yearly averaged composite SSDT fields in the KE region in Contour intervals are 0.1 m. experiments on the assimilation of the T/P data with the bottom topography, we describe the basic mesoscale features and eddy activity in the KE region. We start this section with the method of preparing the T/P data and the definition of composite SSDT in Subsection 4.1. The basic mesoscale features in the KE region are explained with reference to previous papers in Subsection 4.2. We then divide the SSDT in a frequency domain in Subsection 4.3 and show the eddy activity, such as formation, contraction, propagation and eddy-eddy interaction in Subsection Data preparation This study uses the T/P (with ERS-1/2) altimeter data, for a 7-year period from October through October (repeat cycles 2 223). The Corrected Sea Surface Height data set (AVISO, 1996) is based on data from all these satellites. An anomaly is determined on the basis of the average sea surface height calculated from Geosat and T/P altimeter data (Yi, 1995) and edited by the optimal interpolation method, which is suitable for the mesoscale features (spatial resolution of 150 km and temporal resolution of 17 days). A problem exists in calculation of the absolute SSDT from the satellite altimeter data: because the accuracy of the Geoid is still insufficiently well known in the KE region, the absolute SSDT cannot be taken exactly as the sum of the SSDT anomaly and the Geoid. Therefore, the composite SSDT (rather than absolute SSDT) is defined as the sum of the SSDT anomaly and climatology (Ichikawa and Imawaki, 1994). The SSDT climatology is the SSDT relative to a 1000 dbar depth calculated from the temperature and the salinity data in the World Ocean Atlas on 1/4-degree grids (Boyer and Levitus, 1997). The spatial grid of these data is 1/2 degree and the temporal interval is days. These data are given by Ichikawa (2001). 4.2 General characteristics of basic mesoscale features Many previous papers have shown that the mesoscale features in the KE region are influenced by the bathymetric features, especially the Shatsky Rise near 165 E. The bottom topography in the KE region is shown in Fig. 1 with the Shatsky Rise around the center of the domain and the Emperor Seamount Chain near 170 E. The entire domain is divided into nine subdomains for the purpose of describing the mesoscale features (Fig. 1). Figure 2 shows the annual mean of the composite 884 Y. Nishihama and M. Ikeda

7 Fig. 3. Longitude-time diagram of SSDT anomalies at 33 N (left), 35 N (right): contour interval is 0.2 m. Darker (lighter) shaded areas denote positive (negative) SSDT. SSDT in the KE region in The area of the southern recirculation gyre was decreasing in and increasing in Although there were five negative (cyclonic) eddies in 1996 near 34 N at 156 E, 160 E, 164 E, 170 E and 176 E, no obvious negative eddy existed there in the other years. By contrast, there were a couple of positive eddies near 32 N over the Shatsky Rise in A remarkable feature in was a pair of positive and negative eddies near 145 E at 35 N and 34 N, respectively. The overall results are consistent with figure 5 of Qiu (2000), who utilized the annual mean kinetic energy distributions. The longitude-time diagram in Fig. 3 shows the differences in the propagation velocities and the mesoscale features clearly bounded by the Shatsky Rise (near 160 E), especially along 35 N. There were a few standing eddies near 33 N over the Shatsky Rise, which show the effects of the Shatsky Rise. On the other hand, there were standing eddies along 35 N, between 144 E and 155 E, which are considered to be standing Rossby waves or stationary KE meanders in balance between the westward Rossby wave propagation and the eastward KE advection. The propagation direction and velocity in this upstream region are more complex than those in the downstream region, since they are influenced by the latitudinal movement of the KE axis, the eddy-eddy interaction and the eddy-mean flow interaction. 