1. Introduction

East of Taiwan and Luzon, a weak (a few centimeters per second) and shallow (~100 m) zonal current, called the North Pacific Subtropical Countercurrent (STCC), penetrates for thousands of kilometers into the Pacific Ocean. The STCC was predicted by Yoshida and Kidokoro (1967), although the wind-driven mechanism that they proposed is now believed not to be the principal one for the formation of the STCC (Kobashi and Kubokawa 2012). The first observation of the STCC was by Uda and Hasunuma (1969) and later confirmed by White et al. (1978) and others. The portion of the STCC from 130°E to 180° is believed to actually consist of two bands: the northern STCC between 21° and 25°N and the southern STCC between 19° and 21°N (Kobashi et al. 2006) (Fig. 1a). Farther east, between 175°E and 160°W, there is another zonal current along ~26°N, but this latter current will not be considered in this work. These STCCs are supported by subsurface density fronts produced by low potential vorticity (PV) mode waters formed in the Kuroshio and Kuroshio Extension. A review of the STCC is given by Kobashi and Kubokawa (2012).

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(a) Annual-mean sea surface height (m; shading) and velocity (vectors; black for zonal velocity u > 0 and white for u < 0) at the first grid level nearest the surface obtained by diagnostically calculating the corresponding monthly fields using the WOA data from NODC (http://www.nodc.noaa.gov/OC5/WOA05/pr_woa05.html), then averaging over 12 months. Dashed box is the study STCC region. (b) March zonally averaged (130°E–180°), y–z sectional contours of temperature (black lines; °C) and u [shading with positive magenta (negative white) contours every 1 (−2) cm s⁻¹].

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0162.1

The STCC region (defined here as 17°–27°N, 130°E–180°) is populated with eddies that typically have diameters of 150–300 km and sea surface height (SSH or η) anomalies of ±0.1 m (Chelton et al. 2011). Eddies propagate westward at speeds of approximately 0.05 m s⁻¹, and the horizontal velocities u within them have speeds of ~0.1–0.3 m s⁻¹. We define x, y, and z as the zonal, meridional, and vertical (positive upward, z = 0 at the mean sea surface) directions, and the corresponding velocity components are u, υ, and w and primes denote fluctuations. Using altimetry data (1992–97), Qiu (1999, hereafter Q99) showed that eddy kinetic energy (EKE) in the STCC peaks in May and is weakest in November–December. The EKE [(u′² + υ′²)/2] seasonally correlates with a 2-month lag with the vertical shear of the zonal geostrophic current computed from monthly climatological temperature T and salinity S data (Levitus and Boyer 1994). To explain the lag, Qiu used a 2.5-layer quasigeostrophic (QG) model (two active layers overlying a quiescent, infinitely deep layer) with an eastward current in the upper layer of depth H₁ = 150 m and velocity U₁ = 0.03 m s⁻¹ (0.01 m s⁻¹) representing the STCC in spring (fall). The second layer has H₂ = 300 m and U₂ = −0.03 m s⁻¹, the same for spring and fall, representing the westward-flowing North Equatorial Current (NEC). The system is baroclinically unstable with a maximum growth rate σ_max ~ 0.016 day⁻¹ in spring at a wavelength of 2π/k_max ~ 300 km, while the value in fall is lower, σ_max ~ 0.005 day⁻¹, and the corresponding wavelength is longer at ~350 km. Kobashi and Kawamura (2002, hereafter KK02) extended the instability analysis using a 3-layer QG model (with a finite, lowest layer). Despite the different H₁ and H₂ used (125 and 575 m, respectively), a similar maximum growth rate ~0.015 day⁻¹ for March was obtained.

