Editorial Type:
Article
Article Type:
Research Article

Meridional Location and Profile of a Prematurely Separated WBC Extension Jet in a Two-Layer System

Atsushi Kubokawa aFaculty of Environmental Earth Science, Hokkaido University, Sapporo, Japan

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Online Publication:
14 Mar 2023
Print Publication:
01 Apr 2023
DOI:
https://doi.org/10.1175/JPO-D-21-0313.1
Page(s):
995–1010
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Abstract

Western boundary currents (WBCs) under no-slip boundary conditions tend to separate from the coast prematurely (without reaching the intergyre boundary) and form eastward jets. This study theoretically considers the meridional structure and location of a prematurely separated WBC extension jet using a two-layer quasigeostrophic model. Assuming homogenized potential vorticity (PV) regions on both sides of and below the jet, we constructed a simple model for the meridional profiles of the zonal flows in the western subtropical gyre. This work clarifies that the meridional structure can be determined if two variables, such as the strength of the PV front (difference in PV across the jet) and the value of the streamfunction at the jet’s center, are given in addition to the meridional profile of the Sverdrup zonal flow and the vertical stratification. The zonal velocity profiles in both layers agreed well with those obtained by numerical experiments. When the jet is close to the intergyre boundary, the meridional location of the jet depends only on the front’s strength. When the northern recirculation gyre is detached from the intergyre boundary and the local wind effect on the jet is negligible, comparisons with the numerical experiments suggest that the jet’s central streamline connects to the central streamline of the eastward Sverdrup flow. We also found that a downward Ekman pumping velocity shifts the jet southward.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Atsushi Kubokawa, kubok@ees.hokudai.ac.jp

Abstract

Western boundary currents (WBCs) under no-slip boundary conditions tend to separate from the coast prematurely (without reaching the intergyre boundary) and form eastward jets. This study theoretically considers the meridional structure and location of a prematurely separated WBC extension jet using a two-layer quasigeostrophic model. Assuming homogenized potential vorticity (PV) regions on both sides of and below the jet, we constructed a simple model for the meridional profiles of the zonal flows in the western subtropical gyre. This work clarifies that the meridional structure can be determined if two variables, such as the strength of the PV front (difference in PV across the jet) and the value of the streamfunction at the jet’s center, are given in addition to the meridional profile of the Sverdrup zonal flow and the vertical stratification. The zonal velocity profiles in both layers agreed well with those obtained by numerical experiments. When the jet is close to the intergyre boundary, the meridional location of the jet depends only on the front’s strength. When the northern recirculation gyre is detached from the intergyre boundary and the local wind effect on the jet is negligible, comparisons with the numerical experiments suggest that the jet’s central streamline connects to the central streamline of the eastward Sverdrup flow. We also found that a downward Ekman pumping velocity shifts the jet southward.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Atsushi Kubokawa, kubok@ees.hokudai.ac.jp
Keywords: Baroclinic flows; Fronts; Ocean circulation; Jets; Quasigeostrophic models

1. Introduction

Strong western boundary currents (WBCs) in subtropical gyres such as the Kuroshio and the Gulf Stream tend to separate from the coast long before reaching the intergyre boundary (i.e., between the subtropical and subarctic gyres) and then flow eastward as a WBC extension jet. For example, the Kuroshio generally leaves the southern coast of Japan at approximately 35°N, 140°E (e.g., Niiler et al. 2003) and then flows eastward. The latitude of the Kuroshio extension jet (approximately 35°N) is considerably south of the intergyre boundary, which is at approximately 40°–45°N based on the Sverdrup streamfunction (e.g., Aoki and Kutsuwada 2008). The Gulf Stream also separates from the coast south of the intergyre boundary.

Since this separation from a coastal boundary seems to occur at a cape (Cape Inubo for the Kuroshio and Cape Hatteras for the Gulf Stream), researchers have postulated that either the coastline orientation or the bottom topography plays an important role in determining the separation point and the latitude of the extension jet (e.g., Dengo 1993; Özgökmen et al. 1997; Marshall and Tansley 2001). However, using an ocean general circulation model (OGCM), Nakano et al. (2008) showed that the latitude of the eastward extension jet does not change significantly, even in the case of a rectangular basin. Alternatively, they discussed the importance of a pair of recirculations on the northern and southern flanks of the jet embedded in the Sverdrup gyre.

This structure is similar to the phenomenon known as “premature separation” under the no-slip or partial slip boundary condition in a quasigeostrophic (QG) model (e.g., Haidvogel et al. 1992; Verron and Blayo 1996). Under the no-slip boundary condition, the western boundary creates positive vorticity on the western side of the boundary current, while the northward advection of the low potential vorticity (PV) fluid on the eastern side generates negative vorticity (see, e.g., Kiss 2002). These vorticities yield northern cyclonic and southern anticyclonic recirculations and maintain the eastward jet. Using a two-layer QG model, Sue and Kubokawa (2015) conducted a series of numerical experiments to study the dependence of the meridional position of the jet on various parameters, such as the coefficient of viscosity, nonlinearity, partial slip coefficient, bottom stress, and stratification, and they reported that the meridional position of a prematurely separated jet is insensitive to these parameters, although these parameters are important for the occurrence of premature separation. They also showed that the jet location strongly depends on the meridional profile of the wind stress curl. Özgökmen et al. (1997) reported that the coastline orientation can determine where separation occurs, but the latitude of the separated jet varies depending on the meridional profile of the wind stress curl, as well.

The present article revisits the issue of a prematurely separated WBC extension jet with recirculations in a two-layer QG model. Our model differs from that of Sue and Kubokawa (2015) in terms of the viscosity form; the present model adopts biharmonic viscosity, while Sue and Kubokawa’s model uses harmonic viscosity. The meridional width of recirculations tends to be wider in the former than in the latter because the decay rate of mesoscale eddies depends on the viscosity form. In the present paper, we reconsider this problem by constructing a theoretical model for the meridional profile of the western subtropical gyre with an extension jet and recirculations. The theoretical model consists of piecewise uniform PV regions and Sverdrup profiles. The theoretical method adopted in this paper is similar to those of Cessi (1988) and Sheremet (2002), but we also employ the integral constraint derived by Sue and Kubokawa (2015). The first purpose of this approach is to clarify how many variables need to be specified to obtain the profiles, including the meridional location of the jet; then, we discuss how the meridional location of the jet is determined, using numerical experimental results together.

After describing the QG model and numerical experiments in section 2, section 3 constructs the theoretical model and briefly discusses its solutions. Section 4 compares the numerical results and the theory focusing on the meridional location of the jet, and section 5 summarizes and discusses the outcomes.