4.3 Extracting eddy features In this section we propose a method for identifying individual eddies from the T/P data for the purpose of focusing on the mesoscale eddies. In addition, we try to discover the location where eddy-eddy interactions occur, since the interactions seem crucial to the performance of the data assimilation. We consider that SSDT at the point changes mainly due to three mechanisms: 1. steric height from seasonal heating and cooling: 2. propagation of Rossby waves and advection of individual eddies: 3. recirculation gyre deformation associated with the basin-scale wind variability and/or intensity of KE instability. Here we divide them in a frequency domain, assuming that the first mechanism has annual variability, the third one has interannual variability, and the second one is the remainder, with variability higher than annual. We then expect an isolated eddy to be recognized as the same eddy, regardless of the seasonal variability. In general, the first and the third mechanisms are larger in extent, while the second one has a much smaller extent. We first average the SSDT anomaly data spatially, either in the entire domain or the nine subdomains (Fig. 1), and separate it into the following three frequency ranges by the three kinds of running mean: low frequency variability (LFV), middle frequency variability (MFV) which is consistent with the annual signal, and high-fre- Kuroshio Extension Variability Explored through Altimeter Data Assimilation 885

8 Fig. 4. Time series of the LFV, MFV, HFV for the whole domain of the KE region (repeat cycles 2 202): thick, medium and thin lines denote LFV, MFV and HFV, respectively. quency variability (HFV). In the present paper, the LFV, MFV and HFV are defined as follows: the LFV is the 37 cycle-running mean, the MFV is the difference between the 17 cycle-running mean and the 37 cycle-running mean, and the HFV is the difference between 17 cycle-running mean and the spatial mean itself without running mean. We first calculated these three variabilities for the entire domain (Fig. 4). The frequency of the MFV was actually almost one year, and its amplitude was nearly constant during the time series. On the other hand, the LFV decreased from the start to Cycle 130 and increased to the end. We then calculated the same variabilities for the nine subdomains without figures. The LFVs had quite different temporal variations between the northern and the southern sides of the KE. Following this result, we decided to remove the MFV for the entire domain and use the rest. 4.4 Eddy-eddy interactions Mesoscale eddies in the KE region have been defined by a few typical conditions in previous papers. Following them, we recognize generally that: (1) the characteristic diameters of the eddies are in the range km: and (2) eddy formation is usually a rapid process, taking more than 2 weeks, in other words, with a lifetime longer than 2 weeks (e.g. Mizuno and White, 1983; Ebuchi and Hanawa, 2000). The diameters and the evolution of the eddies identified in the present paper need to be consistent with these values. The two terms anomalous point and eddy are defined below. An anomalous point is defined as the grid point at which the absolute SSDT is larger than a certain critical value. Here, one grid is 1/2 degree by 1/2 degree. An eddy is identified as a group of anomalous points satisfying the three conditions stated below. As the first condition, we decide the critical value of the anomalous point to be 0.3 m. An eddy center is set to be a maximum point of the absolute value of SSDT anomaly among a group of anomalous points. As the second condition, the group is composed of the anomalous points within 3.0 from the eddy center. As the last condition, an eddy has more than 10 anomalous points within 3.0 from the eddy center. Judging from widely accepted characteristic natures of an eddy (i.e., strength, closed contour, approximately circular shape and distribution), we are convinced that the critical values listed above are appropriate. What is described is generally an isolated mesoscale eddy. The positive and the negative eddies interact if the two eddies are located close enough to each other. We now estimate eddy-eddy interactions. The eddy radius is set to be larger than 1.0 and smaller than 3.0, and the two eddies interact at a distance of 1.0. Therefore, the two neighboring anomalous points in the two eddies are positive and negative, and there is a steep change, greater than 0.6 m per 1.0 between them. In a typical case, two