The 60-day e-folding growth in late winter has been used by Q99 and others to explain the large EKE observed in spring (through summer). The rationale is that “the 2-month lag is the length of time the unstable waves need to fully grow” (Q99, p. 2479). On the other hand, it has long been known to forecasters (Petterssen 1955; Buzzi and Tibaldi 1978; more recently, for the ocean, see Yin and Oey 2007) that several (2–3) e-folding times are often required for perturbations to grow to matured weather events. In the case of the STCC, a simple estimate suffices to show that the 60-day growth is too slow. Instability analysis calculates growths of infinitesimal initial perturbations (Pedlosky 1979). Suppose the SSH perturbations are |η′| ≈ 10⁻² m, which is approximately 10% of the observed values (see, e.g., Fig. 1 of Q99, showing η′ ~ 0.07–0.2 m in the STCC), then it would require 2–3 e-folding times [i.e., eⁿ ~ (0.07–0.2)/0.01; n = 2–3], or 120–180 days, for the perturbations to grow to the observed eddy amplitude, and the months of peak EKE would be in July or September, contrary to the observed peak month in May. A main goal of this manuscript is to resolve this discrepancy between theory and observation. Pedlosky (1964) noted that discrete normal modes [obtained by assuming a separable solution of the form ψ(z)F(t)e^i(kx+ly), as in the above cited instability analyses] are, in general, incomplete and need to be supplemented by a (generally) continuous spectrum of nonmodal modes. An extreme example is the Couette problem, for which the normal-mode solution is a trivial one (Case 1960; Farrell 1982). For the Eady (1949) problem, Farrell (1982) showed that the nonmodal solution can produce growths exceeding the exponential growth in the early stages and also for wavenumbers near the neutral or decaying Eady modes (Farrell 1982, 1984, 1985). Descamps et al. (2007) demonstrated strong initial growth with a realistic case study and attributed the subsequent decreased growth to nonlinearity (cf. Orlanski and Cox 1973). In this work, instead of a normal-mode approach, we compute the unstable evolution of perturbations by direct numerical integration of an initial-value problem. The resulting growth rates are compared with those calculated using the same background state but based on normal modes. We then examine the relative importance of the sea surface temperature (SST) front on instability growths, as suggested by KK02 and Kobashi and Xie (2012), who pointed out its potential relevance to STCC variability, as well as interannual variability.

Section 2 describes the instability experiments and observational data. Section 3 validates the model’s basic flows. Section 4 describes growth rates, and section 5 describes wavelengths of unstable disturbances. Section 6 discusses interannual variability, and section 7 concludes the manuscript.

2. Numerical model, instability experiments, and observational data

We use the parallel version of the Princeton Ocean Model (POM; Blumberg and Mellor 1987) implemented by Jordi and Wang (2012) and developed by Oey et al. (2013) for the North Pacific Ocean: 16°S–70°N and 98°E–73°W. The horizontal resolution is 0.1° × 0.1° and there are 41 terrain-following (sigma) levels. The model is extensively described in Oey et al. (2013). Smagorinsky (1963) horizontal viscosity is used with the (nondimensional) coefficient = 0.1, which yields values of 10–100 m² s⁻¹ in the STCC; the horizontal diffusivity is made 10 times smaller. All surface fluxes are zero. Monthly World Ocean Atlas (WOA) data from the National Oceanographic Data Center (NODC; http://www.nodc.noaa.gov/OC5/WOA05/pr_woa05.html) for T and S are first used to diagnostically (i.e., T and S are kept fixed) integrate the model to steady state, which yields 12 monthly current fields in balance with the corresponding densities ρ. Starting from these balanced states, the calculation of instability is posed as an initial-value problem for each of the 12 months from January to December. Spatially random (i.e., grid scale) temperature perturbations with an amplitude |T′| of 0.01°C that decay exponentially from the surface with an e-folding depth of 500 m are seeded into each of the 12 balanced fields, and each evolution is tracked prognostically (i.e., with T and S freely evolving) for 360 days. The exponential decay follows the decay of the amplitude function of the fastest growing normal mode (see the appendix), which was also found by Tulloch et al. (2011, their Fig. 4p). However, the growth rates are found to be insensitive to other z-dependent forms (different e-folding depths, constant and sinusoidal) of the initial perturbations. For the purpose of showing faster nonmodal growths, the demonstration of any of these solutions suffices. To separate the effects of the SST front from the total solution, another set of experiments is carried out, as will be discussed in the appropriate subsection below. The above experiments track the unstable (if one exists) evolution of perturbations from a basic state that has energy contained in long but finite wavelengths (approximately >1000 km). Pierrehumbert and Swanson (1995) pointed out that eddies in nature rarely proceed from small perturbations of a nearly undisturbed jet. For a comparison with the more classical approach, another set of experiments, with zonally invariant basic flow, is also conducted in a zonal periodic channel 3°–33°N and 130°E–180° with flow-relaxation sponge zones (T and S fields are relaxed to climatology; Oey and Chen 1992) of 3° latitude width along the northern and southern boundaries, where free-slip conditions are also imposed. Zonally averaged, monthly T and S from WOA are specified, and the model is again integrated until the velocity is in steady balance with the density field. Perturbations are introduced and tracked prognostically, as before, for each of the 12 months. This latter set of experiments closely mimics the conventional instability analysis in which zonally parallel basic flow is perturbed, with the important difference that normal modes are not assumed. The experiment is called “Model//.”