2. Model description

A two-layer QG model on a β plane is used in this paper. The nondimensional equations are
∂q˜i∂t+1βJ(ψi,q˜i)+∂ψi∂x=δ1,iwe−ν∇6ψi−δ2,ir∇2ψ2,
with
q˜i=∇2ψi+(−1)iFi(ψ1−ψ2),
where i is the layer number (i = 1, 2); t denotes time; x and y are the eastward and northward coordinates, respectively; ∇ is the horizontal differential operator; ψi is the streamfunction in the ith layer (i = 1, 2); δi,j is the Kronecker delta function; we is the Ekman pumping velocity; ν is the coefficient of horizontal biharmonic viscosity; and r is the coefficient of bottom Ekman friction.
The equations are nondimensionalized by using the meridional length of a wind-driven gyre L, the velocity scale of the Sverdrup flow U=fWe/β*H1, and the time scale of 1/β*L, where f is the Coriolis frequency at the center of the domain, H1 is the upper-layer thickness, We is the scale of Ekman pumping, and β* is the meridional gradient of the Coriolis frequency. The nondimensional parameters in (1) and (2) are
β=β*L2U,F1=ρ0f2L2ΔρgH1,F2=ρ0f2L2ΔρgH2,
where H2 is the lower-layer thickness, g is the acceleration due to gravity, ρ0 is the mean density of seawater, and Δρ is the density difference between the upper and lower layers. If we define F = F1 + F2, F−1/2 is the nondimensional baroclinic deformation radius Rd/L, and F1=(H2/H)F and F2=(H1/H)F, where H = H1 + H2.
We consider a rectangular basin with a double-gyre system. The nondimensionalized meridional extent of the basin is −1 < y < 1, and the corresponding zonal extent is 0 < x < 2. y = 0 constitutes the gyre boundary. The nondimensional Ekman velocity we is normalized so that the maximum Sverdrup streamfunction is unity, i.e.,
|∫20wedx|max=1.
We adopt the form of we as
we(x,y)={12−xfsin⁡π[(1−α)y+αy3]for xf≤x<20for 0<x<xf,
where α is a parameter controlling the meridional profile of the Sverdrup flow and xf gives the region of the wind forcing. When α = 0, the profile is sinusoidal, and the gyre center moves away from the intergyre boundary as α increases. The meridional profile of we multiplied by xf − 2 for x ≥ xf is the same as that of the upper-layer Sverdrup streamfunction (shown in Figs. 3, 4, and 6) for the cases of α = 0.5, 0, and 0.75. Here, we is set to zero in the western region because we seek to eliminate the local effect of we on the system consisting of the extension jet and the recirculation gyres. Unless otherwise noted, we use we with xf = 1.
To obtain the prematurely separated solution, we set the lateral boundary condition on the western boundary to be no-slip. On the other hand, because we compare the meridional profile of the flow field with an inviscid theory, we set all other boundary conditions to be free-slip. That is,
∂ψi∂η=0,∂3ψi∂η3=0on the western boundary,
∂2ψi∂η2=0,∂4ψi∂η4=0on the other boundaries,
where η is the coordinate normal to the boundary. While ψi is constant along the boundary due to no-normal flow condition, the boundary value of the baroclinic streamfunction, ψ1 − ψ2, varies with time, since the mass conservation in each layer is considered. The numerical methods employed herein are similar to those used in the ocean part of the quasi-geostrophic coupled model (Q-GCM; Hogg et al. 2003).
We set the meridional length of the gyre L to be 45 times the baroclinic Rossby deformation radius Rd; i.e., the meridional extent of the gyre is 2025 km if Rd = 45 km, and the model basin is 90×90Rd2; this yields F = 2025. We set H1/H2 = 1/4 so that F1 = 1620 and F2 = 405. Furthermore, we set β = 5000, which is obtained as follows: if we assume that the Sverdrup transport of the wind-driven gyre UH1L is 32 Sv (1 Sv ≡ 106 m3 s−1), with β*=2×10−11 s−1 m−1, H1 = 103 m, and L ≃ 2 × 106 m, we obtain
β=β*L2U=β*H1L3UH1L≃2×10−11×103×(2×106)332×106=5000.
The horizontal viscosity ν is unknown, and we choose it so that the width of the viscous boundary layer l = ν1/5 becomes 0.3 times the deformation radius (=0.3F−1/2); this yields ν = (0.3/45)5. The bottom friction coefficient r is used to control the strength of the PV front; we adopt values of r = 5.0 × 10−3, 2.5 × 10−3, and 10−3, where r = 10−3 corresponds to a damping time of 286 days. The model grid number is 512 × 512, so the grid size is approximately 0.176F−1/2. The baroclinic adjustment time is 4050 based on the baroclinic Rossby wave speed 1/F. The time needed for the system to reach a statistically steady state is longer than the Rossby wave adjustment time because advection is important. We start the numerical computation from a resting state and run the model to t = 1.5 × 105. Time-mean fields are obtained by averaging the data from t = 105 to 1.5 × 105.

Figure 1 shows an example of the numerical result in which xf = 1, α = 0.5, and r = 10−3. The prematurely separated jets in the upper layer are seen in both the snapshot (Fig. 1a) and the time-mean (Fig. 1c). The lower-layer streamfunction field in the snapshot (Fig. 1b) is filled with mesoscale eddies, and thus, we cannot discern any organized structures, whereas we can detect the recirculation gyres in the time-mean field (Fig. 1d). The eastward jets are associated with strong PV fronts (Fig. 1e), and the meridional gradient of PV in the recirculation regions is small. In the lower layer, the tendency of PV homogenization is observed under the jet (Fig. 1f). The time-mean meridional PV profiles in the experiment spatially averaged for 0.3 < x < 0.6 can be found in Fig. 3 of section 3d, along with theoretical model results, which show a strong upper-layer frontal structure and nearly uniform regions to its north and south, and below.

Fig. 1.
Fig. 1.

A double-gyre circulation accompanying prematurely separated WBC extension jets in a two-layer model. (a) Snapshot of the upper-layer streamfunction, (b) snapshot of the lower-layer streamfunction, (c) upper-layer time-mean streamfunction, (d) lower-layer time-mean streamfunction, (e) upper-layer time-mean potential vorticity, and (f) lower-layer time-mean potential vorticity. Ekman pumping is imposed only for x ≥ 1.0, α = 0.5, and r = 10−3. The details of the model configuration and parameters are given in section 2.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-21-0313.1

3. Theoretical model

In the preceding section, it is shown that the eastward jet is accompanied by recirculation gyres on its northern and southern sides. This jet corresponds to the PV front in the upper layer, and the PV is nearly uniform in the recirculation regions. In the lower layer, a PV plateau exists under the jet. Based on these findings, we simplify the PV structure, as shown in Fig. 2. In the present paper, we consider only the subtropical gyre, that is, −1 < y < 0. The broken curves in this figure represent the PVs in the Sverdrup balance, qs1 = −F1ψs + βy in the upper layer and qs2 = F2ψs + βy in the lower layer with the Sverdrup streamfunction ψs=∫2xwedx in the western region where we is zero. The jet latitude is y3, where the PV jumps; the upper-layer PV is homogenized to Q11 for y1 < y < y3 and Q15 for y3 < y < y5, where y1 and y5 are the outer boundaries of the recirculation gyres; and the lower-layer PV is homogenized to Q23 for y2 < y < y4. We refer to the six regions divided by yi (i = 1, 2, …, 5) as region I–region VI in succession from south to north. The discontinuities in PV at y1, y2, y4, and y5, and distributions of lower-layer PV in regions II and IV are dynamically determined. This structure is similar to that of the two-layer inertial circulation theoretically studied by Cessi (1988), although there is a substantial difference in the existence of the basic flow driven by Ekman pumping we, which is not symmetric to the jet axis.

Fig. 2.
Fig. 2.

Assumed meridional profiles of the PV in the western region for the (a) upper layer and (b) lower layer. The dashed curves represent the PV distributions in the Sverdrup balance, i.e., qs1 = −F1ψs + βy and qs2 = F2ψs + βy with the Sverdrup streamfunction ψs in the western region. Here, y1 is the southern edge of the southern recirculation gyre, and y5 is the northern edge of the northern recirculation gyre. The jet exists at y3. The upper-layer PV is Q11 for y1 < y < y3 and Q15 for y3 < y < y5. The lower-layer PV is Q23 for y2 < y < y4. The Roman numbers from I to VI denote the six regions, and regions II, IV, and VI are shaded.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-21-0313.1

At present, y1, y2, y3, y4, y5, Q11, Q15, and Q23 are unknown variables. We will formulate the relationships among the variables, but some of them will need to be specified to determine others. Our first purpose is to clarify how many variables need to be specified to obtain the meridional structure. The second purpose is to clarify how the meridional location of the extension jet, y3, is determined.