eddies with radii of 1.0 and SSDT of 0.5 m at the eddy centers exist at a distance of 3.0 between the two eddy centers. In this case, a horizontal velocity at the eddy center induced by the other eddy is m s 1 if the eddy did not exist. Thus, our definition of the eddies provides appropriate eddy-eddy interactions in the KE region. Figure 5 shows the time series of the numbers of the anomalous points, with absolute values larger than 0.3 m, in the nine subdomains. The numbers of positive anomalous points were distinct only in the subdomains NW, CW and CC. By contrast, there were large numbers of negative anomalous points in the subdomains CW and SC. They had larger peaks around Cycle 150. In addition, there were almost no anomalous points (positive or negative) in the other subdomains. Figure 6 shows the distribution maps of the cycle numbers where absolute values are larger than 0.3 m. There were many positive anomalous points only in the upstream region west of 164 E, and the maxima existed at 144 E, 155 E and 158 E on the KE axis. On the other hand, many negative anomalous points were distributed around 144 E, 150 E, 153 E, 155 E, 160 E, 165 E and 170 E in the southern side of the KE. White and McCreary (1976) proposed the linear theory of a stationary Rossby lee-wave in uniform eastward flow, which is consistent with these eddies. Mizuno and White (1983) suggested that the two meanders at 144 E and 150 E have the wavelength in consistency with the Rossby lee-wave theory. Figure 7 shows the time series of the numbers of eddies which interact with each other, with a focus on the subdomain CW, where many interactive eddies exist. The numbers were low until Cycle 100, increased to Cycle 150 and decreased to the end. This time series is consist- 886 Y. Nishihama and M. Ikeda

9 Fig. 5. Time series of the points whose absolute values are larger than 0.3 m in the nine subdomains calculated from the data removed MFV for the whole domain. Thick and thin lines denote negative and positive points, respectively. Fig. 6. Distribution maps of the cycle numbers when the absolute values of SSDT are larger than 0.3 m. Values are calculated from the MFV- removed data: Left and right panels show the positive and negative eddies, respectively. Contour intervals are 5 cycles. ent with that of the numbers of the negative eddy points in Fig. 5. That is to say, the numbers of the interactive eddies are nearly proportional to the numbers of the negative eddies in the subdomain CW, but not in the subdomain SC where many negative eddies existed far from the interactive domain, since there are almost no positive eddies. 5. Results of the Data Assimilation 5.1 Cost function In this section we first determine the mean velocity field, in particular, of the middle layer in the three-layer model, by optimizing the assimilation solution which is Fig. 7. Time series of the anomalous point numbers whose neighboring eddies interact with each other in the subdomain CW. Kuroshio Extension Variability Explored through Altimeter Data Assimilation 887

10 Fig. 8. Upper: Sea surface topography (in meters) relative to 1000 dbar calculated from climatology (Boyer and Levitus, 1997). Lower: Longitudinal change of maximum geostrophic velocity calculated from climatology. reconstructed from the T/P data from October through October (repeat cycles 2 223). We then examine the energy transfer from the mean field to the perturbation field. Our investigation is extended to the baroclinic instability, which may be influenced by the bottom topography. A cost function needs to be determined for optimizing the assimilation solution. In other words, we judge whether or not the assimilation field correctly reproduces the true field using the cost function. We define it as the spatial mean of the square of the difference between the observed T/P data and the uppermost-layer streamfunction immediately before the time of data assimilation: ( ) 1 2 COST()= t pobs( xi, yj, t) p1( xi, yj, t), N i j where p obs (x i, y j, t) is the observed SSDT data, and p 1 (x i, y j, t) is the uppermost-layer pressure, or streamfunction in the model. This function is used in many papers such as Smedstad and Fox (1994) and Kubota and Yokota (1999). 