1. Introduction

Recent observations have revealed an important aspect of interannual to decadal extratropical thermocline variability, which is that the sea surface height (SSH) anomaly propagates predominantly westward at all latitudes (Jacobs et al. 1994; Chelton and Schlax 1996), while a substantial part of the thermocline temperature variability follows the gyre circulation (Deser et al. 1996; Zhang and Levitus 1997; White and Cayan 1997). This complexity of thermocline variability motivated us to reexamine the dynamics of planetary waves in the presence of a thermocline circulation. Liu (1998) suggested that the different features of variability may be associated with two types of planetary wave modes in the extratropical ocean: the non-Doppler-shift mode (N mode) and the advective mode (A mode). The N mode resembles the first baroclinic mode and propagates westward regardless of the mean flow. The A mode resembles the second baroclinic mode and tends to follow the mean flow. Furthermore, because of the distinct vertical structure of the two modes, it is speculated that the N mode is mostly responsible for the observed SSH variability, while the A mode is predominantly responsible for observed temperature variability in the upper thermocline.

In this study, we extended the results of Liu (1998) by first generalizing the basic state of the zonal flow to a more realistic thermocline gyre circulation. More importantly, we investigated the response of the thermocline to anomalies of surface dynamic forcing and buoyancy forcing in light of planetary wave dynamics. We found that anomalous Ekman pumping is the most efficient mechanism for generating the N mode with the clearest signal in the SSH field, while anomalous surface buoyancy forcing may be important for producing the A mode with a strong temperature signal in the upper thermocline.

The paper is arranged as follows: Section 2 introduces the model. Section 3 investigates the propagation of planetary waves in a thermocline gyre circulation. Section 4 further studies the generation mechanism of planetary waves by both the surface wind stress forcing and surface buoyancy forcing. Supporting experiments with three numerical models are presented in section 5. A further discussion on the dynamics of the A mode is presented in section 6. In section 7, we provide a summary and some further discussion of the relevance of our results to observations.

2. The model

To highlight the physical mechanisms involved, we used a 2.5-layer QG model as in Liu (1998). The densities for layer 1 (upper thermocline), layer 2 (deeper thermocline), and layer 3 (abyss) are assumed as ρ₁ = ρ − 2Δρ, ρ₂ = ρ − Δρ, and ρ₃ = ρ. The thicknesses for layers 1 and 2 are D₁ and D₂, respectively. Under a surface Ekman pumping w_e and an entrainment velocity w_d between layers 1 and 2, the nondimensional QGPV 2.5-layer model can be derived as (§6.16 of Pedlosky 1987; Liu 1998)

Here q₁ = y + (ψ₂ − ψ₁)/d₁ and q₂ = y + (ψ₁ − 2ψ₂)/d₂ are the potential vorticities of layers 1 and 2, respectively. Note that the relative vorticity is neglected because we are interested only in planetary waves. The nondimensional layer thicknesses are d_n = D_n/(D₁ + D₂) (n = 1, 2). Other nondimensionl variables are represented by their dimensional counterparts (in “*”) as

Here L is the gyre scale, V_s = βL²_D is a typical speed, T = L/V_s is the timescale of gyre advection, and W is the magnitude of Ekman pumping that satisfies the Sverdrup balance as βV_s(D₁ + D₂) = f₀W. In addition, L_D = [g′(D₁ + D₂)]^1/2/f₀ and g′ = gΔρ/ρ are the baroclinic deformation radius and the reduced gravity, respectively. Typical parameters in the midlatitudes are Δρ/ρ = 0.002, f₀ = 10⁻⁴ s⁻¹, β = 1.2 × 10⁻¹¹ m⁻¹ s⁻¹, D₁ + D₂ = 1000 m, which give L_D = 45 km, V_s = 0.025 m s⁻¹, and W = 3 × 10⁻⁶ m s⁻¹. Finally, the equivalent dynamic elevations on the surface and the two interfaces are h₁ = ψ₁, h₂ = ψ₂ − ψ₁, and h₃ = −ψ₂.

