Nonlinear Midlatitude Ocean Adjustment

William K. Dewar Department of Oceanography, The Florida State University, Tallahassee, Florida

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Abstract

Ocean adjustment on annual to interdecadal scales to variable forcing is considered for a more nonlinear general circulation than has previously been studied. The nature of the response is a strong function of forcing frequency and importantly involves the inertial recirculations rather than linear baroclinic waves. The spatial expression of this variability is concentrated near the separation latitudes of the Gulf Stream extension, a model region corresponding to an area in the real ocean of well-known strong ocean–atmosphere buoyancy exchange. “Turn-on” cases, periodically forced cases, and stochastically forced cases are considered. The first set of experiments clarifies the adjustment timescales and dynamics of a nonlinear circulation. The second set examines modifications to that adjustment rendered by time-dependent forcing. The last set is perhaps the most realistic in terms of the atmospheric forcing of the ocean, because wind spectra are not strongly peaked beyond a few weeks. Multidecadal forcing is argued both to excite a novel, rapid mode of adjustment and to resonate with a considerably slower, nonlinear mode. Stochastic forcing seems clearly to excite the fast mode and to contribute to the slower mode, although the latter also derives considerable variance from intrinsic sources. These conclusions are based on a suite of distinct spatial and temporal characteristics of the dominant ocean variability patterns under various forcing scenarios and comment on the ocean dynamics likely to be important to decadal timescale midlatitude climate variability.

Corresponding author address: Dr. William K. Dewar, Department of Oceanography, The Florida State University, Tallahassee, FL 32306-4320. Email: dewar@ocean.fsu.edu

Abstract

Ocean adjustment on annual to interdecadal scales to variable forcing is considered for a more nonlinear general circulation than has previously been studied. The nature of the response is a strong function of forcing frequency and importantly involves the inertial recirculations rather than linear baroclinic waves. The spatial expression of this variability is concentrated near the separation latitudes of the Gulf Stream extension, a model region corresponding to an area in the real ocean of well-known strong ocean–atmosphere buoyancy exchange. “Turn-on” cases, periodically forced cases, and stochastically forced cases are considered. The first set of experiments clarifies the adjustment timescales and dynamics of a nonlinear circulation. The second set examines modifications to that adjustment rendered by time-dependent forcing. The last set is perhaps the most realistic in terms of the atmospheric forcing of the ocean, because wind spectra are not strongly peaked beyond a few weeks. Multidecadal forcing is argued both to excite a novel, rapid mode of adjustment and to resonate with a considerably slower, nonlinear mode. Stochastic forcing seems clearly to excite the fast mode and to contribute to the slower mode, although the latter also derives considerable variance from intrinsic sources. These conclusions are based on a suite of distinct spatial and temporal characteristics of the dominant ocean variability patterns under various forcing scenarios and comment on the ocean dynamics likely to be important to decadal timescale midlatitude climate variability.

Corresponding author address: Dr. William K. Dewar, Department of Oceanography, The Florida State University, Tallahassee, FL 32306-4320. Email: dewar@ocean.fsu.edu

1. Introduction

Le Traon and Minster (1993), Chelton and Schlax (1996), and Cippolini et al. (1997) have all argued from TOPEX/Poseidon observations that first mode baroclinic planetary waves move in the ocean at speeds beyond those predicted by the linear theory of a resting ocean; Zang and Wunsch (1999) argue against such a discrepancy. Interest in this problem stems from the need to understand the timescales over which, and the mechanisms by which, the ocean can adjust to variable forcing. Several essentially linear, theoretical explanations for these observations have recently arisen (Killworth et al. 1997; Qiu et al. 1997; Dewar 1998; White et al. 1998; De Szoeke and Chelton 1999; Tailleux and McWilliams 2002). There is, however, a nonlinear mode of ocean response not obviously connected to linear planetary waves, that is, the transient adjustment of the so-called inertial recirculations. The adjustment of these zones is examined under different forcing scenarios in the present paper with a view toward defining the spatial and temporal characteristic of a nonlinear gyre. Such adjustment is likely to be significant to midlatitude climate variability according to some recent theoretical ideas.

a. Background

Observations and models are clear that the ocean is not entirely linear, even on its larger scales. For the midlatitude general circulation, the most obvious quasi-basin-scale example of this is to be found in the inertial recirculations. In the North Atlantic, these are the zones where the interior transport grows to better than 100 Sv (Sv ≡ 106 m3 s−1), roughly 3 times the expected interior wind-driven transport of 35 Sv. They show up in dynamic height maps of the North Atlantic (Stommel et al. 1978) and have been recognized in models for 40 years (Bryan 1963; Holland 1978). Such areas are highly time dependent and the variability appears to be strongly connected to their dynamics (Hogg 1983, 1985). From long-term averages of numerical experiments, the mean dynamics of such areas can be investigated, and it is routinely found that mean advection of mean relative vorticity is important in inertial recirculation structure, and the leading theories of inertial recirculations emphasize nonlinearity at leading order (Cessi 1990). Inertial recirculations are also located in the northwestern (southwestern) corners of subtropical (subpolar) gyres and constitute recirculations of the local western boundary currents. It is observed both in the Atlantic and Pacific that the maxima in air–sea buoyancy exchange are found in the same areas as the inertial recirculations, due largely to the presence of the warm western boundary current SSTs lying just offshore of large landmasses.

A separate aspect of the inertial recirculations is that they are centers of the generation of intrinsic oceanic variability. This was first recognized in the role of the separated boundary current in generating mesoscale variability through barotropic and baroclinic variability (Holland 1978). More recently, the instabilities of the inertial recirculations themselves, occurring on larger scales and lower frequencies, have been studied (Jiang et al. 1995; Berloff and Meacham 1998; Berloff and McWilliams 1999). McCalpin and Haidvogel (1996), Spall (1996), and Meacham (2000) have emphasized preferred frequencies for self-sustained oscillations resident largely in the mechanics of these zones. In summary, large-scale zones of the midlatitude oceans are essentially nonlinear and intrinsically variable and are found at strategic points in the ocean with regards to the critical issue of ocean–atmosphere buoyancy exchange. They are thus potentially rich sources of climate variability.

b. This paper

The above analyses of ocean intrinsic variability have all taken place under steady forcing. Given the above description of the inertial recirculations and the potential for local midlatitude climate coupling, it becomes critical however to understand their adjustment under variable conditions. Such studies have been conducted for the linear part of the general circulation, where the standard paradigms have been oscillatory forcing (Dewar 1989; Qiu et al. 1997; Dewar and Morris 2000) or the “turn-on” problem (Anderson and Gill 1975; Anderson and Killworth 1977). A smaller number of interesting studies with stochastic forcing (Frankignoul and Müller 1979; Willebrand et al. 1980; Frankignoul et al. 1997; Cessi and Louazel 2001) have also been conducted, but comparable studies of the variably forced inertial recirculations have received little past attention. This and the possible importance of the phenomena to climate variability form the primary motivations for the present work.