5.2 Optimum mean field The uppermost-layer mean velocity (u 1 ) is determined based on the longitudinal mean of the geostrophic velocity calculated from the SSDT climatology data (Boyer and Levitus, 1997). The lowest-layer field is given no mean velocity. The middle-layer mean velocity (u 2 ) is assumed to be proportional to that in the uppermost layer. The ratio (u 2 /u 1 ) is determined from the sensitivity tests by data assimilation within the range of the ratio The assimilation experiments proceed by assimilating 222 times at every 10-day cycle for the total period of 2220 days. The zonally uniform mean flow is used, instead of the climatology in Fig. 8, consisting of an eastward-flowing zonal jet in the uppermost layer, since the assimilation s performance is not greatly affected by the zonal gradient of the mean velocity (not shown here). The jet has a Gaussian velocity profile: y y p1( y)= u 0 1 exp R p2( y)= rp1( y), 2 dy where p 1 (y) and p 2 (y) are the streamfunctions of the uppermost and the middle layers, u is maximum velocity, R is the e-folding width of the current, and r is the ratio of the maximum velocity of the middle layer to that of the uppermost layer. Therefore, the middle layer mean flow is also an eastward jet with the same Gaussian velocity profile. In this paper, u 1 is set to 0.2 m s 1, and R is 270 km. 888 Y. Nishihama and M. Ikeda

11 Fig. 9. Sensitivity test to the mean flow velocity in the middle layer. Fig. 10. Upper: Time series of cost function values for the entire domain with real bottom (EX1) and flat bottom (EX2). Thick and thin lines denote EX1 and EX2, respectively. Lower: As upper, but for normalized cost function values. Next, we need to determine the value of u 2 using sensitivity tests. When the temporal mean of the cost function value is at its smallest value, we judge that the value of u 2 is the most appropriate. As shown in Fig. 9, the cost function value attains its minimum value when u 2 is in the range m s 1. Prior to this long nudging calculation, we perform sensitivity tests on the middle layer velocity with a shorter nudging solution of period 100 days starting from the beginning data cycle No. 2 No In this sensitivity test, u 2 = 0.04 is the best parameter. In this range, the assimilation performance is not greatly affected by the value of r (u 2 ). Therefore, we take the first digit in r, and the values of r and u 2 are fixed to 0.2 and 0.04 m s 1 hereafter. We carry out assimilation experiments using the three layer model with the real bottom topography as the standard case. Moreover, we run the three layer model with the flat bottom topography as a comparison experiment. 5.3 Analysis of baroclinic instability We mainly present the standard case with the real topography (EX1) using the three layer model, along with the comparison case on the flat bottom (EX2) to investigate the bottom topographic effects on baroclinic instability. First, we show both ordinary and normalized cost function values for the entire domain in EX1 and EX2 (Fig. 10). The ordinary cost function value in EX1 is larger than that in EX2. This is because the uppermost-layer eddy field in EX1 is more vigorous than that in EX2 due to baroclinic instability as a result of the bottom topographic effect. Therefore, the activity of the uppermost-layer eddy Kuroshio Extension Variability Explored through Altimeter Data Assimilation 889

12 Fig. 11. Upper: Temporal mean of maximum correlation lag for the real bottom (EX1), where the start cycles are from Cycle 2 to Cycle 187. Lower: As upper figure, except for flat bottom (EX2). Contours are 8.0, 4.0, +4.0 and Darker (lighter) shaded areas denote the positive (negative) maximum correlation lag. Unit +1.0 means the lowest layer streamfunction has maximum lag correlation between the uppermost layer streamfunction at 1/6 grid eastern side than the uppermost layer. field is taken into consideration when evaluating the optimization. The normalized cost function is then defined as follows: NCOST t ()= 1 N µ 1 t 1 N i j i j () COST t ( p1( xi, yj, t) µ 1() t ) ( ) 1 2 pobs( xi, yj, t) µ 2() t N ; i j 1 ()= p ( xi, yj, t) ; 1 µ 2 t pobs xi, yj, t. N ()= ( ) i j Here, µ 1 and µ 2 denote the variance of the uppermost-layer streamfunction and the assimilating data, which describe the