Observational data used for analysis are weekly SSH anomalies (SSHAs) at ⅓° resolution from 1993 to 2010, from the Archiving, Validation, and Interpretation of Satellite Oceanographic data (AVISO; http://www.aviso.oceanobs.com/) project.

3. Validation of the modeled basic flow

The < shaped SSH contour (Fig. 1a) between 19° and 27°N is characteristic of the existence of the STCC; the southern and northern branches are as described above. The simulation also shows other familiar features besides the STCC: poleward shelf flows in the Taiwan Strait and the East China Sea (Isobe 2008); the Kuroshio along the East China Sea and Japan’s southern continental slopes (Taft 1972; Wijffels et al. 1998); recirculation gyres south of Japan and a southwestward countercurrent east of the Ryukyu Islands (Hasunuma and Yoshida 1978); the Kuroshio Extension (Niiler et al. 2003); and the westward-flowing NEC that bifurcates near 13°–14°N off the Philippines (Nitani 1972; Toole et al. 1990).

In the y–z sectional contours of u and T (Fig. 1b), the southern and northern branches of the STCC and their vertical shears are seen. The subtropical density front is primarily controlled by T (e.g., White et al. 1978), although salinity gradients tend to weaken the front (Kobashi and Kubokawa 2012). The eastward current exists in the upper 100 m where isotherms (depth Z_T measured from the free surface) slope upward from south to north, ∂Z_T/∂y > 0. This supports a positive shear by the thermal wind: ∂u/∂z ≈ (N²/f)∂Z_T/∂y, in such a way that u ≈ 0.03 m s⁻¹ near the surface (Aoki et al. 2002). Beneath the eastward current is the broad, westward return flow of the subtropical gyre that merges with the NEC in the south. The main thermocline deepens northward, that is, ∂Z_T/∂y and ∂u/∂z < 0. The core of this subsurface flow is 400–500 m thick, and u ≈ −0.03 m s⁻¹. These features are in good agreement with previous observations (e.g., Kobashi et al. 2006). The NEC is quite strong near the surface, u ≈ −0.16 m s⁻¹, and below it, for z < −300 m, a weak eastward current of a few centimeters per second is simulated; these features are also consistent with observations (e.g., Qiu and Joyce 1992).

The seasonal variation of the STCC has been examined (plots not shown). The surface current is stronger in late winter–early spring because the temperature is vertically more mixed near the surface (50 m) and ∂Z_T/∂y > 0 as mentioned above, creating strong meridional gradients and zonal current in the thermal wind balance. In summer–fall, the near-surface layer is stratified, isotherms flatten, and the surface STCC weakens. The PV (and ∂PV/∂y) distributions are consistent with the previous analyses of hydrographic observations and numerical models (Aoki et al. 2002; KK02; Kobashi et al. 2006; Yamanaka et al. 2008) with characteristic low PV mode waters north of the STCC in the potential density σ_θ ranges of 25.3–25.6 (subtropical mode water) for the northern front and additionally 25.7–26.2 (central model water) for the southern front. The resulting ∂PV/∂y distribution is such that its y–z integral from 19° to 25°N and the 0–400-m layer is ≈0 from February to April, which is close to satisfying the necessary condition for linear instability (Pedlosky 1979), consistent with the peak growth rates σ found during those months (Q99; KK02).