Given the high levels of intrinsic variability and inherent nonlinearity of the region, gauging the adjustment characteristics of the inertial recirculation is subtle. At least at the outset, analytical approaches are unlikely to succeed, suggesting the necessity of the numerical analysis adopted here. A suite of numerical experiments are conducted using a variety of forcing fields. The simplest turn-on problem is first considered to define the limits to which adjustment can proceed. Such experiments also permit the dynamics behind the adjustment to be analyzed. Periodically forced cases, covering the frequency range from annual to interdecadal, are then used to define and distinguish between intrinsic and forced inertial gyre variability. These are followed by the more realistic case of white noise stochastic forcing of a realistic strength.

All experiments discussed here are adiabatic, even though potentially important connections to midlatitude climate involve ocean–atmosphere buoyancy exchange. It is still likely the results are relevant to midlatitude climate variability in the same way adiabatic wave adjustment studies are thought to be. These experiments build upon the linear wave adjustment problems discussed above by their focus on the inertial recirculation adjustment. They also complement the study in Dewar and Morris, where inertial recirculations were present, but were confined in horizontal extent by parameter selection, and where wave propagation in the Sverdrup part of the gyre was emphasized.

Accordingly, it is argued that wave propagation is difficult to recognize in the inertial recirculations, being masked by intrinsic variability and nonlinear adjustment of the inertial recirculation. Simple adjustment of the inertial recirculations exhibits both a fast and a slow phase, the latter requiring roughly 20–30 years to complete. With forcing patterned loosely after that of the Atlantic, it is suggested that a mix of intrinsic and forced variability controls the inertial recirculations out to roughly decadal periods. Variable forcing at longer periods generate dominant changes in the inertial recirculations reflecting forced adjustment. The fast mode of adjustment appears in these experiments, but the slow mode can apparently be achieved by resonance with an intrinsic mode. Stochastically forced experiments exhibit characteristics consistent with the rapid adjustment mode. The slow mode also appears, but is also driven by ocean dynamics as well as forcing.

The model and parameter selections are described in the next section. Results of the turn-on, periodically forced, and stochastically forced experiments and their analysis are given in section 3. Section 4 concludes the paper with a discussion.

2. Model development

All the experiments discussed here employ a three-layer quasigeostrophic (QG) ocean on a β plane, for the reasons that QG dynamics support inertial recirculations and nonlinear ocean features well, have been studied extensively as models of intrinsic ocean variability, and permit extensive investigation. The equations governing the model are
i1520-0485-33-5-1057-e1
expressing the evolution of potential vorticity (qi) in layer i. The quantities ψi represent layer streamfunctions, fo is Coriolis parameter, Hi are layer mean-state thicknesses, hi is interface perturbations [hi = fo(ψiψi+1)/gi with gi being the interfacial reduced gravities], J is the usual Jacobian operator, β is the Coriolis gradient, K are viscosities, and r is related to bottom drag. Subscripted t denotes a time derivative. A schematic appears in Fig. 1.

The numerical implementation of Holland (1978) was employed, grid resolution was 10 km, and the deformation radii were set to Rd = 45.8 and 19.5 km for the first and second modes, respectively. Layer thicknesses were 500, 500, and 4000 m for the upper, middle, and lower layers. A simple, flat-bottomed, rectangular, ocean domain of size 2000 km by 2000 km was employed. Both slip and no slip boundary conditions were studied and K was always set at 300 m2 s−1. The bottom drag decay timescale r−1 ranged from 240 to 1000 days. The experimental plan was to spin the model up from a state of rest for at least 10 years, subject to a specified Ekman pumping field, we. The runs were continued for an additional period of time, ranging from 80 to 200 years, archiving the model upper-layer pressure, subsampled at 30 km, at time intervals from one month to one year, depending on the experiment.

A common assumption in uncoupled and coupled conceptual ocean models is that atmospheric forcing is separable in space and time. Such practice has a long history in the study of ocean circulation subject to steady forcing. Mean and dominant variable atmospheric patterns from the North Atlantic, the latter associated with the North Atlantic Oscillation (NAO), have been discussed by Hurrell (1995) and Czaja and Marshall (2001). These studies argue the amplitude of the NAO curl is comparable to that of the mean field and, second, the anomaly pattern is confined near the zero of the mean wind stress curl. Equations (1) are thus here forced with the zonally independent fields in Fig. 2, meant as a crude model of the North Atlantic setting. The mean Ekman structure is given by
weoμoπyL
throughout the paper, and consists of a classical, asymmetric double gyre pumping. The anomalous NAO-like pumping is proportional to
i1520-0485-33-5-1057-e3
This pumping straddles the zero-mean Ekman pumping line and is latitudinally confined near it. The meridional extent of the anomaly is LN/2 = 500 km on either side of the basin midlatitude. The nominal strength associated with the anomaly μ′ = 2.5 × 10−6 m s−1 is also comparable to that of the mean field μo = 5 × 10−6 m s−1. The choice in (3) reflects interest in the potential role of the midlatitude ocean in midlatitude climate variability.
The total Ekman pumping in (1) is
weweoweweoatŵex,y
where several forms are used for the temporal modulation, a(t), of the NAO Ekman pumping field. The first experiments in section 3 use sinusoidal forcing; that is,
ataoπωtϕo
with frequencies ω from a low of 0 cpy (i.e., the turn-on experiments) to a high of 1 cpy and ϕo a fixed phase. The nondimensional scaling amplitude ao is either 0 or 1. The later stochastically forced experiments use
atκNt
for the time-dependent amplitude of the Ekman pumping. Here N(t) is a random variable drawn from a Gaussian distribution with zero mean and unit variance and κ is a scaling coefficient taking the values 1 or 2. The resulting rms variability in the forcing field is roughly 2.5 × 10−6 m s−1 (5 × 10−6 m s−1) for κ = 1 (2), and thus comparable to the mean pumping, as suggested by data.

Examples are shown in Fig. 3 of mean states representative of the present set of experiments. The upper row contains upper-layer transport in Sverdrups and the lower row total transport also in Sverdrups. The left hand column (middle column) comes from an experiment with ao = 0 (1), ω = 0, and ϕo = π/2 (i.e., both are of oceans subject to steady forcing). The left-hand circulations, averaged over 80 years of model data, are the classical double-gyre structures seen in many previous studies. The total transport of 120 Sv in the present case is similar to the estimates for the total transport of the Gulf Stream system, and is several times larger than the expected linear Sverdrup transport of 35 Sv. The middle column transports, averaged over roughly 150 years of model data, grow to better than 150 Sv in the subpolar gyre, which is larger than any estimate of mean North Atlantic transport in either gyre. On the other hand, the winds in this experiment are composed from both the mean and the anomaly, the latter permanently maintained at an extreme value.