activity of the eddy field. The normalized cost function value in EX1 is less than that in EX2. We therefore judge that EX1 is a more realistic simulation than EX2. Many mesoscale- and smaller-scale eddies moved along the bottom slope of the Shatsky Rise and propagated with effects of the bottom topography. In addition, as discussed later, baroclinic instability is much weaker 2 in EX2, probably due to its more intense barotropic recirculation gyres. In the case when the baroclinic instability is stronger, more energy is transformed into the eddy field and the mesoscale eddy field is more vigorous. Thus, the bottom topography is important in reproducing the ocean, especially the mesoscale field. Before the assimilation experiments are performed, we carried out sensitivity tests by examining the anomalyfield development on the straight mean flow similar to that in the assimilation experiments and an anomaly field for an idealized upstream meander. These experiments are typical of those done by Ikeda (1981). The results of these experiments shows that when the uppermost-layer velocity is too weak (slower than 0.5 m s 1 ) or the middlelayer velocity is too strong (faster than 40% of the uppermost layer), meanders and eddies are fully developed due to the lack of baroclinic instability. Our explanation is that, in the assimilation experiments EX1 and EX2, mesoscale anomalies fully develop only in EX1, while they are not active in EX2. The cost function value for the entire domain is increasing from Cycle 60 to Cycle 160 and decreasing from Cycle 160 to Cycle 210. It is easy to see the time series of the cost functions look similar between EX1 and EX2, with a trend similar to the number of negative eddies in the subdomain CW (Fig. 5), which represents the meridionally central and zonally western region in the nine subdomains, as well as the eddy-eddy interactions (Fig. 7). We suggest that the cost function values depend on the eddy distribution in the subdomain CW along with 890 Y. Nishihama and M. Ikeda

13 Fig. 12. Time series of the baroclinic instability index, defined as the spatial mean of the maximum correlation lag for the nine subdomains. the bottom topographic effects. Our analysis is now extended to baroclinic instability. A typical and necessary characteristic of a baroclinically unstable mesoscale feature associated with an upper-layer jet is the phase shift between the upper and the lower portions: i.e., the lower portion advances downstream relative to the upper portion. We calculate a zonal lag-correlation for a one-year period (36 continuous cycles) between the uppermost- and lowest-layer streamfunctions. The lag at maximum correlation is measured by model grid points. Figure 11 shows the temporal mean of the maximum correlation lag for EX1, where the start cycles are from Cycle 2 to Cycle 187. The baroclinic instability occurs only in the shallower regions (the Shatsky Rise and the Hess Rise) in EX1, and does not occur in EX2. We then calculate the spatial mean of the maximum correlation lag, which is called a baroclinic instability index. The time series of the index are shown in Fig. 12 for the nine subdomains. Baroclinic instability does not occur in the subdomains, NW, SW, SC and SE for either EX1 or EX2. In contrast to stable EX2, the instability occurs in the subdomains CC, CE, NC and NE for EX1. The instability tends to increase from the start to Cycle 40 and then decrease to Cycle 120, after which it stays constant to the end in the subdomain CW. This time series is rather opposite to CC, in which it decreases from the start to Cycle 120, rapidly increases to Cycle 140 and stays high to the end. Our interpretation is that baroclinic instability is present in either the upstream region (CW) or the downstream part of the KE region (CC). This result is compared with the two phases of the southern recirculation gyre, which was decreasing in (Cycles ) and increasing in (Cycles ). During the period , the baroclinic instability decreased in the subdomain CW, as the KE shifted from the elongated to the contracted state. While the instability was weak in , the KE split Kuroshio Extension Variability Explored through Altimeter Data Assimilation 891