Surface dynamic forcing is Ekman pumping in a QG model. Surface buoyancy forcing is parameterized as an entrainment velocity w_d at the interface of layers 1 and 2. This parameterization can be understood as follows. Convection due to surface cooling generates a low PV anomaly that will be ventilated into the lower thermocline (layer 2 in our model). In a planetary geostrophic model that allows explicit outcropping, the cooling forcing can be represented by the motion of the outcrop line (Liu and Pedlosky 1994; Marshall and Marshall 1995); this motion, however, is difficult to represent properly in our QG model. Nevertheless, we speculate that the low PV anomaly along the outcrop line can be crudely simulated by a detrainment w_d < 0 from layer 1 to layer 2, beneath the base of the mixed layer.¹ This speculation is implied from the solution of Liu and Pedlosky (1994). Our speculation has been confirmed by two recent works that use idealized ventilated thermocline models (Schneider et al. 1998; Huang and Pedlosky 1998) and will also be confirmed in a full ocean general circulation model later in section 5.

For small amplitude disturbances ϕ_n, we have ψ_n = Φ_n(x, y) + ϕ_n(x, y, t) (n = 1, 2), where Φ_n is the mean flow streamfunction and ϕ_n ≪ Φ_n. Similarly, the forcing is separated into mean and perturbation components: w_e = W_e + w_e and w_d = W_d + w_d. The linearized perturbation PV equations can therefore be obtained from (2.1a), (2.1b) as

where the mean currents are (U_n, V_n) = (−∂_yΦ_n, ∂_xΦ_n) (n = 1, 2); the mean PVs are Q₁ = y + (Φ₂ − Φ₁)/d₁ and Q₂ = y + (Φ₁ − 2Φ₂)/d₂; and the perturbation PVs are q₁ = (ϕ₂ − ϕ₁)/d₁ and q₂ = (ϕ₁ − 2ϕ₂)/d₂.

Planetary waves will be studied using the WKB method (Lighthill 1979). For waves with spatial scales much smaller than that of the mean flow, the mean flow and PV gradient can be regarded as locally homogeneous in (2.2). This assumption can be applied to perturbations measuring hundreds of kilometers as shown by altimeter observations (Jacobs et al. 1994; Chelton and Schlax 1996) because the gyre width is on the order of thousands of kilometers. For even larger-scale waves, the WKB approximation may not be very accurate and should be explored in detail in a future study. Nevertheless, the WKB method used here should qualitatively shed light on some basic features of planetary waves, as has been seen in previous studies of Rossby waves (e.g. Hoskins and Karoly 1981). Furthermore, the qualitative conclusions from the WKB solution will be substantiated in section 5 with experiments using three numerical models.

Given the perturbation as

^′_ne^iθ(x,y,t)nw_ew^′_ee^iθ(x,y,t)w_dw^′_de^iθ(x,y,t)

where the amplitudes (in prime) are locally constant (the prime will be dropped hereafter without confusion), the WKB solution at the first order yields

where k = ∂_xθ, l = ∂_yθ, σ = −∂_tθ are local wavenumbers and frequency. In addition,

represents the advection on the mean PV field by the perturbation flows. In the absence of forcing, w_e = w_d = 0, (2.3) gives the dispersion relationship for planetary waves

Aσ, k, l(2.4)

where

Here (U_B, V_B) = (d₁U₁ + d₂U₂, d₁V₁ + d₂V₂) is the vertically averaged flow within the thermocline. The structure of a planetary wave mode can be shown to satisfy

aϕ₂ϕ₁kd₁σkd₂(2.6)

3. Planetary wave propagation in a thermocline gyre

We first study the propagation of planetary waves in a gyre circulation. This is important to understand the remote response, or teleconnection, in the upper ocean, as has been studied extensively in the atmosphere (Hoskins and Karoly 1981; Wallace and Gultz 1981). Consider a patch of anomalous forcing, which locally has the form

w_ew^′_ee^{i(mx+ny−ωt)}w_dw^′_de^{i(mx+ny−ωt)}(3.1)

where without loss of generality, ω > 0 is assumed. (The prime will be dropped later without confusion.) This forcing generates planetary waves to radiate away from the forced region. In our 2.5-layer model, the frequency of the two modes should be the same as that of the forcing:

σω(3.2a)

The wavenumber along the edge of the local forcing should also be the same as that of the forcing. Here, for simplicity, we will only focus on the waves that radiate from a meridional section x = x*, such as the eastern boundary or the eastern/western edges of the forcing patch (which is assumed to have a straight meridional section as its eastern/western edges). Therefore, we have the condition of the matching of initial meridional wavenumber as

ln.(3.2b)

The initial zonal wavenumber can be solved from the dispersion relationship (2.4) as

where Δ = {[(1 + d₁ − U₂)ω − nV_B]² − 4(d₁d₂ − U_B)ω(ω − nV₂)}^1/2 using (3.2a,b). The subscripts N and A denote the N mode and the A mode, respectively. The structure of each mode can be derived from (2.6) as

aϕ₂ϕ₁kd₁σkd₂(3.4)

where * = N or A. In this paper, we will limit ourselves to the case of real k in (3.3). In other words, we do not consider the region of very strong westward flow, such as the southwestern part of the subtropical gyre (in the Northern Hemisphere) where planetary waves are destabilized [see Liu (1998) for more details on planetary wave instability].

The propagation of the waves can be studied using the ray tracing theory (Lighthill 1979). Since the mean flow field does not change with time, the frequency of the radiating wave should remain unchanged as in Eq. (3.2a). However, since the mean flow varies in the x as well as the y directions, both k and l will vary along a ray path. By virtue of the dispersion relationship (2.4), the ray tracing equations can be written as

where · = d/dt and D = 2ω − [kU₂ + lV₂ − (1 + d₁)k]. Note that (3.5d) is derived by simply solving the dispersion relationship (2.4). The group velocity can be verified to satisfy C_gx/(ω/k) + C_gy/(ω/l) = 1. For a wave ray that starts from (x₀, y₀) at t = 0, the complete initial conditions for the wave ray are

xx₀yy₀kkln,(3.6)

where * = N for the N-mode wave ray and * = A for the A-mode wave ray; we have also used (3.2b) and (3.3a,b).

We give one example to demonstrate the major features of wave propagation. The basic-state ventilated thermocline is derived as in Hendershoot (1989) under a zonal outcrop line y₀ = const and mean Ekman pumping W_e(y) = W_e0 sin(yπ). The thermocline solution is

Here x_E = 0 is the eastern boundary and x_s(y) = d₁d₂(y₀ − y)/W_e(y) is the shadow zone boundary. The parameters are chosen as W_e0 = −0.2, y₀ = 0.9, and d₁ = 0.4. [See Hendershoot (1989) for more details.] A decadal frequency forcing (ω = 1) is used. The distinctive ray pathways of the two modes can be seen clearly in Figs. 1a and 1b for the wave rays of the N mode and A mode, respectively. Ten wave rays are plotted in each panel. Six rays start near the outcrop line (solid circles) in the ventilated zone where flows originate from the subduction of the mixed layer waters. Four rays originate near the eastern boundary in the shadow zone where there is no flow in the subsurface (Luyten et al. 1983). For comparison, mean flow trajectories are also plotted in Figs. 1c–e following the streamlines of layers 1, 2, and barotropic flow, respectively. In addition, the trajectories that follow the 2-layer wave characteristics (U_B − d₁d₂, V_B) (Rhines 1986; Dewar 1989; Liu 1996) are also plotted in Fig. 1f. These wave characteristics are parallel to the mean subduction flow U₂ in the ventilated zone in a 2.5-layer model (Luyten and Stommel 1986).

Figure 1 shows that, despite the eastward flow in the northern part of the subtropical gyre in all the flow fields (Figs. 1c–e), all the N-mode wave rays propagate westward (Fig. 1a). This is caused by the non-Doppler-shift effect in this mode.² In contrast, all the A-mode wave rays tend to follow the mean circulation due to its advective nature (Fig 1b). A comparison of the A-mode wave rays with other trajectories clearly shows that the path of the A-mode wave rays closely resembles that of the wave characteristics (Fig. 1f) or the subsurface subduction flow in the ventilated zone (Fig. 1d).