The inertial recirculations are located in the western 1000 km or so of the domain and are contained within a zone extending a few hundred kilometers on either side of the jet. These length scales are similar to those observed for the inertial recirculations in both North Atlantic and North Pacific Oceans. The points of this figure are twofold. First, the mean forcing field used here sustains realistic oceanic transports and gyre length scales, especially in the nonlinear regions of the circulation, and thus that the present parameter range is relevant. Second, the structure expected of a circulation fully equilibrated with the combined mean and anomalous forcing, illustrated by the middle column, differs significantly from the unperturbed mean. Their difference is illustrated in the right-hand column, showing the result of subtracting the unperturbed fields from the perturbed ones. The transports change, as does the location of the jet separation and the subsequent pathway of the separated jet.

An evident drawback of the experimental design is that the 2000 km × 2000 km basin is relatively small compared to the Atlantic and Pacific basins. The region occupied by the linear Sverdrup flow is thus truncated relative to the real ocean. This represents a compromise between the desire to integrate for long periods of time while resolving mesoscale dynamics and a focus on the adjustment of the inertial recirculation. The spatial scales of the modeled inertial recirculations, their transports, and the flow speeds contained in them are like those observed in the Northern Hemisphere oceans. On the other hand, the variability of this system will naturally feature, and perhaps overemphasize, that of the recirculation regions.

a. Waves in a nonlinear ocean

The simplest possibility, discussed briefly here, is that the present adjustment is governed by linear wavelike responses. This appears not to be the case, however, as evidenced by Fig. 4, where time–longitude plots from the central latitude of two model runs are shown. Both are subject to an annually pulsating wind stress and no-slip boundary conditions. The difference between the runs is that the left-hand column is from an experiment with the mean flow appearing in the left column in Fig. 3, while on the right are results from a run without a mean flow. To obtain the results on the left, the long-term mean circulation has been removed. On the top left is a composite of the annual cycle, formed by averaging six years of model output at fixed phases relative to the forcing. On the bottom in both columns are two consecutive years worth of data. All plots are of sea surface pressure extracted from the central latitude (i.e., the mean intergyre boundary) of the model. This latitude is studied because recent theoretical ideas about midlatitude coupling focus on intergyre heat exchange as an oceanic process controlling SST. The adjustment mechanisms of this latitude then might be expected to be more important to climate that those elsewhere in the gyres. Further, the strongest SST gradients are associated with the western boundary currents and the separated jet. Intergyre heat transports then naturally are concentrated on the west in the vicinity of the inertial recirculations.

The right column is obviously dominated by simple westward wave propagation at a speed of about −0.042 m s−1, consistent with first-mode long-wave propagation. A similar picture would hold, but with faster propagation rates if a mean flow interacting quasi linearly with the waves were present. In the shown no-mean flow case, the baroclinic cross-gyre propagation timescale is roughly 1.5 yr. Almost no evidence of wavelike behavior is seen on the left; rather, the structure is dominated more by a nearly standing pattern, and evidences of phase propagation in both directions are seen. Complex EOF analysis (Barnett 1983), often used to identify propagating signals in data, also showed no baroclinic wavelike modes. Evidence for a long, propagating barotropic mode was seen, but its fundamental frequency was far higher than that of the forcing. Also, its propagation tendencies were arrested in the separated jet. The reason planetary waves are not seen is most likely due to the strong barotropic flow in the area. In the separated jet it exceeds 0.15 m s−1, thus overwhelming the baroclinic wave speed and effectively canceling the 0.20 m s−1 high-frequency barotropic wave propagation speed. The bottom-left plot is also not periodic in time, indicating the presence of interannual variability not directly forced by the annual wind stress cycle. This is the model internal variability. At other latitudes away from the intergyre boundary and recirculations, evidence of wave propagation is more pronounced, leading to the conclusion that the inertial recirculations and intrinsic variability of the system are largely masking any wavelike behavior at the basin central latitude. The leading order time dependence of this part of the circulation is relatively independent of linear wave dynamics. The remainder of the paper addresses the nature of this variability.

3. Numerical results

Figures 5 and 6 show spatial patterns of the first three EOFs from a run without anomalous pumping (i.e., ao = 0) and no-slip boundary conditions.1 The pattern of the first EOF corresponds to a meridional shifting of the intergyre boundary and that of the second to a simultaneous waxing and waning of the inertial recirculation. These modes explain 57%, 9.2%, and 3.7% of the variance, respectively.

Spectra of the principal components associated with these EOFs appear in the left-hand column of the first figure. Considerable variance exists for at least the first two principal components in the interdecadal band (i.e., beyond 10 years). The first EOF spectrum in particular exhibits a rapid growth in variance beginning at the 10-yr mark. Such variability with red spectral signatures out to decades can confuse the identification of adjustment timescales to altered forcing conditions, which are timescales that may also be on the order of decades.

a. Turn-on experiments

The first results to be discussed come from a so-called turn-on experiment. Here, the applied Ekman pumping was switched from a steady, unperturbed state, like the upper panel in Fig. 2, to a perturbed, but steady, state, like the bottom panel in the same figure. Again, no-slip boundary conditions were used. The transition was initiated after 100 years of unperturbed forcing, after which the model was continued for another 100 years. The last several decades of model output showed little to no drift in several global measures of model behavior, arguing equilibration with the new forcing had been achieved.

Figures 7–10 show various measures of ocean adjustment to such forcing. The data shown here are taken from 2-yr running box car averages of upper-layer monthly model streamfunction output. The top row shows such averaged streamfunctions centered on given years. The second row shows the difference fields of these streamfunctions relative to an arbitrarily chosen 2-yr averaged streamfunction from roughly 12 years prior to the initiation of the anomalous forcing. This reference time is labeled as time zero in the discussion to follow. The bottom row shows the averaging interval used in processing the data relative to the initiation of the anomalous forcing. This format is used in Figs. 7–10, although the model output years vary from plot to plot.

The streamfunction in Fig. 7 (left) comes from before the onset of the forcing anomaly and possesses the north–south mirror symmetry relative to the basin central latitude of the forcing. This changes almost immediately with the onset of the forcing anomaly, as is apparent from the second column in Fig. 7, centered on year 13. The 2-yr averaged streamfunction has a considerably distorted, separated jet. The inclusion of the variable forcing necessarily breaks the symmetry of the steadily forced problem because the anomalous Ekman pumping straddles the zero line of the mean field. As a result, the zero pumping line separating the gyres migrates and the subpolar and subtropical gyres receive unequal potential vorticity forcing. Unbalanced forcing has been studied in the past and shown to result in an undulating western boundary current separation (Cessi et al. 1990). The associated sequence of highs and lows along the separated jet axis shows clearly in the second panel of the streamfunction difference column.

This pattern is slow to change, as appears in Fig. 8, coming from years 14 and 19. Both the mean streamfunction and streamfunction difference fields strongly resemble the right-hand column in the earlier figure, and are dominated by the overshoot and plunge caused by the unequal gyre forcing. Some weak evidence is seen in the right column plots of a smoothing out of the highs and lows along the jet axis.