a. Integral constraints

Before considering the dynamics in each region, let us derive integral constraints. The area integration of (1) from xe (eastern boundary) to x0 (in the western region where we = 0) for x and from −1 (southern boundary) to 0 (intergyre boundary) for y yields
∫−10∫xex0∂qi∂tdxdy+1β∫−10[−∂ψi∂yqi]x=x0dy+1β∫xex0[∂ψi∂xqi]y=0dx=δ1,i∫−10ψs(x0,y)dy−∮[δ2,ir∇ψ2+ν∇∇4ψi]×dr,
where i is the layer number, qi=q˜i+βy, and ψs(x, y) is the Sverdrup streamfunction ψs=∫xexwedx. The last term is line integrals of the bottom friction and the horizontal viscosity, which are carried out along the boundary of the areal integral.
In the present model, β = 5000 and F = 2025, while the Sverdrup velocity is of order unity. Except around the extension jets, the magnitude of the velocity is of order unity, as well. Since the coefficient of the viscosity and bottom friction are very small, the contribution of the viscous and friction terms to (5) is negligible. The third term on the lhs is the PV flux across the intergyre boundary. Since eddy variability is weak and qi is small at the intergyre boundary, it is expected that this term is negligible. To confirm this expectation, we calculated this term with x0 = 0.5 for the case shown in Fig. 1 and obtained that the temporal mean of this term normalized by ∫−10ψs(x0,y)dy is 10−3 for the upper layer and is −3 × 10−3 for the lower layer. Since the contribution of the relative vorticity in the vorticity equation is much smaller than β and the stretching terms except around the eastward jet, we may write PV approximately as
qi=∂2ψi∂y2+(−1)iFi(ψ1−ψ2)+βy,
and then, the second term on the lhs of (5) becomes
−[12β(∂ψi∂y)2]y=−1y=0+(−1)i+1∫−10Fiβ(ψ1−ψ2)∂ψi∂ydy−[yψi]y=−1y=0+∫−10ψidy.
Here, we write ψi(x0, y) and ψs(x0, y) as simply ψi(y) and ψs(y), respectively, and we adopt the same expressions hereafter. The first term of (7) is negligible, since the current is weak except around the jet. If we consider the time-mean state, the third term is also negligible, because of the symmetric double gyre configuration in which ψi ≃ 0 on the boundary (see Fig. 1c). Then, the time-mean version of (5) becomes
(−1)i+1∫−10Fiβ⟨(ψ1−ψ2)∂ψi∂y⟩dy+∫−10⟨ψi⟩dy=δ1,i∫−10ψsdy,
where 〈*〉 means the time-mean of *. The first term on the lhs of (8) is proportional to the volume flux in each layer across x = x0. If we assume that no mass is exchanged between the upper and lower layers in the western region, this term must vanish. Therefore,
∫−10⟨ψi⟩dy=δ1,i∫−10ψsdy.
Sue and Kubokawa (2015) confirmed that this relation is satisfied very well for i = 1 from the results of their numerical model. In our experiment shown in Fig. 1, the author confirmed that this relation is satisfied very well too. In the present paper, to obtain the solution, we employ the baroclinic and barotropic versions of (9):
∫−10⟨ψ^⟩dy=∫−10ψsdy, and
∫−10⟨ψ¯⟩dy=F2F∫−10ψsdy,
where ψ^=ψ1−ψ2 and ψ¯=(F2/F)ψ1+(F1/F)ψ2 are the baroclinic streamfunction and the barotropic streamfunction, respectively. Since we discuss only the time-mean state, we omit 〈*〉, hereafter, and ψ1 and ψ2 now refer to time averages.

b. Equation set giving the recirculation

In the following derivations, we assume

  • ∂/∂x = ∂2/∂x2 = 0,

  • the Rossby deformation radius F−1/2 is small enough to neglect exp⁡(−F1/2|yj−yi|) for i ≠ j, and

  • relative vorticity is important only near the discontinuities of PV.

1) Regions I and VI

Since these regions do not contain recirculations, the flow is the same as that in the linear solution, that is, ψ1 = ψs and ψ2 = 0.

2) Regions II and V

In each of y1 < y < y2 and y4 < y < y5, the recirculation homogenizes the PV in the upper layer. On the other hand, the PV is not homogenized in the lower layer. When the relative vorticity and the damping terms are negligible, the barotropic component of (1) becomes
F2F∂ψ1∂x+F1F∂ψ2∂x=F2Fwe,
irrespective of the distribution of the upper-layer PV. Although we will take the relative vorticity into consideration to satisfy the matching conditions at y = y1 and y5, its relative contribution is of order F−1/2. Therefore, we can assume that the barotropic field is dominated by the Sverdrup flow, i.e.,
ψ¯=F2Fψ1+F1Fψ2=F2Fψs.
Using (13), the upper-layer PV can be written as
Q1=d2ψ1dy2−Fψ1+F2ψs+βy.
Solving (14) under the matching conditions at y = y1 and y5, we can obtain the homogenized values of Q1, which we refer to as Q1j, where j is either 1 or 5. The matching conditions at y = y1 and y5 are
ψ1(yj)=ψs(yj),dψ1dy|yj=dψsdy|yj,
where yj is y1 for region II and y5 for region V. Since F is much larger than unity, the solution of (14) can be obtained as
ψ1=Ajexp⁡(−F1/2|y−yj|)−Q1jF+βyF+F2Fψs(y)+O(F−1).
Substituting (16) into the matching conditions (15), we obtain
A1=F−3/2(β−F1dψsdy|y1),Q11=qs1(y1)+F−1/2(β−F1dψsdy|y1)+O(1),
A5=−F−3/2(β−F1dψsdy|y5),Q15=qs1(y5)−F−1/2(β−F1dψsdy|y5)+O(1),
where qs1(y) = βy − F1ψs(y). That is, the PV jump at y = yj is of order F−1/2 relative to qs1, which is of order F. It should be noted that the homogenized values of the upper-layer PV for y1 < y < y3 and y3 < y < y5 are functions of y1 and y5, respectively. Neglecting the O(F−1) term, the expressions of ψ1(y) in these regions become
ψ1(y)={1F3/2(β−F1dψsdy|y1){exp⁡[−F1/2(y−y1)]−1}−1F[qs1(y1)−βy−F2ψs(y)]fory1≤y≤y21F3/2(β−F1dψsdy|y5){1−exp⁡[F1/2(y−y5)]}−1F[qs1(y5)−βy−F2ψs(y)]fory4≤y≤y5.
The lower-layer streamfunction ψ2 is obtained from (13):
ψ2=F2F1(ψs−ψ1).
Since d2ψs/dy2 is negligible, d2ψ2/dy2 = −(F2/F1)d2ψ1/dy2. Therefore,
q2=−F2F1d2ψ1dy2+F2(ψ1−ψ2)+βy=−F2F1Q1j+FF1βy.

3) Regions III and IV

For y2 < y < y4 (regions III and IV), the lower-layer PV is homogenized to Q23. In these regions, Eq. (6) can be written as
d2ψ1dy2−F1(ψ1−ψ2)+βy=Q13+{12Zfor y>y3−12Zfor y<y3,
d2ψ2dy2−F2(ψ2−ψ1)+βy=Q23,
where Q13=(1/2)(Q11+Q15) and Z = Q15 − Q11. In terms of the baroclinic streamfunction, ψ^=ψ1−ψ2, and the barotropic streamfunction, ψ¯=(F2/F)ψ1+(F1/F)ψ2, (22) and (23) can be rewritten as
d2ψ^dy2−Fψ^=Q^+{−12Zfor y2<y<y312Zfor y3<y<y4,
d2ψ¯dy2+β(y−y3)=ζ¯0+{−F22FZfor y2<y<y3F22FZfor y3<y<y4,
where
Q^=Q13−Q23,ζ¯0=1F(F2Q13+F1Q23)−βy3.
Under the condition that F−1/2 ≪ 1, the baroclinic streamfunction can easily be obtained as
ψ^=−Q^F+{Z2F{1−exp⁡[F1/2(y−y3)]}for y<y3Z2F{exp⁡[−F1/2(y−y3)]−1}for y>y3.
The baroclinic streamfunction (27) does not impose any restrictions on y2, y3, and y4. On the other hand, the barotropic zonal velocity and the streamfunction become
u¯=β2η2−ζ¯0η+u¯0+{F22FZηfor y<y3−F22FZηfor y>y3,
ψ¯=−β6η3+12ζ¯0η2−u¯0η+ψ¯0+{−F2Z4Fη2for y<y3F2Z4Fη2for y>y3,
where η = y − y3, ψ¯0 is ψ¯(y3), and u¯0 is the barotropic velocity at y = y3. Since the matching conditions at y2 and y4 are ψ¯=ψ¯s=(F2/F)ψs and dψ¯/dy=dψ¯s/dy=(F2/F)dψs/dy, we obtain
−β2η22+ζ¯0η2−u¯0−F2F(dψsdy|y2+Z2η2)=0,
−β6η23+12ζ¯0η22−u¯0η2+ψ¯0−F2F[Z4η22+ψs(y2)]=0,
−β2η42+ζ¯0η4−u¯0−F2F(dψsdy|y4−Z2η4)=0,
−β6η43+12ζ¯0η42−u¯0η4+ψ¯0+F2F[Z4η42−ψs(y4)]=0,
where η2 = y2 − y3 and η4 = y4 − y3.