The relationship between the wave rays and the mean-flow trajectories is seen more clearly in Figs. 2 and 3. Figure 2a shows two wave rays (the two marked curves), one for the N mode and the other for the A mode, which originate in the ventilated zone near the outcrop line. For comparison, we also plotted the trajectories that follow the mean flow of layer 1 (curve with *) and layer 2 (curve with ∘). In addition, the wave characteristics (U_B − d₁d₂, V_B) (curve with ×) are also plotted. It is clear in Fig. 2a that the N-mode wave ray does not follow the circulation at all. The A-mode wave ray, on the other hand, follows closely the characteristics (U_B − d₁d₂, V_B) or subsurface ventilation flow U₂, but deviate significantly from the surface flow U₁. This independence of the A-mode wave speed from the surface flow was noted in Liu (1998). Furthermore, the wavenumbers k (solid), l (curve with ∘), and the wave structure (a = ϕ₂/ϕ₁) are also shown in Figs. 2b and 2c for the N mode and A mode, respectively. The vertical structure of the wave remains as a_N > 0 for the N mode and a_A < 0 for the A mode. Thus, the perturbation flows of the two layers are in the same direction for the N mode, but are in the opposite directions for the A mode. Therefore, the N mode resembles the first mode, while the A mode resembles the second mode, as discussed in Liu (1998). Figures 2b and 2c also show that the meridional wavenumber l changes significantly, while the zonal wavenumber k does not. This is to be expected because the variation of the mean flow is much stronger in the y direction than in the x direction. This may imply that our WKB approximation is less accurate in the y direction than in the x direction.

To illustrate the trajectory from a shadow zone, we plot trajectories in Fig. 3 the same as in Fig. 2 but for trajectories originating in the shadow zone. The most noticeable difference from Fig. 2 is that, despite the absence of the layer-2 mean flow in the shadow zone, the A mode still propagates roughly along the wave characteristics. Therefore, the characteristics (U_B − d₁d₂, V_B) are a more general indicator for the wave rays of the A mode than the subsurface flow. This also indicates a difference between the A mode and a passive tracer. A passive tracer, if started in the shadow zone, will stay in the initial position because of the absence of subsurface flow; this relationship will be further discussed later in section 6.

The conclusions above are not very sensitive to model parameters. Sensitivity experiments were carried out on the forcing frequency ω, the initial meridional wavenumber n, and the mean stratification d₁. For a typical thermocline stratification (d₁ ∼ d₂), our conclusions above regarding the wave ray, especially in the A mode, remain robust. This is also reinforced in an analytical study outlined in appendix C. Figure 4 shows a sensitivity test of the frequency for a wave ray originating in the ventilated zone (Figs. 4a,b) and the shadow zone (Figs. 4c,d). In both zones, the direction of the N-mode wave ray changes only modestly with the frequency (Figs. 4a,c), and the direction of the A-mode wave ray shows virtually no change. In a typical thermocline gyre, the slope of the A-mode ray can be shown to follow that of (U_B − d₁d₂, V_B) within 10%–20% (see appendix C).

Finally, the wave propagation features discussed above can be proven analytically in the limit of low-frequency forcing ω → 0. In this limit, we have from (2.4) or (3.3a,b) the wavenumber solutions for the two modes as

The group velocities and wave structures for the two modes can therefore be derived from (3.5a,b) and (3.4). Using (3.8a), the N mode has at the leading order the group velocity

C_gxNC_gyNF_NF_NV_BV₂(3.9a)

and the wave structure

ϕ_2Nϕ_1NV₂V₁(3.9b)

With (3.8b), the A mode has the leading order representation of the group velocity

C_gxAC_gyAU_Bd₁d₂V_BF_A(3.10a)

where F_A = V_B/[V₂(U_B − d₁d₂) + V_B(1 + d₁ − U₂)] and wave structure

ϕ_2Aϕ_1Ad₁d₂(3.10b)