4. Growth rates

These are shown in Fig. 2 for different cases. In order to estimate growth rate from satellite data (σ_AVISO), eddies are first identified based on the Okubo–Weiss parameter W (Okubo 1970; Weiss 1991) using the velocity (u, υ) calculated from weekly SSHA from 1993 to 2010 assuming geostrophy:

Here, s_n = ∂u/∂x − ∂υ/∂y and s_s = ∂υ/∂x + ∂u/∂y are the normal and shear components of strain, respectively, and ω = ∂υ/∂x − ∂u/∂y is the relative vorticity. Eddies are connected regions within which the vorticity dominates the two strain terms, that is, W < 0. The identification method follows Isern-Fontanet et al. (2006), who define a vortex as a region where W is smaller than −0.2 times the standard deviation of W. Various eddy properties are then calculated:

Using E is similar to computing EKE by first high-passing SSHA to filter out long wavelengths and then using the resulting geostrophic velocities; it yields slightly higher EKE than the Fig. 5 of Q99 but the same seasonal variation with a maximum in May–June and a minimum in December–January. The growth rate is then computed from (e.g. Farrell 1982)

and composited from 1993 to 2010 to obtain the solid curve in Fig. 2. Since at any time the observed eddies are of finite amplitude, σ_AVISO does not correspond to the linear growth rate in the mathematically defined sense (Pedlosky 1979). One expects a weaker growth rate (Orlanski and Cox 1973; Descamps et al. 2007) than the linear growth rate σ_KK02 from KK02 (shown as dotted line connected by triangles in Fig. 2); nonetheless σ_AVISO shows a seasonal peak in March and minimum and negative in fall, in remarkable consistency with σ_KK02.

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Monthly growth rates (day⁻¹) averaged in 19°–25°N and 130°E–180° (cf. Q99 and KK02) from various data labeled AVISO, KK02 (their Table 3, dotted line triangles; negative values for July–November are not shown), Model (solid red) and Model// (zonal–parallel basic flow, gray) averaged from 0 to 400 m (equivalent to KK02), and Model_0–100m (red dashed line) and Model_NF (no SST front, green dashed line) averaged from 0 to 100 m. Growth rate curve from normal-mode instability is labeled NM. The thin horizontal line at 0.0152 day⁻¹ is 12-month mean Eady growth rate averaged in the surface 400 m.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0162.1

The Eady (1949) (see also Gill 1982) maximum growth rate

where Ri = N²/[(∂u/∂z)² + (∂υ/∂z)²] is the Richardson number and N² = −g(∂ρ/∂z)/ρ_o is the buoyancy frequency, is computed from the monthly basic flow fields and averaged in the STCC region from the surface to z = −400 m. The resulting σ_Eady shows a statistically barely significant seasonal variation with a maximum in February and a minimum in August; its 12-month mean (0.0152 ± 10⁻³ day⁻¹) is shown in Fig. 2 as the thin horizontal line, which is close to the peak growth rate of 0.015–0.016 day⁻¹ computed by Q99 and KK02. The Eady model assumes constant shear and N so the good agreement is fortuitous; nonetheless the classic model anticipates many of the gross instability properties detailed in Q99, for example, dependence of the growth rate on stratification and shear. The Eady model is included here for comparison only as in Smith (2007) and Tulloch et al. (2011).

The modeled growth rate is calculated using the Okubo–Weiss vortex energy method described above for AVISO but applied to the prognostic simulations for each of the 12 calendar months, then averaged over the STCC region and from z = 0 to −400 m, as well as for the first 60 days. The resulting σ_Model (red solid curve in Fig. 2) is insensitive to averaging periods less than ~60 days before finite-amplitude disturbances begin to appear. Q99 (his Fig. 10) estimated shears from observations such as ΔU = U at the surface minus U at z = −400 m for instability analysis, while KK02 (their Fig. 6) used the surface minus “the central depth of the westward flow” that lies slightly shallower than z = −400 m (their Fig. 7a) (see appendix; Fig. A2). The 400-m averaging depth is also based on Roemmich and Gilson (2001), who, with regard to observed temperature anomalies T′, concluded that “...the features slant westward with decreasing depth…on average from 400 m up to the sea surface.” It also agrees with the analysis of Argo data by Qiu and Chen (2010), which shows significant coherence of T′ confined in the top 400 m (see their Fig. 7), consistent with Roemmich and Gilson (2001). The σ_Model is further checked by computing the eddy kinetic energy plus eddy potential energy (EPE), as well as the barotropic (BT) and baroclinic (BC) conversion terms, as in Oey (2008):