It should be noted that the location of the lower-layer jet axis does not necessarily coincide with that of the upper-layer jet axis, y3. Their positions coincide if we set ζ¯0 in (26) to zero. However, there is no reason for the relative vorticity of the barotropic flow to be zero at y = y3. Although we do not specify ζ¯0, the lower-layer jet location is very close to y3.

c. Solution method

The solution of the problem, i.e., ψ1 and ψ2, is obtained if y1, y2, y3, y4, y5, u¯0,ψ¯0, and Q23 (or Q^) are specified because Q11 and Q15 are functions of y1 and y5. The equations that must be satisfied are the four matching conditions for the barotropic components given by (30)–(33) and the two integral constraints given in section 3a. Using (13), (16), (27), and (29), we can rewrite the integral constraints on the baroclinic and barotropic streamfunctions, (10) and (11), as
−1F1[Q11(y2−y1)+Q15(y5−y4)−β2(y22−y12+y52−y42)]+1F[(Q23−Q11)(y3−y2)+(Q23−Q15)(y4−y3)]=∫y1y5ψsdy, and
−β24(η44−η24)+ζ¯06(η43−η23)−u¯02(η42−η22)+ψ¯0(η4−η2)+F2Z12F(η43−η23)=F2F∫y2y4ψsdy.
Since there are six equations for eight unknown variables, two variables need to be specified to obtain the solution. As will be shown, there are cases in which the northern recirculation region extends to the intergyre boundary. In this case, one of the variables, y5, is automatically set to y5 = 0, and we may specify only one additional variable to obtain the solution. Although we refer to y1, y2, y3, y4, y5, u¯0,ψ¯0, and Q23 (or Q^) as variables, we can choose other variables, such as the difference in PV across the jet Z to solve the theoretical model. If another variable is added, we introduce an equation showing the relation between the newly chosen variable and the other variables. Because the nonlinear Eqs. (30)–(35) cannot be solved analytically, we solve them numerically by using the Levenberg–Marquardt method using MINPACK (More et al. 1980). Once these variables are determined, ψ1 and ψ2 are extracted by using (19), (20), (27), (29), and ψs(y).

d. Examples of the solution

In this subsection, we discuss two examples of the solution: one is the case for which we need to specify two variables, and the other is the case in which the northern recirculation attaches to the intergyre boundary (y5 = 0).

Figure 3 shows the theoretical solution for the case illustrated in Fig. 1. We superimpose the theoretically obtained meridional profiles (black solid curves) onto the experimental results (blue dotted–dashed curves). The experimental profiles are created by zonally averaging the time-mean model outputs for 0.3 < x < 0.6. In this case, since the recirculation is detached from the intergyre boundary, two parameters are needed to determine the solution. Here, we introduce the PV difference across the jet Z, which represents the jet’s strength, and the jet latitude y3 into the theoretical model, since we want to see how well the theoretical model reproduces the experimental result. The uniform PV values, namely, Q11, Q15, and Q23, agree well with the experimental values (Figs. 3a,d). The lower layer PV in regions II and V also agrees well with the experiment. Although the jet speed is twice the experimental value (Fig. 3c), the transport is the same (Fig. 3b).

Fig. 3.
Fig. 3.

Meridional profiles for the case of α = 0.5 and r = 10−3: (a) upper-layer PV, (b) upper-layer streamfunction, (c) upper-layer zonal velocity, (d) lower-layer PV, (e) lower-layer streamfunction, and (f) lower-layer zonal velocity. Black solid curves denote the theoretical results, blue dotted–dashed curves denote the experimental results, which are created by zonally averaging the time-mean model outputs for 0.3 < x < 0.6, and red dashed curves denote those in the Sverdrup balance. The shade denotes regions II, IV, and VI.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-21-0313.1

The theoretical model captures the characteristics of the lower layer zonal flow distribution, too (Fig. 3f). In the lower layer, an eastward jet exists below the upper-layer PV front, and westward flows occur on its northern and southern sides. In addition, eastward flows are again observed outside these westward flows. This profile is similar to that discussed by Holland and Rhines (1980) for the case of the WBC extension jet along the intergyre boundary. From the theoretical model, the outer eastward flows are interpreted as being generated by the eddy-driven upper-layer westward flow under the Sverdrup balance.

Figure 4 shows an example of the theoretical solution in the case where y5 = 0 with Q15 = 0. This is the case with α = 0 and r = 1.0 × 10−3. We introduce only the PV difference across the jet Z based on the experimental result to calculate the theoretical solution. Although we input only Z, the meridional profiles obtained by the theoretical model generally agree well with the experimental results, including the position of the PV front y3. From the integral constraints (9), we can derive
∫−10(q1−qs)dy=0,
where q1 = ∂2ψ1/∂y2 − F1(ψ1 − ψ2) + βy and qs = −F1ψs + βy. Here, we neglect ∂ψ1/∂y|y=−1 and ∂ψ1/∂y|y=0 as in section 3a. This equation means that if Q15 and Q11 are given, y3 is automatically determined; this equation also implies that the larger the PV difference Z is, the more southward the jet location becomes.
Fig. 4.
Fig. 4.

As in Fig. 3, but for the case of α = 0 and r = 10−3.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-21-0313.1

e. Lower-layer PV homogenized region

Sue and Kubokawa (2015) reported that when the lower layer is much thicker than the upper layer, premature separation does not occur and the jet occurs along the northern boundary. Furthermore, the prematurely separated jet always accompanied lower-layer flow in their model.

The lower-layer flow is due to the lower-layer PV homogenized region, where positive shear and negative shear exist on the southern and northern sides of the lower-layer eastward jet. For this vorticity structure (d3ψ2/dy3 > 0) to occur in the uniform PV region, (23) implies
F2dψ^dy+β<0.
This is the same as the necessary condition for baroclinic instability. When using Z, (37) becomes
Z>Zc=2F1/2βF2.
If this condition is not satisfied, the PV homogenized region in the lower layer vanishes. Since the barotropic component coincides with that of the Sverdrup streamfunction in this case, only the integral of the baroclinic streamfunction (10) constrains the solution. Since there are three unknown variables, y1, y3, and y5, two of them should be given to obtain the solution as the case where (38) is satisfied.
Although the theoretical model indicates nothing about the importance of the lower-layer PV homogenized region, numerical experiments suggest its importance, as shown in appendix A, where we carried out three additional experiments varying F2 and adding a bottom topographic beta. When the bottom topographic beta βTy exists, (27) becomes
ψ^=−Q^F−βTyF+{Z2F{1−exp⁡[F1/2(y−y3)]}for y<y3,Z2F{exp⁡[−F1/2(y−y3)]−1}for y>y3.
Then, (38) becomes
Z>Zc=2F1/2F2(β+F1βTF).
As shown in appendix A, when (40) is not satisfied, the extension jet fluctuates greatly, and the temporal mean PV front is weakened. Since Sue and Kubokawa (2015) set α to 0, the jet was located close to the northern boundary of the subtropical gyre. Therefore, it is thought that the highly variable jet was attached to and trapped by the northern boundary when the lower layer was deep.

4. Latitude of the extension jet

In this section, we consider what determines the meridional location of the extension jet in the present model. Theory states that jet location y3 is determined if two parameters are given. We adopt the PV difference across the jet Z as one of the parameters because the jet’s strength does not appear to be determined internally but by an external forcing. The relation between Z and the other variables is Z = Q15 − Q11. To control Z, we conduct several experiments with the bottom friction varying from r = 1.0 × 10−3 to r = 5.0 × 10−3, which varies the PV difference across the jet from 2900 to 4500 (from 0.58 × β to 0.9 × β). Additionally, we conduct experiments with varying values of α, where α = 0, 0.25, 0.5, and 0.75. As mentioned in the preceding section, when α is small and the jet is strong, the northern edge of the region in which the upper-layer PV is homogenized, y5, coincides with gyre boundary y = 0. In this case, Z determines the solution. However, when y5 < 0, we take the upper-layer streamfunction of jet center ψ1(y3) as an additional parameter because the heavy dependence of the jet location on the meridional profile of the Sverdrup streamfunction suggests that ψ1(y3) could be constant, independent of other variables. The relation between ψ1(y3) and the other variables is
ψ1(y3)=−F1(Q13−Q23)F2+ψ¯0.