The most striking feature is that, regardless of the mean circulation structure, the N-mode wave ray is purely zonal as shown in (3.9a), while the A-mode wave ray is exactly parallel to the wave characteristic (U_B − d₁d₂, V_B) as in (3.10a). Furthermore, the N mode has an equivalent barotropic wave structure as shown in (3.9b), while the A mode has a pure baroclinic wave structure (with a zero barotropic transport) as indicated by (3.10b). The wave ray directions and wave structures are consistent with the nature of the two modes that are discussed regarding Figs. 1–4. More specifically, since in the subtropical gyre V_B, V₂ < 0 (unless in the shadow zone where V₂ = 0), we have F_N > 0 in (3.9a). Thus, the N mode propagates purely westward, as expected from the non-Doppler-shift nature of the N mode. However, the propagation speed could vary with the mean flow. In the ventilated zone (3.7a), we have the speed factor F_N = d₂. For the A mode, in both the ventilated zone (3.7a) and the shadow zone (3.7b), one can show that the speed factor is F_A = 1/(1 + d₁) < 1. Thus, the wave ray is somewhat slower than the wave characteristics as seen in Figs. 1–3. We will return to the wave speed later in section 6.

In short, the N mode propagates westward regardless of the mean flow, while the A mode follows closely (U_B − d₁d₂, V_B), which in the ventilated zone is parallel to the mean ventilation flow in layer 2. The conclusions above are not limited to a ventilated thermocline; see a similar discussion regarding the Rhines–Young pool in appendix B. Furthermore, similar conclusions can be obtained in a 3.5-layer model (appendix D).

4. Local response under wind and buoyancy forcing

We further investigate the forcing mechanism of the planetary wave modes. We will study the structure of the locally forced response to anomalies of Ekman pumping and surface buoyancy forcing by projecting the local response onto the two dynamic modes. Using forcing of the form of (3.1), the forced local response is derived from (2.3) as

where A is given in (2.5). This local response is projected onto the two dynamic modes (3.3a,b) as

ϕ_1Fϕ_1Nϕ_1Aϕ_2Fϕ_2Nϕ_2A(4.2)

With the aid of (3.4), the modal projection is uniquely determined from (4.2) as

and ϕ_2N = a_Nϕ_1N, ϕ_2A = a_Aϕ_1A.

Some general results of this forced response can be speculated according to the vertical structures of the wave modes and the surface forcing. On the wave side, within the thermocline, the N mode tends to have a barotropic flow structure, while the A mode has a baroclinic structure, as seen in Liu (1998) and Figs. 2b,c and 3b,c. Furthermore, the strongest amplitude occurs in SSH for the N mode and in thermocline temperature for the A mode. On the forcing side, Ekman pumping has a strong projection component that is barotropic within the thermocline, while entrainment shows a purely baroclinic forcing within the thermocline. Therefore, the vertical structure seems to match between the N mode and Ekman pumping, and between the A mode and entrainment forcing. This suggests a distinct modal response: Ekman pumping most efficiently generates the N mode, while entrainment forcing most effectively forces the A mode.

This modal response can also be seen in a more generalized thermocline circulation. Indeed, in a region of weak subsurface flow such as a shadow zone, we have (see appendix A)

As an example, Fig. 5 plots the ratio of the amplitude of the N mode over the A mode under spatially uniform Ekman pumping and entrainment forcing in the basin. Under Ekman pumping forcing (Figs. 5a,b), the SSH is overwhelmed by the N mode, as in (4.4a), over almost the entire basin. In the shadow zone, the ratio |h_1N/h_1A| (=|ϕ_1N/ϕ_1A| in the QG model here) exceeds 5 (the unstable region in the southwest is not discussed here). In the ventilated zone, a very large |h_1N/h_1A| occurs in a belt that extends from the northwest into the midwest of the gyre. This high |h_1N/h_1A| belt is caused by a virtually vanishing h_1A (not shown). The ratio of the two h₂ does not show such a dramatic difference except in a belt that extends from the northeast to the mid west.