14 Fig. 13. Yearly averaged streamfunction maps of all layers in for the real bottom (EX1); contour intervals are 0.8, 0.2 and 0.1 in the uppermost, middle and lowest layer. Darker (lighter) shaded areas indicate p 1 > 1.6 (p 1 > 0.8), p 2 > 0.4 (p 2 > 0.2) and p 3 < 0.2 (p 3 < 0.1). with a pair of positive and negative eddies near 145 E at 36 N and 34 N, respectively. The instability in the subdomain CC jumped as the gyre expanded in Ocean field affected by bottom topography Figures 13 and 14 show the yearly averaged streamfunction maps of all layers in for EX1 and EX2, respectively. In EX2, the southern recirculation gyre was formed consistently in the middle and lowest layers. This barotropic gyre was not formed in EX1 because it was obstructed by the bottom topography, mainly the Shatsky Rise. All mesoscale eddies in the subdomains CW, CC and SC had strong barotropic structures in EX2, whereas they had a variety of structures in EX1. A negative mesoscale eddy cannot exist in the lowest layer over a shallow region such as the Shatsky Rise and the Emperor Seamount chain. Therefore, in these shallow regions of EX1, baroclinic eddies were trapped on the topography in the lowest layer. Barotropic negative eddies were formed in the deep regions near 150 E, 33 N and 165 E, 33 N in In addition to these persistent baroclinic and barotropic eddies, transient eddies near these regions moved and disappeared quickly due to the topographic Rossby waves and advection by eddy-eddy interaction. Qualitatively consistent features were reproduced in the variational assimilation model (Ikeda and Yamada, 2006). 6. Summary and Discussion In this paper, the T/P altimeter data were assimilated into the three-layer QG model on the realistic bottom topography using the nudging method. The uppermost layer was given a Gaussian jet as a mean eastward-flowing component, along with assimilation of the anomalies in the altimeter data. No mean flow was given to the lowest layer, while the recirculation gyres were induced over the entire depth by the assimilation. The middle-layer mean 892 Y. Nishihama and M. Ikeda

15 Fig. 14. As Fig. 13, but for flat bottom (EX2). Fig. 15. Time series of baroclinic instability index in subdomain CW (thick line) and SSH difference between southern and northern boundaries in subdomains CW and CC (thin line). jet was determined by optimizing the difference between the assimilation solutions and the T/P data. The assimilation solutions were analyzed in particular for baroclinic instability associated with the KE, where the instability index was calculated from the phase shift from the uppermost- to the lowest-layer anomalies. The main comparison is between the states of the KE and the baroclinic instability index. The two states were identified as the elongated and the contracted ones. The southern recirculation gyre length reduced in and spread in This evolution corresponds to the shift from the elongated state to the contracted state Kuroshio Extension Variability Explored through Altimeter Data Assimilation 893

16 and the following opposite shift (Qiu, 2002). He attributed the evolution to basin-wide external wind forcing, along with the nonlinear dynamics associated with the southern recirculation gyre. We attempted to relate the evolution with the baroclinic instability. We could imagine two types of relationship: the KE states determine the intensity of baroclinic instability (e.g., more intense instability occurs with the elongated state), or else the baroclinic instability determines the KE states (e.g., the elongated-tocontracted shift occurs due to more intense instability). We compared the low-frequency variability in the KE transport in the subdomain CW and CC with the baroclinic instability in the subdomain CW (Fig. 15). When the KE shifted from the elongated state to the contracted state during Cycles ( ), strong baroclinic instability occurred in the upstream region of the KE with a peak in Cycles Opposite to this period, from Cycle 140 to the end (in ), the baroclinic instability did not occur, and the KE shifted from the contracted to the elongated state. Therefore, the baroclinic instability played some significant roles in the low-frequency variability in the KE by weakening the KE flow. In addition, the cost function value is large when