Figure 5 summarizes the theoretical solutions and the results of the numerical experiments. The horizontal axis is Z, and the vertical axis is y3. The solid curves show the theoretical results for α = 0, 0.25, 0.5, and 0.75 with ψ1(y3) = 0.5ψsmax when y5 ≠ 0, where ψsmax = 1. The term ψ1(y3) = 0.5 signifies that the streamline at the center of the jet connects to the central streamline of the eastward Sverdrup flow. We hypothetically take this condition. The curves of ψ1(y3) = 0.45 (dotted–dashed curve) and ψ1(y3) = 0.55 (dotted curve) are also plotted to show the range of ±10%. The theoretical curve with ψ1(y3) = 0.5 for α = 0 shows a gentle slope for small Z, reaches a small peak at approximately Z = 3300 and then displays a rather steep slope afterward. The gentle slope for Z < 3000 is caused by the negative curvature of ψs(y) near ψs = 0.5, as pointed out by Sue and Kubokawa (2015); in addition, the PV jump at the outer edge of the homogenized PV region creates a small peak at Z ≃ 3300 [Q15 is not zero for y5 < 0 but is zero for y5 = 0, as shown in (18)]. In the steep slope region for Z > 3300, an increase in Z broadens the northern homogenized PV region and pushes y3 southward. The theoretical curve for α = 0.25 has the same characteristics as α = 0. The theoretical curve for α = 0.5 is almost flat because the curvature of ψs is almost zero near ψs = 0.5. For Z > 4650, y3 decreases because y5 = 0. The theoretical curve for α = 0.75 shows that y3 slightly increases with Z. This slight increase occurs because the curvature of ψs is positive near ψs = 0.5.

Fig. 5.
Fig. 5.

Dependence of the meridional location of the jet y3 on the strength of the PV front Z (the PV difference across the jet) for α = 0, 0.25, 0.5, and 0.75. The solid curves are the theoretical solutions assuming that ψ1(y3)=(1/2)ψsmax when y5 ≠ 0, where ψsmax = 1 is the maximum value of the Sverdrup streamfunction. The dotted–dashed curves and dashed curves are the theoretical curves of ψ1(y3)=(1/2)ψsmax×0.9 and ψ1(y3)=(1/2)ψsmax×1.1, respectively. Symbols denote the experimental results shown in Table 1: asterisks correspond to α = 0, diamonds to 0.25, circles to 0.5, and triangles to 0.75. The open circles and open triangles are the results forced by (4) with xf = 1.0, whereas the closed circles and closed triangles are the results forced by (4) with xf = 1.5. The closed diamonds are the results of Z increased by assuming Q15 = 0.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-21-0313.1

The symbols plotted in Fig. 5 are the experimental results summarized in Table 1. The jet latitude of the experiment, yjet, is defined as the y coordinate where the upper-layer eastward velocity reaches the maximum. The Z is subjectively estimated based on the meridional PV profile. Because the PV profiles are the temporal and zonal means of the numerical results, the PV front is not sharp, in contrast to the theory, and the homogenized PVs on the northern and southern sides are not perfect and instead are often wavy. Such differences between the experiments and theory make it difficult to estimate Z. There is a case of a positive Q15, but since such values are not theoretically allowed, we estimate Z by setting Q15 = 0 for such a case. The meridional profiles of the numerical experiments are shown in appendix B with the theoretical profiles based on the estimated Z and ψ1(y3) = 0.5. The values of the streamfunction at the jet in the experiments, ψ1(yjet), as shown in Table 1, are distributed at approximately 0.5. However, we cannot discuss further details because ψ changes considerably within one grid interval (Δx = 1/256), approximately 0.08 in the experiments with Z ≃ 4000.

Table 1

Experimental results plotted in Fig. 5. Here, xf is the x coordinate of the western edge of the forcing region, and Q11 and Q15 are estimated subjectively; Z = Q15 − Q11 for Q15 ≤ 0, and Z = −Q11 for Q15 > 0.
Table 1

The asterisks denote the cases with α = 0, which have Z values that are sufficiently large for the northern homogenized PV region to extend to the intergyre boundary. Hence, y5 = 0, and the solution is determined only by Z. The experimental results with α = 0 (experiments a1, a2, and a3) agree well with the theoretical curve with y5 = 0, although the jet locations are somewhat closer to the intergyre boundary than theoretically predicted. In particular, result a2 with r = 2.5 × 10−3 exhibits a relatively large deviation from the theory. This deviation is perhaps caused by the positive Q15 of the experiment. For α = 0.25, the case with the smallest Z is on the theoretical curve with ψ1(y3) = 0.5, while the other two cases (open diamonds) are more south of the theoretical curve with y5 = 0. This difference could be attributed to the fact that the Q15 values for these cases are negative even though Z is large enough for y5 to be zero. When we set Q15 = 0 for these two cases (i.e., Z = −Q11), the experimental results (closed diamonds) are almost on the theoretical curve. These findings suggest that the theory can qualitatively reproduce the experimental results by the strength of PV front Z when the northern recirculation gyre attaches to the intergyre boundary.

For the cases where α = 0.5 and 0.75, open and closed symbols are plotted in Fig. 5. The open symbols denote the results obtained with the wind forcing given by (4) with xf = 1.0, whose forcing region is 1.0 ≤ x ≤ 2.0. The meridional locations y3 of the smallest Z for each α = 0.5 and α = 0.75 agree well with the theory with ψ1(y3) = 0.5, as does that for α = 0.25 (experiment b1). However, with increasing Z, the jet location shifts southward. That is, to reproduce the experimental result, we must increase ψ1(y3) with Z. To evaluate this dependence on Z, we reexamine Fig. 1, which shows the result of the case with α = 0.5 and r = 10−3. In this case, the eastern edge of the jet extends into the forced region of x ≥ 1, where we is not zero. Although the mechanism involved remains unclear, we tends to dampen the eastward jet, and the jet disappears soon after intruding into the forced region in this case. The top row of Fig. 6 (experiments c2, c3, d2, and d3) shows the meridional profiles of the streamfunctions at the western edge of the forced region, x = 1.0, for open symbols. The structure of the extension jet can be seen in experiments c3 and d3, whose meridional locations of the extension jet are shifted southward. Because PV is conserved under advection, the southward Sverdrup flow, ∂ψs/∂x = we, may advect the extension jet southward. To reduce the direct effect of we on the extension jet, we conduct additional experiments driven by we that are nonzero only for x ≥ 1.5, i.e., xf = 1.5 in (4).

Fig. 6.
Fig. 6.

Meridional profiles of the streamfunction at the western edge of the forced region: (top) experiments c2, c3, d2, and d3 for the case of xf = 1.0, and (bottom) experiments e2, e3, f2, and f3 for the case of xf = 1.5. The black solid curves are the profiles at x = xf, the red dotted curves are those of the Sverdrup streamfunction, and the blue dotted–dashed curves in the bottom row are those at x = 1.0. The thin curves with small magnitudes show the streamfunctions in the lower layer.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-21-0313.1

The bottom row of Fig. 6 shows the meridional profiles at x = 1.0 (blue dotted–dashed curves) and x = 1.5 (black solid curves) in the case of xf = 1.5. The jets at x = 1.0 are much stronger than those of xf = 1.0 cases (top row), suggesting that the we forcing on the jet damps its upstream (west) jet, as well. Since the jet-recirculation system decays eastward through eddy PV flux (e.g., Waterman and Jayne 2011), we forcing on the jet perhaps changes the upstream eddy field. Although the mechanism needs further study, details of the direct we effects on the jet are beyond the scope of the present article. On the other hand, the profiles at the western edge of the forced region (x = 1.5) are almost the same as those of the Sverdrup streamfunction, suggesting that the extension jet considerably decays before reaching the forced region in the case of xf = 1.5. The closed circles and closed triangles in Fig. 5 denote the (Z, yjet) values of these cases. The cases with r = 10−3 driven by (4) with xf = 1.5 (experiments e3 and f3) are closer to the theoretical curves with ψ1(y3) = 0.5 than those driven by (4) with xf = 1.0 (experiments c3 and d3), while the differences between the cases with r = 2.5 × 10−3 are nonsignificant. From these results, we may conclude that the southward Sverdrup flow tends to shift the jet southward, but if the southward Sverdrup flow is negligible, ψ1(y3) = 0.5 can reproduce the experimental result to some degree when y5 ≠ 0. However, there is no clear reason why the streamline of the jet center must connect to the central streamline of the eastward Sverdrup flow.