Then,

The two methods yield growth rates that are within ±5% of each other. In the latitudinal range from 17° to 27°N, we find that BC ≫ BT, which supports the hypothesis of Q99 and KK02 that baroclinic instability (BCI) dominates in the STCC.

The same methods of computing growth rates are applied to the instability simulations of the zonally parallel basic flow. The corresponding σ_Model// is shown as the dashed–dotted line connected by triangles in Fig. 2. Also, the red dashed line with open circles, σ_{Model0–100m}, is simply σ_Model, but computed for the upper 100 m. Finally, σ_NM is the growth rate computed based on normal modes using the same monthly WOA data used in Model// (see the appendix).

It is apparent that during the period of large growth (0–400 m) from January through April σ_Model// > σ_Model > σ_Eady > σ_KK02 ≈ σ_NM. We interpret the stronger growths (σ_Model or σ_Model//) as resulting from the existence of nonmodal, rapidly growing solutions that are absent from the normal-mode formulation of Q99 and KK02, but which are consistent with the results of Farrell (1982, 1984, 1985) and Descamps et al. (2007). This interpretation for σ_Model// > σ_NM is clear, since the difference between them is that σ_NM is from the normal mode, while σ_Model// is from the initial-value integration. On the other hand, the interpretation of σ_Model// > σ_KK02 (≈σ_Q99) is less clear, because the larger σ_Model// could arise from differences in the climatologies being used. In the appendix, we show that the background states used in Q99 and KK02 are comparable to that used to compute σ_Model//. Therefore, differences in the background states cannot account for the fact that σ_Model// > σ_KK02 (≈σ_Q99).

It is interesting that during strong growth months in winter, σ_Model is bounded by σ_Model//, though the difference is insignificant (it barely exceeds the ±5% uncertainty between the different methods of calculating σ’s). From April to August, the σ_Model// drops more rapidly than the σ_Model; the reason is unclear.

In March, the σ_Model (=0.023 day⁻¹) exceeds σ_KK02 and σ_NM by about 50%, which may be compared with the 38%–45% increase in the initial growths calculated by Farrell (1982) for the Eady problem and 50% increase (also compared to σ_Eady) calculated by Descamps et al. (2007) for the initial growth of a realistic winter cyclone across the North Atlantic Ocean. It is clear that σ_Model is a more realistic estimate (than σ_NM); however, the resulting e-folding time of approximately 43 days is still slow. It is clear from Fig. 2 that the most rapid growth is near the surface where (σ_{Model0–100m})⁻¹ ≈ 28 days in March. The 2–3 e-folding times because of this rapid surface growth are consistent with the 2–3-month period that is required for perturbations in late winter to grow into the observed EKE in late spring and summer.