Figure 9, with data approximately 17 and 26 years after the forcing transition, shows a continuation of this trend. In particular, the difference fields begin to lose their characterization as a string of mesoscale highs and lows, and to resemble instead a persistent zone of anticyclonic anomaly located north of the basin central latitude. The forcing perturbation consists of a strengthening of the Ekman downwelling over the central basin latitude, in effect selecting the extant subtropical anticyclone over the subpolar cyclone for strengthening. This is becoming apparent in the right-hand plots of Fig. 9. However, clear evidence of quasi equilibration with the perturbed forcing is not found until Fig. 10, which presents data from years 41 and 48 (i.e., 29 and 36 years after the forcing transition). A persistent anticyclone, robust in spatial structure, appears and the difference fields come to resemble the anomaly field in Fig. 3. Two-year means extracted from later periods in the run also strongly resemble the means seen here. These data, from roughly 30 years after the initiation of the forcing anomaly, date the period required for nearly complete nonlinear ocean adjustment as about three decades.

The dynamics of the adjustment can be studied by examining the transitions of the potential vorticity field. This quantity, unlike streamfunction, is materially conserved in the lower layers, aside from relatively weak lateral viscous and bottom drag effects. Modifications to the mean potential vorticity field of the lowest layer, since it is not directly forced, must be due to eddy-driven potential vorticity fluxes.

Figure 11 shows potential vorticity transects using 2-yr-averaged data from south to north across the basin. The data comes from a longitude 450 km east of the western basin boundary, and the several curves are of data 8 years apart, starting at year 7 in Fig. 7 and ending at year 47. Upper-layer potential vorticity (q) appears in the upper panel and third layer q in the bottom panel.

Upper-layer q is dominated by the front between the gyres, representing the convergence of potential vorticity along the jet axis. Note that this drifts north in stages, nearing completion by year 39. The bottom layer lacks the strong jet front but also shows distinct q structure. Plan views of potential vorticity show the third-layer structure is dominated by a pool of well-mixed q under the separated jet. Figure 11 shows this pool drifts northward over the 40 years encompassed by the plot. Potential vorticity flux divergence when low-pass filtered2 accounts for the noted mean potential vorticity changes. Such modification necessarily reflects eddy processes operating in the lowest layer. Specifically, the short term adjustment sets up a modified gyre which results in a modified eddy field generated by instabilities. These modified eddies then begin a slower adjustment of the lower layers by rearranging subsurface potential vorticity fields. This last step is important to the total adjustment process, generating a major change in the character of the circulation. The most noticeable aspect of this is the loss of the mesoscale wave train along the jet axis and the establishment of the persistent, strong anticyclonic anomaly.

b. Periodic forcing

The above experiment is useful to define the gross elements of nonlinear ocean adjustment. To probe this problem further, periodic forcing at specified frequencies has been used. This is not meant to imply periodic forcing is a good model of atmospheric forcing. It is adopted here in order to aid the examination of system response on specific timescales.

The equilibrated ocean structures in Fig. 3 represent the extreme cases of unperturbed and perturbed steady forcing. One may anticipate that for sufficiently slow, periodic, forcing, the ocean proceeds through a sequence of quasi-(statistical) equilibrium states, settling eventually on an extreme “mean” state like the middle column in Fig. 3, and then reversing. Consider now the character of the ocean variability of the periodically forced case versus that of the steadily forced case. For unperturbed steady forcing, the ocean circulation exhibits modes of intrinsic variability, the leading three of which appear in Figs. 5 and 6. If variable forcing at specified frequencies is included, it is anticipated by the preceding argument that sufficiently low frequency forcing will result in different leading order EOFs, such that some of the new modes will be related to the spatial differences between the two states appearing in Fig. 3. An earmark of their principal components should be a strong, nearly delta function, spectral peak at the forcing frequency. At higher-frequency variable forcing, the comparison between the leading EOFs and the expected spatial difference will degrade. Performing such comparisons at a number of frequencies will bound the adjustment timescales of the circulation, and further analysis can comment on the adjustment physics. This is the strategy employed here, where sinusoidal forcing at specified frequencies is applied to (1) in order to identify the preferred timescale for nonlinear oceanic adjustment to anomalous forcing.

1) Annual forcing

A partial look at an experiment with annual forcing appears in the time–longitude plots of Fig. 4. Now consider the EOFs emerging from that experiment in comparison to the EOFs from the experiment with no anomalous forcing. On the left in Fig. 12 are the spectra of the first three principal components and on the right are the expressions of the first EOF in each of the layers. Similar spatial representations of the next two modes appear in Fig. 13.

The second EOF is analogous in its spatial structure to the leading EOF from the unperturbed experiment and corresponds to a meridional shifting of the jet. The spectrum of its principal component also displays a sudden growth in variance. This was seen for the steadily forced case, although there the jump began near the 10-yr mark, and in the present case, the jump begins at 5 yr. Nonetheless, no unusual spectral structure in the second principal component occurs at 1 yr. Given the spatial similarity of the EOF and the overall spectral structure of its associate principal component, it is plausible to identify the second mode as essentially intrinsic rather than forced. The third EOF from the annually forced experiment bears a qualitative resemblance to the previous second mode. Both measure strengthening and weakening of the inertial recirculations, and the two associated spectra also have some similarity. Again, no unusual variance appears at 1 yr. Identifying the third EOF as belonging to the intrinsic variability also seems defensible.

Clearly however, the first mode in the annually forced experiment is quite distinct from any of the leading, steadily forced EOFs. It corresponds rather well with the anomaly fields occurring after a few years from the turn-on experiments (see Fig. 7) in that it is dominated by a sequence of highs and lows under the mean jet axis. In addition, the first mode is the only one exhibiting a peak at the annual period. Due to these characteristics, the first mode is identified as a result of the variable forcing and as representing an oceanic adjustment to that forcing. Its spatial structure measures the oceanic overshoot and plunge associated with unequal potential vorticity forcings of the two gyres, a process that clearly belongs to a very rapid adjustment mechanism as the timescale of the variability is very short. This is consistent with the turn-on experimental results. The spatial structure of the mode is concentrated almost exclusively in the region of the mean inertial recirculation and further is relatively depth independent. The spatial amplitudes in each layer do decrease with depth, but the relative positions of the highs and lows are coincident. Thus this mode of response is composed jointly of barotropic and baroclinic structure, but it does exhibit considerable vertical coherence.

EOFs have also been extracted from each layer individually (as opposed to collectively as was done in Figs. 12 and 13). The leading mode from each layer was found to correspond to the expression of the leading three-dimensional mode in that same layer. Also, the principal component spectra of the latter EOFs were peaked at the annual period. The principal components of each of these latter modes are compared in Fig. 14. They are all in-phase, supporting this adjustment mode as essentially depth independent. Finally, examining the principal components of the EOFs of the model barotropic and baroclinic modes revealed the leading barotropic modes where the only ones displaying significant variance in the annual band.