5. Summary and remarks

The present study attempted to clarify what factors determine the latitude of a prematurely separated WBC extension jet in a two-layer QG system. To solve this problem, we constructed a theoretical model that can capture the meridional structure of the western subtropical gyre. The theoretical model requires two parameter values to obtain the solution for a specified Sverdrup flow and stratification. We chose the strength of the jet (the difference in PV across the jet) as one of these parameters. When the northern recirculation attaches to the intergyre boundary, one of the other parameters is automatically specified, allowing the solution to be uniquely determined. On the other hand, when the northern recirculation was detached from the intergyre boundary, we chose the streamfunction at the jet center as the other parameter and compared the theory with numerical experiments. Then, we found that setting the streamfunction to coincide with that at the center of the eastward Sverdrup flow yields a solution that agrees well with the experimental results when the southward Sverdrup flow is not within the jet region. We also discovered that the southward Sverdrup flow shifts the jet southward. Although not the main focus of the present article, it was found that lower-layer PV homogenization is important to stabilize the jet path (appendix A).

Although the dynamics determining the jet latitude are not fully understood, if we accept the experimentally obtained relation between the streamfunction of the jet center and the Sverdrup flow, the theoretical model has made it possible to discuss some details regarding the extension jet and the meridional distribution of zonal flows. However, we still need to address many problems before discussing real oceanic extension jets based on the present theory. For example, we should quantify the effect of the meridional Sverdrup flow on the jet. In addition, the Ekman pumping imposed only on the eastern portion of the jet affects the jet latitude in the western portion, and a Rossby wave adjustment may occur (e.g., Sasaki and Schneider 2011). Furthermore, if the upper-layer meridional flow is essential, the meridional overturning circulation could play a significant role in determining the latitude.

Another issue concerns the model dependency of the jet latitude. Nakano et al. (2008) studied the extension jet using an ocean general circulation model (OGCM). In their model, the extension jet is located close to the center of the Sverdrup gyre, and the value of the barotropic streamfunction at the jet center seems much larger than half the Sverdrup streamfunction maximum. The authors proposed interpreting their result as the southern edge of the southern recirculation coinciding with the center of the Sverdrup gyre. In our model, the meridional location of the recirculation’s southern edge depends on the strength of the PV front. Therefore, this hypothesis is not consistent with our two-layer model. One possible cause is the abovementioned effect of the southward Sverdrup flow. The other is an effect of the vertical resolution. Idealized thermocline theories have shown that the central latitude of the subtropical gyre where the zonal flow vanishes shifts southward with decreasing depth (e.g., Young and Rhines 1982; Luyten et al. 1983). In the actual ocean, the same structure can also be seen (Kimizuka et al. 2015). Therefore, the upper-layer eastward flow is wider than expected from the Sverdrup streamfunction, and its center is shifted southward. If the jet connects to the streamline of the center of the wind-driven upper-layer eastward flow in such a situation, the jet is located more toward the south than predicted by the present model. The outcropping of the density surfaces may also affect the solution.

In the present article, we could not give any reason why ψ1(y3) should be the half of the maximum Sverdrup streamfunction when the northern recirculation gyre detached from the intergyre boundary. The system treated here is highly nonlinear and the basin is filled with eddies. Eastward decay of the jet and recirculation gyres is governed by northward eddy PV flux and Rossby waves are radiated at the eastern edge of the jet (e.g., Waterman and Jayne 2011). Mizuta (2012) showed that potential vorticity redistribution by the Rossby waves is essentially important in the transition from the narrow jet to the interior broad flow. A clue to understanding the mechanism determining the jet latitude might be found in such a local transition process, or in seeking some other global constraint such as some form of energy that takes minimum.

Acknowledgments.

The author thanks two anonymous reviewers for their valuable comments, which have greatly helped the author to improve the manuscript. This work was supported by JSPS KAKENHI Grant JP19K03962. GFD-DENNOU Library was used for drawing figures.

Data availability statement.

The author wrote the numerical codes used in the present research, including the QG-model, for private use, and the source codes are not publicly archived but can be made available through direct requests to the author. MINPACK, LAPACK, and FFTPACK Libraries used in the codes are available at https://netlib.org.

APPENDIX A

Importance of the Lower Layer PV Homogenization

As shown in section 3e, for the lower-layer PV homogenized region to occur, Z must be larger than critical value Zc, which is a function of F2, and the topographic beta βT shown by (40). In the present appendix A, we show four experimental results with different F2 and βT.

The black curves in Fig. A1 represent the PV and upper layer eastward velocity profiles for the standard case shown in Figs. 1 and 3. In this case, F2/F = 0.2 and β = 5000 such that Zc = 1111. Since Z ≃ 4000, the lower layer has a homogenized PV region according to the theory. The navy dotted–dashed curves are those for a case with F2/F = 0.1 and βT = 0. These values yield Zc = 2222. The navy dotted–dashed curve in the upper layer almost coincides with that of the standard case (black solid curve) with a sharp front and strong jet. The red dotted curves are those for a case with F2/F = 0.05 and βT = 0, and the blue dashed curves are those with F2/F = 0.1 and βT = βF/F1. The critical values Zc for both the red dotted curves and the blue broken curves are 4444, which are higher than Z in the standard case. In these cases, the mean upper-layer PV front is rather gentle, the time-mean eastward jet is broad and weak, and the mean jet location shifts northward. The above results shows that there are two different cases of the strength of the front, and we may conclude that the difference between strong and weak front cases is due to the presence or absence of a PV homogenized region in the lower layer.

Fig. A1.
Fig. A1.

Meridional profiles of the PV and the eastward velocity in the experiments with different F2 and βT created by zonally averaging the time-mean model outputs for 0.3 < x < 0.6: (a) upper-layer PV profiles, (b) lower-layer PV profiles divided by F2, and (c) upper-layer eastward velocity. The black curves are the results for F2/F = 0.2 and βT = 0, the navy dotted–dashed curves are for F2/F = 0.1 and βT = 0, the red dotted curves are for F2/F = 0.05 and βT = 0, and the blue dashed curves are for F2/F = 0.1 and βT = βF/F1.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-21-0313.1

Figure A2 shows the paths of the eastward jet in the four cases and shows that the differences between the time-mean profiles of the cases with Z > Zc and those with Z < Zc are caused by the difference in the variability of the path. This indicates that the lower-layer PV homogenization tends to stabilize the upper-layer jet. Although the mechanism is unknown, this tendency is similar to the relation between the strength and variability of the Kuroshio Extension jet found by Qiu and Chen (2005), who reported that the jet’s path is highly variable when the sea surface height (SSH) difference across the jet is low, while the jet’s path tends to be stabilized when the SSH difference is high.

Fig. A2.
Fig. A2.

Paths of the jet defined by ψ1 = 0.5 contours for four cases with different F2 and βT: (a) F2/F = 0.2 and βT = 0, (b) F2/F = 0.1 and βT = 0, (c) F2/F = 0.05 and βT = 0, and (d) F2/F = 0.1 and βT = βF/F1. Here, 50 paths are plotted every 1000 after T > 100 000.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-21-0313.1

The mean latitudes of the jets in the weak front case are roughly in line with the theoretical latitude with ψ1(y3) = 0.5. This may be because the jet is too short to be directly affected by the wind forcing.

APPENDIX B

Meridional PV Profiles of the Experiments

Because we subjectively estimate the difference in PV across the jet Z, we show all the PV profiles listed in Table 1. In Fig. B1, we show the results of experiments a1–d3. The red curves are the profiles of the theoretical model. In a1, Z is in the region of y5 = 0, but the PV in the northern recirculation is slightly negative. In a2, the PV in the northern recirculation is positive, but we set Q15 to zero to estimate the value of Z, that is, Z = −Q11. In a3, the PV on the northern side of the jet is wavy, and we estimate the Q15 in this case to be zero. In b2 and b3, the PV in the northern recirculation is negative, but we calculate the theoretical curves superimposed onto these panels by setting Q15 = 0 (i.e., Z = −Q11). In the other cases, roughly estimated values of Z and ψ1(y3) = 0.5 are used to obtain the theoretical profiles. Figure B2 shows the profiles for the cases in which the direct effect of the Sverdrup southward flow on the jet is reduced. The experimental parameters α and r of experiments e2, e3, f2, and f3 are the same as those of c2, c3, d2, and d3, respectively. The theoretical profiles for experiments e3 and f3 are closer to those of the experiments than are the profiles for c3 and d3.

Fig. B1.
Fig. B1.