the positive and negative eddies interact with each other in the subdomain CW, and it is small when many negative eddies exist in the subdomain SC. Therefore, the value does not depend on the strength of the baroclinic instability but rather on eddy-eddy interactions (Figs. 7 and 10). We found that the strong baroclinic instability occurred in the upstream region near the KE path where many positive and negative eddies existed, and the region tended to spread in and to contract in This evolution indicates that many eddies are formed after the energy is transformed from the mean field to the eddy field. No baroclinic instability occurs with the flat bottom topography, as the vertical shear becomes too weak to induce the instability. From these results we conclude that KE variability is influenced by: (1) the low frequency variability of the KE states (elongated or contracted): (2) the bottom topography (mainly the Shatsky Rise) effects which block the eastward flow below the KE: and (3) the interaction between the positive and the negative eddies. The flow structures in the lowest layer have been compared with the observed data (Joyce and Schmitz, 1988). There were two recirculation gyres, composed of the eastward flow (less than 0.01 m s 1 ) in the deep layer (near 4000 m) under the KE path (34 N 37 N) along with two westward components (less than 0.03 m s 1 ) in the northern and the southern regions of the KE path at 165 E east of Shatsky Rise. There is a very week eastward flow (less than 0.01 m s 1 ) in the deep layer at 175 W. Judging from the gradient of the streamfunction in the lowest layer, the temporal mean flow in the lowest layer shows qualitatively similar structures to the observed data. The temporal mean velocity is of the order of 0.02 m s 1, which is consistent with the observed data. Acknowledgements Much appreciation is given to Drs. Youichi Ishikawa and Nobumasa Komori concerning the successful technique for assimilation into the model. We express our gratitude to Prof. Atsushi Kubokawa, Dr. K. V. Valsala and Mr. J. J. Carriere for their fruitful discussions. We also thank Prof. Kaoru Ichikawa for producing the TOPEX/POSEIDON altimeter data set interpolated onto the 0.5 latitude-longitude grid. This study was partly supported by grants from the Japanese Ministry of Education, Culture, Sports, Science and Technology. All the figures were produced by the GFD-DNNOU Library. References Aoki, S., S. Imawaki and K. Ichikawa (1995): Baroclinic disturbances propagating westward in the Kuroshio Extension region as seen by a satellite altimeter and radiometers. J. Geophys. Res., 15, AVISO (1996): AVISO Handbook for Merged TOPEX/ POSEIDON Products, AVI-NT CN, Edition 3.0. Boyer, T. P. and S. Levitus (1997): Objective Analysis of Temperature and Salinity for the World Ocean on 1/4 Degree Grid. NOAA Atlas NESDIS, 11. Ebuchi, N. and K. Hanawa (2000): Mesoscale eddies observed by TOLEX/ADCP and TOPEX/POSEIDON altimeter in the Kuroshio recirculation region south of Japan. J. Oceanogr., 56, Haines, K. (1991): A direct method for assimilating sea surface height data into ocean models with adjustments to the deep circulation. J. Phys. Oceanogr., 21, Hsieh, W. W. (1985): Modal bias in sea level and sea surface temperature, with applications to remote sensing. J. Phys. Oceanogr., 15, Ichikawa, K. (2001): Variation of the Kuroshio in the Tokara Strait induced by meso-scale eddies. J. Oceanogr., 57, Ichikawa, K. and S. Imawaki (1994): Life history of a cyclonic ring detached from the Kuroshio Extension as seen by the Geosat altimeter. J. Geophys. Res., 99(C8), Ikeda, M. (1981): Meanders and detached eddies of a strong eastward-flowing jet using a two-layer quasi-geostrophic model. J. Phys. Oceanogr., 11, Ikeda, M. and A. Yamada (2006): Mean circulation induced over bottom topography by mesoscale variabilities in the Kuroshio Extension. J. Oceanogr., 62, Jacobs, G. A., H. E. Hurlburt, J. C. Kindle, E. J. Metzger, J. L. Mitchell, W. J. Teague and A. J. Wallcraft (1994): Decadescale trans-pacific propagation and warming effects of an El Niño warming anomaly. Nature, 370, Joyce, T. M. and W. J. Schmitz, Jr. (1988): Zonal velocity structure and transport in the Kuroshio Extension. J. Phys. 894 Y. Nishihama and M. Ikeda