KK02 (paragraphs 26–29 and their Fig. 6; see also Kobashi and Xie 2012) pointed out that in addition to being maintained by the subsurface front, the STCC also depends on the strength of the SST front in the mixed layer (Fig. 3a, black contours). The SST front is strong from December to April and weak from June to October, as seen in the contours of ∂SST/∂y in Fig. 3b. In particular, in February and March, the region of large |∂SST/∂y| extends southward into the main portion of the STCC. As a result, KK02 shows that the contribution of the SST front to the vertical shear of the STCC is large in late winter to spring and is weak in other months, especially from summer to fall (Fig. 3c, black). To study how the presence of the SST front may change the growth rate, we design a set of experiments in which the SST front is removed from the monthly climatology and the above perturbed simulations and growth rate calculations are repeated. For a given month, the frontal-removal procedure “flattens” the isotherms by merging them with the September values in the region 19°–30°N from the surface to z ≈ −100 m, but leaving temperatures unchanged outside this region; an example for March is shown in Fig. 3a (gray contours). September is chosen as the reference month because it is when the surface front is weakest (Fig. 3b). In the absence of the SST front, the vertical shear weakens, especially in late winter and spring (Fig. 3c, gray). The growth rate for the “no SST front case” σ_ModelNF is computed for the surface 100 m (shown in Fig. 2 as the green dashed line with open circles) to compare with σ_{Model0–100m}. There is about a 30% drop in the growth rate when the SST front is removed. The drop is anomalously large for March. An examination of the eddy energetics maps [using (4.6) and (4.7)] reveals that this is caused by the anomalously stronger SST front (Fig. 3b) and BC (not shown) in the region 19°–25°N, 135°–150°E.

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(a)The y–z section (at 150°E) isotherms (°C) in March for WOA climatology (black) and for WOA climatology with SST front removed in the near 100 m (gray; see text); abscissa is latitude (°N) and ordinate is depth (m). The latter isotherms are also close to the isotherms in September. (b) Zonally averaged (140°E–180°) contours of ∂SST/∂y from WOA (°C m⁻¹); abscissa is calendar month and ordinate is latitude (°N). White contours are −5 and −8 × 10⁻⁶ °C m⁻¹. Dotted lines indicate 17°N and 27°N. (c) Monthly u_0m − u_−400m (cm s⁻¹) averaged from 19° to 25°N and 130°E to 180°, computed using the WOA climatology (black) and using WOA climatology with SST front removed in the near 100 m (gray; see text); bars are their difference with scale (cm s⁻¹) on right.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0162.1

5. Spatial scales

Q99 deduced a zonal wavelength λ_x|_Q99 ≈ 300 km for the most unstable wave assuming an isotropic eddy field. KK02 found using observed EKE spectra from altimetry that the meridional scales are generally smaller than their zonal scales. Monthly zonal wavelengths of the most unstable waves computed from the normal-mode analysis of KK02 are reproduced in Fig. 4c (dotted line), showing λ_x|_KK02 ≈ 330 km in March, increasing to a maximum ≈370 km in June. In the present case, unstable perturbations grow into eddies during each of the model 360-day run for each calendar month. Figure 4a gives an example for the March case using W and currents at z = −25 m, showing eddies with 100–300-km diameters and current speeds of 0.1–0.2 m s⁻¹ in agreement with observations (Chelton et al. 2011). To determine the dominant wavelengths (λ_x, λ_y), spectral energies E_W of 10-day-averaged maps of W_ω (=W with signed vorticity within each eddy) in zonal and meridional directions are calculated. Then (following KK02),

The 360-day evolution of λ_x for each of the 12 calendar months is shown in Fig. 4b, and 0–60- and 60–120-day averages of λ_x are compared with λ_x|_KK02 in Fig. 4c [The term λ_y is similar (not shown), and as in KK02 its values are generally smaller. Also, the same calculation is repeated on the W using the AVISO data, and good agreements with Chelton et al. (2011, section 4.2) are found.]. During the initial growing stage (t < 60 days), λ_x ≈ 225–270 km irrespective of the months, though waves with the fastest growth rates in February–March tend to be shorter (Fig. 4b). These wavelengths are notably shorter than those of the most unstable waves of Q99 and KK02 using normal modes and are in fact close to their short-wave cutoff wavelengths: λ_min|_Q99 ≈ λ_min|_KK02 ≈ 224–273 km (spring–fall; see Fig. 11b of Q99 and Fig. 11a of KK02). For comparison, for the Eady problem (Pedlosky 1979), λ_min|_Eady = 2π (NH/f)/2.4 ≈ 200 km using N ≈ 10⁻² s⁻¹ (averaged in the STCC from 0 to 400 m using WOA), H ≈ 400 m, and f ≈ 5 × 10⁻⁵ s⁻¹. In the Eady problem, Farrell (1982, 1984) found growing nonmodal solutions near the neutral normal modes. In addition to faster growths, the shorter wavelength λ_x near the neutral normal modes of Q99 and KK02 found here during the initial growth stage is more evidence of the existence of the nonmodal solution.