Meridional profiles of the PV in the experiments from a1 to d3. The black curves are the experimental results created by zonally averaging the time-mean model outputs for 0.3 < x < 0.6, and the red curves are the theoretical model’s results. The solid curves correspond to the upper layer, while the dashed curves correspond to the lower layer.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-21-0313.1

Fig. B2.
Fig. B2.

As in Fig. B1, but for experiments e2, e3, f2, and f3.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-21-0313.1

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  • Mizuta, G., 2012: Role of the Rossby waves in the broadening of an eastward jet. J. Phys. Oceanogr., 42, 476–494, https://doi.org/10.1175/JPO-D-11-070.1.

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  • More, J. J., B. S. Garbow, and K. E. Hillstrom, 1980: User guide for MINPACK-1 [In FORTRAN]. Tech. Rep. ANL-80-74, Argonne National Laboratory, 258 pp., https://doi.org/10.2172/6997568.

  • Nakano, H., H. Tsujino, and R. Furue, 2008: The Kuroshio Current system as a jet and twin “relative” recirculation gyres embedded in the Sverdrup circulation. Dyn. Atmos. Oceans, 45, 135–164, https://doi.org/10.1016/j.dynatmoce.2007.09.002.

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  • Niiler, P. P., N. A. Maximenko, G. G. Panteleev, T. Yamagata, and D. B. Olson, 2003: Near-surface dynamical structure of the Kuroshio extension. J. Geophys. Res., 108, 3193, https://doi.org/10.1029/2002JC001461.

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  • Özgökmen, T. M., E. P. Chassignet, and A. M. Paiva, 1997: Impact of wind forcing, bottom topography, and inertia on midlatitude jet separation in a quasigeostrophic model. J. Phys. Oceanogr., 27, 2460–2476, https://doi.org/10.1175/1520-0485(1997)027<2460:IOWFBT>2.0.CO;2.

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  • Qiu, B., and S. Chen, 2005: Variability of the Kuroshio extension jet, recirculation gyre, and mesoscale eddies on decadal time scales. J. Phys. Oceanogr., 35, 2090–2103, https://doi.org/10.1175/JPO2807.1.

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  • Sasaki, Y. N., and N. Schneider, 2011: Decadal shifts of the Kuroshio extension jet: Application of thin-jet theory. J. Phys. Oceanogr., 41, 979–993, https://doi.org/10.1175/2010JPO4550.1.

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  • Sheremet, V. A., 2002: Inertial gyre driven by a zonal jet emerging from the western boundary. J. Phys. Oceanogr., 32, 2361–2378, https://doi.org/10.1175/1520-0485(2002)032<2361:IGDBAZ>2.0.CO;2.

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  • Sue, Y., and A. Kubokawa, 2015: Latitude of eastward jet prematurely separated from the western boundary in a two-layer quasigeostrophic model. J. Phys. Oceanogr., 45, 737–754, https://doi.org/10.1175/JPO-D-13-058.1.

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  • Verron, J., and E. Blayo, 1996: The no-slip condition and separation of western boundary currents. J. Phys. Oceanogr., 26, 1938–1951, https://doi.org/10.1175/1520-0485(1996)026<1938:TNSCAS>2.0.CO;2.

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  • Waterman, S., and S. R. Jayne, 2011: Eddy-mean flow interactions in the along-stream development of a western boundary current jet: An idealized model study. J. Phys. Oceanogr., 41, 682–707, https://doi.org/10.1175/2010JPO4477.1.

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  • Young, W. R., and P. B. Rhines, 1982: A theory of the wind-driven circulation. II. Gyres with western boundary layers. J. Mar. Res., 40, 849–872.

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  • Kiss, A. E., 2002: Potential vorticity “crises”, adverse pressure gradients, and western boundary current separation. J. Mar. Res., 60, 779–803, https://doi.org/10.1357/002224002321505138.

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  • Marshall, D. P., and C. E. Tansley, 2001: An implicit formula for boundary current separation. J. Phys. Oceanogr., 31, 1633–1638, https://doi.org/10.1175/1520-0485(2001)031<1633:AIFFBC>2.0.CO;2.

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  • Mizuta, G., 2012: Role of the Rossby waves in the broadening of an eastward jet. J. Phys. Oceanogr., 42, 476–494, https://doi.org/10.1175/JPO-D-11-070.1.

    • Search Google Scholar
    • Export Citation

  • More, J. J., B. S. Garbow, and K. E. Hillstrom, 1980: User guide for MINPACK-1 [In FORTRAN]. Tech. Rep. ANL-80-74, Argonne National Laboratory, 258 pp., https://doi.org/10.2172/6997568.

  • Nakano, H., H. Tsujino, and R. Furue, 2008: The Kuroshio Current system as a jet and twin “relative” recirculation gyres embedded in the Sverdrup circulation. Dyn. Atmos. Oceans, 45, 135–164, https://doi.org/10.1016/j.dynatmoce.2007.09.002.

    • Search Google Scholar
    • Export Citation

  • Niiler, P. P., N. A. Maximenko, G. G. Panteleev, T. Yamagata, and D. B. Olson, 2003: Near-surface dynamical structure of the Kuroshio extension. J. Geophys. Res., 108, 3193, https://doi.org/10.1029/2002JC001461.

    • Search Google Scholar
    • Export Citation

  • Özgökmen, T. M., E. P. Chassignet, and A. M. Paiva, 1997: Impact of wind forcing, bottom topography, and inertia on midlatitude jet separation in a quasigeostrophic model. J. Phys. Oceanogr., 27, 2460–2476, https://doi.org/10.1175/1520-0485(1997)027<2460:IOWFBT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Qiu, B., and S. Chen, 2005: Variability of the Kuroshio extension jet, recirculation gyre, and mesoscale eddies on decadal time scales. J. Phys. Oceanogr., 35, 2090–2103, https://doi.org/10.1175/JPO2807.1.

    • Search Google Scholar
    • Export Citation

  • Sasaki, Y. N., and N. Schneider, 2011: Decadal shifts of the Kuroshio extension jet: Application of thin-jet theory. J. Phys. Oceanogr., 41, 979–993, https://doi.org/10.1175/2010JPO4550.1.

    • Search Google Scholar
    • Export Citation

  • Sheremet, V. A., 2002: Inertial gyre driven by a zonal jet emerging from the western boundary. J. Phys. Oceanogr., 32, 2361–2378, https://doi.org/10.1175/1520-0485(2002)032<2361:IGDBAZ>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Sue, Y., and A. Kubokawa, 2015: Latitude of eastward jet prematurely separated from the western boundary in a two-layer quasigeostrophic model. J. Phys. Oceanogr., 45, 737–754, https://doi.org/10.1175/JPO-D-13-058.1.

    • Search Google Scholar
    • Export Citation

  • Verron, J., and E. Blayo, 1996: The no-slip condition and separation of western boundary currents. J. Phys. Oceanogr., 26, 1938–1951, https://doi.org/10.1175/1520-0485(1996)026<1938:TNSCAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Waterman, S., and S. R. Jayne, 2011: Eddy-mean flow interactions in the along-stream development of a western boundary current jet: An idealized model study. J. Phys. Oceanogr., 41, 682–707, https://doi.org/10.1175/2010JPO4477.1.