The response is completely different under entrainment forcing (Figs. 5c,d). The ratio of the h₂ response is dominated by the A mode, as in (4.4b) near the outcrop line and in the shadow zone. The SSH, however, does not show such a strong difference in modal response.

These modal responses are not very sensitive to model forcing parameters. Figures 6 and 7 show some sensitivity tests in which the forcing has a spatial structure of m = 3, n = 3. In addition, three forcing frequencies are used, ω = 0.1, 1, and 10, corresponding to centennial, decadal, and annual times respectively. Under Ekman pumping forcing, the modal response of SSH is shown along the eastern boundary and the outcrop line in Figs. 6a and 6c, respectively. As in the case of Fig. 5, the SSH response is overwhelmed by the N mode (|h_1N/h_1A| ≫ 1) for all the frequencies (except in the very western part of the outcrop line). Relatively speaking, the upper thermocline temperature variation |h_2N/h_2A| is less dominated by the N mode (Figs. 6c,d).

By comparison, under entrainment forcing (Figs. 7b,d), the A mode has the strongest response in the upper thermocline temperature (|h_2N/h_2A| ≪ 1) along the eastern boundary and along the outcrop line (note the vertical scale is one order smaller in Fig. 7 than in Fig. 6). Nevertheless, the A mode becomes almost comparable to the N mode in SSH (Figs. 7a,c). These features agree with our earlier discussions (see Figs. 5c,d).

Based on the discussions in the last two sections, we offer a speculative synthesis of thermocline variability from the viewpoint of planetary waves. Ekman pumping is efficient in generating the N mode, which propagates westward rapidly along all latitudes and has the clearest signal in the SSH field. Surface buoyancy forcing (or entrainment forcing here), however, tends to generate a strong A mode that tends to follow the subsurface circulation, or more generally, the characteristics (U_B − d₁d₂, V_B), and has the strongest signature in the thermocline temperature.

5. Numerical experiments

To support our WKB analysis, we present numerical experiments in three models: a 3-layer eddy-resolving QG model in an idealized sector basin, a 2.5-layer primitive equation model for the North Pacific, and a 13-layer Miami Isopycnal Model (MICOM, Bleck et al. 1992) in an idealized North Atlantic basin. In each model, two experiments are presented, one forced by anomalous wind stress, and the other by anomalous surface buoyancy (interfacial entrainment) forcing.

The QG model (Fig. 8) has a double-gyre mean circulation. The model parameters are D₁ = 300 m, D₂ = 700 m, D₃ = 4000 m, g^′₁₂ = g^′₂₃ = 2 cm s⁻², f₀ = 0.73 × 10⁻⁴ s⁻¹, β = 2 × 10⁻¹¹ m⁻¹ s⁻¹, biharmonic mixing coefficient 10¹⁰ m⁴ s⁻¹, resolution 25 km × 25 km, and domain size 4000 km × 6000 km. The Ekman pumping is slightly asymmetrical with a meridional width of 2100 km and 1900 km for the subtropical and subpolar gyres, respectively. Figures 8a–c plot the case forced by a local Ekman pumping of a 3-yr period. The first mode of a 3-yr free wave has about 2 wavelengths across the basin. The SSH variability shows strong N-mode amplitude (Fig. 8a) that propagates westward rapidly as seen in the phase diagram (Fig. 8c). The A mode is visible but relatively weak and propagates southwestward slowly. The general pattern of the two wave rays agree well with the WKB analysis (appendix B, Fig. B1). In comparison to the SSH response in Fig. 8a, the upper thermocline temperature field (Fig. 8b) has a weaker signal in the N mode. Under entrainment forcing, however, the opposite occurs as shown in Figs. 8d–f. The A mode becomes dominant, especially in the upper thermocline temperature field (Fig. 8e), while the N mode is visible only in SSH (Fig. 8d).