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(a) Okubo–Weiss W (s⁻²) and u (m s⁻¹) at z = −25 m for model’s March calculation at day 360 (note different zonal and meridional scales); (b) zonal wavelength λ_x (km) from W_ω for 12 calendar months (abscissa) and days 0–360 (ordinate); (c) 0–60- and 60–120-day-averaged monthly λ_x (km; ordinate) and comparison with KK02 values (dotted line with triangles from their Table 3).

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0162.1

Because of the idealized nature of the instability experiments, the time-asymptotic eddy state for each of the 12 independent calendar months does not have an observed correspondence. For example, because of the faster growth, March eddies tend to be the most energetic and numerous, but in reality they evolve into May as observed. Nonetheless, Fig. 4b shows that with time, eddies with larger scales emerge, as can be anticipated from the theory of geostrophic turbulence (Rhines 1979). The asymptotic scales of around 300–315 km are comparable to the longest of the eddy scales estimated by Chelton et al. (2011) from altimetry data.

6. Interannual variation

Qiu and Chen (2010) suggested that in the STCC, EKE is modulated by vertical geostrophic shear due to interannually varying Ekman convergence that acts on the meridional surface density gradients. The corresponding expression can be derived by integrating the heat equation across the surface layer from z = −δ_E to 0 (Chang et al. 2010):

where u₃ = (u, υ, w) is velocity and Q is the vertical diffusive heat flux. We assume that δ_E ≈ constant (or varies slowly)¹ and that in the thin surface layer u₃ can be approximated using the wind-driven Ekman theory, then (6.1) becomes

where k is the z-unit vector, τ_o is the kinematic wind stress, u_E = −k × τ_o/(fδ_E) is the Ekman depth-averaged velocity, Q_sδ_E is the surface heat flux (positive upward, i.e., heat loss from the ocean), and α_NTδ_E parameterizes the corresponding heat flux at z = −δ_E, that is, across the base of the layer, as a Newtonian cooling term with coefficient α_N. The second term on the RHS is due to Ekman pumping and will be neglected here, assuming that the surface layer is well mixed, ∂T/∂z ≈ 0, and/or the scale of the STCC front is much smaller than the scale of the wind, so that the second term is smaller in magnitude than the first term. Take the “grad” of (6.2):

We call E_kcon the Ekman forcing term and G_s the geostrophic shear vector through the thermal wind equation −∇T = f/(gα)k × ∂U_o/∂z, where U_o is the geostrophic velocity and α is the thermal expansion coefficient. In the STCC, |∂/∂x| ≪ |∂/∂y|, and setting Q_s = 0 and α_N = 0 leads to

which is (3) of Qiu and Chen (2010), where for brevity we use the same symbols for scalar quantities: G_s = −∂T/∂y and E_kcon = −∂(υ_EG_s)/∂y the Ekman convergence due to the dominance of ∂υ_E/∂y at interannual time scales (Qiu and Chen 2010). As seen from (6.4), (6.5) remains valid if Q_s is also included (cf. Qiu and Chen 2013).

Using reanalysis wind and SST data, Qiu and Chen (2010) calculated E_kcon and showed that it leads EKE (from AVISO) by approximately 9 months:

where the notation →_{9 mo} means “leads by 9 months.” Qiu and Chen (2010, p. 221) concluded that the 9-month lead “represents the time required for the STCC-NEC shear to adjust to the time-varying Ekman forcing plus the time for instability…to fully grow.” On the other hand, we note that (6.5) implies that E_kcon leads G_s by π/2 or 10 ± 3 months, since the period of E_kcon is 3.3 ± 1 yr (see Fig. 10a of Qiu and Chen 2010)². Thus,

From (6.6) and (6.7), since both use the same E_kcon data, we deduce that

where means “in phase with.”

This implies a very rapid growth that, while it may be consistent with the existence of nonmodal solutions and SST front, evidence below argues for a different balance than (6.5).