2) Multiyear forcing

Consider now lower-frequency oscillatory forcing and the associated response of the ocean circulation. Several frequencies have been used in these experiments, but the present focus will be on the results for 10- and 30-yr cycles. Here results are shown from free-slip experiments. The characteristics of the decadal forcing problem appear in Figs. 15 and 16, where the presentation is the same as for the annually forced case. Note the spectra of the first three principal components show clear evidence that the first mode is a direct result of the forcing and argues the third mode is relatively independent of the forcing. The spatial structure of the first mode also is analogous to that of the first mode in the annually forced experiment, arguing that the rapid response mode seen previously corresponds to the first mode in this decadally forced case.

The second mode is somewhat hybrid in its properties. There is an elevation in its spectral levels beginning at 5 yr, which is tempting to interpret as sensitivity to the decadal forcing, but comparable increases are found in the annually forced and steadily forced cases. Further, the spectra does not exhibit a pronounced spike at the decadal period. The spatial structure of the second mode is similar to the spatial structure of the second mode in the annually forced experiment, although the comparison is not as strong as occurred between the annual and steadily forced experiments. The change in structure captured by this mode is mostly that of meridional shifts in the jet axis, although other gyre modifications reside in it as well.

The third mode gives no spectral indication that it is responding to the decadal forcing, but it does have a fairly novel spatial structure. It appears in this model that a decade is insufficient for the ocean to come into quasi equilibrium with variable forcing, although the ocean is clearly sensitive to these timescales.

Experiments with longer-period variability have been conducted, in particular 20 and 30 years. The EOF results from the latter set appear in Figs. 17 and 18. The spectra of the first three principal components appear in the former figure, where it is seen that the first and second modes now clearly exhibit sensitivity to the variable forcing period. The third mode displays no spectral peak at 30 years, suggesting this mode is not a direct result of the Ekman pumping variability. The spatial structure of the first EOF appears on the right in the former figure and those of the second and third EOFs appear in the latter figure. Here a major change is noted between these results and those of the previous cases. The most significant change is that the first and second EOFs have switched places, with the leading mode now capturing the shifting of the jet and the second mode the jet overshoot and plunge.

The current third leading mode, which might thus be expected to correspond to the third mode in the decadally forced case, has instead adopted a spatial structure like the second mode from the steadily forced case. Clearly the spatiotemporal character of the variability is quite novel for this experiment.

Comparisons of these results with those of a 20-yr variable forcing experiment show many similarities. In particular, the first two principal component spectra are peaked at the appropriate period, although not quite as strongly as in the 30-yr forcing case. The spatial fields of the first three EOFs are also very similar between the 20- and 30-yr forcing experiments.

The composite of the two leading modes from the 30-yr forced experiment appears in Fig. 19. The resulting field takes on several characteristics of the spatial difference of the two equilibrated mean fields in Fig. 3. These are characteristics that each individual mode independently does not possess. The important features retained in the composite include an extended zonal reach, combined with a few undulations, along the separated jet axis. The biggest distinction between the composite and the mean field difference in Fig. 3 is that the latter is displaced north of the central basin latitude. This is due to the lack of symmetry in the perturbed forcing in that experiment (i.e., the anomalous downwelling is maintained). The present forcing when averaged over a cycle does not possess this same asymmetry; hence a bias either north or south of the central basin latitude is unexpected. These results are consistent with the idea that the two modes represent quasi-equilibrated oceanic adjustment to variable forcing. A variable forcing cycle of 30 years implies a forcing anomaly of one sign lasts for 15 years, and this is significantly shorter than the 20–30 years required in the turn-on experiment for equilibration to occur. Nonetheless, the timescales appear to be sufficiently slow that significant adjustment to the forcing can occur throughout the water column.

The temporal structure of the new leading mode is also somewhat distinct in behavior from previous results. EOFs were extracted from each layer individually, rather than collectively, and compared to the above results. The upper two layers yielded leading EOFs analogous to the upper-layer expressions of the leading EOF in Fig. 17. The third-layer leading EOF, however, was distinct both spatially and temporally from the third-layer expression in that figure. Most importantly, the spectrum of the principal component did not exhibit a peak on the 30-yr period; rather, the second mode was the one that displayed the peak (see Fig. 20). The spatial structure of that mode however only weakly resembled the third-layer structure in Fig. 17 (see Fig. 21). The strongest point of comparison between these two fields is in their joint tendency to develop an extremum under the jet axis. Comparable patterns were less obvious in the analogous fields computed from the 20-yr variable forcing experiments.

It is also interesting to compare the timings of the individual layer EOFs; this appears in Figs. 22 and 23. The former plot compares the first two principal components shown in Figs. 17 and 18. The second of these modes is essentially in phase with the forcing cycle and leads the first principal component by roughly 7.5 yr. The latter plot compares correlations between the first- and second-layer leading principal components and the first-layer leading and third-layer second principal component. The correlation values for the latter case are generally lower than for the former. The oscillations are roughly in phase, but the third layer tends to lead the upper layer by a few years.

These various results invite the interpretation that nonlinear adjustment can be broken into barotropic and baroclinic pieces. The second EOF in Fig. 18 essentially represents a fast barotropic adjustment, occurring in near lockstep with the forcing, and with a spatial structure that is highly vertically coherent. The leading EOF in that figure represents a much slower adjustment that distinguishes between the layers, with the lowest layer comprised of a mix of the fast and slow modes.3 Presumably, longer period forcing cycles would result in leading third-layer EOFs with peaks on the forcing period, and with spatial structures similar to those seen when all the layers are handled simultaneously.

An interesting comparison can be made between these last results and those from the steadily forced case. The potential vorticity field EOFs, rather than the streamfunction field EOFs, of the 30-yr forced experiment were extracted from the model results. The third-layer mean field (upper panel), the leading potential vorticity EOF (middle panel), and a composite plot (bottom panel) from this exercise appear in the left-hand column of Fig. 24. The right-hand column contains the same fields extracted from the steadily forced case. The strong similarity between the two sets of fields is obvious. For both, the mean field is dominated by a pool of homogenized potential vorticity under the jet axis. Structure of this sort is classical, and is now largely explained as a result of eddy potential vorticity flux. The middle panels show the leading EOF of the third layer. This mode is decidedly large scale, covering an area comparable to that of the inertial recirculation, and is dominated by a single extremum.

The effect of this EOF on the potential vorticity field appears in the lower panels. These are plots of the third-layer meridional potential vorticity structure from a longitude 450 km east of the western basin boundary. The line labeled “mean” is the mean q structure. The remaining two curves are composites of the mean field and the leading EOF at this same longitude. The composite consists of the mean with the EOF structure, multiplied by the fractional variance explained by the EOF, both added to and subtracted from the mean field. Clearly, these EOFs are capturing the north–south migration of the well-mixed pool. Such migration is essential for the interpretation of almost complete ocean adjustment of the ocean to apply to the 30-yr experiment.