The basic features in the QG model are reproduced in a 2.5-layer primitive equation model with realistic North Pacific geometry (Zhang 1998) (Fig. 9). The model is first spun up to the steady state under a mean wind field that resembles the observed annual mean wind field. Then a 10-yr periodic Ekman pumping anomaly is added in the midlatitudes of the central North Pacific (marked in Fig. 9c). The experiment is continued for another 80 years with the data of the last 60 years used for harmonic analyses (Figs. 9a–c). The SSH amplitude is dominated by a westward tongue all the way to the Kuroshio (Fig. 9a). The phase diagram further shows that this tongue reaches the western boundary in about 5 years (Fig. 9c). This is clearly the N mode. The A mode has little signal in SSH (Fig. 9a) but becomes visible in the upper thermocline temperature field as a weak amplitude tongue that extends southwestward (Fig. 9b). The response under entrainment forcing (Figs. 9d–f) is in sharp contrast to that under wind forcing. With an anomalous entrainment forcing, the dominant signal in the upper thermocline temperature field (Fig. 9e) is a tongue of the A mode that first extends southward to about 20°N, and then extends southwestward toward western boundary at 15°N. The phase diagram (Fig. 9f) further shows that this wave path takes more than 10 years to reach the western boundary. The N-mode signal also appears as a weak amplitude tongue that extends westward in the SSH field (Fig. 9d).

The two models above, although relatively simple to understand, have several serious limitations. First, the vertical resolution is too coarse. Second, there is not a surface mixed layer. Third, surface buoyancy forcing anomaly is indirectly represented by an interfacial entrainment. To improve these aspects, a 13-layer MICOM model will be used. The model has a domain of 0° to 60° in longitude and 20°S to 60°N in latitude with a resolution of 2° in both longitude and latitude. There are 12 isopycnal layers and an active surface mixed layer. The surface wind stress forcing is the zonal mean Atlantic zonal wind stress; the surface temperature and salinity are restored toward the zonal mean Levitus (1982) SST and SSS with a restoring time of 90 days. A full annual cycle is used in both the wind stress and restoring SST and SSS. The model is first spun up for 30 years before the anomalous forcing is imposed.

Figures 10 and 11 present the anomalous annual mean response 10 yr after a steady anomalous wind stress forcing is imposed. The anomalous wind curl has a uniform positive curl within a disk centered at (45°N, 35°E) of a radius of 10°, but has no wind curl outside the disk³ (see Fig. 10a and the caption). The barotropic streamfunction shows the zonally enlongated anomalous cyclonic gyre spanning between 25° and 45°N. The SST change is small (less than 0.5°C, not shown). The SSH field (Fig. 10c) is dominated by a negative westward extending tongue, which can also be identified as the westward tongue of negative isopycnal depth anomalies in the upper (Fig. 10d, 25.5σ_T), middle (Fig. 10e, 26.25σ_T), and lower (Fig. 10f, 27σ_T) thermocline. The coherent vertical structure can also be observed in the deep cold temperature anomalies along the meridional section at x = 22°E (between 30° and 50°N in Fig. 11a) and the zonal section at y = 37°N (west of 40° in Fig. 11b). These features are consistent with the N-mode structure. [Notice that the positive temperature (or depth) anomaly near the eastern boundary in the middle latitude is caused by the strong Ekman downwelling forced by the anomalous meridional wind stress along the eastern boundary (see Fig. 10a).] In addition to the deep, westward N-mode anomaly, a shallow cold temperature anomaly follows the subduction flow and extends from the forcing region southwestward, as shown in the SSH (Fig. 10c), the upper thermocline (Fig. 10d), and the meridional temperature field (Fig. 11a). The vertical structure of this shallow anomaly resembles that of the A mode because it has its maximum in the upper thermocline and has a deep temperature anomaly out of phase with the upper layer. The magnitude of this A mode is, however, weaker than that of the deep N mode. This is similar to the QG and 2.5-layer model experiments (Figs. 8a–c, 9a–c). In all the cases, the wind stress curl anomaly forces predominantly the N-mode response, but also forces a significant A-mode response.