It was shown above that σ_AVISO estimated from (4.4) yields the correct seasonal variation, even though the value is underestimated (Fig. 2). Interannual variations of the growth rate arise because in some years the seasonal conditions are more favorable for instability than other years (Qiu and Chen 2010). Figure 5a shows σ_AVISO and E [EKE computed using (4.3)] averaged in 19°–25°N and 130°E–180°. Years of large (weak) E from 1995 to 1998 and 2003 to 2007 (1993–95, 1999–2002, and after 2008) agree with the EKE in Fig. 3b of Qiu and Chen (2010; see also Fig. 15a of Qiu and Chen 2013). For convenience, we temporarily set E = EKE of Qiu and Chen (2010), as this does not change the logic. The lagged correlation³ Corr(σ_AVISO, EKE, 10) = 0.67 is maximum when σ_AVISO leads EKE by 10 months (Fig. 5a):

Since STCC eddies are produced by baroclinic instability, for which growth rate and shear are approximately in phase (e.g., Qiu and Chen 2010), we obtain

Clearly, relations (6.8) and (6.10) cannot both be true.

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(a) The 360-day low-passed σ_AVISO (black line) averaged in 19°–25°N and 130°E–180° from AVISO geostrophic velocity anomaly and EKE [red and blue bars, = E of (4.3)] nondimensionalized by their standard deviations: 5 × 10⁻⁴ day⁻¹ and 4 × 10⁻³ m² s⁻², respectively, and E_kcon (10⁻¹⁴ °C s⁻¹ m⁻¹; gray). Their various correlations and 95% significance are shown in parentheses. (b) EKE (contours, black for values >0.04 m² s⁻²) and σ_AVISO (color; day⁻¹) plotted with the abscissa as years from 1993 to 2010 (18 discrete values, intervals between years have no meaning) and ordinate as calendar months to show season.

Citation: Journal of Physical Oceanography 44, 3; 10.1175/JPO-D-13-0162.1

The contradiction arises because (6.5) (which leads to 6.7) is incomplete and should be replaced by (6.3):

The Newtonian cooling term with α_N in per second parameterizes (crudely) the effects of unresolved eddies and turbulence arising from, for example, mixing due to surface waves (e.g., Polton and Belcher 2007), enhanced turbulence in fronts (D’Asaro et al. 2011), or from the effects of rapid submesoscale dynamics (Boccaletti et al. 2007). At interannual time scales |∂/∂t| ≪ α_N, then instead of (6.7), we have

Physically, small-scale turbulent eddies are effective and geostrophic shear responds instantaneously (compared to interannual time scales) to surface forcing by Ekman convergence and/or heat flux. Using (6.12), we obtain using (6.9)

and the observed relations (6.6) and (6.10) are now consistent.

To verify relation (6.12), we compare E against σ_AVISO and E_kcon that is independently estimated by Qiu and Chen (2010; their Fig. 10a) using National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) monthly reanalysis data. They are plotted in Fig. 5a and their maximum lagged correlations are shown; they indicate that

Thus, E_kcon leads σ_AVISO (↔G_s) by a short but nonzero time of about 1 month, and the latter in turn leads E by another 10 months. Theory (6.3) and the data are now self-consistent.

7. Conclusions

This manuscript revisits the problem of the instability of the STCC. We argue that the previously estimated peak growth rates of 60 day⁻¹ in March are too slow in explaining the large EKE in May. Instead of the normal-mode approach, we obtain faster growths by posing an initial-value instability problem and numerically track the evolution of unstable perturbations. The resulting stronger growths near the short-wave cutoff wavelengths of neutral normal modes agree well with nonmodal growths previously found in the literature. Additionally, we show that the existence of the SST front in late winter also leads to strong growth rates. At interannual time scales, we formulate a self-consistent theory of modulation of EKE by BCI based on the idea that the STCC geostrophic shear rapidly adjusts to surface forcing, leading to growths of longer-lasting, larger, and more energetic eddies. Finally, we note that years of strong BCI and EKE do not necessarily coincide and suggest that near-surface dissipation may play a role.