Comparable analyses were performed with the 10-yr and 20-yr variable forcing results. The 10-yr third-layer q EOF bore almost no resemblance to the leading EOF shown here, while the 20-yr EOF exhibited some of the noted structure. Most important in the latter results were that the central extremum was emerging as the dominant structure of the leading EOF, although there was considerable structure aside from the extremum. Again, these results support the interpretation that nonlinear ocean adjustment is a process taking decades to complete and is a process that essentially reflects fluxing of potential vorticity by the variability.

An interesting result is that the variably forced system is apparently achieving this equilibrium by resonating with a natural mode of the system. This is strongly suggested by the similarities in the spatial mode structures and in the more dynamical potential vorticity mode structures, as seen in the previous figure. The right-hand column comes from an experiment without variable forcing, and thus represents a mode belonging to the steady system. Its spectra show multidecadal timescales. The anomalous forcing fields promote asymmetry between the two gyres, and the realization of this comes by exciting the gravest asymmetric mode. This does not happen at higher-frequency forcings, probably because of the mismatch between the time scales of the variable forcing and the response timescale of the intrinsic mode.

3) Stochastically forced experiments

This section concludes with a description of some stochastically forced experiments. The motivation here is that periodic forcing is not a very good model of the real climatic setting. One might well imagine that the nature of the ocean response to stochastic forcing, in which no single period is emphasized, would differ from that described above. It is also of interest to examine if the apparent resonance noted in the previous subsection survives into a stochastic setting.

Figures 25 and 26 show the principal component spectra and the three leading EOFs from a stochastically forced experiment. The leading and second EOFs spatial structures show many similarities with the first two EOFs of the 30-yr forced experiment. In particular, the present second EOF, consisting of a train of highs and lows situated on the separated jet axis, compares extremely well with the mesoscale mode whose existence seems clearly tied to variable forcing. This supports the contention that the forced scenario described in the periodically forced experiments translates to the present case. The present leading mode also captures the north–south jet migration, as does the leading mode in the 30-yr case. For that experiment, the connection to the anomalous forcing was clear because of the spike in principal component associated with the EOF. Given that the steadily forced case possesses a leading EOF with essentially the same structure as the present EOF, the origin of this mode requires further analysis.

The question is whether the leading few modes arise in response to the anomalous forcing or if they originate in the intrinsic variability of the steadily forced system. To address this point, it is useful to consider the coherence between the anomalous forcing time series and the principal components of the EOFs. Figure 27 shows graphs of the coherence between the principal components and the anomalous forcing, along with a measure of the uncertainty in the estimate. The second EOF emerges as coherent in the near-decadal band, and also in the interannual band. The leading EOF also exhibits some coherence with the forcing in the lower frequencies, although considerable variance is clearly not accounted for. The principal variance of the leading EOF is found in the lower frequencies. The frequency structure of the second EOF coherence is consistent with the prior results, which have shown that the mesoscale mode corresponds to a rapid adjustment process. Further, the spectrum of the second EOF (not shown) is close to white and exhibits considerable energy in the coherent band. Thus, it appears that this stochastically forced experiment exhibits a forced response, at least in this principal component. The leading EOF spectrum is decidedly red, with approximately a factor of 50–80 times more variance in the interdecadal band than in the interannual band. Given that the energetic band for this mode shows apparently significant, but weak, coherence, the likely interpretation is that the leading EOF is driven by a mix of forced and intrinsic sources.

The dependence of these results on the level of stochastic forcing has also been tested. In particular, the rms Ekman pumping was increased from 2.5 × 10−6 m s−1, as in the above experiments, to 5 × 10−6 m s−1. One change was that the two leading modes switched order with the mesoscale mode elevated to the leading spot. The coherence structure associated with each of these modes, however, did not change appreciably. The mesoscale mode retained coherence in the annual-to-decadal band, and the second mode, with a red spectrum, was weakly coherent with the forcing near the decadal band. Intrinsic sources remained important for that mode.

4. Summary

The nonlinear adjustment of the general ocean circulation to anomalous Ekman pumping patterns loosely based on observations has been studied. It is argued that the main features of this adjustment involve the inertial recirculations of the gyres, rather than waves, and importantly involve eddy turbulent potential vorticity transport. Further, the adjustment can be broken roughly into two parts. The first is a rapid, barotropic adjustment that establishes a sequence of mesoscale highs and lows along the separated jet axis. These reflect the unequal potential vorticity forcing of the gyres and the migration of the zero Ekman pumping line inherent in the assumed structure of the forcing. The second, slower adjustment is baroclinic and reflects deep potential vorticity redistributions driven by the eddy field of the modified circulation. These eventually establish a much larger scale, uniform circulation anomaly consistent with the large-scale forcing anomaly. Comparisons between the turn-on experiments and the periodically forced experiments suggest periodic forcing can resonate with one of the intrinsic modes to drive the baroclinic adjustment. Both experiment sets identify the timescale of this change to be a few decades.

Stochastically forced experiments have also been conducted. The variability in these experiments exhibits a mixed of forced and intrinsic variability. Recalling that North Atlantic wind stress forcing is comparable to the mean field in strength, it is possible that the leading EOF can be either of the forced or intrinsic variety. In either case, the high-frequency mode clearly survives into the stochastically forced case. The slower mode, capable of resonating with an intrinsic mode, apparently can also respond to forcing, but also partly remains the product of intrinsic circulation dynamics.

The physics of nonlinear circulation adjustment is quite different from that ascribed to the classical adjustment problem driven by linear waves. In the latter case, the cross-basin transit time of low-mode baroclinic waves set the fundamental timescale of adjustment, and reflects the reestablishment of the basic pressure field of the circulation. In the present case, waves appear not to play a measurable role. Rather, advective processes in the inertial recirculations dominate the rapid timescale and geostrophic, turbulent transport processes dominate the slow, baroclinic timescale. While the physics is different, a common point is that both adjustment models predict the general circulation will lag the forcing by a reasonably quantifiable interval. This seems a central point as ocean adjustment is likely to be important in the evolution of the coupled climate system and many conceptual climate models are built on delayed oscillator concepts (Jin 1997; Marshall et al. 2001). Further, the separated western boundary jet usually is associated with the strongest gradients in sea surface temperature, and variability in these gradients is likely to be significant to midlatitude climate variability. Inertial recirculation variability can be expected to contribute significantly to this. Thus the basic delayed oscillator paradigm may apply as well to a nonlinearly adjusting ocean even though the wave model underlying many delayed oscillator models might not.

Acknowledgments

WKD is supported by NSF Grant ATM-9818628 and NASA Grants NAG5-7630 and NAG5-8291, the latter awarded in support of the NASA Seasonal to Interannual Prediction Project. The author wishes to acknowledge a number of interesting conversations with Tony Sturges. Ms. J. Jimeian assisted with computational issues. This is a contribution of the Climate Institute, a Center of Excellence support by the Research Foundation of The Florida State University.