    • Search Google Scholar
    • Export Citation

  • Young, W. R., and P. B. Rhines, 1982: A theory of the wind-driven circulation. II. Gyres with western boundary layers. J. Mar. Res., 40, 849–872.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    A double-gyre circulation accompanying prematurely separated WBC extension jets in a two-layer model. (a) Snapshot of the upper-layer streamfunction, (b) snapshot of the lower-layer streamfunction, (c) upper-layer time-mean streamfunction, (d) lower-layer time-mean streamfunction, (e) upper-layer time-mean potential vorticity, and (f) lower-layer time-mean potential vorticity. Ekman pumping is imposed only for x ≥ 1.0, α = 0.5, and r = 10−3. The details of the model configuration and parameters are given in section 2.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Assumed meridional profiles of the PV in the western region for the (a) upper layer and (b) lower layer. The dashed curves represent the PV distributions in the Sverdrup balance, i.e., qs1 = −F1ψs + βy and qs2 = F2ψs + βy with the Sverdrup streamfunction ψs in the western region. Here, y1 is the southern edge of the southern recirculation gyre, and y5 is the northern edge of the northern recirculation gyre. The jet exists at y3. The upper-layer PV is Q11 for y1 < y < y3 and Q15 for y3 < y < y5. The lower-layer PV is Q23 for y2 < y < y4. The Roman numbers from I to VI denote the six regions, and regions II, IV, and VI are shaded.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Meridional profiles for the case of α = 0.5 and r = 10−3: (a) upper-layer PV, (b) upper-layer streamfunction, (c) upper-layer zonal velocity, (d) lower-layer PV, (e) lower-layer streamfunction, and (f) lower-layer zonal velocity. Black solid curves denote the theoretical results, blue dotted–dashed curves denote the experimental results, which are created by zonally averaging the time-mean model outputs for 0.3 < x < 0.6, and red dashed curves denote those in the Sverdrup balance. The shade denotes regions II, IV, and VI.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    As in Fig. 3, but for the case of α = 0 and r = 10−3.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Dependence of the meridional location of the jet y3 on the strength of the PV front Z (the PV difference across the jet) for α = 0, 0.25, 0.5, and 0.75. The solid curves are the theoretical solutions assuming that ψ1(y3)=(1/2)ψsmax when y5 ≠ 0, where ψsmax = 1 is the maximum value of the Sverdrup streamfunction. The dotted–dashed curves and dashed curves are the theoretical curves of ψ1(y3)=(1/2)ψsmax×0.9 and ψ1(y3)=(1/2)ψsmax×1.1, respectively. Symbols denote the experimental results shown in Table 1: asterisks correspond to α = 0, diamonds to 0.25, circles to 0.5, and triangles to 0.75. The open circles and open triangles are the results forced by (4) with xf = 1.0, whereas the closed circles and closed triangles are the results forced by (4) with xf = 1.5. The closed diamonds are the results of Z increased by assuming Q15 = 0.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Meridional profiles of the streamfunction at the western edge of the forced region: (top) experiments c2, c3, d2, and d3 for the case of xf = 1.0, and (bottom) experiments e2, e3, f2, and f3 for the case of xf = 1.5. The black solid curves are the profiles at x = xf, the red dotted curves are those of the Sverdrup streamfunction, and the blue dotted–dashed curves in the bottom row are those at x = 1.0. The thin curves with small magnitudes show the streamfunctions in the lower layer.

  • <sc>Fig</sc>. A1.
    View in gallery
    Fig. A1.

    Meridional profiles of the PV and the eastward velocity in the experiments with different F2 and βT created by zonally averaging the time-mean model outputs for 0.3 < x < 0.6: (a) upper-layer PV profiles, (b) lower-layer PV profiles divided by F2, and (c) upper-layer eastward velocity. The black curves are the results for F2/F = 0.2 and βT = 0, the navy dotted–dashed curves are for F2/F = 0.1 and βT = 0, the red dotted curves are for F2/F = 0.05 and βT = 0, and the blue dashed curves are for F2/F = 0.1 and βT = βF/F1.

  • <sc>Fig</sc>. A2.
    View in gallery
    Fig. A2.

    Paths of the jet defined by ψ1 = 0.5 contours for four cases with different F2 and βT: (a) F2/F = 0.2 and βT = 0, (b) F2/F = 0.1 and βT = 0, (c) F2/F = 0.05 and βT = 0, and (d) F2/F = 0.1 and βT = βF/F1. Here, 50 paths are plotted every 1000 after T > 100 000.

  • <sc>Fig</sc>. B1.
    View in gallery
    Fig. B1.

    Meridional profiles of the PV in the experiments from a1 to d3. The black curves are the experimental results created by zonally averaging the time-mean model outputs for 0.3 < x < 0.6, and the red curves are the theoretical model’s results. The solid curves correspond to the upper layer, while the dashed curves correspond to the lower layer.

  • <sc>Fig</sc>. B2.
    View in gallery
    Fig. B2.

    As in Fig. B1, but for experiments e2, e3, f2, and f3.

Fig. 1.

A double-gyre circulation accompanying prematurely separated WBC extension jets in a two-layer model. (a) Snapshot of the upper-layer streamfunction, (b) snapshot of the lower-layer streamfunction, (c) upper-layer time-mean streamfunction, (d) lower-layer time-mean streamfunction, (e) upper-layer time-mean potential vorticity, and (f) lower-layer time-mean potential vorticity. Ekman pumping is imposed only for x ≥ 1.0, α = 0.5, and r = 10−3. The details of the model configuration and parameters are given in section 2.
Fig. 2.

Assumed meridional profiles of the PV in the western region for the (a) upper layer and (b) lower layer. The dashed curves represent the PV distributions in the Sverdrup balance, i.e., qs1 = −F1ψs + βy and qs2 = F2ψs + βy with the Sverdrup streamfunction ψs in the western region. Here, y1 is the southern edge of the southern recirculation gyre, and y5 is the northern edge of the northern recirculation gyre. The jet exists at y3. The upper-layer PV is Q11 for y1 < y < y3 and Q15 for y3 < y < y5. The lower-layer PV is Q23 for y2 < y < y4. The Roman numbers from I to VI denote the six regions, and regions II, IV, and VI are shaded.
Fig. 3.

Meridional profiles for the case of α = 0.5 and r = 10−3: (a) upper-layer PV, (b) upper-layer streamfunction, (c) upper-layer zonal velocity, (d) lower-layer PV, (e) lower-layer streamfunction, and (f) lower-layer zonal velocity. Black solid curves denote the theoretical results, blue dotted–dashed curves denote the experimental results, which are created by zonally averaging the time-mean model outputs for 0.3 < x < 0.6, and red dashed curves denote those in the Sverdrup balance. The shade denotes regions II, IV, and VI.
Fig. 4.

As in Fig. 3, but for the case of α = 0 and r = 10−3.
Fig. 5.

Dependence of the meridional location of the jet y3 on the strength of the PV front Z (the PV difference across the jet) for α = 0, 0.25, 0.5, and 0.75. The solid curves are the theoretical solutions assuming that ψ1(y3)=(1/2)ψsmax when y5 ≠ 0, where ψsmax = 1 is the maximum value of the Sverdrup streamfunction. The dotted–dashed curves and dashed curves are the theoretical curves of ψ1(y3)=(1/2)ψsmax×0.9 and ψ1(y3)=(1/2)ψsmax×1.1, respectively. Symbols denote the experimental results shown in Table 1: asterisks correspond to α = 0, diamonds to 0.25, circles to 0.5, and triangles to 0.75. The open circles and open triangles are the results forced by (4) with xf = 1.0, whereas the closed circles and closed triangles are the results forced by (4) with xf = 1.5. The closed diamonds are the results of Z increased by assuming Q15 = 0.
Fig. 6.

Meridional profiles of the streamfunction at the western edge of the forced region: (top) experiments c2, c3, d2, and d3 for the case of xf = 1.0, and (bottom) experiments e2, e3, f2, and f3 for the case of xf = 1.5. The black solid curves are the profiles at x = xf, the red dotted curves are those of the Sverdrup streamfunction, and the blue dotted–dashed curves in the bottom row are those at x = 1.0. The thin curves with small magnitudes show the streamfunctions in the lower layer.
Fig. A1.

Meridional profiles of the PV and the eastward velocity in the experiments with different F2 and βT created by zonally averaging the time-mean model outputs for 0.3 < x < 0.6: (a) upper-layer PV profiles, (b) lower-layer PV profiles divided by F2, and (c) upper-layer eastward velocity. The black curves are the results for F2/F = 0.2 and βT = 0, the navy dotted–dashed curves are for F2/F = 0.1 and βT = 0, the red dotted curves are for F2/F = 0.05 and βT = 0, and the blue dashed curves are for F2/F = 0.1 and βT = βF/F1.
Fig. A2.

Paths of the jet defined by ψ1 = 0.5 contours for four cases with different F2 and βT: (a) F2/F = 0.2 and βT = 0, (b) F2/F = 0.1 and βT = 0, (c) F2/F = 0.05 and βT = 0, and (d) F2/F = 0.1 and βT = βF/F1. Here, 50 paths are plotted every 1000 after T > 100 000.
Fig. B1.

Meridional profiles of the PV in the experiments from a1 to d3. The black curves are the experimental results created by zonally averaging the time-mean model outputs for 0.3 < x < 0.6, and the red curves are the theoretical model’s results. The solid curves correspond to the upper layer, while the dashed curves correspond to the lower layer.
Fig. B2.

As in Fig. B1, but for experiments e2, e3, f2, and f3.
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