In contrast, Figs. 12 and 13 show the anomalous thermocline response to a cold surface forcing anomaly. The cold Gaussian shape restoring SST anomaly is localized in the region of 30° to 50°E, 30° to 40°N with a maximum of 2°C. The resulted anomalous SST has a maximum of about 1.8°C (Fig. 12a). Different from the wind curl forcing case, the dominant signal in the subtropical gyre is a southwestward extending advective signal in all the anomaly fields including the negative SSH (Fig. 12b), negative upper thermocline depth (26σ_T, Fig. 12c), and the positive deep thermocline depth (27σ_T, Fig. 12d). The lower thermocline temperature anomaly tends to be weaker and out of phase with the upper thermocline temperature. This baroclinic vertical structure can be observed by comparing the negative upper thermocline anomaly in Fig. 12c with the positive lower thermocline anomaly in Fig. 12d and can also be seen clearly in the meridional section at x = 22°E (Fig. 13a) and the zonal section at y = 23°N (Fig. 13b). This advective anomaly resembles closely to the A-mode structure in our theory, and the subduction resembles closely the observed North Pacific subduction anomaly (Deser et al. 1996; Zhang and Liu 1999 hereafter ZL; Schneider et al. 1998). Therefore, the surface cooling produces predominantly the A-mode response. This is consistent with our theory, the QG experiment (Figs. 8d–f) and the 2.5-layer model experiment (Figs. 9d–f).

Similar sensitivity experiments are also carrried out in the GFDL MOM1 model with a horizontal resolution of 1° × 1°. The wind curl forcing case is similar to the MICOM simulation. However, the cooling experiment produces a much weaker advective signal, probably due to the strong diapycnal diffusion in the model. Overall, our numerical experiments strongly support our WKB analyses in section 3 and section 4 as well as our speculative synthesis at the end of section 4.

6. Advective mode or passive tracer

The A mode shows some similarity to a passive tracer. A passive tracer in the subsurface will follow the subsurface flow perfectly (in the absence of diffusion). The A mode, as discussed above, tends to follow the circulation and, therefore, resembles a passive tracer. More specifically, in a ventilated zone, the wave path of an A mode follows closely to that of the subsurface ventilation flow path U₂ (Figs. 1b,d and 2a). Moreover, in a Rhines–Young pool where the subsurface PV is homogeneous (Rhines and Young 1982), the A mode propagates as a perfect passive tracer, following the subsurface flow trajectory of U₂ exactly [see (B.1) and appendix B].

In spite of some apparent similarity, it is more advantageous to treat the A mode as a dynamic mode, instead of a passive tracer. First as seen in section 4, the A mode has a distinct dynamic response to surface forcing: the A mode is most efficiently excited by surface buoyancy forcing. This forcing mechanism can only be understood by considering the dynamic nature of the A mode, rather than treating it as a passive tracer.

There are also important differences between the propagation of the A mode and a passive tracer. One obvious difference has been discussed in section 2. In a shadow zone where no mean subsurface flow exists, a passive tracer would remain at its initial position, while the A mode propagates along the wave characteristics (Figs. 1b, 3a).

The wave amplitude could also differ significantly between the A mode and a passive tracer. One example is the planetary wave instability found in the southwestern part of a subtropical gyre (Liu 1998), where the A mode is destabilized by coupling with the N mode. This may imply an amplification of the wave amplitude during propagation.⁴ A passive tracer, in contrast, can never become unstable and therefore its magnitude can never exceed its initial value. Finally, even in the stable region, the amplitude of the A mode, which varies according to the dynamics, may be substantially different from that of a passive tracer. This issue remains to be studied in the future.

7. Summary and discussion

The response of a thermocline gyre to surface wind and buoyancy forcing was studied with the WKB method in a 2.5-layer QG model. The theory is substantiated by numerical experiments. A unified theory is proposed for the planetary waves in the thermocline. Ekman pumping is the most efficient mechanism to generate the N mode, which has its clearest signal in the SSH field and the lower thermocline. This mode propagates westward rapidly even in the presence of a gyre circulation, because of the non-Doppler-shift effect. In contrast, surface buoyancy forcing generates the strongest response in the A mode, with its largest signature in the upper thermocline temperature field. This mode is strongly advected by the subsurface gyre circulation, but may propagate at a different speed in the central to eastern part of the gyre where it is dominated by the ventilated zone and the shadow zone.