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Fig. 1.
Fig. 1.

Model schematic. A three-layer, flat-bottom, β-plane model is used. Model notation is as shown in the figure and the system is forced by Ekman pumping

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 2.
Fig. 2.

Model Ekman pumping. Cross sections from south to north through the mean field and the perturbation field. This structure is based loosely on the structure of the wind stress curl field associated with the North Atlantic Oscillation

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 3.
Fig. 3.

Mean states from steadily forced experiments. The left-hand column is from an experiment with μo = 5 × 10−6 m s−1 and ao = 0, and the middle column employs the same μo, but ao = 1. The upper row is upper-layer transport and the bottom row is total transport, all in Sverdrups. The right column is the difference between the two earlier columns

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 4.
Fig. 4.

Waves in a nonlinear ocean. Time–longitude plots from the central latitude of a periodically forced QG model run are shown. On the left are results from a model with the mean flow shown in Fig. 3; on the right is a model with no mean flow. The upper-left panel is a composite annual cycle obtained by averaging six years of output. The bottom panel is two consecutive annual periods. Little clear evidence of wave propagation is seen on the left

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 5.
Fig. 5.

Spectra and EOF patterns from an experiment with no anomalous forcing. The spectra (left side, with 95 confidence intervals) all show roll-offs to white spectra at frequencies between 10 and 30 yr. The spatial patterns in each layer of the first EOF appear in the right-hand column

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 6.
Fig. 6.

The spatial patterns in each layer of the second and third leading EOFs from an experiment with no anomalous forcing.

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 7.
Fig. 7.

Two-year averages over the upper layer of the three-layer model appear in the upper row. The second row shows the streamfunction difference between the present model output and that occurring 12 years before the transition in forcing (labeled as year 0 in the plots). The bottom row shows the 2-yr averaging period relative to the initiation of the forcing anomaly, indicated by the sudden jump in the bottom graph from zero to a finite value. The left column uses data centered on year 8, and the right-column data are centered on year 13

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 8.
Fig. 8.

As in Fig. 7 but for years 14 and 19

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 9.
Fig. 9.

As in Fig. 7 but for years 29 and 38.

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 10.
Fig. 10.

As in Fig. 7 but for years 41 and 48.

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 11.
Fig. 11.

Potential vorticity transects 450 km east of the western basin boundary. The upper (lower) panel shows upper- (third-) layer potential vorticity, and the various curves are from years 7, 15, 23, 31, 39, and 47. Only the first and last are labeled for clarity of the plot. The transition between the earliest and latest plots is monotonic

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 12.
Fig. 12.

Annually forced experiment results. On the left are the spectra of the principal components of the first three EOFs obtained from an experiment with annual forcing. On the right are the expressions of the first EOF in each of the layers. Note only the first EOF exhibits a peak on the annual cycle

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 13.
Fig. 13.

The second and third EOFs from the annually forced experiment

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 14.
Fig. 14.

Comparison of the first-mode principal components for the annually forced experiment, obtained from each layer independently. They are in phase over the duration of the record, arguing the adjustment of the ocean on the annual timescale is essentially barotropic

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 15.
Fig. 15.

Results of a model forced by decadal Ekman pumping. The left side shows the spectra of the first three EOFs from the experiment. EOF 1 shows clear evidence of the decadal forcing. The right-hand column shows the layer expressions of the first EOF. It is similar in structure to the first EOF computed for the annually forced case

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 16.
Fig. 16.

The second and third EOFs from the decadally forced experiment. The second mode has an analog in the annually forced and steadily forced cases; the third mode is somewhat novel in structure

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 17.
Fig. 17.

Thirty-year variable forcing results. On the left are spectra from the first three principal components and the layer expressions of the first EOF appears in the right column

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 18.
Fig. 18.

Second and third EOFs from an experiment with 30-yr variable forcing

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 19.
Fig. 19.

A composite of the two leading EOFs from the 30-yr experiment. The resulting spatial structure resembles the difference field between the two equilibrated fields in Fig. 3

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 20.
Fig. 20.

Principal component spectra from layer three analyzed alone. The second mode exhibits a peak at the 30-yr period

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 21.
Fig. 21.

Second EOF from layer three analyzed alone. The spatial structure shown here is different from the third-layer spatial structure in Fig. 17

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 22.
Fig. 22.

The first two principal components from the 30-yr experiment. Mode 2 is essentially in phase with the forcing cycle, and it is seen that the first mode lags the second in time by about 7.5 yr.

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 23.
Fig. 23.

Correlations versus lag between the leading upper-layer EOF and the leading EOF in the second layer (labeled “Upper-Second Layer”). Also shown is the correlation vs lag of the upper-layer leading EOF and the third-layer second EOF. These two associated principal components are roughly in phase, with the lower layer leading by a few years. The correlations are lower by roughly a factor of 2 in comparison with the former case

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 24.
Fig. 24.

Third-layer potential vorticity. The left column contains data from the 30-yr forced experiment, and the right column is from the steadily forced experiment. The upper panel is the mean field, dominated by a pool of uniform potential vorticity under the jet axis. The second panel is the third-layer leading EOF of potential vorticity. The bottom panel contains three plots of meridional third-layer potential vorticity structure from a longitude 450 km east of the western basin boundary. The line labeled “mean” is the mean potential vorticity structure. The remaining two lines are composites of the mean structure plus (minus) the leading EOF structure (adjusted for the percent variance explained by the leading mode) at the same longitude. Note the effect of the leading EOF is to shift the uniform pool north and south

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 25.
Fig. 25.

Spectra of the leading three PCs and the leading EOF from a stochastically forced experiment

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 26.
Fig. 26.

The second and third EOFs from a stochastically forced experiment

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

Fig. 27.
Fig. 27.

Coherence as a function of frequency between the principal components associated with the first and second EOFs of the stochastically forced experiment and the anomalous Ekman pumping time series. Also shown is the spurious coherence generated between two independent random time series of equal length and processed in the same manner as the model data. The second EOF is significantly coherent with the anomalous Ekman pumping in the multiyear band, while the first EOF is not significantly coherent anywhere in the considered frequency domain

Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<1057:NMOA>2.0.CO;2

1

These EOFs have been computed by collapsing streamfunction data from all three layers at a given time step into a single vector. The temporal covariance matrix is computed from the resulting two-dimensional (space, time) dataset and time principal components are obtained. Forming the inner product of the data with the eigenvectors and expanding them back to the original three-layer format yields the essentially three-dimensional spatial EOFs discussed in this paper. EOFs of the barotropic and two baroclinic modes of the model have been studied and yield results analogous to those shown. Complex EOFs (Barnett 1983) have also been computed, for the layers collectively and individually and for the model modes, to comment on propagative signals.

2

Low-pass filtering was performed by Fourier transforming the potential vorticity flux divergences, setting all frequencies higher than 1 cycle (2 yr)−1 to zero and inverting the transform.

3

Consistent with these results, the leading EOFs of the two baroclinic modes exhibit peaks at the forcing frequency. Their spatial structure resembles those of the layer EOFs. Complex EOF analysis shows no propagative tendencies in any layer or mode.

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  • Anderson, D., and A. Gill, 1975: Spin-up of a stratified ocean with application to up-welling. Deep-Sea Res., 22 , 583596.

  • Anderson, D., and P. Killworth, 1977: Spin-up of a stratified ocean, with topography. Deep-Sea Res., 24 , 709732.

  • Barnett, T., 1983: Interaction of the monsoon and Pacific trade wind system at interannual time scales. Part I: The equatorial zone. Mon. Wea. Rev., 111 , 756773.

    • Search Google Scholar
    • Export Citation
  • Berloff, P., and S. Meacham, 1998: The dynamics of a simple baroclinic model of the wind-driven circulation. J. Phys. Oceanogr., 28 , 361388.

    • Search Google Scholar
    • Export Citation
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  • Fig. 1.

    Model schematic. A three-layer, flat-bottom, β-plane model is used. Model notation is as shown in the figure and the system is forced by Ekman pumping

  • Fig. 2.

    Model Ekman pumping. Cross sections from south to north through the mean field and the perturbation field. This structure is based loosely on the structure of the wind stress curl field associated with the North Atlantic Oscillation

  • Fig. 3.

    Mean states from steadily forced experiments. The left-hand column is from an experiment with μo = 5 × 10−6 m s−1 and ao = 0, and the middle column employs the same μo, but ao = 1. The upper row is upper-layer transport and the bottom row is total transport, all in Sverdrups. The right column is the difference between the two earlier columns

  • Fig. 4.

    Waves in a nonlinear ocean. Time–longitude plots from the central latitude of a periodically forced QG model run are shown. On the left are results from a model with the mean flow shown in Fig. 3; on the right is a model with no mean flow. The upper-left panel is a composite annual cycle obtained by averaging six years of output. The bottom panel is two consecutive annual periods. Little clear evidence of wave propagation is seen on the left

  • Fig. 5.

    Spectra and EOF patterns from an experiment with no anomalous forcing. The spectra (left side, with 95 confidence intervals) all show roll-offs to white spectra at frequencies between 10 and 30 yr. The spatial patterns in each layer of the first EOF appear in the right-hand column

  • Fig. 6.

    The spatial patterns in each layer of the second and third leading EOFs from an experiment with no anomalous forcing.

  • Fig. 7.

    Two-year averages over the upper layer of the three-layer model appear in the upper row. The second row shows the streamfunction difference between the present model output and that occurring 12 years before the transition in forcing (labeled as year 0 in the plots). The bottom row shows the 2-yr averaging period relative to the initiation of the forcing anomaly, indicated by the sudden jump in the bottom graph from zero to a finite value. The left column uses data centered on year 8, and the right-column data are centered on year 13

  • Fig. 8.

    As in Fig. 7 but for years 14 and 19

  • Fig. 9.

    As in Fig. 7 but for years 29 and 38.

  • Fig. 10.

    As in Fig. 7 but for years 41 and 48.

  • Fig. 11.

    Potential vorticity transects 450 km east of the western basin boundary. The upper (lower) panel shows upper- (third-) layer potential vorticity, and the various curves are from years 7, 15, 23, 31, 39, and 47. Only the first and last are labeled for clarity of the plot. The transition between the earliest and latest plots is monotonic

  • Fig. 12.

    Annually forced experiment results. On the left are the spectra of the principal components of the first three EOFs obtained from an experiment with annual forcing. On the right are the expressions of the first EOF in each of the layers. Note only the first EOF exhibits a peak on the annual cycle

  • Fig. 13.

    The second and third EOFs from the annually forced experiment

  • Fig. 14.

    Comparison of the first-mode principal components for the annually forced experiment, obtained from each layer independently. They are in phase over the duration of the record, arguing the adjustment of the ocean on the annual timescale is essentially barotropic

  • Fig. 15.

    Results of a model forced by decadal Ekman pumping. The left side shows the spectra of the first three EOFs from the experiment. EOF 1 shows clear evidence of the decadal forcing. The right-hand column shows the layer expressions of the first EOF. It is similar in structure to the first EOF computed for the annually forced case

  • Fig. 16.

    The second and third EOFs from the decadally forced experiment. The second mode has an analog in the annually forced and steadily forced cases; the third mode is somewhat novel in structure

  • Fig. 17.

    Thirty-year variable forcing results. On the left are spectra from the first three principal components and the layer expressions of the first EOF appears in the right column

  • Fig. 18.

    Second and third EOFs from an experiment with 30-yr variable forcing

  • Fig. 19.

    A composite of the two leading EOFs from the 30-yr experiment. The resulting spatial structure resembles the difference field between the two equilibrated fields in Fig. 3

  • Fig. 20.

    Principal component spectra from layer three analyzed alone. The second mode exhibits a peak at the 30-yr period

  • Fig. 21.

    Second EOF from layer three analyzed alone. The spatial structure shown here is different from the third-layer spatial structure in Fig. 17

  • Fig. 22.

    The first two principal components from the 30-yr experiment. Mode 2 is essentially in phase with the forcing cycle, and it is seen that the first mode lags the second in time by about 7.5 yr.

  • Fig. 23.

    Correlations versus lag between the leading upper-layer EOF and the leading EOF in the second layer (labeled “Upper-Second Layer”). Also shown is the correlation vs lag of the upper-layer leading EOF and the third-layer second EOF. These two associated principal components are roughly in phase, with the lower layer leading by a few years. The correlations are lower by roughly a factor of 2 in comparison with the former case

  • Fig. 24.

    Third-layer potential vorticity. The left column contains data from the 30-yr forced experiment, and the right column is from the steadily forced experiment. The upper panel is the mean field, dominated by a pool of uniform potential vorticity under the jet axis. The second panel is the third-layer leading EOF of potential vorticity. The bottom panel contains three plots of meridional third-layer potential vorticity structure from a longitude 450 km east of the western basin boundary. The line labeled “mean” is the mean potential vorticity structure. The remaining two lines are composites of the mean structure plus (minus) the leading EOF structure (adjusted for the percent variance explained by the leading mode) at the same longitude. Note the effect of the leading EOF is to shift the uniform pool north and south

  • Fig. 25.

    Spectra of the leading three PCs and the leading EOF from a stochastically forced experiment

  • Fig. 26.

    The second and third EOFs from a stochastically forced experiment

  • Fig. 27.

    Coherence as a function of frequency between the principal components associated with the first and second EOFs of the stochastically forced experiment and the anomalous Ekman pumping time series. Also shown is the spurious coherence generated between two independent random time series of equal length and processed in the same manner as the model data. The second EOF is significantly coherent with the anomalous Ekman pumping in the multiyear band, while the first EOF is not significantly coherent anywhere in the considered frequency domain

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